[    {        "title": "0807.5009v1.Scattering_Theory_of_Gilbert_Damping.pdf",        "content": "arXiv:0807.5009v1  [cond-mat.mes-hall]  31 Jul 2008Scattering Theory of Gilbert Damping\nArne Brataas,1,∗Yaroslav Tserkovnyak,2and Gerrit E. W. Bauer3\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamics of a single domain ferromagnet i n contact with a thermal bath\nis studied by scattering theory. We recover the Landau-Lift shitz-Gilbert equation and express the\neffective fields and Gilbert damping tensor in terms of the sca ttering matrix. Dissipation of magnetic\nenergy equals energy current pumped out of the system by the t ime-dependent magnetization, with\nseparable spin-relaxation induced bulk and spin-pumping g enerated interface contributions. In\nlinear response, our scattering theory for the Gilbert damp ing tensor is equivalent with the Kubo\nformalism.\nMagnetization relaxation is a collective many-body\nphenomenon that remains intriguing despite decades of\ntheoretical and experimental investigations. It is im-\nportant in topics of current interest since it determines\nthe magnetization dynamics and noise in magnetic mem-\nory devices and state-of-the-art magnetoelectronic ex-\nperiments on current-induced magnetization dynamics\n[1]. Magnetization relaxation is often described in terms\nof a damping torque in the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation\n1\nγdM\ndτ=−M×Heff+M×/bracketleftBigg˜G(M)\nγ2M2sdM\ndτ/bracketrightBigg\n, (1)\nwhereMis the magnetization vector, γ=gµB//planckover2pi1is the\ngyromagnetic ratio in terms of the gfactor and the Bohr\nmagnetonµB, andMs=|M|is the saturation magneti-\nzation. Usually, the Gilbert damping ˜G(M) is assumed\nto be a scalar and isotropic parameter, but in general it\nis a symmetric 3 ×3 tensor. The LLG equation has been\nderived microscopically [2] and successfully describes the\nmeasured response of ferromagnetic bulk materials and\nthin films in terms of a few material-specific parameters\nthatareaccessibletoferromagnetic-resonance(FMR) ex-\nperiments [3]. We focus in the following on small fer-\nromagnets in which the spatial degrees of freedom are\nfrozen out (macrospin model). Gilbert damping pre-\ndicts a striclylinear dependence ofFMR linewidts on fre-\nquency. This distinguishes it from inhomogenous broad-\nening associated with dephasing of the global precession,\nwhich typically induces a weaker frequency dependence\nas well as a zero-frequency contribution.\nThe effective magnetic field Heff=−∂F/∂Mis the\nderivative of the free energy Fof the magnetic system\nin an external magnetic field Hext, including the classi-\ncal magnetic dipolar field Hd. When the ferromagnet is\npart of an open system as in Fig. 1, −∂F/∂Mcan be\nexpressed in terms of a scattering S-matrix, quite anal-\nogous to the interlayer exchange coupling between ferro-\nmagnetic layers [4]. The scattering matrix is defined in\nthe space of the transport channels that connect a scat-\ntering region (the sample) to thermodynamic (left andleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath via metallic normal metal leads (N).\nright) reservoirs by electric contacts that are modeled by\nideal leads. Scattering matrices also contain information\nto describe giant magnetoresistance, spin pumping and\nspin battery, and current-induced magnetization dynam-\nics in layered normal-metal (N) |ferromagnet (F) systems\n[4, 5, 6].\nIn the following we demonstrate that scattering the-\nory can be also used to compute the Gilbert damping\ntensor˜G(M).The energy loss rate of the scattering re-\ngion can be described in terms of the time-dependent\nS-matrix. Here, we generalize the theory of adiabatic\nquantum pumping to describe dissipation in a metallic\nferromagnet. Our idea is to evaluate the energy pump-\ningoutoftheferromagnetandtorelatethistotheenergy\nloss of the LLG equation. We find that the Gilbert phe-\nnomenology is valid beyond the linear response regime of\nsmall magnetization amplitudes. The only approxima-\ntion that is necessary to derive Eq. (1) including ˜G(M)\nis the (adiabatic) assumption that the frequency ωof the\nmagnetization dynamics is slow compared to the relevant\ninternal energy scales set by the exchange splitting ∆.\nThe LLG phenomenology works so well because /planckover2pi1ω≪∆\nsafely holds for most ferromagnets.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from spin-orbit interaction in\ncombinationwith impurityscatteringthattransfersmag-\nnetic energy to itinerant quasiparticles [3]. The subse-\nquent drainage of the energy out of the electronic sys-\ntem,e.g.by inelastic scattering via phonons, is believed\nto be a fast process that does not limit the overall damp-\ning. Our key assumption is adiabaticiy, meaning that\nthe precession frequency goes to zero before letting the\nsample size become large. The magnetization dynam-\nics then heats up the entire magnetic system by a tiny2\namount that escapes via the contacts. The leakage heat\ncurrent then equals the total dissipation rate. For suf-\nficiently large samples, bulk heat production is insensi-\ntive to the contact details and can be identified as an\nadditive contribution to the total heat current that es-\ncapes via the contacts. The chemical potential is set\nby the reservoirs, which means that (in the absence of\nan intentional bias) the sample is then always very close\nto equilibrium. The S-matrix expanded to linear order\nin the magnetization dynamics and the Kubo linear re-\nsponse formalisms should give identical results, which we\nwill explicitly demonstrate. The role of the infinitesi-\nmal inelastic scattering that guarantees causality in the\nKubo approach is in the scattering approach taken over\nby the coupling to the reservoirs. Since the electron-\nphonon relaxation is not expected to directly impede the\noverall rate of magnetic energy dissipation, we do not\nneed to explicitly include it in our treatment. The en-\nergy flow supported by the leads, thus, appears in our\nmodel to be carried entirely by electrons irrespective of\nwhethertheenergyisactuallycarriedbyphonons, incase\nthe electrons relax by inelastic scattering before reaching\nthe leads. So we are able to compute the magnetization\ndamping, but not, e.g., how the sample heats up by it .\nAccording to Eq. (1), the time derivative of the energy\nreads\n˙E=Heff·dM/dτ= (1/γ2)˙ m/bracketleftBig\n˜G(m)˙ m/bracketrightBig\n,(2)\nin terms of the magnetization direction unit vector m=\nM/Msand˙ m=dm/dτ. We now develop the scatter-\ning theory for a ferromagnet connected to two reservoirs\nby normal metal leads as shown in Fig. 1. The total\nenergy pumping into both leads I(pump)\nEat low tempera-\ntures reads [11, 12]\nI(pump)\nE= (/planckover2pi1/4π)Tr˙S˙S†, (3)\nwhere˙S=dS/dτandSis the S-matrix at the Fermi\nenergy:\nS(m) =/parenleftbiggr t′\nt r′/parenrightbigg\n. (4)\nrandt(r′andt′) are the reflection and transmissionma-\ntrices spanned by the transport channels and spin states\nfor an incoming wave from the left (right). The gener-\nalization to finite temperatures is possible but requires\nknowledge of the energy dependence of the S-matrix\naround the Fermi energy [12]. The S-matrix changes\nparametrically with the time-dependent variation of the\nmagnetization S(τ) =S(m(τ)). We obtain the Gilbert\ndamping tensor in terms of the S-matrix by equating the\nenergy pumping by the magnetic system (3) with the en-\nergy loss expression (2), ˙E=I(pump)\nE. Consequently\nGij(m) =γ2/planckover2pi1\n4πRe/braceleftbigg\nTr/bracketleftbigg∂S\n∂mi∂S†\n∂mj/bracketrightbigg/bracerightbigg\n,(5)which is our main result.\nThe remainder of our paper serves three purposes. We\nshow that (i) the S-matrix formalism expanded to linear\nresponseis equivalentto Kubolinearresponseformalism,\ndemonstrate that (ii) energy pumping reduces to inter-\nface spin pumping in the absence ofspin relaxationin the\nscattering region, and (iii) use a simple 2-band toy model\nwith spin-flip scattering to explicitly show that we can\nidentify both the disorder and interface (spin-pumping)\nmagnetization damping as additive contributions to the\nGilbert damping.\nAnalogous to the Fisher-Lee relation between Kubo\nconductivity and the Landauer formula [15] we will now\nprove that the Gilbert damping in terms of S-matrix (5)\nis consistent with the conventionalderivation of the mag-\nnetization damping by the linear response formalism. To\nthis end we chose a generic mean-field Hamiltonian that\ndepends on the magnetization direction m:ˆH=ˆH(m)\ndescribes the system in Fig. 1. ˆHcan describe realistic\nband structures as computed by density-functional the-\nory including exchange-correlation effects and spin-orbit\ncouplingaswell normaland spin-orbitinduced scattering\noff impurities. The energy dissipation is ˙E=/angb∇acketleftdˆH/dτ/angb∇acket∇ight,\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value for the non-\nequilibriumstate. Inlinearresponse,weexpandthemag-\nnetization direction m(t) around the equilibrium magne-\ntization direction m0,\nm(τ)=m0+u(τ). (6)\nThe Hamiltonian can be linearized as ˆH=ˆHst+\nui(τ)∂iˆH, where ˆHst≡ˆH(m0) is the static Hamilto-\nnian and∂iˆH≡∂uiˆH(m0), where summation over re-\npeated indices i=x,y,zis implied. To lowest order\n˙E= ˙ui(τ)/angb∇acketleft∂iˆH/angb∇acket∇ight, where\n/angb∇acketleft∂iˆH/angb∇acket∇ight=/angb∇acketleft∂iˆH/angb∇acket∇ight0+/integraldisplay∞\n−∞dτ′χij(τ−τ′)uj(τ′).(7)\n/angb∇acketleft.../angb∇acket∇ight0denotes equilibrium expectation value and the re-\ntarded correlation function is\nχij(τ−τ′) =−i\n/planckover2pi1θ(τ−τ′)/angbracketleftBig\n[∂iˆH(τ),∂jˆH(τ′)]/angbracketrightBig\n0(8)\nin the interaction picture for the time evolution. In order\nto arrive at the adiabatic (Gilbert) damping the magne-\ntization dynamics has to be sufficiently slow such that\nuj(τ)≈uj(t) + (τ−t) ˙uj(t). Since m2= 1 and hence\n˙ m·m= 0 [7]\n˙E=i∂ωχij(ω→0)˙ui˙uj, (9)\nwhereχij(ω) =/integraltext∞\n−∞dτχij(τ)exp(iωτ). Next, we use\nthe scattering states as the basis for expressing the\ncorrelation function (8). The Hamiltonian consists of\na free-electron part and a scattering potential: ˆH=\nˆH0+ˆV(m). We denote the unperturbed eigenstates of3\nthe free-electron Hamiltonian ˆH0=−/planckover2pi12∇2/2mat en-\nergyǫby|ϕs,q(ǫ)/angb∇acket∇ight, wheres=l,rdenotes propagation\ndirection and qtransverse quantum number. The po-\ntentialˆV(m) scatters the particles between these free-\nelectron states. The outgoing (+) and incoming wave\n(-) eigenstates |ψ(±)\ns,q(ǫ)/angb∇acket∇ightof the static Hamiltonian ˆHst\nfulfill the completeness conditions /angb∇acketleftψ(±)\ns,q(ǫ)|ψ(±)\ns′,q′(ǫ′)/angb∇acket∇ight=\nδs,s′δq,q′δ(ǫ−ǫ′) [10]. These wave functions can be ex-\npressed as |ψ(±)\ns(ǫ)/angb∇acket∇ight= [1 +ˆG(±)\nstˆVst]|ϕs(ǫ)/angb∇acket∇ight, where the\nstatic retarded (+) and advanced (-) Green functions are\nˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1andηis a positive infinites-\nimal. By expanding χij(ω) in the basis of the outgo-\ning wave functions |ψ(+)\ns/angb∇acket∇ight, the low-temperature linear re-\nsponse leads to the followingenergydissipation (9) in the\nadiabatic limit\n˙E=−π/planckover2pi1˙ui˙uj/angbracketleftBig\nψ(+)\ns,q|∂iˆH|ψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′|∂jˆH|ψ(+)\ns,q/angbracketrightBig\n,\n(10)\nwith wave functions evaluated at the Fermi energy ǫF.\nIn order to compare the linear response result, Eq.\n(10), withthat ofthe scatteringtheory, Eq. (5), weintro-\nduce the T-matrix ˆTasˆS(ǫ;m) = 1−2πiˆT(ǫ;m), where\nˆT=ˆV[1 +ˆG(+)ˆT] in terms of the full Green function\nˆG(+)(ǫ,m) = [ǫ+iη−ˆH(m)]−1. Although the adiabatic\nenergy pumping (5) is valid for any magnitude of slow\nmagnetization dynamics, in order to make connection to\nthe linear-response formalism we should consider small\nmagnetization changes to the equilibrium values as de-\nscribed by Eq. (6). We then find\n∂τˆT=/bracketleftBig\n1+ˆVstˆG(+)\nst/bracketrightBig\n˙ui∂iˆH/bracketleftBig\n1+ˆG(+)\nstˆVst/bracketrightBig\n.(11)\ninto Eq. (5) and using the completeness of the scattering\nstates, we recover Eq. (10).\nOur S-matrix approach generalizes the theory of (non-\nlocal) spin pumping and enhanced Gilbert damping in\nthin ferromagnets [5]: by conservation of the total an-\ngular momentum the spin current pumped into the\nsurrounding conductors implies an additional damping\ntorque that enhances the bulk Gilbert damping. Spin\npumping is an N |F interfacial effect that becomes impor-\ntant in thin ferromagnetic films [14]. In the absence of\nspin relaxation in the scattering region, the S-matrix can\nbe decomposed as S(m) =S↑(1+ˆσ·m)/2+S↓(1−ˆσ·\nm)/2, where ˆσis a vector of Pauli matrices. In this case,\nTr(∂τS)(∂τS)†=Ar˙ m2, whereAr= Tr[1−ReS↑S†\n↓]\nand the trace is over the orbital degrees of freedom only.\nWe recover the diagonal and isotropic Gilbert damping\ntensor:Gij=δijGderived earlier [5], where\nG=γMsα=(gµB)2\n4π/planckover2pi1Ar. (12)\nFinally, we illustrate by a model calculation that\nwe can obtain magnetization damping by both spin-\nrelaxationandinterfacespin-pumpingfromtheS-matrix.We consider a thin film ferromagnet in the two-band\nStoner model embedded in a free-electron metal\nˆH=−/planckover2pi12\n2m∇2+δ(x)ˆV(ρ), (13)\nwhere the in-plane coordinate of the ferromagnet is ρ\nand the normal coordinate is x.The spin-dependent po-\ntentialˆV(ρ) consists of the mean-field exchange interac-\ntion oriented along the magnetization direction mand\nmagnetic disorder in the form of magnetic impurities Si\nˆV(ρ) =νˆσ·m+/summationdisplay\niζiˆσ·Siδ(ρ−ρi),(14)\nwhich are randomly oriented and distributed in the film\natx= 0. Impurities in combination with spin-orbit cou-\npling will give similar contributions as magnetic impuri-\nties to Gilbert damping. Our derivation of the S-matrix\nclosely follows Ref. [8]. The 2-component spinor wave\nfunction can be written as Ψ( x,ρ) =/summationtext\nk/bardblck/bardbl(x)Φk/bardbl(ρ),\nwhere the transverse wave function is Φ k/bardbl(ρ) = exp(ik/bardbl·\nρ)/√\nAfor the cross-sectional area A. The effective one-\ndimensional equation for the longitudinal part of the\nwave function is then\n/bracketleftbiggd2\ndx2+k2\n⊥/bracketrightbigg\nck/bardbl(x) =/summationdisplay\nk′\n/bardbl˜Γk/bardbl,k′\n/bardblck/bardbl(0)δ(x),(15)\nwhere the matrix elements are defined by ˜Γk/bardbl,k′\n/bardbl=\n(2m//planckover2pi12)/integraltext\ndρΦ∗\nk/bardbl(ρ)ˆV(ρ)Φk′\n/bardbl(ρ)and the longitudinal\nwave vector k⊥is defined by k2\n⊥= 2mǫF//planckover2pi12−k2\n/bardbl. For\nan incoming electron from the left, the longitudinal wave\nfunction is\nck/bardbls=χs√k⊥/braceleftBigg\neik⊥xδk/bardbls,k′\n/bardbls′+e−ik⊥xrk/bardbls,k′\n/bardbls′,x<0\neik⊥xtk/bardbls,k′\n/bardbls′,x>0,\n(16)\nwheres=↑,↓andχ↑= (1,0)†andχ↓= (0,1)†. Inver-\nsion symmetry dictates that t′=tandr=r′. Continu-\nity of the wave function requires 1+ r=t. The energy\npumping (3) then simplifies to I(pump)\nE=/planckover2pi1Tr/parenleftbig˙t˙t†/parenrightbig\n/π.\nFlux continuity gives t= (1 +iˆΓ)−1, whereˆΓk/bardbls,k′\n/bardbls′=\nχ†\nsˆΓk/bardbls,k′\n/bardbls′χs′(4k⊥k⊥)−1/2.\nIn the absence of spin-flip scattering, the transmis-\nsion coefficient is diagonal in the transverse momentum:\nt(0)\nk/bardbl= [1−iη⊥σ·m]/(1+η2\n⊥), whereη⊥=mν/(/planckover2pi12k⊥).\nThe nonlocal (spin-pumping) Gilbert damping is then\nisotropic,Gij(m) =δijG′,\nG′=2ν2/planckover2pi1\nπ/summationdisplay\nk/bardblη2\n⊥\n(1+η2\n⊥)2. (17)\nIt can be shown that G′is a function of the ratio be-\ntween the exchange splitting versus the Fermi wave vec-\ntor,ηF=mν/(/planckover2pi12kF).G′vanishes in the limits ηF≪14\n(nonmagnetic systems) and ηF≫1 (strong ferromag-\nnet).\nWe include weak spin-flip scattering by expanding the\ntransmission coefficient tto second order in the spin-\norbit interaction, t≈/bracketleftbigg\n1+t0iˆΓsf−/parenleftBig\nt0iˆΓsf/parenrightBig2/bracketrightbigg\nt0, which\ninserted into Eq. (5) leads to an in general anisotropic\nGilbert damping. Ensemble averaging over all ran-\ndom spin configurations and positions after considerable\nbut straightforward algebra leads to the isotropic result\nGij(m) =δijG\nG=G(int)+G′(18)\nwhereG′is defined in Eq. (17). The “bulk” contribution\nto the damping is caused by the spin-relaxation due to\nthe magnetic disorder\nG(int)=NsS2ζ2ξ, (19)\nwhereNsis the number of magnetic impurities, Sis the\nimpurity spin, ζis the average strength of the magnetic\nimpurity scattering, and ξ=ξ(ηF) is a complicated ex-\npression that vanishes when ηFis either very small or\nvery large. Eq. (18) proves that Eq. (5) incorporates the\n“bulk” contribution to the Gilbert damping, which grows\nwith the number of spin-flip scatterers, in addition to in-\nterface damping. We could have derived G(int)[Eq. (19)]\nas well by the Kubo formula for the Gilbert damping.\nThe Gilbert damping has been computed before based\non the Kubo formalism based on first-principles elec-\ntronic band structures [9]. However, the ab initio appeal\nis somewhat reduced by additional approximations such\nas the relaxation time approximation and the neglect of\ndisorder vertex corrections. An advantage of the scatter-\ningtheoryofGilbertdampingisitssuitabilityformodern\nab initio techniques of spin transport that do not suffer\nfrom these drawbacks [16]. When extended to include\nspin-orbit coupling and magnetic disorder the Gilbert\ndamping can be obtained without additional costs ac-\ncording to Eq. (5). Bulk and interface contributions can\nbe readily separated by inspection of the sample thick-\nness dependence of the Gilbert damping.\nPhononsareimportantforthe understandingofdamp-\ning at elevated temperatures, which we do not explic-\nitly discuss. They can be included by a temperature-\ndependent relaxation time [9] or, in our case, structural\ndisorder. A microscopic treatment of phonon excitations\nrequires extension of the formalism to inelastic scatter-\ning, which is beyond the scope of the present paper.\nIn conclusion, we hope that our alternative formal-\nism of Gilbert damping will stimulate ab initio electronic\nstructure calculations as a function of material and dis-\norder. By comparison with FMR studies on thin ferro-\nmagnetic films this should lead to a better understanding\nof dissipation in magnetic systems.This work was supported in part by the Re-\nsearch Council of Norway, Grants Nos. 158518/143 and\n158547/431, and EC Contract IST-033749 “DynaMax.”\n∗Electronic address: Arne.Brataas@ntnu.no\n[1] For a review, see M. D. Stiles and J. Miltat, Top. Appl.\nPhys.101, 225 (2006) , and references therein.\n[2] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta-\ntus Solidi 23, 501 (1967); V. Kambersky, Can. J. Phys.\n48, 2906 (1970); V. Korenman, and R. E. Prange, Phys.\nRev. B6, 2769 (1972); V. S. 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Bauer, Phys. Rev. B 49, 14684\n(1994).\n[9] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[10] P. A. Mello and N. Kumar, Quantum Transport in Meso-\nscopic Systems , Oxford University Press (New York,\n2005).\n[11] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys.\nRev. Lett., 87, 236601 (2001).\n[12] M. Moskalets and M. B¨ uttiker, Phys. Rev. B 66, 035306\n(2002);Phys. Rev. B 66, 205320 (2002).\n[13] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404(R) (2002); X. Wang,\nG. E. W. Bauer, B. J. van Wees, A. Brataas, and Y.\nTserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n[14] B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A.Brataa s,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett 90,\n187601 (2003);M. V. Costache, M. Sladkov, S. M. Watts,\nC. H. van der Wal, and B. J. van Wees, Phys. Rev.\nLett.97, 216603 (2006); G. Woltersdorf, O. Mosendz, B.\nHeinrich, and C. H. Back, Phys. Rev. Lett 99, 246603\n(2007).\n[15] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).\n[16] M. Zwierzycki et al., Phys. Stat. Sol. B 245, 623 (2008)"    },    {        "title": "2209.07908v1.Pseudo_PT_symmetric_Dirac_equation___effect_of_a_new_mean_spin_angular_momentum_operator_on_Gilbert_damping.pdf",        "content": "arXiv:2209.07908v1  [quant-ph]  16 Sep 2022Pseudo- PTsymmetric Dirac equation : effect of a new mean\nspin angular momentum operator on Gilbert damping\nY. Bouguerra, S. Mehani, K. Bechane and M. Maamache\nLaboratoire de Physique Quantique et Syst` emes Dynamiques ,\nFacult´ e des Sciences, Universit´ e Ferhat Abbas S´ etif 1, S ´ etif 19000, Algeria\nP. -A. Hervieux\nInstitut de Physique et Chimie des Mat´ eriaux de Strasbourg ,\nCNRS and Universit´ e de Strasbourg BP 43, F-67034 Strasbour g, France\n(Dated: September 19, 2022)\nAbstract\nThe pseudo- PTsymmetric Dirac equation is proposed and analyzed by using a non-unitary\nFoldy-Wouthuysen transformations. A new spin operator PTsymmetric expectation value (called\nthe mean spin operator) for an electron interacting with a ti me-dependent electromagnetic field is\nobtained. Weshowthatspinmagnetization -whichisthequan tityusuallymeasuredexperimentally\n- is not described by the standard spin operator but by this ne w mean spin operator to properly\ndescribe magnetization dynamics in ferromagnetic materia ls and the corresponding equation of\nmotion is compatible with the phenomenological model of the Landau-Lifshitz-Gilbert equation\n(LLG).\nPACS numbers: Pseudo PT Symmetry; Non-Hermitian Dirac equation ; Foldy-Wouthuysen transformation;\nLandau-Lifshitz-Gilbert equation\n1In the field of micromagnetism ,which provides the physical framework for understanding\nand simulating ferromagnetic materials ,there is a fundamental unsolved problem which is\nthe microscopic origin of the intrinsic Gilbert damping. However, this d amping mechanism\nhas been introduced phenomenologically by T. L. Gilbert in 1955 for de scribing the spatial\nand temporal evolution of the magnetization (known as the LLG equ ation), a vector field\nwhich determines the properties of ferromagnetic materials on the sub-micron length scale\n[1]. Let us stress that this equation leads to the conservation of th e magnetization modulus.\nThis phenomenological model has since been validated by numerous e xperimental data and\nconstitutes the foundation of micromagnetism [2]. Moreover, magn etic damping plays a\ncrucial role in the operation of magnetic devices .The scattering theory can be used to\ncompute the Gilbert damping tensor [3].\nSpinisaquantumconcept [4]thatarisesnaturallyfromtheDiracth eoryandisassociated\nwith the operator ˆΣD≡1\n2iα∧α=\nσ0\n0σ\nwhereα=\n0σ\nσ0\nandσare the usual\n2×2 Pauli matrices [5]. For a classical version of the spin (see Supplementary\nMaterials). Usually, spin magnetization M(r,t) (the quantity which is experimentally\nmeasured) is defined as the expectation value of the spin angular mo mentum given by\nµB/angbracketleftBig\nΨD|ˆΣD|ΨD/angbracketrightBig\nwithµB≡e/planckover2pi1\n2mthe Bohr’s magneton ( e <0) and where ΨDisasolution\nof the Dirac equation. Indeed, in most magnetic materials the orbita l moment is quenched\nand therefore magnetism is only due to the spins [6].\nWhile for a free electron the spin angular momentum in the Heisenberg picture is not a\nconstant of motion/parenleftBigg\ndˆΣD\ndt/ne}ationslash= 0/parenrightBigg\n[5], there exists another spin operatorˆΣD\n,considered to\nbe a constant of motion ,(called the mean spin operator [7])\ndˆΣD\ndt= 0\n. In the presence\nof an electromagnetic field, which is relevant for exploring the micros copic origin of the\nLandau-Lifshitz-Gilbert (LLG) equation, a satisfactory result ha s not yet given.\nKnowingthatthespinorsintheDiractheoryconsistoffourcompon ents, itisimportantto\ncheck whether the Dirac equation yields physically reasonable result s in the non-relativistic\nexpansion case and to show that the Dirac equation reproduces th e two-component Pauli\nequation. We transform the Hamiltonian in such a way that all operat ors of the type αthat\ncouple the large to the small components will be removed. This can be achieved by a Foldy\nWouthuysen transformation [7–9] which is a non-relativistic expans ion of the Hamiltonian\n2into series of the particle ′s Compton wave lengths λC≡h\nmc.\nHickey and Moodera [10] have proposed that the spin-orbit interac tion, which arises from\nthe non-relativistic expansion of the Dirac equation, may be respon sible for the intrinsic\nferromagnetic line width. In their work, the term containing the curl of the electric field\nwhen coupled to Maxwell’s equations lead to a time-varying magnetic ind uction,and the\ntheoretical methods employed involve previously developed formalis ms in which an effective\nnon-Hermitian and time-dependent Hamiltonian is used. However, th e non-Hermiticity of\nthe Hamiltonian imposes new rules which are modified with respect to th ose of standard\nquantum mechanics. This fact was not explicitly taken into account b y the authors of [10]\nand therefore, their derivation of the intrinsic damping process is u nfortunately incorrect.\nMoreover, there is another fundamental issue which emerges fro m this work [10] concerning\nhow to properly perform the coupling between the classical Maxwell equations and the\nquantum evolution resulting from the non-relativistic limit of the Dirac equation. In what\nfollows, we show how to overcome this difficulty by using the well-known correspondence\nprinciple. Intheref. [11], themain goalwastodemonstratethatt hereisawaytoderivethe\nLLG equation coming from a non Hermitian quantum mechanics and to s park a discussion\nabout the connection between quantum and classical spin dynamics . Unfortunately, the\nquantum Heisenberg equation for a non-Hermitian Hamiltonian opera tor describing the\ndamping process is not compatible with the time-evolution operators for non-Hermitian\nHamiltonian operator.\nFromthe relativistic Diracequation, performing a Foldy-Wouthuyse n transformationand\nusing the Heisenberg equation of spin motion, Mondal et al [12–14] derive general relativistic\nexpressions for the Gilbert damping, but the term involving the cros s-product between the\nmagnetisationandthetime-derivativeofthemagneticfieldispurelyim aginary,andtherefore\nappears not to correspond to damping.\nIn the seminal work made by Dirac on relativistic quantum mechanics, the corresponding\nHamiltonian would be Hermitian. We stress that this property is very u seful to have in a\nphysical system, however we argue that the same features can b e achieved when starting\nfromnon-HermitianHamiltonian systems. These features canalso b eobtainedfromtheories\nbased on non-Hermitian Hamiltonians that have been considered in diff erent contexts. And\nwe distinguish three separate regimes: i) The PT-symmetric regime where the eigenval-\nues are real, ii) The spontaneously broken PTregime where the eigenvalues are complex\n3conjugate pairs and iii) The regime with complex, unrelated, eigenvalu es in which the PT\n-broken regime.\nOur objective here is to derive the LLG equation based on a non-Her mitian Dirac Hamil-\ntonian when compared to the most common standard approaches [ 12–14].\nTherefore, the Dirac equation in its fundamental representation is not unique to either\nHermitian quantum mechanics orquantum field theory. By relaxing the assumption of Her-\nmiticity andadoptinginstead the principles of P0T0-symmetry quantum mechanics outlined\nin the following paragraph, we will not make any modifications to the Dir ac equation. By\nP0T0-symmetry we mean reflection in space, with a simultaneous reversa l of time. The\nfundamental representation of the Dirac equation emerges comp letelyintact,identical in\nevery aspect to the Dirac equation derived from Hermitian theory. Before constructing the\nanalogous 4-d representation using the principles of P0T0quantum mechanics, let us briefly\nrecall the notion of P0T0-symmetry.\nThe Hermiticity of quantum Hamiltonians depends on the choice of the inner product\nof the states in the physical Hilbert space. This point was first point ed out by Bender et\nal [16, 17]. They showed that a wide class of Hamiltonians that respec tP0T0-symmetry\ncan exhibit entirely real spectra. Since then P0T0-symmetry has been a subject of intense\ninterest in the field of quantum mechanics.\nWhile any evidence of P0T0-symmetry has remained out of reach due to the hermitian\nnatureofthequantummechanics theory,opticshave providedafe rtilegroundforobservation\nof this property- P0T0-symmetry-since this field mainly relies on the presence of gain and\nloss.Note that even though HandP0T0commute, they do not continuously have identical\neigenvectors, as a result of the anti-linearity of the P0T0operator. If HandP0T0don’t have\nthe same eigenvectors, we say that the P0T0−symmetry is broken. The parity operator P0\neffects the momentum operator pand the position operator ras (P0:r→ −r,p→\n−p). This parity transformation has the following effect on the various vector potentials\nP0A(r,t)(P0)−1=−A(r,t) andP0Φ(r,t)(P0)−1= Φ(r,t), expressing thus their scalar\nand vector nature.\nThe anti-linear time reversal operator T0has the effect of changing the sign of the mo-\nmentum operator p,the pure imaginary complex quantity iand the time t(T0:r→r,\np→ −p, i→ −i, t→ −t). Since A(r,t) is generated by currents, which reverse s\nsignswhen the sense of time is reversed, it holds that T0A(r,t)(T0)−1=−A(r,t)\n4andT0Φ(r,t)(T0)−1= Φ(r,t).The two reflection operators commute with each other:\nP0T0=T0P0.\nTherefore, it is natural to introduce a modified Hilbert space, which is now endowed with\nP0T0-inner product, for the P0T0-symmetric nonself-adjoint theories. In such a Hilbert\nspace, the time evolution becomes unitary as the Hamiltonian is self- P0T0-adjoint and\nthe eigenfunctions form a complete set of orthonormal functions . But the norms of the\neigenfunctions have alternate signs even in the new Hilbert space en dowed with the P0T0-\ninnerproducts. Infact, anytheoryhavinganunbroken P0T0-symmetryitexistsasymmetry\nof the Hamiltonian associated with the fact that there are equal nu mbers of positive-norm\nand negative-norm states [17]:\n/an}b∇acketle{tψm,ψn/an}b∇acket∇i}htP0T0=/integraldisplay\ndx/bracketleftbig\nP0T0ψn(x)/bracketrightbig\nψm(x) =/integraldisplay\ndxψ∗\nn(−x)ψm(x)\n= (−1)nδmn (1)\nThe situation here is analogous to the problem that Dirac encounter ed in formulating the\nspinor wave equation in relativistic quantum theory [18] .\nThis again raises an obstacle in probabilistic interpretation in spite of t he systembeingin\nanunbroken P0T0phase. Afterwards, a new symmetry C, inherent to all P0T0-symmetric\nnon-Hermitian Hamiltonians, has been introduced [17]. Ccommutes with both HandP0T0\nand fixes the problem of negative norms of the eigenfunctions when the inner products are\ntaken with respect to CP0T0-adjoint.\nDoes aP0T0-symmetric Hamiltonian Hspecify a physical quantum theory in which the\nnorms of states are positive and time evolution is unitary? The answe r is that if Hhas an\nunbroken P0T0symmetry, then it has another symmetry represented by a linear o perator\nC.Thereforewecanconstruct atime-independent inner productwit hapositive-definite norm\nin terms of C.\nAnother possibility to explain the reality of the spectrum is making use of the\npseudo/quasi-Hermiticity transformations which do not alter the e igenvalue spectra. It was\nshown by Mostafazadeh [19] that P0T0-symmetric Hamiltonians are only aspecific class\nof the general families of thepseudo-Hermitian operators. A Hamiltonian is said to be\nη-pseudo-Hermitian if:\nH†=ηHη−1, (2)\n5whereηisametric operator . Theeigenvaluesofpseudo-HermitianHamiltoniansareeither\nreal or appear in complex conjugate pairs whilethe eigenfunctions satisfy bi-orthonormality\nrelations in the conventional Hilbert space. Due to this reason, suc h Hamiltonians do not\npossessacomplete set of orthogonal eigenfunctions in the conventional Hilb ert space and\nhence the probabilistic interpretation and unitarity of time evolution have not been satisfied\nby these pseudo-Hermitian Hamiltonians.\nHowever, like the case of P0T0-symmetric non-Hermitian systems, thepresence of the\nadditional operator ηin the pseudo-Hermitian theories allows usto define a new inner\nproduct in the fashion\n/an}b∇acketle{tφ|ψ/an}b∇acket∇i}htη=/an}b∇acketle{tηφ|ψ/an}b∇acket∇i}ht=/integraldisplay\n(ηφ(x))ψ(x)dx=/an}b∇acketle{tφ|ηψ/an}b∇acket∇i}ht. (3)\nLateranovel concept ofthepseudoparity-time (pseudo -P0T0)symmetry wasintroduced\nin [15] to connect the non-Hermitian Hamiltonian Hto its Hermitian conjugate H†\nH†=/parenleftbig\nP0T0/parenrightbig\nH/parenleftbig\nP0T0/parenrightbig−1(4)\nwhere in the expression of the inner product (3), the metric ηis replaced by P0T0. We now\nturn our attention to the main topic of interest, the spatial reflec tion and the time-reversal\ninvariance of the Dirac equation. The complete spatial reflection (p arity) transformation\nfor spinors and the complete time-inversion operator are denoted asP=γ0P0andT=\n−iα1α3T0=iγ1γ3T0such that\nPγ0P−1=γ0,PγiP−1=−γi\nTγ0T−1=γ0,TαT−1=−α (5)\nwhere the Hermitian matrices β=γ0andαsatisfy the ‘Dirac algebra’ {αi,αj}= 2δij;\n{αi,β}= 0.\nInwhat follows, we will denote by\nˆHD(t) =i/parenleftbig\ncα.(ˆp+ieA(r,t))−eΦ(r,t)+mc2β/parenrightbig\n(6)\nthe non-Hermitian DiracHamiltonian for a single electron in thepresence of a classical time-\ndependent external electromagnetic field defined by ( A(r,t),Φ(r,t)). The associated non-\nHermitian Dirac equation for a single electron in thepresence of a classical time-dependent\n6external electromagnetic field reads\ni/planckover2pi1∂ΨD(r,t)\n∂t=ˆHD(t)ΨD(r,t)\n=i/parenleftBig\nˆT+ˆV+mc2β/parenrightBig\nΨD(r,t), (7)\nwhere ΨD(r,t) = col(u(r,t),v(r,t)),is bispinors verifying the pseudo-othogonality relation\n/angbracketleftbig\nΨD|ΨD/angbracketrightbig\nPT=I,ˆV≡ −eΦ andˆT≡cα.ˆπ=cα.(ˆp+ieA(r,t)) is the kinetic energy which\nproduces a coupling between small and large components of the Dirac wavefun ction ΨD,\nα,βandγ5(see Eq.(15)) are the Dirac matrices [20]. The PTsymmetry condition\n(PT)ˆHD(t)(PT)−1=ˆH†D(t), (8)\nconnects the non-Hermitian Dirac Hamiltonian ˆHD(t) to its Hermitian conjugate\nˆH†D(t).This observation leads us to introduce a novel concept of the pseu do-parity-time\n(pseudo-PT) symmetry, where ( PT) is interpreted as a metric. Thus as the case of pseudo-\nhermiticity, the bispinor ΨD(r,t) = col(u(r,t),v(r,t)),verifiesthe pseudo-othogonality\nrelation/angbracketleftbig\nΨD|ΨD/angbracketrightbig\nPT=I. Note that, there is another situation which differs from that\ndescribed above ,also called pseudo- PTsymmetry which means that the system can have\na real eigenvalues whether or not the original system is PT- symmetric [21, 22].\nAs we are dealing with the non-relativistic expansion of the Dirac equa tion, the following\ndecomposition ΨD(r,t) = Π(r,t)×χD(us,t) [23] can be used, where χD(us,t) is the time-\ndependent bi-spinor representing the spin state oriented in the dir ection defined by usand\nΠ(r,t) thescalarpart of the wave function.\nWe argue that, in the non-relativistic limit, the operatorˆΣD\nis the one which must be\ninterpreted as the spin operator in the Pauli theory and used to de fine the magnetization as\nM(r,t)≡µB/angbracketleftbigg\nχD|ˆΣD\n|χD/angbracketrightbigg\nPT=µB/angbracketleftbigg\nχD|ˆU−1ˆΣFWˆU|χD/angbracketrightbigg\nPT=µB/an}b∇acketle{tχFW|ˆΣFW|χFW/an}b∇acket∇i}htPTwhere\nχFWis the spin part of the Dirac bi-spinor wave function ΨDin a non-relativistic expansion,\nobtained by using the Foldy-Wouthuysen (FW) transformation [7] andˆUis the associated\noperator (see the definition in the following). It worth mentioning th at,according to the\nabove definition ,expanding the spin operator (to a consistent order in h/mc) and using the\nDirac representation of the wave function is equivalent to expand ingthe wave function (also\nto a consistent order in h/mcusing the FW transformation) and keep ingthe original spin\noperator in the Dirac representation. In this work we have chosen to expand the mean value\noperator. The classical magnetization, M(r,t), is obtained by using the correspondence\n7principle. Indeed, we will show in what follows that the equation of mot ion of the mean\nspin operator for an electron interacting with a time-dependent ele ctromagnetic field leads\nto the LLG equation of motion revealing thus its microscopic origin.\nIna seminal work, FoldyandWouthuysen (FW)solved theproblem of finding a canonical\ntransformation thatallows to obtain a two-component theory in the low-energy limit (Pauli\napproximation) ,in the case of the Dirac equation coupled to an electromagnetic field [7 ].\nUnfortunately, contrarytothefree-electroncase, thesolutio ncannotbeexpressed inaclosed\nform. However, FW showed how to obtain successive approximation s of this transformation\nas a power series expansion in powers of the Compton wave length of the particle λC≡\nh\nmc. This procedure, generally restricted to the second-order in 1 /m, is presented in many\ntextbookson relativistic quantum mechanics [8, 20, 24, 25] and has been exte nded to fifth\norder in powers of 1 /m[26].\nIn what follows, the symbol [ ˆC,ˆD] ({ˆC,ˆD}) denotes the commutator (anticommutator)\nof the operators ˆCandˆD. We shall also use the following notations: ˆX≡X(r,t) and\nˆY≡Y(r,t).\nIn the Hermitian Dirac representation\nˆHhD(t) =/parenleftbig\ncα.(ˆp−eA(r,t))+eΦ(r,t)+mc2β/parenrightbig\n, (9)\nthe Heisenberg equation of motion for the spin operator reads as f ollows\ndˆΣD\ndt=i\n/planckover2pi1/bracketleftBig\nˆHhD,ˆΣD/bracketrightBig\n=−2c\n/planckover2pi1[α∧(ˆp−eA(r,t)]. (10)\nIt is well established that the expectation value onto/vextendsingle/vextendsingleΨD/angbracketrightbig\nof the above equation does\nnot lead to the LLG equation for the magnetization. However, as it w illbeshown in the\nfollowing, the latter can be obtained by using the non unitary FW tran sformation and the\nnew definitionof themagnetizationasanexpectation valueof means pinoperator. Thefirst-\nand second-order terms of the FW expansion in powers of 1 /mcorrespond, respectively, to\nthe precessional motion of the magnetization around an effective m agnetic field and its\ndamping.\nSince the Hamiltonian ˆHD(t) has a similar structure to the one in the Dirac case, by\nanalogy with the latter, we use for ˆU(t) the form eˆS(t)whereˆSis a non self-adjoint\n8operator. Therefore, the transformation ΨFW(r,t) =eˆS(t)ΨD(r,t)≡ˆU(t)ΨD(r,t) leads to a\nnew Hamiltonian\nˆHFW(t) =eˆS(t)/parenleftbigg\nˆHD(t)−i/planckover2pi1∂\n∂t/parenrightbigg\ne−ˆS(t), (11)\nwhereˆSis a non self-adjoint operator. More generally, any operator inthe FW\nrepresentation ,that is not explicitly time dependent, ˆOFWwill be transformed in the Dirac\nrepresentation as ˆOD(t) =ˆU−1(t)ˆOFWˆU(t).\nThe most natural extension oftheEhrenfest equation tonon-He rmitian pseudo- PTsym-\nmetric systems is by replacing a Hermitian ˆHhDwitha non-Hermitian one. The structure\nof the Ehrenfest equation does not change, having assumed that part of the action of T\n(i.e.T0) is to send t→ −t, i.e.;we show that it anticommute swith the operator ∂/∂t,\nconsequently, the operator PTcommuteswithi∂/∂t.Which immediately leads us to\ndeduce the Ehrenfest equation of motion for the diagonal matrix e lement of an operator\nˆOD(t)\nd\ndt/angbracketleftBig\nΨD|ˆOD|ΨD/angbracketrightBig\nPT=/angbracketleftBigg\nΨD|i\n/planckover2pi1/bracketleftBig\nˆHD,ˆOD/bracketrightBig\n+∂ˆOD\n∂t|ΨD/angbracketrightBigg\nPT. (12)\nIt’s straightforward to show that the equation of motion (12) lead s to\nd\ndt/angbracketleftBig\nΨFW|ˆOFW|ΨFW/angbracketrightBig\nPT=/angbracketleftbigg\nΨFW|i\n/planckover2pi1/bracketleftBig\nˆHFW,ˆOFW/bracketrightBig\n|ΨFW/angbracketrightbigg\nPT. (13)\nBy expanding ˆS=ˆS1m−1+ˆS2m−2+ˆS3m−3+...withˆS1=β\n2c(α.ˆπ),ˆS2=−i/planckover2pi1e\n4c3(α.E),\nˆS3=−β\n8c6/parenleftBig\n4c3\n3(α.ˆπ)(α.ˆπ)(α.ˆπ)+iec/planckover2pi12(α.∂tE)/parenrightBig\nandE=−∇Φ−∂A\n∂t[26], the mean spin\noperator is computed using the inverse FW transformation of the s pin operator asˆΣD\n=\ne−(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣDe(ˆS1/m+ˆS2/m2+ˆS3/m3)and may be expanded in power series of (1 /m)\nleading toˆΣD\n=ˆΣD\n0+ˆΣD\n1m−1+ˆΣD\n2m−2+ˆΣD\n3m−3+...\nˆΣD\n0=ˆΣD,\nˆΣD\n1=−iβ\nc(α׈π),\nˆΣD\n2=1\n8c2/parenleftBig\n−4ie/planckover2pi1B+2e/planckover2pi1/parenleftBig\nˆΣD×B/parenrightBig\n−4/parenleftBig\nˆπ×/parenleftBig\nˆΣD׈π/parenrightBig/parenrightBig/parenrightBig\n−e/planckover2pi1\n2c3(α×E),\nˆΣD\n3=β\n48c3/bracketleftBig\n(α.ˆπ),/bracketleftBig\n(α.ˆπ),/bracketleftBig\n(α.ˆπ),ˆΣD/bracketrightBig/bracketrightBig/bracketrightBig\n+β\n6c3/bracketleftBig\n(α.ˆπ)(α.ˆπ)(α.ˆπ),ˆΣD/bracketrightBig\n−e/planckover2pi12β\n4c5(α×(∂tE)), (14)\n9whereB=∇ ×A.The free case which is investigated in [7] (may be obtained in closed\nform in this case) is recovered from the above formula by substitut ingE=B= 0 ,iβ→β\nandiˆπ→ˆpleading to\nˆΣD\n=ˆΣD−iβ(α∧ˆp)\nEp−/parenleftBig\nˆp∧/parenleftBig\nˆΣD∧ˆp/parenrightBig/parenrightBig\nEp(Ep+mc)\nwhereEp=/radicalbig\nm2c2+p2.\nThe pseudo- PTequation of motion (12) for the mean spin operator is\nd\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT\n=/angbracketleftbig\nΨD/vextendsingle/vextendsingleeβ\nm/parenleftBig\nˆΣD∧B/parenrightBig\n−e/planckover2pi1\n4m2c2/parenleftBig\nˆΣD∧∂tB/parenrightBig\n−ie\n2m2c2/parenleftBig\nˆΣD∧(E∧ˆπ)/parenrightBig\n+1\n4/planckover2pi1m2c/parenleftBig/bracketleftBig\n(α.π)ˆΣD(α.π),(α.π)/bracketrightBig\n−/bracketleftbig\n(α.π)(α.π)(α.π),ΣD/bracketrightbig/parenrightBig\n+ϑ(m−3)/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT.(15)\n(In order) to check this result, we have applied to the above equat ion the direct FW trans-\nformation. It leads to\nd\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT=d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsinglee(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣD\ne−(ˆS1/m+ˆS2/m2+ˆS3/m3)/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT\n≡d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsingleˆΣ/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT=/angbracketleftbig\nΨFW/vextendsingle/vextendsinglei\n/planckover2pi1/bracketleftbigg\nˆHFW,ˆΣFW/bracketrightbigg/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT(16)\nwhereˆΣFW≡ˆΣand the expression of ˆHFWis given by\nˆHFW=iβmc2−ieˆΦ+iβˆπ2\n2m+iβˆπ4\n8m3c2+βe/planckover2pi12\n16m3c4{ˆπ,∂tE}\n−βe/planckover2pi1\n2mˆΣ./bracketleftbigg\nB−β/planckover2pi1\n4mc2/parenleftbigg\n(∇∧E)+2i\n/planckover2pi1E∧ˆπ/parenrightbigg\n+i/planckover2pi1\n8m2c4(∂tE∧ˆπ)+ ˆπ∧∂tE)/bracketrightbigg\n−βe/planckover2pi1\n8m3c2/braceleftBig\nˆπ2,ˆΣ.B/bracerightBig\n−iβ/parenleftbigge/planckover2pi1\n2m/parenrightbigg2B2\n2mc2+ie/planckover2pi12\n8m2c2∇.E+ϑ(m−4). (17)\nNow, the equation of motion for the mean spin operator (15) can be written in a simpler\nway being obtained as the pseudo- PTexpectation value a spin Dirac states |χD/an}b∇acket∇i}htassociated\ntoˆHD, one gets\nd\ndt/an}b∇acketle{tχD|ˆΣD\n|χD/an}b∇acket∇i}htPT\n=eβ\nm/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧B−e/planckover2pi1\n4m2c2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧∂tB\n−e\n2mc2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧/parenleftbigg\nE∧/an}b∇acketle{tχD|i\nmˆπ|χD/an}b∇acket∇i}htPT/parenrightbigg\n+ϑ(m−3). (18)\n10The above expression has been obtained by using/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT=/an}b∇acketle{tχD|ˆΣD\n|χD/an}b∇acket∇i}htPT. It\nis of interest to mention that the non-diagonal terms which appear at thesecondline of\n(15) are due to the Zitterbewegung phenomenon [5]. They cancel o ut when they are pseudo\naveraged out in a Dirac states.\nLet us note that, it is more convenient to use the FW representatio n to find the evolution\nof spin, (indeed) from Eq.(7) whered\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT≡d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsingleˆΣFW/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT, we get\nd\ndt/an}b∇acketle{tχFW|ˆΣ|χFW/an}b∇acket∇i}htPT=eβ\nm/an}b∇acketle{tχFW|ˆΣ∧/parenleftbigg\nB+1\n2c2/bracketleftbigg\nE∧iβ\nmˆπ/bracketrightbigg/parenrightbigg\n−e/planckover2pi1\n4m2c2/bracketleftBig\nˆΣ∧(∇∧E)/bracketrightBig\n|χFW/an}b∇acket∇i}htPT\n(19)\nConcerning the damping process, as previously explained, the only t erm of importance\nin the expression (15) is −e/planckover2pi1\n4m2c2/parenleftBig\nˆΣD∧∂tB/parenrightBig\nwhich has been obtained from the commutator\n[(α×E),(α.ˆπ)] coming from/bracketleftBig\nˆHD,ˆΣD/bracketrightBig\nand the partial derivative with respect to time\nof the mean spin angular momentum operator at second order in 1 /m[27] given in Eq.\n(14). In addition, classical Maxwell equations have been also employ ed. Similarly to the\nBreit Hamiltonian ,which is obtained from the classical Darwin Lagrangian (which also\noriginates from Maxwell equations) by using the correspondence p rinciple (CP) [25] ,we\nhere resort to the same procedure (in its inverse form, from quan tum to classical) for the\nMaxwell equations. According to this principle, the quantum counte rpartsˆf, ˆgof classical\nobservables f,gsatisfy/angbracketleftBig/bracketleftBig\nˆf,ˆg/bracketrightBig/angbracketrightBig\n=i/planckover2pi1{f,g}p,qwhere/angbracketleftBig/bracketleftBig\nˆf,ˆg/bracketrightBig/angbracketrightBig\nis the expectation value of\nthe commutator and the symbol {}p,qdenotes the Poisson bracket [28–31]. Let’s take for\ninstance the Maxwell-Faraday equation, we have\n∇∧E(r,t) =−∂B(r,t)\n∂t(20)\nwhich can be rewritten as\nǫijk{pi,Ej}p,qek=−∂B(r,t)\n∂t, (21)\nand using the CP one gets\nǫijk[ˆpi,Ej]\ni/planckover2pi1ek=−∂B(r,t)\n∂t≡ −∂tB. (22)\nConsequently, by using our definition of the magnetization M (r,t)≡\nµB/angbracketleftbigg\nχD|ˆΣD\n|χD/angbracketrightbigg\nPT≡µB/angbracketleftBig\nχFW|ˆΣ|χFW/angbracketrightBig\nPTand the CP, the equation of motion (18)\n11may be rewritten for the electron part as\ndM(r,t)\ndt=e\nmM(r,t)∧B(r,t)−e\n4m2c2M(r,t)∧∂tB(r,t)\n+e\n2mc2M(r,t)∧(E(r,t)∧v)+ϑ(m−3). (23)\nThe above equation constitutes the main result of this work.\nMoreover, if the electron is embedded in a magnetically polarizable med ium,defined\nby its magnetic polarizability χm,then∂M\n∂tgenerates a time-dependent magnetic induction\naccording to the relation ∂tB(r,t) =1\nχm∂M\n∂tand the equation (23) can be rewritten as\ndM(r,t)\ndt=−γM(r,t)∧Beff(r,t)−αG\nM/parenleftbigg\nM(r,t)∧∂M(r,t)\n∂t/parenrightbigg\n(24)\nwithγ=−e\nm>0 the gyromagnetic ratio for an isolated electron, Beff≡B−1\n2c2v∧E\nandαG≡eM\n4m2c2χm. The first term describes the precessional motion of the magnetiz ation\nvector around the direction of the effective magnetic field and the s econd term represents\nits damping ,characterized by the Gilbert’s constant αG.\nLet us stress that the first term in the right hand side of equation ( 24) can be retrieved\nfrom the non-relativistic expansion of the Bargmann-Michel-Telegd i’s equation [24, 32, 33]\nwhichrepresents the relativistic equation of motion of a classical magnetic dipole momen t\n[34]. However, the damping term cannot be obtained from this classic al description due to\nits quantum origin.\nIn summary, the mean spin angular momentum operator introduced for the first time\nby Foldy and Wouthuysen for the case of a free electron has been e xtended to the non\n-Hermitian or precisely to a pseudo PT-symmetric case of an electron interacting with\na time-dependent electromagnetic field. The expectation equation of the motion of the\nlatter leads to the Landau-Lifshitz-Gilbert equation revealing thus its microscopic origin.\nWe therefore argue that the expectation value of the pseudo-me an spin operator with the\nnew definition of PT-inner product must be used instead of the usual one to properly\ndescribe the dynamics of the spin magnetization.\n[1] T. L. Gilbert, IEEE Transactions on magnetics 40, 3443 (2 004). In this paper published in\nClassics in Magnetics, the key results of the Gilbert’s thes is are reproduced.\n12[2] H. Kronm¨uller and M. F¨ahnle, ”Micromagnetism and the M icrostructure of Ferromagnetic\nSolids”, Cambridge (2003).\n[3] Arne Brataas, Yaroslav Tserkovnyak, and Gerrit E. W. Bau er, Phys. Rev. Lett. 101, 037207\n(2008).\n[4] Jean-Marc L´ evy-Leblond, Commun. math. Phys. 6, 286-311 (1967).\n[5] J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wes ley (1967).\n[6] S. Blundell, Magnetism in Condensed Matter, Oxford Univ ersity Press, (2001).\n[7] L. Foldy and S. Wouthuysen, Phys. Rev. 78, 29 (1950).\n[8] W. Greiner, Relativistic quantum mechanics. - Wave equa tions, Springer, (2000).\n[9] J. D. Bjorken and S.D. Drell, Relativistic Quantum Mecha nics. McGraw-Hill Book Company,\nNew York (1964).\n[10] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 1376 01 (2009).\n[11] R. Wieser, Phys. Rev. Lett. 110, 147201 (2013).\n[12] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer, P hys. Rev. B 96, 024425 (2017).\n[13] R. Mondal, Marco Berritta, Peter M. Oppeneer, J. Phys.: Condens. Matter 30, 165801 (2018).\n[14] R. Mondal, Peter M. Oppeneer, J. Phys.: Condens. Matter 32, 455802 (2020).\n[15] Naima Mana and Mustapha Maamache, International Journ al of Modern Physics A, 35, No.1,\n2075001 (2020)\n[16] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5234 ( 1998).\n[17] C. M. Bender, Dorje C. Brody, and Hugh F. Jones, Phys. Rev . Lett.89, 270401 (2002).\n[18] P. A. M. Dirac, Proc. R. Soc. Lond. A 180,1 (1942).\n[19] A. Mostafazadeh, J. Math. Phys. 43, 205 (2002).\n[20] P. Strange, Relativistic Quantum Mechanics, Cambridg e University Press, (2005).\n[21] X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar an d C. Lee, Phys. Rev. Lett. 110,\n243902 (2013).\n[22] M. Maamache, S. Lamri and O. Cherbal, Annals Phys. 378, 1 50 (2017).\n[23] J. S. Roman, L. Roso and L. Plaja, J. Phys. B 37, 435 (2004).\n[24] C. Itzykson, J. -B. Zuber, Quantum Field Theory, McGraw -Hill (1985).\n[25] M. Reiher and A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH (2009).\n[26] Y. Hinschberger and P.-A. Hervieux, Physics Letters A 3 76, 813 (2012).\n13[27]ˆBof/parenleftBig\nˆΣD׈B/parenrightBig\nin Eq. (14) also originates from a commutator [ˆ πi,ˆπj].\n[28] R. L. Liboff, Foundations of Physics 17, 981 (1987).\n[29] L´ evy-Leblond, J.M. The pedagogical role and epistemological significance of gro up theory in\nquantum mechanics . Riv. Nuovo Cim. 4, 99–143 (1974)\n[30] Hove, L´ eon Van. “ Sur le probl` eme des relations entre les transformations un itaires de la\nm´ ecanique quantique et les transformations canoniques de la m´ ecanique classique .” (1951).\n[31] D. Sen, S. K. Das, A. N. Basu and S. Sengupta, Current Scie nce, Vol. 80, No.4,536-541\n(2001).\n[32] V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Let t. 2, 435 (1959).\n[33] J. D. Jackson, Classical Electrodynamics, John Wiley ( 1998).\n[34] In order to be compatible with our description which doe s not include QED effects, the Land´ e\nfactor must be g= 2.\nSupplementary Materials\nIn terms of the conjugate variable ( q,p) the classical spin− →Sis desribed by [1, 2]\n\n\nSx=/radicalbig\nS2−p2cosq\nSy=/radicalbig\nS2−p2sinq\nSz=p(25)\nthe Poisson brackets {Si,Sj}=εijkSk(i,j,karex,yorz) are analogous to the same rela-\ntionships one has with spin components and commutators in quantum mechanics.\nSuppose we have the following Hamiltonian\nH=− →B.− →S (26)\nwhich is formally identical to the Hamiltonian for a spin 1 /2 system in a uniform magnetic\nfield. We can calculate the evolution of the vector components using the standard Hamil-\ntonian techniques and The motion of spin− →Son the sphere (phase space) with (conserved)\nradiusS=/vextendsingle/vextendsingle/vextendsingle− →S/vextendsingle/vextendsingle/vextendsinglegenerated by (26), can be obtained by regarding H(26) as classical hamil-\ntonian . It may be confirmed that Hamilton’s equation reproduce exa ctly what spin does in\na magnetic field i.e,− →·\nS=− →B∧− →S.\nThe two-level spin system can be written as a classical model if we em ploy the anticom-\nmuting Grassmann variables [3–6]− →ζwhich are transformed to the spin operator after the\n14quantization∧− →ζ=∧− →S/√\n2 . Unlike the classical spin defined in the equation ((25)) which\ndoes not tranformed into a spin operator after the quantization∧− →S/ne}ationslash=− →S.\n[1] M. V. Berry, in ”Fundamental Aspects of Quantum” (Edited by V. Gorini and A. Frigerio),\nPlenum, Nato ASI series vol. 144, 267-278 (1986)).\n[2] M. Maamache, exact solution and geometic Angle for the cl assical spin system, Phys. Scr. 54,\n21 (1996).\n[3] R. Casalbuoni, On the quantization of systems with antic ommuting variables, Nuovo Cimento\nA 33, 115 (1976).\n[4] F.A. Berezin and M.S. Marinov, Particle Spin Dynamics as the Grassmann Variant of Classical\nMechanics, Ann. Phys. (N.Y) 104, 336 (1977).\n[5] E. Gozzi and W. D. Thacker, Classical adiabatic holonomy in a Grassmannian system, Phys.\nRev. D 35, 2388 (1987).\n[6] M. Maamache and O. Cherbal, Evolution of Grassmannian in variant-angle coherent states and\nnonadiabatic Hannay’s angle, Eur. Phys. J. D 6, 145 (1999).\n15"    },    {        "title": "2311.16268v2.Gilbert_damping_in_two_dimensional_metallic_anti_ferromagnets.pdf",        "content": "Gilbert damping in two-dimensional metallic anti-ferromagnets\nR. J. Sokolewicz,1, 2M. Baglai,3I. A. Ado,1M. I. Katsnelson,1and M. Titov1\n1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands\n2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden\n(Dated: March 29, 2024)\nA finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional\nmetallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert\ndamping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb\nmagnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three\nregimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength,\nElliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong\nspin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane\nGilbert damping component. We also show that anisotropy of Gilbert damping persists for any\nfinite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector.\nIsotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon\nwavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems.\nI. INTRODUCTION\nMagnetization dynamics in anti-ferromagnets con-\ntinue to attract a lot of attention in the context\nof possible applications1–4. Various proposals utilize\nthe possibility of THz frequency switching of anti-\nferromagnetic domains for ultrafast information storage\nand computation5,6. The rise of van der Waals magnets\nhas had a further impact on the field due to the pos-\nsibility of creating tunable heterostructures that involve\nanti-ferromagnet and semiconducting layers7.\nUnderstanding relaxation of both the N´ eel vector and\nnon-equilibrium magnetization in anti-ferromagnets is\nrecognized to be of great importance for the function-\nality of spintronic devices8–13. On one hand, low Gilbert\ndamping must generally lead to better electric control of\nmagnetic order via domain wall motion or ultrafast do-\nmain switching14–16. On the other hand, an efficient con-\ntrol of magnetic domains must generally require a strong\ncoupling between charge and spin degrees of freedom due\nto a strong spin-orbit interaction, that is widely thought\nto be equivalent to strong Gilbert damping.\nIn this paper, we focus on a microscopic analysis of\nGilbert damping due to Dyakonov-Perel and Elliot-Yafet\nmechanisms. We apply the theory to a model of a two-\ndimensional N´ eel anti-ferromagnet with a honeycomb\nmagnetic lattice.\nTwo-dimensional magnets typically exhibit either\neasy-plane or easy-axis anisotropy, and play crucial\nroles in stabilizing magnetism at finite temperatures17,18.\nEasy-axis anisotropy selects a specific direction for mag-\nnetization, thereby defining an axis for the magnetic or-\nder. In contrast, easy-plane anisotropy does not select a\nparticular in-plane direction for the N´ eel vector, allowing\nit to freely rotate within the plane. This situation is anal-\nogous to the XY model, where the system’s continuous\nsymmetry leads to the suppression of out-of-plane fluc-\ntuations rather than fixing the magnetization in a spe-\ncific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can\ngrow uncontrollably large to destroy any long-range mag-\nnetic order, according to the Mermin-Wagner theorem21.\nRecent density-functional-theory calculations for\nsingle-layer transition metal trichalgenides22, predict the\nexistence of a large number of metallic anti-ferromagnets\nwith honeycomb lattice and different types of magnetic\norder as shown in Fig. 1. Many of these crystals may\nhave the N´ eel magnetic order as shown in Fig. 1a and are\nmetallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3,\nFeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3,\nFeSnSe 3, NiSnS 3. Apart from that it has been predicted\nthat anti-ferromagnetism can be induced in graphene by\nbringing it in proximity to MnPSe 323or by bringing it\nin double proximity between a layer of Cr 2Ge2Te6and\nWS224.\nPartly inspired by these predictions and recent\ntechnological advances in producing single-layer anti-\nferromagnet crystals, we propose an effective model to\nstudy spin relaxation in 2D honeycomb anti-ferromagnet\nwith N´ eel magnetic order. The same system was studied\nby us in Ref. 25, where we found that spin-orbit cou-\npling introduces a weak anisotropy in spin-orbit torque\nand electric conductivity. Strong spin-orbit coupling was\nshown to lead to a giant anisotropy of Gilbert damping.\nOur analysis below is built upon the results of Ref. 25,\nand we investigate and identify three separate regimes\nof spin-orbit strength. Each regime is characterized by\nqualitatively different dependence of Gilbert damping on\nspin-orbit interaction and conduction electron transport\ntime. The regime of weak spin-orbit interaction is dom-\ninated by exchange field relaxation of electron spin, and\nthe regime of moderate spin-orbit strength is dominated\nby Elliot-Yafet spin relaxation. These two regimes are\ncharacterized also by a universal factor of 2 anisotropy\nof Gilbert damping. The regime of strong spin-orbit\nstrength, which leads to substantial splitting of electron\nFermi surfaces, is characterized by Dyakonov-Perel relax-\nation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2  [cond-mat.dis-nn]  28 Mar 20242\nFIG. 1. Three anti-ferromagnetic phases commonly found\namong van-der-Waals magnets. Left-to-right: N´ eel, zig-zag,\nand stripy.\nlaxation of the perpendicular-to-the-plane Gilbert damp-\ning which leads to a giant damping anisotropy. Isotropic\nGilbert damping is restored only for finite magnon wave\nvectors such that the magnon wavelength is smaller than\nthe spin-orbit length.\nGilbert damping in a metallic anti-ferromagnet can be\nqualitatively understood in terms of the Fermi surface\nbreathing26. A change in the magnetization direction\ngives rise to a change in the Fermi surface to which the\nconduction electrons have to adjust. This electronic re-\nconfiguration is achieved through the scattering of elec-\ntrons off impurities, during which angular momentum is\ntransferred to the lattice. Gilbert damping, then, should\nbe proportional to both (i) the ratio of the spin life-time\nand momentum life-time of conduction electrons, and (ii)\nthe electric conductivity. Keeping in mind that the con-\nductivity itself is proportional to momentum life-time,\none may conclude that the Gilbert damping is linearly\nproportional to the spin life-time of conduction electrons.\nAt the same time, the spin life-time of localized spins is\ninversely proportional to the spin life-time of conduc-\ntion electrons. A similar relation between the spin life-\ntimes of conduction and localized electrons also holds\nfor relaxation mechanisms that involve electron-magnon\nscattering27.\nOur approach formally decomposes the magnetic sys-\ntem into a classical sub-system of localized magnetic mo-\nments and a quasi-classical subsystem of conduction elec-\ntrons. A local magnetic exchange couples these sub-\nsystems. Localized magnetic moments in transition-\nmetal chalcogenides and halides form a hexagonal lat-\ntice. Here we focus on the N´ eel type anti-ferromagnet\nthat is illustrated in Fig. 1a. In this case, one can de-\nfine two sub-lattices A and B that host local magnetic\nmoments SAandSB, respectively. For the discussion of\nGilbert damping, we ignore the weak dependence of both\nfields on atomic positions and assume that the modulus\nS=|SA(B)|is time-independent.\nUnder these assumptions, the magnetization dynamics\nof localized moments may be described in terms of two\nfields\nm=1\n2S\u0000\nSA+SB\u0001\n,n=1\n2S\u0000\nSA−SB\u0001\n, (1)\nwhich are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the\nmean-field approach, the vector fields yield the equations\nof motion\n˙n=−Jn×m+n×δs++m×δs−, (2a)\n˙m=m×δs++n×δs−, (2b)\nwhere dot stands for the time derivative, while δs+and\nδs−stand for the mean staggered and non-staggered non-\nequilibrium fields that are proportional to the variation of\nthe corresponding spin-densities of conduction electrons\ncaused by the time dynamics of nandmfields. The en-\nergy Jis proportional to the anti-ferromagnet exchange\nenergy for localized momenta.\nIn Eqs. (2) we have omitted terms that are propor-\ntional to easy axis anisotropy for the sake of compact-\nness. These terms are, however, important and will be\nintroduced later in the text.\nIn the framework of Eqs. (2) the Gilbert damping can\nbe computed as the linear response of the electron spin-\ndensity variation to a time change in both the magneti-\nzation and the N´ eel vector (see e. g. Refs.25,28,29).\nIn this definition, Gilbert damping describes the re-\nlaxation of localized spins by transferring both total and\nstaggered angular momenta to the lattice by means of\nconduction electron scattering off impurities. Such a\ntransfer is facilitated by spin-orbit interaction.\nThe structure of the full Gilbert damping tensor can be\nrather complicated as discussed in Ref. 25. However, by\ntaking into account easy axis or easy plane anisotropy we\nmay reduce the complexity of relevant spin configurations\nto parameterize\nδs+=α∥\nm˙m∥+α⊥\nm˙m⊥+αmn∥×(n∥×˙m∥),(3a)\nδs−=α∥\nn˙n∥+α⊥\nn˙n⊥+αnn∥×(n∥×˙n∥), (3b)\nwhere the superscripts ∥and⊥refer to the in-plane\nand perpendicular-to-the-plane projections of the corre-\nsponding vectors, respectively. The six coefficients α∥\nm,\nα⊥\nm,αm,α∥\nn,α⊥\nn, and αnparameterize the Gilbert damp-\ning.\nInserting Eqs. (3) into the equations of motion of\nEqs. (2) produces familiar Gilbert damping terms. The\ndamping proportional to time-derivatives of the N´ eel vec-\ntornis in general many orders of magnitude smaller than\nthat proportional to the time-derivatives of the magneti-\nzation vector m25,30. Due to the same reason, the higher\nharmonics term αmn∥×(n∥×∂tm∥) can often be ne-\nglected.\nThus, in the discussion below we may focus mostly on\nthe coefficients α∥\nmandα⊥\nmthat play the most important\nrole in the magnetization dynamics of our system. The\nterms proportional to the time-derivative of ncorrespond\nto the transfer of angular momentum between the sub-\nlattices and are usually less relevant. We refer to the\nresults of Ref. 25 when discussing these terms.\nAll Gilbert damping coefficients are intimately related\nto the electron spin relaxation time. The latter is rel-\natively well understood in non-magnetic semiconductors3\nwith spin-orbital coupling. When a conducting electron\nmoves in a steep potential it feels an effective magnetic\nfield caused by relativistic effects. Thus, in a disordered\nsystem, the electron spin is subject to a random magnetic\nfield each time it scatters off an impurity. At the same\ntime, an electron also experiences precession around an\neffective spin-orbit field when it moves in between the\ncollisions. Changes in spin direction between collisions\nare referred to as Dyakonov-Perel relaxation31,32, while\nchanges in spin-direction during collisions are referred to\nas Elliot-Yafet relaxation33,34.\nThe spin-orbit field in semiconductors induces a char-\nacteristic frequency of spin precession Ω s, while scalar\ndisorder leads to a finite transport time τof the con-\nducting electrons. One may, then, distinguish two limits:\n(i) Ω sτ≪1 in which case the electron does not have\nsufficient time to change its direction between consec-\nutive scattering events (Elliot-Yafet relaxation), and (ii)\nΩsτ≫1 in which case the electron spin has multiple pre-\ncession cycles in between the collisions (Dyakonov-Perel\nrelaxation).\nThe corresponding processes define the so-called spin\nrelaxation time, τs. In a 2D system the spin life-time\nτ∥\ns, for the in-plane spin components, appears to be dou-\nble the size of the life-time of the spin component that\nis perpendicular to the plane, τ⊥\ns32. This geometric ef-\nfect has largely been overlooked. For non-magnetic 2D\nsemiconductor one can estimate35,36\n1\nτ∥\ns∼(\nΩ2\nsτ,Ωsτ≪1\n1/τ, Ωsτ≫1, τ∥\ns= 2τ⊥\ns. (4)\nA pedagogical derivation and discussion of Eq. 4 can\nbe found in Refs. 35 and 36. Because electrons are con-\nfined in two dimensions the random spin-orbit field is\nalways directed in-plane, which leads to a decrease in the\nin-plane spin-relaxation rate by a factor of two compared\nto the out-of-plane spin-relaxation rate as demonstrated\nfirst in Ref. 32 (see Refs. 36–40 as well). The reason is\nthat the perpendicular-to-the-plane component of spin is\ninfluenced by two components of the randomly changing\nmagnetic field, i. e. xandy, whereas the parallel-to-the-\nplane spin components are only influenced by a single\ncomponent of the fluctuating fields, i. e. the xspin pro-\njection is influenced only by the ycomponent of the field\nand vice-versa. The argument has been further general-\nized in Ref. 25 to the case of strongly separated spin-orbit\nsplit Fermi surfaces. In this limit, the perpendicular-to-\nthe-plane spin-flip processes on scalar disorder potential\nbecome fully suppressed. As a result, the perpendicular-\nto-the-plane spin component becomes nearly conserved,\nwhich results in a giant anisotropy of Gilbert damping in\nthis regime.\nIn magnetic systems that are, at the same time, con-\nducting there appears to be at least one additional energy\nscale, ∆ sd, that characterizes exchange coupling of con-\nduction electron spin to the average magnetic moment of\nlocalized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc-\ntionselectron and the localized magnetic moment of d\norfelectron on an atom.) This additional energy scale\ncomplicates the simple picture of Eq. (4) especially in the\ncase of an anti-ferromagnet. The electron spin precession\nis now defined not only by spin-orbit field but also by\n∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1\nmay easily coexist. This dissolves the distinction between\nElliot-Yafet and Dyakonov-Perel mechanisms of spin re-\nlaxation. One may, therefore, say that both Elliot-Yafet\nand Dyakonov-Perel mechanisms may act simultaneously\nin a typical 2D metallic magnet with spin-orbit coupling.\nThe Gilbert damping computed from the microscopic\nmodel that we formulate below will always contain both\ncontributions to spin-relaxation.\nII. MICROSCOPIC MODEL AND RESULTS\nThe microscopic model that we employ to calculate\nGilbert damping is the so-called s–dmodel that couples\nlocalized magnetic momenta SAandSBand conducting\nelectron spins via the local magnetic exchange ∆ sd. Our\neffective low-energy Hamiltonian for conduction electrons\nreads\nH=vfp·Σ+λ\n2\u0002\nσ×Σ\u0003\nz−∆sdn·σΣzΛz+V(r),(5)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli\nmatrices acting on sub-lattice, spin and valley space,\nrespectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ, and a random im-\npurity potential V(r).\nThe Hamiltonian of Eq. (5) can be viewed as the\ngraphene electronic model where conduction electrons\nhave 2D Rashba spin-orbit coupling and are also cou-\npled to anti-ferromagnetically ordered classical spins on\nthe honeycomb lattice.\nThe coefficients α∥\nmandα⊥\nmare obtained using linear\nresponse theory for the response of spin-density δs+to\nthe time-derivative of magnetization vector ∂tm. Impu-\nrity potential V(r) is important for describing momen-\ntum relaxation to the lattice. This is related to the an-\ngular momentum relaxation due to spin-orbit coupling.\nThe effect of random impurity potential is treated pertur-\nbatively in the (diffusive) ladder approximation that in-\nvolves a summation over diffusion ladder diagrams. The\ndetails of the microscopic calculation can be found in the\nAppendices.\nBefore presenting the disorder-averaged quantities\nα∥,⊥\nm, it is instructive to consider first the contribution\nto Gilbert damping originating from a small number of\nelectron-impurity collisions. This clarifies how the num-\nber of impurity scattering effects will affect the final re-\nsult.\nLet us annotate the Gilbert damping coefficients with\nan additional superscript ( l) that denotes the number\nof scattering events that are taken into account. This4\n01234\u0016\u000b(i)\n?[\"\u001c]\n\u0016\u000b(0)\n?\u0016\u000b(1)\n?\u0016\u000b(2)\n? \u0016\u000b(1)\n?\n10\u0000210\u00001100101\n\u0015\u001c01234\u0016\u000b(i)\nk[\"\u001c]\n\u0016\u000b(0)\nk\u0016\u000b(1)\nk\u0016\u000b(2)\nk\u0016\u000b(1)\nk\nFIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green)\nlines correspond to the results of the numerical evaluation of\n¯α(l)\nm,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The\ndashed (orange) line corresponds to the diffusive (fully vertex\ncorrected) results for ¯ α⊥,∥.\nm.\nmeans, in the diagrammatic language, that the corre-\nsponding quantity is obtained by summing up the ladder\ndiagrams with ≤ldisorder lines. Each disorder line cor-\nresponds to a quasi-classical scattering event from a sin-\ngle impurity. The corresponding Gilbert damping coeffi-\ncient is, therefore, obtained in the approximation where\nconduction electrons have scattered at most lnumber\nof times before releasing their non-equilibrium magnetic\nmoment into a lattice.\nTo make final expressions compact we define the di-\nmensionless Gilbert damping coefficients ¯ α∥,⊥\nmby extract-\ning the scaling factor\nα∥,⊥\nm=A∆2\nsd\nπℏ2v2\nfS¯α∥,⊥\nm, (6)\nwhere Ais the area of the unit cell, vfis the Fermi ve-\nlocity of the conducting electrons and ℏ=h/2πis the\nPlanck’s constant. We also express the momentum scat-\ntering time τin inverse energy units, τ→ℏτ.\nLet us start by computing the coefficients ¯ α∥,⊥(l)\nm in the\nformal limit ∆ sd→0. We can start with the “bare bub-\nble” contribution which describes spin relaxation without\na single scattering event. The corresponding results read\n¯α(0)\nm,⊥=ετ1−λ2/4ε2\n1 +λ2τ2, (7a)\n¯α(0)\nm,∥=ετ\u00121 +λ2τ2/2\n1 +λ2τ2−λ2\n8ε2\u0013\n, (7b)\nwhere εdenotes the Fermi energy which we consider pos-\nitive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1,\nwhile the parameter λτmay in principle be arbitrary. For\nλτ≪1 the disorder-induced broadening of the electron\nFermi surfaces exceeds the spin-orbit induced splitting.\nIn this case one basically finds no anisotropy of “bare”\ndamping: ¯ α(0)\nm,⊥= ¯α(0)\nm,∥. In the opposite limit of substan-\ntial spin-orbit splitting one gets an ultimately anisotropic\ndamping ¯ α(0)\nm,⊥≪¯α(0)\nm,∥. This asymptotic behavior can be\nsummarized as\n¯α(0)\nm,⊥=ετ(\n1 λτ≪1,\n(λτ)−2λτ≫1,(8a)\n¯α(0)\nm,∥=ετ(\n1 λτ≪1,\n1\n2\u0000\n1 + (λτ)−2\u0001\nλτ≫1,(8b)\nwhere we have used that ε≫λ.\nThe results of Eq. (8) modify by electron diffusion. By\ntaking into account up to lscattering events we obtain\n¯α(l)\nm,⊥=ετ(\nl+O(λ2τ2) λτ≪1,\n(1 +δl0)/(λτ)2λτ≫1,(9a)\n¯α(l)\nm,∥=ετ(\nl+O(λ2τ2) λτ≪1,\n1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b)\nwhere we have used ε≫λagain.\nFrom Eqs. (9) we see that the Gilbert damping for\nλτ≪1 gets an additional contribution of ετfrom each\nscattering event as illustrated numerically in Fig. 2. This\nleads to a formal divergence of Gilbert damping in the\nlimit λτ≪1. While, at first glance, the divergence looks\nlike a strong sensitivity of damping to impurity scatter-\ning, in reality, it simply reflects a diverging spin life-time.\nOnce a non-equilibrium magnetization mis created it\nbecomes almost impossible to relax it to the lattice in\nthe limit of weak spin-orbit coupling. The formal diver-\ngence of α⊥\nm=α∥\nmsimply reflects the conservation law\nfor electron spin polarization in the absence of spin-orbit\ncoupling such that the corresponding spin life-time be-\ncomes arbitrarily large as compared to the momentum\nscattering time τ.\nBy taking the limit l→ ∞ (i. e. by summing up the\nentire diffusion ladder) we obtain compact expressions\n¯α⊥\nm≡¯α(∞)\nm,⊥=ετ1\n2λ2τ2, (10a)\n¯α∥\nm≡¯α(∞)\nm,∥=ετ1 +λ2τ2\nλ2τ2, (10b)\nwhich assume ¯ α⊥\nm≪¯α∥\nmforλτ≫1 and ¯ α⊥\nm= ¯α∥\nm/2\nforλτ≪1. The factor of 2 difference that we observe\nwhen λτ≪1, corresponds to a difference in the elec-\ntron spin life-times τ⊥\ns=τ∥\ns/2 that was discussed in the\nintroduction32.\nStrong spin-orbit coupling causes a strong out-of-plane\nanisotropy of damping, ¯ α⊥\nm≪¯α∥\nmwhich corresponds to5\na suppression of the perpendicular-to-the-plane damping\ncomponent. As a result, the spin-orbit interaction makes\nit much easier to relax the magnitude of the mzcompo-\nnent of magnetization than that of in-plane components.\nLet us now turn to the dependence of ¯ αmcoefficients on\n∆sdthat is illustrated numerically in Fig. 3. We consider\nfirst the case of absent spin-orbit coupling λ= 0. In\nthis case, the combination of spin-rotational and sub-\nlattice symmetry (the equivalence of A and B sub-lattice)\nmust make Gilbert damping isotropic (see e. g.25,41). The\ndirect calculation for λ= 0 does, indeed, give rise to the\nisotropic result ¯ α⊥\nm= ¯α∥\nm=ετ(ε2+∆2\nsd)/2∆2\nsd, which is,\nhowever, in contradiction to the limit λ→0 in Eq. (10).\nAt first glance, this contradiction suggests the exis-\ntence of a certain energy scale for λover which the\nanisotropy emerges. The numerical analysis illustrated\nin Fig. 4 reveals that this scale does not depend on the\nvalues of 1 /τ, ∆sd, orε. Instead, it is defined solely by\nnumerical precision. In other words, an isotropic Gilbert\ndamping is obtained only when the spin-orbit strength\nλis set below the numerical precision in our model.\nWe should, therefore, conclude that the transition from\nisotropic to anisotropic (factor of 2) damping occurs ex-\nactly at λ= 0. Interestingly, the factor of 2 anisotropy is\nabsent in Eqs. (8) and emerges only in the diffusive limit.\nWe will see below that this paradox can only be re-\nsolved by analyzing the Gilbert damping beyond the in-\nfinite wave-length limit.\nOne can see from Fig. 3 that the main effect of finite\n∆sdis the regularization of the Gilbert damping diver-\ngency ( λτ)−2in the limit λτ≪1. Indeed, the limit of\nweak spin-orbit coupling is non-perturbative for ∆ sd/ε≪\nλτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1,\nthe results of Eqs. (10) are no longer valid. Assuming\n∆sd/ε≪1 we obtain the asymptotic expressions for the\nresults presented in Fig. 3 as\n¯α⊥\nm=1\n2ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1\nλ2τ2 λτ≫∆sd/ε,(11a)\n¯α∥\nm=ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(11b)\nwhich suggest that ¯ α⊥\nm/¯α∥\nm= 2 for λτ≪1. In the op-\nposite limit, λτ≫1, the anisotropy of Gilbert damping\ngrows as ¯ α∥\nm/¯α⊥\nm= 2λ2τ2.\nThe results of Eqs. (11) can also be discussed in terms\nof the electron spin life-time, τ⊥(∥)\ns = ¯α⊥(∥)\nm/ε. For the\ninverse in-plane spin life-time we find\n1\nτ∥\ns=\n\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ≪1,\n1/τ 1≪λτ,(12)\nthat, for ∆ sd= 0, is equivalent to the known result of\nEq. (4). Indeed, for ∆ sd= 0, the magnetic exchange\n10\u0000310\u0000210\u00001100101\n\u0015\u001c10\u00001101103105\u0016\u000bm;k;?[\"\u001c]\n\u0001sd=\"= 0:1\u0001sd=\"= 0\u0016\u000bm;k\n\u0016\u000bm;?FIG. 3. Numerical results for the Gilbert damping compo-\nnents in the diffusive limit (vertex corrected)as the function\nof the spin-orbit coupling strength λ. The results correspond\ntoετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic\nexpressions of Eq. (11). Three different regimes can be dis-\ntinguished for ¯ α∥\nm: i) spin-orbit independent damping ¯ α∥\nm∝\nε3τ/∆2\nsdfor the exchange dominated regime, λτ≪∆sd/ε, ii)\nthe damping ¯ α∥\nm∝ε/λ2τfor Elliot-Yafet relaxation regime,\n∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥\nm∝ετfor the\nDyakonov-Perel relaxation regime, λτ≫1. The latter regime\nis manifestly absent for ¯ α⊥\nmin accordance with Eqs. (12,13).\nplays no role and one observes the cross-over from Elliot-\nYafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax-\nation.\nThis cross-over is, however, absent in the relaxation of\nthe perpendicular spin component\n1\nτ⊥s= 2(\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ,(13)\nwhere Elliot-Yafet-like relaxation extends to the regime\nλτ≫1.\nAs mentioned above, the factor of two anisotropy in\nspin-relaxation of 2 Dsystems, τ∥\ns= 2τ⊥\ns, is known in the\nliterature32(see Refs.36–38as well). Unlimited growth of\nspin life-time anisotropy, τ∥\ns/τ⊥\ns= 2λ2τ2, in the regime\nλτ≪1 has been described first in Ref. 25. It can be qual-\nitatively explained by a strong suppression of spin-flip\nprocesses for zspin component due to spin-orbit induced\nsplitting of Fermi surfaces. The mechanism is effective\nonly for scalar (non-magnetic) disorder. Even though\nsuch a mechanism is general for any magnetic or non-\nmagnetic 2D material with Rashba-type spin-orbit cou-\npling, the effect of the spin life-time anisotropy on Gilbert\ndamping is much more relevant for anti-ferromagnets. In-\ndeed, in an anti-ferromagnetic system the modulus of m\nis, by no means, conserved, hence the variations of per-\npendicular and parallel components of the magnetization\nvector are no longer related.\nIn the regime, λτ≪∆sd/εthe spin life-time is de-\nfined by exchange interaction and the distinction between\nDyakonov-Perel and Elliot-Yafet mechanisms of spin re-\nlaxation is no longer relevant. In this regime, the spin-\nrelaxation time is by a factor ( ε/∆sd)2larger than the\nmomentum relaxation time.\nLet us now return to the problem of emergency of the6\n10\u00006410\u00005410\u00004410\u00003410\u00002410\u000014\n\u0015\u001c12\u0016\u000bk=\u0016\u000b?n= 32\nn= 64n= 96\nn= 128\nFIG. 4. Numerical evaluation of Gilbert damping anisotropy\nin the limit λ→0. Isotropic damping tensor is restored only\nifλ= 0 with ultimate numerical precision. The factor of 2\nanisotropy emerges at any finite λ, no matter how small it\nis, and only depends on the numerical precision n, i.e. the\nnumber of digits contained in each variable during computa-\ntion. The crossover from isotropic to anisotropic damping can\nbe understood only by considering finite, though vanishingly\nsmall, magnon qvectors.\nfactor of 2 anisotropy of Gilbert damping at λ= 0. We\nhave seen above (see Fig. 4) that, surprisingly, there ex-\nists no energy scale for the anisotropy to emerge. The\ntransition from the isotropic limit ( λ= 0) to a finite\nanisotropy appeared to take place exactly at λ= 0. We\ncan, however, generalize the concept of Gilbert damping\nby considering the spin density response function at a\nfinite wave vector q.\nTo generalize the Gilbert damping, we are seeking a\nresponse of spin density at a point r,δs+(r) to a time\nderivative of magnetization vectors ˙m∥and ˙m⊥at the\npoint r′. The Fourier transform with respect to r−r′\ngives the Gilbert damping for a magnon with the wave-\nvector q.\nThe generalization to a finite q-vector shows that the\nlimits λ→0 and q→0 cannot be interchanged. When\nthe limit λ→0 is taken before the limit q→0 one\nfinds an isotropic Gilbert damping, while for the oppo-\nsite order of limits, it becomes a factor of 2 anisotropic.\nIn a realistic situation, the value of qis limited from\nbelow by an inverse size of a typical magnetic domain\n1/Lm, while the spin-orbit coupling is effective on the\nlength scale Lλ= 2πℏvf/λ. In this picture, the isotropic\nGilbert damping is characteristic for the case of suffi-\nciently small domain size Lm≪Lλ, while the anisotropic\nGilbert damping corresponds to the case Lλ≪Lm.\nIn the limit qℓ≪1, where ℓ=vfτis the electron mean\n\u00002 0 2\nk[a.u.]\u00002:50:02:5energy [a.u.]\u0015=\u0001sd= 4\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 2\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 1FIG. 5. Band-structure for the effective model of Eq. (5)\nin a vicinity of Kvalley assuming nz= 1. Electron bands\ntouch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to\nspin-orbit band inversion. The band structure in the valley\nK′is inverted. Our microscopic analysis is performed in the\nelectron-doped regime for the Fermi energy above the gap as\nillustrated by the top dashed line. The bottom dashed line\ndenotes zero energy (half-filling).\nfree path, we can summarize our results as\n¯α⊥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n1\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1\n2λ2τ2 λτ≫∆sd/ε,, (14a)\n¯α∥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n2\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(14b)\nwhich represent a simple generalization of Eqs. (11).\nThe results of Eqs. (14) correspond to a simple behav-\nior of Gilbert damping anisotropy,\n¯α∥\nm/¯α⊥\nm=(\n1 λτ≪qℓ,\n2\u0000\n1 +λ2τ2\u0001\nqℓ≪λτ,(15)\nwhere we still assume qℓ≪1.\nIII. ANTI-FERROMAGNETIC RESONANCE\nThe broadening of the anti-ferromagnet resonance\npeak is one obvious quantity that is sensitive to Gilbert\ndamping. The broadening is however not solely defined\nby a particular Gilbert damping component but depends\nalso on both magnetic anisotropy and anti-ferromagnetic\nexchange.\nTo be more consistent we can use the model of Eq. (5)\nto analyze the contribution of conduction electrons to an\neasy axis anisotropy. The latter is obtained by expanding\nthe free energy for electrons in the value of nz, which has\na form E=−Kn2\nz/2. With the conditions ε/λ≫1 and\nε/∆sd≫1 we obtain the anisotropy constant as\nK=A\n2πℏ2v2(\n∆2\nsdλ 2∆sd/λ≤1,\n∆sdλ2/2 2∆ sd/λ≥1,(16)7\nwhere Ais the area of the unit cell. Here we assume\nboth λand ∆ sdpositive, therefore, the model natu-\nrally gives rise to an easy axis anisotropy with K > 0.\nIn real materials, there exist other sources of easy axis\nor easy plane anisotropy. In-plane magneto-crystalline\nanisotropy also plays an important role. For example,\nN´ eel-type anti-ferromagnets with easy-axis anisotropy\nare FePS 3, FePSe 3or MnPS 3, whereas those with easy\nplane and in-plane magneto-crystalline anisotropy are\nNiPS 3and MnPSe 3. Many of those materials are, how-\never, Mott insulators. Our qualitative theory may still\napply to materials like MnPS 3monolayers at strong elec-\ntron doping.\nThe transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in\nEq. (16) corresponds to the touching of two bands in the\nmodel of Eq. (5) as illustrated in Fig. 5.\nAnti-ferromagnetic magnon frequency and life-time in\nthe limit q→0 are readily obtained by linearizing the\nequations of motion\n˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a)\n˙m=Kn×n⊥+n×(ˆαn˙n), (17b)\nwhere we took into account easy axis anisotropy Kand\ndisregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m).\nWe have also defined Gilbert damping tensors such as\nˆαm˙m=α∥\nm˙m∥+α⊥\nm˙m⊥, ˆαn˙n=α∥\nn˙n∥+α⊥\nn˙n⊥.\nIn the case of easy axis anisotropy we can use the lin-\nearized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we\nget the energy of q= 0 magnon as\nω=ω0−iΓ/2, (18)\nω0=√\nJK, Γ =Jα∥\nn+Kα∥\nm (19)\nwhere we took into account that K≪J. The expression\nforω0is well known due to Kittel and Keffer42,43.\nUsing Ref. 25 we find out that α∥\nn≃α⊥\nm(λ/ε)2and\nα⊥\nn≃α∥\nm(λ/ε)2, hence\nΓ≃α∥\nm\u0012\nK+J/2\nε2/λ2+ε2τ2\u0013\n, (20)\nwhere we have simply used Eqs. (10). Thus, one may\noften ignore the contribution Jα∥\nnas compared to Kα∥\nm\ndespite the fact that K≪J.\nIn the context of anti-ferromagnets, spin-pumping\nterms are usually associated with the coefficients α∥\nnin\nEq. (3b) that are not in the focus of the present study.\nThose coefficients have been analyzed for example in Ref.\n25. In this manuscript we simply use the known results\nforαnin Eqs. (17-19), where we illustrate the effect of\nboth spin-pumping coefficient αnand the direct Gilbert\ndamping αmon the magnon life time. One can see from\nEqs. (19,20) that the spin-pumping contributions do also\ncontribute, though indirectly, to the magnon decay. The\nspin pumping contributions become more important in\nmagnetic materials with small magnetic anisotropy. The\nprocesses characterized by the coefficients αnmay also be\n10\u0000310\u0000210\u00001100101\n\u0015\u001c0:000:010:021=\u0016\u000bk\nm\u0015=\"= 0:04\n\u0015=\"= 0:02\n\u0015=\"= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping\n1/¯α∥\nmas a function of the momentum relaxation time τ. The\ninverse damping is peaked at τ∝1/λwhich also corresponds\nto the maximum of the anti-ferromagnetic resonance quality\nfactor in accordance with Eq. (21).\ninterpreted in terms of angular momentum transfer from\none AFM sub-lattice to another. In that respect, the spin\npumping is specific to AFM, and is qualitatively differ-\nent from the direct Gilbert damping processes ( αm) that\ndescribe the direct momentum relaxation to the lattice.\nAs illustrated in Fig. 6 the quality factor of the anti-\nferromagnetic resonance (for a metallic anti-ferromagnet\nwith easy-axis anisotropy) is given by\nQ=ω0\nΓ≃1\nα∥\nmr\nJ\nK. (21)\nInterestingly, the quality factor defined by Eq. (21) is\nmaximized for λτ≃1, i. e. for the electron spin-orbit\nlength being of the order of the scattering mean free path.\nThe quantities 1 /√\nKand 1 /¯α∥\nmare illustrated in\nFig. 6 from the numerical analysis. As one would ex-\npect, the quality factor vanishes in both limits λ→0\nandλ→ ∞ . The former limit corresponds to an over-\ndamped regime hence no resonance can be observed. The\nlatter limit corresponds to a constant α∥\nm, but the reso-\nnance width Γ grows faster with λthan ω0does, hence\nthe vanishing quality factor.\nIt is straightforward to check that the results of\nEqs. (20,21) remain consistent when considering systems\nwith either easy-plane or in-plane magneto-crystalline\nanisotropy. Thus, the coefficient α⊥\nmnormally does not\nenter the magnon damping, unless the system is brought\ninto a vicinity of spin-flop transition by a strong external\nfield.\nIV. CONCLUSION\nIn conclusion, we have analyzed the Gilbert damping\ntensor in a model of a two-dimensional anti-ferromagnet\non a honeycomb lattice. We consider the damping mech-\nanism that is dominated by a finite electron spin life-time8\ndue to a combination of spin-orbit coupling and impu-\nrity scattering of conduction electrons. In the case of a\n2D electron system with Rashba spin-orbit coupling λ,\nthe Gilbert damping tensor is characterized by two com-\nponents α∥\nmandα⊥\nm. We show that the anisotropy of\nGilbert damping depends crucially on the parameter λτ,\nwhere τis the transport scattering time for conduction\nelectrons. For λτ≪1 the anisotropy is set by a geo-\nmetric factor of 2, α∥\nm= 2α⊥\nm, while it becomes infinitely\nlarge in the opposite limit, α∥\nm= (λτ)2α⊥\nmforλτ≫1.\nGilbert damping becomes isotropic exactly for λ= 0, or,\nstrictly speaking, for the case λ≪ℏvfq, where qis the\nmagnon wave vector.\nThis factor of 2 is essentially universal, and is a geomet-\nric effect: the z-component relaxation results from fluctu-\nations in two in-plane spin components, whereas in-plane\nrelaxation stems from fluctuations of the z-component\nalone. This reflects the subtleties of our microscopic\nmodel, where the mechanism for damping is activated\nby the decay of conduction electron momenta, linked to\nspin-relaxation through spin-orbit interactions.\nWe find that Gilbert damping is insensitive to mag-\nnetic order for λ≫∆sd/ετ, where ∆ sdis an effective\nexchange coupling between spins of conduction and local-\nized electrons. In this case, the electron spin relaxation\ncan be either dominated by scattering (Dyakonov-Perel\nrelaxation) or by spin-orbit precession (Elliot-Yafet re-\nlaxation). We find that the Gilbert damping component\nα⊥\nm≃ε/λ2τis dominated by Elliot-Yafet relaxation irre-\nspective of the value of the parameter λτ, while the other\ncomponent crosses over from α∥\nm≃ε/λ2τ(Elliot-Yafet\nrelaxation) for λτ≪1, to α∥\nm≃ετ(Dyakonov-Perel re-\nlaxation) for λτ≫1. For the case λ≪∆sd/ετthe spin\nrelaxation is dominated by interaction with the exchange\nfield.\nCrucially, our results are not confined solely to the N´ eel\norder on the honeycomb lattice: we anticipate a broader\napplicability across various magnetic orders, including\nthe zigzag order. This universality stems from our focus\non the large magnon wavelength limit. The choice of the\nhoneycomb lattice arises from its unique ability to main-\ntain isotropic electronic spectra within the plane, coupled\nwith the ability to suppress anisotropy concerning in-\nplane spin rotations. Strong anisotropic electronic spec-\ntra would naturally induce strong anisotropic in-plane\nGilbert damping, which are absent in our results.\nFinally, we show that the anti-ferromagnetic resonance\nwidth is mostly defined by α∥\nmand demonstrate that the\nresonance quality factor is maximized for λτ≈1. Our\nmicroscopic theory predictions may be tested for systems\nsuch as MnPS 3monolayer on Pt and similar heterostruc-\ntures.ACKNOWLEDGMENTS\nWe are grateful to O. Gomonay, R. Duine, J. Sinova,\nand A. Mauri for helpful discussions. This project has\nreceived funding from the European Union’s Horizon\n2020 research and innovation program under the Marie\nSklodowska-Curie grant agreement No 873028.\nAppendix A: Microscopic framework\nThe microscopic model that we employ to calculate\nGilbert damping belongs to a class of so-called s–dmod-\nels that describe the physical system in the form of a\nHeisenberg model for localized spins and a tight-binding\nmodel for conduction electrons that are weakly coupled\nby a local magnetic exchange interaction of the strength\n∆sd.\nOur effective electron Hamiltonian for a metallic\nhexagonal anti-ferromagnet is given by25\nH0=vfp·Σ+λ\n2[σ×Σ]z−∆sdn·σΣzΛz,(A1)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli-\nmatrices acting on sub-lattice, spin and valley space re-\nspectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ.\nTo describe Gilbert damping of the localized field n\nwe have to add the relaxation mechanism. This is pro-\nvided in our model by adding a weak impurity potential\nH=H0+V(r). The momentum relaxation due to scat-\ntering on impurities leads indirectly to the relaxation of\nHeisenberg spins due to the presence of spin-orbit cou-\npling and exchange couplings.\nFor modeling the impurity potential, we adopt a delta-\ncorrelated random potential that corresponds to the\npoint scatter approximation, where the range of the im-\npurity potential is much shorter than that of the mean\nfree path (see e.g. section 3.8 of Ref. 44), i.e.\n⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2)\nwhere the dimensionless coefficient α≪1 characterizes\nthe disorder strength. The corresponding scattering time\nfor electrons is obtained as τ=ℏ/παϵ , which is again\nsimilar to the case of graphene.\nThe response of symmetric spin-polarization δs+to the\ntime-derivative of non-staggered magnetization, ∂tm, is\ndefined by the linear relation\nδs+\nα=X\nβRαβ|ω=0˙mβ, (A3)\nwhere the response tensor is taken at zero frequency25,45.\nThe linear response is defined generally by the tensor\nRαβ=A∆2\nsd\n2πSZdp\n(2πℏ)2\nTr\u0002\nGR\nε,pσαGA\nε+ℏω,pσβ\u0003\u000b\n,(A4)9\nwhere GR(A)\nε,pare standing for retarded(advanced) Green\nfunctions and the angular brackets denote averaging over\ndisorder fluctuations.\nThe standard recipe for disorder averaging is the diffu-\nsive approximation46,47that is realized by replacing the\nbare Green functions in Eq. (A4) with disorder-averaged\nGreen functions and by replacing one of the vertex op-\nerators σxorσywith the corresponding vertex-corrected\noperator that is formally obtained by summing up ladder\nimpurity diagrams (diffusons).\nIn models with spin-orbit coupling, the controllable dif-\nfusive approximation for non-dissipative quantities may\nbecome, however, more involved as was noted first in\nRef. 48. For Gilbert damping it is, however, sufficient to\nconsider the ladder diagram contributions only.\nThe disorder-averaged Green function is obtained by\nincluding an imaginary part of the self-energy ΣR(not\nto be confused here with the Pauli matrix Σ 0,x,y,z) that\nis evaluated in the first Born approximation\nIm ΣR= 2παv2\nfZdp\n(2π)2Im1\nε−H0+i0. (A5)\nThe real part of the self-energy leads to the renormaliza-\ntion of the energy scales ε,λand ∆ sd.\nIn the first Born approximation, the disorder-averaged\nGreen function is given by\nGR\nε,p=1\nε−H0−iIm ΣR. (A6)\nThe vertex corrections are computed in the diffusive\napproximation. The latter involves replacing the vertex\nσαwith the vertex-corrected operator,\nσvc\nα=∞X\nl=0σ(l)\nα, (A7)\nwhere the index lcorresponds to the number of disorder\nlines in the ladder.\nThe operators σ(l)\nαcan be defined recursively as\nσ(l)\nα=2ℏv2\nf\nετZdp\n(2π)2GR\nε,pσ(l−1)\nαGA\nε+ℏω,p, (A8)\nwhere σ(0)\nα=σα.\nThe summation in Eq. (A7) can be computed in the\nfull operator basis, Bi={α,β,γ}=σαΣβΛγ, where each\nindex α,βandγtakes on 4 possible values (with zero\nstanding for the unity matrix). We may always normalize\nTrBiBj= 2δijin an analogy to the Pauli matrices. The\noperators Biare, then, forming a finite-dimensional space\nfor the recursion of Eq. (A8).\nThe vertex-corrected operators Bvc\niare obtained by\nsumming up the matrix geometric series\nBvc\ni=X\nj\u00121\n1− F\u0013\nijBj, (A9)where the entities of the matrix Fare given by\nFij=ℏv2\nf\nετZdp\n(2π)2Tr\u0002\nGR\nε,pBiGA\nε+ℏω,pBj\u0003\n.(A10)\nOur operators of interest σxandσycan always be de-\ncomposed in the operator basis as\nσα=1\n2X\niBiTr (σαBi), (A11)\nhence the vertex-corrected spin operator is given by\nσvc\nα=1\n2X\nijBvc\niTr(σαBi). (A12)\nMoreover, the computation of the entire response tensor\nof Eq. (A4) in the diffusive approximation can also be\nexpressed via the matrix Fas\nRαβ=α0ετ\n8ℏX\nij[TrσαBi]\u0014F\n1− F\u0015\nij[TrσβBj],(A13)\nwhere α0=A∆2\nsd/πℏ2v2\nfSis the coefficient used in\nEq. (6) to define the unit of the Gilbert damping.\nIt appears that one can always choose the basis of\nBioperators such that the computation of Eq. (A13)\nis closed in a subspace of just three Bioperators with\ni= 1,2,3. This enables us to make analytical computa-\ntions of Eq. (A13).\nAppendix B: Magnetization dynamics\nThe representation of the results can be made some-\nwhat simpler by choosing xaxis in the direction of the\nin-plane projection n∥of the N´ eel vector, hence ny= 0.\nIn this case, one can represent the result as\nδs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n,\nwhere ndependence of the coefficients cimay be param-\neterized as\nc1=r11−r22−r31(1−n2\nz)/(nxnz)\n1−n2z, (B1a)\nc2=r11−r31(1−n2\nz)/(nxnz), (B1b)\nc3=r33, (B1c)\nc4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d)\nThe analytical results in the paper correspond to the\nevaluation of δs±up to the second order in ∆ sdusing\nperturbative analysis. Thus, zero approximation corre-\nsponds to setting ∆ sd= 0 in Eqs. (A1,A5).\nThe equations of motion on nandmare given by\nEqs. (2),\n∂tn=−Jn×m+n×δs++m×δs−, (B2a)\n∂tm=m×δs++n×δs−, (B2b)10\nIt is easy to see that the following transformation leaves\nthe above equations invariant,\nδs+→δs+−ξn, δ s−→δs−−ξm, (B3)\nfor an arbitrary value of ξ.\nSuch a gauge transformation can be used to prove that\nthe coefficient c4is irrelevant in Eqs. (B2).\nIn this paper, we compute δs±to the zeroth order in\n|m|– the approximation which is justified by the sub-\nlattice symmetry in the anti-ferromagnet. A somewhat\nmore general model has been analyzed also in Ref. 25 to\nwhich we refer the interested reader for more technical\ndetails.\nAppendix C: Anisotropy constant\nThe anisotropy constant is obtained from the grand po-\ntential energy Ω for conducting electrons. For the model\nof Eq. (A1) the latter can be expressed as\nΩ =−X\nς=±1\nβZ\ndε g(ε)νς(ε), (C1)\nwhere β= 1/kBTis the inverse temperature, ς=±is\nthe valley index (for the valleys KandK′),GR\nς,pis the\nbare retarded Green function with momentum pand in\nthe valley ς. We have also defined the function\ng(ε) = ln (1 + exp[ β(µ−ε)]), (C2)\nwhere µis the electron potential, and the electron density\nof states in each of the valleys is given by,\nνς(ε) =1\nπZdp\n(2πℏ)2Im Tr GR\nς,p, (C3)\nwhere the trace is taken only over spin and sub-lattice\nspace,\nIn the metal regime considered, the chemical potential\nis assumed to be placed in the upper electronic band.\nIn this case, the energy integration can be taken only for\npositive energies. The two valence bands are always filled\nand can only add a constant shift to the grand potential\nΩ that we disregard.\nThe evaluation of Eq. (C1) yields the following density\nof states\nντ(ε) =1\n2πℏ2v2\nf\n\n0 0 < ε < ε 2\nε/2 +λ/4ε2< ε < ε 1,\nε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum\npoints (zero velocity) for the electronic bands. These\nenergies, for each of the valleys, are given by\nε1,ς=1\n2\u0000\n+λ+p\n4∆2+λ2−4ς∆λnz\u0001\n, (C5a)\nε2,ς=1\n2\u0000\n−λ+p\n4∆2+λ2+ 4ς∆λnz\u0001\n(C5b)\nwhere ς=±is the valley index.\nIn the limit of zero temperature we can approximate\nEq. (C1) as\nΩ =−X\nς=±1\nβZ∞\n0dε(µ−ε)νς(ε). (C6)\nThen, with the help of Eq. (C1) we find,\nΩ =−1\n24πℏ2v2\nfX\nς=±\u0002\n(ε1,ς−µ)2(4ε1,ς−3λ+ 2µ)\n+(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ)\u0003\n. (C7)\nBy substituting the results of Eqs. (C5) into the above\nequation we obtain\nΩ =−1\n24πℏ2v2\nfh\n(4∆2−4nz∆λ+λ2)2/3\n+(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i\n.(C8)\nA careful analysis shows that the minimal energy cor-\nresponds to nz=±1 so that the conducting electrons\nprefer an easy-axis magnetic anisotropy. 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Titov, EPL 111,\n37004 (2015)."    },    {        "title": "1202.3450v1.Current_induced_motion_of_a_transverse_magnetic_domain_wall_in_the_presence_of_spin_Hall_effect.pdf",        "content": "Current -induced motion of a transverse magnetic domain \nwall in the presence of  spin Hall effect  \n \nSoo-Man Seo1, Kyoung-Whan Kim2, Jisu Ryu2, Hyun -Woo Lee2,a), and Kyung -Jin Lee1,3,4,b) \n \n1Department of M aterials Science and Engineering, Korea University , Seo ul 136 -701, Korea  \n2PCTP and Department of Physics, Pohang University of Science and Technology , Kyungbuk 790 -\n784, Korea  \n3Center for Nanoscale Science and Technology , National Institute of Standards and Technology, \nGaithersburg, Maryland 20899 -8412, USA   \n4Maryland Nanocenter, University of Maryland, College Park, MD 20742, USA  \n \nWe theoretically  study  the current -induced dynamics of a transverse magnetic domain \nwall in bi-layer nanowire s consisting of a ferromagnet  on top of a nonmagnet having \nstrong spin-orbit coupling . Domain wall  dynamics is characterized  by two threshold \ncurrent densities, \nWB\nthJ  and \nREV\nthJ , where \nWB\nthJ  is a threshold for the chirality switching \nof the domain wall  and \nREV\nthJ  is another threshold for the reversed domain wall motion  \ncaused by spin Hall effect . Domain wall s with a certain chirality may move opposite to \nthe electron -flow direction with high speed in the current range \nWB\nthREV\nth JJ J  for the \nsystem designed to satisfy the conditio ns \nWB REV\nth thJJ  and \n , where \n  is the \nGilbert damping constant and \n  is the  nonadiabaticity of spin torque . Micromagnetic \nsimulation s confirm the validity of analytical results. 1 \n                                                      \na)Electronic mail:  hwl@postech.ac.kr.  \nb)Electronic mail:  kj_lee@korea.ac.kr .    Electric  manipulati on of domain wall s (DW s) in magnetic nanowire s can be  realized by  \nthe spin-transfer torque (STT) due to the coupling  between local  magnetic moment s of the \nDW and spin-polarized current s1,2. Numerous studie s on this subject  have addressed  its \nfundamental physic s3-5, and to explore  its potential in application s such as data storage and \nlogic devices6. Up until now, h owever, most studies have focused on the effect of the spin \ncurrent that i s polarized by a ferromagnet ic layer . \nAnother  way to generate a spin curren t is the spin Hall effect (SHE)7,8. In ferromagnet \n(FM) |nonmagnet (NM) bi -layer system s, an in-plane  charge current density (Jc) passing \nthrough the NM is converted into a perpendic ular spin current density (Js) owing to the SHE . \nThe ratio of Js to Jc is parameterized by spin  Hall angle . This spin current caused by SHE \nexerts  a STT (= SHE -STT)  on the FM and consequently modifies its m agnetization dynamics. \nDuring the last decade , most studies on the SHE  have focused on measuring the spin  Hall \nangle9-14. Recently the magnetization  switching15 and the modulation of propagating spin \nwaves  by SHE -STT were investigated16-18. However, the effect of SHE -STT on current -\ninduced DW dynamics has not been treated . \nIn this Letter, we study DW dynamics including  all current -induced STT s in a nanowire \nconsisting of FM/NM bi-layers  (Fig. 1) , where FM has a n in-plane magnetic anisotropy and \nNM has strong spin-orbit coupling ( SOC ) responsible for  the SHE. A charge current passing \nthrough the FM generates  conventional adiabatic and nonadiabatic STTs19-21, whereas a \ncharge current flowing through the NM experiences SHE and generates SHE -STT on the FM . \nFor the current running in the x axis, t he modified Landau-Lifshitz -Gilbert equation including \nall the STTs  is given by  \n),ˆ ( y cxbxbt tJ SH J J \n\n\nm mmmmm mmm Hmm\neff    \n   (1) where m is the unit vector along the magnetization, α is the Gilbert damping constant, bJ  \n) 2 (S F B eM PJg\n is the magnitude of adiabatic STT, β is the nonadiabatic ity of STT, \nJ SHc\n( 2 )SH N S FJ eM t\n is the magnitude of SHE -STT, θSH is an effective spin Hall angle \nfor the bi -layer system , γ is the gyromagnetic ratio,  g is the Land é g-factor, μB is the Bohr \nmagneton, P is the spin polarization in the FM, e is the electron charge, MS is the saturation \nmagnetization of the FM, and JF (JN) is the current density in the FM (NM). JF and JN are \ndetermined by a simple circuit model; i.e.,  \n) /() (0 N N FF F N F F t t ttJ J    and \n) /() (0 N N FF N N F N t t ttJ J  \n, where J0 is the total current density in the bi -layer \nnanowire , σF (σN) is the conductivity  of the FM (NM), and tF (tN) is the thickness of the FM \n(NM).  We assume that θSH is smaller than 1  as is usually the case experimentally . \nFor a nanowire with an in-plane magnetic anisotropy, a  net effective field is given by  \n2\n22ˆ ,mH x Heff K x d\nSAHmMx  \n                      (2) \nwhere A is the exchange stiffness constant, HK is the easy axis anisotropy field along the x \naxis, and \ndH  is the magnetostatic field  given by \n)() (~r )(3rmrr rH Nd MS d , where \nthe components of the tensor \nN~  are given by \n3 2 2/]/31[ r rx Nxx , \n5/3 rxy Nxy  [22]. \nOther components are defined in a  similar way . For a one-dimensional  DW as shown in Fig. \n1, the spatial profile of the magnetization is described by  \n)sin sin, cos sin, (cos m, \nwhere \n  ) ( sech sin Xx , \n  ) ( tanh cos Xx , \n)(tX  is the  DW position, \n)(t  is \nthe DW tilt angle , and λ is the DW  width . By using the procedure developed by Thiele23, we \nobtain the equation s of motion for the two collective coordinates \nX  and \n  in the rigid DW \nlimit,  \nsin(2 ),2d\nJH Xbtt    \n                       (3) ,eff JXbtt                               (4) \nwhere \n) 2(S d d MK H , \n  sin 1SH eff B , \n/2SH SH N F FB J t PJ  , and Kd is the \nhard-axis anisotropy energy density . From Eqs. (3) and (4), one finds that the effect of SHE -\nSTT on DW d ynamics is captured by replacing β by βeff. Assuming that FM is Permall oy (Py: \nNi80Fe20) and NM is Pt, for the parameters of  tF = 4 nm, tN = 3 nm, σF = σN, θSH = 0.1, β ≈ \n0.01 to 0.03 [ 24], P = 0.7, and λ = 30 nm, we find \nSHB  ≈ 18 to 56, which is not small . \nTherefore, βeff can be much larger than β unless  sin is extremely small. Furthermore, it is \npossible that βeff is even negative  if \n1 sinSHB . \nTo get an insight into the effect of SHE -STT on DW dynamics, we derive several \nanalytical solutions from  Eqs. (3) and (4). It is known that DW dynamics in a nanowire can \nbe classified into two regimes; i.e., below and above the Walker breakdown25. Below the \nWalker breakdown, \n  increases in the initial time stage and then becomes saturat ed to a \ncertain value over time.  In this limit (\n0 t  as \nt ), we obtain  \n,) (22sin\neffd\nJHb\n\n                             (5) \nThreshold adiabatic STT for the Walker breakdown  (\nWB\nJb ) is obtained from the maximum \nvalue of  the right -hand -side of  Eq. ( 5); i.e., \n ) (22sin maxeff dWB\nJ H b    . Note that \nWB\nJb\n is not simply \n2( )d effH      because  \neff  also includes \n . When  BSH = 0, Eq. \n(5) reduces to \n   2dWB\nJ H b , reproducing the previ ous result  [26] in the absence  of \nSHE.  \n For \nWB\nJJbb  (below t he Walker breakdown) and using  the small -angle approximation , \nDW velocity  (vDW) is given by \n,) (1\n\n\n\n\nSHJ dJ\nSH J DWBb HbB b v\n\n                (6) \nwhere  the sign “+” and “” in the parenthesis corresponds  to the initial tilt angle s \n0  \nand \n00 , respectively.  This \n0  dependen ce of  \nDWv  originates from the fact that SHE -\nSTT acts like a damping or an anti -damping term depending on \n0 . When  \n , \nJ DW b v\n so that vDW does not depend  on SHE -STT. However, this condition is hardly \nrealized  in the bi -layer system that we consider since the strong SOC in NM increases the \nintrinsic α of FM  through the spin pumping effect27. When BSH = 0, \nJ DW b v  , \nconsistent with the DW velocity in the absence of  SHE26. Note that in our sign convention, a \nnegative bJ corresponds to the electron -flow in + x direction and a positive \nDWv  corresponds \nto the DW motion along the electron -flow direction.  Therefore, when the term in the \nparenthesis of Eq. ( 6) is negative, the DW moves against the electron -flow direction instead \nof along it. Threshold adiabatic STT for this re versed DW motion (\nREV\nJb ) is given by  \n.REV d\nJ\nSHHbB\n                                     (7) \n   For \nWB\nJJbb\n  (far above the Walker breakdown) , the time-average d values of \nsin  \nand \n2sin  can be set to zero because of the precession of \n . In this limit, \nDWv  is \ndetermined by Eq. (3) and becomes bJ so that the DW moves along the electron -flow \ndirection and its motion does not depend on SHE -STT. \n   Based on the above investigations , there are  two interesting effects of SHE on  current -\ninduced DW dynamics. First, current -induced DW dynamics is determined  by two thre sholds, WB\nJb and \nREV\nJb . Whe n \nWB\nJ JREV\nJ b b b  , the DW can move against the electron -flow \ndirection. Note that the existence of such bJ range implicitly assumes  \nWB\nJREV\nJ b b . When \nthis inequality is not satisfied, the DW always moves along the electron -flow direction.  For \nall cases, \nDWv  can be larger than \nJb  depending on the parameters (see Eq. ( 6)). \nSecond , vDW is asymmet ric against  the initial tilt angle \n0  for a fixed current polarity. A \nsimilar argument is also valid for a fixed \n0  but with varying the current polarity;  i.e., \nDWv  \nis asymmetric with respect to the current polarity  for a fixed \n0 . This behavior follows \nbecause SHE -STT acts like a damping term for one sign of the current but acts like an anti -\ndamping term for the other sign. Therefore, although the condi tion of \nREV WB\nJ J Jb b b  is \nsatisfied, the reversed DW motion is expected to be observed  only for one current polarity.  \nTo verify the analytical results , we perform a one-dimensional micromagnetic simulation \nby numerically solving Eq. (1). We con sider a Py/Pt bi -layer  nano wire of (length × width × \nthickness ) = (2000 n m × 80 nm × 4 nm (Py) and 3 nm (Pt)) (Fig. 1) . Py m aterial parameters \nof MS = 800 kA/m, A = 1.3×10−11 J/m, P = 0.7, α = 0.02, and  β = 0.01  to 0.03 are used . The \ncrystalline anisotropy and the temperature are assumed to be zero. Conductivit ies of both \nlayers are assumed to be the same as σPy = σPt = 6.5 (μΩm)1, and thus J0 = JF = JN. For all \ncases, the in itial DW tilt angle \n0  is set to zero.  \nAnalytical and numerical results are compared in Fig. 2. DW velocity (\nDWv ) and DW tilt \nangle (\nDW ) as a function of the total current density of the bi -layer (J0) for three values of \nθSH (= +0.1, 0.0, 0.1) and β = 0.01  (thus  > ) are shown in Fig. 2(a) and (b) , respectively . \nDWv\n is estimated from  the terminal velocity . Here, we test both positive and negative values \nof θSH since the spin Hall angle can have e ither sign .  Current dependence s of \nDWv  (Fig. 2(a)) and \nDW  (Fig. 2(b)) show close correlation , \nmeaning that the DW tilting plays  a crucial role for the effect of SHE on DW dynamics  as \ndemonstrated analytically . In Fig. 2(a) , the numerical  results (symbols) are in agreement with \nthe results obtained from Eq. (6)  (lines) . For θSH = 0, \nDWv  is linearly proportional to  J0 and \nthe DW always move s along the electron -flow direction . However, for 0.5×1012 ≤ J0 ≤ \n1.0×1012 A/m2 with θSH = 0.1 (1.0×1012 A/m2 ≤ J0 ≤ 0.5×1012 A/m2 with θSH = 0.1 ), \nDWv  \nhas the same polarity as the current . Thus, the DW moves along  the current -flow direction  for \nthese ranges of the current . The threshold for the  reversed DW motion  is consistent with the \nanalytical solution of Eq. (7); i.e., \nREV\nJb  = ±26.6 m/s corresponding to J0 = ±0.52×1012 A/m2. \nThe maximum \nDWv  is obtained at J0 = ±1 .0×1012 A/m2 immediately before the DW \nexperiences Walker breakdown and switches  its chirality . As shown in the Fig. 2  (c), the \nnormalized y-component of the magnetization at the DW center  (my) abruptly change s from \n1\n to \n1  for J0 = \n 1.0×1012 A/m2 and θSH = \n 0.1. This current density is consist ent \nwith the threshold for Walker breakdown (\nWB\nJb ); i.e.,  \nWB\nJb  = ±53 m/s corresponding to J0 = \n±1.04 5×1012 A/m2. At this current density, vDW is enhanced by a factor of 5  compared to the \ncase for θSH = 0.  \nFig. 3 (a) and (b) show  \nDWv  and \nDW  as a function of J0 for three values of θSH (= +0.1, \n0.0, 0.1) and β = 0.0 3 (thus \n ). Similar ly to the cases for β = 0.0 1, \nDWv  is closely \ncorrelated to \nDW  and significantly enhance d near \nWB\nJb . In this case, in contrast to the case \nfor \n , reversed DW motion is not observed . It is because the sign of  the \n()  term \nin Eq. (6) is negative  in this case , and thus the overall sign of \nDWv  corresponds to the DW \nmotion along the electron -flow direction.  We find that the current -induced Oersted field has \nonly a negligible  effect  on \nDWv  (not shown).  Thus, the numerical results confirm the validity of the analytical solutions; the DW moves along the current -flow directio n at the limited \nrange of the current (i.e., \nREV WB\nJ J Jb b b ) when  \n . In addition this reversed DW \nmotion appears only for one current polarity . \nFinally, we remark the effect of  SHE on DW dynamics  in the nanowire with a \nperpendicular anisotropy . It was experimentally reported that the DW moves along the \ncurrent -flow direction with a high \nDWv  (≈ 400 m/s ) in the perpendicularly magnetized \nnanowire consisting of Pt/Co/AlO x [28, 29 ]. We n ote that  this D W dynamics cannot be \nexplained by the SHE only. Considering the materials parameter s in Ref. [ 29] as MS = 1090 \nkA/m, K = 1.2×106 J/m3, A = 1.3×10−11 J/m, α = 0.2, P = 0.7, λ = 5 nm, and assuming θSH = \n0.1 and β = 0.1, we find BSHλ = 18.8 that is comparable  to the value for  the Py/Pt  bi-layer \ntested in this work . For Pt/Co/AlO x, however,  \nREV\nJb  and \nWB\nJb are respectively 1.5 and 3 \nm/s (corresponding to J0 = 0.4×1011 and 0.8×1011 A/m2). These thresholds  are much \nsmaller t han th ose of the Py/Pt bi-layer since Hd of DW in a perpendicular system is smaller \nthan in an in-plane system  (i.e., Hd = 848  mT for the system  of Py/Pt  in this w ork, 33 mT  for \nthe system in Ref. [29])22. Note that the maximum DW velocity moving along  the current -\nflow direction (\nREV\nDWv ) is obtained at \nWB\nJJbb . The \nWB\nJb (= 3 m/s)  in Pt/Co/AlO x system is \ntoo small to allow such a high \nREV\nDWv  (≈ 400 m/s) . Indeed, the numerically obtained \nmaximum  \nREV\nDWv  is 8.2 m/s at J0 = 0.71×1011 A/m2 (bJ = 2.64 m/s ) (not shown) , which is \nmuch smaller than the experimentally obtained value, 400 m/s . More importantly, in the \nPt/Co/AlO x system, the reversed DW motion was observ ed at both current polarities31 \nwhereas the SHE allows the reversed motion at only one current polarity. On the other hand, \nwe theoretically demonstrated that the DW dynamics reported in Ref. [ 28, 29] can be \nexplained by STTs caused by Rashba SOC32. We als o remark that one of us reported the effect of SOC on current -driven DW motion recently33. In Ref. [33], h owever, the effect of \nSOC within FM  was investigated , in contrast to the present work where the effect of SOC in \nNM of the FM /NM bi -layer system is investigated . \nTo conclude, we present the analytical model for current -induced  DW motion  in the \npresence of SHE . We demonstrate  that DW dynamics is significantly affected by the SHE . In \nparticular, for the case of \n , the SHE  enables t he reversed DW motion with high speed \nat one current polarity when the system is designed to satisfy the condition of \nWB\nJREV\nJ b b  \nand the current density is selected to be in the range between the two thresholds. Our result \ndemo nstrates that the engineering of SOC and thus the SHE provides an  important \nopportunity for an efficient operation of spintronic devices.  \nThis work was supported by the NRF ( 2010 -0014109, 2010 -0023798, 2011 -0009278, \n2011 -0028163 , 2011 -0030046 ) and the MKE/KEIT (2009 -F-004-01). K.J.L. acknowledges \nsupport under the Cooperative Research Agreement between the University of Maryland and \nthe National Institute of Standards and Technology Center for Nanoscale Science and \nTechnology, Award 70NANB10H193, through the University of Maryland.  REFERENCE  \n \n[1] J. C. Slonczewski, J. Magn. Mag. Mater. 159, L1 (1996) . \n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).  \n[3] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, \n077205 (2004).  \n[4] M.  Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature (London) 428, 539 (2004).  \n[5] M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. \nHeyderman, F. Nolting, and U. Rüdiger, Phys. Rev. Lett. 94, 106601 (2005).  \n[6] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).  \n[7] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).  \n[8] S. Zhang, Phys. Rev. Lett. 85, 393 (2000 ). \n[9] S. O. Valenzuela and M. 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Auffret,  B. Rodmacq, A. Schul, S. \nPizzini, J. V ogel, and M. Bonfim, Appl. Phys. Lett. 93, 262504 (2008); ibid 95, 179902 \n(2009).  \n[29] I. M. Miron , T. Moore, H. Szambolics, L. D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, S. \nPizzini, J. V ogel, M. Bonfim, A. Schul, and  G. Gaudin,  Nat. Mater. 10, 189 (2011).  \n[31] I. M. Miron, private communication.  \n[32] K.-W. Kim, S. -M. Seo, J. Ryu, K. -J. Lee, and H. -W. Lee, arXiv:1111.3422v2.  \n[33] A. Manchon and K. -J. Lee, Appl. Phys. Lett. 99, 022504 (2011); ibid 99, 229905 (2011).  FIGURE  CAPTION  \n \nFIG. 1. (Color online)  Schematic s of FM/NM bi-layer nanowire . (top) Structure. (lower left) \nSpatial profile of DW. The color ed contour  shows x component  of the magnetization  for 2 -D \nmicromagnetics . (lower rig ht) Width -averaged magnetization components .  \n \nFIG. 2. (Color online)  Domain wall velocity  for  > . (a) DW velocity (\nDWv ) as a function of  \nthe total current density of bi -layer ( J0) for three  values of θSH (= +0.1, 0 .0, 0.1) and β = 0.01  \n( = 0.02) . Symbols are modeling results, whereas solid lines correspond to Eq. (6) . (b) DW \ntilt angle (\nDW ) as a function of J0. Filled  symbols above the chiral switching threshold (| J0| = \n1.1×1012 A/m2) are shifted from their original values by –180° (filled green triangles) and \n+180° (filled red circles) . (c) Normalized y component of the magnetization at the DW center \n(my) as a function of J0.  \n \nFIG. 3. (Color online) Domain wall velocity for  < . (a) DW velocity (\nDWv ) as a function of \nthe total current density of bi -layer ( J0) for three values of θSH (= +0.1, 0.0, 0.1) and β = 0.0 3 \n( = 0.02) . Symbols are modeling results, whereas solid lines correspond to Eq. (6). (b) DW \ntilt angle (\nDW ) as a function of J0. Filled  symbols represent the cases that the chirality of \nDW switches from its  initial tilt angle \n00 . (c) N ormalized y component of the \nmagnetization at the DW center (my) as a  function of J0.  \n  \n \n \n \n \nFIG. 1. Seo et al.  \n \n \n \n \nFIG. 2. Seo et al. \n \n  \n \n \n \n \nFIG. 3. Seo et al. \n \n \n \n "    },    {        "title": "1209.3669v2.Nonlinear_emission_of_spin_wave_caustics_from_an_edge_mode_of_a_micro_structured_Co2Mn0_6Fe0_4Si_waveguide.pdf",        "content": "arXiv:1209.3669v2  [cond-mat.mes-hall]  12 Dec 2012Nonlinear emission of spin-wave caustics from an edge mode o f a\nmicro-structured Co 2Mn0.6Fe0.4Si waveguide\nT.Sebastian∗and T.Br¨ acher\nFachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern,\n67663 Kaiserslautern, Germany and\nGraduate School Materials Science in Mainz,\nGottlieb-Daimler-Straße 47, 67663 Kaiserslautern, Germa ny\nP.Pirro, A.A.Serga, and B.Hillebrands\nFachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\nT.Kubota\nWPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nKatahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan\nH.Naganuma, M.Oogane, and Y.Ando\nDepartment of Applied Physics, Graduate School of Engineer ing,\nTohoku University, Aoba-yama 6-6-05, Sendai 980-8579, Jap an\n(Dated: August 30, 2018)\nAbstract\nMagnetic Heusler materials with very low Gilbert damping ar e expected to show novel magnonic\ntransport phenomena. We report nonlinear generation of hig her harmonics leading to the emission\nofcausticspin-wave beamsinalow-damping, micro-structu redCo 2Mn0.6Fe0.4SiHeuslerwaveguide.\nThe source for the higher harmonic generation is a localized edge mode formed by the strongly\ninhomogeneous field distribution at the edges of the spin-wa ve waveguide. The radiation character-\nistics of the propagating caustic waves observed at twice an d three times the excitation frequency\nare described by an analytical calculation based on the anis otropic dispersion of spin waves in a\nmagnetic thin film.\n1In the last years nonlinear spin dynamics in magnetic microstructure s made of metallic\nferromagnetic thin films or layer stacks have gained large interest.[1 –5] The intrinsically\nnonlinear Landau-Lifshitz and Gilbert equation (LLG), that govern s the spin dynamics,\ngives rise to a variety of nonlinear effects.[6, 7]\nAmong the metallic ferromagnets, the class of Cobalt-based Heusle r materials is very\npromising for future magnon spintronics devices and the observation of new phenomena of\nmagnonic transport. The reasons for the interest in these mater ials are the small magnetic\nGilbert damping, the high spin-polarization, and the high Curie temper ature.[8, 9]\nAs shown recently, the full Heusler compound Co 2Mn0.6Fe0.4Si (CMFS) is a very suitable\nmaterial to be used as a micro-structured spin-wave waveguide du e to the increased decay\nlength which was observed for wave propagation in the linear regime.[ 10] The reason for\nthis observation is the low Gilbert damping of α= 3×10−3of CMFS compared to Ni 81Fe19\nwithα= 8×10−3, which is the material commonly used in related studies.[9] The decre ased\nmagnetic losses not only lead to an increase of the decay length but a lso to large precession\nangles of the magnetic moments and, thus, to the occurrence of n onlinear effects. Regarding\nfuture applications, the investigation and the thorough understa nding of phenomena related\nto the spin-wave propagation in the nonlinear regime in Heusler compo unds is crucial.\nIn this Letter, we report nonlinear higher harmonic generation fro m a localized edge\nmode [11, 12] causing the emission of caustic spin-wave beams [13, 14 ] in a micro-structured\nHeusler waveguide. Spin-wave caustics are characterized by the s mall transversal aperture\nof a beam, which practically does not increase during propagation, a nd the well-defined\ndirection of propagation.\nThe investigated sample is a 5 µm wide spin-wave waveguide structured from a 30nm\nthick film of the Heusler compound CMFS. Details about the fabricatio n and the material\nproperties can be found in Refs.15 and 16. The microfabrication of the waveguide was per-\nformedusing electron-beam lithography andion-milling. Forthe excit ation ofspin dynamics\nin the waveguide the shortened end of a coplanar waveguide made of copper was placed on\ntop of it. The Oersted field created by a microwave current in this an tenna structure can be\nused to excite spin dynamics in the Gigahertz range. The antenna ha s a thickness of 400nm\nand a width of ∆ x= 1µm.\nAll observations have been carried out using Brillouin light scattering microscopy\n(µBLS).[17] µBLS is a powerful tool to investigate spin dynamics in microstructur es with a\n2FIG. 1. (Color online) Sketch of the sample design. The short ened end of a coplanar waveguide\nis used as an antenna structure to excite spin dynamics in a 5 µm wide CMFS waveguide with a\nthickness of 30nm. The waveguide is positioned in the x-y-pl ane with its long axis pointing along\nthe x-direction. The external magnetic field is applied tran sversely to the waveguide in y-direction.\nThe figure includes a µBLS spectrum taken at a distance of 4.5 µm from the antenna in the center\nof the waveguide (see laser beam in the sketch) for an excitat ion frequency of fe= 3.5GHz, a\nmicrowave power of 20mW, and an external field µ0Hext= 48mT.\nspatial resolution of about 250nm and a frequency resolution of up to 50MHz.\nIn the following description, the waveguide is positioned in the x-y-pla ne with the long\naxis pointing in x-direction. The origin of the coordinate system is give n by the position\nof the antenna between x=−1µm andx= 0µm. An external magnetic field of µ0Hext=\n48mT was applied transversely to the waveguide in y-direction result ing in Damon-Eshbach\ngeometry [18] forspin waves propagatingalong thewaveguide. Ask etch of thesample layout\nis shown in Fig.1.\nIn addition, Fig.1 includes a spectrum taken by µBLS for an excitation frequency of\nfe= 3.5GHz and a microwave power of 20mW at a distance of 4.5 µm from the antenna in\nthe center of the waveguide. The spectrum shows not only a peak a tfe= 3.5GHz but also\nat 2fe= 7.0GHzand 3 fe= 10.5GHz. Furthermore, the intensity of the directly excited spin\nwaveat3.5GHzislowerthanforthehigherharmonicsatthepointofo bservation. Aswewill\nsee, the higher harmonics are excited resonantly by nonlinear magn on-magnon interactions\nand the intensity distribution is a consequence of the different prop agation characteristics\nof the observed spin-wave modes.\nA two-dimensional intensity distribution for the detection frequen cyfd=fe= 3.5GHz\nas well as the calculated dispersion relation [19] for the center of th e CMFS waveguide are\n3012345\ny position ( µm)123456789x position ( µm)fd=fe\n0123456\nwavevector ( µm−1)3456789101112frequency (GHz)\nfe2fe3fe\n(a) (b)\nFIG. 2. (Color online) (a) µBLS intensity distribution for the detection frequency fdbeing equal\nto the excitation frequency fe= 3.5GHz. The observed intensity of the edge mode is maximal near\nthe edges of the CMFS waveguide. Please note that the excitin g antenna is positioned between\nx=−1µm andx= 0µm. The blue dots in the graph indicates the position of the mea surement\npresented in Fig.1. (b) Calculated dispersion relation for the center of the CMFS waveguide\naccording to Ref.19 as well as excitation frequency and high er harmonics (dashed lines).\nshown in Fig.2. Figure2(a) reveals a strong localization of the intens ity at the edges of the\nwaveguide. The non-vanishing intensity close to the antenna and be tween the edges of the\nwaveguide ( y= 1−4µm) can be attributed to nonresonant, forced excitation by the Oe rsted\nfield created by the microwave current. Figure2(b) shows the spin -wave dispersion for the\nCMFS waveguide calculated according to Ref.19 assuming a homogen eous magnetization\noriented in y-direction by the external field. The material paramet ers used in all our calcula-\ntions were determined experimentally on the unstructured film via fe rromagnetic resonance\n(MS= 1003kA/m and Hani= 1kA/m) and via µBLS in the micro-structured waveguide\n(Aex= 13pJ/m) following a method described in [9]. The effective field of µ0Heff= 46mT\nused in the calculations was obtained by micromagnetic simulations. As can be seen, the\nlower cut-off frequency f= 6.9GHz is well above the excitation frequency of fe= 3.5GHz\n(see dashed line in Fig.2(b)). In the center of the waveguide, the a ssumption of a homoge-\nneous magnetizationis a very goodapproximation. Close to theedge s, demagnetizing effects\nare responsible for a strongly decreased effective field and an inhom ogeneous magnetization\nconfiguration. The inhomogeneity of the magnetization does not allo w for a quantitative\n45 10 15 20\nmicrowave power (mW)BLS intensity (arb. u.)fd=fe\nfd=2fe\nfd=3fe\nFIG. 3. (Color online) Power dependence of the BLS intensity for the detection frequencies fd=\n3,5GHz, 7GHz, and 10.5GHz for the fixed excitation frequency fe= 3.5GHz. Please note the\nlog-log presentation of the data. The lines in the graph corr espond to fits according to Eq.1. A\nleast square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3.\nmodeling of spin dynamics close to edges.[20] However, from previous work it is known that\nthis field and magnetization configuration allow for the existence of lo calized spin waves -\ncommonly referred to as edge modes - energetically far below the sp in-wave dispersion for\npropagating modes in the center.[11, 12] Therefore, we conclude t hat the spin-wave mode\natfd=fe= 3.5GHz is excited resonantly by the microwave field.\nAs can be seen from the spectrum in Fig.1, the applied microwave pow er of 20mW is\nsufficiently high to observe the nonlinear generation of higher harmo nics of the excitation\nfrequency fein theµBLS spectra. These resonant frequency multiplications to f= 2fe=\n7.0GHz and f= 3fe= 10.5GHz result in the excitation of propagating spin-wave modes in\nthe waveguide energetically above the cut-off frequency of the dis persion shown in Fig.2(b).\nFigure3 shows the dependence of the directly excited mode and the higher harmonics\non the applied microwave power. This data has been acquired close to the position of the\nedge mode and near the antenna at x= 0.7µm andy= 0.8µm. The data is presented on\na log-log scale with fits according to\nIn(p) =Anpsn+b, (1)\nwhereInis theµBLS intensity, Ana coupling parameter, pthe applied microwave power,\nandbthe noise-level in our measurement. As expected, these process es do not show a\n5threshold power level, but reliable detection on the background of t he noise is not possible\nfor powers below 5mW for fd= 3fe= 10.5GHz. The different slopes of the curves for the\ndifferent spin-wave modes nare caused by the different power-laws specified by the exponent\nsn. A least square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3.\nThese experimental findings close to the integer values 1, 2, and 3 a re in accordance with\nboth reported experimental data and theoretical predictions fo r the nonlinear generation of\nhigher harmonics.[3, 7]\nThe observation of the second harmonic can be understood qualita tively by considering\nthe strong demagnetizing fields caused by the out-of-plane compo nentmz(t) during the\nmagnetization precession. Due to the demagnetizing fields, the mag netization precession\nM(t) around the y-direction (defined by the effective field) follows an ellip tical trajectory\nrather than a circular one. In contrast to the case of a circular pr ecession, the resulting\nprojection of Mon the y-axis is time dependent and oscillating with the frequency 2 fe. The\nresulting dynamic dipolar field |hy(t)| ∝m2\nx−m2\nzcan be regarded as the source for the\nfrequency doubling. Similar considerations lead to the observation o f higher harmonics. A\nfull quantitative derivation of higher harmonic generation and othe r nonlinear effects based\non the expansion of the LLG in terms of the dynamic magnetization ca n be found in Ref.7.\nFigures4(a)and(b)showintensity mapsforthedetectionfreque nciesfd= 2fe= 7.0GHz\nandfd= 3fe= 10.5GHz, respectively. In both cases the intensity radiated from the\nposition of the edge mode is strongly directed, has a small transver sal aperture, and shows\nnondiffractive behavior. These radiation characteristics recorde d for the three spin-wave\nmodes at f= 3.5, 7.0 and 10.5GHz presented in Figs.2(a) and 4 are responsible for t he\nintensity distribution shown in the spectrum in Fig.1. The position of t his measurement\nis indicated in the corresponding intensity maps with a circle. Since the spin wave at\nfe= 3.5GHz is localized at the edges of the waveguide, its intensity is compar ably weak in\nthe center. In contrast, the higher harmonics have frequencies above the cut-off frequency of\nthe spin-wave dispersion and can propagate in the center of the wa veguide. The propagation\nanglesofthese modessupport theintensity distributionrecorded inourmeasurement. While\nan increased intensity can be found already for fd= 2fe, forfd= 3fethe two beams starting\nfrom both edges of the spin-wave waveguide even intersect at the measurement position\nresultinginthehighestintensityatthispoint. Becauseofthewell-de finedpropagationangles\nof the higher harmonics and the localization of the edge mode, the int ensity distribution is\n6012345\ny position ( µm)123456789x position ( µm)θ=78.0◦\nHextvG\nθfd=2fe\n012345\ny position ( µm)123456789x position ( µm)θ=67.0◦\nHextvG\nθfd=3fe\n(a) (b)\nFIG. 4. (Color online) µBLS intensity distribution for (a) fd= 2fe= 7.0GHz and (b) fd=\n3fe= 10.5GHz. Both intensity maps show strongly directed spin-wave beams along the angle\nθ=∠(Hext,vG). The lines in the maps are guides to the eye to identify the pr opagation angle θ.\nThe blue dots in the graphs indicate the position of the measu rement presented in Fig.1.\nstrongly depending on the measurement position.\nThis observation of spin-wave beams with small transversal apert ure is reminiscent of the\nresults in Refs.21 and 22, where nonlinear three-magnon scatter ing in yttrium iron garnet\nis reported. However, in that case, the propagation direction of t he nonlinearly generated\nspin wave is given by momentum conservation in the scattering proce ss. In contrary, in\nour case, due to the strong localization of the edge mode, the assu mption of a well-defined\ninitial wavevector and, thus, a strict momentum conservation is no t justified. In particular,\nit is not possible to find an initial wavevector that allows for the nonline ar generation of the\nsecond and third harmonic at the same time still respecting momentu m conservation.\nIn the following, we will describe the observed propagationcharact eristics using the prop-\nerties of the anisotropic spin-wave dispersion in a magnetic thin film.[13 , 14, 19] Because\nof this anisotropy, the direction of the flow of energy, which is given by the direction of\nthe group velocity vG= 2π∂f(k)/∂kof the investigated spin waves, can differ significantly\nfrom the direction of its wavevector k. To estimate the relevant range in k-space in our\nexperiment, we have to consider the lateral dimensions of the sour ce for the nonlinear pro-\ncesses. Since the excitation by the oscillating Oersted field is most effi cient directly below\nthe antenna, the edge mode has the highest intensity in this region g iven by the width of\nthe antenna of ∆ x= 1µm. The spread of the edge mode in y-direction can be estimated\n7from the intensity map in Fig.2(a) to be smaller than 1 µm. Because of this localization, the\nedge mode in our measurement at f= 3.5GHz can be regarded as a source for the nonlinear\nemission of the higher harmonics with lateral dimensions of approxima tely 1×1µm2. As a\nfirst approximation, the Fourier transformation of this geometry lets us estimate the maxi-\nmum wavevector that can be excited by the edge mode to be kmax≈6.3µm−1. As we will\nsee, the direction of the group velocity can be assumed to be const ant for most wavevectors\nthat can be excited. This finally leads to the formation of the caustic s in our experiment.\nFor given frequency and external field, the iso-frequency curve f(kx,ky) =constcan be\ncalculated analytically from the dispersion relation. Calculations for f= 2fe= 7.0GHz and\nf= 3fe= 10.5GHz are illustrated in Fig.5(a), where kyis shown as a function of kx. Using\nthis data, we calculate the direction θof the flow of energy of the spin waves relative to the\nexternally applied field by:\nθ=∠(Hext,vG) = arctan( vx/vy) = arctan( dky/dkx). (2)\nFigure5(b)showsthecalculatedpropagationangle θintheCMFSwaveguideasafunction\nofky. The most important feature in the trend of θis the small variation of ∆ θ≤2◦in the\nrange of ky= 2−7µm−1for both frequencies f= 2feandf= 3fe. While the wavevector\nkchanges, the direction of vG- and, thus, the flow of energy - keeps almost constant as\na function of k. In this range, which includes the maximum wavevector kmax≈6.3µm−1\nthat can be emitted from the edge mode (see considerations above ), the calculations yield\nθcalc(2f) = 79◦andθcalc(3f) = 66◦as mean values, respectively. The dash-dotted lines in\nFig.5(b) represent the propagation angles θof the spin-wave beams observed experimentally\nas shown in Fig.4 ( θexp(2f) = 78◦andθexp(3f) = 67◦). The comparison of experimental\nfindings and analytical calculations shows an agreement within the ex pected accuracy of\nour measurement setup and is, therefore, supporting our conclu sion. Higher harmonics with\nky≤2µm−1are emitted with strongly varying directions from the edge mode and can be\nregarded as a negligible background in our measurement.\nIn summary, we reported nonlinear higher harmonic generation fro m a localized source\nin a micro-structured CMFS waveguide leading to the emission of stro ngly directed spin-\nwave beams or caustics. This observation results from the complex interplay of different\nphenomena in magnonic transport in magnetic microstructures. Th e localization of an edge\nmode due to demagnetizing fields in the waveguide leads to the format ion of a source for\n80 2 4 6 8\nkx (µm−1)02468ky (µm−1)\nk\nvG\nHextθf=2fe\nf=3fe\n0 2 4 6 8\nky (µm−1)657075808590θ (degree)f=2fe\nf=3fe\n(a) (b)\nFIG. 5. (Color online) Analytical calculations according t o Ref.19. (a) Iso-frequency curves\nfconst=f(kx,ky) forf= 2fe= 7.0GHz and f= 3fe= 10.5GHz. Based on these calculations ex-\nemplary directions for k,vG,Hextand the propagation angle θ=∠(Hext,vG) = arctan( dky/dkx)\nare shown in the graph. (b) Radiation direction θcalculated from the iso-frequency curves shown\nin (a). Dash-dotted lines correspond to the angles θexpobserved in the experiment.\nthe following nonlinear processes. The nonlinear higher harmonic gen eration results in the\nresonant excitation and emission of propagating spin waves at 2 feand 3fein a wavevector\nrange corresponding to the localization of the edge mode. The expe rimentally observed\npower dependencies of the different spin-wave modes show the exp ected behavior for direct\nresonant excitation and nonlinear higher harmonic generation. As s hown by our calculation,\nthe anisotropic spin-wave dispersion yields a well-defined direction of the flow of energy\nof the emitted spin waves in the relevant range in k-space. The calcu lation is not only\nqualitatively in accordance with our experimental findings but does a lso show quantitative\nagreement.\nWe gratefully acknowledge financial support by the DFG Research U nit 1464 and the\nStrategic Japanese-German Joint Research from JST: ASPIMATT . Thomas Br¨ acher is sup-\nported by a fellowship of the Graduate School Materials Science in Ma inz (MAINZ) through\nDFG-fundingoftheExcellence Initiative(GSC266). Wethankourco lleaguesfromthe Nano\nStructuring Center of the TU Kaiserslautern for their assistance in sample preparation .\n∗tomseb@physik.uni-kl.de\n[1] H.Schultheiss, X.Janssens, M. van Kampen, F.Ciubotaru , S.J.Hermsdoerfer, B.Obry,\n9A.Laraoui, A.A.Serga, L.Lagae, A.N.Slavin, B.Leven, B.Hi llebrands, Phys. Rev. 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B79, 144428 (2009).\n11"    },    {        "title": "2102.07712v2.Magnetodynamic_properties_of_dipole_coupled_1D_magnonic_crystals.pdf",        "content": " \n1 \n Magnetodynamic properties of dipole -coupled 1D magnonic crystals  \nSuraj Singh1*, Xiansi Wang1, Ankit Kumar2, Alireza Qaiumzadeh1, Peter  Svedlindh2, Thomas \nTybell ,3 and Erik Wahlström1 \n1Center for Quantum Spintronics, De partment of Ph ysics, NTNU - Norwegian University of \nScience and Technology, NO -7491 Trondheim, Norway  \n2Department of Materials Sciences and Engineering, Uppsala University, Box 516, SE -75121 \nUppsala, Sweden  \n3Department of Electronic Systems, NTNU - Norwegian University of Science and Technology, \nNO-7491 Trondheim, Norway  \n*Corresponding author e -mail : rsinghsuraj1992@gmail.com  \n \nAbstract  \nMagnonic crystals  are magnetic meta material s, that provide a pr omising way  to manipulate \nmagnetodynamic properties by controlling the geometry of the patterned structures . Here , we \nstudy the  magnetodynamic properties  of 1D magnonic crystals  consist ing of parallel NiFe  strip s \nwith different strip  widths and separations . The strips couple via dipole -dipole interaction s. As \nan alternative to experiments and/or micromagnetic simulations , we investigate the accuracy of  \na simple macrospin model . For the case of simple strips , a model with a single free parameter to \naccount for an overestimation of the out of plane demagn etization of the magnonic lattice  is \ndescribed . By adjusting  this parameter , a good fit with experimental as well as micromagnetic \nresults  is obtained . Moreove r, the  Gilbert damping is found independent of lattice constant  \nhowever  the inhomogeneous linewidth broadening  found to increase with decreasing stripe \nseparation .  \nKeywords:  Permalloy,  Magnonic Crystals, Dipole coupling, Ferromagnetic resonance   \n   \n2 \n 1. Introduction  \nSpin dynamics in nanostructured material s have attracted attention due to  interesting underlying \nphysics and potential for technological applications. Magnonic crystals (MCs)  are a class of \nartificial magnetic media  that offer a promising way to manipulate the magnetodynamic \nproperties  in microwave frequ ency by exploiting the pattern ed geometry1,2. Due to their \ninteresting magnetic properties, MCs find applications in a wide range of magnetic devices such \nas advanced magnetic storage, data processing , and spin logic gates3,4,5. As a consequence, MCs \nhave been studied extensively both theoretically and experimentally in numerous magnetic \nsystem s in order to explore the impact of MCs control parameters on its static and \nmagnetodynamic properties , and for their potential application in novel magnonic devices6-\n8,9,10 ,11,12,13,14. \nAdvances in lithography technique s make it possible to fabricate  nanometer -sized MCs \nwith  narrow spacing s. 1D MCs , with  a periodic magnetic strip pattern along one direction ha ve \nattracted considerable attention due to  their simple geometry , convenient for studying the \nimpact of lattice confinement on the magnetodynamic properties at the nanoscale  7,10,6. In such \nsystem s, the dipolar coupli ng of magnetic strips plays an important role in the magnetodynamic \nproperties. When the strips form a closely packed array, the fundamental mode of individual \nstrips couples via a dynamic dipolar interactio n resulting in formation of  collective spin -wave \nexcitation s6,7,15 ,16,17,18. This i s the result of  the dynamic dipolar magnetic field remov ing the \ndegeneracy between the discrete energy levels of the different magnetic elements.  The collective \ndynamics stemming from the  magnetodynamic  dipolar interaction affect the writing time in \nclosely packed storage media, the synchronization of spin -torque oscillators and most \nimportantly the spin -wave dynamics in MCs19. The spacing between adj acent magnetic elements  \nin such system s is a central  parameter governing the interstrip dipolar coupling . Thus, \ninvestigating the effect of dipolar coupling on the magnetodynamic properties gives valuable \ninformation on the  underlying physics and for poten tial application of MCs in magnonic devices.  \nMost of the previous investigations have been focused on  Brillion Light Scattering  (BLS) \nstud ies of dipolarly coupled 1D MCs, where the interplay of dipolar coupling on collective mode  \n3 \n excitation s and the forma tion of magnonic bandgap s have been studied extensively7,11,16. \nFerromagnetic resonance (FMR) is a sensitive nondestructive technique allowing to study the \nmagn etodynamic properties of the MCs . However, there are only a few report s of FMR stud ies \nof dipole -coupled MCs 20,21, which is especially tru e for  detailed stud ies of important  control \nparameter s such as size and separation of the building blocks making up the MC s and their impact \non the spin dynamics. Also, the impact of dipolar coupling on magnetic damping , which is  \nimportant to  lower the power of magnonic devices , is poorly understood.  In this paper, we \npresent a study of  magnetodynamic properti es of dipolarly coupled 1D MCs by FMR  \nspectroscopy. The 1D MCs consist of parallel Permalloy (Py) strip s prepared using electron beam \nlithography. We report the effect of strip  width  and lattice constant on the resonance field  and \ndescribe a simple macrosp in model that can be used to predict resonance behavior of 1D MCs . \nAlso, the impact  of the MC structure  on the FMR linewidth  has been investigated by broadband \nFMR spectroscopy.  \n2. Experimental Details  \nThe MCs consisted  of Py (Ni 80Fe20) deposited on silicon substrate s by e -beam evaporation . \nElectron beam lithography and lift -off techniques  were  used to fabricate the 1D strip -based  MCs,  \nhaving variable strip width  𝑤 and inter -strip separation  𝑠. The lattice constant of the MCs is  𝜆=\n𝑤+𝑠. Fig. 1 (a) shows the SEM image of a sample described by 𝜆=100 nm and 𝑤=50 nm. \nThe total area of each MC is  11 mm2.  A constant deposition rate was used for all samples to \nensure approximately the same thickness, 𝑑 of 14±3 nm. \nThe magnetodynam ic properties of the MCs were investigated by two complementary \nFMR techniques . Cavity FMR measurements were carried out in a commercial  X-band electron \nparamagnetic resonance (EPR) setup with a fixed microwave frequency of 9.4 GHz (Bruker Bio -\nspin ELEXSYS 500, with a cylindrical TE -011 microwave cavity).  The setup is equipped with a \ngoniometer allowing to rotate the sample 360° in -plane as well as out -of-plane . A schematic of \nthe sample rotation including the magnetization and magnetic field vectors is  shown in fig. 1 (b) . \nA microwave field is applied to the cavity and an applied dc magnetic field is swept to record the \nmicrowave absorption. The measurements were performed with low amplitude modul ation of  \n4 \n static field with lock -in detection to enhance the signal to noise ratio. To extract the resonant \nfield, the measured FMR absorption was fitted to a sum of the derivative of symmetric and \nantisymmetric Lorentzian functions, and the line -shape para meters such as resonant field and \nlinewidth were extracted22. \nFor broadband FMR measurements , a microwave signal generator  FMR setup with a \ncoplanar waveguide (CPW) and  lock-in amplifier  detection technique  was emplo yed. A pair of \nhomemade Helmholtz coils generating a low -frequency (211.5 Hz) and low -amplitude magnetic \nfield (0.25 mT) was used to modulate the microwave signal, which was detected by the lock -in \namplifier.  The FMR spectra  were recorded sweeping  the dc magnetic field at constant microwave \nfrequency. The measurements were taken  at various frequenc ies ranging from 5 to 16 GHz in \nsteps of 0.5 GHz with the dc magnetic field applied parallel  and perpendicular to the magnetic \nstrips of the  MCs23. \n3. Results and Discussion  \n3.1 Resonant field  \nTo investigate  the magnetodynamic properties , we studied  MCs with lattice constants ranging \nfrom 100  nm to 550 nm  with  a fixed width 𝑤=50 nm. In this subsection, all the experiments \nwere done using a 9.4 GHz cavity . The obtained resonant field s as a function of the in-plane \nrotation angle for different  lattice constants  are shown in fig. 2(a). For each sample, two modes \ncan be observed for a magnetic field applied perpendicular to the strips - along the 𝑥-axis (𝜙=\n0°). The two modes shift towards each other as the applied field direction rotates away from 𝜙=\n0°, and at 𝜙= ±15° the modes merge into a single -mode before disappearing.  The two modes \ncorrespond to two equilibrium magnetization directions. No modes are observed for a  wide \nrange of field directions around  the 𝑦-direction i.e., along the strip s. The frequency of the easy -\naxis mode falls outside the detectable range of cavity FMR measurements . The resonant field  \ndecrease s with decreasing  𝜆, wh ich is due to the increasing dipolar  interaction s. We then fix the \nmagnetic field along 𝐇∥𝒙̂ (𝜙=0°) and plot the higher resonant field  versus the lattice constant \n𝜆 in fig. 2(b). The resonant field  increases with increasing 𝜆.  \n5 \n To understand the observed magnetodynamic behaviors, we develop a macrospin \nanalytical model for MCs, and verify its validity  by micromagnetic simulations. Each magnetic \nstrip of the  MCs is considered as a macrospin  with  synchron ized precession of the spin s. The \nmagnetodynamics of the MC is governed by the Landau -Lifshitz -Gilbert (LLG) equation,  \n𝜕𝒎\n𝜕𝑡=−𝛾𝒎×𝑯eff+𝛼𝒎×𝜕𝒎\n𝜕𝑡, \nwhere 𝒎 is the unit vector along the magnetization direction of each strip , 𝛾=𝑔𝜇𝐵\nℏ is the \ngyromagnetic  ratio ,  𝑔 is the La ndé g-factor, µ𝐵 is the Bohr magneton, and ℏ is the reduced \nPlanck’s constant.   There are different values of 𝛾 corresponding to 𝑔=2.00 to 2.17 used in \nliterature24. Here we use 𝛾=176 Grad/s/T corresponding to 𝑔=2.00. The specifi c value of 𝛾 \ndoes not qualitatively  affect the physics we discuss. The term  𝛼 is the Gilbert damping  parameter  \nand 𝑯eff is the  total  effective field , \n𝑯eff=𝐻𝒙̂−𝑀𝑠(𝑁𝑥𝑚𝑥𝒙̂+𝑁𝑦𝑚𝑦𝒚̂+𝑁𝑧𝑚𝑧𝒛̂)+𝑯int, \nwhere 𝐻𝒙̂ is the applied external field along the 𝑥-direction  and −𝑀𝑠(𝑁𝑥𝑚𝑥𝒙̂+𝑁𝑦𝑚𝑦𝒚̂+\n𝑁𝑧𝑚𝑧𝒛̂) is the demagnetization field of the strip with 𝑀𝑆 being the saturation magnetization and \n𝑁𝑥,𝑦,𝑧 the demagnetization components  of each strip which  can be  calculated analytically  from \nthe strip dimensions (width 𝑤, thickness 𝑑 and the length of the strip25). The magneti c damping \nwas neglected when calculating the resonant field  of the modes for the sake of simplicity. 𝑯int is \nthe inter -strip dipolar  interaction  field . To calculate 𝑯int, we consider a strip with two \nneighbo uring strips as illustrated in fig. 2(c) and calculate the dipolar field from the neighbo uring \nstrip s at the center point of the middle  strip . The magnetic charge density on the left and right \nsurfaces (orange surface s in fig. 2(c))  are −𝑀𝑠𝑚𝑥 and 𝑀𝑠𝑚𝑥, respectively . \nThe total field from the left surface (fig.(2c)) can then be estimated by  \n𝐻left=−𝑀𝑠𝑚𝑥\n4𝜋∫∫1\n(𝑥2+𝑦2+𝑧2)−𝑥\n√𝑥2+𝑦2+𝑧2𝑑𝑧𝑑𝑦𝑑\n2\n−𝑑\n2∞\n−∞=arctan𝑑\n2𝑥\n𝜋𝑀𝑠𝑚𝑥, \nSimilarly, for the right surface  \n𝐻right=−arctan𝑑\n2𝑥\n𝜋𝑀𝑠𝑚𝑥,  \n6 \n where  𝑥 is the distance from the center point to the surface.  \nFor the 𝑛th strip, the distance is 𝑥=±𝑤\n2+𝑛𝜆. Thus, the total dipolar field on the center of a \nstrip reads  \n𝐻int𝑥=𝑀𝑠𝑚𝑥\n𝜋∑[arctan𝑑\n(−𝑤+2𝑛𝜆)−arctan𝑑\n(𝑤+2𝑛𝜆)]+∞\n𝑛=−∞, \nSince we have already conside red the shape anisotropy of the center strip, the 𝑛=0 term should \nbe omitted. This term can be absorbed in 𝑁𝑥 (𝑁𝑥′=𝑁𝑥−2\n𝜋∑[arctan𝑑\n(−𝑤+2𝑛𝜆)−+∞\n𝑛=1\narctan𝑑\n(𝑤+2𝑛𝜆)]). On the other hand, the magnetic charge density on top and bottom surfaces \n(blue surfaces) are 𝑀𝑠𝑚𝑧 and −𝑀𝑠𝑚𝑧, respectively. The 𝑥 component cancels out due to the \nMCs symmetry, and the 𝑧 component  at the center is  \n𝐻top𝑧=2𝑀𝑠𝑚𝑧\n4𝜋∫∫1\n(𝑥2+𝑦2+𝑑2/4)−𝑑/2\n√𝑥2+𝑦2+𝑑2/4𝑑𝑥𝑑𝑦𝑛𝜆+𝑤\n2\n𝑛𝜆−𝑤\n2∞\n−∞, \nand the total 𝑧-componet is  \n𝐻int𝑧=−𝑀𝑠𝑚𝑧\n𝜋∑[arctan(𝑤+2𝑛𝜆)\n𝑑−arctan(−𝑤+2𝑛𝜆)\n𝑑]+∞\n𝑛=−∞ \nThis term can be absorbed in 𝑁𝑧 (𝑁𝑧′=𝑁𝑧+2\n𝜋∑[arctan(𝑤+2𝑛𝜆)\n𝑑−arctan(−𝑤+2𝑛𝜆)\n𝑑]+∞\n𝑛=1 ). \nFor all the MCs that we have prepared , 𝑁𝑦<𝑁𝑥′<𝑁𝑧′. Thus, the equilibrium magnetization \ntends to be in-plane and along the 𝑦-direction. When applying the external field along the 𝑥-\ndirection, the equilibrium magnetization is tilted  with respect to the 𝑦-direction  satisfying  𝒎∥\n𝑯eff, so that  \n𝐻−𝑁𝑥′𝑀𝑆cos𝜙𝑚\n−𝑁𝑦𝑀𝑆sin𝜙𝑚=cos𝜙𝑚\nsin𝜙𝑚, \nwhere 𝜙𝑚 is the azimuthal angle of 𝒎. The largest 𝐻 for the above equation to have a solution \n𝜙𝑚=arccos𝐻\n(𝑁𝑥′−𝑁𝑦)𝑀𝑠 is 𝐻=(𝑁𝑥′−𝑁𝑦)𝑀𝑠. Above this value, 𝒎 is saturated along  the 𝑥- \n7 \n direction. To obtain the eigenfrequency of the MC s, we expand 𝒎 around its equilibrium \ndirection, assume the small precessional component to have a harmonic form 𝑒𝑖𝜔𝑡, and keep \nonly  linear  terms . The result is the well -known Smit -Beljers formula26, \n𝜔=\n{    𝛾√(𝑁𝑧′𝑀𝑠−𝑁𝑥′𝑀𝑠)(𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻2\n𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠)       𝐻<(𝑁𝑥′−𝑁𝑦)𝑀𝑠  \n𝛾√(𝑁𝑧′𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻)(𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻)                      𝐻≥(𝑁𝑥′−𝑁𝑦)𝑀𝑠 (1) \nwhere 𝜔 = 2𝜋𝑓 is the eigen frequency of the FMR mode . We plot the eigenfrequency versus 𝐻 \nin fig. 2(d) to understand the behavior observed in the f ig. 2(a). The horizontal line at 9.4 GHz \ncorresponds to the frequency of t he cavity, and its intersections with the dispersion curve \ncorre spond  to the two experimentally observed resonant modes.  The low field mode is an \nunsaturated mode around  an intermediate equilibrium magnetization state between the long \nand short axes of the strips . The high field mode is attri buted to the uniform precession in the \nfully satu rated state along  the 𝑥-direction .  \n Concentrating on the saturated mode , the fig. 2(b) depicts the resonance field of the \nmode vs lattice constant . The red solid line is the result from Eq. (1) with 𝑤=50 nm and fitting \nparameter 𝑑=14.0 nm. The result agrees with the experimental data within ±3% error . To be \nmore accurate, and to further verify the validity of the macrospin model , we perform \nmicromagnetic simulation s using  the open -source package Mumax 327. The simulation s were \nperformed using 1×1×𝑑 nm3 meshes. Limited by the computational capacity, the system size \nwas set to be 1024×1024×1 meshes  with periodic boundary conditions.  The exchange \nconstant 𝐴=1.3×10−11 J/m.  For same geometry, the simulation results of the r esonance fields \nare larger than that obtained by the macrospin model.  This fact has also been observed in other \nstudies13. Thus, to fit the experimental data, we have to use a smaller thickness 𝑑=12.5 nm. \nThe simulation results are shown by black squares  in the f ig. 2(b). The reason why results of \nanalytical model based on macrospin approximation differ from the simulations is as follows. T he \nmagnetization throughout each strip is not homogeneous as assumed in the macrospin model.  \nAt the two edges, the precessi onal amplit ude is larger, while near the center the amplitude is \nsmaller  [see inset of the f ig. 2(b) as well as fig. S3(b) of the Supplemental Material ]. The first  \n8 \n consequence is that  the total effective field is also inhomogeneous in the strips28. There is a \ndipolar interaction  field near the center, so a larger external field is necessary to reach the \nresonance than  in the macrospin model. The second consequence is that the demagnetization \nfactor in  the 𝑧-direction  (𝑁𝑧′) is significantly overes timated in the macrospin model, while 𝑁𝑥′ and \n𝑁𝑦 are not affected much. Nevertheless, the macrospin model qualitatively reproduces the \nexperimental results, showing that the main reason of the increasing resonant field is the \ndecrea sing inter -strip dipolar interaction when the separation becomes larger. To compensate \nthe overestimated 𝑁𝑧, we introduce an empirical  dimensionless parameter  𝜂<1 to renormalize  \n𝑁𝑧 in Eq. (1) (for the saturated peak 𝐻≥(𝑁𝑥′−𝑁𝑦)𝑀𝑠), \n𝜔=𝛾√(𝜂𝑁𝑧′𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻)(𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻), (2) \nFor 𝑤=50 nm and 𝑑=12.5 nm obtained from the simulation , we find that 𝜂=0.82 fits the \nsimulation results best , as shown by the green dashed line in  the fig. 2(b). The gray area between \nthe two black dashed lines means the range of 𝐻 from Eq. (2) with 𝑤=50±2 nm and 𝑑=\n12.5±0.3 nm, in which the errors are estimated from the limited precision of lithography and \nthe e -beam evaporation techniques. The experimen tal data are well in the range  indicated by the \ngray area in  the fig. 2(b) .  \nWe also prepared MC s samples with larger strip width, in which the macrospin \napproximation obviously  fails. Fig . 3(a) shows the derivative FMR absorption  of a 𝑤=200 nm, \n𝜆=250 nm sample when rotating the applied field in -plane. We can observe that for any 𝜙 \nangle, there is a t least one resonant mode. When the field is along the 𝑥-direction, there are \nthree resonant modes , instead of two in the previous samples. To under stand  the three modes, \nwe performed micromagnetic simulations for 𝑯∥𝒙̂. After Fourier transform, the result is shown \nin Fig. 3(b). Three peaks can be observed, which is consistent with the experiment. More details \ncan be found in Supplemental Material, si nce the main topic of the paper concerns  the narrow -\nstrip samples.  \n \n  \n9 \n 3.2. Ma gnetic Damping  \nTo investigate the impact of dipolar coupling on magnetic damping,  the recorded  FMR signal was \nfitted to the  sum of  derivative s of the symmetric and antisymmetric Lorentzian functions to \ndetermine the lineshape parameters: resonance field ( Hr) and the full width of half maxima \n(𝛥𝐻)22. The fitting determined linewidth versus frequency data is fitted to the model:  \n𝜇0∆𝐻(𝑓)= 𝜇0∆𝐻0+4𝜋𝛼𝑒𝑓𝑓\n𝛾𝑓 (3) \nwhere 𝛼𝑒𝑓𝑓 is the effective damping parameter  including the Gilbert damping and the eddy \ncurrent contributions29 and ∆𝐻0  is the linewidth at zero frequency also known as  \ninhomogeneous line width  broadening30.  \nThe linewidth broadening was measur ed for the magnetic  field applied parallel and \nperpendicular to the strips.  Table 1  summarizes  the characteristics of the FMR linewidth at \ndifferent frequencies when the field is applied along and perpendicular to the strips ( 𝑦-direction)  \n(spec ific data can be found in Fig. S6 in supplemental materials) . In total , the effective damping \nparameter  𝛼𝑒𝑓𝑓 = 0.0045±0.0005 , and is almost independent of the strip separation. The \ninhomogeneous linewidth broadening decreases with increasing lattice consta nt (or strip \nseparation).  The same trend is obtained when the field is perpendicular to the strips ( 𝑥-direction). \nThe inhomogeneous broadening is small for a reference thin film sample without  the MCs \nstructure, which is within expectation because the thi n film is much more homogeneous .  An \ninteresting observation is that the effective Gilbert damping  is very different for 𝐇∥𝑦̂, 𝐇∥𝑥̂ and \nthin film experiments.  It is noted that different multi -magnon scattering processes31 and metallic \nelectromagnetic effects such as eddy curren ts32 can contribute to such effects.   \n4. Discussions and Summary  \nA lot of effort has been made to understand the dynamics of MCs8,12 -14,31,33,34. For the FMR mode, \none important aspect is to understand the deviation from the macrospin model. Since the \ndimensions of the magnetic strips are usually much l arger than the exchange length, 𝑙𝑒𝑥=√2𝐴\n𝜇0𝑀𝑠2 \n, a deviation is natural  due to an inhomogeneous magnetization profile. As expected, the  \n10 \n micromagnetic simulations shows that, when the mesh size is comparable to strip size, the \nmacrospin model is recover ed. When reducing the mesh size, the resonant field becomes lower, \nuntil the mesh size is as small as  𝑙𝑒𝑥 and the result converges to the “genuine” value. This \nobservation is also applicable when 𝐦 is allowed to be inhomogeneous in the thickness direc tion. \nFor thin strips whose thickness is much smaller than the width, an analytical formula for boundary \nconditions has been derived which can be used to solve the nonuniform magnetization profile35. \nHowever, in our case the thickness and the width are of the same order of magnitude. So, we \nintroduce an empirical parameter 𝜂 which depends only on the bar dimensions, but independent \nof the bar separation, as a correction to the macrospin model. Recen tly, there is a very \ncomprehensive study on the FMR mode of MCs34. Frequency -sweeping FMR was modeled and \nstudied in that paper. Here, we consider field sweeping FMR, and we prov ide an intuitive picture \nfor the deviation and failure of macrospin model. For 50 nm strips, the strip edge and center have \ndifferent oscillation amplitudes but same phase, so the macrospin model is still qualitatively \ncorrect (i.e. emergence of two modes at two equilibrium 𝐦 directions), and the quantitative \nresults can be recovered by introducing a compensation factor. For 200 -nm strips, the oscillations \nat strip edge and center are out -of-phase, so the macrospin model fails and full micromagnetic \nsimula tion is necessary.  \nWe observe a strong anisotropy in the line broadening. Both the interception (the \ninhomogeneous broadening) and the slope (the effective damping) are anisotropic. The effective \ndamping includes the Gilbert damping and dissipation by the eddy current. The Gilbert damping \nis usually isotropic30, but the eddy current is anisotropic because it is related to the geometry of \nthe sample. This is confirmed by our numerical simulation, where we assume an isotropic Gilbert \ndamping and we do not obs erve the anisotropy in the effective damping. Therefore, we attribute \nthe anisotropic effective damping to the eddy current effect29,32. The inhomogeneous line \nbroadening contains the contribution from external sources such as the multi -magnon scattering, \nanisotropy, and scattering due to roughness and defects. These effects can be strongly \nanisotropic31,36, and are not considered in the simulation. So, we suppose that they could be the \nreason for the o bserved anisotropic Δ𝐻0. We also observe that Δ𝐻 is very small for a film but \nshows a decreasing trend when increasing the strip separation. This may relate to the  \n11 \n inhomogeneity of the whole sample. The inhomogeneous broadening is positively related to t he \ninhomogeneity of the sample37. \nIn summary , we have investigated the  magnetodynamic properties of 1D MCs  with different \nlattice constants . The resonant field found to increas e with increasing lattice constant because \nof decrea sing inter -strip dipolar coupling. The experimental results are qualitatively explained by \na macrospin model when the strips are narrow . The accuracy of the macrospin model can be \nquantitatively  improved b y renormalizing the out -of-plane demagnetization factor fitted by \nmicromagnet ic simulation s. Obvious difference in linewidth slopes was found for different field \ndirections.  \n \nAcknowledgements  \nThis work was partly supported by the Research Counc il of Norway through its Centre  of \nExcellence funding scheme, project number 262633, “QuSpin” . S. S. acknowledge s partial funding \nobtained from the Norwegian PhD Network on Nanotechnology for Microsystems, which is \nsponsored by the Research Council of Norway, Di vision for Science, under contract no. \n221860/F40.  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Crowell,  Phys \nRev B 101 (13), 134430 ( 2020).  \n37 Ernst Schlömann,  Physical Review 182 (2), 632 (1969).  \n   \n14 \n Table 1  The Gilbert damping parameters estimated for MCs  and the reference P y thin film  \n \n \n \n \n \n  Sample  H along y -axis H along x -axis \nα ΔH (Oe)  α ΔH(Oe)  \n𝝀=𝟏𝟎𝟎  nm, 𝒘=𝟓𝟎 nm 0.004  97 0.010  72 \n𝝀=𝟏𝟓𝟎  nm, 𝒘=𝟓𝟎 nm 0.004  58 0.012  35 \n𝝀=𝟐𝟓𝟎  nm, 𝒘=𝟓𝟎 nm 0.005  37 - - \nPy thin-film 0.007  5 0.007  5  \n15 \n Figure captions  \nFig. 1   (a) SEM image of a MC with 𝜆=100 nm and 𝑤=𝑠=50 nm. (b) Schematic diagram of \nthe coordinate system defining geometry of the field.  \nFig. 2   (a) The polar plot of resonant field extracted for samples of different 𝜆. (b) The measured \nhigher resonant fields when the applied field is along x direction for different samples \n(blue dots). The red line is the macrospin model result (Eq. 1) for 𝑑=14.0 nm and 𝑤=\n50 nm. The simulation results for 𝑑=12.5 nm and 𝑤=50 nm are shown as black \nsquares. The green dashed line is effective model (Eq. 2)  for  𝑑=12.5 nm and 𝑤=50 \nnm with empirical coefficient 𝜂=0.82. The gray area indicates the range of resonant \nfield for 𝑑=12.5±0.3 nm and 𝑤=50±2 nm. The inset schematically illustrates th e \namplitude of  magnetization tilting during the precession for the macrospin model and \nthe actual simulation.  (c) Schematics depicting the geometry of magnetic strips used in \nanalytical calculations and micromagnetic simulations. (d) The dispersion curves  for 𝜆=\n100 nm  and 𝜆=550 nm. The dashed horizontal line is the cavity frequency 9.4 GHz  \nFig. 3   (a) Experimental results of angle  (𝜙)-dependent differential FMR absorption for the 𝜆=\n250 nm, 𝑤=200 nm sample at RT. (b) Fourier amplitude of the average magnetization \nestimated by micromagnetic simul ations on  the 𝜆=250 nm, 𝑤=200 nm sample for \nfield along 𝜙=0. Three peaks can be identified, which is consistent with the experiment \n(indicated by the ar rows).  \n   \n16 \n  \n \n \n \n \n \n \n \nFig. 1  \n  \nM \nH \n𝛉𝐌 \n𝜃 \n𝜙 \n𝑦Ԧ \n𝑥Ԧ \n𝑧Ԧ \n (b) \n (a) \n \n17 \n  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2  \n  \n(b) \n(c) \n (d) \n(a) \n \n18 \n Fig. 3  \n  \n(a) \n (b)  \n19 \n Supplementary Information  \nMapping the dipolar coupling and magnetodynamic  properties of \ndipole coupled 1D magnonic crystals  \nSuraj Singh1, Xiansi Wang1, Ankit Kumar2, Alireza Qaiumzadeh1, Peter  Svedlindh2, Thomas \nTybell ,3 and Erik Wahlström1 \n1Center for Quantum Spintronics, De partment of Ph ysics, NTNU - Norwegian University of \nScience and Technology, NO -7491 Trondheim, Norway  \n2Department of Materials Sciences and Engineering, Uppsala University, Box 516, SE -75121 \nUppsala, Sweden  \n3Department of Electronic Systems, NTNU - Norwegian University of Science and Technology, \nNO-7491 Trondheim, Norway  \n   \n20 \n Figure S1(a) shows the raw cavity FMR spectra recorded on the magnonic  crystals. Two modes \nwere observed for the magnetic applied perpe ndicular to strips. Fig. S1(b) shows the angle -\ndependent FMR measur ements . The two modes found shifting towards the each other as the \nmagnetic field is rotated away from the angle (φ = 0°). The resona nce field of low field mode \nincreases as a fu nction of angle (φ) whereas the it decreases for the high field mode. The modes \nmerge into a single -mode at around φ= ± 15° and then disappears . \n \n \n \n \n \n \n \n \n \n \n \n  Fig. S1 (a) The FMR lineshape recorded on 𝜆=100 nm and 𝑤=50 nm sample for the magnetic \nfield applied perpendicular to the strips. S2(b) The angle -dependent FMR spectra recorded rotating \nthe magnetic field in the plane of the sample from angle  φ = 0° to 360°.  \n(a) \n (b) \nH \nφ=0° \n H  \n21 \n The micromagnetic simulations performed on the magnonic lattices were used to \ninvestigate  the equilibrium magnetization  profile at the observed modes . Figures. S2(a) and S2(b) \ndepict the equilibrium magnetization profiles of the low -field  mode  and the high -field  mode  \nobserved in 𝜆=100  nm, 𝑤=50 nm sample  for the magnetic field applied along x -axis.  As \ndescribed in the main paper, the high field mode occurs at 𝐻=3.34×103 Oe when the \nmagnetization is homogeneously  magnetized  along the x direction. The low field  occurs at 𝐻=\n1×103 Oe when 𝒎 is oriented about 63°  with respect to the x -axis in the bars .  Although 𝒎 is \nnot uniform over the bar, the variation is not large .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(a) \n (b) \n(c) \nFig. S2 The equilibrium profile of static magnetization at the low field mode in (a) and high field \nmode in (b) ) for  𝜆=100 nm, 𝑤=50 nm sample . (c) The resonant field  plotted as function \nfrequency for the low field mode .   \n22 \n  \nKnowing the equilibrium magnetization profile, time -dependence of the 𝑚𝑧 component at \nresonance and its behaviors at the edge and the center of the strip  can be plotted, see  figs. S3(a) \nand S 3(b). It can be seen that  the edge and the center precess  in-phase at both peaks . At the first \npeak, the amplitudes at the edge and center are similar, while at the second peak the edge has a \nlarger amplitude than the center.  \n \n  \nFig. S3 The amplitude of the magnetization precession at the low field mode in (a) and at high \nfield mode in (b).   \n(a) \n (b)  \n23 \n The equilibrium magnetization profiles at the three peaks from low field to high field for the  \n𝜆=250 nm, 𝑤=200 nm sample  for the magnetic field applied along x -axis are shown in figs. \nS4(a), S4(b), and S4(c). The magnetic fields are at the resonant peaks shown in Fig. 3(b) in the \nmain text. The magnetization pofile is much different than the 𝜆=100 nm, 𝑤=50 nm sample  \nbecau se the bars are much wider  and the spatial variation of 𝒎 is much more significant in the \ntwo low -field peaks.  The time dependence of 𝑚𝑧 oscillation from the three peaks is shown figs. \nS5(a), S5(b), and S5(c).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nAt the first peak, the edge and the center have a 𝜋-phase difference. Since the energy absorption \nis measured, the energy absorptions at the edge and the center cancel with each other. So \nalthough the oscillation amplitude is large, the peak intensity is small. At the second peak, the \ncenter magnetization is almost along the x direction, but edge magnetization is significantly Fig. S4 The equilibrium magnetization profiles at three peaks observed in 𝜆=250 nm, 𝑤=50 \nnm sample from low to high in (a), (b), and (c) respectively.  \n(a) \n(b) \n(c) \n \n24 \n tilted. The oscillations also have a 𝜋-phase difference, but at the center the amplitude is much \nlarger. Furthermore, the center magnetization has a larger proportion in the whole ba r. So the \nenergy absorption is dominated by the central part, and has the largest intensity. At the third \npeak, the magnetization is fully aligned along the x direction. The oscillations have a 𝜋\n2-phase \ndifference, but the edge has a much larger amplitude . The energy absorption is dominated by \nthe edge part.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFor both  𝜆=100 nm, 𝑤=50 nm and 𝜆=250 nm, 𝑤=200  samples  the amplitudes at the \nedges are larger than that at the center for the saturated peaks. But for other peaks the Fig. S5 The amplitude of the magnetization precession at the three peaks from low to high in (a), \n(b) and (c) respectively.  \n(a) \n (b) \n(c)  \n25 \n comparison is complicated. This can be understood as follows. At equilibrium, the local effective \nfield is paral lel to the local magnetization for each point. The larger magnitude the effective field \nhas, the harder the magnetization precesses. For the saturated peaks, the edges bear larger \ndemagnetization field (because they are closer to the surface magnetic charg es). The \ndemagnetization field is antiparallel to 𝒎, so the total effective field at the edges is smaller than \nthe center. For other non -saturated peaks, the situation is very complicated, so there is no simple \npicture for the observed amplitude difference . \n \nThe details of the measur ements of magnetic damping are summarized in fig . S6. \n \nFig. S6 The linewidth versus frequency data fitted to a straight line for the magnetic field applied \nalong and perpendicular to the magnetic strips for 𝜆 with  𝑤=50 nm samples in 5(a) & 5(b) and \nfor the reference thin -film in fig. 5(c). The filled colored symbols the experimental data points \nwhereas the solid red line shows the fitted data.  \n(b) \n (a) \n(c) "    },    {        "title": "2306.04617v2.Helicity_dependent_optical_control_of_the_magnetization_state_emerging_from_the_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "1  Helicity-dependent optical control of the magnetization state emerging from the Landau-Lifshitz-Gilbert equation  Benjamin Assouline, Amir Capua*  Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel   *e-mail: amir.capua@mail.huji.ac.il  Abstract:   It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, 𝑯, and the Gilbert damping, 𝜶. Therefore, in ultrashort optical pulses, where 𝑯 can temporarily be extremely large, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for typical ultrashort pulses, the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced to the LLG equation, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by 𝜼=𝜶𝜸𝑯/𝒇𝒐𝒑𝒕, where 𝒇𝒐𝒑𝒕 and 𝜸 are the optical frequency and gyromagnetic ratio. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids.  2  The ability to control the magnetization order parameter using ultrashort circularly polarized (CP) optical pulses has attracted a great deal of attention since the early experiments of the all-optical helicity dependent switching (AO-HDS) [1-4]. This interaction was found intriguing since it appears to have all the necessary ingredients to be explained by a coherent transfer of angular momentum, yet it occurs at photon energies of 1\t−\t2\t𝑒𝑉, very far from the typical resonant transitions in metals. The technological applications and fundamental scientific aspects steered much debate and discussion [5,6], and the experiments that followed found dependencies on a variety of parameters including material composition [7-9], magnetic structure [10-12], and laser parameters [1,3,13], that were often experiment-specific [4]. Consequently, a multitude of mechanisms that entangle photons [14,15], spins [16,17], and phonons [18,19] have been discovered. References [4,20] provide a state of the art review of the theoretical and experimental works of the field. Ferromagnetic resonance (FMR) experiments are usually carried out at the 𝐺𝐻𝑧 range. In contrast, optical fields oscillate much faster, at ~\t400−800\t𝑇𝐻𝑧. Therefore, it seems unlikely that such fast-oscillating fields may interact with magnetic moments. However, the amplitude of the magnetic field in ultrashort optical pulses can, temporarily, be very large such that the magnetization may respond extremely fast. For example, in typical experiments having 40\t𝑓𝑠−1\t𝑝𝑠 pulses at 800\t𝑛𝑚, with energy of 0.5\t𝑚𝐽 that are focused to a spot size of ~0.5\t𝑚𝑚$, the peak magnetic flux density can be as high as ~\t5\t𝑇, for which the corresponding Gilbert relaxation time reduces to tens of picoseconds in typical ferromagnets. Here we show that ultrashort optical pulses may control the magnetization state by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. We find that the strength of the interaction is determined by 𝜂=𝛼𝛾𝐻/𝑓%&', where 𝑓%&' and 𝛼 are the angular optical frequency and the Gilbert damping, respectively, and 𝛾 is the gyromagnetic ratio. Moreover, we show that for circularly polarized (CP) pulses, the polarity of the optically induced torque is determined by the optical helicity. From a quantitative analysis, we find that a sizable effective out-of-plane field is generated which is comparable to that measured experimentally in ferromagnet/heavy-metal (FM/HM) material systems. 3  The LLG equation is typically not applied in the optical limit, and hence requires an alternative mathematical framework whose principles we adopt from the Bloch equations for semiconductor lasers [21,22]. We exploit the analogy between the magnetization state and the Bloch vector of a two-level system (TLS) [23,24] by transforming the LLG equation under a time-varying magnetic field excitation to the dynamical Maxwell-Bloch (MB) equations in the presence of an electrical carrier injection. In this transformation, the reversal of the magnetization is described in terms of population transfer between the states. The paper is organized as follows: We begin by transforming the LLG equation to the density matrix equations of a TLS. We then identify the mathematical form of a time-dependent magnetic field in the LLG equation, 𝐻AA⃗&()&↓↑, that is mapped to a time-independent carrier injection rate into the TLS. Such excitation induces a population transfer that varies linearly in time and accordingly to a magnetization switching profile that is also linear in time. The mathematical 𝐻AA⃗&()&↓↑ field emerges naturally as a temporal impulse-like excitation. We then show that when 𝛼 is sizable, 𝐻AA⃗&()&↓↑ acquires a CP component whose handedness is determined by the direction of the switching. By substituting 𝐻AA⃗&()&↓↑ for an experimentally realistic picosecond CP Gaussian optical magnetic pulse, we show that it can also exert a net torque on the magnetization. In this case as well, the helicity determines the polarity of the torque. Finally, we present a quantitative analysis that is based on experimental data. The LLG equation describing the dynamics of the magnetization, 𝑀AA⃗, where the losses are introduced in the Landau–Lifshitz form is given by [25]: 𝑑𝑀AA⃗𝑑𝑡=\t−𝛾1+𝛼$𝑀AA⃗×𝐻AA⃗−𝛾𝛼1+𝛼$1𝑀,𝑀AA⃗×𝑀AA⃗×𝐻AA⃗.(1) Here 𝑀, and 𝐻AA⃗ are the magnetization saturation and the time dependent externally applied magnetic field, respectively. We define 𝐻AA⃗-.. by: 𝐻AA⃗-..≜K𝐻AA⃗−\t𝛼𝑀,𝐻AA⃗×𝑀AA⃗\tL,(2) and in addition, 𝜅≜/012!O𝐻-..\t4−𝑗𝐻-..\t5Q/2 and 𝜅6\t≜/012!𝐻-..\t7, where 𝜅 and 𝜅6 can be regarded as effective AC and DC magnetic fields acting on 𝑀AA⃗, respectively. We 4  transform 𝑀AA⃗ to the density matrix elements of the Bloch state in the TLS picture and compare it to the Bloch equations describing a semiconductor laser that is electrically pumped [26]:  ⎩⎪⎨⎪⎧𝜌̇00=𝛬0−𝛾0𝜌00+𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇$$=𝛬$−𝛾$𝜌$$−𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇0$=\t−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗(𝜌00−𝜌$$)𝑉0$\t.\t\t(3) In this reference model, 𝛬0 and 𝛬$ are injection rates of carriers to the ground and excited states of the TLS, respectively. They are assumed to be time independent and represent a constant injection of carriers from an undepleted reservoir [27]. 𝛾0 and 𝛾$ are the relaxation rates of the ground and excited states, and 𝛾;<= is the decoherence rate due to an inhomogeneous broadening. 𝑉0$ is the interaction term and 𝜔89: is the resonance frequency of the TLS. Figure 1(a) illustrates schematically the analogy between the magnetization dynamics and the electrically pumped TLS. We find the connection between the LLG equation expressed in the density matrix form and the model of the electrically pumped TLS: ]𝛬0−𝛾0𝜌00+[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t−𝑗𝜅𝜌$0+𝑐.𝑐.𝛬$−𝛾$𝜌$$−[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t𝑗𝜅𝜌$0+𝑐.𝑐.−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗𝑀7𝑉0$=\t−𝑗𝜅6𝜌0$\t+𝑗𝜅𝑀7.(4) The pumping of the excited and ground states by the constant 𝛬0 and 𝛬$ rates implies that the reversal of the magnetization along the ∓\t𝑧̂ direction is linear in time. Using Eq. (4) we find 𝜅, and hence a field 𝐻AA⃗, that produces such 𝛬0 and 𝛬$. We define this field as 𝐻AA⃗&()&↓↑: 𝐻AA⃗&()&↓↑=\t±𝛬&𝑀,$−𝑀7$f𝑀5−\t𝑀40g.(5) 𝐻AA⃗&()&↓↑ depends on the temporal state of 𝑀AA⃗ while 𝛬&=𝛾𝛬0/(1+𝛼$) is the effective field strength parameter. 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ induce a linear transition of 𝑀AA⃗ towards the –𝑧̂ and +𝑧̂ direction, respectively. 5  Figure 1(b) presents the outcome of the application of 𝐻AA⃗&()&↓↑ by numerically integrating the LLG equation. The Figure illustrates 𝐻AA⃗(𝑡), 𝑀7(𝑡), and the 𝑧̂ torque, O−𝑀AA⃗×𝐻AA⃗Q7, for alternating 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ that switch 𝑀AA⃗ between ∓𝑀,𝑧̂. The magnitude of 𝛬& determines the switching time, 𝛥𝜏↓↑, chosen here to describe a femtosecond regime. Equation (4) yields 𝛥𝜏↓↑=(1+𝛼$)𝑀,/(\t𝛾𝛬&)≈𝑀,/𝛾𝛬& in which 𝑀7 is driven from 𝑀7=0 to 𝑀7≅±𝑀, (for derivation, see Supplemental Material Note 1). It is seen that O−𝑀AA⃗×𝐻AA⃗Q7 is constant when 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑ are applied so that the switching profile of 𝑀7 is linear in time. It is also seen that 𝐻AA⃗&()&↓↑ requires that m𝐻AA⃗m diverge as 𝑀7 approaches ±𝑀,, which is not experimentally feasible. To account for a more realistic excitation, in Fig. 1(c) we simulated a pulse whose trailing edge was taken as a reflection in time of 𝐻AA⃗&()&↓↑, and that is shorter by an order of magnitude as compared to the leading edge. In this case 𝑀AA⃗ remains in its final state when 𝐻AA⃗ is eventually turned off. The polarization state of 𝐻AA⃗&()&↓↑ is determined from the polarization state of the transverse components of 𝑀AA⃗. Next, we show that for larger 𝛼, 𝑀5(𝑡) becomes appreciable such that 𝐻AA⃗&()&↓↑ acquires an additional CP component. This result emerges naturally from the Bloch picture: we recall that the transverse components of 𝑀AA⃗ are expressed by the off-diagonal density matrix element. According to Eq. (3), 𝜌0$ oscillates at 𝜔89: and decays at the rate 𝛾;<=, whereas the sign of 𝜔89: determines the handedness of the transverse components of 𝑀AA⃗. Namely, the ratio between 𝜔89: and 𝛾;<= determines the magnitude of the circular component in the n𝑀4(𝑡),𝑀5(𝑡)o trajectory. Under the application of 𝐻AA⃗&()&↓↑, Eq. (4) yields 𝜔89:=±𝛾𝛬&𝛼𝑀,/[(𝑀,$−𝑀7$)(1+𝛼$)] and 𝛾;<==∓𝛾𝛬&𝑀7/[(𝑀,$−𝑀7$)(1+𝛼$)] readily showing that |𝜔89:/𝛾;<=|=𝛼𝑀,/𝑀7 increases with 𝛼, so that 𝐻AA⃗&()&↓↑ acquires an additional CP component (see Supplemental Note 2 for full derivation). Figure 2 illustrates these results. Panel (a) presents the components of 𝑀AA⃗(𝑡) for the same simulation in Fig. 1(b). It is seen that 𝑀5(𝑡) is negligible and thus 𝐻AA⃗&()&↓↑ remains linearly polarized. When 𝛼 is increased, an elliptical trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane emerges, while the constant transition rate of 𝑀7 persists as illustrated in Fig. 2(b). In this case, 6  𝐻AA⃗&()&↓↑ acquires a right-CP (RCP) or left-CP (LCP) component depending on the choice of 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑. The coupling between the handedness and reversal direction in a femtosecond excitation is reminiscent of the switching reported in AO-HDS experiments and emerges naturally in our model. These results call to examine the interaction of the CP magnetic field of a short optical pulse with 𝑀AA⃗. Figure 3(a) presents the calculation for experimental conditions [4]. The results are shown for an 800\t𝑛𝑚 optical magnetic field of an RCP Gaussian optical pulse 𝐻AA⃗%&'(𝑡). The pulse has a duration determined by 𝜏&, an angular frequency 𝜔%&', and a peak amplitude 𝐻&->? that is reached at 𝑡=𝑡&->?. In our simulations 𝜏&=3\t𝑝𝑠 and 𝑡&->?=10\t𝑝𝑠. The pulse energy was ~\t5\t𝑚𝐽 and assumed to be focused to a spot size of ~\t100\t𝜇𝑚$, for which 𝐻&->?=8⋅10@\t𝐴/𝑚. Here we take 𝛼=0.035 [28,29]. For such conditions, the Gilbert relaxation time corresponding to 𝐻&->? is 𝜏2=02/A\"#$%≈16\t𝑝𝑠 [30]. It is readily seen that for such 𝜏2 the magnetization responds within the duration of the optical pulse indicating that the interaction between the optical pulse and 𝑀AA⃗ becomes possible by the LLG equation. Following the interaction, 𝑀7=−5×10BC⋅𝑀:, namely a sizable net longitudinal torque results. In agreement with the prediction of the TLS model, pulses of the opposite helicity induce an opposite transition as shown in Fig. 3(b). The results are compared to the measured data discussed in Supplemental Material Note 3.  To this end we simulate the same conditions of the measurements including optical intensity and sample parameters. Accordingly, we find from our calculations an effective field which is of the same order of magnitude as measured.   For a given pulse duration, we define the interaction strength parameter 𝜂=2𝜋𝛼𝛾𝐻&->?/𝜔%&', which expresses the ratio between 𝜏2 and the optical cycle and is 2.5⋅10BC in Fig. 3(a). The principles of the interaction can be better understood at the limit where 𝜂→1 and for which the interaction can be described analytically. To this end, we set 𝜂=1. The higher optical magnetic fields required for this limit are achievable using conventional amplified femtosecond lasers, for example by focusing a ~\t5\t𝑚𝐽 pulse into a spot size of ~\t1\t𝜇𝑚$. Figure 3(c) illustrates the results for an RCP 𝐻AA⃗%&' pulse of a duration of 20\t𝑓𝑠 determined by the full width at half-maximum of the 7  intensity. The Figure reveals the different stages of the interaction. During the leading edge, for 𝑡<~\t40\t𝑓𝑠, the relative phase between 𝐻AA⃗%&' and 𝑀AA⃗ seems arbitrary. As 𝑡&->? is reached, the Gilbert relaxation time becomes as short as the optical cycle allowing 𝑀AA⃗ to follow 𝐻AA⃗%&' until it is entirely locked to 𝐻AA⃗%&'. In this case, 𝑀AA⃗ undergoes a right-circular trajectory about 𝑧̂. The switching of 𝑀AA⃗ takes place at the final stage of the interaction: During the trailing edge of the pulse, the amplitude of 𝐻AA⃗%&' reduces and 𝜏2 extends, thereby releasing the locking between 𝑀AA⃗ and 𝐻AA⃗%&'. In this case, the switching profile of 𝑀7 is monotonic linear-like in time, closely resembling the transition stemming from a constant carrier injection rate in the Bloch picture. The optically induced transition can be described analytically following the calculation presented in Supplemental Note 4, from which we find the transition rate:  𝛤/𝑀,=∓32√2𝑙𝑛K43L1𝜏&}~𝑙𝑛K𝐻&->?0.27𝐻'=L−~𝑙𝑛𝐻&->?𝐻'=/√2,(6) where 𝐻'==D&\"'$E/2 is the value of 𝐻&->? at 𝜂=1. The rate 𝛤/𝑀, is plotted as well in Fig. 3(c) and reproduces the numerical calculation. 𝛤 depends on the ratio between 𝐻&->? and 𝐻'= and is only weekly dependent on 𝐻&->?. Namely, when 𝐻&->?≫𝐻'=, the circular trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane persists longer after 𝑡&->?, but as the amplitude of the pulse decays below 𝐻'=/√2, 𝑀AA⃗ is driven out of the 𝑥−𝑦 plane and the reversal takes place (see Supplemental Material Note 5). This analysis also holds for LCP pulses, which result in an opposite reversal of 𝑀AA⃗, as shown in Fig. 3(d).  To summarize, in this work we demonstrated that the control of the magnetization by an optical field arises from first principles by introducing the magnetic part of the optical radiation to the LLG equation. This was seen from the comparison between the case where 𝜂≪1 and the case of 𝜂=1. Using the TLS model, we demonstrated the coupling between the optical helicity state and the polarity of the longitudinal torque. A quantitative analysis of the optically induced torque revealed that it can be comparable to that observed in experiments.   8  Figure 1 \n       Fig. 1. (a) Left panel: Illustration of 𝑴AAA⃗ on the Bloch sphere. Right panel: Illustration of the electrically pumped TLS. (b) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ of Eq. (5). The Figure illustrates the temporal plots of 𝑴𝒛/𝑴𝒔, 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑,𝒚 and O−𝑴AAA⃗×𝑯AAA⃗Q𝒛 normalized to unity. (c) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ and a more realistic trailing edge, for the same conditions in (b). Full lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and dashed lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.    \n9  Figure 2 \n Fig. 2. Temporal evolution of the components of 𝑴AAA⃗ under the influence of alternating 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑ for (a) small and (b) large damping. Black dashed lines indicate the alternation between 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.   \n10   \n11  Fig. 3. (a) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟐.𝟓⋅𝟏𝟎B𝟒. Top and middle panels depict the temporal evolution of the 𝒙 and 𝒚 components of 𝑴AAA⃗ and 𝑯AAA⃗𝒐𝒑𝒕\tin normalized units. Bottom panel depicts 𝑴𝒛/𝑴𝒔. (b) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. (c) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟏. Top panel presents the temporal behavior of m𝑯AAA⃗𝒐𝒑𝒕m and 𝝉𝜶, where 𝑯𝒄𝒓𝒊𝒕=𝑯𝒕𝒉/√𝟐 and 𝑯𝟏/𝟐=𝟎.𝟐𝟕𝑯𝒕𝒉. (d) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. In (c) and (d), black solid lines represent the analytical solution of 𝜞/𝑴𝒔.   References  [1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, \"All-Optical Magnetic Recording with Circularly Polarized Light\", Physical Review Letters 99, 047601 (2007). [2] J. Hohlfeld, C. D. Stanciu, and A. Rebei, \"Athermal all-optical femtosecond magnetization reversal in GdFeCo\", Applied Physics Letters 94, 152504 (2009). [3] D. Steil, S. Alebrand, A. Hassdenteufel, M. Cinchetti, and M. 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"    },    {        "title": "1505.00522v2.High_topological_number_magnetic_skyrmions_and_topologically_protected_dissipative_structure.pdf",        "content": "High-topological-number magnetic skyrmions and topologically protected dissipative structure\nXichao Zhang,1, 2Yan Zhou,1, 2,\u0003and Motohiko Ezawa3,y\n1Department of Physics, University of Hong Kong, Hong Kong, China\n2School of Electronics Science and Engineering, Nanjing University, Nanjing 210093, China\n3Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan\n(Dated: January 6, 2016)\nThe magnetic skyrmion with the topological number of unity ( Q= 1) is a well-known nanometric swirling\nspin structure in the nonlinear \u001bmodel with the Dzyaloshinskii-Moriya interaction. Here, we show that magnetic\nskyrmion with the topological number of two ( Q= 2) can be created and stabilized by applying vertical spin-\npolarized current though it cannot exist as a static stable excitation. Magnetic skyrmion with Q= 2 is a\nnonequilibrium dynamic object, subsisting on a balance between the energy injection from the current and the\nenergy dissipation by the Gilbert damping. Once it is created, it becomes a topologically protected object against\nfluctuations of various variables including the injected current itself. Hence, we may call it a topologically\nprotected dissipative structure. We also elucidate the nucleation and destruction mechanisms of the magnetic\nskyrmion with Q= 2 by studying the evolutions of the magnetization distribution, the topological charge\ndensity as well as the energy density. Our results will be useful for the study of the nontrivial topology of\nmagnetic skyrmions with higher topological numbers.\nPACS numbers: 75.70.-i, 75.78.-n, 85.70.-w, 85.75.-d\nI. INTRODUCTION\nThere is a long history of skyrmions from the particle\nphysics to the condensed matter physics [1–4]. Originally,\nSkyrme introduced \"skyrmions\" in the three-dimensional (3D)\nspace to describe nucleons as elementary particles possess-\ning nontrivial topological numbers [1]. Subsequently, Belavin\nand Polyakov (BP) applied the concept to the two-dimensional\n(2D) ferromagnetic (FM) system and predicted a magnetic ex-\ncitation carrying a nontrivial topological number, that is, the\nPontryagin number [2]. The BP-skyrmion is an exact solution\nof the nonlinear \u001bmodel. However this solution has a scale\ninvariance, and hence the BP-skyrmion has no definite radius.\nIt is necessary for a dynamically stable and physical skyrmion\nto have a definite radius. Consequently, a skyrmion is char-\nacterized by two types of stabilities: the topological stability\nbased on the conservation of a nontrivial Pontryagin number,\nand the dynamical stability having a finite radius.\nSome interactions are required in order to break the scale\ninvariance. For instance, the dipole-dipole interaction breaks\nthe scale invariance. However, since it is a long-distance\nforce, the skyrmion size becomes very large in general [5, 6].\nRecently, the Dzyaloshinskii-Moriya interaction (DMI) has\nattracted much attention to provide the skyrmion with a finite\nradius, where the resultant radius is of the order of nanome-\nters [7–16]. It is called the magnetic skyrmion. Magnetic\nskyrmions might be suitable for building next-generation non-\nvolatile memory devices based on their topological stabil-\nity [10]. Furthermore, it has also been proposed to utilize them\nin logic computing [17].\nStrictly speaking, the topological number is defined only\nin the continuum field theory with an infinitely large space.\n\u0003yanzhou@hku.hk\nyezawa@ap.t.u-tokyo.ac.jpAlthough the underlying system has a finite size and a lat-\ntice structure in the condensed matter physics, the topological\nnumber is well-defined, provided that the skyrmion spin tex-\nture is sufficiently smooth and sufficiently far away from the\nedge. Thus it is an intriguing concept in the condensed matter\nphysics possessing the aspect of the continuum theory and that\nof the lattice theory. We can actually leverage these properties\nfor practical applications. It is possible to create or destroy\nmagnetic skyrmions which are topologically stable [5, 6, 18–\n23].\nAs far as the topological analysis is concerned, a magnetic\nskyrmion with any topological number Qis possible. Let\nus call such a skyrmion with Q\u00152a high-Q skyrmion. It\nwill be realized when the in-plane component of the spin ro-\ntates by 2\u0019Qand the skyrmion acquires a high helicity Q.\nHowever, only the magnetic skyrmion with Q= 1 has so far\nbeen realized. This is because that a static magnetic skyrmion\nwithQ\u00152is unstable since the DMI cannot prevent it from\nshrinking to a point, which we shall prove later. In this paper,\nemploying a dynamical breaking of the scale invariance [21],\nwe create a high-Q skyrmion with Q= 2by applying a spin-\npolarized current perpendicular to the FM nanodisk with the\nDMI. We investigate in details how a high-Q skyrmion with\nQ= 2is created from a magnetic bubble with Q= 0through\nsuccessive creations of two Bloch points.\nOnce the spin texture of a high-Q skyrmion becomes suf-\nficiently smooth with respect to the lattice spacing, the topo-\nlogical protection becomes active. Namely, the resultant spin\ntexture is topologically robust against fluctuations of various\nvariables including the injected current itself. This is due to\nthe fact that the topological number cannot change continu-\nously from its quantized value. However, when the current is\nswitched off, the high-Q skyrmion quickly shrinks to the order\nof the lattice scale and dissipates. The equilibrium and dissi-\npation processes are well-described in terms of the Rayleigh\ndissipation function composed of the energy injection and dis-\nsipation terms [24]. The dissipation is found to spread all overarXiv:1505.00522v2  [cond-mat.mes-hall]  5 Jan 20162\nFIG. 1. (Color online) The ordinary skyrmion ( Q= 1) and the high-Q skyrmion ( Q= 2). (a) Our system consists of a 1-nm-thick FM\nnanodisk with a thickness of 150 nm, where the spin current polarized along the \u0000z-direction is injected into the central 30-nm-diameter\nregion. The out-of-plane magnetization mz, the in-plane magnetization ( mx;my) and the topological charge density qare illustrated for the\nskyrmion (Q= 1) and the high-Q skyrmion ( Q= 2). (b) The topological number Qas a function of time t. We create a skyrmion with\nQ= 1(Q= 2) by applying an external magnetic field perpendicular to the nanodisk plane pointing along the +z-direction with an amplitude\nofBz= 50 mT (Bz= 250 mT), as depicted in blue (red). The DMI constant is D= 2 mJ m\u00002. The spin current with a current density\nofj= 3\u00021012A m\u00002is switched on and off at t= 0 andt= 5 ns. When the DMI is set to be D= 0 mJ m\u00002, no skyrmion is formed\n(Q= 0), as depicted in green. A skyrmion with Q= 1remains stable as it is even after the spin current is switched off when D= 4mJ m\u00002\nandBz= 0 mT, as depicted in cyan, which is consistent with the previous result shown in Ref. 10. (c) A magnetic bubble with Q= 0 has a\npair (two pairs) of blue and red areas, when it develops into a skyrmion with Q= 1 (Q= 2). The cones represent the magnetization, while\nthe color denotes the topological charge density.\nlike a burst at the moment of the skyrmion generation and de-\nstruction [25].\nThe high-Q skyrmion is a topologically protected dissipa-\ntive structure. One might think that the concept is a self-\ncontradictory proposition. However, this is not the case. We\nmay suggest an analogy of the quantum Hall (QH) state. It\nis well-known that the QH state at an integer filling factor is\nrobust against impurities and also against the change of the ex-\nternal magnetic field. Indeed the latter develops QH plateaux.\nThe robustness is due to the fact that the state is protected\nby the conservation of the topological number, that is, the\nChern number. Nonetheless, when the external magnetic field\nis switched off, the topological robustness is lost and the QH\nstate collapses. Clearly, the external magnetic field in the QH\nsystem corresponds to the injected spin current in the topolog-\nically stabilized dissipative structure of this work.\nII. NUMERICAL RESULTS ON THE HIGH-Q SKYRMION\nA. Nucleation of the high-Q skyrmion\nOur system is composed of a FM nanodisk and a spin-\npolarized current injection region with a radius of rc, as il-\nlustrated in Fig. 1(a) (see the Appendix for modeling details).\nThe development of the topological number Qis shown in\nFig. 1(b). Soon after the spin current is injected, Qsuddenly\nincreases to 1or2from 0, and remains stable, in the pres-\nence of the DMI. On the other hand, when the spin current\nis switched off, Qdecreases to 0. It should be noted that the\nhigh-Q skyrmion cannot be created dynamically without theDMI.\nWe are interested in the process how a magnetic bubble\nwithQ= 0 is converted into a magnetic skyrmion with\nQ6= 0. Upon the application of the spin current, there is a\nlarge energy injected into the core through the spin transfer\ntorque (STT). The spins are forced to reverse within the core\nupon the spin-polarized current injection. Due to the DMI, the\nspins are twisted around the core. The topological number is\nzero for such a state. This is a magnetic bubble with Q= 0.\nWe point out that the seed of a magnetic skyrmion is al-\nready present in the magnetic bubble. In Fig. 1(c), we show\nthe densities of the in-plane components of the magnetization\n(mx(x);my(x)) and the topological charge density q(x)of a\nmagnetic bubble before the nucleation to a magnetic skyrmion\nwithQ= 1 or2. We clearly observe a pair (two pairs) of\nblue and red areas indicating negative and positive topologi-\ncal charge densities, respectively.\nIn Fig. 2(a), we show the time evolution of the topological\nnumber, the total energy, the DMI energy, the average mag-\nnetization (mx,my,mz), and the Rayleigh dissipation func-\ntionWfor a high-Q skyrmion with Q= 2 (see Ref. 26 for\nSupplementary Movie 1). The selected top-views are shown\nin Fig. 2(b). The spin component mzmeasures the size of\nthe skyrmion, while mxandmycontribute to the topologi-\ncal charge density. First, mzstarts to decrease, implying that\nthe spins are inverted in the disk region. However, the topo-\nlogical number remains zero, since the in-plane spin compo-\nnents point along the same direction, as shown in Fig. 2(b) at\nt= 0:2ns.\nThe Rayleigh dissipation function gives us a vivid informa-\ntion on how the dissipation occurs in the dissipative system.3\nFIG. 2. (Color online) Time evolution of the high-Q skyrmion ( Q= 2). (a) Time evolution of the total energy Etotal, the DMI energy EDMI,\nthe topological number Q, the in-plane ( mx;my) and out-of-plane mzcomponents of magnetization averaged over the simulation area, and\nthe dissipation functions W. A high-Q skyrmion with Q= 2 is created nearly at t= 0:38ns. The DMI constant D= 2 mJ m\u00002. The spin\ncurrent density j= 3\u00021012A m\u00002. The external magnetic field Bz= 250 mT. (b) Top-views of the magnetization distributions mx,my\nandmzof the FM nanodisk, the corresponding topological charge density distribution qand the Rayleigh dissipation function Wat selected\ntimes. The green circle indicates the spin current injection region. The nucleation of a high-Q skyrmion with Q= 2occurs nearly at t= 0:38\nns, where the dissipation function spreads all over the FM nanodisk, implying that the spin wave propagates. The DMI is turned off at t= 10\nns. It is remarkable that the high-Q skyrmion remains stable even if the DMI is switched off, which demonstrates the topological protection\nagainst the change of a variable, that is, the DMI.\nIn the present system, the energy is injected into the core and\ndissipated in its outer side steadily before and after the nu-\ncleation of a magnetic skyrmion. However, the dissipation\nspreads around all over the FM nanodisk like a burst at the\ntransition moment of the topological number from Q= 0 to\nQ= 2(see Fig. 2(b) at t= 0:38ns).\nWe show a close-up of the change of the topological num-\nberQaroundt= 0:38ns in Fig. 3(a). Clearly, there are two\nsuccessive jumps of Qas0!1!2. We also show the\ntopological charge density, the energy density, and the spin\ndistribution around t= 0:38ns in Fig. 4.\nBased on these we obtain the following picture of the nu-\ncleation process. We focus on the case where there are two\npairs of blue and red areas in the domain boundary region of\nthe magnetic bubble (Fig. 1(c)). Both the topological charge\ndensityq(x)and the energy density \u000f(x)are large in these\nareas. In particular, there is a chance that \u000f(x)becomes large\nin a lattice-scale area so that the area has almost Q=\u00001.\nThis happens when two spins are antiparallel with one down-\nspin site between them, at t= 372:2ps in Fig. 4. A single\nor a few spins make a large rotation in order to decrease the\nenergy of the area. Indeed, the antiparallel spins become par-\nallel att= 372:8ps in Fig. 4. In this process the topological\nnumberQ=\u00001is lost, which is possible in the lattice theory.\nThis phenomenon would be viewed as a generation of a Bloch\npoint in the continuum theory. This makes clearer the role of\nthe Bloch point presented by Sampaio et al. in Ref. 10. Thereexists still a pair of blue and red areas. Now, there is a chance\nthat another lattice-scale area develops which has Q=\u00001in\nthe spin texture with Q= 1. (Note that there are two spins\nantiparallel at t= 406 ps, which become parallel at t= 407:6\nps.) It corresponds to the emergence of another Bloch point.\nAs a result, it turns out that two Bloch points emerge suc-\ncessively in a single magnetic bubble. When the spin texture\nbecomes sufficiently smooth, it yields a high-Q skyrmion with\nQ= 2.\nWe have also numerically observed that a high-Q skyrmion\ncan be successfully created in a wide range of parameters as\nwell as the spin current injection size, as shown for instance\nin Fig. 5. Fig. 5(a) shows the phase diagram of the high-Q\nskyrmion creation with respect to the external magnetic field\nBzand time, where the spin current is injected into a circle\nregion with a radius of rc= 15 nm. Fig. 5(b) shows the phase\ndiagram of the high-Q skyrmion creation with respect to the\nspin current injection region radius rcand the time t, where\nthe spin current is injected into a circle region with a radius of\nrc.\nMoreover, as shown in Fig. 6, the high-Q skyrmion is cre-\nated and maintained even at finite temperature. Fig. 6(a) il-\nlustrates the phase diagram of the high-Q skyrmion creation\nwith respect to the temperature T and time. Fig. 6(b) shows\nthe topological number Qas a function of the time at T = 0\nK and T = 100 K, where the high-skyrmion with Q= 2 is\ncreated shortly after the spin current is switched on. The struc-4\nFIG. 3. (Color online) (a) Nucleation and (b) annihilation pro-\ncesses of the high-Q skyrmion with Q= 2. A sudden change of\nthe topological number Qoccurs twice successively when the high-Q\nskyrmion with Q= 2is created or destroyed. The topological charge\ndensityq(x), the energy density \u000f(x), and the spin-component distri-\nbutionmz(x)of the state indexed by (t;Q)in (a) and (b) are shown\nin Fig. 4 and Fig. 9, respectively.\nture of the high-Q skyrmion is deformed at finite temperature\n(see Fig. 6(b) insets). Its topological number is almost 2but\nfluctuates slightly because the continuity of the spin texture of\na deformed skyrmion is broken at finite temperature.\nB. Evolution of the high-Q skyrmion\nThe continuity of the spin texture is recovered since a\nsmooth texture has a lower energy. Then the topological pro-\ntection becomes active. The dissipation function decreases\nrapidly and oscillates around zero. The system is relaxed to\na steady state around t= 5 ns. Both the topological charge\ndensity and the Rayleigh dissipation function are almost zero\noutside the domain wall encircling the high-Q skyrmion (see\nFig. 2).\nOscillations in various variables occur due to the DMI. It is\ninstructive to switch off the DMI in numerical simulations. In\nFig. 2, we switch off the DMI at t= 10 ns. First of all, the\nhigh-Q skyrmion remains stable. Second, the oscillations in\nthe energy, the magnetization components, and the dissipation\nfunctions disappear. After the relaxation at t= 20 ns, the in-\nplane magnetization components and the dissipation functions\nbecomes exactly zero (see Ref. 26 for Supplementary Movie\n2). The spin texture is described precisely by the magnetiza-\ntion components of the domain wall.We show how the balance holds between the energy injec-\ntion from the spin current and the dissipation by the Gilbert\ndamping in Fig. 7. The energy injection occurs in the vicinity\nof the edge of the current injection region ( r.rc) while the\ndissipation occurs mostly within the domain wall encircling\nit (r\u0018r0) together with energy flow from the inner to outer\nregions. This is due to the fact that spins precess within the\ndomain wall.\nC. Topological protection of the high-Q skyrmion\nIt is important to point out that this energy balance takes\nplace automatically so as to keep the topological number un-\nchanged. To check this, we change the spin current stepwise,\nwhich is shown in Fig. 8. The topological number remains to\nbe2when the spin current intensity changes even more than\ntwice. The energy injection due to the STT becomes larger\nas the spin current density increases. Accordingly, the energy\ndissipation due to the Gilbert damping increases. As a result,\nthe Rayleigh dissipation function remains zero in average al-\nthough it is oscillating. The total energy increases stepwise\nbut remains almost constant for each current strength. The to-\ntalmzdecreases as the spin current density increases, which\nimplies that the magnetic skyrmion expands for larger cur-\nrent density. We have shown that the magnetic skyrmion is\ntopologically robust against a considerable change of the spin\ncurrent injection.\nAs we have stated, once it is created, the high-Q skyrmion\nremains stable even if the DMI is switched off (see Fig. 2).\nFurthermore, it is stable against the fluctuations of various\nvariables. This is because a small change can induce only a\nsmall change of the topological number Q, but this is impos-\nsible sinceQis a quantized quantity. This property is called\nthe topological protection.\nD. Destruction of the high-Q skyrmion\nWhen the spin current is switched off at t= 5ns, the topo-\nlogical number remains as Q= 2 untilt\u00186:4ns and sud-\ndenly decreases to Q= 0, as shown in Fig. 1(b). The radius\nof the magnetic skyrmion shrinks since the skyrmion core is\nfixed by the spin current against the shrinking force due to\nthe kinetic energy as well as the external magnetic field (see\nRef. 26 for Supplementary Movie 3).\nA close examination shows that the collapse of the topo-\nlogical number occurs in two steps as Q= 2!1!0as in\nFig. 3(b). We may understand how the destruction of a mag-\nnetic skyrmion with Q= 2 occurs by investigating the time\nevolution of the topological charge density, the energy den-\nsity, and the spin distribution around t= 6:4ns as shown in\nFig. 9.\nIn the first step the energy density is localized almost on one\nlattice site, where a Bloch point is generated and the topolog-\nical number changes from Q= 2 toQ= 1. The size of the\nmagnetic skyrmion with Q= 1shrinks almost to the order of\nthe lattice site.5\nFIG. 4. (Color online) Nucleation process of the high-Q skyrmion with Q= 2. Snapshots of the topological charge density q(x), the energy\ndensity\u000f(x), the spin-component distribution mz(x), and its close up at sequential times. The DMI constant D= 2 mJ m\u00002. The spin\ncurrent density j= 3\u00021012A m\u00002. The external magnetic field Bz= 250 mT. In the simulation, each cell corresponds to one spin, and\nthe cell size is 1:5nm\u00021:5nm\u00021nm. In the spin distribution panels, each arrow stands for four spins, while it stands for one spin in the\ninsets. The nucleation process of the high-Q skyrmion with Q= 2 is found to occur in two steps. First, it starts when a high-energy-density\npart is localized to a lattice-scale area, which possesses almost Q=\u00001. A few spins rotate by large angles in this area, making the topological\nnumber of the area almost zero. The resultant spin texture has Q= 1. Second, a similar phenomenon occurs, yielding the high-Q skyrmion\nwithQ= 2after the relaxation. See Ref. 26 for Supplementary Movie 4.\nThe second step has some new features. The topologi-\ncal stability of the magnetic skyrmion is guaranteed by the\nfact that the core spin points in the direction opposite to the\nFM background. However, such a core spin disappears at\nt= 6407 ps since there is no lattice site at the core spin.\nAs a result, the spin texture becomes a vortex structure. Ac-\ncordingly, all the spins point along upward direction and the\nmagnetic skyrmion disappears. This is possible since the sys-\ntem is on the lattice and such a transition never happens for\nthe continuum system. In this process, the spiral spin wave is\ngenerated, as seen obviously in the topological density as well\nas the energy density at t= 6410 ps in Fig. 9.\nA comment is in order with respect to the stability of a mag-\nnetic skyrmion with Q= 1when the spin current is turned off\n(see Fig. 1(b)). The stability diagram has been explored in\nRef. 10 in the absence of the spin current. For instance, it is\nstable forD= 4 mJ m\u00002andBz= 0 mT, while it is unsta-\nble forD= 2 mJ m\u00002andBz= 50 mT. Indeed, when the\nspin current is switched off, a magnetic skyrmion with Q= 1\nremains stable or is destroyed according to these parameter\nchoices, as depicted in cyan or in blue in Fig. 1(b).III. THEORETICAL ANALYSIS OF THE HIGH-Q\nSKYRMION\nA. Hamiltonian\nThe Hamiltonian of the system is given by\nH=\u0000JX\nhi;jimi\u0001mj+X\nhi;jiD\u0001(mi\u0002mj)\n+KX\ni[1\u0000(mz\ni)2] +BzX\nimz\ni+HDDI;(1)\nwhere mirepresents the local magnetic moment orientation\nnormalized asjmij= 1 at the sitei, andhi;jiruns over all\nthe nearest neighbor sites in the FM layer. The first term rep-\nresents the FM exchange interaction with the FM exchange\nstiffnessJ. The second term represents the DMI with the\nDMI vector D. The third term represents the perpendicular\nmagnetic anisotropy (PMA) with the anisotropic constant K.\nThe fourth term represents the Zeeman interaction. The fifth\ntermHDDIrepresents the dipole-dipole interaction. Although\nwe have included the dipole-dipole interactions in all numer-\nical calculations, the effect is negligible since the size of a\nmagnetic skyrmion is of the order of nanometers.6\nFIG. 5. (Color online) (a) Phase diagram of the high-Q skyrmion\ncreation with respect to the external magnetic field Bzand timet.\nThe spin current density j= 3\u00021012A m\u00002, which is injected\ninto a circle region with a radius of rc= 15 nm. The DMI constant\nD= 2mJ m\u00002. (b) Phase diagram of the high-Q skyrmion creation\nwith respect to the spin current injection region radius rcand time\nt. The spin current density j= 3\u00021012A m\u00002, which is injected\ninto a circle region with a radius of rc. The DMI constant D= 2\nmJ m\u00002. The external magnetic field Bz= 250 mT. The color scale\nindicates the topological number Q.\nB. Topological number\nThe classical field m(x)is introduced for the spin texture\nin the FM system by considering the zero limit of the lattice\nconstant, that is, a!0. The ground-state spin texture is\nm= (0;0;1). We employ the continuum theory when we\nmake an analytic study of the system.\nA magnetic skyrmion is a spin texture which has a topolog-\nical number. Spins swirl continuously around the core, where\nspins point downward, and approach the spin-up state asymp-\ntotically. The magnetic skyrmion is characterized by the topo-\nlogical number known as the Pontryagin number. It is given\nbybQ=R\nd2xq(x)with the density\nq(x) =\u00001\n4\u0019m(x)\u0001(@xm(x)\u0002@ym(x)): (2)\nThe spin configuration of a magnetic skyrmion is\nparametrized as\nmx= cos\u001e(') sin\u0012(r); my= sin\u001e(') sin\u0012(r);\nmz= cos\u0012(r); (3)\nwhere'is the azimuthal angle and ris the radius in the polar\ncoordinate. The topological charge density q(x)is shown to\nbe a total derivative, and hence the topological number is a\nboundary value. It is explicitly calculated as\nbQ=1\n4\u0019[cos\u0012(1)\u0000cos\u0012(0)][\u001e(2\u0019)\u0000\u001e(0)]; (4)\nwhich does not depend on the detailed profile of cos\u0012(r)\nand\u001e('). The boundary conditions cos\u0012(0) =\u00001and\ncos\u0012(1) = 1 are imposed for any skyrmion at the skyrmion\ncenter (r= 0) and at infinity ( r=1). When\u001e=Q'+\u001f,\nthe topological number is Q, where\u001fstands for the helicity.\nFIG. 6. (Color online) (a) Phase diagram of the high-Q skyrmion\ncreation with respect to the temperature T and the time t. The DMI\nconstantD= 2 mJ m\u00002. The spin current density j= 3\u00021012\nA m\u00002. The external magnetic field Bz= 250 mT. The color scale\nindicates the topological number Q. (b) The topological number Q\nas a function of time tat T = 0 K and T = 100 K. The insets show\nthe snapshots of the high-Q skyrmion at T = 0K and T = 100 K.\nQmust be an integer for the single-valuedness. In general, \u0012\nand\u001fare functions of time t. The latter interpolates the Néel-\ntype (\u001f= 0;\u0019) or Bloch-type ( \u001f=\u0019=2;3\u0019=2) skyrmion.\nWe show the spin configurations of the magnetic skyrmions\nwithQ= 1 andQ= 2 in Fig. 1(a). Spins rotate Qtimes as\n'changes from '= 0to'= 2\u0019for the magnetic skyrmion\nwithQ. That is to say, when going around the spin texture\nof the magnetic skyrmion, the in-plane component of the spin\nrotates by 2\u0019Q.\nC. Dzyaloshinskii-Moriya interaction\nThe DMI is the Néel-type or Bloch-type depending on\nwhether it is introduced from the surface or bulk. We take\nthe interface-induced Néel-type DMI,\nHDM=D?Z\nd2x[nzdivn\u0000(n\u0001r)nz]: (5)\nWe substitute the magnetic skyrmion configuration equa-\ntion (3) into the DMI Hamiltonian, and we find\nHDM=D?Z\nrdrd'1\n2rcos [(Q\u00001)'+\u001f]\n\u0002(Qsin 2\u0012(r) + 2r@r\u0012(r)): (6)\nForQ6= 1, by integrating over ', we findHDM= 0. As a\nresult, the DMI does not prevent a static magnetic skyrmion7\nFIG. 7. (Color online) Profile of mzand the Rayleigh dissipation function Wof the high-Q skyrmion with Q= 2 in the presence and\nabsence of the DMI. (a) The radius of the FM nanodisk equals 75nm, and the spin-polarized current is injected into a circle region with a\nradius ofrc= 15 nm. The simulated skyrmion radius r0is equal to 24:55nm, which is defined as the radius of the circle where mz= 0. The\nDMI constant D= 2 mJ m\u00002. The spin current density j= 3\u00021012A m\u00002. The external magnetic field Bz= 250 mT. (b) The form of\nmz(r)is fitted by the domain wall solution equation (14) with the use of \u0015= 4:54nm, which is in good agreement with the theoretical value\n\u0015=p\nJ=K = 4:33nm. The functions W\u000bandWSTTin (a) are also well fitted by the same domain-wall solution.\nfrom shrinking to a point unless Q= 1. Hence, there is no\nstatic magnetic skyrmion stabilized by the DMI for Q6= 1.\nD. Rayleigh dissipation function\nThe system contains an energy injection by the spin-\npolarized current and an energy dissipation by the Gilbert\ndamping. They cannot be analyzed in the framework of the\nHamiltonian formalism, where the energy is a constant of mo-\ntion. It is described by the generalized Lagrangian formalism\nincluding the Rayleigh dissipation function.\nThe Rayleigh dissipation function consists of two terms,\nW=W\u000b+WSTT, the Gilbert damping term [27],\nW\u000b=~\u000b\u0012dm\ndt\u00132\n=~\u000b\u0010\n_\u00122+_\u001e2sin2\u0012\u0011\n; (7)\nwith the Gilbert damping constant \u000b, and the STT term,\nWSTT=~j\rjuz\u0001(_m\u0002m) =\u0000~j\rju_\u001esin2\u0012; (8)\nwhereudescribes the injection of the spin-polarized current,\nu(r) =j~\n\u00160ejP\n2dMSj(r)withj(r)representing the injected\ncurrent and z= (0;0;1). We takeu(r) =u0forr <rcand\nu(r) = 0 forr >rc. We note that W\u000b>0, whileWSTTcan\nbe positive or negative depending on the direction of the spin\ncurrent. We use the first (second) equations in equation (7)\nand equation (8) for numerical (analytical) calculations.The generalized Lagrange equation reads [24]\nd\ndt\u000eL\n\u000e_Q\u0000\u000eL\n\u000eQ=\u0000\u000eW\n\u000e_Q; (9)\nwhereLis the Lagrangian and Qis the generalized coor-\ndinate. By taking masQ, the generalized Lagrange equa-\ntion yields the Landau-Lifshitz-Gilbert-Slonczewski (LLGS)\nequation,\ndm\ndt=\u0000j\rjm\u0002Beff+\u000bm\u0002dm\ndt+j\rjum\u0002(z\u0002m);\n(10)\nwith ~Beff=\u0000@H=@m.\nThe energyEchanges in the presence of the energy injec-\ntion and dissipation, dE=dt =\u00002R\nd2xW6= 0, in general.\nNevertheless, when we take the time average, we should have\nZ\nd2xhWi= 0; (11)\nbecause this is necessary for a dynamically stabilized mag-\nnetic skyrmion. We may call it the weak stationary condition.\nE. Skyrmion solution\nWe substitute equation (3) in the LLGS equation (10),\nwhich leads to a set of two equations for \u0012(t;r;' )and8\nFIG. 8. (Color online) The high-Q skyrmion with Q= 2 under\nstepwise increasing of the spin-polarized current injection. We show\nhow the topological number Q, the total energy Etotal, the averaged\nout-of-plane magnetization mz, and the dissipation functions ( W\u000b,\nWSTT,W) change, when the injected spin current density jis in-\ncreased. The DMI constant D= 2 mJ m\u00002. The external magnetic\nfieldBz= 250 mT. The topological number Qremains as it is when\nwe change the spin current density. The average of Wis zero, which\nimplies the energy is balanced in average. It demonstrates that the\ntopological protection against the change of a variable, that is, the\nspin current injection.\n\u001f(t;r;' ). They are too complicated to solve, reflecting\ncomplicated behaviors revealed by numerical solutions (see\nRef. 26 for Supplementary Movies 1-3). However, when we\nsetD?= 0, simple behaviors have been revealed by numeri-\ncal simulations. Hence, we solve them by setting D?= 0as\nthe unperturbed system.\nWe search for a solution such that _\u0012= 0. By substituting\nequation (3) in the LLGS equation (10), and by setting _\u0012= 0,\nwe obtain\n\u0000J(@2\nr\u0012+@r\u0012\nr) + (JQ2\n2r2+K) sin 2\u0012\n+Bzcos\u0012+1\n\u000bu(r) sin\u0012= 2D?F(r;\u0012;' );(12)\nwhere\nF=\u0000Q\nrsin\u0012cos[(1\u0000Q)'+\u001f]\n\u00001\n\u000b@r\u0012sin[(1\u0000Q)'+\u001f] sin\u0012: (13)\nThe role of the injected spin-polarized current ( u6= 0) is to\nimpose the boundary condition cos\u0012=\u00001atr= 0.\nWhenQ= 1, sinceF=\u0000sin\u0012=r, equation (12) is numer-\nically solvable with respect to \u0012with the boundary condition\ncos\u0012=\u00001atr= 0andcos\u0012= 1atr=1. The azimuthalangle is given by '0(t) = (\ru=\u000bQ )t. The equations of mo-\ntion are well approximated by J@2\nr\u0012=Ksin 2\u0012for anyQ\nasymptotically. This equation has the domain-wall solution,\ncos\u0012= tanhr\u0000r0\n\u0015; (14)\nwith the domain-wall width \u0015=p\nJ=K and the skyrmion\nradiusr0.\nOur major interest is the case of Q6= 1. Since\u0012and'\nare coupled, it is not straightforward to solve for a magnetic\nskyrmion. Let us require the weak stationary condition by\ntaking the time average of equation (13). We find that hFi= 0\nunlessQ= 1. Then, equation (12) is solvable with respect\nto\u0012with the boundary condition cos\u0012=\u00001atr= 0 and\ncos\u0012= 1atr=1.\nThe profile of mz(r)is given by equation (3) and equa-\ntion (14). The Gilbert damping term W\u000b(r)and the STT term\nWSTT(r)are given by equations (7) - (8) together with _\u0012= 0,\n_\u001e=constant and equation (14) or sin2\u0012= sec2[(r\u0000r0)=\u0015].\nThe theory and numerical simulation lead to identical results\nwhich overlap within the precision of numerical simulation as\nshown in Fig. 7(b).\nOn the other hand, when the injected spin-polarized current\nis switched off, that is, u= 0, there is no skyrmion solution\nand we recover the FM ground state cos\u0012= 1.\nIV . CONCLUSIONS\nWe have analyzed the nucleation, the stability, and the de-\nstruction of the high-Q skyrmion with Q= 2in ferromagnets.\nIt is realized when the in-plane component of the spin rotates\nby2\u0019Qand the magnetic skyrmion has acquired a high helic-\nityQ. In the presence of the DMI, although there exist static\nmagnetic skyrmions with Q= 1, there exist no static high-Q\nskyrmions. Nevertheless, a high-Q skyrmion can be created\nand stabilized dynamically by injecting the spin-polarized cur-\nrent. The DMI plays a crucial role in the creation mechanism\nby twisting spins and generating fluctuations of the energy\ndensity and the topological charge density. Indeed, we have\nnumerically verified that a high-Q skyrmion cannot be created\ndynamically without the DMI.\nWe have also observed numerically that a high-Q skyrmion\ncan be created in a wide range of parameters as well as the\nspin current injection size. Furthermore, the high-Q skyrmion\nhas been found to be created and maintained even at finite\ntemperature, although its structure is deformed due to thermal\nfluctuations.\nThe nucleation process of a high-Q skyrmion is revealed\nby investigating the magnetization distribution, the topologi-\ncal charge density, and the energy density. It occurs in two\nsteps. First, it so happens that the high density part is local-\nized to a lattice-scale area in the boundary of a magnetic bub-\nble withQ= 0. The topological number jumps from Q= 0\ntoQ= 1 by making a large spin rotation in this lattice-scale\narea within a few picoseconds. This phenomenon would be\nviewed as an emergence of a Bloch point in the continuum\ntheory. Second, a similar process occurs to make a jump from9\nFIG. 9. (Color online) Annihilation process of the high-Q skyrmion with Q= 2. Snapshots of the topological charge density q(x), the\nenergy density \u000f(x), the spin-component density mz(x), and its close up at sequential times. The DMI constant D= 2mJ m\u00002. The external\nmagnetic field Bz= 250 mT. The spin current density is switched off at t= 5000 ps. In the simulation, each cell corresponds to one spin,\nand the cell size is 1:5nm\u00021:5nm\u00021nm. In the spin distribution panels, each arrow stands for four spins, while it stands for one spin in\nthe insets. In the first step, the topological number changes from Q= 2toQ= 1, where the skyrmion size remains almost unchanged. Then,\nthe magnetic skyrmion with Q= 1 shrinks to the size of the lattice scale. In the second step, the topological number changes from Q= 1 to\nQ= 0, and the magnetic skyrmion disappears. See Ref. 26 for Supplementary Movie 5.\nQ= 1toQ= 2. The dissipation spreads over the sample like\na burst at the moment of the birth of a magnetic skyrmion.\nThe continuity of the spin texture is recovered since a\nsmooth texture has a lower energy. Once a sufficiently smooth\nhigh-Q skyrmion is generated, it remains stable even if we\nswitch off the DMI or even if we change the current density\nof the injected spin current considerably. We have explained\nits stability as a topologically protected dissipative structure.\nWhen the spin current injection is switched off, the high-Q\nskyrmion is destroyed. The destruction process occurs also\nin two steps as in the case of the nucleation process. How-\never, the detailed mechanism is different. The first step occurs\nby way of a Bloch point, where the continuum picture is still\ngood. In the second step, the lattice structure becomes impor-\ntant since the skyrmion size is so small, where the skyrmion\nspin texture turns into the vortex spin texture and disappears.\nThe dissipation spreads over the sample like a burst at the mo-\nment of the destruction of a magnetic skyrmion.\nIt is a hard problem to solve the nucleation or destruction\nprocess analytically since it is a highly nonlinear process in-\nvolving Bloch points. Furthermore, the lattice structure plays\na key role in these processes at the microscopic level. We hope\nthis work provides useful guidelines in searching new type\nof skyrmions and will afford a new dimension towards fully\nunderstanding the nontrivial topology of magnetic skyrmions\nwith higher topological numbers.ACKNOWLEDGMENTS\nY .Z. acknowledges the support by National Natural Sci-\nence Foundation of China (Project No. 1157040329), the Seed\nFunding Program for Basic Research and Seed Funding Pro-\ngram for Applied Research from the HKU, ITF Tier 3 fund-\ning (ITS/171/13 and ITS/203/14), the RGC-GRF under Grant\nHKU 17210014, and University Grants Committee of Hong\nKong (Contract No. AoE/P-04/08). M.E. thanks the sup-\nport by the Grants-in-Aid for Scientific Research from MEXT\nKAKENHI (Grant Nos. 25400317 and 15H05854). M.E. is\nvery much grateful to N. Nagaosa for many helpful discus-\nsions on the subject. X.Z. was supported by JSPS RONPAKU\n(Dissertation Ph.D.) Program. X.Z. greatly appreciates on-\ngoing discussions with J. Xia.\nAPPENDIX: SIMULATION AND MODELING\nThe micromagnetic simulation is carried out with the\nwell-established Object Oriented MicroMagnetic Framework\n(OOMMF) software (1.2a5 release) [28]. The OOMMF ex-\ntensible solver (OXS) extension module of the interface-\ninduced DMI, that is, the Oxs_DMExchange6Ngbr class, is\nincluded in the simulation [29]. The OXS extension module of\nthe thermal fluctuation, that is, the Xf_ThermSpinXferEvolve10\nclass, is employed to simulate the finite-temperature system.\nThe 3D time-dependent magnetization dynamics at zero tem-\nperature is determined by the LLGS equation [28, 30], while\na highly irregular fluctuating field representing the irregular\ninfluence of temperature is added into the LLGS equation for\nsimulating the magnetization dynamics at finite temperature.\nThe average energy density of the system contains the ex-\nchange energy, the anisotropy energy, the applied field (Zee-\nman) energy, the DMI energy, and the magnetostatic (demag-\nnetization) energy terms.\nIn the simulation, we consider a 1-nm-thick FM nanodisk\nwith a diameter of 150nm, which is attached to a heavy-metal\nsubstrate. The material parameters used by the simulation pro-\ngram are adopted from Refs. 9 and 10: the Gilbert damping\ncoefficient\u000b= 0:01, the gyromagnetic ratio \r=\u00002:211\u0002\n105m A\u00001s\u00001, the saturation magnetization MS= 580 kA\nm\u00001, the exchange stiffness J= 15 pJ m\u00001, the interface-induced DMI constant D= 0\u00183mJ m\u00002, and the PMA\nconstantK= 0:8MJ m\u00003. The polarization rate of the verti-\ncal spin current applied in the simulation is fixed at P= 0:4.\nThe simulated model is discretized into tetragonal cells with\nthe optimum cell size of 1:5nm\u00021:5nm\u00021nm, which gives\na good trade-off between the computational accuracy and effi-\nciency. The finite-temperature simulation is performed with a\nfixed time step of 1\u000210\u000014s, while the time step in the simu-\nlation with zero temperature is adaptive ( \u00186\u000210\u000014s). 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Safko, Classical Mechanics, 3rd ed.\n(Addison-Wesley, New York, 2002), Chap. 1, Sec. 5.\n[25] M. Ezawa, Phys. Lett. A 375, 3610 (2011).\n[26] See Supplemental Material at [URL] for supplementary movies\non the nucleation, maintenance, and annihilation of the high-Q\nskyrmion.\n[27] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213\n(2008); J. Phys. Soc. Jpn. 77, 031003 (2008).\n[28] M. J. Donahue and D. G. Porter, OOMMF, User’s Guide, Inter-\nagency Report NISTIR 6376, NIST Gaithersburg, MD (1999)\nhttp://math.nist.gov/oommf.\n[29] S. Rohart and A. Thiaville, Phys. Rev. B 88, 184422 (2013).\n[30] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405\n(2004)."    },    {        "title": "1311.7620v1.Magnon_radiation_by_moving_Abrikosov_vortices_in_ferromagnetic_superconductors_and_superconductor_ferromagnet_multilayers.pdf",        "content": "arXiv:1311.7620v1  [cond-mat.supr-con]  29 Nov 2013Magnon radiation by moving Abrikosov vortices in ferromagn etic superconductors\nand superconductor-ferromagnet multilayers\nA. A. Bespalov,1,2A. S. Mel’nikov,1,3and A. I. Buzdin2\n1Institute for Physics of Microstructures, Russian Academy of Sciences, GSP-105, 603950, Nizhny Novgorod, Russia\n2Universit´ e Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talenc e, France\n3Nizhny Novgorod State University, 22 Gagarin av., 603950, N izhny Novgorod, Russia\nIn systems combining type-II superconductivity and magnet ism the non-stationary magnetic field\nof moving Abrikosov vortices may excite spin waves, or magno ns. This effect leads to the appear-\nance of an additional damping force acting on the vortices. B y solving the London and Landau-\nLifshitz-Gilbert equations we calculate the magnetic mome nt induced force acting on vortices in\nferromagnetic superconductors and superconductor/ferro magnet superlattices. If the vortices are\ndrivenby a dc force, magnon generation due to the Cherenkov r esonance starts as the vortexvelocity\nexceeds some threshold value. For an ideal vortex lattice th is leads to an anisotropic contribution\nto the resistivity and to the appearance of resonance peaks o n the current voltage characteristics.\nFor a disordered vortex array the current will exhibit a step -like increase at some critical voltage.\nIf the vortices are driven by an ac force with a frequency ω, the interaction with magnetic moments\nwill lead to a frequency-dependent magnetic contribution ηMto the vortex viscosity. If ωis below\nthe ferromagnetic resonance frequency ωF, vortices acquire additional inertia. For ω > ω Fdissipa-\ntion is enhanced due to magnon generation. The viscosity ηMcan be extracted from the surface\nimpedance of the ferromagnetic superconductor. Estimates of the magnetic force acting on vortices\nfor the U-based ferromagnetic superconductors and cuprate /manganite superlattices are given.\nPACS numbers: 74.25.Uv, 75.30.Ds, 74.25.nn\nI. INTRODUCTION\nWithin the last 13 years a number of fascinating com-\npounds has been discovered, revealing the coexistence\nof ferromagnetism and superconductivity in the bulk.1–4\nThese compounds are U-based ferromagnets which be-\ncome superconducting at temperatures ∼1K under ap-\nplied pressure, or even at atmospheric pressure. Experi-\nmental investigation of magnetic properties of these ma-\nterials in the superconducting state is hampered by the\nMeissner effect, making static measurements inefficient.\nHowever, important parameters can be extracted from\ndynamical measurements of the spin-wave (magnon)\nspectrum, which can be determined, e. g., by microwave\nprobing5,6or using Abrikosov vortex motion.7,8\nA number of papers has been devoted to theoreti-\ncal investigation of the magnon spectrum in magnetic\nsuperconductors.5,6,9–13Buzdin9determined the magnon\nspectrum in a superconducting antiferromagnet with an\neasy-axis anisotropy. Different types of spin waves in fer-\nromagnetic superconductors in the Meissner state have\nbeen studied by Braude, Sonin and Logoboy,5,6,10,11in-\ncluding surface waves and domain wall waves.\nExperimental measurements of the ac magnetic sus-\nceptibility ofsuperconductingferromagnetsrevealedthat\nthe screening of the magnetic field created by mag-\nnetic moments in these materials is incomplete.2,3This\nindicates that the superconducting transition in the\nU-compounds occurs in the spontaneous vortex state.\nOnly two papers so far have addressed the influence of\nAbrikosov vortices on the magnon spectrum in ferromag-\nnetic superconductors. In Ref. 12 coupled magnetic\nmoment-vortex dynamics has been studied in the limitof long wavelength λw≫a, whereais the inter-vortex\ndistance. Later,13this analysis has been extended to the\ncaseλw/lessorsimilara. It has been demonstrated that in the pres-\nence of a vortex lattice the magnon spectrum acquires a\nBloch-like band structure.\nTo study the spin wave spectrum experimentally two\nsimple procedures have been proposed. The first method\nconsistsindirectexcitationofmagnonsbyanelectromag-\nneticwaveincidentatthesample.5,6,13Then, information\nabout the spin wave spectrum can be extracted from the\nfrequency dependent surface impedance Z. Note that\nthis procedure can be applied also to ordinary ferromag-\nnets, but inallcasesthehigh qualitycrystallinesurfaceis\nrequired. The surface impedance has been calculated for\na ferromagnetic superconductor in the Meissner state5,6\nand in the mixed state13for a static vortex lattice.\nThe second method is based on the indirect magnon\nexcitation: an external source of current sets in motion\nthe Abrikosov vortices, which start to radiate magnons\nwhen the Cherenkov resonance condition is satisfied.7,8\nHere, the current-voltage characteristics yield informa-\ntion about the magnon spectrum. Since this method in-\nvolves Abrikosov vortices, it is specific for superconduct-\ning materials. Different phenomena arising from vortex-\nmagnetic moment interaction in magnetic superconduc-\ntors have been studied by Bulaevskii et al.7,8,14–17In\nRefs. 7 and 14 the dissipation power due to magnon\ngeneration by a moving with a constant velocity vortex\nlattice in a superconducting antiferromagnet has been\ncalculated. In Ref. 8 this result has been generalized for\nthe case of a vortex lattice driven by a superposition of\nac and dc currents. In Refs. 15 and 16 a polaronic mech-\nanism of self-induced vortex pinning in magnetic super-2\nconductors is discussed. The motion of the vortex lattice\nunder the action of dc15and ac currents16has been stud-\nied. Finally, in Ref. 17it has been predicted that the flux\nflow should lead to the creation of domain walls in sys-\ntems with slow relaxation of the magnetic moments.\nIn the present paper, by solving the phenomenological\nLondon and Landau-Lifshitz equations, we analyze the\nproblem of magnon generation by moving Abrikosovvor-\ntices in ferromagnetic superconductors and superconduc-\ntor/ferromagnet (SF) multilayers. Theoretical investiga-\ntion of the latter systems is relevant in view of the recent\nsuccess in fabrication and characterization of cuprate-\nmanganite superlattices.18–20Also, recently an experi-\nmental study of the flux-flow resistivity in Nb/PdNi/Nb\ntrilayers has been reported.21Our consideration of bulk\nferromagnetic superconductors, on the other hand, is rel-\nevant to the U-based compounds mentioned above. In\nthis aspect, the present work complements the preceding\npapers,7,8,14–17which concetrated mainly on antiferro-\nmagnetic materials. As we show, the presence of ferro-\nmagnetism introduces its own specifics, as the magnon\nspectrum in ferromagnets differs from the antiferromag-\nnetic spectrum. Our results also include the comparison\nof the cases of a disordered and regular vortex lattice.\nThe outline of the paper is as follows. In Sec. II we\ngiveamodeloftheferromagneticsuperconductorandde-\nrive a general equation for the magnetic moment induced\nforcefMacting on vortices in ferromagnetic supercon-\nductors. In Sec. III this force is calculated for a vortex\nlattice and disordered vortex array moving under the ac-\ntionofadctransportcurrent. Here, thedifferencesinthe\ndependence of fMvs. vortex velocity for ferromagnetic\nand antiferromagnetic materials are discussed. Section\nIV is devoted to vortex motion under the action of an ac\ndriving force. The magnetic contributions to the vortex\nviscosity and vortex mass are determined. In Sec. V it\nis shown how the force fMcan be estimated experimen-\ntally by measuringthe surfaceimpedance. In Sec. VI the\ngeneralization of our calculations for the SF multilayers\nis discussed. In the conclusion a summary of our results\nis given.\nII. THE INTERACTION FORCE BETWEEN\nVORTICES AND MAGNETIC MOMENTS:\nGENERAL EQUATIONS\nIn the London approximation the free energy of the\nferromagnetic superconductor in the mixed state can be\ntaken in the form\nF=/integraltext/bracketleftBig\n1\n8πλ2/parenleftbig\nA+Φ0\n2π∇θS/parenrightbig2+(rotA−4πM)2\n8π\n+α\n2/parenleftBig\n∂M\n∂xi∂M\n∂xi/parenrightBig\n+KM2\n⊥\n2−BHe\n4π/bracketrightBig\nd3r.(1)\nHereλis the London penetration depth, Ais the vector\npotential, Φ 0is the flux quantum (Φ 0=π/planckover2pi1c/|e|>0),\nθSis the superconducting order parameter phase, Mis\nthe magnetization, and αis a constant characterizingCompound UGe2UCoGe URhGe\nλ, nm 10001200 3450\nL, nm 13,6 45 900\nHan, T∼100∼10∼10\nµU1,4µB0,07µB0,3µB\nωF, Hz∼1013∼1010∼1011\nVth, cm/s ∼107∼105∼107\nK=Han/M∼104∼104∼103−104\nTABLE I. Parameters of some ferromagnetic superconduc-\ntors.L=/radicalbig\nα/Kis the effective domain wall width, Hanis\nthe anisotropy field, µUis the magnetic moment per U atom,\nµBis the Bohr magneton, ωFis the ferromagnetic resonance\nfrequency, and Vthis the critical vortex velocity for magnon\nradiation (see Sec. III). The data have been taken from Refs.\n1, 22–24.\nthe exchangeinteraction. The U-based ferromagnetic su-\nperconductors, listed in Table I, have a strong easy-axis\nmagnetocrystalline anisotropy, which is accounted for by\nthe term KM2\n⊥/2, where Kis an anisotropy constant,\nM⊥=M−(e·M)e, andeis a unit vector along the\nanisotropy axis. The term BHe/4πin Eq. (1) accounts\nfor a uniform external field He. All terms in the right-\nhand side of Eq. (1) are integrated over the whole space,\nexceptforthefirstterm, containing λ, whichisintegrated\nover the sample volume. Certainly, outside the sample\nM= 0. In the sample the magnetization modulus is\nconstant.\nFirst, we determine the equilibrium state by minimiz-\ningFwith respect to M, and then with respect to A\nandθS. We note that anisotropy field Han=KMis\ntypically very large (see Table I): it is comparable to or\ngreater than the upper critical field. This means that the\ninequality B≪Hanholds for any internal field Bthat\ndoes not suppress superconductivity. Then the trans-\nverse component of the magnetization M⊥can be esti-\nmated as M⊥/lessorsimilarB/K≪M. SinceK≫1, in a zero\napproximation with respect to K−1we can neglect the\ntransverse magnetization (even in the anisotropy energy,\nwhich appears to be proportional to K−1). Then\nF(A,θS)≈/integraltext/bracketleftBig\n1\n8πλ2/parenleftbig\nA+Φ0\n2π∇θS/parenrightbig2\n+B2\n8π−BM0−BHe\n4π+2πM2/bracketrightBig\nd3r,(2)\nwhereM0=Me. Foranarbitraryshapedsamplefurther\nminimization can not be performed analytically. Here\nwe assume the ferromagnetic superconductor to be an\nellipsoid. The results derived below should be also valid\nin the extreme cases of slabs and long cylinders. It is\nreasonable to assume that the average internal magnetic\nfieldB0inanellipsoidalsamplewillbe uniform(compare\nwith a dielectric ellipsoid in a uniform external field - see\nRef. 25). Denoting the superconductor volume as V, we3\ncan rewrite the free energy as\nF=V/parenleftbig\nfS(B0)−M0B0−B0He\n4π/parenrightbig\n+/integraltext\nr/∈V/bracketleftBig\nB2\n8π−BHe\n4π/bracketrightBig\nd3r+const, (3)\nwhere the constant does not depend on the magnetic in-\nductionB, andfSis the free energy density of the vortex\nlattice:\nfS(B0) =/angbracketleftBigg\n1\n8πλ2/parenleftbigg\nA+Φ0\n2π∇θS/parenrightbigg2\n+B2\n8π/angbracketrightBigg\n.(4)\nAveragingis performed overa volumethat is much larger\nthan the inter-vortex distance. The function fS(B0) can\nbe determined explicitly by solving the London equation\n(11) with a given vortex lattice density, corresponding\nto the average field B0. To transform the integral in\nEq. (3) we introduce several quantities: the self-field of\nthe sample BS=B−He, the magnetization MSdue\nto supercurrents, the effective full magnetization Meff=\nM0+MS, and the effective H-field Heff=BS−4πMeff.\nThen the integral can be transformed as\n/integraltext\nr/∈V/bracketleftBig\nB2\n8π−BHe\n4π/bracketrightBig\nd3r=/integraltext\nr/∈VB2\nS\n8πd3r−/integraltext\nr/∈VH2\ne\n8πd3r\n=/integraltextH2\neff\n8πd3r−/integraltext\nr∈VH2\neff\n8πd3r+const\n=V\n2Meff/parenleftBig\nˆN−ˆN2\n4π/parenrightBig\nMeff+const.\nHereˆNis the demagnetizing tensor, connecting the effec-\ntive magnetization and effective field inside the sample:\nHeff=−ˆNMeff. Analytical and numerical values of ˆN\ncan be found in Ref. 26. Finally, if we eliminate Meff\nusing the relation\nMeff= (4π−ˆN)−1(B0−He),\nwe obtain\nF\nV=fS(B0)−M0B0−B0He\n4π\n+1\n8π(B0−He)ˆN(4π−ˆN)−1(B0−He)+const .(5)\nHere, the only variable is the internal field B0, which\nshould be determined from the equation\n∂F\n∂B0= 0. (6)\nEquations (5) and (6) completely define the equilibrium\nstate of the ferromagnetic superconductor.\nNow we proceed from statics to coupled vortex and\nmagnetization dynamics. We focus on two systems, for\nwhich the derivation of the force acting on vortices is\nalmost identical: a bulk ferromagnetic superconductor,\nand an SF multilayer (see Fig. 1), where S is an or-\ndinary type-II superconductor, and F is a ferromagnet\nwith a strong easy-axis anisotropy: K≫1. For the\nmultilayer system the same expression (1) for the free\nenergy is used with M= 0 in the superconductor and\nFIG. 1. A scheme of the SF multilayer system. The dashed\nlines denote vortices.\nλ=∞in the ferromagnet. We neglect the Joseph-\nson coupling between neighboring S layers. This is jus-\ntified for /greaterorsimilar10 nm thick ferromagnets: in the case of\nan ordinary (non-triplet) proximity effect, superconduct-\ning correlations decay exponentially on a scale of several\nnanometers in the ferromagnet.27\nLet the vortices be aligned along the z-axis (which is\nperpendicular to the S/F interface in the multilayer sys-\ntem). They may form a regular or disordered lattice.\nWhen the vortices are set in motion by a dc or ac trans-\nport current, their time-dependent positions are given by\nthe vectorfunctions Ri(z,t), lyingin the xy-plane, where\ni= 1..Nv, andNvis the number of vortices. In our cal-\nculations we will assume the vortices to be straight, i. e.\nRidoes not depend on z.\nAs vortices move, the magnetic moments start to fluc-\ntuate. We describe the magnetization dynamics using\nthe Landau-Lifshitz-Gilbert equation28\n∂M\n∂t=γ/parenleftbigg\nM×δF\nδM/parenrightbigg\n+ν\nM2/parenleftbigg\nM×∂M\n∂t/parenrightbigg\n,(7)\nwhereγis the gyromagnetic ratio, νis a dissipation con-\nstant, and the free energy Fis given by Eq. (1).\nThe force acting on a single vortex per unit length of\nthe vortex equals\nfi=−1\nLv∂F\n∂Ri. (8)\nwhereLvis the vortex length. Averaging fiover all vor-\ntices, we obtain the average force\nf=−1\nLvNv/summationdisplay\ni∂F\n∂Ri, (9)\nWhen we considered the equilibrium state, the mag-\nnetization component perpendicular to the easy axis e\nhas been neglected. Now we have to abandon this ap-\nproximation, as it would lead to a vanishing force acting4\non the vortices from the side of the magnetic moments.\nWe putM=M0+m, wherem≈M⊥,|m| ≪M, and\nlinearize Eq. (7) with respect to m:\n∂m\n∂t=−γM0×/parenleftbig\nα∇2m−Km+B/parenrightbig\n+ν\nM2M0×∂m\n∂t.\n(10)\nFrom this equation it is evident that magnetization fluc-\ntuations are excited if the vortex field is not parallel to\nthe magnetization easy-axis. In a ferromagnetic super-\nconductor this may be achieved by applying an exter-\nnal field at an angle to the magnetization easy axis or\nby choosing an appropriate sample geometry (for exam-\nple, anellipsoidalsamplewiththemagnetizationdirected\nalong neither of the principal axes).\nThe magnetic induction inside the superconductor\nshould be determined from the London equation\nδF\nδA= 0,or\n−∇2B+B\nλ2=Φ0\nλ2z0/summationtext\niδ(2)(ρ−Ri)+4π\ncrotrotm,(11)\nwherez0is a unit vector along the z-axis. In the case of\nthe multilayer system (see Fig. 1), Maxwell’s equations\ninside the F-layers read\nrotB= 4πrotM,divB= 0.(12)\nOn the SF-interface appropriate boundary conditions\nmust be imposed:\nBz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nF=Bz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nS, H x,y/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nF=Hx,y/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nS(13)\n∂m\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nF= 0. (14)\nThe last condition follows directly from Eq. (7), if no\nsurface term is present in the free energy (1).\nWe present the magnetic field as the sum of the vortex\nfieldhand the magnetization field bMdefined by\n−∇2h+h\nλ2=Φ0\nλ2z0/summationdisplay\niδ(2)(ρ−Ri),(15)\n−∇2bM+bM\nλ2= 4πrotrotm (16)\ninside the superconductor, and\nroth= 0,divh= 0, (17)\nrotbM= 4πrotm,divbM= 0 (18)\nin the ferromagnetic layers.\nIn Eqs. (11) - (18) we neglected the magnetic field in-\nduced by normal currents. These are given by j=σE,\nwhereσis the normal conductivity, and Eis the elet-\nric field. We will first estimate the contribution of the\nnormal currents flowing in the F-layers of the multilayersystemtothe magneticfield. UsingbothMaxwell’sequa-\ntions for rot Band rotE, we obtain\nrotrotB=4πσF\ncrotE=−4πσF\nc2∂B\n∂t,\nor\n∂2B\n∂z2+∇2\nρB−4πσF\nc2∂B\n∂t= 0, (19)\nwhereσFis the conductivity in the magnetic layers. As-\nsuming\n∂B\n∂t≈ −(VL∇ρ)B,\nwhereVLis the flux velocity, we can see that the in-\nfluence of the normal currents on the magnetic field is\nnegligible, if the inequality\n4πσF\nc2lVL≪1\nholds, where lis the characteristic in-plane length scale\nof the problem. Similar arguments can be applied to the\nS-layers. Then, for the multilayer system we find the\nfollowing constraint on the vortex velocity:\nVL≪c2\n4πmax(σn,σF)l, (20)\nwhereσnis the normal-state conductivity of the super-\nconductor. In the case of a bulk ferromagnetic supercon-\nductor, we have to demand\nVL≪c2\n4πσnl.(21)\nAs we will see, the main length scales of the problem are\nthe inter-vortex distance aand the length L=/radicalbig\nα/K,\nwhich is of the order of the domain wall width of the\nferromagnet (or ferromagnetic superconductor). Further\non we assume that Eqs. (20) and (21) with l= min(a,L)\nare satisfied. Then, we may not take into account the\nnormal currents in Eqs. (11) - (18).\nIn the free energy (1) the interaction of the vortices\nwith magnetizationoriginatesfrom the Zeeman-liketerm\n(a brief explanation is given in Appendix A)\nFZ=−/integraldisplay\nMhd3r.\nThen, the magnetic moment induced force fMacting on\nthe vortices is\nfM=−1\nLvNv/integraldisplay\nmz∇hzd3r. (22)\nHere it has been assumed that the perpendicular to the\nz-axis component of the field his negligible. In the case5\nof SF multilayers, this is true for a sufficiently small pe-\nriod of the structure. To draw a parallel with preced-\ning works,7,8,14–17where the susceptibility formalism has\nbeen used, we note that fMcan be written in the form\nfM=−1\nLvNv/integraldisplay\n(ˆχzzhz)∇hzd3r,\nwhere ˆχzzis the susceptibility operator. If desired, the\nexplicit form of ˆ χzzcan be easily derived from Eq. (29),\ngiven below.\nAll further calculationsin this section andSecs. III, IV\nandVarecarriedoutforaferromagneticsuperconductor.\nIn Section VI we discuss how our results can be extended\nto the case of the multilayer system.\nIn the Fourier representation Eq. (22) reads\nfM=4π2\nNvi/integraldisplay\nqmqzh∗\nqzd2q, (23)\nwhere for any function X(ρ) its Fourier transform is de-\nfined as\nXq=1\n(2π)2/integraldisplay\nX(ρ)e−iqρd2ρ.\nBy Fourier transforming Eqs. (10), (15), and (16), as-\nsuming that all quantities do not depend on z, we obtain\n∂mq\n∂t=−γM0×/parenleftbig\n−(K+αq2)mq+bMq+hq/parenrightbig\n+ν\nM2M0×∂mq\n∂t, (24)\nhqz=Φ0\n4π2(1+λ2q2)/summationdisplay\nie−iqRi(t).(25)\nbMq=−4πq×(q×mq)\nq2+λ−2. (26)\nIt can be seen that the absolute value of the term bMq\nin Eq. (24) is much smaller than |Kmq|. Further on we\nwill neglect the magnetization field bMq.\nEquation (24) is an inhomogeneous linear differential\nequationwithconstantcoefficientswithrespectto mq. It\ncan be solved using standard methods. We are interested\nin thez-component of the magnetization, which equals\nmqz=γMi\n2sin2θ/integraltextt\n−∞hqz(t′)/braceleftBig/parenleftbig\n1+iν\nM/parenrightbig−1\n×exp/bracketleftBig\n−/parenleftbig\n1+iν\nM/parenrightbig−1iω(q)(t−t′)/bracketrightBig\n−/parenleftbig\n1−iν\nM/parenrightbig−1exp/bracketleftBig/parenleftbig\n1−iν\nM/parenrightbig−1iω(q)(t−t′)/bracketrightBig/bracerightBig\ndt′,(27)\nwhereθis the angle between eandz0, and\nω(q) =γM(K+αq2) =ωF(1+L2q2) (28)\ngives the magnon dispersion law in an ordinary ferro-\nmagnet, if the dipole-dipole interaction is not taken intoaccount (see Ref. 29). Here, ωF=γMKis the ferro-\nmagnetic resonance frequency. In the small dissipation\nlimit,ν≪M, we have\nmqz=γMi\n2sin2θ/integraltextt\n−∞hqz(t′)/braceleftbig\nexp/bracketleftbig/parenleftbig\n−i−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig\n×/parenleftbig\n1−iν\nM/parenrightbig\n−exp/bracketleftbig/parenleftbig\ni−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig/parenleftbig\n1+iν\nM/parenrightbig/bracerightbig\ndt′.\n(29)\nThen, the force fMtakes the form\nfM=2π2γM\nNvsin2θ/integraltext\nd2q/integraltextt\n−∞hqz(t′)h∗\nqz(t)\n/braceleftbig\nexp/bracketleftbig/parenleftbig\ni−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig/parenleftbig\n1+iν\nM/parenrightbig\n−exp/bracketleftbig/parenleftbig\n−i−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig/parenleftbig\n1−iν\nM/parenrightbig/bracerightbig\nqdt′.(30)\nIII. MAGNON RADIATION BY VORTICES\nMOVING WITH A CONSTANT VELOCITY\nLet us consider the motion of vortices under the action\nof a constant external force (e. g., spacially uniform and\ntime-independent transport current). Then the positions\nof individual vortices are given by\nRi(t) =Ri0+VLt+∆Ri(t). (31)\nHere the vectors Ri0denote the vortexpositions in a reg-\nular lattice, VLis the average flux velocity, and ∆ Ri(t)\nis responsible for fluctuations of vortices due to interac-\ntions with pinning cites ( /angb∇acketleft∆Ri(t)/angb∇acket∇ight= 0). It should be\nstressed here that we do not take into account the influ-\nence of pinning on the flux velocity. The effect that is\nimportant for us is the vortex lattice distortion caused\nby impurities, which strongly influences the efficiency of\nmagnon generation.\nThe product of magnetic fields under the integral in\nEq. (30) is\nhqz(t′)h∗\nqz(t) =/parenleftbiggΦ0\n4π2(1+λ2q2)/parenrightbigg2\neiqVL(t−t′)K,(32)\nK=/summationtext\ni,jexp[iq(Rj0−Ri0)+iq(∆Rj(t)−∆Ri(t′))].\nBelow we consider the cases of a perfect vortex lattice\nand a disordered vortex array.\nA. A perfect vortex lattice.\nThe approximation used in this section is valid for suf-\nficiently weak pinning, when we can put\n/angbracketleftBig\neiq(∆Rj(t)−∆Ri(t′))/angbracketrightBig\n≈1,\nwhere the averaging is over i. To ensure the fulfillment\nof this condition it is sufficient to demand\n∆Rq≪1, (33)6\nwhere ∆ Ris the characteristic displacement of vortices\nfrom their positions in a perfect lattice. The inequality\n(33)mustholdforall qgivingaconsiderablecontribution\nto the integral in Eq. (30). In the end of Sec. IIIB it\nwill be shown that this leads to the condition\n∆R≪L. (34)\nWhen (34) holds, we have\nK=4π2NvB0\nΦ0/summationdisplay\nGδ(q−G), (35)\nwhereGare the vectors of the lattice, reciprocal to the\nvortex lattice. After integration over qandt′the mag-\nnetic force takes the form\nfM= Φ0B0γMsin2θ/summationtext\nGG\n(1+λ2G2)2\n×iω(G)+ν\nMGVL\nω2(G)−(VLG)2−2iν\nMVLGω(G).\nWhen the terms corresponding to Gand−Gare com-\nbined, this can be written as\nfM=−γνB0Φ0sin2θ/summationtext\nGG(GVL)\n(1+λ2G2)2\n×(GVL)2+ω2(G)\n[ω2(G)−(GVL)2]2+4ν2\nM2(GVL)2ω2(G),(36)\nwhere small terms of the order of ν/Min the numerator\nhave been droped. From this it follows that the force has\nlocal maxima when for some G=G0the condition\nω(G0)≈VLG0 (37)\nis satisfied. This relation presents the well-known\nCherenkov resonance condition. When Eq. (37) holds,\nmagnons with the wave vector G0are effectively gener-\nated. When the vortex velocity is close to a resonance\nvalue, in the sum in Eq. (36) we can drop all terms ex-\ncept the two resonant terms corresponding to G0and\n−G0. Then\nfM≈ −γνB0Φ0sin2θG0\n(1+λ2G2\n0)2\n×ω(G0)\n(ω(G0)−VLG0)2+ν2\nM2ω2(G0).(38)\nIt can be seen that the fMvs.VLdependence for a given\nvortex velocity direction exhibits a Lorentzian-like peak\nwith the width\n∆VL=ν\nMω(G0)VL\nG0VL.\nThe maximum value of fMis\n|fM|max=γM2B0Φ0G0sin2θ\n(1+λ2G2\n0)2νω(G0). (39)\nAnother remarkeable feature is that the force is directed\natsomeangletothevelocityofthevortices: fMisparallel\ntoG0, and not VL. The angle between fMandVL\nmay range from 0◦to 90◦. This effect also follows fromEquation (3) in Ref. 15, though the authors did not\nmention it, because it has been assumed that VLandfM\nare always parallel.\nLet us discuss how the Cherenkov resonances influence\nthecurrent-voltagecharacteristics. Abrikosovvortexmo-\ntion in a superconductor is governed by the equation\nΦ0\ncj×z0=−f (40)\nThe term in the left-hand side represents the Lorentz\nforce with jbeing the macroscopic supercurrent density.\nAll other forces are represented by the term f. We take\ninto account two contributions to f: the viscous drag\nforce−ηVLandfM. Here,ηis the viscosity due to order\nparameter relaxation processes and normal current flow-\ning through the vortex core.30Taking the cross product\nof Eq. (40) and z0, we obtain the expression for the\ncurrent\nj=−cη\nΦ0VL×z0+c\nΦ0fM(VL)×z0.(41)\nThe relation between jandEis the established via\nE=−1\nc(VL×B), (42)\nwhichfollowsfromFaraday’slaw. Accordingto Eq. (41),\nthe vortex-magnetic moment interaction leads to an in-\ncrease ∆ jof the current density at a given electric field\nE:\n∆j=c\nΦ0fM/parenleftbiggc\nB0E×z0/parenrightbigg\n×z0.\nAccording to Eq. (38), near the Cherenkov resonance we\nhave\n∆j=γνB0csin2θz0×G0\n(1+λ2G2\n0)2\n×ω(G0)/bracketleftBig\nω(G0)−c\nB0(z0×G0)E/bracketrightBig2\n+ν2\nM2ω2(G0).(43)\nThisrelationindicatesthat theI-Vcurveexhibitsaseries\nof peaks corresponding to the resonance electric fields\ngiven by\nω(G)−c\nB0(z0×G)E≈0 (44)\nMoreover, close to the resonance the additional current\n∆jis directed along the vector z0×G0and notE. The\nangle between ∆ jandEmay range from 0◦to 90◦.\nThus, locally the resistance is anisotropic. Considering\nmacroscopic ferrromagnetic superconductors and multi-\nlayer systems, care should be taken when applying Eq.\n(43) to the whole sample: it is known that even a small\nconcentration of pinning cites destroysthe long-rangeor-\nder in the vortex lattice.31In fact, vortex lattice domains\nare formed in large superconducting samples – see Ref.\n32 and references therein. The vortex nearest-neighbor\ndirections are typically linked to crystal axes. Hence, in7\nmonocrystalline samples there are only few energetically\nfavorable orientations of the vortex lattices. This fact\nallows us to put forward a qualitative argument. Let us\ndenote as Gthe set of all reciprocal lattice vectors for all\nvortex lattice domains. Since there are only few possible\norientations of the domains, the set Gconsists of isolated\npoints. We claim that when the applied electric field sat-\nisfies Eq. (44) for some G∈ G, the enhancement of the\ncurrent should be observable. Hence, even if there are\nseveral vortex lattice domains, the peaks on the current-\nvoltage characteristics are present. The measurement of\nthe peak voltages at different applied magnetic fields al-\nlows to probe the magnon spectrum ω(q).\nB. A disordered vortex array\nIn this section we analyze the opposite extreme case of\nchaotically placed vortices. This situation may be real-\nized in weak magnetic fields, B0/lessorsimilarΦ0/λ2, when vortex-\nvortex interaction is weak and the lattice is easily de-\nstroyed by defects and thermal fluctuations.\nWe assume\n/angbracketleftBig\neiq(∆Rj(t)−∆Ri(t′))/angbracketrightBig\n≈0. (45)\nwhereqis not too small, and the averaging is over i/negationslash=j.\nNote that the behavior of Kat small qalmost does not\ninfluence the force fM, sinceKenters the integral in Eq.\n(30) with a factor q. Fori=jwe have\n/angbracketleftBig\neiq(∆Ri(t)−∆Ri(t′))/angbracketrightBig\n= 1\nwhent=t′, and\n/angbracketleftBig\neiq(∆Ri(t)−∆Ri(t′))/angbracketrightBig\n=/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\neiq∆Ri(t)/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\n≪1\nwhen|t−t′| → ∞. To derive some qualitative results,\nwe make the following assumption:\n/angbracketleftBig\neiq(∆Ri(t)−∆Ri(t′))/angbracketrightBig\n=e−|t−t′|/τ(q),\nwhere the the time τ(q) is chosen so that\n/angb∇acketleft|q(∆Ri(t)−∆Ri(t′))|/angb∇acket∇ight ∼1 att−t′=τ(q). Then\nK=Nve−|t−t′|/τ(q), (46)\nand after integration over t′Eq. (30) yields\nfM=γMΦ2\n0sin2θ\n8π2/integraltextqd2q\n(1+λ2q2)2/bracketleftBig\n1+iν\nM\nτ−1(q)+ν\nMω(q)−iω(q)−iqVL\n−1−iν\nM\nτ−1(q)+ν\nMω(q)+iω(q)−iqVL/bracketrightBig\n. (47)\nIt can be seen here that in the case of fast vortex fluc-\ntuations, τ−1(q)≫ω(q), magnon generation is strongly\nsuppressed, as the integral is proportional to τ(q). Wewillanalyzeindetailtheoppositelimitingcase, τ−1(q)≪\nω(q). Then\nfM≈γMΦ2\n0sin2θ\n4π2/integraltextqd2q\n(1+λ2q2)2\n×iω(q)\nω2(q)−(qVL)2−2i(qVL)τ−1\n1(q), (48)\nwhereτ−1\n1(q) =τ−1(q)+νω(q)/M, and in the numerator\nterms proportional to ν/Mhave been droped. The main\ncontribution to the integral comes from qlying in the\nvicinity of two circles in the q-plane, given by ω(q) =\n±qVL(this equation specifies the Cherenkov resonance\ncondition). Near the circle ω(q) =qVLwe make the\nfollowing transformation:\nω2(q)−(qVL)2−2i(qVL)τ−1\n1(q)\n≈2ω(q)(ω(q)−qVL−iτ−1\n1(q)).\nForthecircle ω(q) =−qVLthetransformationsareanal-\nogous. Then\nfM≈γMΦ2\n0sin2θ\n4π2/integraldisplayqd2q\n(1+λ2q2)2ℜi\nω(q)−qVL−iτ−1\n1(q).\n(49)\nThe last fraction in the right-hand side resembles the\nexpression\nℜi\nf(x)−iǫ,\nwhich reduces to δ(f(x)) when ǫ→+0. Hence, the last\nfactor in Eq. (49) also can be replaced by a δ-function,\nwhenτ−1\n1(q) issufficientlysmall. Toderivethelimitation\nonτ−1\n1(q) we direct the qx-axis along VLand rewrite the\ndenominator of the large fraction in Eq. (49) as follows:\nωF(1+L2q2)−qxVL−iτ−1\n1(q)\n=ωF/parenleftBig\n1−V2\nL\nV2\nth/parenrightBig\n−iτ−1\n1(q)+ωFL2/bracketleftbigg/parenleftBig\nqx−VL\n2L2ωF/parenrightBig2\n+q2\ny/bracketrightbigg\n,\nwhereVth= 2ωFL. Now it is evident that the δ-function\ncan be introduced in Eq. (49) when\nτ−1\n1(q)≪ωF/vextendsingle/vextendsingle/vextendsingle/vextendsingleV2\nL\nV2\nth−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nThen\nfM≈ −γMΦ2\n0sin2θ\n4π/integraldisplayqd2q\n(1+λ2q2)2δ(ω(q)−qVL).(50)\nHere, two points should be noted: (i) the expression for\nfMdoes not depend on the dissipation rate and on the\nartificially introduced time τ(q); (ii) Equation (50) can\nbe derived from Eq. (36) in the limit of an extremely\nsparse vortex lattice, when summation can be replaced\nby integration.\nTechnical details of integration in Eq. (50) are given\nin Appendix B. The final result is\nfM=−γMΦ2\n0sin2θ\n8λ4ω2\nF/parenleftBigg\n1+/parenleftbiggVL\nλωF/parenrightbigg2/parenrightBigg−3/2\nΘ(VL−Vth)VL\n(51)8\nforλ≫L. Equation (51) asserts that the quantity Vthis\nthe magnon generation threshold velocity. The maximal\nvalue offMis reached at VL=λωF√\n2≫Vth:\n|fM|max=Φ2\n0γMsin2θ\n8√\n2λ3ωF33/2. (52)\nThe influence of the magnetic force fMon the current-\nvoltage characteristics in general has been discussed in\nthe previous section. According to Eqs. (41) and (42),\nat the electric field E=VthB0/cthe average current\ndensity should exhibit a stepwise increase by\n∆j=c\nΦ0fM(Vth) =γMΦ0csin2θ\n8λ4ω2\nF/parenleftBigg\n1+/parenleftbiggVth\nλωF/parenrightbigg2/parenrightBigg−3/2\nVth.\nThe maximum enhancement of the current density due\ntovortex-magneticmomentinteractionisreachedat E=√\n2λωFB0/cand equals\n∆jmax=Φ0cγMsin2θ\n8√\n2λ3ωF33/2.\nIn Ref. 14 it has been predicted that in antiferromag-\nnetic superconductors in the sparse lattice limit the cur-\nrent enhancement ∆ jis proportional to√VL−Vc(at\nVL> Vc), where Vcis some critical velocity. This result\nis in contrast with ours: we found that ∆ j∼Θ(VL−Vth)\nnear the magnon generation threshold. This difference\nis due to different magnon spectra in ferromagnets and\nantiferromagnets – see Fig. 2. In an antiferromagnet\nω(q) =/radicalbig\nω2\n0+s2q2, whereω0is a gap frequency, and sis\nthe short-wavelength magnon velocity. As the vortex ve-\nlocity is increased, the resonance condition ω(q) =VLq\nisfirstsatisfiedatinfinitely large q. However,at q≫ξ−1,\nwhereξis the coherence length, the Fourier components\nhqzareexponentially small. Magnongenerationbecomes\nefficient at q∼ξ, which is reached at a critical velocity\nthat roughly equals Vc=/radicalbig\nω2\n0ξ2+s2. In short, the gen-\neration threshold in antiferromagnetic superconductors\ncorresponds to an intersection of the curves ω=ω(q)\nandω=VLqatq∼ξ−1(see Fig. 2a), yielding a ∆ j∼√VL−Vcdependence. On the contrary, in ferromag-\nnetic superconductors at VL=Vththe curves ω=ω(q)\nandω=VLqtouch each other at q=L−1< ξ−1(see\nFig. 2b). This fact leads to a stepwise increase of the\ncurrent at the threshold vortex velocity.\nFinally, we need to make a remark concerning the con-\ndition (33), providing that the ideal lattice approxima-\ntion can be used. It follows from Fig. 2b that near\nthe generation threshold magnons with wave numbers\nq≈L−1are generated. This means that for VL/greaterorsimilarVth\nthe main contribution to the integral in Eq. (30) comes\nfromq∼L−1. Thus, the condition (34) should be im-\nposed to ensure the applicability of the perfect lattice\napproximation./s113\n/s45/s49/s48\n/s45/s49/s32/s40/s50\n/s48/s43 /s115/s50\n/s113/s50\n/s41/s49/s47/s50\n/s32/s115/s113\n/s32/s86\n/s76/s113\n/s76/s45/s49/s113/s98\n/s70/s32\n/s70/s40/s49/s43 /s76/s50\n/s113/s50\n/s41\n/s32/s86\n/s76/s113/s97\nFIG. 2. The magnon spectra in an (a) antiferromagnet and\n(b) ferromagnet. The dash-dotted line is given by ω=VLq,\nwhereVLis the vortex velocity at which magnon generation\nbecomes efficient ( VL=Vcfor antiferromagnetic supercon-\nductors and VL=Vthfor ferromagnetic superconducors).\nC. Estimates of the threshold vortex velocity in\nferromagnetic superconductors and SF multilayers.\nLet us check whether it is possible to observe the\nfeatures connected with the Cherenkov resonances on\nthe current-voltage characteristics of ferromagnetic su-\nperconductors and SF multilayers. To satisfy the condi-\ntion (37) sufficiently large vortex velocities VL> Vthare\nrequired. Estimates of the theshold velocity for known\nferromagnetic superconductors are given in Table I. One\ncanseethatthe valuesof Vthareverylarge. Thequestion\narises if such velocities are compatible with superconduc-\ntivity in the U-based superconductors. To investigate\nthis question we will estimate the supercurrent density\njthwhich is sufficient to accelerate the vortices up to the\nvelocityVth. Equation (41) yields\njth≈cη\nΦ0Vth (53)\nFor the viscosity ηwe use the Bardeen and Stephen\nexpression33(which is a good estimate for relatively slow9\nprocesses)34\nη= Φ0Hc2σn/c2, (54)\nwhereHc2= Φ0/(2πξ2) is the upper critical field. For\nthe normal state conductivity we use Drude’s estimate\nσn∼e2nℓ\nmVF.\nHerenis the concentration of charge carriers, mis their\nmass,ℓis the mean free path, and VFis the Fermi veloc-\nity. Then\njth∼e2nℓHc2Vth\nmcVF. (55)\nThis value should be compared with the depairing cur-\nrent density which is given within the BCS theory by\njcr∼en∆\nmVF,\nwhere ∆ is the superconducting gap. We demand jth≪\njcr. Using the relation ∆ ∼/planckover2pi1VF/ξ(valid for clean super-\nconductors) we can rewrite the inequality above as\nVth≪ξ\nℓVF. (56)\nIn the U-based compounds the coexistence of supercon-\nductivity and ferromagnetism appears in clean samples\nwithℓ/greaterorsimilarξ. The Fermi velocities are of the order of\n108cm/sin UGe 2and 105cm/sin UCoGe and URhGe –\nsee Refs. 35–37. Thus, the inequality (56) is satisfied in\nneither of these compounds, and our model breaks down\nat vortex velocities below Vth. This is a consequence\nof the high magnetic anisotropy and large quasiparticle\nmass in the U-compounds.\nThe situation seems to be more optimistic in SF su-\nperlattices. Certainly, we should consider if Eq. (54) is\nvalid for multilayers. A study of the vortex viscosity in\nsuperconductor/normal metal multilayers is presented in\nRefs. 38 and 39. It has been shown the Bardeen-Stephen\nviscosity (54) may be significantly modified for vortices\ninclined with respect to the z-axis, or for strongly con-\nducting normal metal layers. Still, in our case Eq. (54)\nis a good order-of magnitude estimate for dS∼dFand\nσF/lessorsimilarσn, where dSanddFare the thicknesses of the\nsuperconducting and ferromagnetic layers (see Fig. 1),\nrespectively.\nRecently, a number of experimental papers18–20\nhave reported successful fabrication of high-quality\nYBa2Cu3O7/La2/3Ca1/3MnO3superlattices. In Ref. 40\nthe value Han= 1200Oefor La 0.7Ca0.3MnO3is given,\nthough it is noted that the anisotropy is significantly in-\nfluenced by strain. The measured domain wall width in\nthe same compound, denoted as δin Ref. 41, is 12 nm.\nAssuming γ∼µB//planckover2pi1, whereµBis the Bohr magneton,\nwe obtain the following estimate for the vortex threshold\nvelocity:\nVth= 2γHanL∼104cm/s. (57)The Fermi velocity in YBa 2Cu3O7is of the order of or\ngreater than 107cm/s.42Thus, the condition (56) can\nsurely be satisfied in the cuprate/manganite superlat-\ntices.\nIV. MAGNON RADIATION BY A\nHARMONICALLY OSCILLATING VORTEX\nLATTICE\nAs it has been shown in Sec. IIIC, magnon genera-\ntion in U-based ferromagnetic superconductors by a vor-\ntex array moving with constant velocity seems problem-\natic due to the extremely high required vortex veloci-\nties. In this section we study a more feasible approach\nto magnon generation in magnetic superconductors, an-\nalyzing the case of a harmonic external current acting on\nthe vortices. Experimentally, the oscillating current in\nthe superconductor can be created using the microwave\ntechnique (for example, see Ref. 21). Then, the surface\nimpedance yields information about the high-frequency\nproperties of the sample – see Sec. V.\nSubjected to the action of a harmonic force, in the\nlinear regime the vortices oscillate harmonically:\nRi(t) =R′\ni0+Re−iωt+R∗e−iωt.(58)\nHereR′\ni0are the equilibrium positions of the vortices,\nwhich are defined by vortex-vortex interaction as well as\npinning. The vectors R′\ni0do not necessarily form a reg-\nular lattice, unlike Ri0.Ris the amplitude of vortex\noscillations. We will consider frequencies of the order of\nthe ferromagnetic resonance frequency in ferromagnetic\nsuperconductors, ωF∼100GHz. This frequency is sev-\neral orders of magnitude larger than the typical depin-\nning frequency.43This fact allowsto neglect the influence\nof the pinning force on vortex motion and to assume that\nthe oscillation amplitudes of all vortices are equal to R.\nThe product of the magnetic fields in Eq. (30) equals\nhqz(t′)h∗\nqz(t) =/parenleftBig\nΦ0\n4π2(1+λ2q2)/parenrightBig2\nK′eiq(Ri(t)−Ri(t′))\n≈/parenleftBig\nΦ0\n4π2(1+λ2q2)/parenrightBig2\nK′[1+iq(Ri(t)−Ri(t′))],(59)\nwhere\nK′=/summationdisplay\ni,je−iqR′\ni0+iqR′\nj0=Nv/angbracketleftBigg/summationdisplay\nje−iqR′\ni0+iqR′\nj0/angbracketrightBigg\n.\n(60)\nHere, the averaging is over i. The linear with respect to\nRcontribution to the force takes the form\nfM=γMΦ2\n0\n8π2Nvsin2θ/integraltext\nd2q/integraltextt\n−∞iK′qR\n(1+λ2q2)2(e−iωt−e−iωt′)\n×/braceleftBig\ne[iω(q)−ν\nMω(q)](t−t′)−e[−iω(q)−ν\nMω(q)](t−t′)/bracerightBig\nqdt′+c.c.\n≈γMΦ2\n0\n4π2Nvsin2θe−iωt/integraltextd2qK′(q)qR\n(1+λ2q2)2\n×/bracketleftBig\nω(q)\nω2(q)−ω2−2iν\nMωω(q)−ω−1(q)/bracketrightBig\nq+c.c.(61)10\n/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51/s45/s49/s44/s48/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50/s48/s44/s48\n/s32/s32/s40\n/s77/s41\n/s66\n/s48/s76/s50\n/s47\n/s48/s32 /s48/s46/s53\n/s70\n/s32 /s48/s46/s54\n/s70\n/s32 /s48/s46/s55\n/s70\n/s32 /s48/s46/s56\n/s70\n/s32 /s48/s46/s57\n/s70\nFIG. 3. The ℑ(ηM) vs. magnetic field dependence at fre-\nquencies below the ferromagnetic resonance frequency for\nan ideal triangular vortex lattice (see Eq. (66)). η0=\nγMΦ2\n0sin2θ/(2λ4ω2\nF).\nHerec.c.denotes the complex conjugate. Like before, we\nneglected small terms of the order of ν/M.\nTo proceed further, the explicit form of K′(q) is re-\nquired. Again, we will consider the cases of a pefect\nvortex lattice and a disordered array.\nA. A perfect vortex lattice.\nLet us assume that pinning is sufficiently weak, so that\nq∆R≪1, (62)\nwhere ∆ R∼ |R′\ni0−Ri0|is the characteristic deviation\nof the vortices from their positions in a perfect lattice.\nThe inequality (62) should hold for all qgiving a con-\nsiderable contribution to the integral in Eq. (61). The\ncharacteristic value of qwill be estimated below.\nForq∆R≪1 we have\nK′=4π2NvB0\nΦ0/summationdisplay\nGδ(q−G). (63)\nSubstituting Eq. (63) into Eq. (61), assuming that the\nvortex lattice is either square or regular triangular, we\nobtain\nfM=iωηMRe−iωt+c.c., (64)\nηM=−iγMΦ0B0\n2ωsin2θ/summationtext\nGG2\n(1+λ2G2)2\n×/bracketleftBig\nω(G)\nω2(G)−ω2−2iν\nMωω(G)−ω−1(G)/bracketrightBig\n.(65)\nHere, wehaveintroducedthe complexquantity ηM, play-\ning the role of a generalized vortex viscosity. Indeed,\nwhenηMis purely real, the magnetic force is simply\nfM=−ηMdRi/dt. In our system there is a phase shiftbetween the vortex velocity and fM, and the more gen-\neral expression (64) is valid. Further on we will call ηM\nthe magnetic viscosity.\nThe ideal vortex lattice is likely to form when vortex-\nvortex interaction is sufficiently strong, or the inter-\nvortex distance is suffiently small. Let this distance be\nmuch smaller than the London penetration depth, which\nmeansB0≫Φ0/λ2. ThenλG≫1 for all G/negationslash= 0, and\nηM≈ −iγMΦ0B0ω\n2λ4sin2θ/summationtext\nG/negationslash=0G−2ω−1(G)\n×/bracketleftbig\nω2(G)−ω2−2iν\nMωω(G)/bracketrightbig−1.(66)\nNow we consider the behavior of ηMin different fre-\nquency ranges. First, let the frequency be below the\nferromagnetic resonance frequency ( ω < ω M). Then\nmagnon generation is inefficient. However, if we put\nν= 0, the force fMwill not vanish below the generation\nthreshold, unlike in the case of constant vortex velocity.\nInstead, the magnetic viscosity will be purely imaginary,\nsignifying that there are no magnetic losses. In Fig. 3\nwe plot the imaginary part of ηvs. magnetic field B0de-\npendencies for different frequencies (below ωF) and for a\nfixed angle θ.\nReturning to the condition (62), one can see that for\nω < ω Fthe characterisitic values of Gin Eq. (66) are\nof the order of L−1. Hence, ∆ R≪Lis required for Eq.\n(66) to be valid.\nAt frequencies above the ferromagnetic resonance fre-\nquency magnetic dissipation can not be neglected, and\nthe real part of ηMbecomes significant. In Fig. 4 we\nplot theηMvs.B0dependencies for different frequencies\nand for a fixed angle θand dissipation rate ν/M= 0.02.\nThegraphsexhibit asequenceofLorentzian-like( ℜ(ηM))\nandN-shaped( ℑ(ηM))features,locatedatsomeresonant\nfield values, BR, which are determined from the relation\nω(G) =ω. (67)\nFor small fields these features may overlap, but the reso-\nnance correspondingto the highest field remains well dis-\ntinguishable. For a triangular vortex lattice the largest\nresonance field equals\nBR=√\n3\n8π2Φ0\nL2/parenleftbiggω\nωF−1/parenrightbigg\n,\nand for a square lattice\nBR=1\n4π2Φ0\nL2/parenleftbiggω\nωF−1/parenrightbigg\n.\nSolving Eq. (67) with respect to G, we obtain\nG=L−1/radicalbiggω−ωF\nωF. (68)\nHence, the peaks on the ηMvs.B0dependences must\nbe observable if the characteristic deviation ∆ Rof the\nvortices from their positions in an ideal lattice satisfies\n∆R≪G−1=L/radicalbiggωF\nω−ωF. (69)11\n/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s45/s49/s48/s49/s50/s51\n/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s32/s61/s49/s46/s53\n/s70/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\n/s32/s32/s32\n/s32/s32/s47\n/s48\n/s32/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\n/s32\n/s32/s32/s61/s49/s46/s49\n/s70\n/s32/s61/s50/s46/s48\n/s70/s32 /s32/s32\n/s66\n/s48/s76/s50\n/s47\n/s48/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\nFIG. 4. The ηMvs. magnetic field dependencies for fre-\nquencies above the ferromagnetic resonance frequency (see\nEq. (66)). η0=γMΦ2\n0sin2θ/(2λ4ω2\nF). The vortices form an\nideal triangular lattice.\nNote that for frequencies close to ωFthis condition is\nweaker than ∆ R≪L.\nWe conclude this section by giving a numeric estimate\nof the magnetic viscosity. When the resonance condition\n(67) is satisfied, we obtain from Eq. (66)\nηM∼γMΦ0B0\nλ4G2ω2M\nν.\nSinceB0G−2∼Φ0, and the lowest allowable value of ω\nisωF=γMK, we have\nηM/lessorsimilarΦ2\n0\nKλ4ωFM\nν. (70)\nThen, according to Eq. (54), the ratio of ηMtoηis\nηM/η/lessorsimilarM\nνξ2c2\nKλ4ωFσn. (71)\nWe will make the numeric estimate for UCoGe, the ferro-\nmagnetic superconductor with the lowest ferromagnetic\nresonancefrequency. In Ref. 3 we find the value 12 µΩcm\nfor the normal resistivity, and the maximal value 200 ˚A/s48/s44/s48 /s48/s44/s53 /s49/s44/s48 /s49/s44/s53 /s50/s44/s48/s45/s49/s48/s45/s53/s48/s53/s49/s48/s48\n/s70/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\nFIG. 5. The frequency dependence of the magnetic vis-\ncosity,ηM, for a disordered vortex array – see Eq. (76).\nThe value ln( λ/L) = 4.3 of UGe 2has been used. η0=\nγMΦ2\n0sin2θ/(2λ4ω2\nF).\nfor the coherence length. Using Table I, we obtain\nηM/η∼M\nν3×10−5. (72)\nData on the ratio M/νare not available yet. The small\nfactor 10−5in Eq. (72) appears due to the large magne-\ntocrystalline anisotropy of UCoGe: it can be seen from\nEq. (71) that ηM/ηis proportional to K−2, sinceωF=\nγMK. Hence, to increase the ratio of ηMtoη, com-\npounds (or multilayer systems) with a lower anisotropy\nare preferable.\nB. A disordered vortex array.\nNow let us assume that due to relatively strong pin-\nning the vortices are placed chaotically, i. e., there is no\ncorrelation between R′\ni0andR′\nj0fori/negationslash=j. Then\nK′=Nv/parenleftBig/integraltextB0\nΦ0e−iqR′\ni0+iqR′\nj0d2R′\nj0+1/parenrightBig\n=Nv/parenleftBig\n4π2B0\nΦ0δ(q)+1/parenrightBig\n. (73)\nIn a real vortex lattice there is a short-rangeorder, which\nleads to the smearing of the delta-function on a scale of\nthe order of the inverse inter-vortex distance. However,\nthe behavior of K′at small qis not important, since K′\nenters the integral in Eq. (61) with a factor q2. For\nclarity, we stress here that the product hqz(t′)h∗\nqz(t) does\nnot decay with increasing t−t′, unlike in the case of a\nconstant driving force – see Eqs. (32) and (46). This is\nexplained by the fact that the vortices oscillate close to\ntheir equilibrium positions and do not travel from one\npinning cite to another. Thus, the positions Ri(t) and\nRi(t′) of a single vortex are always well correlated, cor-\nresponding to an infinite correlation time τ(q).12\nWithK′givenby Eq. (73) the magneticviscositytakes\nthe form\nηM=−iγMΦ2\n0\n4πωsin2θ/integraltext∞\n0q3dq\n(1+q2λ2)2\n×/bracketleftBig\nω(q)\nω2(q)−ω2−iǫ−ω−1(q)/bracketrightBig\n. (74)\nHere, like in Sec. IIIB, we assume that the imaginary\nterm−iǫ(ǫ >0) in the denominator is an infinitesimal.\nTo simplify the expression in the right-hand side of Eq.\n(74), we note that the contribution to the integral from\nsmallq(q/lessorsimilarλ−1) can be neglected in the λ≫Llimit.\nThen we can put 1+ λ2q2≈λ2q2, and cut the integral\noff atq=λ−1:\nηM=−iγMΦ2\n0\n4πωλ4/integraltext∞\nλ−1dq\nq\n×/bracketleftBig\nω(q)\n(ω+ω(q))(ω(q)−ω−iǫ)−ω−1(q)/bracketrightBig\n.(75)\nFurther integration should not present difficulties. For\nλ≫Lwe obtain\nηM=γMΦ2\n0sin2θ\n8πωωFλ4/braceleftBig\nπωF\n2(ω−ωF)Θ(ω−ωF)−i/bracketleftBig\n2ω2\nω2\nF−ω2lnλ\nL\n+ωF\n2(ω+ωF)lnω+ωF\nωF+ωF\n2(ωF−ω)ln/vextendsingle/vextendsingle/vextendsingleωF−ω\nωF/vextendsingle/vextendsingle/vextendsingle/bracketrightBig/bracerightBig\n.(76)\nLike in the previous section, below the ferromagnetic res-\nonance frequency the magnetic viscosity is purely imagi-\nnary. However, unlike in the case of a perfect fortex lat-\ntice, now the viscosity does not depend on the magnetic\nfield. It should be also noted that in the limit B0→0\nEq. (65) after summation transforms into (76), i. e., the\ncases of isolated vortices and chaotically placed vortices\nare equivalent, like in Sec. III. The ηMvs.ωdependence\nis depicted in Fig. 5.\nC. Vortex mass.\nAs we have seen, at ω < ω Fthe magnetic viscosity is\nimaginary. Moreover, at ω≪ωFthe viscosity is propor-\ntional to ω. This signifies that the vortex can be ascribed\na mass per unit length, Mv, so that the equation of mo-\ntion becomes\nMvd2Ri\ndt2=fext, (77)\nwherefextincludes all forces, except for the force fM.\nThe mass is defined by\nMv=iηM\nω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0. (78)\nBefore we give explicit expressions for Mv, we should\ncomment on the connection between the vortex mass en-\nhancement and the self-induced polaronic pinning mech-\nanism, studied in Refs. 15 and 16. In the mentioned\npapers it has been assumed that the magnetization dy-\nnamics is purely dissipative, i. e., ν/M≫1, which is\nin contrast to our case. In fact, ν/M≫1 is a necessarycondition for the formationof polaronlikevortices. Thus,\nthe polaronic pinning mechanism contributes rather to\nthe real part of ηMthan to its imaginary part, an it is\nnot related to the vortex mass enhancement discussed\nhere.\nUsing Eq. (66), we find that the magnetic contribution\nto the vortex mass for a perfect lattice is\nMv=γMΦ0B0\n2λ4sin2θ/summationdisplay\nG/negationslash=0G−2ω−3(G) (79)\nwhenB0≫Φ0/λ2. For a disordered array we obtain\nfrom Eq. (74)\nMv=γMΦ2\n0\n4πω3\nFsin2θ/integraltext∞\n0q3dq\n(1+q2λ2)2(1+L2q2)3\n≈γMΦ2\n0sin2θ\n16πω3\nFλ4/parenleftbig\n4lnλ\nL−5/parenrightbig\n(λ≫L).(80)\nLet us estimate the characteristic magnetic contribution\nMvto the vortexmass andcompareit with the electronic\ncontribution (see, for example, Ref. 44), which is present\nin any superconductor:\nMe=2\nπ3m2VF\n/planckover2pi1. (81)\nWe give estimates for the ferromagnetic superconductor\nURhGe. The valuesof ωF=γMKandλcanbe found in\nTable I. The electron mass and Fermi velocity for one of\nthe Fermi surface pockets of URhGe have been measured\nin Ref. 37. The values given there are m= 22meand\nVF= 4.4×105cm/s, wheremeis the free electron mass.\nThen\nMv∼γMΦ2\n0\n16πω3\nFλ4≈10−24g/cm, M e∼10−20g/cm.\nIt can be seen that the magnetic contribution to the vor-\ntex mass is negligible for URhGe. Estimates for UGe 2\nand UCoGe yield the same result. This happens due to\nthe very large ferromagnetic resonance frequency ωFin\nthese compounds: note that the right-hand side of Eq.\n(80) contains ω−3\nF. The situation is the same as for the\nmagnetic viscosity – see Eqs. (71) and (72). Thus, the\nmagneticmass Mvshould be detectablein materialswith\na smaller ferromagnetic resonance frequency.\nV. DISCUSSION OF THE MAGNETIC\nVISCOSITY MEASUREMENT\nAsimple experimentalmethod to studyvortexdynam-\nics in type-II superconductors consists in the measure-\nment of the surface impedance. A possible geometry for\nsuch experiment is depicted in Fig. 6. We consider the\nsimplestsituation, whenthevorticesareperpendicularto\nthesamplesurface,andtheprobingelectromagneticwave\nwith the amplitude heis normally incident on this sur-\nface. Then, foranon-magneticsuperconductor( M0= 0)13\nheHe\nB0\nM0Lx\nLy\nLzx\nyzFerromagnetic\nsuperconductor\nFIG. 6. The geometry for the measurement of the surface\nimpedance of a ferromagnetic superconductor. The dashed\nlines denote vortices.\ntheory30,45predicts that in a wide range of parameters\nthe surface impedance Z(ω) equals\nZ(ω) =/parenleftbigg−iωµρf\n4π/parenrightbigg1/2\n, (82)\nwhereµis the static differential magnetic permeability,\nµ=dB0z\ndHez,\nandρfis the flux-flow resistance,\nρf=BΦ0\nc2η.\nThus, the experimental value of the surface impedance\nprovides information about the viscosity coefficient η.\nWe will prove that for a ferromagnetic superconductor\na range of parameters exists, where Eq. (82) can be ap-\nplied, if the magnetic viscosity is taken into account: η\nshould be replaced by η+ηM.\nFirst, we outline the applicability conditions of Eq.\n(82) for an ordinary superconductor. Within the con-\ntinuous medium approximation used in Ref. 45 an alter-\nnating external field heexcites a long-wavelength and a\nshort-wavelength mode in the superconductor. For con-\nvenience, we will call these modes type-1 and type-2, and\ndenote the z-projections of their wave vectors as k1and\nk2, respectively. These quantities are explicitly defined\nbyEquation(24)inRef. 45. Tousethesimpleexpression\n(82) fortheimpedance, threeconditionsmustbefulfilled:\n(i)|k1|λ≪1, (ii)|k1| ≪ |k2|, and (iii) |k2|Lz≫1 (Lz\nis the sample thickness – see Fig. 6). According to Ref.\n45, the conditions (i) and (ii) are satisfied, if\nω≪ωC=Φ0C∗\n44\nB0λ2η, (83)whereC∗\n44is an elastic modulus of the vortex lattice.\nThis inequality presents a limitation on the frequency.\nWe would like to note that in the limit Hc1≪B0≪Hc2,\nwhereHc1is the lower critical field, the condition (83)\ncan be weakened, namely\nω≪ωB=Φ0B0\n4πλ2η(ωB≫ωC).(84)\nThis follows directly from Equation (22) in Ref. 45.\nLetusturn tothecaseofaferromagneticsuperconduc-\ntor. We assume the sample is a slab with dimensions Lx,\nLyandLz, whereLz≪Lx,Ly– see Fig. 6. The x-axis,\nparallel to the large surface of the sample, is the magne-\ntization easy-axis. In fact, the slab geometry is not a key\npoint for us, but the equilibrium magnetization must be\nparallel to one of the sample surfaces. By applying an\nexternal field we can provide that the internal field B0is\nparallel to the z-axis. In the slab geometry the compo-\nnents of the demagnetizing tensor are Nxx≈0,Nyy≈0,\nNzz≈4π. Then, according to Eq. (5), if the external\nfield isHe= (−4πM,0,Hez), the internal field equals\nB0= (0,0,Hez).\nNow we discuss the surface impedance of a ferromag-\nnetic superconductor. Compared to the case of a conven-\ntional superconductor, an additional complication arises\ndue to the presence of new degrees of freedom. These\nare connected with magnetization dynamics and lead to\nthe appearance of new magnon-like modes. Such modes\ncan be directly excited by an electromagnetic wave even\nin the absence of vortices,5,6and they may significantly\ninfluence the surface impedance. However, in our geome-\ntry the excitation of these modes can be avoided, as will\nbe demonstrated below.\nIf the frequency is not too close to the ferromag-\nnetic resonance frequency ( |ω−ωF|/ωF≫K−1) we\ncan neglect the magnetostatic interaction in the Landau-\nLifshitz equation when analyzing the additional magnon-\nlike modes, as we have done in Sec. II (where the term\nbMqhas been dropped). Then, in the limit of small dis-\nsipation, Eq. (10) takes the form\n∂m\n∂t=−γM0×/parenleftbigg\nα∂2m\n∂z2−Km/parenrightbigg\n.(85)\nThis yields two modes, which we label as type-3 and 4:\nm= (z0∓iy0)m3,4eik3,4z,\nk3=L−1/radicalBig\nω\nωF−1, k 4=iL−1/radicalBig\nω\nωF+1,(86)\nwherem3andm4are scalar amplitudes. Now suppose\nthat the magnetic field hein the probing electromagnetic\nwave oscillates along the x-axis, i. e., along the equilib-\nrium magnetization (see Fig. 6). We assume that inside\nthe sample the alternating magnetic induction /angb∇acketleftb/angb∇acket∇ight, aver-\naged over the xy-plane, is also parallel to the x-axis. It\nwill be shown that this statement is self-consistent. In-\ndeed, for /angb∇acketleftb/angb∇acket∇ightparallel to M0we see from Eqs. (10) and\n(14) that ∂/angb∇acketleftm/angb∇acket∇ight/∂t= 0. This means that the magnon-\nlike type-3 and 4 modes are not excited. In the type-114\nand 2 modes /angb∇acketleftm/angb∇acket∇ight= 0, but /angb∇acketleftb/angb∇acket∇ight /negationslash= 0. Hence, these modes\ndiffer from their analoguesin non-magneticsuperconduc-\ntors only by the presence of the magnetic contribution to\ntheviscosity, ηM, whichisduetotheFourier-components\nmqwithq/negationslash= 0. Then, according to Ref. 45, the internal\nfield/angb∇acketleftb/angb∇acket∇ightwill be parallel to the probing field he(which\nfollows from the London equation (11), if the deforma-\ntion of the vortex lattice is taken into account). Thus,\nwe have proved the validity of our assumption, having\nshown in addition that only the type-1 and 2 modes are\nexcited.\nStrictly speaking, the effective viscosity for the long-\nwavelength type-1 mode differs from η+ηM, because the\nvortices are not straight. However, since |k1| ≪λ−1, the\nradius of curvature of the vortices is sufficiently large to\nmake this difference negligible.\nAn electromagnetic wave polarized in the y-direction\n(he=heyy0) requires separate treatement, which is out-\nsidethescopeofthispaper. Here,themagnon-likemodes\nof type 3 and 4 must be taken into account. For a study\nof the surface impedance in the case he⊥M0(in a differ-\nent geometry) see Ref. 13.\nVI. MAGNON EXCITATION IN SF\nMULTILAYERS.\nIn this section it is shown how our results can be ex-\ntended to the case of SF multilayers with S and F be-\ning an ordinary type-II superconductor and ordinary fer-\nromagnet, respectively. We consider structures with a\nsufficiently small period d(see below) and with vortices\noriented perpendicular to the layer surfaces – see Fig. 1.\nThen, the generalization of the results from Secs. II - IV\nis straightforward, if two points are taken into account:\n(i) Since the magnetic moments now occupy only a frac-\ntion of the sample, the force fMis reduced by a factor of\nd/d′\nF, whered′\nF≤dFis the effective thickness of the fer-\nromagnetic layer. Formally, all expressions for fM, start-\ning with Eq. (23), should be multiplied by d′\nF/d. The\nquantities d′\nFanddFcoincide, if the mutual influence\nof the superconducting and magnetic orders is negligi-\nble. However, this is not the case for cuprate/manganite\nsuperlattices. Experimental papers report giant super-\nconductivity induced modulation of the magnetization19\nand the suppression of magnetic order in the mangan-\nite layer close to the SF interface.20In the latter case,\nd′\nF< dF, but both quantities are of the same order of\nmagnitude.\n(ii) Due to the fact that the structure is only partially\nsuperconducting, the in-plane London penetration depth\nnow equals λeff=λ(d/dS)1/2– see Ref. 46, for example.\nThe expression for the single vortex field\nhqz≈Φ0\n4π2(1+q2λ2\neff)(87)\ncan be used if the period dof the structure is muchsmaller than the characteristic in-plane length scale of\nthe problem. To apply our results for the case of a\nconstant driving force, we have to demand d≪L, ac-\ncording to Sec. III. The constraint is somewhat weaker\nin the case of the harmonic driving current. Indeed,\nforω > ω Fthe main contribution to fMcomes from\nq≈L−1/radicalbig\nω/ωF−1, hence, the limitation on the period\nof the structure is\nd≪L/radicalbiggωF\nω−ωF.\nThus, for ( ω−ωF)/ωF≪1 the thickness dmay be of\nthe order of or larger than the domain wall width.\nFinally, we will discuss briefly a recent paper by\nTorokhtii et al.,21where the flux-flow resistivity in\nNb/PdNi/Nb trilayers has been measured. It has been\nreported that in the presence of the magnetic PdNi\nlayer the flux-flow resistivity in Nb exceeds the Bardeen-\nStephen estimate,33as if the vortex viscosity is reduced\nbythe interactionwith magneticmoments. At first sight,\nthis seems to contradict our prediction. However, this\nexperiment can not be interpreted in the framework of\nthe model used here, since the ferromagnetic alloy PdNi\ndoes not posess a well-defined magnetic anisotropy, and\nthe magnon modes can not be characterized by a wave\nvectorqduetothe lackoftranslationalsymmetry. More-\nover, the dependence of the critical temperature ofNb on\nthe PdNi layer thickness signifies strong influence of the\nmagnetic order on superconductivity. We suppose that\nthe explanation of the viscosity reduction in the men-\ntioned experiment requires a more complicated micro-\nscopic treatment.\nVII. CONCLUSION\nWe have calculated the magnetic moment in-\nduced force fMacting on moving Abrikosov vor-\ntices in ferromagnetic superconductors and supercon-\nductor/ferromagnet multilayers. When the vortices are\ndriven by a dc transport current, magnons are efficiently\ngenerated when the vortex velocity exceeds the value\nVth= 2ωFL. As a result, narrow peaks appear on the\ncurrent-voltage characteristics of the superconductor, if\nthe vortices form a regular lattice. Within a vortex lat-\ntice domain the current may be not parallel to the elec-\ntric field. For a disordered vortex array a step-like fea-\ntureshouldappearonthe current-voltagecharacteristics.\nThis behavior is in contrast with antiferromagnetic su-\nperconductors, where the increase of the current at the\nmagnongenerationthresholdisproportionalto√U−Uc,\nwhereUis the voltage, and Ucis some threshold value.14\nAccording to our estimates, the transport current re-\nquired to reach the vortex velocity Vthin the U-based\nferromagnetic superconductors is of the order the depair-\ning current due to the large magnetic anisotropy of these\ncompounds. On the other hand, in cuprate/manganite15\nmultilayers18–20the requiredcurrent is well below the de-\npairingcurrent,sothementionedfeaturesmaybeobserv-\nable on the current-voltage characteristics of such sys-\ntems.\nIf the vortices are driven by an ac current, the interac-\ntion with magnetic moments results in the appearance of\nacomplexmagneticcontribution ηMtothevortexviscos-\nity. We determined this quantity for the cases of an ideal\nvortex lattice and a disordered vortex array. For low fre-\nquencies, ω≪ωF, the magnetic contribution to the vor-\ntex mass has been estimated. From the ηMvs. magnetic\nfield and frequency dependencies the magnon spectrum\nin the ferromagnetic superconductor can be extracted.\nExperimentally, ηMcan be determined by measuring the\nsurface impedance of the sample in the geometry, where\nthe equlibrium magnetization isparallelto the oscillating\nexternal magnetic field.\nACKNOWLEDGEMENTS\nWe are grateful to L. Bulaevskii for useful discussions\nand valuable comments. This work was supported in\npart by the Russian Foundation for Basic Research, Eu-\nropean IRSES program SIMTECH (contract n.246937),\nthe French ANR program ”electroVortex” and LabEx\n”Amadeus” program.\nAppendix A\nIn this appendix we will prove that the magnetic mo-\nment induced force acting on vortices can be written as\n(22). Wehavetocalculatethe variationofthefreeenergy\nwhen all vortices are shifted by an equal vector, and the\nmagnetization is kept fixed. To simplify the calculations\nwe use the fact that the free energy acquires the same\nvariation if the vortices are kept fixed, and the magneti-\nzation is shifted in the opposite direction. Then\nδF=/integraldisplay/parenleftbiggδF\nδAδA+δF\nδMδM/parenrightbigg\nd3r.\nAccording to the London equation δF/δA= 0 the first\nterm in the right-hand side vanishes. Also, the terms in\nEq. (1) which depend only on M(e. g., the exchange en-\nergy) are not affected by the magnetization shift. Hence,\nonly the term\nδF=−/integraldisplay\nBδMd3r, (A1)\nremains, and the force acting on a vortex is\n(fM)xi=1\nNvLv/integraldisplay\nB∂M\n∂xid3r=−1\nNvLv/integraldisplay∂B\n∂xiMd3r\nPresenting the magnetic field as B=h+bM, we have\nfM=fM1+fM2. (A2)(fM1)xi=−1\nNvLv/integraltext∂bM\n∂xiMd3r,\n(fM2)xi=−1\nNvLv/integraltext∂h\n∂xiMd3r.\nNote that the term fM1does not depend on the vortex\npositions. Hence, to calculate this term we can place the\nvortices anywhere in the superconductor. Let us posi-\ntion the vortices in an area with uniform magnetization\n(M= const). Then, fM2vanishes, and fM=fM1. On\nthe other hand, in the area with homogenous magneti-\nzationbM= 0 inside the superconductor (in the fer-\nromagnetic superconductor this happens due to London\nscreening, and in the SF multilayer system the field bM\nis simply confined to the ferromagnetic layers). Hence,\nthe magnetization has no influence on the magnetic field\nand supercurrent in the vortex region, and the force fM\nvanishes. Then, fM1= 0, and for any vortex positions\nfM=fM2. From this follows Eq. (22).\nAppendix B\nIn this appendix we show how the integral in Eq. (50)\ncan be evaluated. We introduce the dimensionless quan-\ntitiesl=L/λ,lv=VL/(ωFλ) andg=λq, and direct\nthegx-axis along VL. ThenfMy= 0, and\nfMx=−γMΦ2\n0sin2θ\n4πλ3ωF/integraltextgxd2g\n(1+g2)2δ(1+l2g2−lvgx)\n=−γMΦ2\n0sin2θ\n4πλ2VL/integraltext(1+l2g2)δ(1+l2g2−lvgx)\n(1+g2)2d2g\n=−γMΦ2\n0sin2θ\n4πλ2VL/bracketleftBig\nl2/integraltextδ(1+l2g2−lvgx)\n1+g2d2g\n+(1−l2)/integraltextδ(1+l2g2−lvgx)\n(1+g2)2d2g/bracketrightBig\n. (B1)\nNow we make a coordinate shift, redesignating gx−\nlv/(2l2) bygx:\nfMx=−γMΦ2\n0sin2θ\n4πλ2VL/braceleftbigg/integraltextδ(g2−g2\n0)d2g\n1+g2y+(gx+lv\n2l2)2\n+(l−2−1)/integraltextδ(g2−g2\n0)d2g/bracketleftBig\n1+g2y+(gx+lv\n2l2)2/bracketrightBig2/bracerightBigg\n,(B2)\nwhere\ng2\n0=l−2/parenleftbiggl2\nv\n4l2−1/parenrightbigg\n.\nFurther we assume that VL> Vth, so that g2\n0>0 (at\nVL< VthfM= 0). Integration over the modulus of gis\nnow straightforward. 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Klemm, Layered Superconductors (Oxford University\nPress, Oxford, England, 2012), Vol. 1."    },    {        "title": "0910.1477v2.Fast_domain_wall_propagation_under_an_optimal_field_pulse_in_magnetic_nanowires.pdf",        "content": "arXiv:0910.1477v2  [cond-mat.mtrl-sci]  2 Feb 2010Fast domain wall propagation under an optimal field pulse in m agnetic nanowires\nZ. Z. Sun and J. Schliemann\nInstitute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany\n(Dated: December 2, 2018)\nWeinvestigatefield-drivendomainwall (DW)propagation in magneticnanowiresintheframework\nof the Landau-Lifshitz-Gilbert equation. We propose a new s trategy to speed up the DW motion\nin a uniaxial magnetic nanowire by using an optimal space-de pendent field pulse synchronized with\nthe DW propagation. Depending on the damping parameter, the DW velocity can be increased by\nabout two orders of magnitude compared to the standard case o f a static uniform field. Moreover,\nunder the optimal field pulse, the change in total magnetic en ergy in the nanowire is proportional\nto the DW velocity, implying that rapid energy release is ess ential for fast DW propagation.\nPACS numbers: 75.60.Jk, 75.75.-c, 85.70.Ay\nRecently the study of domain wall (DW) motion in\nmagnetic nanowires has attracted a great deal of atten-\ntion, inspired both by fundamental interest in nanomag-\nnetism as well as potential industrial applications. Many\ninteresting applications like memory bits[1, 2] or mag-\nnetic logic devices[3] involve fast manipulation of DW\nstructures, i.e. a large magnetization reversal speed.\nIngeneral,themotionofaDWcanbedrivenbyamag-\nnetic field [4–6] and/or a spin-polarized current[7–11].\nAlthough the DW dynamics in systems of higher spatial\ndimension can be very complicated, some simple but im-\nportant results were obtained by Schryer and Walker for\neffectively one-dimensional (1D) situations[12]: At low\nfield (or current density), the DW velocity vis linear in\nthe field strength HuntilHreaches a so-called Walker\nbreakdown field Hw[12]. Within this linear regime, DW\npropagates as a rigid object. For H > H w, the DW\nloses its rigidity and develops a complex time-dependent\ninternal structure. The velocity can even oscillate with\ntime due to the “breathing” of the DW width. The time-\naveraged velocity ¯ vdecreases with the increase of H, re-\nsulting in a negative differential mobility. ¯ vcan be again\nlinear with Happroximately when H≫Hw. The pre-\ndictedv-Hcharacteristic is in a good agreement with\nexperimental results on permalloy nanowires[4–6]. Re-\ncently a general definition of the DW velocity proper for\nany types of DW dynamics has been also introduced[13].\nFor a single-domain magnetic nanoparticle (called\nStoner particle), an appropriate time-dependent but spa-\ntially homogeneous field pulse can substantially lower\nthe switching field and increase the reversal speed\nsince it acts as an energy source enabling to overcome\nthe energy barrier for switching the spatially constant\nmagnetization[14, 15]. In the present letter, we inves-\ntigate the dynamics of a DW in a magnetic nanowire\nunder a field pulse depending both on time and space.\nAs a result, such a pulse, synchronized with the DW\npropagation, can dramatically increase the DW velocity\nby typically two orders compared with the situation of a\nconstant field. Moreover, the total magnetic energy typi-\ncally decreases with a rate being proportional to the DWvelocity, i.e. the external field source can even absorb\nenergy from the nanowire.\nA\nBzx\ny\nz\nx\nDWy\nFIG. 1: A schematic diagram of two dynamically equivalent\n1D magnetic nanowire structures. (A) Easy axis is along the\nwire axis (z-axis); (B) Easy axis (z-axis) ⊥the wire axis (x-\naxis). The region between two dashed lines denotes the DW\nregion.\nA magnetic nanowirecan be described as an effectively\n1D continuum of magnetic moments along the wire axis\ndirection. Magnetic domains are formed due to the com-\npetitionbetweentheanisotropicmagneticenergyandthe\nexchangeinteractionamongadjacentmagneticmoments.\nLet us first concentrate on the case of a uniaxial mag-\nnetic anisotropy: Two dynamically equivalent configura-\ntions of 1D uniaxial magnetic nanowires are schemati-\ncally shown in Fig. 1. Type A shows the wire axis to\nbe also the easy-axis (z-axis). Type B shows the easy\naxis (z-axis) is perpendicular to the wire axis (x-axis).\nAlthough our results described below apply to both con-\nfigurations, we will focus in the following on type B.\nThe spatio-temporal dynamics of the magnetization den-\nsity/vectorM(x,t) is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation[16]\n∂/vectorM\n∂t=−|γ|/vectorM×/vectorHt+α\nMs/parenleftBigg\n/vectorM×∂/vectorM\n∂t/parenrightBigg\n,(1)\nwhere|γ|= 2.21×105(rad/s)/(A/m) the gyromag-\nnetic ratio, αthe Gilbert damping coefficient, and Ms\nis the saturation magnetization density. The total ef-\nfective field /vectorHtis given by the variational derivative of\nthe total energy with respect to magnetization, /vectorHt=2\n−(δE/δ/vectorM)/µ0, whereµ0the vacuum permeability. The\ntotal energy E=/integraltext∞\n−∞dxε(x) can be written as an inte-\ngral over an energy density (per unit section-area),\nε(x) =−KM2\nz+J/bracketleftBigg/parenleftbigg∂θ\n∂x/parenrightbigg2\n+sin2θ/parenleftbigg∂φ\n∂x/parenrightbigg2/bracketrightBigg\n−µ0/vectorM·/vectorH,\n(2)\nwherexis the spatial variable in the wire direc-\ntion. Here K,Jare the coefficients of energetic\nanisotropy and exchange interaction, respectively, and\n/vectorHis the external magnetic field. Moreover, we have\nadopted the usual spherical coordinates, /vectorM(x,t) =\nMs(sinθcosφ,sinθsinφ,cosθ) where the polar angle\nθ(x,t) andthe azimuthalangle φ(x,t) depend onposition\nand time.\nHence, the total field /vectorHtconsists of three parts: the\nexternal field /vectorH, the intrinsic uniaxial field along the\neasy axis /vectorHK= (2KMz/µ0)ˆz, and the exchange field\n/vectorHJwhich reads in spherical coordinates as[12, 17],\nHJ\nθ=2J\nµ0Ms∂2θ\n∂x2−Jsin2θ\nµ0Ms/parenleftbigg∂φ\n∂x/parenrightbigg2\n,\nHJ\nφ=2J\nµ0Mssinθ∂\n∂x/parenleftbigg\nsin2θ∂φ\n∂x/parenrightbigg\n. (3)\nFollowing Ref. [12], let us focus on DW structures ful-\nfilling∂φ/∂x= 0, i.e. all the magnetic moments rotate\naround the easy axis synchronously. Then the dynamical\nequations take the form\nΓ˙θ=α/parenleftbigg\nHθ−KMs\nµ0sin2θ+2J\nµ0Ms∂2θ\n∂x2/parenrightbigg\n+Hφ,\nΓsinθ˙φ=αHφ−Hθ+KMs\nµ0sin2θ−2J\nµ0Ms∂2θ\n∂x2,(4)\nwhere we have defined Γ ≡(1 +α2)|γ|−1, andHi(i=\nr,θ,φ) are the three components of the external field\nin spherical coordinates. In the absence of an exter-\nnal field, an exact solution for a static DW is given by\ntanθ(x)\n2= exp(x/∆)where∆ =/radicalbig\nJ/(KM2s)isthewidth\nof the DW. We note that a static DW can exist in a con-\nstant field only if the field component along the easy axis\nis zero,Hz= 0. In fact, according to Eqs. (4) static solu-\ntions need to fulfill Hφ= 0 [implying φ= tan−1(Hy/Hx)\nis spatially constant] and\n2J\nµ0Ms∂2θ\n∂x2−KMs\nµ0sin2θ+Hθ= 0 (5)\nor, upon integration,\nJ\nµ0Ms/parenleftbigg∂θ\n∂x/parenrightbigg2\n+KMs\n2µ0cos2θ+Hr(θ) = constant .(6)\nConsidering the two boundaries at θ= 0(x→ −∞) and\nθ=π(x→+∞) for the DW, we conclude Hr(0) =Hr(π), which requires Hz= 0. In this case, the station-\nary DW solutions under a transverse field are described\nasx=/integraltext\n[/radicalBig\n(KM2ssin2θ−µ0MsHsinθ)/J]−1dθ.\nThus, when an external field with a component along\nthe easy axis is applied to the nanowire, the DW is ex-\npected to move. We use a travelling-wave ansatzto de-\nscribe rigid DW motion[12],\ntanθ(x,t)\n2= exp/parenleftbiggx−vt\n∆/parenrightbigg\n, (7)\nwhere the DW velocity vis assumed to be constant. Sub-\nstituting this trial function into Eq. (4), the dynamic\nequations become\nΓsinθv=−∆(αHθ+Hφ),Γsinθ˙φ=αHφ−Hθ.(8)\nEq. (8) describes the dependence of the linear velocity v\nand the angular velocity ˙φon the external field /vectorH. Our\nfollowing results discussion will be based on Eqs. (8).\nLet us firstturn to the caseofastatic field caseapplied\nalong the easy axis (z-axis in type B of Fig. 1), Hθ=\n−Hsinθ,Hφ= 0. Here we recover the well-known static\nsolution for a uniaxial anisotropy[18],\nv=|γ|∆H\nα+α−1, (9)\nwhere the azimuthal angle φ(t) =φ(0)+|γ|Ht/(1+α2) is\nspatially constant (i.e. ∂φ/∂x= 0) and increases linearly\nwith time.\nLet us now allow the applied external field to depend\nboth on space and time. Our task is to design, under a\nfixed field magnitude H, an optimal field configuration\n/vectorH(x,t) to increase the DW velocity as much as possible.\nFrom Eqs. (8), we find a manifold of solutions of specific\nspace-time field configurations described by a parameter\nu,\nHr(x,t) =Hcosθ, H θ(x,t) =−Hsinθ//radicalbig\n1+u2,\nHφ(x,t) =−uHsinθ//radicalbig\n1+u2.(10)\nThe velocities vand˙φreads\nv=|γ|∆H\n1+α2α+u√\n1+u2,˙φ=|γ|H\n1+α21−αu√\n1+u2.(11)\nThe previous static field case is recovered for u= 0. The\nmaximum of the velocity vmwith regard to uus reached\nforu= 1/α,\nvm=|γ|∆H√\n1+α2, (12)\nwhere the angular velocity is zero, ˙φ= 0. On the other\nhand,˙φattains a maximum for u=−α, where, in turn,\nthe linear velocity vanishes. In Fig. 2 we have plotted3\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s61/s50/s48/s110/s109\n/s72/s61/s49/s48/s48/s79/s101\n/s32/s32/s118/s32/s40/s109/s47/s115/s41\n/s117/s32 /s61/s48/s46/s49\n/s32 /s61/s48/s46/s50\n/s32 /s61/s48/s46/s53\n/s32 /s61/s48/s46/s56\nFIG. 2: (Color online) The DW propagation velocity vver-\nsus the parameter uat the different damping values α=\n0.1,0.2,0.5,0.8. The other parameters are chosen as ∆ =\n20nmandH= 100Oe.\nthe dependence of the velocity on the parameter ufor\ndifferent damping strengths and typical values for the\nDW width ∆ and the magnitude Hof the external field.\nTo understand the physical meaning of the maximum\nvelocityvm, we note that, according to Eqs. (8), the field\ncomponents HθandHφare required to be proportional\nto sinθto ensure the constant velocity under the rigid\nDW approximation. Moreover,at u= 1/αwe haveHθ=\nαHφ, and from the identity\n(αHθ+Hφ)2+(αHφ−Hθ)2= (1+α2)(H2−H2\nr),(13)\nwe conclude that the term ( αHθ+Hφ) is maximal re-\nsulting in a maximal velocity according to Eqs. (8). As\na result, the new velocity under the optimal field pulse is\nlargerby a factor of vm/v=√\n1+α2/α≈1/αcompared\nto a constant field with the same field magnitude. To\ngive a practical example, the typical value for the damp-\ning parameter in permalloy is α= 0.01 which results in\nan increase of the DW velocity by a factor of 100.\nIt is instructive to also analyze the optimal field pulse\naccording to Eq. (10) with u= 1/αin its cartesian com-\nponents,\nHx(x,t) =Hsin2θ(1−α//radicalbig\n1+α2)/2,\nHy(x,t) =−Hsinθ//radicalbig\n1+α2, (14)\nHz(x,t) =H(cos2θ+αsin2θ//radicalbig\n1+α2),\nwhereθfollowsthewave-likemotiontanθ(x,t)\n2= exp(x\n∆−\n|γ|H√\n1+α2t). In Fig. 3 we plotted these quantities at t= 0\naround the DW center where the main spatial variation\nof the pulse occurs. Note that the space-dependent field\ndistribution should move with the same speed vmsyn-\nchronizedwiththeDWpropagation. NeartheDWcenter\nthe components HxandHzare (almost) zero whereas a\nlargetransversecomponent Hyis required to achievefast\nDW propagation. Qualitatively speaking, the transverse\nfield causes a precessionof the magnetization resulting inits reversal. This finding is consistent with recent micro-\nmagnetic simulations showing that the DW velocity can\nbe largely increased by applying an additional transverse\nfield[19].\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48\n/s61/s48/s46/s48/s49\n/s61/s50/s48/s110/s109\n/s72/s61/s49/s48/s48/s79/s101\n/s32/s32/s72\n/s120/s44/s121/s44/s122/s32/s40/s79/s101/s41\n/s120/s32/s40/s110/s109/s41/s32/s72\n/s120\n/s32/s72\n/s121 \n/s32/s72\n/s122\nFIG. 3: (Color online) The x,y,zcomponents of the optimal\nfield pulse. The parameters are chosen as α= 0.01, ∆ =\n20nmfor permalloy[6]. The field magnitude is H= 100Oe.\nIt is also interesting to study the energy variation un-\nder the optimal field pulse,\ndE\ndt=−µ0/integraldisplay+∞\n−∞dx/parenleftBigg\n∂/vectorM\n∂t·/vectorHt+/vectorM·∂/vectorH\n∂t/parenrightBigg\n≡Pα+Ph.\n(15)\nThe first term Pαis the intrinsic damping power due to\nall kinds of damping mechanisms described by the phe-\nnomenologicalparameter α. According to the LLG equa-\ntionPα=−µ0α\n|γ|Ms/integraltext+∞\n−∞dx(∂M\n∂t)2is always negative[14],\nimplying an energy loss. Phis the external power due to\nthe time-dependent external field. From Eq. (11), both\npowers are obtained as\nPα=−2α\n1+α2µ0|γ|Ms∆H2, (16)\nPh=2α(√\n1+u2−1)−2u\n(1+α2)√\n1+u2µ0|γ|Ms∆H2,(17)\nsuch that the total energy change rate is\ndE\ndt=−2µ0MsHv=−2(α+u)µ0|γ|Ms∆H2\n(1+α2)√\n1+u2.(18)\nNote that the intrinsic damping power is independent of\nthe parameter uand always negative, whereas the total\nenergy change rate is proportional to the negative DW\nvelocity. Thus, for positive velocities ( u >−α) the total\nmagnetic energy decreases while it grows for negative ve-\nlocities (u <−α). In the former case energy is absorbed\nby the external field source while in the latter case the\nfield source provides energy to the system. The optimal\nfield source helps to rapidly release or gain magnetic en-\nergywhichisessentialforfastDWmotion. Thisaspectis\nverydifferent from the reversalof a Stoner particle where\nthe time-dependent field is always needed to provide en-\nergy to the system to overcome the energy barrier[14].4\nMoreover, our new strategy of employing space-\ndependent field pulses can also be applied to uniaxial\nanisotropies of arbitrary type: Let w(θ) be the uniaxial\nmagnetic energy density. The static DW solution in the\nabsence of an external field reads x=/integraltext\nχ−1(θ)dθ, where\nχ(θ) =/radicalbig\n[w(θ)−w0]/J. (19)\nHerew0is the minimum energy density for magneti-\nzation along the easy axis. By performing analogous\nsteps as before, we obtain the the optimal velocity as\nvm=|γ|H√\n1+α2χmax, whereχmaxdenotes the maximum of\nχ(θ) throughout all θ.\nOn the other hand, our approach is not straightfor-\nwardlyextended tothecaseofamagneticwirewith biax-\nial anisotropy. To see this, consider, a biaxial anisotropy\nεi=−KM2\nz+K′M2\nxwhere the coefficients K,K′cor-\nrespond to the easy and hard axis, respectively[12]. The\nLLG equations read\nΓ˙θ=α/parenleftbigg\nHθ−KMs\nµ0sin2θ−K′Ms\nµ0sin2θcos2φ\n+2J\nµ0Ms∂2θ\n∂x2/parenrightbigg\n+Hφ+K′Ms\nµ0sinθsin2φ,\nΓsinθ˙φ=αHφ−Hθ+KMs\nµ0sin2θ−2J\nµ0Ms∂2θ\n∂x2\n+K′Ms\nµ0sin2θcos2φ+αK′Ms\nµ0sinθsin2φ.(20)\nLet us assume φ(x,t) =φ0is a constant deter-\nmined by the applied field. Substituting the travelling-\nwaveansatztanθ(x,t)\n2= exp/parenleftbigx−vt\n∆/parenrightbig\n, where now ∆ =/radicalbig\nJ/(K+K′cos2φ0)/Ms, into Eqs. (20) we obtain\nΓsinθv=−∆(αHθ+Hφ+K′Mssinθsin2φ0/µ0),\n(21)\nαK′Mssinθsin2φ0/µ0+(αHφ−Hθ) = 0.(22)\nFor a static field along z-axis Hθ=−Hsinθ,Hφ= 0, the\nsolution is just the Walker’s result v=|γ|∆H/α(Note\nhere ∆ also depends on H)[12]. To implement our new\nstrategy, we need to find the maximum of the right-hand\nside of Eq. (21) under two constraints of Eq. (22) and\nEq. (13) with HθandHφbeing proportional to sin θ.\nThe unique solution to this problem is indeed a constant\nfield along the z-axis which is thus the optimal field con-\nfiguration.\nIn summary, our theory is general and can be applied\nto a magnetic nanowire with a uniaxial anisotropy which\ncan be from shape, magneto-crystalline or the dipolar\ninteraction. The experimental challenge of our proposal\nis obviously the generation of a field pulse focused on\nthe DW region and synchronized with its motion. How-\never, the field sourcesynchronizationvelocity can be pre-\ncalculated from the material parameters. As for the re-\nquired localized field (See Fig. 3), we propose to employa ferromagnetic scanning tunneling microscope (STM)\ntip to produce a localized field perpendicular to the wire\naxis[20] and use a localized current to produce an Oer-\nsted field along the wire axis[21]. Moreover, such re-\nquired localized fields may also be produced by nano-\nferromagnetswithstrongferromagnetic(orantiferromag-\nnetic) coupling to the nanowire. We also point out that,\nalthough the field source typically does not consume en-\nergy but gain energy from the magnetic nanowire, the\npulse source may still require excess energy to overcome\neffects such as defects pinning, which is not included in\nour model. At last, the generalization of the strategy be-\nyond the rigid DW approximation, and to DW motion\ninduced by spin-polarized current will also be attractive\ndirection of future research.\nZ.Z.S. thanks the Alexander von Humboldt Founda-\ntion (Germany) for a grant. This work has been sup-\nported by Deutsche Forschugsgemeinschaft via SFB 689.\n[1] R. P. Cowburn, Nature (London) 448, 544 (2007).\n[2] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[3] D. A. Allwood, et al., Science 309, 1688 (2005).\n[4] T. Ono, et al., Science, 284, 468 (1999).\n[5] D. Atkinson, et al., Nature Mater. 2, 85 (2003).\n[6] G. S. D. Beach, et al., Nature Mater. 4, 741 (2005); G.\nS. D. Beach, et al., Phys. Rev. Lett. 97, 057203 (2006);\nJ. Yang, et al., Phys. Rev.B 77014413 (2008).\n[7] M. Klaui, et al., Phys. Rev. Lett. 94, 106601 (2005).\n[8] M. Hayashi, et al., Phys. Rev. Lett. 96, 197207 (2006);\nL. Thomas, et al., Nature (London) 443, 197 (2006); M.\nHayashi, et al., Phys. Rev. Lett. 98, 037204 (2007).\n[9] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[10] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[11] A. Thiaville, et al., Europhys. Lett. 69, 990 (2005).\n[12] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[13] X.R.Wang, P.Yan, andJ.Lu, Europhys.Lett. 86, 67001\n(2009); X. R. Wang, et al., Ann. Phys. (N.Y.) 324, 1815\n(2009).\n[14] Z. Z. Sun and X.R. Wang, Phys. Rev. Lett. 97, 077205\n(2006); Phys. Rev. B 73, 092416 (2006); ibid.74, 132401\n(2006).; X. R. Wang and Z. Z. Sun, Phys. Rev. Lett. 98,\n077201 (2007).\n[15] X. R. Wang, et al., Europhys. Lett. 84, 27008 (2008).\n[16] Z. Z. Sun and X. R. Wang, Phys. Rev. B 71, 174430\n(2005), and references therein.\n[17] M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008).\n[18] A. P. Malozemoff and J. C. Slonczewski, Domain Walls\nin Bubble Materials , (Academic, New York, 1979).\n[19] M. T. Bryan, et al., J. Appl. Phys. 103, 073906 (2008).\n[20] T. Michlmayr, et al., J. Appl. Phys. 99, 08N502 (2006).\n[21] T. Michlmayr, et al., J. Phys. D: Appl. Phys. 41, 055005\n(2008)."    },    {        "title": "1810.07020v4.Superfluid_spin_transport_in_ferro__and_antiferromagnets.pdf",        "content": "Super\ruid spin transport in ferro- and antiferromagnets\nE. B. Sonin\nRacah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel\n(Dated: March 25, 2019)\nThis paper focuses on spin super\ruid transport, observation of which was recently reported in\nantiferromagnet Cr 2O3[Yuan et al. , Sci. Adv. 4, eaat1098 (2018)]. This paper analyzes the role of\ndissipation in transformation of spin current injected with incoherent magnons to a super\ruid spin\ncurrent near the interface where spin is injected. The Gilbert damping parameter in the Landau{\nLifshitz{Gilbert theory does not describe dissipation properly, and the dissipation parameters are\ncalculated from the Boltzmann equation for magnons scattered by defects. The two-\ruid theory is\ndeveloped similar to the two-\ruid theory for super\ruids. This theory shows that the in\ruence of\ntemperature variation in bulk on the super\ruid spin transport (bulk Seebeck e\u000bect) is weak at low\ntemperatures. The scenario that the results of Yuan et al. are connected with the Seebeck e\u000bect at\nthe interface between the spin detector and the sample is also discussed.\nThe Landau criterion for an antiferromagnet put in a magnetic \feld is derived from the spectrum\nof collective spin modes. The Landau instability starts in the gapped mode earlier than in the\nGoldstone gapless mode, in contrast to easy-plane ferromagnets where the Goldstone mode becomes\nunstable. The structure of the magnetic vortex in the geometry of the experiment is determined.\nThe vortex core has the skyrmion structure with \fnite magnetization component normal to the\nmagnetic \feld. This magnetization creates stray magnetic \felds around the exit point of the vortex\nline from the sample, which can be used for experimental detection of vortices.\nI. INTRODUCTION\nThe concept of spin super\ruidity is based on the anal-\nogy of the equations of magnetodynamics with the equa-\ntions of super\ruid hydrodynamics.1. The analogy led to\nthe suggestion that in magnetically ordered media persis-\ntent spin currents are possible, which are able to trans-\nport spin on macroscopical distances without essential\nlosses.2\nThe phenomenon of spin super\ruidity has been dis-\ncussed for several decades.2{15We de\fne the term super-\n\ruidity in its original meaning known from the times of\nKamerlingh Onnes and Kapitza: transport of some phys-\nical quantity (mass, charge, or spin) over macroscopical\ndistances without essential dissipation. This requires a\nconstant or slowly varying phase gradient at macroscopic\nscale with the total phase variation along the macroscopic\nsample equal to 2 \u0019multiplied by a very large number.\nSpin super\ruidity assumes the existence of spin current\nproportional to the gradient of the phase (spin super-\ncurrent). In magnetically ordered media the phase is an\nangle of rotation in spin space around some axis (further\nin the paper the axis z). In contrast to the dissipative\nspin-di\u000busion current proportional to the gradient of spin\ndensity, the spin supercurrent is not accompanied by dis-\nsipation.\nSpin super\ruidity require special topology of the order\nparameter space. This topology is realized at the pres-\nence of the easy-plane magnetic anisotropy, which con-\n\fnes the magnetization of the ferromagnet or sublattice\nmagnetizations of the antiferromagnet in an easy plane.\nIn this case one may expect that the current state is sta-\nble with respect to phase slips, which lead to relaxation of\nthe supercurrent. In the phase slip event a vortex with\n2\u0019phase variation around it crosses streamlines of thesupercurrent decreasing the total phase variation across\nstreamlines by 2 \u0019. The concept of the phase slip was\nintroduced by Anderson16for super\ruid4He and later\nwas used in studying spin super\ruidity.2,3\nPhase slips are suppressed by energetic barriers for vor-\ntex expansion. But these barriers disappear when phase\ngradients reach critical values determined by the Landau\ncriterion. The physical meaning of the Landau criterion\nis straightforward: the current state becomes unstable\nwhen there are elementary excitations with negative en-\nergy. So, to check the Landau criterion one must know\nthe full spectrum of collective modes.\nSometimes any presence of spin current proportional\nto the phase gradient is considered as a manifestation\nof spin super\ruidity.17,18However, spin current propor-\ntional to the spin phase gradient is ubiquitous and ex-\nists in any spin wave or domain wall, also in the ground\nstate of disordered magnetic media. In all these cases\nthe total variation of the phase is smaller, or on the or-\nder of\u0019. Connecting these cases with spin super\ruid-\nity makes this phenomenon trivial and already observed\nin old experiments on spin waves in the middle of the\n20th Century. One may call the supercurrent produced\nby the total phase variation of the order or less than\n2\u0019microscopical supercurrent, in contrast to persistent\nmacroscopical supercurrents able to transport spin over\nmacroscopical distances.\nThe analogy with usual super\ruids is exact only if\nthe spin space is invariant with respect to spin rotation\naround the hard axis normal to the easy plane. Then\nthere is the conservation law for the spin component\nalong the hard axis. In reality this invariance is bro-\nken by in-plane anisotropy. But this anisotropy is usu-\nally weak, because it originates from the spin-orbit in-\nteraction, which is relativistically small compared to thearXiv:1810.07020v4  [cond-mat.mes-hall]  22 Mar 20192\nexchange interaction, i.e., inversely proportional to the\nspeed of light.19Macroscopical spin supercurrents are\nstill possible if the energy of supercurrents exceeds the\nin-plane anisotropy energy. Thus, one cannot observe\nmacroscopical spin supercurrents not only at large cur-\nrents as in usual super\ruids, but also at small currents.2\nFrom the time when the concept of spin super\ruidity\n(in our de\fnition of this term) was suggested2, it was\ndebated about whether the super\ruid spin current is a\n\\real\" transport current. As a response to these con-\ncerns, in Ref 2 a Gedanken (at that time) experiment\nfor demonstration of reality of super\ruid spin transport\nwas proposed. The spin is injected to one side of a mag-\nnetically ordered layer of thickness dand spin accumula-\ntion is checked at another side. If the layer is not spin-\nsuper\ruid, then the spin is transported by spin di\u000busion.\nThe spin current and the spin density exponentially de-\ncay at the distance of the spin di\u000busion length, and the\ndensity of spin accumulated at the other side decreases\nexponentially with growing distance d. However, if the\nconditions for spin super\ruidity are realized in the layer,\nthen the super\ruid spin current decays much slower, and\nthe accumulated spin density at the side opposite to the\nside where the spin is injected is inversely proportional\ntod+C, whereCis some constant.\nThe interest to long-distance spin transport, especially\nto spin super\ruid transport, revived recently. Takei and\nTserkovnyak7carried out a microscopic analysis of in-\njection of spin to and ejection of spin out of the spin-\nsuper\ruid medium in an easy-plane ferromagnet justify-\ning the aforementioned scheme of super\ruid spin trans-\nport. Takei et al.8extended this analysis to easy-plane\nantiferromagnets. Finally Yuan et al.20were able to real-\nize the suggested experiment in antiferromagnetic Cr 2O3\nobserving spin accumulation inversely proportional to the\ndistance from the interface where spin was injected into\nCr2O3.\nPreviously Borovik-Romanov et al.21reported evi-\ndence of spin super\ruidity in the Bphase of super\ruid\n3He. They detected phase slips in a channel with su-\nper\ruid spin current close to its critical value. It was\nimportant evidence that persistent spin currents are pos-\nsible. But real long-distance transportation of spin by\nthese currents was not demonstrated. Moreover, it is\nimpossible to do in the nonequilibrium magnon Bose{\nEinstein condensate, which was realized in the Bphase\nof3He super\ruid6and in yttrium-iron-garnet magnetic\n\flms.22The nonequilibrium magnon Bose{Einstein con-\ndensate requires pumping of spin in the whole bulk for\nits existence. In the geometry of the aforementioned spin\ntransport experiment this would mean that spin is per-\nmanently pumped not only by a distant injector but also\nall the way up the place where its accumulation is probed.\nThus, the spin detector measures not only spin coming\nfrom a distant injector but also spin pumped close to\nthe detector. Therefore, the experiment does not prove\nthe existence of long-distance spin super\ruid transport.\nThere were also reports on experimental detection ofspin super\ruidity in magnetically ordered solids17,18, but\nthey addressed microscopical spin supercurrent.23As ex-\nplained above, \\super\ruidity\" connected with such cur-\nrents was well proved by numerous old experiments on\nspin waves and does not need new experimental con\fr-\nmations. The work of Yuan et al.20was the \frst report\non long-distance super\ruid spin transport with spin ac-\ncumulation decreasing with distance from the injector as\nexpected from the theory. Long distance super\ruid spin\ntransport was also recently reported in a graphene quan-\ntum antiferromagnet.24\nThe experiment on super\ruid spin transport20has put\nto rest another old dispute about the spin super\ruidity\nconcept. At studying spin super\ruidity in the Bphase\nof super\ruid3He, it was believed4that spin super\ruidity\nis possible only if there are mobile carriers of spin and\na counter\row of carriers with opposite spins transports\nspin. If so, then spin super\ruidity is impossible in insu-\nlators. Moreover, Shi et al.25argued that it is a critical\n\raw of spin-current de\fnition if it predicts spin currents\nin insulators. Since Cr 2O3is an insulator the experiment\nof Yuan et al.20rules out this presumption.\nBoosted by the super\ruid spin transport experiment20\nthis paper addresses some issues deserving further inves-\ntigation. It is especially needed because Lebrun et al.26\nmade an experiment in an antiferromagnetic iron oxide\nsimilar to that of Yuan et al.20and observed similar de-\npendence of spin accumulation on the distance from the\ninjector. However, Lebrun et al.26explain it not by spin\ntransport from the distant injector but by the Seebeck\ne\u000bect at the detector, which is warmed by the heat \row\nfrom the injector. We shall compare these two interpre-\ntations in Sec. VIII.\nWe analyzed the role of dissipation in the super\ruid\nspin transport. A widely used approach to address dis-\nsipation in magnetically ordered solids is the Landau{\nLifshitz{Gilbert (LLG) theory with the Gilbert damp-\ning parameter. But we came to the conclusion that\nthe Gilbert damping does not provide a proper descrip-\ntion of dissipation processes in easy-plane ferromagnets.\nThe Gilbert damping is described by a single parame-\nter, which scales alldissipation processes independently\nfrom whether they do violate the spin conservation law,\nor do not. Meanwhile, the processes violating the spin\nconservation law, the Bloch spin relaxation in particular,\noriginate from spin-orbit interaction and must be rela-\ntivistically small as explained above. This requires the\npresence of a small factor in the intensity of the Bloch\nspin relaxation, which is absent in the Gilbert damping\napproach. So we determined the dissipation parameters\nfrom the Boltzmann equation for magnons scattered by\ndefects. Dissipation is possible only in the presence of\nthermal magnons, and we developed the two-\ruid theory\nfor easy-plane ferromagnets similar to that in super\ruid\nhydrodynamics for the clamped regime, when the gas of\nquasiparticles cannot freely drift without dissipation in\nthe laboratory frame.\nAs mentioned above, to check the Landau criterion for3\nsuper\ruidity, one must calculate the spectrum of collec-\ntive modes and check whether some modes have nega-\ntive energies. The Landau critical gradient is determined\nby easy-plane crystal anisotropy and was known qualita-\ntively both for ferro- and antiferromagnets long ago.2For\neasy-plane ferromagnets the Landau critical gradient was\nrecently determined quantitatively from the spin-wave\nspectrum in the analysis of ferromagnetic spin-1 BEC\nof cold atoms.15But Cr 2O3, which was investigated in\nthe experiment,20has no crystal easy-plane anisotropy,\nand an \\easy plane\" necessary for spin super\ruidity is\nproduced by an external magnetic \feld. The magnetic\n\feld should exceed the spin-\rop \feld, above which mag-\nnetizations of sublattices in antiferromagnet are kept in\na plane normal to the magnetic \feld. We analyze the\nmagnon spectrum in the spin current states in this situ-\nation. The analysis has shown that the Landau critical\ngradient is determined by the gapped mode, but not by\nthe Goldstone gapless mode as in the cases of easy-plane\nferromagnets.\nWithin the two-\ruid theory the role of spatial temper-\nature variation was investigated. This variation produces\nthe bulk Seebeck e\u000bect. But the e\u000bect is weak because it\nis proportional not to the temperature gradient, but to a\nhigher (third) spatial derivative of the temperature.\nThe transient processes near the interface through\nwhich spin is injected were also discussed. Conversion\nfrom spin current of incoherent thermal magnons to co-\nherent (super\ruid) spin transport is among these pro-\ncesses. The width of the transient layer (healing length),\nwhere formation of the super\ruid spin current occurs,\ncan be determined by di\u000berent scales at di\u000berent condi-\ntion. But at low temperatures it is apparently not less\nthan the magnon mean-free-path.\nIn reality the decay of super\ruid currents starts at val-\nues less than the Landau critical value via phase slips\nproduced by magnetic vortices. The di\u000berence in the\nspectrum of collective modes in ferro- and antiferromag-\nnets leads to the di\u000berence in the structure of magnetic\nvortices. In the past magnetic vortices were investi-\ngated mostly in ferromagnets (see Ref. 15 and references\ntherein). The present work analyzes a vortex in an anti-\nferromagnet. The vortex core has a structure of skyrmion\nwith sublattice magnetizations deviated from the direc-\ntion normal to the magnetic \feld. At the same time\ninside the core the total magnetization has a component\nnormal to the magnetic \feld. In the geometry of the\nCr2O3experiment this transverse magnetization creates\nsurface magnetic charges at the point of the exit of the\nvortex line from the sample. Dipole stray magnetic \felds\nproduced by these charges hopefully can be used for de-\ntection of magnetic vortices experimentally.\nSection II reminds the phenomenological model of\nRef. 2 describing the spin di\u000busion and super\ruid spin\ntransport. Section III reproduces the derivation of the\nspectrum of the collective spin mode and the Landau\ncriterion in a spin current state of an easy-plane ferro-\nmagnet known before15. This is necessary for compari-son with the spectrum of the collective spin modes and\nthe Landau criterion in a spin current state of an easy-\nplane antiferromagnet derived in Sec. IV. Thus, Sec. III,\nas well as Sec. II, do not contain new results, but were\nadded to the paper to make it self-su\u000ecient and more\nreadable. In Sec. V we address two-\ruid e\u000bects and dis-\nsipation parameters (spin di\u000busion and second viscosity\ncoe\u000ecients) deriving them from the Boltzmann equation\nfor magnons. The section also estimates the bulk See-\nbeck e\u000bect and shows that it is weak. Section VI ana-\nlyzes the transient layer near the interface through which\nspin is injected and where the bulk super\ruid spin cur-\nrent is formed. Various scales determining the width of\nthis layer (healing length) are discussed. In Sec. VII the\nskyrmion structure of the magnetic vortex in an anti-\nferromagnets is investigated. The concluding Sec. VIII\nsummarizes the results of the work and presents some\nnumerical estimations for the antiferromagnetic Cr 2O3\ninvestigated in the experiment. The Appendix analyzes\ndissipation in the LLG theory with the Gilbert damping.\nIt is argued that this theory predicts dissipation coe\u000e-\ncients incompatible with the spin conservation law.\nII. SUPERFLUID SPIN TRANSPORT VS SPIN\nDIFFUSION\nHere we remind the simple phenomenological model of\nspin transport suggested in Ref. 2 (see also more recent\nRefs. 5, 7, and 8). The equations of magnetodynamics\nare\ndMz\ndt=\u0000r\u0001J\u0000M0\nz\nT1; (1)\nd'\ndt=\u0000\rM0\nz\n\u001f+\u0010r2': (2)\nHere\u001fis the magnetic susceptibility along the axis z,\n'is the angle of rotation (spin phase) in the spin space\naround the axis z, andM0\nz=Mz\u0000\u001fHis a nonequilib-\nrium part of the magnetization density along the mag-\nnetic \feldHparallel to the axis z. The time T1is the\nBloch time of the longitudinal spin relaxation. The term\n/r2'in Eq. (2) is an analog of the second viscosity in\nsuper\ruid hydrodynamic.27,28The magnetization density\nMzand the magnetization current Jdi\u000ber from the spin\ndensity and the spin current by sign and by the gyromag-\nnetic factor \r. Nevertheless, we shall call the current J\nthe spin current to stress its connection with spin trans-\nport. The total spin current J=Js+Jdconsists of the\nsuper\ruid spin current\nJs=Ar'; (3)\nand the spin di\u000busion current\nJd=\u0000DrMz: (4)4\nJzxJLzxSpin injection\nSpin injectionSpin injectionMedium withoutspin superfluidityMedium withspin superfluiditymxzmz\n0\na)\nb)Spin detection\ndyxzPtPtCr2O3c)H\nc)PtPtCr2O3Hzdxy\nFIG. 1. Long distance spin transport. (a) Spin injection to\na spin-nonsuper\ruid medium. (b) Spin injection to a spin-\nsuper\ruid medium. (c) Geometry of the experiment by Yuan\net al.20. Spin is injected from the left Pt wire and \rows along\nthe Cr 2O3\flm to the right Pt wire, which serves as a detector.\nThe arrowed dashed line shows a spin-current streamline. In\ncontrast to (a) and (b), the spin current is directed along\nthe same axis zas a magnetization parallel to the external\nmagnetic \feld H.\nThe pair of the hydrodynamical variables ( Mz;') is a\npair of conjugate Hamiltonian variables analogous to\nthe pair \\particle density{super\ruid phase\" in super\ruid\nhydrodynamics.1\nThere are two kinds of spin transport illustrated in\nFig. 1. In the absence of spin super\ruidity ( A= 0) there\nis no super\ruid current. Equation (2) is not relevant, and\nEq. (1) describes pure spin di\u000busion [Fig. 1(a)]. Its solu-\ntion, with the boundary condition that the spin current\nJ0is injected at the interface x= 0, is\nJ=Jd=J0e\u0000x=L d; M0\nz=J0r\nT1\nDe\u0000x=L d;(5)\nwhere\nLd=p\nDT1 (6)\nis the spin-di\u000busion length. Thus the e\u000bect of spin injec-\ntion exponentially decays at the scale of the spin-di\u000busion\nlength.However, if spin super\ruidity is possible ( A6= 0), the\nspin precession equation (2) becomes relevant. As a re-\nsult of it, in a stationary state the magnetization M0\nz\ncannot vary in space (Fig. 1b) since according to Eq. (2)\nthe gradient rM0\nzis accompanied by the linear in time\ngrowth of the gradient r'. The requirement of constant\nin space magnetization Mzis similar to the requirement\nof constant in space chemical potential in super\ruids, or\nthe electrochemical potential in superconductors. As a\nconsequence of this requirement, spin di\u000busion current is\nimpossible in the bulk since it is simply \\short-circuited\"\nby the super\ruid spin current. Only in AC processes\nthe oscillating spin injection can produce an oscillating\nbulk spin di\u000busion current coexisting with an oscillating\nsuper\ruid spin current.\nIn the super\ruid spin transport the spin current can\nreach the other boundary opposite to the boundary where\nspin is injected. We locate it at the plane x=d. As a\nboundary condition at x=d, one can use a phenomeno-\nlogical relation connecting the spin current with the mag-\nnetization: Js(d) =M0\nzvd, wherevdis a phenomenologi-\ncal constant. This boundary condition was derived from\nthe microscopic theory by Takei and Tserkovnyak7. To-\ngether with the boundary condition Js(0) =J0atx= 0\nthis yields the solution of Eqs. (1) and (2):\nM0\nz=T1\nd+vdT1J0; Js(x) =J0\u0012\n1\u0000x\nd+vdT1\u0013\n:(7)\nThus, the spin accumulated at large distance dfrom the\nspin injector slowly decreases as the inverse distance 1 =d\n[Fig. 1(b)], in contrast to the exponential decay /e\u0000d=Ld\nin the spin di\u000busion transport [Fig. 1(a)].\nIn Figs. 1(a) and 1(b) the spin \rows along the axis\nx, while the magnetization and the magnetic \feld are\ndirected along the axis z. In the geometry of the experi-\nment of Yuan et al.20the spin \rows along the magnetiza-\ntion axiszparallel to the magnetic \feld. This geometry is\nshown in Fig. 1c. The di\u000berence between two geometries\nis not essential if spin-orbit coupling is ignored. In this\nsection we chose the geometry with di\u000berent directions of\nthe spin current and the magnetization in order to stress\nthe possibility of the independent choice of axes in the\nspin and the con\fgurational spaces. But in Sec. VII ad-\ndressing a vortex in an antiferromagnet we shall switch\nto the geometry of the experiment because in this case\nthe di\u000berence between geometries is important.\nWithout dissipation-connected terms, the phenomeno-\nlogical theory of this section directly follows from the\nLLG theory. For ferromagnets the LLG equation is\ndM\ndt=\r[Heff\u0002M]; (8)\nwhere\nHeff=\u0000\u000eH\n\u000eM=\u0000@H\n@M+rj@H\n@rjM(9)\nis the e\u000bective \feld determined by the functional deriva-\ntive of the Hamiltonian H. For a ferromagnet with uni-5\naxial anisotropy the Hamiltonian is\nH=GM2\nz\n2+AriM\u0001riM\u0000MzH: (10)\nHereHis an external constant magnetic \feld parallel to\nthe axisz, and the exchange constant Adetermines sti\u000b-\nness with respect to deformations of the magnetization\n\feld. In the case of easy-plane anisotropy the anisotropy\nparameter Gis positive and coincides with the inverse\nsusceptibility: G= 1=\u001f.\nSince the absolute value Mof the magnetization is\na constant, one can describe the 3D magnetization vec-\ntorMonly by two Hamiltonian conjugate variables: the\nmagnetization zcomponent Mzand the angle 'of rota-\ntion around the zaxis. Then the LLG theory yields two\nequations\n_Mz=\u0000r\u0001Js; (11)\n_'=\u0000\r\u0016; (12)\nwith the Hamiltonian in new variables\nH=M2\nz\n2\u001f+AM2\n?r'2\n2+AM2(rMz)2\n2M2\n?\u0000MzH: (13)\nHereM?=p\nM2\u0000M2z, and the spin \\chemical poten-\ntial\" and the super\ruid spin current are\n\u0016=\u000eH\n\u000eMz=@H\n@Mz\u0000rj@H\n@rjMz;Js=\r@H\n@r':(14)\nAfter substitution of explicit expressions for functional\nderivatives of the Hamiltonian (13) the equations become\n_Mz\n\r=\u0000r\u0001(AM2\n?r'); (15)\n_'\n\r=\u0000Mz\u00141\n\u001f\u0000A(r')2\u0000AM2(rMz)2\nM4\n?\u0015\n+AM2\nM2\n?r2Mz+H: (16)\nThe equations (1) and (2) without dissipation terms fol-\nlow from Eqs. (15) and (16) after linearization with re-\nspect to small gradients r'and nonequilibrium magne-\ntizationM0\nz=Mz\u0000\u001fHand ignoring the dependence\nof the spin chemical potential \u0016onrMz. ThenA=\n\rAM2\n?, andM?is determined by its valuep\nM2\u0000\u001f2H2\nin the equilibrium.\nIII. COLLECTIVE MODES AND THE LANDAU\nCRITERION IN EASY-PLANE FERROMAGNETS\nTo check the Landau criterion one should know the\nspectrum of collective modes. In an easy-plane ferromag-\nnet the collective modes (spin waves) are determined byEqs. (15) and (16) linearized with respect to weak pertur-\nbations of stationary states. Further the angle variable \u0012\nwill be introduced instead of the variable Mz=Msin\u0012.\nLet us consider a current state with constant gradient\nK=r'and constant magnetization\nMz=Msin\u0012=\u001fH\n1\u0000\u001fAK2: (17)\nTo derive the spectrum of collective modes, we consider\nweak perturbations \u0002 and \b of this state: \u0012!\u0012+ \u0002,\n'!'+ \b. Equations (15) and (16) after linearization\nare:\n_\u0002\u00002\rMzAK\u0001r\u0002 =\u0000\rAM cos\u0012r2\b;\n_\b\u00002\rMzAK\u0001r\b =\n\u0000\rMcos\u0012\n\u001f\u0000\n1\u0000\u001fAK2\u0001\n\u0002 +\rAM cos\u0012r2\u0002:(18)\nFor plane waves/eik\u0001r\u0000i!tthese equations describe the\ngapless Goldstone mode with the spectrum:13,15\n(!+w\u0001k)2= ~c2\nsk2: (19)\nHere\n~cs=r\u001f\n~\u001fcs; (20)\n~\u001f=\u001f\n1\u0000\u001fA\u0010\nK2\u0000M2k2\nM2\n?\u0011; (21)\nand\ncs=\rM?s\nA\n\u001f(22)\nis the spin-wave velocity in the ground state without any\nspin current. In this state the spectrum becomes\n!=csks\n1 +\u001fAM2k2\nM2\n?: (23)\nThe velocity\nw= 2\rMzAK; (24)\ncan be called Doppler velocity because its e\u000bect on the\nmode frequency is similar to the e\u000bect of the mass ve-\nlocity on the mode frequency in a Galilean invariant\n\ruid (Doppler e\u000bect). But our system is not Galilean\ninvariant,13and the gradient Kis present also in the\nright-hand side of the dispersion relation (19).\nIn the long-wavelength hydrodynamical limit magnons\nhave the sound-like spectrum linear in k. Quadratic cor-\nrections/k2become important at k\u0018M?=Mp\u001fA[see\nEq. (23)]. These corrections emerge from the terms in6\nthe Hamiltonian, which depend on rMz. So the hydro-\ndynamical approach is valid at scales exceeding\n\u00180=M\nM?p\n\u001fA; (25)\nwhich can be called the coherence length, in analogy with\nthe coherence length in the Gross{Pitaevskii theory for\nBEC. Also in analogy with BEC, the coherence length\ndiverges at M?!0, i.e., at the second-order phase tran-\nsition from the easy-plane to the easy-axis anisotropy.\nThe same scale determines the Landau critical gradient\nand the vortex core radius. Telling about hydrodynamics\nwe bear in mind hydrodynamics of a perfect \ruid without\ndissipation. Later in this paper we shall discuss hydro-\ndynamics with dissipation. In this case the condition\nk\u001c1=\u00180is not su\u000ecient, and an additional restriction\non using hydrodynamics is determined by the mean-free\npath of magnons.\nAccording to the Landau criterion, the current state\nbecomes unstable at small kwhenkis parallel to wand\nthe frequency !becomes negative. This happens at the\ngradientKequal to the Landau critical gradient\nKc=M?p4M2\u00003M?1p\u001fA\u00181\n\u00180: (26)\nSpin super\ruidity becomes impossible at the phase tran-\nsition to the easy-axis anisotropy ( M?= 0). In the oppo-\nsite limit of small Mz\u001cMthe pseudo-Doppler e\u000bect is\nnot important, and the Landau critical gradient Kcis de-\ntermined from the condition that the spin-wave velocity\n~csvanishes at small k:\nKc=1p\u001fA=\rM\n\u001fcs: (27)\nExpanding the Hamiltonian (13) with respect to weak\nperturbations \u0002 and \b up to the second order one obtains\nthe energy of the spin wave mode per unit volume,\nEsw=M?!(k)\n\rp~\u001fAkj\u0002kj2; (28)\nwherej\u0002kj2is the squared perturbation of the angle \u0012\nwith the wave vector kaveraged over the wave period.\nIn the quantum theory the energy density Eswcorre-\nsponds to the magnon density\nn(k)\nV=Esw\n~!(k)=M?j\u0002kj2\n~\rp~\u001fAk; (29)\nwheren(k) is the number of magnons in the plane-wave\nmode with the wave vector kandVis the volume of the\nsample. Summing over the whole kspace, the averaged\nsquared perturbation is\nh\u00022i=X\nkj\u0002kj2=~\rp\nA\nM?Zp\n~\u001fn(k)kd3k\n(2\u0019)3:(30)Further we proceed within the hydrodynamical ap-\nproach neglecting quadratic corrections to the spectrum.\nThere are quadratic in spin-wave amplitudes corrections\nto the spin super\ruid current and to the spin chemical\npotential:\nJsjsw=\u0000\rM?A(M?h\u00022iK+ 2Mzh\u0002r\bi);(31)\n\u0016jsw=\u0000A(Mzh(r\b)2i+ 2M?K\u0001h\u0002r\bi):(32)\nUsing Eq. (30) and the relation\nr\b =\u0002p\u001fAk\nk; (33)\nwhich follows from the equations of motion (18), one ob-\ntains:\nJsjsw=\u0000\u001f2~c3\ns\n\rM2\n?Z\nn(k)\u0012\nK+2\rMz\n\u001fcsk\nk\u0013\nkd3k\n(2\u0019)3;\n(34)\n\u0016jsw=\u0000\u001f~c2\ns\n\rM2\n?Z\nn(k)\u0012\rMz\n\u001fcs+2K\u0001k\nk\u0013\nkd3k\n(2\u0019)3:\n(35)\nIV. COLLECTIVE MODES AND THE LANDAU\nCRITERION IN ANTIFERROMAGNETS\nFor ferromagnetic state of localized spins the deriva-\ntion of the LLG theory from the microscopic Heisenberg\nmodel was straightforward.29The quantum theory of the\nantiferromagnetic state even for the simplest case of a\ntwo-sublattice antiferromagnet, which was widely used\nfor Cr 2O3, is more di\u000ecult. This is because the state\nwith constant magnetizations of two sublattices is not\na well de\fned quantum-mechanical eigenstate.29Never-\ntheless, long time ago it was widely accepted to ignore\nthis complication and to describe the long-wavelength dy-\nnamics by the LLG theory for two sublattices coupled via\nexchange interaction:30\ndMi\ndt=\r[Hi\u0002Mi]; (36)\nwhere the subscript i= 1;2 points out to which sublattice\nthe magnetization Mibelongs, and\nHi=\u0000\u000eH\n\u000eMi=\u0000@H\n@Mi+rj@H\n@rjMi(37)\nis the e\u000bective \feld for the ith sublattice determined by\nthe functional derivative of the Hamiltonian H. For an\nisotropic antiferromagnet the Hamiltonian is\nH=M1\u0001M2\n\u001f+A(riM1\u0001riM1+riM2\u0001riM2)\n2\n+A12rjM1\u0001rjM2\u0000H\u0001(M1+M2):(38)7\nIn the uniform ground state without the magnetic \feld H\nthe two magnetizations are antiparallel, M2=\u0000M1, and\nthe total magnetization M1+M2vanishes. At H6= 0 the\nsublattice magnetizations are canted, and in the uniform\nground state the total magnetization is parallel to H:\nm=M1+M2=\u001fH: (39)\nThe \frst term in the Hamiltonian (38), which determines\nthe susceptibility \u001f, originates from the exchange inter-\naction between spins of two sublattices. This is the sus-\nceptibility normal to the staggered magnetization (anti-\nferromagnetic vector) L=M1\u0000M2. Since in the LLG\ntheory absolute values of magnetizations M1andM2are\n\fxed the susceptibility parallel to Lvanishes.\nIn the uniform state only the uniform exchange en-\nergy/1=\u001fand the Zeeman energy (the \frst and the\nlast terms) are present in the Hamiltonian, which can be\nrewritten as\nH=\u0000L2\u0000m2\n4\u001f\u0000H\u0001m=\u0000M2\n\u001f+m2\n2\u001f\u0000mHm;(40)\nwhereHm= (H\u0001m)=mis the projection of the mag-\nnetic \feld on the direction of the total magnetization m.\nMinimizing the Hamiltonian with respect to the absolute\nvalue of m(at it \fxed direction, i.e., at \fxed Hm) one\nobtains\nH=\u0000M2\n\u001f\u0000\u001fH2\nm\n2=\u0000M2\n\u001f\u0000\u001fH2\n2+\u001fH2\nL\n2;(41)\nwhereHL= (H\u0001L)=Lis the projection of the magnetic\n\feld on the staggered magnetization L. The \frst two\nterms are constant, while the last term plays the role of\nthe easy-plane anisotropy energy con\fning Lin the plane\nnormal to H. 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sha1_base64=\"iO+t6s0e5FdT3FzK2f3PzRc30JA=\">AAAB/XicbVBNSwMxEJ31s9avqkcvwSJ4KrtF0GPRixehgv2AdilJmm1Dk+ySZIWyFH+DVz17E6/+Fo/+E9N2D7b1wcDjvRlm5pFEcGN9/9tbW9/Y3Nou7BR39/YPDktHx00Tp5qyBo1FrNsEGya4Yg3LrWDtRDMsiWAtMrqd+q0npg2P1aMdJyyUeKB4xCm2Tmp1iUT3vWqvVPYr/gxolQQ5KUOOeq/00+3HNJVMWSqwMZ3AT2yYYW05FWxS7KaGJZiO8IB1HFVYMhNms3Mn6NwpfRTF2pWyaKb+nciwNGYsieuU2A7NsjcV//M6qY2uw4yrJLVM0fmiKBXIxmj6O+pzzagVY0cw1dzdiugQa0ytS2hhC5ETl0mwnMAqaVYrgV8JHi7LtZs8nQKcwhlcQABXUIM7qEMDKIzgBV7hzXv23r0P73PeuublMyewAO/rFxoXlX8=</latexit>\nFIG. 2. Angle variables \u0012and\u00120for the case when the both\nmagnetizations are in the plane xz('0='= 0).\nWe introduce the pairs of angle variables \u0012i,'ideter-\nmining directions of the sublattice magnetizations:\nMix=Mcos\u0012icos'i; Miy=Mcos\u0012isin'i;\nMiz=Msin\u0012i:(43)\nThe equations of motion in the angle variables are\ncos\u0012i_\u0012i\n\r=1\nM\u0012@H\n@'i\u0000r@H\n@r'i\u0013\n;\ncos\u0012i_'i\n\r=\u00001\nM\u0012@H\n@\u0012i\u0000r@H\n@r\u0012i\u0013\n: (44)\nIn the further analysis it is convenient to use other angle\nvariables:\n\u00120=\u0019+\u00121\u0000\u00122\n2; \u0012=\u0019\u0000\u00121\u0000\u00122\n2;\n'0='1+'2\n2; '='1\u0000'2\n2: (45)\nIn these variables the Hamiltonian becomes\nH=\u0000M2\n\u001f(cos 2\u00120cos2'\u0000cos 2\u0012sin2')\u00002HM cos\u0012sin\u00120\n+AM2[(1 + cos 2\u00120cos 2\u0012)r'2\n0+r'2\n2\u0000sin 2\u00120sin 2\u0012r'0\u0001r'+r\u00122\n0+r\u00122]\n+A12M2f(cos 2\u0012sin2'+ cos 2\u00120cos2')(r\u00122\n0\u0000r\u00122)\u0000cos 2\u00120+ cos 2\u0012\n2cos 2'(r'2\n0\u0000r'2)\n\u0000sin 2'[sin 2\u0012(r\u00120\u0001r'0+r\u0012\u0001r') + sin 2\u00120(r\u0012\u0001r'0+r\u00120\u0001r')]g: (46)\nThe polar angles \u0012for the staggered magnetization Land\nthe canting angle \u00120are shown in Fig. 2 for the case when\nthe both magnetizations are in the plane xz('0='=\n0).\nIn the uniform ground state \u0012= 0,'= 0,mz=\n2Msin\u00120=\u001fH, while the angle '0is an arbitrary con-stant. Since we consider \felds Hweak compared to the\nexchange \feld, \u00120is always small. In the state with con-\nstant current K=r'0the magnetization along the\nmagnetic \feld is\nmz=\u001fH\n1\u0000\u001fA\u0000K2=2; (47)8\nwhereA\u0006=A\u0006A12.\nIn a weakly perturbed current state small but nonzero\n\u0012and'appear. Also the angles \u00120and'0di\u000ber from\ntheir values in the stationary current state: \u00120!\u00120+ \u0002,\n'0!'0+ \b. Linearization of the nonlinear equations\nof motion with respect to weak perturbations \u0002, \b, \u0012,\nand'yields decoupled linear equations for two pairs of\nvariables (\u0002 ;\b) and (\u0012;'):\n_\u0002\n\r\u0000A\u0000mzK\u0001r\u0002 =\u0000A\u0000M?r2\b;\n_\b\n\r\u0000A\u0000mzK\u0001r\b =\u0000\u0012\n1\u0000\u001fA\u0000K2\n2\u00132M?\n\u001f\u0002\n+(A+A12cos 2\u00120)\ncos\u00120Mr2\u0002;(48)\n_\u0012\n\r\u0000A+mzK\u0001r\u0012\n=\u00002M?\n\u001f\u0000\n1 +\u001fA12K2\u0001\n'+A+M?r2';\n_'\n\r\u0000A+mzcos\u00120K\u0001r'\n=m2\nz\n2\u001fM?(1 +\u001fA12K2)\u0012\u0000A\u0000K2M?\u0012\n\u0000A\u0000A12cos 2\u00120\ncos\u00120Mr2\u0012: (49)\nFor plane waves/eik\u0001r\u0000i!tEq. (48) describes the gapless\nGoldstone mode with the spectrum:\n(!+\rmzA\u0000K\u0001k)2\n=c2\ns\u0014\n1\u0000\u001fA\u0000K2\n2+\u001f(A+A12cos 2\u00120)k2\n2 cos2\u00120\u0015\nk2:(50)\nHere\ncs=\rM?s\n2A\u0000\n\u001f(51)\nis the spin-wave velocity in the ground state without spin\ncurrent. Apart from quadratic corrections k2to the fre-\nquency, the gapless mode in an antiferromagnet does not\ndi\u000ber from that in a ferromagnet, if one replaces in all\nexpressions for the ferromagnet AbyA\u0000=2 and the pa-\nrameterMby 2M.\nEquation (49) describes the gapped mode with the\nspectrum\n(!+\rmzA+K\u0001k)2=\u0012\n1 +\u001fA12K2+\u001fA+k2\n2\u0013\n\u0002\u0014(1 +\u001fA12K2)\r2m2\nz\n\u001f2\u0000c2\nsK2\n+2\r2M2(A\u0000A12cos 2\u00120)k2\n\u001f\u0015\n:(52)Without spin current and neglecting the term /A+k2\nthe spectrum is\n!=s\n\r2m2z\n\u001f2+c2sk2: (53)\nThis spectrum determines a new correlation length\n\u0018=M\nHs\n2A\u0000\n\u001f=cs\n\rH; (54)\nwhich is connected with the easy-plane anisotropy energy\n(42) and determines the wave vector k= 1=\u0018at which\nthe gap and the kdependent frequency become equal.\nApplying the Landau criterion to the gapless mode one\nobtains the critical gradientp\n2=\u001fA\u0000similar to the value\n(27) obtained for a ferromagnet. But in contrast to a fer-\nromagnet where the susceptibility \u001fis connected with\nweak anisotropy energy, in an antiferromagnet the sus-\nceptibility\u001fis determined by a much larger exchange\nenergy and is rather small. As a result, in an antiferro-\nmagnet the gapless Goldstone mode becomes unstable at\nthe very high value of K. But at much lower values of\nKthe gapped mode becomes unstable. According to the\nspectrum (52), the gap in the spectrum vanishes at the\ncritical gradient\nKc=1\n\u0018=\rH\ncs=\rmz\n\u001fcs: (55)\nV. TWO-FLUID EFFECTS AND DISSIPATION\nFROM THE BOLTZMANN EQUATION FOR\nMAGNONS\nKnowledge of the spectrum of collective modes allows\nto derive the dynamical equations at \fnite temperatures\ntaking into account the presence of thermal magnons.\nFurther we follow the procedure of the derivation of the\ntwo-\ruid hydrodynamics in super\ruids.27We address the\nhydrodynamical limit when all parameters ( Mz,K,T)\nof the system slowly vary in space and time.\nWe shall focus on ferromagnets. The equilibrium\nPlanck distribution of magnons in a ferromagnet with\na small spin current /Kis\nnK=1\ne~!(k)=T\u00001\u0019n0(!0)\u00002\u001fc2\nsMz\n\rM2\n?@n0(!0)\n@!0K\u0001k;\n(56)\nwhere!0=cskand\nn0(!0) =1\ne~!0=T\u00001(57)\nis the Planck distribution in the state without spin cur-\nrent.\nIn the theory of super\ruidity the Plank distribution\nof phonons in general depends not only on density and\nsuper\ruid velocity (analogs of our MzandK) but also on9\nthe normal velocity, which characterizes a possible drift of\nthe gas of quasiparticles with respect to the laboratory\nframe of coordinates. This drift is possible because of\nthe Galilean invariance of super\ruids. In our case the\nGalilean invariance is broken by possible interaction of\nmagnons with defects, and in the equilibrium the drift of\nthe quasiparticle gas is impossible. The case of broken\nGalilean invariance, when the normal velocity vanishes,\nwas also investigated for super\ruids in porous media or\nin very thin channels, when the Galilean invariance is\nbroken by interaction with channel walls. It was called\nthe clamped regime.31,32\nSubstituting the Planck distribution (56) into Eqs. (34)\nand (35) one obtains the contribution of equilibrium\nmagnons to the spin current and the spin chemical po-\ntential:\nJsjeq=\r@\n@K=\u0000\u00192\u001f2T4\n30\rM2\n?~3csK\u0012\n1 +16M2\nz\n3M2\n?\u0013\n;(58)\n\u0016jeq=@\n@Mz=\u00192MzT4\n30~3c3sM2\n?; (59)\nwhere\n\n =TZ\nln(1\u0000e\u0000~!(k)=T)d3k\n(2\u0019)3: (60)\nis the thermodynamical potential for the magnon Bose-\ngas. The contribution (58) decreases the super\ruid spin\ncurrent at \fxed phase gradient K, similarly to the de-\ncrease of the mass super\ruid current after replacing the\ntotal mass density by the lesser super\ruid density.\nYuan et al.20used in their experiment very thin \flm at\nlow temperature, when de Broglie wavelength of magnons\nexceeds \flm thickness, and it is useful to give also the\ntwo-\ruid corrections for a two-dimensional case. Repeat-\ning our calculations after replacing integralsR\nd3k=(2\u0019)3\nby integrals WR\nd2k=(2\u0019)2, one obtains:\nJsjeq=\u0000\u0010(3)\u001f2T3\n\u0019W\rM2\n?~2K\u0012\n1 +6M2\nz\nM2\n?\u0013\n; (61)\n\u0016jeq=\u0010(3)MzT3\n\u0019W~2c2sM2\n?; (62)\nwhere the value of the Riemann zeta function \u0010(3) is\n1.202 andWis the \flm thickness.\nThe next step in derivation of the two-\ruid theory at \f-\nnite temperatures is the analysis of dissipation. A widely\nused approach of studying dissipation in magnetically or-\nder systems is the LLG theory with the Gilbert damp-\ning term added. However, this approach is incompatible\nwith the spin conservation law. This law, although being\napproximate, plays a key role in the problem of spin su-\nper\ruidity. Therefore, we derived dissipation parameters\nfrom the Boltzmann equation for magnons postponing\ndiscussion of the LLG theory with the Gilbert damping\nto the Appendix.Dissipation is connected with nonequilibrium correc-\ntions to the magnon distribution. At low temperatures\nthe number of magnons is small, and magnon-magnon\ninteraction is weak. Then the main source of dissipa-\ntion is scattering of magnons by defects. The Boltzmann\nequation with the collision term in the relaxation-time\napproximation is\n_n+@!\n@k\u0001rn\u0000r!\u0001@n\n@k=\u0000n\u0000nK\n\u001c: (63)\nIf parameters, which determine the magnon distribution\nfunctionn, vary slowly in space and time one can substi-\ntute the equilibrium Planck distribution nKinto the left-\nhand side of the Boltzmann equation (63). This yields:\n@n0\n@!_!+@n0\n@T\u0012\n_T+@!\n@k\u0001rT\u0013\n=\u0000n\u0000n0\n\u001c; (64)\nWe consider small gradients Kwhen the di\u000berence be-\ntweennKandn0is not important. But weak depen-\ndence of!onKis important at calculation of _ !. One\ncan see that at the constant temperature Tin any sta-\ntionary state the left-hand side vanishes, and there is\nno nonequilibrium correction to the magnon distribution.\nCorrespondingly, there is no dissipation. This is one more\nillustration that stationary super\ruid currents do not de-\ncay.\nIn nonstationary cases time derivatives are determined\nby the equations of motions. The equations of motion for\nMzandKare not su\u000ecient, and the equation of heat bal-\nance is needed for \fnding _T. In general the heat balance\nequation is rather complicated since it must take into ac-\ncount interaction of magnons with other subsystems, e.\ng., phonons. Instead of it we consider a simpler case,\nwhen magnons are not important in the heat balance,\ni.e., the temperature does not depend on magnon pro-\ncesses. In other words we consider the isothermal regime\nwhen _T= 0. But we allow slow temperature variation in\nspace.\nThe temporal variation of the frequency !emerges\nfrom slow temporal variation of MzandK, and at small\nK\n_!=@!\n@Mz_Mz+@!\n@K_K=\u0000Mz\nM2\n?\u0012\ncsk_Mz+2\u001fc2\ns\n\rk\u0001_K\u0013\n:\n(65)\nThe partial derivatives @!=@Mzand@!=@Kwere deter-\nmined from the spectrum (19), while the time derivatives\nofMzandKwere found from the linearized equations\n(15) and (16) assuming that r'=Kis small and ig-\nnoring gradients of Mzin the right-hand side of Eq. (16),\nwhich are beyond the hydrodynamical limit. Then\n_!=Mz\nM2\n?c2\ns\u0014\u001f\n\rcskr\u0001K+ 2(k\u0001r)Mz\u0015\n: (66)\nEventually the nonequilibrium correction to the magnon10\ndistribution function is\nn0=n\u0000n0=\u0000Mz\nM2\n?cs\u0014\u001fcs\n\rkr\u0001K\n+2(k\u0001r)Mz\u0000M2\n?\nMzT(k\u0001r)T\u0015\n\u001c@n0\n@k(67)\nSubstituting n0into Eqs. (34) and (35) one obtains dis-\nsipation terms in the spin current and the spin chemical\npotential:\nJd=\u0000D\u0012\nrMz\u00001\n2TM2\n?\nMzrT\u0013\n; (68)\n\u0016d=\u0000\u0010\n\rr\u0001K; (69)\nwhere\nD=\u00002\u001f~c3\ns\n3\u00192M2\nz\nM4\n?Z\n\u001c@n0\n@kk4dk;\n\u0010=\u0000\u001f~c3\ns\n2\u00192M2\nz\nM4\n?Z\n\u001c@n0\n@kk4dk: (70)\nIn addition to the spin di\u000busion current, the dissipative\nspin current Jdcontains also the current proportional\nto the temperature gradient. This is the bulk Seebeck\ne\u000bect. Estimation of the integral in these expressions\nrequires knowledge of possible dependence of the relax-\nation time \u001con the energy. Under the assumption that\n\u001cis independent from the energy,\nD=8\u00192\u001c\r2T4M2\nz\n45~3c3sM2\n?; \u0010=2\u00192\u001c\r2T4M2\nz\n15~3c3sM2\n?; (71)\nor for the two-dimensional case,\nD=16\u0010(3)\u001c\r2T3M2\nz\n3\u0019W~2c2sM2\n?; \u0010=4\u0010(3)\u001c\r2T3M2\nz\n\u0019W~2c2sM2\n?:(72)\nAlthough in antiferromagnets the Landau critical gra-\ndient is connected with the gapped mode, at small phase\ngradients the gapless Goldstone mode has lesser energy,\nand at low temperatures most of magnons belong to this\nmode. Since the Goldstone modes in ferromagnets and\nantiferromagnets are similar, our estimation of dissipa-\ntion coe\u000ecients for ferromagnets is valid also for antifer-\nromagnets after replacing AbyA\u0000=2 andMby 2M.\nThe microscopic analysis of this section agrees with\nthe following phenomenological equations similar to the\nhydrodynamical equations for super\ruids in the clamped\nregime:\n_Mz=\u0000r\u0001Js\u0000@R\n@\u0016+r@R\n@r\u0016; (73)\n_'=\u0000\r\u0016+@R\n@(r\u0001Js); (74)where the spin chemical potential and the super\ruid spin\ncurrent,\n\u0016=\u000eF\n\u000eMz;Js=\r@F\n@r'; (75)\nare determined by derivatives of the free energy\nF=H+ \n\u0000TS: (76)\nThe spin conservation law forbids the term @R=@\u0016 in the\ncontinuity equation (73), because it is not a divergence of\nsome current. Thus, the dissipation function is compati-\nble with the spin conservation law if it depends only on\nthe gradient of the spin chemical potential \u0016, but not on\n\u0016itself. This does not take place in the LLG theory with\nthe Gilbert damping discussed in the Appendix. The\nanalysis of this section assumed the spin conservation\nlaw and corresponded to the dissipation function\nR=\u001fD\n2r\u00162\u0000D\n2TM2\n?\nMzr\u0016\u0001rT+\u0010\n2\rAM2\n?(r\u0001Js)2:\n(77)\nIn general the dissipation function contains also the term\n/rT2responsible for the thermal conductivity. But it\nis important only for the heat balance equation, which\nwas not considered here.\nIf the temperature does not vary in space, then the only\ntemperature e\u000bect is a correction to the spin chemical\npotential. This does not a\u000bect the basic feature of super-\n\ruid spin transport: there is no gradient of the chemical\npotential in a stationary current state, and all dissipation\nprocesses are not e\u000bective except for the relativistically\nsmall spin Bloch relaxation. If there is spatial variation of\ntemperature, then the spin chemical potential also varies\nin space. One can \fnd its gradient by exclusion of r\u0001Js\nfrom Eqs. (73) and (74):\nr\u0016=rMz\n\u001f=\u0000D\u0010\n2\r2AMzTr(r2T): (78)\nNote that the spin chemical potential gradient is propor-\ntional not to the \frst but to the third spatial derivative\nof the temperature. The constant temperature gradi-\nent does not produce spatial variation of the chemical\npotential. This is an analog of the absence of thermo-\nelectric e\u000bects proportional to the temperature gradients\nin superconductors.33Naturally the e\u000bect produced by\nhigher derivatives of the temperature is weaker than pro-\nduced by the \frst derivative.\nThe nonuniform correction to the spin chemical po-\ntential strongly depends on temperature. Assuming the\nT4dependence of the dissipation parameters Dand\u0010in\nEq. (71) the coe\u000ecient before the temperature-gradient\nterm in Eq. (78) is proportional to T8. Now the spin dif-\nfusion current\u0000\u001fDr\u0016does not disappear in the equa-\ntion (73) of continuity for the spin, but it is proportional\ntoT12.\nEarlier Zhang and Zhang34used the Boltzmann equa-\ntion for derivation of the spin di\u000busion coe\u000ecient and11\nthe Bloch relaxation time in an isotropic ferromagnet in\na constant magnetic \feld. We derived the spin di\u000busion\nand the second viscosity coe\u000ecients in an easy-plane fer-\nromagnet with di\u000berent spin-wave spectrum. Two-\ruid\ne\u000bects in easy-plane ferromagnets were investigated by\nFlebus et al.35. They solved the Boltzmann equation\nusing the equilibrium magnon distribution function with\nnonzero chemical potential of magnon (do not confuse it\nwith the spin chemical potential introduced in the present\npaper). In contrast, we assumed complete thermaliza-\ntion of the magnon distribution when the magnon chem-\nical potential vanishes. The thermalization assumption\nis questionable in the transient layer near the interface\nthrough which spin is injected, and in this layer the ap-\nproach Flebus et al.35may become justi\fed. The tran-\nsient layer is discussed in the next section.\nVI. TRANSIENT (HEALING) LAYER NEAR\nTHE INTERFACE INJECTING SPIN\nInjection of spin from a medium without spin super\ru-\nidity to a medium with spin super\ruidity may produce\nnot only a super\ruid spin current but also a spin cur-\nrent of incoherent magnons. But at some distance from\nthe interface between two media, which will be called the\nconversion healing length, the spin current of incoherent\nmagnons (spin di\u000busion current) must inevitably trans-\nform to super\ruid spin current, as we shall show now.\nWe return back to Eqs. (1) and (2) but now we neglect\nthe relativistically small Bloch spin relaxation (the term\n/1=T1). In Sec. II we considered the stationary solution\nof the these equations with constant magnetization and\nabsent spin di\u000busion current. But it is not the only sta-\ntionary solution. Another solution is an evanescent mode\nM0\nz/r'/e\u0000x=\u0015, where\n\u0015=s\n\u001fD\u0010\n\rA(79)\nis the conversion healing length. We look for superposi-\ntion of two solutions, which satis\fes the condition that\nthe injected current J0transforms to the spin di\u000busion\ncurrent, while the super\ruid current vanishes at x= 0:\nJ0=\u0000DrxM0\nz(0);rx'(0) = 0: (80)\nThis superposition is\nM0\nz(x) =M0\nz+\u0015J0\nDe\u0000x=\u0015;rx'(x) =J0\nA(1\u0000e\u0000x=\u0015);\n(81)\nwhereM0\nzin the right-hand side is a constant magneti-\nzation far from the interface x= 0. Thus, at the length\n\u0015the spin di\u000busion current Jddrops from J0to zero,\nwhile the super\ruid spin current grows from zero to J0\nand remains at larger distances constant.\nAs pointed out in the end of Sec. II, the phenomenolog-\nical equations (1) and (2) were derived assuming that thespin chemical potential \u0016=M0\nz=\u001f\u0000Hdoes not depend\non gradients rMz. However, the dissipation coe\u000ecients\nDand\u0010decrease very sharply with temperature, and\nthe conversion healing length eventually becomes much\nsmaller than the scale \u00180[see Eq. (25)], when the depen-\ndence of the free energy and the spin chemical potential\non the gradients rMzbecomes important. But in fact\naddingrMz-dependent terms into the expression for \u0016,\n\u0016=Mz\n\u001f\u0000H\u0000AM2r2Mz\nM2\n?; (82)\ndoes not a\u000bect the expression (79) for the healing length.\nThe generalization of the analysis reduces to replacing of\nM0\nzin Eqs. (1), (2), and (81) by \u001f\u0016.\nTransformation of the injected incoherent magnon spin\ncurrent to the super\ruid spin current is not the only tran-\nsient process near the interface between media with and\nwithout spin super\ruidity. Even in the absence of spin\ncurrent the interface may a\u000bect the equilibrium mag-\nnetic structure. For example, the interface can induce\nanisotropy di\u000berent from easy-plane anisotropy in the\nbulk. Then the crossover from surface to bulk anisotropy\noccurs at the healing length of the order of the correla-\ntion length \u00180determined by Eq. (25) in ferromagnets, or\nthe correlation length \u0018determined by Eq. (54) in anti-\nferromagnets. The similar healing length was suggested\nfor ferromagnets by Takei and Tserkovnyak7and for an-\ntiferromagnets by Takei et al.8although using di\u000berent\narguments.\nThe expression (79) for \u0015was derived within hydrody-\nnamics with dissipation. At distances shorter than the\nmean-free path incoherent magnons are in the ballistic\nregime and cannot converge to the super\ruid current,\nsince conversion is impossible without dissipation. Alto-\ngether this means that the real healing length at which\nthe bulk super\ruid spin current state is formed cannot\nbe less than the longest from three scales: \u0015,\u00180, and the\nmagnon mean-free path cs\u001c. Apparently at low tempera-\ntures and weak magnetization Mzthe latter is the longest\none from three scales. However, close to the phase transi-\ntion to the easy-axis anisotropy ( Mz=M) the coherence\nlength\u00180diverges and becomes the longest scale.\nSolving the Boltzmann equation we assumed complete\nthermalization of the magnon distribution. At low tem-\nperatures when magnon-magnon interaction is weak the\nlength at which thermalization occurs essentially exceeds\nthe mean-free path on defects. It could be that the heal-\ning length would grow up to the thermalization length.\nThis requires a further analysis.\nVII. MAGNETIC VORTEX IN AN\nEASY-PLANE ANTIFERROMAGNET\nLet us consider structure of an axisymmetric vortex in\nan antiferromagnet with one quantum of circulation of\nthe angle'0of rotation around the vortex axis. Now\nwe consider the geometry of the experiment20when the12\nPtPtzxyCr2O3H\n\u0001\u0002\u0001\u0002\u0001\u0002\u0002\u0003\u0001\u0002\u0001\u0004\u0001\u0003\u0002\u0004\u0002\u0003\u0003\u0004\u0002\u0004\u0001\u0006\u0001\u0002\u0002\u0006\u0001\u0002\u0004\u0006\u0001\u0002\u0003\u0006\u0003\u0004\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0004\u0006\u0001\u0002\u0002\u0006\u0001\u0002\u0001\u0006\u0002\u0004\u0005\u0006\u0002\u0003\u0002\u0004\u0001\u0003\u0001\u0004\u0001\u0002\u0001\u0003\u0001\u0004\u0003\u0001a)\nb)\nFIG. 3. Precession of magnetization maround the direction\nof the magnetic \feld Halong the path around the vortex\naxis. (a) The geometry of the experiment20with the magnetic\n\feld (the axis z) in the plane of the Cr 2O3\flm. The vortex\naxis is normal to the \flm (the axis y). (b) Precession of the\nmagnetization mis shown in the plane xz(the plane of the\n\flm). The path around the vortex axis (dashed lines) is inside\nthe vortex core where the total magnetization is not parallel\ntoH(\u0012 6= 0).\nmagnetic \feld H(the axisz) is in the \flm plane. The\nvortex axis is the axis ynormal to the \flm plane (Fig, 3a).\nThe azimuthal component of the angle '0gradient is\nr'0=1\nr: (83)\nAt the same time '= 0 and\u00120is small. Then the Hamil-\ntonian (46) transforms to\nH=2M2\n\u001f\u00122\n0\u00002HM cos\u0012\u00120+A\u0000M2\u0012cos2\u0012\nr2+r\u00122\u0013\n:\n(84)\nMinimization with respect to small \u00120yields\n\u00120=\u001fHcos\u0012\n2M; (85)\nand \fnally the Hamiltonian is\nH=\u0000\u001fH2cos2\u0012\n2+A\u0000M2\u0012cos2\u0012\nr2+r\u00122\u0013\n:(86)The Euler{Lagrange equation for this Hamiltonian de-\nscribes the vortex structure in polar coordinates:\nd2\u0012\ndr2+1\nrd\u0012\ndr\u0000sin 2\u0012\n2\u00121\n\u00182\u00001\nr2\u0013\n= 0; (87)\nwhere the correlation length \u0018is given by Eq. (54) and\ndetermines the size of the vortex core.\nThe vortex core has a structure of a skyrmion, in which\nthe total weak magnetization deviates from the direction\nof the magnetic \feld H(\u00126= 0). The component of\nmagnetization transverse to the magnetic \feld is\nm?=\rHsin 2\u0012\n2: (88)\nThe transverse magnetization creates stray magnetic\n\felds at the exit of the vortex line from the sample. Fig-\nure 3 shows variation of the magnetization inside the core\nalong the path around the vortex axis parallel to the axis\ny. Along the path the magnetization mrevolves around\nthe direction of the magnetic \feld forming a cone. The\nprecession in space creates an oscillating ycomponent\nof magnetization my=m?(r) sin\u001e, where\u001eis the az-\nimuthal angle at the circular path around the vortex line.\nThis produces surface magnetic charges 4 \u0019myat the exit\nof the vortex to the boundary separating the sample from\nthe vacuum. These charges generate the curl-free stray\n\feldh=r . At distances from the vortex exit point\nmuch larger that the core radius the stray \feld is a dipole\n\feld with the scalar potential\n (R) =\u0019\u001fH\n2(R\u0001n)\nR3Z1\n0sin 2\u0012(r)r2dr\n= 1:2\u0019\u001fH\u00183(R\u0001n)\nR3=1:2\u0019\u001fc3\ns\n\r3H2(R\u0001n)\nR3: (89)\nHereR(x;y;z ) is the position vector with the origin in\nthe vortex exit point and nis a unit vector in the plane\nxzalong which the surface charge is maximal ( \u001e=\u0019=2).\nIn our model the direction of nis arbitrary, but it will\nbe \fxed by spin-orbit interaction or crystal magnetic\nanisotropy violating invariance with respect to rotations\naround the axis z. These interactions were ignored in our\nmodel. In principle, the stray \feld can be used for detec-\ntion of vortices nucleated at spin currents approaching\nthe critical value.\nVIII. DISCUSSION AND SUMMARY\nThe paper analyzes the long-distance super\ruid spin\ntransport. The super\ruid spin transport does not require\na gradient of the spin chemical potential (as the electron\nsupercurrent in superconductors does not require a gra-\ndient of the electrochemical potential). As result of it,\nmechanisms of dissipation are suppressed except for weak\nBloch spin relaxation. Other dissipation mechanisms af-\nfect the spin transport only at the transient (healing)13\nlayer close to the interface through which spin is injected,\nor in nonstationary processes.\nThe paper calculates the Landau critical spin phase\ngradient in a two-sublattice antiferromagnet when the\neasy-plane topology of the magnetic order parameter is\nprovided not by crystal magnetic anisotropy but by an\nexternal magnetic \feld. This was the case realized in\nthe experiment by Yuan et al.20. For this goal it was\nnecessary to derive the spectrum of collective modes (spin\nwaves) in spin current states. The Landau instability\ndestroying spin super\ruidity sets on not in the Goldstone\ngapless mode as in easy-plane ferromagnets but in the\ngapped mode, despite that at small spin currents the\nlatter has energy larger than the Goldstone mode.\nThe paper analyzes dissipation processes determining\ndissipation parameters (spin di\u000busion and second viscos-\nity coe\u000ecients) by solving the Boltzmann equation for\nmagnons scattered by defects. The two-\ruid theory sim-\nilar to the super\ruid two-\ruid hydrodynamics was sug-\ngested. It is argued that the LLG theory with the Gilbert\ndamping parameter is not able to properly describe dissi-\npation in easy-plane magnetic insulators. Describing the\nwhole dissipation by a single Gilbert parameter one can-\nnot di\u000berentiate between strong processes connected with\nhigh exchange energy (e.g., spin di\u000busion) and weak pro-\ncesses connected with spin-orbit interaction (Bloch spin\nrelaxation), which violate the spin conservation law.\nThe formation of the super\ruid spin current in the\ntransient (healing) layer near the interface through which\nspin is injected was investigated. The width of this layer\n(healing length) is determined by processes of dissipation,\nand at low temperatures can reach the scale of relevant\nmean-free paths of magnons including those at which the\nmagnon distribution is thermalized.\nThe structure of the magnetic vortex in the geometry\nof the experiment on Cr 2O3is investigated. In the vortex\ncore there is a magnetization along the vortex line, which\nis normal to the magnetic \feld. This magnetization pro-\nduces magnetic charges at the exit of the vortex line from\nthe sample. The magnetic charges create a stray dipole\nmagnetic \feld, which probably can be used for detection\nof vortices.\nWithin the developed two-\ruid theory the paper ad-\ndresses the role of the temperature variation in space on\nthe super\ruid spin transport. This is important because\nin the experiment of Yuan et al.20the spin is created\nin the Pt injector by heating (the Seebeck e\u000bect). Thus\nthe spin current to the detector is inevitably accompa-\nnied by heat \row. The temperature variation produces\nthe bulk Seebeck e\u000bect, which is estimated to be rather\nweak at low temperatures. However, it was argued26that\nprobably Yuan et al.20detected a signal not from spin\ncoming from the injector but from spin produced by the\nSeebeck e\u000bect at the interface between the heated anti-\nferromagnet and the Pt detector. Such e\u000bect has already\nbeen observed for antiferromagnet Cr 2O3.36If true, then\nYuan et al.20observed not long-distance spin transport\nbut long-distance heat transport. It is not supported bythe fact that Yuan et al. observed a threshold for super-\n\ruid spin transport at low intensity of injection, when ac-\ncording to the theory5violation of the approximate spin\nconservation law becomes essential. Investigation of su-\nper\ruid spin transport at low-intensity injection is more\ndi\u000ecult both for theory and experiment. But the exis-\ntence of the threshold is supported by extrapolation of\nthe detected signals from high-intensity to low-intensity\ninjection. According to the experiment, the signal at the\ndetector is not simply proportional to the squared elec-\ntric current j2responsible for the Joule heating in the\ninjector, but to j2+a. The o\u000bset ais evidence of the\nthreshold, in the analogy with the o\u000bset of IVcurves in\nthe mixed state of type II superconductors determining\nthe critical current for vortex deepening. With all that\nsaid, the heat-transport interpretation cannot be ruled\nout and deserves further investigation. According to this\ninterpretation, one can see the signal observed by Yuan\net al.20at the detector even if the Pt injector is replaced\nby a heater, which produces the same heat but no spin.\nAn experimental check of this prediction would con\frm\nor reject the heat-transport interpretation.\nLet us make some numerical estimations for Cr 2O3us-\ning the formulas of the present paper. It follows from\nneutron scattering data37that the spin-wave velocity is\ncs= 8\u0002105cm/sec. According to Foner38, the magne-\ntization of sublattices is M= 590 G and the magnetic\nsusceptibility is \u001f= 1:2\u000210\u00004. Then the total magne-\ntizationmz=\u001fHin the magnetic \feld H= 9 T used\nin the experiment is about 10 G, and the canting an-\ngle\u00120=mz=2M\u00190:01 is small as was assumed in our\nanalysis. The correlation length (54), which determines\nvortex core radius, is about \u0018\u00190:5\u000210\u00006cm. The stray\nmagnetic \feld produced by magnetic charges at the exit\nof the vortex line from the sample is 10( \u00183=R3) G, where\nRis the distance from the vortex exit point. The task to\ndetect such \felds does not look easy, but it is hopefully\npossible with modern experimental techniques.\nACKNOWLEDGMENTS\nI thank Eugene Golovenchits, Wei Han, Mathias Kl aui,\nRomain Lebrun, Allures Qaiumzadeh, Victoria Sanina,\nSo Takei, and Yaroslav Tserkovnyak for fruitful discus-\nsions and comments.\nAppendix: Dissipation in the LLG theory\nFor ferromagnets the LLG equation taking into ac-\ncount dissipation is\ndM\ndt=\r[Heff\u0002M] +\u000b\nM\u0014\nM\u0002dM\ndt\u0015\n; (A.1)\nwhere\u000bis the dimensionless Gilbert damping parameter.\nFor small\u000bthis equation is identical to the equation with14\nthe Landau{Lifshitz damping term:\n1\n\rdM\ndt=\u0014\nM\u0002\u000eH\n\u000eM\u0015\n+\u000b\nM\u0014\nM\u0002\u0014\nM\u0002\u000eH\n\u000eM\u0015\u0015\n:\n(A.2)\nTransforming the vector LLG equation to the equations\nfor two Hamiltonian conjugate variables, the zcompo-\nnentMzof magnetization and the angle 'of rotation\naround the zaxis, one obtains Eqs. (73) and (74) without\nthe term r(@R=@r\u0016) and with the dissipation function\nR=\u000b\rM2\n?\n2M\u00162+\u000bM\n2M2\n?(r\u0001Js)2; (A.3)\nwhich depends on the spin chemical potential \u0016itself,\nbut not on its gradient. Meanwhile, according to the\ntwo-\ruid theory of Sec. V, the r\u0016-dependent term in the\ndissipation function was responsible for the spin-di\u000busion\nterm in the continuity equation for Mz. Indeed, at deriva-\ntion of the continuity equation (1) from the LLG theory\nunder the assumption that \u0016\u0019M0\nz=\u001f=Mz=\u001f\u0000Hthe\nspin di\u000busion term /Ddoes not appear. The term\ndoes appear only if \u0016in the dissipation function (A.3)\nis determined by the more general expression (82) taking\ninto account the dependence on rMz. Then one obtains\nEqs. (1) and (2) with the equal spin di\u000busion and spin\nsecond viscosity coe\u000ecients\nD=\u0010=\u000b\rMA; (A.4)\nand the inverse Bloch relaxation time\n1\nT1=\u000b\rM2\n?\n\u001fM: (A.5)\nThe outcome looks bizarre. The spin di\u000busion emerges\nfrom the\u0016-dependent term in the dissipation function,\nwhich is incompatible with the spin conservation law, asif the spin di\u000busion is forbidden by the spin conserva-\ntion law. Evidently this conclusion is physically incor-\nrect. Moreover, in the analogy of magnetodynamics and\nsuper\ruid hydrodynamics the magnetization Mzcorre-\nsponds to the \ruid density. In hydrodynamics the \ruid\ndensity gradients are usually not taken into account in\nthe Hamiltonian and in the chemical potential since they\nbecome important only at small scales beyond the hydro-\ndynamical approach. This does not rule out the di\u000busion\nprocess. Similarly, one should expect that it is possible\nto ignore the magnetization gradients in the spin chemi-\ncal potential either. It is strange that the spin di\u000busion\nbecomes impossible in the hydrodynamical limit.\nAccording to the Noether theorem the total magnetiza-\ntion along the axis zis conserved if the Hamiltonian is in-\nvariant with respect to rotations around the axis zin the\nspin space. The Landau{Lifshitz theory of magnetism19\nis based on the idea that the spin-orbit interaction, which\nbreaks rotational symmetry in the spin space and there-\nfore violates the spin conservation law, is relativistically\nsmall compared to the exchange interaction because the\nformer is inversely proportional to the speed of light. So,\nalthough the spin conservation law is not exact, it is a\ngood approximation (see Sec. I). 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Otxoa de Zuazola\nHitachi Cambridge Laboratory, J. J. Thomson Avenue,\nCB3 OHE, Cambridge, United Kingdom and\nDonostia International Physics Center, 20018 San Sebasti´ an, Spain\n(Dated: April 20, 2020)\nAbstract\nWe study theoretically the spin wave emission from a moving d omain wall in a ferromagnet.\nIntroducing a deformation mode describing a modulation of t he wall thickness in the collective\ncoordinate description, we show that thickness variation c ouples to the spin wave linearly and\ninducesspinwave emission. Thedominant emitted spinwave t urnsout tobepolarized in theout-of\nwall plane ( φ)-direction. The emission contributes to the Gilbert dampi ng parameter proportional\nto/planckover2pi1ωφ/K, the ratio of the angular frequency ωφofφand the easy-axis anisotropy energy K.\n1I. INTRODUCTION\nSpin wave (magnon) is an excitation playing essential roles in the tran sport phenomena\nin magnets, and its control, magnonics, is a hot recent issue. Beside s application interest\nfor devices, behaviours of spin waves have been drawing interests from fundamental science\nview points. Many theoretical studies have been carried out on gen eration of spin waves\nby dynamic magnetic objects such as a domain wall [1–6]. The subject is highly nontrivial\nbecause the wall is a soliton, which is stable in the absence of perturb ation, meaning that\nit couples to fluctuations, spin waves, only weakly in the ideal case, w hile in reality, various\nperturbations and dynamics leads to strong emission of spin waves. There are several pro-\ncesses that lead to the emission, and it is not obvious which is the domin ant process and\nhow large is the dissipation caused by the emission.\nThe low energy behavior of a domain wall in a ferromagnet is described in terms of\ncollective coordinates, its center of mass position Xand angle of the wall plane, φ0[7]. In\nthe absence of a pinning potential, a displacement of the wall costs n o energy owing to the\ntranslational invariance, and it is thus natural to regard Xas a dynamic variable X(t).\nThis is in fact justified mathematically; X(t) is a collection of spin waves that corresponds\nto the translational motion of the wall [8, 9]. It turns out that the c anonical momentum of\nthe ferromagnetic domain wall is the angle φ0. This is because the translational motion of\ncollective spins requires a perpendicular spin polarization, i.e., a tilting o f the wall plane.\nMathematically this is a direct consequence of the spin algebra, and is straightforwardly\nderived based on the equation of motion for spin (Landau-Lifshitz( -Gilbert) equation) [7] or\non the Lagrangian formalism [10]. In the absence of hard-axis anisot ropy energy, φ0is also\na zero mode. As zero modes, X(t) andφ0do not have linear coupling to the fluctuation,\nspin wave, and thus emission of spin wave does not occur to the lowes t order. In this case,\nthe second-order interactions to the spin wave give rise to the dom inant effect. In Ref. [1],\nthe coupled equations of motion for the wall and spin wave modes wer e solved classically\nand demonstrated that a damping indeed arises from the quadratic interaction. In the case\nof a strong hard-axis anisotropy, the plane of the wall is constrain ed near the easy-plane,\nφ0is frozen, resulting in a single variable system described solely by X(t) [9, 11]. The spin\nwave coupling and dissipation in this limit was discussed in Ref. [11].\nIn real materials, hard-axis anisotropy and pinning potential exist , andX(t) andφ0\n2are not rigorously zero modes. In other words, wall dynamics induc es a deformation and\nemission of spin wave is possible due to linear couplings. It was argued in Ref. [2] that there\nemerges a linear coupling when the wall driven by a spin-transfer tor que has a velocity ˙X\ndifferent from the steady velocity determined by the spin-transfe r torque, and the damping\ndue to spin wave emission was discussed. Numerical analysis of Ref. [3 ] revealed that spin\nwave emission occurs by the modulation of the wall thickness during t he dynamics. The\ncoupling to the wall velocity and second order in the spin wave was stu died analytically in\ndetail and dissipation was estimated in Ref. [6]. The energy dissipation proportional to the\nsecond-order in the wall velocity was found.\nIn this paper, we study the spin wave emission extending convention al collective coor-\ndinate representation of the wall [10]. As the domain wall is a soliton, t here is no linear\ncoupling of its center of mass motion to the spin wave field if deformat ion is ignored. We\nthus introduce a deformation mode of the wall, a change of the thick nessλ. This is a natural\nvariable in the presence of the hard-axis anisotropy energy, as th e thickness depends on the\nangleφ0as pointed out in Refs. [12, 13]. Following the prescription of spin wave expansion\n[9], we derive the Lagrangian for the three collective coordinates, t he center of mass position\nX(t), the angle of the wall plane φ0(t) and thickness λ(t), including the spin waves to the\nsecond order. It turns out that Xandφ0and their time-derivatives do not have linear\ncoupling to the spin wave, while ˙λdoes. This result is natural as Xandφ0are (quasi) zero\nmodes, and consistent with numerical observation [3]. It is shown th at the emitted spin\nwave is highly polarized; The dominant emission is the fluctuation of ang leφ, while that\nofθis smaller by the order of the Gilbert damping parameter α. The forward emission of\nwavelength λ∗∝v−1\nw, wherevwis the domain wall velocity, is dominant. The modulation of\nλis induced by the dynamics of φ0, and the contribution to the Gilbert damping parameter\ndue to the spin wave emission from this process is estimated from the energy dissipation\nrate. It was found to be of the order of αφ\nsw≃λ\na/planckover2pi1ωφ\nK, whereωφis the angular frequency of\nthe modulation of φ0,Kis the easy-axis anisotropy energy and ais the lattice constant.\nThis damping parameter contribution becomes very strong of the o rder of unity if /planckover2pi1ωφis\ncomparable to the spin wave gap, K, as deformation of the wall becomes significant in this\nregime.\n3II. COLLECTIVE COORDINATES FOR A DOMAIN WALL\nWe consider a one-dimensional ferromagnet along the x-axis with easy and hard axis\nanisotropy energy along the zandyaxis, respectively. The Lagrangian in terms of polar\ncoordinates ( θ,φ) of spin is\nL=LB−HS (1)\nwhere\nLB=/planckover2pi1S\na/integraldisplay\ndx˙φ(cosθ−1)\nHS=S2\n2a/integraldisplay\ndx/bracketleftbig\nJ[(∇θ)2+sin2θ(∇φ)2]+Ksin2θ(1+κsin2φ)/bracketrightbig\n(2)\nare the kinetic term of the spin (spin Berry phase term) and the Ham iltonian, respectively,\nJ >0,K >0andκK≥0being theexchange, easy-axis anisotropyandhardaxisanisotro py\nenergies, respectively, abeing the lattice constant. A static domain wall solution of this\nsystem is\ncosθ= tanhx−X\nλ0,φ= 0 (3)\nwhereλ0≡/radicalbig\nJ/Kis the wall thickness at rest. The dynamics of the wall is described\nby allowing the wall position Xandφas dynamic variables. This corresponds to treat\nthe energy zero mode of spin waves (zero mode) describing a trans lational motion and its\nconjugate variable φas collective coordinates [9]. This treatment is rigorous in the absenc e\nof pinning and hard-axis anisotropy but is an approximation otherwis e. Most previous\nstudies considered a rigid wall, where the wall thickness is a constant λ0. Here we treat\nthe wall thickness as a dynamic variable λ(t) to include a deformation and study the spin-\nwave emission. This treatment was applied in Ref. [13], but only static s olution of λwas\ndiscussed.\nAs demonstrated in Ref. [9], the spin wave around a domain wall in ferr omagnet is\nconveniently represented using\nξ=e−u(x,t)+iφ0(t)+η(x−X(t),t)(4)\nwhereφ0(t) is the angle of the wall,\nu(x,t) =x−X(t)\nλ(t)(5)\n4θ\nφ\nFIG. 1. Fluctuation corresponding to the real and imaginary part of the spin wave variable\n˜η= ˜ηR+i˜ηI. (a): The profile of ˜ ηantisymmetric with respect to the wall center, which turns\nout to be dominant excitation. (b): The real part ˜ ηRdescribes the deformation within the wall\nplane, i.e., modulation of θ, while the imaginary part ˜ ηIdescribes the out-of plane ( φ) fluctuation\nas shown in (c). Transparent arrows denotes the equilibrium spin configuration.\nandη(x−X,t) describes thespin-wave viewed inthemoving frame. Asitis obvious f romthe\ndefinition, the real and imaginary part of ηdescribe the fluctuation of θandφ, respectively.\nThe fluctuations antisymmetric with respect to the wall center, sh own in Fig. 1, turns out\nto be dominant. The ξ-representation of the polar angles are\ncosθ=1−|ξ|2\n1+|ξ|2, sinθsinφ=−iξ−ξ\n1+|ξ|2. (6)\nA. Domain wall dynamic variables\nWe first study what spin-wave mode the new variable λ(t) couples to, by investigating\nthe ’kinetic’ term of the spin Lagrangian, LB, which is written as\nLB=2i/planckover2pi1Sλ\na/integraldisplay\nduIm[ξ˙ξ]\n1+|ξ|2. (7)\nUsing Eq. (6) and\n∂tu=−1\nλ/parenleftBig\n˙X+u˙λ/parenrightBig\n, ∂ tξ=/parenleftbigg1\nλ/parenleftBig\n˙X+u˙λ/parenrightBig\n+i˙φ0+(∂t−˙X∇x)η/parenrightbigg\nξ,(8)\nwe have\n2iIm[ξ˙ξ] = 2i( ˙ηI+˙φ0−˙X∇xηI)|ξ|2(9)\n5sδ\nFIG. 2. Schematic figure showing the effect of asymmetric perpe ndicular spin polarization δs\ndue to the spin wave mode ϕ. The asymmetric torque (curved arrows) induced by asymmetr icδs\nrotates the spins within the wall plane, resulting in a compr ession of the wall, i.e., to ˙λ.\nwhereηi≡Im[η]. The kinetic term is expanded to the second order in the spin wave as\n(using integral by parts)\nLB=2/planckover2pi1S\na[φ0˙X+ϕ˙λ]+L(2)\nB (10)\nwhere\nϕ≡/integraldisplay\nduu\ncoshu˜ηI, (11)\nrepresents an asymmetric configuration of ˜ ηIand\nL(2)\nB≡2/planckover2pi1Sλ\na/integraldisplay\ndu/bracketleftbigg\n˜ηR↔\n∂t˜ηI−˙X˜ηR↔\n∇x˜ηI−2\nλtanhu/parenleftBig\n2˙X+u˙λ/parenrightBig\n˜ηR˜ηI/bracketrightbigg\n,(12)\nwhere ˜η≡η/(2coshu).\nWhen deriving Eq. (10), the orthogonality of fluctuation and the ze ro-mode,\n/integraldisplay\ndu˜η\ncoshu= 0, (13)\nwas used. Equation (10) indicates that ϕis the canonical momentum of λ. In fact, it\nrepresents the asymmetric deformation of angle φ, as the imaginary part of the spin wave,\n˜ηI, corresponds to fluctuation of φas seen in the definition, Eq. (4). Such an asymmetric\nconfiguration of φexerts a torque that induces a compression or expansion of the do main\nwall (Fig. 2), andthisiswhy ϕandλareconjugatetoeach other. Thecoupling ϕ˙λdescribes\nthe spin wave emission when thickness changes, as we shall argue lat er. The second term\nproportional to ˙Xin the bracket in Eq. (12) represents a magnon current induced in t he\nmoving frame (Doppler shift).\n6The Hamiltonian of the system is similarly written in terms of spin wave va riables to the\nsecond order as\nHS=KS2λ\na/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+1+κsin2φ0/bracketrightBigg\n+2KS2λ\na/integraldisplay\ndutanhu\ncoshu˜ηR/bracketleftBigg\n−/parenleftbiggλ0\nλ/parenrightbigg2\n+1+κsin2φ0/bracketrightBigg\n+H(2)\nS,\n(14)\nwhere\nH(2)\nS≡2KS2λ\na/integraldisplay\ndu/bracketleftbigg\nλ2\n0[(∇˜ηR)2+(∇˜ηI)2]\n+ ˜ηR2/bracketleftbigg\n−λ2\n0\nλ2/parenleftbigg\n1−1\ncosh2u/parenrightbigg\n+/parenleftbigg\n2−3\ncosh2u/parenrightbigg\n(1+κsin2φ0)/bracketrightbigg\n+ ˜ηI2/bracketleftbiggλ2\n0\nλ2/parenleftbigg\n1−2\ncosh2u/parenrightbigg\n+κcos2φ0/bracketrightbigg\n+2κ˜ηR˜ηItanhusin2φ0/bracketrightbigg\n(15)\nIn the case of small κandλ≃λ0, the spin waves are described by a simple Hamiltonian as\nHsw≡2KS2λ\na/integraldisplay\ndu/bracketleftbigg\nλ2[(∇˜ηR)2+(∇˜ηI)2]+( ˜ηR2+ ˜ηI2)/parenleftbigg\n1−2\ncosh2u/parenrightbigg/bracketrightbigg\n+HD,(16)\nwhere\nHD≡2/planckover2pi1Sλ\na˙X/integraldisplay\ndu˜ηR↔\n∇x˜ηI, (17)\nis the Doppler shift term. For a constant wall velocity ˙X, it simply shifts the wave vector\nof the spin wave. Without the Doppler shift, the eigenfunction of th is Hamiltonian (16) is\nlabeled by a wave vector kas\nφk(u) =1√2π˜ωk(−ikλ+tanhu)eikλu, (18)\nwhere\n˜ωk≡1+(kλ)2(19)\nis the dimensionless energy of spin wave.\nDissipation function is\nW=α/planckover2pi1S\n2a/integraldisplay\ndx(˙θ2+sin2θ˙φ2)\n=/planckover2pi1Sλ\n2a\nα/parenleftBigg˙X\nλ/parenrightBigg2\n+α˙φ02+αλ/parenleftBigg˙λ\nλ/parenrightBigg2\n, (20)\n7whereαis the Gilbert damping parameter and αλ≡α/integraltext\nduu2\ncosh2u=π2\n12α.\nAsdrivingmechanisms ofadomainwall, weconsider amagneticfieldandc urrent-induced\ntorque (spin-transfer torque) [9, 14, 15]. A magnetic field applied a long the negative easy\naxis is represented by the Hamiltonian ( γ=e/mis the gyromagnetic ratio)\nHB=/planckover2pi1Sγ\naBz/integraldisplay\ndxcosθ. (21)\nUsing Eqs. (6)(13), we obtain\nHB=−2/planckover2pi1Sγ\naBz/parenleftbigg\nX+λ/integraldisplay\ndutanhu˜η2\nR/parenrightbigg\n. (22)\n(The first term is derived evaluating a diverging integral/integraltext\ndx1\n1+e2u(x)carefully introducing\nthe system size Las/integraltextL/2\n−L/2dx1\n1+e2u(x)and dropping a constant.) The magnetic field therefore\nexerts a force2/planckover2pi1Sγ\naBzon the domain wall.\nThe spin-transfer effect induced by injecting spin-polarized electr ic current is represented\nby a Hamiltonian having the same structure as the spin Berry’s phase termLB[9, 15]\nHSTT=−/planckover2pi1S\navst/integraldisplay\ndxcosθ(∇xφ), (23)\nwherevst≡aP\n2eSjis a steady velocity of magnetization structure under spin polarized current\nPj(Pis the spin polarization and jis the applied current density (one-dimensional)). The\nspin wave expression is\nHSTT=2/planckover2pi1S\navst/bracketleftbigg\nφ0+2/integraldisplay\ndx/parenleftbigg\n˜ηR∇x˜ηI+1\nλtanhu˜ηR˜ηI/parenrightbigg/bracketrightbigg\n. (24)\nAs has been known, a spin-transfer torque contributing to the wa ll velocity and does not\nwork as a force, as the applied current or vstcouples to φ0and not to X.\nThe equation of motion for the tree domain wall variables is therefor e obtained from Eqs.\n(10) (14) (20) and driving terms (22)(24) as\n˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0ζ+vst\n˙φ0+α˙X\nλ=˜Bz\nαλ˙λ\nλ=KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n−˙ϕ−2KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+(1+κsin2φ0)/bracketrightBigg\nζ,\n(25)\n8wherevc≡KSκ\n2/planckover2pi1λ,˜Bz≡γBzandϕ(Eq. (11)) and\nζ≡/integraldisplay\ndutanhu\ncoshu˜ηR, (26)\nare contributions linear in spin wave.\nIII. SPIN WAVE EMISSION\nIn this section we study the spin wave emission due to domain wall dyna mics. The\nemission is described by the linear coupling between the spin wave field a nd the domain wall\nin Eqs.(10) (14). Moreover, dynamic second-order couplings in Eqs . (12)(15) leads to spin\nwave excitation. In the first linear process, the momentum and ene rgy of the spin wave is\nsupplied by the dynamic domain wall, while the second process present s a scattering of spin\nwaves where the domain wall transfer momentum and energy to the incident spin wave.\nA. Linear emission\nWe here discuss the emission due to the linear interactions in Eqs.(10) (14) in the labo-\nratory (rest) frame. The laboratory frame is described by replac ingη(x−X(t),t) byη(x,t)\nin the derivation in Sec. IIA. It turns out that the Lagrangian Eq.(1 2) in the laboratory\nframe has no Doppler shift term and the term ˙X˜ηR˜ηIis half. The emitted wave has an\nangular frequency shifted by the Doppler shift from the moving wall. Using the equation of\nmotion, Eq. (25), the spin wave emission arises from the thickness c hange. The interaction\nHamiltonian reads in the complex notation ˜ η= ˜ηR+i˜ηI\nH(1)\nη(t) =˙λ(t)/integraldisplay\ndx(g˜η+g˜η), (27)\nwhere\ng(x)≡2/planckover2pi1S\na1\ncoshx−X(t)\nλ/parenleftbigg\n−αλtanhx−X(t)\nλ+ix−X(t)\nλ/parenrightbigg\n(28)\nLet us study here the emission treating λas a constant as its dynamics is taken account in\nthe first factor in the interaction Hamiltonian (27). The Fourier tra nsform of the interaction\n9is calculated using\n/integraldisplay∞\n−∞duei˜kuu\ncoshu=iπ2\n2sinhπ\n2˜k\ncosh2π\n2˜k\n/integraldisplay∞\n−∞duei˜kutanhu\ncoshu=π˜k\ncoshπ\n2˜k(29)\nas\nH(1)\nη(t) =−π2\n2λ˙λ(t)/summationdisplay\nk1\ncoshπ\n2kλeikX(t)/parenleftbigg\n˜ηIk(t)tanhπ\n2kλ+2\nπαλkλ˜ηRk(t)/parenrightbigg\n,(30)\nWe consider the case where the wall is approximated by a constant v elocityvw, i.e.,X(t) =\nvwt. The frequency representation of time-integral of Eq. (30) is\n/integraldisplay\ndtH(1)\nη(t) =−π2\n2/integraldisplaydΩ\n2π/integraldisplaydω\n2πλ˙λ(Ω)/summationdisplay\nk1\ncoshπ\n2kλ/parenleftbigg\n˜ηIk(t)tanhπ\n2kλ+2αλ\nπkλ˜ηRk(t)/parenrightbigg\nδ(ω−(kvw+Ω)),\n(31)\nIt is seen that the angular frequency of the emitted spin wave ( ω) iskvw+ Ω, i.e., that of\nthe thickness variation ˙λwith a Doppler shift due to the wall motion. The Doppler shift of\nangular frequency, δν≡kvw, is expected tobesignificant; For k= 1/λwithλ= 10−100nm\nandvw= 100 m/s, we have δν= 10−1 GHz. The function g(x) represents the distribution\nof the wave vector k, which has a broad peak at k= 0 with a width of the order of λ−1.\nTo have a finite expectation value ∝angb∇acketleft˜η∝angb∇acket∇ight, the angular frequency ωand wave vector kneeds\nto match the dispersion relation of spin wave, ω=ωk, i.e.,\nkvw+/planckover2pi1Ω =KS(1+(kλ)2). (32)\nThe angular frequency Ω is determined by the equation for λin Eq. (25), and is of the\norder of the angular frequency of φ0,ωφ. (See Sec. VA for more details.) Equation (32) has\nsolution for a velocity larger than the threshold velocity vth≡2KS\n/planckover2pi1λ/radicalBig\n1−/planckover2pi1ωφ\nKS. The emitted\nwave lengths k∗are (plotted in Fig. 4)\nk∗λ=/planckover2pi1vw\n2KSλ\n1±/radicalBigg\n1−/parenleftbiggvth\nvw/parenrightbigg2\n. (33)\nThe sign of k∗(direction of emission) is along the wall velocity, meaning that the emis sion is\ndominantly in the forward direction. The group velocity of the emitte d wave is of the same\n10vwλ∗\nFIG. 3. Schematic figure showing the spin wave emission from a domain wall with thickness\noscillation ( ˙λ) moving with velocity vw. The linear coupling leads to a forward emission of spin\nwave with wave length λ∗≡2π/k∗, wherek∗is defined by Eq. (33).\nvwk*\n•\n•\n•ω~=0.2\nω~=0.5\nω~=0.8\nFIG. 4. Plot of the wave length k∗of the emitted spin wave as function of wall velocity vwfor\n˜ω≡/planckover2pi1ωφ/KS= 0.2,0.5 and 0.8. Dotted line is k∗=/planckover2pi1\nKSλ2vw. Threshold velocity for the emission\nvthis denoted by circles.\norder as the wall velocity;\ndωk\ndk/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=k∗=2KS\n/planckover2pi1λ2k∗=vw\n1±/radicalBigg\n1−/parenleftbiggvth\nvw/parenrightbigg2\n. (34)\nThe dominant spin wave emission considered here is the antisymmetric excitation of the\nimaginary part ˜ ηIrepresenting the fluctuation of angle φ. The antisymmetric excitation of\nφis a natural excitation arising from the intrinsic property, the aniso tropy energy. The\neasy-axis anisotropy energy acts as a local potential VKfor each spin in the wall as in Fig.\n5. When the wall moves to the right, the spins ahead of the wall are d riven towards the\nhigh energy state, while the spins behind (left in Fig. 5) are towards lo w energy states. This\nasymmetry leads to an asymmetric local “velocity” of angle θ, and its canonical momentum\n11FIG. 5. The local potential VKfor spins in a domain wall arising from the easy axis anisotro py\nenergy,K. When the wall moves to the right, the spins right (left) of th e wall rotates towards\nhigh (low) energy states, resulting in an asymmetric local v elocity of rotation, exciting the angle\nφasymmetrically with respect to the wall center.\nφ. This role of Kto induce asymmetric φis seen in the equations of motion for polar angles\n[9]: Focusing on the contribution of the easy axis anisotropy, the ve locity of the in-plane\nspin rotation, sin θ˙φ=−KSsinθcosθis asymmetric with the wall center θ=π/2. Faster\nrotation in the left part of the wall (π\n2< θ < π) than the right part (0 < θ <π\n2) indicates\nthat the wall becomes thinner. In the equation of motion for λ(Eq. (25)), this effect is\nrepresented by the term −˙ϕon the right-hand side, meaning that asymmetric deformation\nmodeϕtends to compress the wall.\nB. Green’s function calculation\nWe present microscopic analysis of the spin wave emission using the Gr een’s function.\nWe consider here the slow domain wall dynamics limit compared to the sp in-wave energy\nscale and neglect the time-dependence of the variable uarising from variation of ˙X. The\ncalculation here thus corresponds to the spin wave effects in the mo ving frame with the\ndomain wall. The amplitude of the spin wave, ∝angb∇acketleft˜η∝angb∇acket∇ight, is calculated using the path-ordered\nGreen’s function method as a linear response to the source field ˙λ. The amplitude is\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−i/integraldisplay\nCdt′˙λ(t′)/integraldisplay\ndu′g(u′)/angbracketleftbig\nTC˜η(u,t)˜η(u′,t′)/angbracketrightbig\n(35)\nwhereCdenotes the contour for the path-ordered (non-equilibrium) Gre en’s function in the\ncomplex time and TCdetnoes the path-ordering. Evaluating the path-order, we obta in the\n12real-time expression of\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=/integraldisplay∞\n−∞dt′˙λ(t′)/integraldisplay\ndu′g(u′)Gr\nη(u,t,u′,t′) (36)\nwhere\nGr\nη(u,t,u′,t′)≡ −iθ(t−t′)/angbracketleftbig\n[˜η(u,t),˜η(u′,t′)]/angbracketrightbig\n(37)\nthe retarded Green’s function of ˜ η. The Green’s function is calculated expressing ˜ ηin terms\nof the orthogonal base for spin wave wave function [9] as\n˜η(u,t) =/summationdisplay\nkηk(t)φk(u), (38)\nwhereφkis the eigenfunction of Eq. (18) and ηkis the annihilation operator satisfying\n[ηk,ηk′] =δk,k′. The time-development of the operator is ηk(t) =e−iωktηk(0), where ωk≡\nKS˜ωkis the energy of spin wave. The retarded Green’s function thus is\nGr\nη(u,t,u′,t′) =−iθ(t−t′)/summationdisplay\nke−iωk(t−t′)φk(u)φk(u′)≡/integraldisplaydω\n2πe−iω(t−t′)Gr\nη(u,u′,ω) (39)\nwhere\nGr\nη(u,u′,ω) =/summationdisplay\nk1\nω−ωk+i0φk(u)φk(u′) (40)\nis the Fourier transform, + i0 denoting the small positive imaginary part. The Green’s\nfunction has a nonlocal nature in space, as seen from the overlap o f the spin wave function\n/summationdisplay\nkφk(u)φk(u′) =a\n2πλ/bracketleftbigg\nδ(u−u′)−1\n2/parenleftBig\ne−|u−u′|(1−tanhutanhu′)+sinh(u−u′)(tanhu−tanhu′)/parenrightBig/bracketrightbigg\n.\n(41)\nHere we use low-frequency approximation, namely, consider the eff ect of high-frequency\nmagnon compared to the wall dynamics and use Gr\nη(u,u′,ω)≃ −/summationtext\nk1\nωkφk(u)φk(u′). The\nretarded Green’s function then becomes local in time as Gr\nη(u,t,u′,t′) =δ(t−t′)Gr\nη(u,u′,ω).\nWe thus obtain\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)/summationdisplay\nk1\nωkφk(u)/integraldisplay\ndu′g(u′)φk(u′) (42)\n13withuandu′havingX(t) of the equal time t. The integral/integraltext\ndu′g(u′)φk(u′) describing the\noverlap of spin-wave wave function and the domain wall is calculated u sing\n/integraldisplay\ndutanhu\ncoshuφk(u) =1√2π˜ωkπ\ncoshπ\n2kλ˜ωk\n2/integraldisplay\nduu\ncoshuφk(u) =1√2π˜ωkπ\ncoshπ\n2kλ(43)\nas\n/integraldisplay\ndug(u)φk(u) =2/planckover2pi1S\na1√2π˜ωkπ\ncoshπ\n2kλ/parenleftbigg\ni−αλ˜ωk\n2/parenrightbigg\n(44)\nThe spin wave amplitude emitted by the wall dynamics is therefore\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)2/planckover2pi1\nKa/summationdisplay\nk1√2π˜ωkπ\ncoshπ\n2kλ1\n˜ωkφk(u)/parenleftbigg\ni−αλ˜ωk\n2/parenrightbigg\n(45)\nThe integral/summationtext\nk(˜ωk)−β1\ncoshπ\n2kλφk(u) (β=1\n2,3\n2) is real, and thus Re[ ∝angb∇acketleft˜η∝angb∇acket∇ight]/Im[∝angb∇acketleft˜η∝angb∇acket∇ight]≃α. As\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ightis odd in u, the emitted spin wave is an antisymmetric fluctuation of the angle\nφwith respect to the wall center (Fig. 1). (Because of low frequenc y approximation in\nderiving Eq. (42), the nonlocal nature (Eq. (41)) is smeared out in the result Eq. (45). )\nThe quantities representing the effects of spin wave emission on the wall dynamics in Eq.\n(25) are\nζ=/integraldisplay\ndutanhu\ncoshuRe[˜η] =˙λπ/planckover2pi1\n4Kaαλ/summationdisplay\nk1\ncosh2π\n2kλ≡αµζ˙λ\nλ\nϕ=/integraldisplay\nduu\ncoshuIm[˜η] =−˙λπ/planckover2pi1\nKa/summationdisplay\nk1\n˜ω2\nk1\ncosh2π\n2kλ≡µϕ˙λ\nλ(46)\nwhereµζ≡π3/planckover2pi1λ\n48Ka/summationtext\nk1\ncosh2π\n2kλandµϕ≡π/planckover2pi1λ\nKa/summationtext\nk1\n˜ω2\nk1\ncosh2π\n2kλ. The first integral is evaluated as\n/summationtext\nk1\ncosh2π\n2kλ=a/integraltextdk\n2π1\ncosh2π\n2kλ=2a\nπ2λand the second one is/summationtext\nk1\n˜ω2\nk1\ncosh2π\n2kλ≡2a\nπ2λγϕ, whereγϕ\nis a constant of the order of unity. The constants are therefore\nµζ=π/planckover2pi1\n24K\nµϕ=−2/planckover2pi1γϕ\nπK. (47)\nFrom Eq. (46), the averaged amplitude of the imaginary part of the emitted spin wave is\nof the order of/planckover2pi1˙λ\nKλ(the real part is a factor of αsmaller). As seen from Eq. (25), the time\nscale ofλdynamics is K//planckover2pi1, and thus the emitted spin wave amplitude can be of the order\nof unity if the modulation of λis strong, resulting in a significant damping. (See Eq. (61)\nbelow.)\n14C. Spin wave excitation due to second order interaction\nBesides emission due to the linear order interaction discussed above , spin waves are\nexcited also due to the second order interaction in Eqs.(10) (14) wh en the wall is dynamic.\nHere we focus on the effect of a dynamic potential in the Hamiltonian ( Eq. (16))\nV(x,t)≡4KS2\na1\ncosh2x−X(t)\nλ(48)\nand calculate the excited spin wave density in the laboratory frame b y use of linear response\ntheory. For a constant wall velocity, X(t) =vwt, the Fourier representation of the potential\nis\nVq(Ω) = 8π2KS2λ\naqλ\nsinhπ\n2qλδ(Ω−qvw), (49)\nThe potential thereforeinduces Dopplershift of qvwintheangular frequency ofthescattered\nspin wave. This dynamic potential induces an excited spin wave densit y asδn(x,t) =\niG<\nη(x,t,x,t), where G<\nηis the lesser Green’s function of spin wave. The linear response\ncontribution in the Fourier representation is\nδn(q,Ω) =i/summationdisplay\nk/integraldisplaydω\n2πVq(Ω)(n(ω+Ω)−n(ω))gr\nkωga\nk+q,ω+Ω (50)\nInthisprocess, theexcitedspinwave density hasthesamewavelen gth andangularfrequency\nof the driving potential Vq(Ω). This means that the excitation moves together with the\ndomain wall, and thus this is not an emission process. For slow limit, q≪kand Ω≪ω,\nusingn(ω+Ω)−n(ω) =n(ω+qvw)−n(ω)≃qvwn′(ω), we obtain a compact expression of\nδn(q,Ω) =i4πKS2\navw(qλ)2\nsinhπ\n2qλδ(Ω−qvw)/integraldisplaydω\n2π/summationdisplay\nkn′(ω)|gr\nkω|2(51)\nand the real space profile is\nδn(x,t) =δn0vw\nvatanhx−vwt\nλ\ncosh2x−vwt\nλ(52)\nwhereδn0=−4\nπ(KS)2/integraltextdω\n2π/summationtext\nkn′(ω)|gr\nkω|2andva≡Kλ//planckover2pi1is a velocity scale determined\nby magnetic anisotropy energy. The induced spin wave density has t hus an antisymmetric\nspatial profile with respect to the wall center and propagate with a domain wall velocity in\nthe present slowly varying limit. It is not therefore a spin wave emissio n, but represents the\ndeformation of the wall asymmetric with respect to the center.\n15IV. EQUATION OF MOTION OF THREE COLLECTIVE COORDINATES\nTheequationofmotion(25)including thespinwave emission effectsex plicitly istherefore\n˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0αµζ˙λ\nλ+vst (53)\n˙φ0+α˙X\nλ=˜Bz\nαλ˙λ\nλ=KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n−µϕ¨λ\nλ−2KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+(1+κsin2φ0)/bracketrightBigg\nαµζ˙λ\nλ.\n(54)\nThe spin-wave contribution of the first equation, the second term of the right-hand side, is of\nthe order αsmaller than the first term and is neglected. From the equations, we see that the\ndynamics of Xandφare not strongly coupled to the variation of the width. In particular ,\nwhenκis small, the dynamics of the wall center ( Xandφ) governed by the energy scale of\nK⊥=κKis much slower than that of a deformation mode λ, which is of the energy scale\nofK, and thus it is natural that the two dynamics are decoupled. Then κis not small, λ\naffects much the wall center dynamics.\nFor static case of λ, we have\nλ=λ0/radicalbig\n1+κsin2φ0, (55)\nas was argued in Refs. [12, 13]. Using this relation assuming slow dynam ics to estimate the\nspin-wave contribution in the equation for λ, we obtain\nµϕ¨λ+ ˜αλ˙λ=KS\n/planckover2pi1λ/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n, (56)\nwhere ˜αλ≡αλ/parenleftbig\n1+2S\nπ/parenrightbig\n=π2\n12α/parenleftbig\n1+2S\nπ/parenrightbig\nis the effective damping for the width. The mass\nforλ,µϕ, was induced by the imaginary part of the spin-wave.\nV. DISSIPATION DUE TO SPIN WAVE EMISSION\nConsidering the action, which is a time-integral of the Lagrangian, a nd by use of integral\nby parts with respect to time, the linear interaction Hamiltonian, Eq. (27), is equivalent to\nH(1)\nη=−λFλ, where\nFλ≡2/integraldisplay\nduRe[g˙˜η], (57)\n16is a generalized force for variable λ. Using Eqs. (45)(43), it reads\nFλ=−¨λfλ, (58)\nwhere (neglecting the order of α2)\nfλ≡π/planckover2pi12S\nKa2/summationdisplay\nk1\n˜ω2\nk1\ncosh2π\n2kλ=2/planckover2pi12S\nπKλaγϕ. (59)\nThe energy dissipation rate due to the spin wave emission is therefor e\ndEsw\ndt≡ −˙λFλ=fλ\n2d\ndt˙λ2, (60)\nand thus Esw=fλ\n2˙λ2. As is seen from Eq. (56), the intrinsic energy scale governing the\ndynamics of λisK, and thus the intrinsic scale of ˙λ/λis of the order of K//planckover2pi1. The energy\ndissipation by an intrinsic spin-wave emission is estimated roughly as Ei\nsw≃Kλ\na, which is\nthe typical spin wave energy multiplied by the number of spin waves ex cited in the wall.\nThe quantitydEi\nsw\ndtcorresponds to a dissipation function Wi\nswinduced by the intrinsic spin\nwave emission. Considering the intrinsic frequency of λof the order of K//planckover2pi1, the Gilbert\ndamping parameter induced by the intrinsic emission is\nαi\nsw≃2aλ\n/planckover2pi1SfλK\n/planckover2pi1=4γϕ\nπ. (61)\nThis value is of the order of unity ( γϕis a constant), meaning that spin wave emission from\nthe thickness change is very efficient in dissipating energy from the w all. This result may\nnot be surprising if one notices that the intrinsic energy scale of thic kness change is that of\neasy-axis anisotropy energy K, which is the energy scale where significant deformation of\nthe wall is induced.\nA. Modulation of λdue toφ0dynamics\nIn most cases, the dynamics of λis driven by the time-dependence of φ0as seen in Eq.\n(56). Let us consider this case of a forced oscillation. We consider b y simplyfying φ0grows\nlinear with time, φ0=ωφt,ωφbeing a constant. Linearizing Eq. (56) using λ=λ+δλ,\nwhereλ≡λ0//radicalbig\n1+κ/2 is the average thickness, we have an equation of motion of a force d\noscillation,\nµϕ¨δλ+ ˜αλ˙δλ+µϕ(Ωλ)2δλ=KS\n2/planckover2pi1λκcos2ωφt, (62)\n17where Ω λ=K\n/planckover2pi1/radicalBig\nπS\nγϕ/parenleftbig\n1+κ\n2/parenrightbig\nis an intrinsic angular frequency of δλ. The solution having an\nexternal angular frequency of 2 ωφis\nδλ=δλcos(2ωφt−εφ), (63)\nwhere\nδλ≡κλπS\n4γϕ(K//planckover2pi1)2\n/radicalBig\n(Ω2\nλ−4ω2\nφ)2+4(˜αλωφ\nµϕ)2(64)\nis the amplitude of the forced oscillation and εφ≡tan−12˜αλωφ\nµϕ\nΩ2\nλ−4ω2\nφis a phase shift. A resonance\noccurs for ωφ= Ωλ/2. The energy dissipation rate for the emission due to forced oscillat ion\ninduced by dynamics of φis\ndEφ\nsw\ndt≃λ\na/parenleftbiggδλ\nλ/parenrightbigg2ω3\nφ\nK. (65)\nThe contribution to the Gilbert damping parameter is obtained from t he relationdEφ\nsw\ndt=\nαφ\nsw(˙λ/λ)2as\nαφ\nsw≃λ\na/planckover2pi1ωφ\nK. (66)\nLet us focus on the periodic oscillation of φ0, realized for large driving forces, namely, for\nBz> αKSκ\n2/planckover2pi1γ≡BW(γBz> αvc) for the field-driven case or j >eS2\n/planckover2pi1Pλ\naKκ≡ji(vst> vc) for\nthe current-driven case ( BWis the Walker’s breakdown field and jiis the intrinsic threshold\ncurrent [10]). The solution of the equation of motion (54) then read s\nφ0≃ωφt, (67)\nwhere (jis defined in one-dimension to have the unit of A=C/s)\nωφ≃˜Bz+αvst\nλ=γBz+aP\n2eSλαj. (68)\nTheGilbertdampingconstant duetospinwaveemission, Eq. (66), th usgrowslinearlyinthe\ndriving fields in this oscillation regime. Using current-induced torque f or a pinned domain\nwall would be straightforward for experimental observation of th is behaviour, although the\ncontribution to the Gilbert damping is proportional to αand not large, αφ\nsw≃α/planckover2pi1P\nej\nK(for\nS∼1,P∼1).\n18VI. SUMMARY\nWestudiedspinwaveemissionfromamovingdomainwallinaferromagne tbyintroducing\na deformation mode of thickness modulation as a collective coordinat e. It was shown that\nthe time-derivative of the thickness ˙λhas a coupling linear in the spin wave field, resulting\nin an emission, consistent with previous numerical result [3]. The domin ant emitted spin\nwave is in the forward direction to the moving domain wall and is strong ly polarized in the\nout-of plane direction, i.e., it is a fluctuation of φ. The dynamics of λis induced by the\nvariation of the angle of the wall plane, φ0, as has been noted [12, 13]. For a φ0with an\nangular frequency of ωφ, the Gilbert damping parameter as a result of spin wave emission\nisαφ\nsw≃λ\na/planckover2pi1ωφ\nK, whereKis the easy-axis anisotropy energy ( ais the lattice constant).\nThe present study is in the low energy and weak spin wave regime, and treating the\nhigher energy dynamics with strong spin wave emission is an important future subject.\nACKNOWLEDGMENTS\nGT thanks Y. Nakatani for discussions. This work was supported b y a Grant-in-Aid\nfor Scientific Research (B) (No. 17H02929) from the Japan Societ y for the Promotion of\nScience and a Grant-in-Aid for Scientific Research on Innovative Ar eas (No.26103006) from\nThe Ministry of Education, Culture, Sports, Science and Technolog y (MEXT), Japan.\n[1] D. Bouzidi and H. Suhl, Phys. Rev. Lett. 65, 2587 (1990).\n[2] Y. L. Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B 79, 174404 (2009).\n[3] X. S. Wang, P. Yan, Y. H. Shen, G. E. W. Bauer, and X. R. Wang,\nPhys. Rev. Lett. 109, 167209 (2012).\n[4] X. S. Wang and X. R. Wang, Phys. Rev. B 90, 014414 (2014).\n[5] N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, A. N. Kuch ko, and V. V. Kruglyak,\nPhys. Rev. B 96, 064415 (2017).\n[6] S. K. Kim, O. Tchernyshyov, V. Galitski, and Y. Tserkovny ak,\nPhys. Rev. B 97, 174433 (2018).\n[7] J. C. Slonczewski, Int. J. Magn. 2, 85 (1972).\n19[8] R. Rajaraman, Solitons and Instantons (North-Holland, 1982) p. Chap. 8.\n[9] G. Tatara, H. Kohno, and J. Shibata, Physics Reports 468, 213 (2008).\n[10] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).\n[11] H.-B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996).\n[12] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974).\n[13] A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier,\nJournal of Applied Physics 95, 7049 (2004), https://doi.org/10.1063/1.1667804.\n[14] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[15] G. Tatara, Physica E: Low-dimensional Systems and Nano structures 106, 208 (2019).\n20"    },    {        "title": "1310.7657v1.Observational_Study_of_Large_Amplitude_Longitudinal_Oscillations_in_a_Solar_Filament.pdf",        "content": "arXiv:1310.7657v1  [astro-ph.SR]  29 Oct 2013Nature of Prominences and their role in Space Weather\nProceedings IAU Symposium No. 300, 2014\nB. Schmieder, JM. Malherbe & S. Wu, eds.c/circlecopyrt2014 International Astronomical Union\nDOI: 00.0000/X000000000000000X\nObservational Study of Large Amplitude\nLongitudinal Oscillations in a Solar Filament\nKalman Knizhnik1,2,Manuel Luna3, Karin Muglach2,4\nHolly Gilbert2, Therese Kucera2, Judith Karpen2\n1Department of Physics and Astronomy\nThe Johns Hopkins University, Baltimore, MD 21218\nemail: kalman.knizhnik@nasa.gov\n2NASA/GSFC, Greenbelt, MD 20771, USA\n3Instituto de Astrof´ ısica de Canarias, E-38200 La Laguna, T enerife, Spain\n4ARTEP, Inc., Maryland, USA\nAbstract. On 20 August 2010 an energetic disturbance triggered damped large-amplitude lon-\ngitudinal (LAL) oscillations in almost an entire filament. I n the present work we analyze this\nperiodic motion in the filament to characterize the damping a nd restoring mechanism of the\noscillation. Our method involves placing slits along the ax is of the filament at different angles\nwith respect to the spine of the filament, finding the angle at w hich the oscillation is clearest,\nand fitting the resulting oscillation pattern to decaying si nusoidal and Bessel functions. These\nfunctions represent the equations of motion of a pendulum da mped by mass accretion. With\nthis method we determine the period and the decaying time of t he oscillation. Our preliminary\nresults support the theory presented by Luna and Karpen (201 2) that the restoring force of LAL\noscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism\nis the ongoing accumulation of mass onto the oscillating thr eads. Following an earlier paper, we\nhave determined the magnitude and radius of curvature of the dipped magnetic flux tubes host-\ning a thread along the filament, as well as the mass accretion r ate of the filament threads, via\nthe fitted parameters.\nKeywords. solar prominences, oscillations, magnetic structures\n1. Procedure\nLAL oscillations consist of periodic motions of the prominence thread s along the mag-\nnetic field that are disturbed by a small energetic event close to the filament (see Luna\net al. paper in this volume). Luna and Karpen (2012) argue that pro minence oscillations\ncan be modeled as a damped oscillating pendulum, whose equation of mo tion satis-\nfies a zeroth-order Bessel function. In their model, a nearby trig ger event causes quasi-\nstationary preexisting prominence threads sitting in the dips of the magnetic structure\nto oscillate back and forth, with the restoring force being the proj ected gravity in the\ntubes where the threads oscillate (e.g. Luna et al. (2012)). In this paper, we report pre-\nliminary results of comparisonsof observations of prominence oscilla tions with the model\npresented by Luna and Karpen (2012). More details will be available in the forthcoming\npaper by Luna et al. (2013).\nIn this analysis, we place slits along the filament spine and measure the intensity along\neach slit as a function of time. Fig. 1 (left) shows the filament in the AI A 171˚A filter\nwith the slits overlaid. Each slit is then rotated in increments of 0.5◦from 0◦to 60◦\nwith respect to the filament spine. We select the best slit according t o the following\ncriteria: (a) continuity of oscillations, (b) amplitude of the oscillation is maximized, (c)\nclear transition from dark to bright regions, (d) maximum number of cycles.\nThe oscillation for a representative slit is shown in Figure 1 (right), wh ich corresponds\nto the grey slit in Figure 1 (left). We identify the position of the cente r of mass of the\n12 K. Knizhnik, M. Luna, K. Muglach, H. Gilbert, T. Kucera, J. Karpen\nFigure 1. Left: Filament seen in AIA 171 with best slits overlaid. Right: An intensity distance–\ntime slit, showing an oscillation with the Bessel fit (white c urve) to equation (2.1) in Luna et\nal. (this volume). The sinusoidal fit was not as good as the Bes sel fit and is not shown.\nthread by finding the intensity minimum along the slit, indicated by black crosses in\nFigure 1 (right). These points are then fit to equation (2.1) of Luna et al. (this volume),\nand the resulting fit is shown in white.\n2. Results\nFitting our data to equation (2.1) of Luna et al. (this volume) yields va lues ofχ2\nranging between 1-13. Using equation (2.2) of Luna et al. (this volum e), we find the\naverageradiusofcurvatureofthe magneticfield dips that suppor t the oscillatingthreads.\nWe find it to be approximately 60 Mm. We also calculate a threshold value for the field\nitself that would allow it to support the observedthreads. Using equ ation (3.1) of Luna et\nal.(this volume), wefind anaveragemagneticfield of ∼20G, assumingatypicalfilament\nnumber density of 1011cm−3, in good agreement with measurements (e.g. Mackay et al.\n2010). On average, the oscillations form an angle of ∼25owith respect to the filament\nspine, and have a period of ∼0.8 hours. To explain the very strong damping mass must\naccrete onto the threads at a rate of about 60 ×106kg/hr.\n3. Conclusions\nWe conclude that the observedoscillationsarealongthe magneticfie ld, which formsan\nangle of∼25owith respect to the filament spine (Tandberg-Hanssen & Anzer, 19 70). We\nfind that both the curvature and the magnitude of the magnetic fie ld are approximately\nuniform on different threads. Both the Bessel and sinusoidal func tions are well fitted,\nindicating that mass accretion is a likely damping mechanism of LAL oscilla tions, and\nthat the restoring force is the projected gravity in the dips where the threads oscillate.\nThe mass accretion rate agrees with the theoretical value (Karpe n et al., 2006, Luna,\nKarpen, & DeVore, 2012).\nReferences\nKarpen, J. T., Antiochos, S. K., Klimchuk, J. A. 2006, ApJ, 63 7, 531\nLuna, M., Karpen, J. T., & Devore, C. R. 2012a, ApJ, 746, 30\nLuna, M., & Karpen, J. 2012, ApJ, 750, L1\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., this volume , 2014\nLuna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., ApJ, 2013,in prep.\nMackay, D., Karpen, J., Ballester, J., Schmieder, B., Aulan ier, G. 2010, Sp. Sci. Rev. , 151, 333\nTandberg-Hanssen, E. and Anzer, U. 1970, Solar Physics 15, 158T"    },    {        "title": "1409.6900v2.Dissipationless_Multiferroic_Magnonics.pdf",        "content": "arXiv:1409.6900v2  [cond-mat.mes-hall]  17 Apr 2015Dissipationless Multiferroic Magnonics\nWei Chen1and Manfred Sigrist2\n1Max-Planck-Institut f ¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, G ermany\n2Theoretische Physik, ETH-Z¨ urich, CH-8093 Z¨ urich, Switz erland\n(Dated: October 15, 2018)\nWe propose that the magnetoelectric effect in multiferroic i nsulators with coplanar antiferromag-\nnetic spiral order, such as BiFeO 3, enables electrically controlled magnonics without the ne ed of\na magnetic field. Applying an oscillating electric field in th ese materials with frequency as low\nas household frequency can activate Goldstone modes that ma nifests fast planar rotations of spins,\nwhose motion is essentially unaffected by crystalline aniso tropy. Combining with spin ejection mech-\nanisms, such a fast planar rotation can deliver electricity at room temperature over a distance of\nthe magnetic domain, which is free from energy loss due to Gil bert damping in an impurity-free\nsample.\nPACS numbers: 85.75.-d, 72.25.Pn, 75.85.+t\nIntroduction.- A primary goal of spintronic research\nis to seek for mechanisms that enable electric ( E) field\ncontrolled spin dynamics, since, in practice, Efields are\nmuch easier to manipulate than magnetic ( B) fields. As\nspinsdonotdirectlycoupleto Efield, incorporatingspin-\norbit coupling seems unavoidable for this purpose. Along\nthis line came the landmark proposals such as spin field\neffect transistor [1] and spin-orbit torque [2–5], the real-\nizations of which suggest the possibility of spin dynamics\nwith low power consumption. On the other hand, in an-\nother major category of spintronics, namely magnonics,\nwhich aims at the generation, propagation, and detection\nof magnons, a mechanism that enables electrically con-\ntrolled magnonics without the aid of a magnetic field has\nyet been proposed.\nRaman scattering experiments [6, 7] on the room tem-\nperature multiferroic BiFeO 3(BFO) shed light on this\nissue. The magnetic order of BFO is a canted antiferro-\nmagnetic(AF) spiralontheplanespannedbythe electric\npolarization Palong[111]andoneofthethreesymmetry-\nequivalent wave vectors on a rhombohedral lattice [8, 9].\nThe spins have only a very small out-of-plane component\n[10, 11]. Applying a static Efield∼100kV/cm signif-\nicantly changes the cyclon (in-plane) and extra-cyclon\n(out-of-plane) magnons because of the magnetoelectric\neffect [7]. Indeed, spin-orbit coupling induced magneto-\nelectric effects are a natural way to connect Efield to\nthe spin dynamics of insulators [12, 13]. Motivated by\nthe Raman scattering experiments on BFO, in this Let-\nter we propose that applying an oscillating Efield to a\ncoplanar multiferroic insulator (CMI) that has AF spiral\norder can achieve electrically controlled dissipationless\nmagnonics, which can deliver electricity with frequency\nas low as household frequency up to the range of mag-\nnetic domains. Compared to the magnonics that uses B\nfield, microwave, or spin torques to generate spin dynam-\nics in prototype Y 3Fe5O12(YIG) [14–16], the advantage\nof using CMI is that a single domain sample up to mm\nsize is available [17], and Raman scattering data indicatewell-defined magnons in the absence of Bfield [7], so an\nexternal Bfield is not required in the proposed mecha-\nnism.\nSpin dynamics in CMI.- We start from the AF spiral\non a square lattice shown in Fig. 1 (a), described by\nH=/summationdisplay\ni,αJSi·Si+α−Dα·(Si×Si+α) (1)\nwhereα={a,c}are the unit vectors defined on\nthexz-plane,J >0, andDα=Dαˆ y>0 is the\nDzyaloshinskii-Moriya (DM) interaction. The staggered\nmoment ( −1)iSiin the ground state shown in Fig. 1\n(a) is characterized by the angle θα=Q·α=\n−sin−1/parenleftig\nDα/˜Jα/parenrightig\nbetweenneighboringspins,where ˜Jα=\n/radicalbig\nJ2+D2α. The DM interaction\nDα=D0\nα+wE×α (2)\ncan be controlled by an Efield [18], where D0\nαrepresents\nthe intrinsic value due to the lack of in version symmetry\nof theα-bond. In the rotated reference frame S′defined\nby\nS′z\ni=Sz\nicosQ·ri+Sx\nisinQ·ri,\nS′x\ni=−Sz\nisinQ·ri+Sx\nicosQ·ri,(3)\nandS′y\ni=Sy\ni, the Hamiltonian is\nH=/summationdisplay\ni,α˜Jα/parenleftbig\nS′x\niS′x\ni+α+S′z\niS′z\ni+α/parenrightbig\n+JS′y\niS′y\ni+α.(4)\nSince˜Jα>J, the spins have collinear AF order and all\nS′z\ni= (−1)iSlie inxz-plane.\nThespin dynamicsin the absenceof Bfieldisgoverned\nby the Landau-Lifshitz-Gilbert (LLG) equation\ndS′\ni\ndt=∂H\n∂S′\ni×S′\ni+αGS′\ni×/parenleftbigg∂H\n∂S′\ni×S′\ni/parenrightbigg\n(5)2\nexpressed in the S′frame, where αGis the phenomeno-\nlogical damping parameter. Eq. (5) can be solved by the\nspin wave ansatz for the even ( e) and odd ( o) sites [19]\n/parenleftbiggS′x\ne,o\nS′y\ne,o/parenrightbigg\n=/parenleftbiggux\ne,o\nvy\ne,o/parenrightbigg\nei(k·re,o−ωt). (6)\nIgnoringthe dampingterm inEq.(5) yieldseigenenergies\nω±\nk\n2S=\n/parenleftigg/summationdisplay\nα˜Jα±γα−(k)/parenrightigg2\n−/parenleftigg/summationdisplay\nαγα+(k)/parenrightigg2\n1/2\n,(7)\nwhereγα±(k) =/parenleftig\n˜Jα/2±J/2/parenrightig\ncosk·α. Their eigenval-\nues and eigenvectors near k= (0,0) andk= (π,π) are\nsummarized below\n/braceleftig\nω+\nk→(0,0),ω−\nk→(π,π)/bracerightig\n= 2S/radicalbig\n2(D2a+D2c),\n\nue\nve\nuo\nvo\n∝\n0\n1\n0\n∓1\n+O/parenleftbiggD\nJ/parenrightbigg\n.\n/braceleftig\nω−\nk→(0,0),ω+\nk→(π,π)/bracerightig\n= 0,\nue\nve\nuo\nvo\n∝\n1\n0\n∓1\n0\n.(8)\nThe in-plane magnon dS′\ni/dt= (dS′x\ni/dt,0,0) is gapless,\nwhile the out-of-plane magnon dS′\ni/dt= (0,dS′y\ni/dt,0)\ndevelops a gap, as displayed in Fig. 1 (c). Even includ-\ning the damping term in Eq. (5), the in-plane magnons\nvery near the Goldstone modes ω−\nk→(0,0)andω+\nk→(π,π)re-\nmainunchangedanddamping-free. Awayfromthe Gold-\nstone limit, the eigenenergies become complex, hence the\nmagnons are subject to the damping and decay within a\ntime scale set by α−1\nG.\nSpin dynamics induced by oscillating Efield.-We an-\nalyze now the spin dynamics in the damping-free in-\nplane magnonchannel induced bymagnetoelectriceffects\n(Eq. (8)). Unlike the spin injection by using the spin Hall\neffect (SHE) to overcome the damping torque [16], our\ndesign does not require an external Bfield, and is fea-\nsible over a broad range of frequencies. Consider the\ndevice shown in Fig. 2, where an oscillating electric field\nE=E0cosωtis applied parallel to the ferroelectric mo-\nment over a region of length L=Na, such that the DM\ninteraction in Eq. (2) oscillates in this region. Thus, the\nwave length of the spiral changes with time yielding an\noscillation of the number of spirals inside this region,\nnQ=L\n2π/|Q|≈N\n2πJ/bracketleftbig\nD0\na+wE0acosωt/bracketrightbig\n,(9)\nassumingDa=D0\na+wE0a≪J,Dc= 0, and E⊥a.\nSuppose the spin S0at one boundary is fixed by, for\nFIG. 1: (color online) Schematics of 2D AF spiral in the (a)\noriginalS-frame and the (b) rotated S′-frame. Red and blue\narrows indicate the spins on the two sublattices. (c) Spin\nwave dispersion ω+\nk(dashed line) and ω−\nk(solid line) solved\nin theS′frame, with Da/J= 0.14,Dc= 0. Inserts show\ntheir eigen modes in the S′frame near k= (0,0) and (π,π),\nwhere the spin dynamics dS′\ni/dtis indicated by black arrows\nor symbols.\ninstance, surface anisotropy because of specific coating.\nThenSNat the other boundary rotates by\n∂θN\n∂t=−N\nJwE0aωsinωt, (10)\nbecause whenever the number of waves nQchanges by 1,\nSNrotates 2πin orderto to wind or unwind the spin tex-\nture in the Efield region. The significance of this mecha-\nnism is that although the Efield is driven by a very small\nfrequencyω, the spin dynamics ∂tθNat the boundary is\nmanyordersofmagnitude enhanced because of the wind-\ning process. The rotation of SNserves as a driving force\nfor the spin dynamics in the field-free region from SNto\nSN+M. As long as the spin dynamics is slower than the\nenergy scale of the DM interaction ∂tθi<|D0|//planckover2pi1∼THz,\none can safely consider the Efield region as adiabatically\nchanging its wave length but remaining in the ground\nstate. The spins in the field-free region rotate coherently\n∂tθN=∂tθN+1=...=∂tθN+M, synonymous to exciting\ntheω−\nk→(0,0)mode in Eq.(8), hence the spin dynamics\nin the field-free region remains damping-free in an ideal\nsituation.\nIn real materials, crystalline anisotropy and impuri-\nties are the two major sources to spoil the spin rota-\ntional symmetry implicitly assumed here. In the supple-\nmentary material[20], their effects are discussed by draw-\ning analogy with similar situations in the atom absorp-\ntion on periodic substrates and the impurity pinning of\ncharge density wave states. It is found that crystalline3\nquantity symbol magnitude\nlattice constant a nm\ns−dexchange Γ 0.1eV\ns−dexchange time τex 10−14s\nspin relaxation time τsf 10−12s\nspin diffusion length λN 10nm\nspin density n01027/m3\nspin Hall angle θH 0.1\nintrinsic DM D0\nα10−3eV\nsuperexchange J 0.1eV\nEq. (2) w 10−19C\nelectric flux quantum ˜Φ0\nE 1V\nTABLE I: List of material parameters and their order of mag-\nnitude values.\nanisotropy remains idle because of the long spiral wave\nlength and the smallness of crystalline anisotropy com-\npared to exchange coupling. The impurities that tend to\npin the spins alongcertaincrystalline directionopen up a\ngap in the Goldstone mode and cause energy dissipation,\nwhich nevertheless do not obstruct the coherent rotation\nof spins generated by Eq. (10).\nFIG. 2: (color online) Experimental proposal of using oscil -\nlatingEfield to induce spin dynamics in CMI. The AF spiral\norder is shown in the S′frame. The Efield is applied between\nS′\n0andS′\nN, causing dynamics in the whole spin texture. Two\nways for spin ejection out of S′\nN+Mare proposed: (a) Using\nSHE to converted it into a charge current. (b) Using time-\nvarying spin accumulation and inductance.\nSpin ejection and delivery of electricity.- We now ad-\ndress the spin ejection from the CMI to an attached\nnormal metal (NM). A spin current is induced in the\nNM when a localized spin Siat the NM/CMI inter-\nface rotates [16, 21]. Defining the conduction electron\nspinm(r,t) =−∝an}b∇acketle{tσ∝an}b∇acket∇i}ht/2, thes-dcoupling at the interface\nHsd= Γσ·Sidefines a time scale τex=/planckover2pi1/2S|Γ|, withΓ<0 [21]. The Bloch equation in the NM reads\n∂m\n∂t+∇·Js=1\nτexm׈Si−δm\nτsf(11)\nwhereJs=JNM\ns/varotimesσ/planckover2pi1/2 is the spin current tensor, and\nτsfis the spin relaxation time in the NM. In equilib-\nrium, we assume mhybridizes with each Sion the spi-\nral texture locally. If the dynamics of Siis slow com-\npared to 1/τex, which is true for the proposed mechanism\nand also for other usual means such as ferromagnetic\nresonance[16], mfollows−ˆSiat any time with a very\nsmall deviation m=m0+δm=−n0ˆSi+δm, wheren0\nis the local equilibrium spin density. The spin current\ntensorJs=−D0∇δmis obtained from the diffusion of\nδm, whereD0is the spin diffusion constant. Under such\nan adiabatic process, the small deviation is[21]\nδm=τex\n1+ξ2/braceleftigg\n−ξn0∂ˆSi\n∂t−n0ˆSi×∂ˆSi\n∂t/bracerightigg\n,(12)\nwhereξ=τex/τsf<1 so one can drop the first term on\nthe right hand side, and replace ˆSi×∂tˆSi→δ(r)ˆSi×∂tˆSi\nsinceˆSiis located at the NM/CMI interface r= 0 (ras\ncoordinate perpendicular to the interface). The resulting\nequation solves the time dependence of δm. Away from\nr= 0, Eq. (11) yields D0∇2δm=δm/τsf, which solves\nthe spatial dependence of δm. The spin current caused\nby a particular Sithen follows\nJNM\nsδˆm=δmD0\nλN=−τexn0D0\n(1+ξ2)λNˆSi×∂ˆSi\n∂te−r/λN,(13)\nwhereλN=/radicalbig\nD0τsf, similar to results obtained previ-\nously[16]. Ifonlythein-planeGoldstonemodeisexcited,\nas shown in Fig. 2, it is equivalent to a global rotation of\nspinsˆSi= (−1)i(sin(θ(t) +Q·ri),0,cos(θ(t) +Q·ri))\nin the field-free region. Thus the time dependence in\nEq. (13), ˆSi×∂tˆSi=ˆy∂θ/∂t, is that described by\nEq. (10), and is the same for every Siat the NM/CMI\ninterface, even though each Sipoint at a different polar\nangle. In other words, the spin current ejected from each\nSiof the AF spiral, described by Eq. (13), is the same,\nso a uniform spin current flows into the NM.\nWe propose two setups to convert the ejected spin cur-\nrent into an electric signal. The first device uses inverse\nSHE[16]inaNMdepositedatthesideofthespiralplane,\nyieldingδˆmperpendicular to JNM\nsand consequently a\nvoltage in the transverse direction, as shown in Fig. 2\n(a). The second design ejects spin into a NM film de-\nposited on top of the spiral plane, as shown in Fig. 2\n(b), causing δˆmparallel to JNM\ns. A spin accumulation\nin the NM develops and oscillates with time, producing\nan oscillatingmagnetic flux Φ Bthrougha coil that wraps\naround the NM, hence a voltage E=−∂ΦB/∂t.\nExperimental realizations.- TheRamanscatteringdata\non BFO [7] show that applying |E| ∼100kV/cm can4\nchange the spin wave velocity by δv0/v0∼1%. We\ncan make use of this information to estimate the field-\ndependence win Eq. (2). The ω−\nkmode in Eq. (7) near\nk= (0,0) is\nω−\nk→0= 2√\n2SJka/bracketleftbigg\n1+5\n16/parenleftbiggD2\nak2\na+D2\nck2\nc\nJ2k2/parenrightbigg/bracketrightbigg\n= (v0+δv0)k, (14)\nwherev0= 2√\n2SJais the spin wave velocity in the\nabsence of DM interaction. Assuming Da∝ne}ationslash= 0,Dc=\n0, andE⊥a, the Raman scattering data gives w∼\n10−19C∼ |e|. We remark that a coplanar magnetic order\ncan be mapped into a spin superfluid [36, 37] ψiby\n∝an}b∇acketle{tSi∝an}b∇acket∇i}ht=S(sinθi,0,cosθi) =√v(Imψi,0,Reψi),(15)\nwherevis the volumeofthe 3Dunit cell. Within this for-\nmalism, the Efield can induce quantum interference of\nthe spin superfluid via magnetoelectric effect, in which\nthe electric flux vector ΦE=/contintegraltext\nE×dlis quantized\n[24, 25]. The flux quantum is ˜Φ0\nE= 2πJ/w, which is\n˜Φ0\nE∼1V for BFO, close to that ( ∼10V) obtained from\ncurrent-voltage characteristics of a spin field-effect tran-\nsistor [24], indicating that strong spin-orbit interaction\nreduces the flux quantum to an experimentally accessi-\nble regime. For instance, BFO has a spiral wave length\n2π/Q∼100nm, so in a BFO ring of µm size, the num-\nber of spirals at zero field is nQ∼10, and applying\n|E| ∼1kV/cm can change nQby 1. Besides changing\nthe winding number, we remarkthat the magnetoelectric\neffect can also be used to affect the topological proper-\nties of a magnet in a different respect[26]. Table I lists\nthe parameters and their order of magnitude values by\nassuming CMI has similar material properties as other\nmagnetic oxide insulators such as YIG, and we adopt\nlattice constant a∼1nm for both CMI and the NM for\nsimplicity.\nFor the device in Fig. 2, consider the field |E0| ∼\n100kV/cm oscillating with a household frequency ω∼\n100Hz is applied to a range L∼1mm. This region covers\nN=L/a∼106sites with a number of spirals nQ∼104\nat zero field. The Efield changes the number of spi-\nrals tonQ∼105within time period 1 /ω∼0.01s, so\nthe spins at the boundary SNwind with angular speed\n∂tθN∼107sinωtwhich is enhanced by 5 orders of mag-\nnitude from the driving frequency ω. To estimate the\nejected spin current in Eq. (13), we use the typical spin\nrelaxation time τsf∼10−12s and length λN∼10nm\nfor heavy metals [16]. The s-dcoupling can range be-\ntween [16] 0 .01eV to 1eV. We choose Γ ∼0.1eV, which\ngivesτex∼10−14s. The spin Hall angle θH∼0.1 has\nbeen achieved [27, 28]. To estimate n0, we use the fact\nthat thes-dhybridization Γ σ·Siis equivalent to ap-\nplying a magnetic field H= 2ΓSi/µ0gµBlocally at the\ninterface atomic layer of the NM. Given the typical mo-\nlar susceptibility χm∼10−4cm3/mol and molar volumeVm∼10cm3/mol, the interface magnetization of the NM\nisn0µB=χmH/Vm∼104C/sm, thus n0∼1027/m3.\nTheoscillating Efieldgives ˆSi×∂tˆSi=∂tθNˆy∼ˆy107Hz,\nso the ejected spin current is JNM\ns∼1024/planckover2pi1/m2s. Using\nthe design in in Fig. 2(a) to convert JNM\nsinto a charge\ncurrent via inverse SHE yields JNM\nc∼104A/m2, hence\na voltage ∼µV oscillating with ωin a mm-wide sample.\nTo use the setup in Fig. 2(b), a NM film of area ∼1 mm2\nand thickness ∼10nm yields E ∼mV oscillating with ω.\nIn summary, we propose that for multiferroics that\nhave coplanar AF spiral order, such as BFO, applying\nan oscillating Efield with frequency as low as house-\nhold frequency generates a coherent planar rotation of\nthe spin texture whose frequency is many orders of mag-\nnitude enhanced. This coherent rotation activates the\nGoldstone mode of multiferroic insulators that remains\nunaffected by the crystalline anisotropy. The Goldstone\nmode can be used to deliver electricity at room tempera-\nture up to the extensions of magnetic domains, in a way\nthat is free from the energy loss due to Gilbert damping\nif the sample is free from impurities. The needlessness\nofBfield greatly reduces the energy consumption and\nincreases the scalability of the proposed device, pointing\nto its applications in a wide range of length scales.\nWe thank exclusively P. Horsch, J. Sinova, H. Naka-\nmura, Y. Tserkovnyak, D. Manske, M. Mori, C. Ulrich,\nJ. Seidel, and M. Kl¨ aui for stimulating discussions.\nSupplementary material\nI. Crystalline anisotropy in multiferroics\nFirst we demonstrate that because the wave length of\nthe spiral order in multiferroics is typically 1 ∼2 orders\nlonger than the lattice constant, and the exchange cou-\npling is typically few orders larger than the crystalline\nanisotropy energy, the spiral order remains truly incom-\nmensurate and very weakly affected by the crystalline\nanisotropy. For simplicity, we consider a spiral state with\nwave vector Q∝ba∇dbl(1,0) and translationally invariant per-\npendicular to Qsuch that the geometry can be reduced\nto a 1D problem. The classical elastic energy for a 1D\nantiferromagnetic (AF) spiral is\nE0=/summationdisplay\nn−˜JaS2cos(θn+1−θn−θa)\n≈ −N˜JaS2+/summationdisplay\nn1\n2˜JaS2(θn+1−θn−θa)2,(16)\nwhereθn=Q·rnis the angle relative to the staggered\nspin (−1)iSi, andθa=Q·ais the natural pitch an-\ngle between neighboring spins ( a= (a,0)). The square\nlattice symmetry of our model yields a 4-fold degener-\nate crystalline spin anisotropy[38], leading to the total5\nenergy\nE=/summationdisplay\nn1\n2˜JaS2(θn+1−θn−θa)2\n+/summationdisplay\nnVani(1−cos4θn), (17)\nwhereVaniis the anisotropy energy per site. This is the\nwell-known Frenkel-Kontorowa(FK) model[30, 31] that\nhas been discussed extensively owing to its rich physics.\nFIG. 3: (color online) Schematics of mapping the AF spi-\nral order in the presence of crystalline anisotropy into FK\nmodel. The angles θiof staggered spins ( −1)iSi(blue ar-\nrows) are mapped into displacements xiof particles (orange\ndots). The width of the 4-fold degenerate pinning potential\nV(1−cos2πxi/b) isb=π/2, and the spacing of particles in\nthe absence of the pinning potential is a0=Q·a.\nWe consider the limit of weak anisotropy V=\nVani/˜JaS2a2≪1 and the case of long wavelength of\nthe spiral, θa≪π/2 whereπ/2 is the angle between\ntwo minima of the anisotropy potential. In the spirit of\nRef.[32, 33] we assume now that there are prime num-\nbers,MandLwithM˜θa=Lπ/2 andM≫Lwhich is\nthe average pitch in the ground state of Eq.(17). Then\nwe introduce the parametrization\nθn=n˜θa+ϕn\n4(18)\nand the misfit parameter δ= 4(θa−˜θa). Turning to\nthe continuous limit one can derive the effective en-\nergy functional based on expanding the first harmonic\napproximation[32–34],\n˜E[ϕ] =/integraldisplay\ndx/bracketleftigg\n1\n2/parenleftbiggdϕ\ndx−δ/parenrightbigg2\n+VMcos(Mϕ)/bracketrightigg\n(19)\nwithVM∼VMwhich can become extremely small for\nM≫1. The commensurate-incommensurate transition\nhappens ifδis large enough to stabilize the formation of\nsolitonsδ > δc(M)∼4√VM/π. Deep inside the incom-\nmensurate phase, ϕ(x)≈δxsuch thatθn≈θanfollows\nesentially the natural spiral pitch.\nIn our system, BFO, the spiral wave length ℓ≈\n60nm∼100awhich yields M∼100/4 = 25, i.e. every\n25thspin could be pinned along one of the 4 anisotropy\nminima (assuming L= 1). Typical anisotropy ener-\ngies for ferrites[35] lead to Vani∼10−3eV while theexchange energy is Ja∼0.1eV, from which we obtain\nV∼Vani/Ja∼10−2and consequently VM∼10−50is\na negligible number. The misfit parameter may be as\nlarge asδ= 4(θa−˜θa)∼π/M2such thatδ≫δc(M) is\nwell satisfied, even if by an electrical field Mshrinks by\none order of magnitude. Thus, the electric field-driven\noscillations of the spin spiral remains most likely unaf-\nfected by the spin anisotropy. The small VMrenders\nthe energy gap due to the anisotropy energy irrelevant,\nhence the in-plane magnon mode remains essentially un-\ndamped. Another important consequence of this analy-\nsis is that although the concept of spin superfluidity, i.e.,\ntreating the spin texture as a quantum condensate, has\nbeen proposed long ago, its realization in collinear mag-\nnets is problematic because of the crystalline anisotropy\nand subsequently the formation of domain walls. We\ndemonstrate explicitly that multiferroics are not sub-\nject to these problems because of the noncollinear or-\nder, hence a room temperature macroscopic condensate\nof mm size can be realized.\nII. Phase-pinning impurities in multiferroics\nWe proceed to show that dilute, randomly distributed\nimpurities, exist either in the bulk of the multiferroic or\nat the metal/multiferroic interface, do not obstruct the\nproposed electrically controlled multiferroic magnonics.\nDrawing analogyfrom the FK model, impurities that pin\nthe spins along certain crystalline directions, denoted by\nphase-pinningimpurities, arethe impuritiesto be consid-\nered because they tend to impede the coherent motion of\nspins[39]. Since we propose to use an oscillating Efield\nto drive the spin rotation from the boundary, each cross\nsection channel is equivalent, which reduces the problem\nfrom 2D to 1D. This leads us to consider the following\n1D classical model similar to Eq. (17) for the field-free\nregion (S′\nN+1toS′\nN+Min the Fig. 2 of the main text).\nE=/summationdisplay\ni1\n2˜JaS2(θi+a−θi−θa)2−/summationdisplay\ni∈impVimpcos4θi,(20)\nwhereVimp>0 is the pinning potential, and/summationtext\ni∈imp\nsums over impurity sites. The total length of the chain is\nL′=MawithManinteger. Inthepresenceofoscillating\nEfield that causes the winding of boundary spins ( S′\nN\nin the Fig. 2 of the main text), the angle of spins in the\ndisordered field-free region has three contributions\nθi=θ0\ni+∆θi+ηi, (21)\nwhereθ0\nirepresents the spiral texture in the unstretched\ncleanlimitsatisfying θ0\ni+a−θ0\ni−θa= 0,∆θiisthestretch-\ning of the spin texture caused by winding of boundary\nspins, and ηiis the distortion due to impurities. Only\nthe later two contribute to the elastic energy, so Eq. (20)6\nbecomes\nE=/summationdisplay\ni1\n2˜JaS2(∆θi+a−∆θi+ηi+a−ηi)2\n−/summationdisplay\ni∈impVimpcos4θi. (22)\nIn this analysis we consider weak impurities Vimp≪\n˜JaS2, andassumethatthe windingofthe boundaryspins\nis slow such that the winding spreads through the whole\nfield-free region evenly, causing every pair of neighboring\nspins to stretch by the same amount ∆ θi+a−∆θi= ∆θ.\nFor the electrically driven magnonics proposed in the\nmain text, which can achieve winding of boundary spins\nbyθN∼nQ∼105within half-period, a field-free region\nof lengthL′∼mm has ∆θ∼0.1, so our numerics is done\nwith ∆θlimited within this value.\nIn the weak impurity limit, the length scale L0over\nwhichηichanges by O(1) can be calculated in the fol-\nlowing way. The elastic energy part in Eq. (22) within\nL0is, in the continuous limit,\nK(L0) =1\na/integraldisplayL0\n0dx1\n2˜JaS2a2/parenleftbigg∆θ\na+∂xη/parenrightbigg2\n=L0\n2a˜JaS2∆θ2+˜JaS2∆θ\nα1+˜JaS2a\n2α0L0,(23)\nwhereα0andα1are numerical constants of O(1), and\nare set to be unity without loss of generality. Denoting\nimpurity density as nimp=Nimp/L′whereNimpis the\ntotal number of impurities in the sample, the impurity\npotential energy within L0is calculated by\nV(L0) =−VimpRe\n/summationdisplay\ni∈impe4i(θ0\ni+∆θ+η)\n\n=−Vimp/radicalbig\nnimpL0. (24)\nNote that the contribution comes not from the zeroth\norder impurity averaging, but its fluctuation that mimics\na random walk in the complex plane[40]. The phase ηis\nassumed to be constant within L0and chosen to give\nEq. (24) and hence the total energy E(L0) =K(L0) +\nV(L0) withinL0. Minimizing the total energy per site\nE(L0)/L0gives the most probable pinning length L0. In\nthe unstretched case ∆ θ= 0,\nL0=/parenleftigg˜JaS2a\nα0Vimpn1/2\nimp/parenrightigg2/3\n(25)\nis similar to the Fukuyama-Lee-Rice (FLR) length that\ncharacterizes the impurity pinning of a charge density\nwave ground state[40, 41]. Putting Eq. (25) back to\nEqs. (24) and (23), the corresponding E(L0)a/L0<0\ncan be viewed as the pinning energy per site that im-\npedes the coherent rotation of spins, and equivalently\nrepresents the gap opened at the Goldstone mode.00.020.040.060.08 0.10246810\n/CapDΕltaΘLog10/LParen1L0/Slash1a/RParen1/LParen1a/RParen1 Aimp/EΘual10/Minus5\n10/Minus4\n10/Minus3\n10/Minus2\n00.020.040.060.08 0.1/Minus1/Minus0.500.51\n/CapDΕltaΘΕ/Multiply103/LParen1b/RParen1\nAimp/EΘual10/Minus4\n10/Minus3\n5/Multiply10/Minus3\n10/Minus2\nFIG. 4: (color online) (a) The logrithmic of the dimensionle ss\nFLR length log10(L0/a) versus winding angle per site ∆ θ, in\nseveral values of the dirtiness parameter Aimp. Dashed line\nindicates the threshold when L0∼mm. (b) The dimensionless\npinning energy ǫversus winding per site.\nIn the presence of the stretching ∆ θ, the expression\nofL0is rather lengthy. It is convenient to define two\ndimensionless parameters\nAimp=Vimp\n˜JaS2/radicalbigg\nNimpa\nL′,\nǫ=E(L0)\n˜JaS2/parenleftbigga\nL0/parenrightbigg\n, (26)\nwhereAimp(the ”dirtiness parameter”) is the impurity\npotential measured in unit of the elastic constant times\nthe square root of the impurity density, and ǫis the to-\ntal energy per site measured in unit of the elastic con-\nstant. Figure 4 shows the logrithmic of the dimensionless\npinning length L0/aand the dimensionless total energy\nǫ, plotted as functions of the stretching ∆ θ. There are\ntwo evidences showing that the spin texture, originally\npinned by impurities with the pinning length in Eq. (25),\nis depinned by the stretching ∆ θ: Firstly, the pinning\nlengthL0increases as increasing ∆ θ. For a particular\nsample size, for instance L′∼mm, the spin texture is\ndepinned when the pinning length exceeds the sample\nsizeL0> L′, or equivalently when ∆ θis greater than\na certain threshold (intercept of the dashed line and the\ncoloredlinesinFig.4(a)). Secondly, thepinningenergy ǫ\nbecomes positive at large ∆ θ, indicating that the elastic\nenergy from stretching overcomes the impurity pinning\nenergy, so the spin texture is depinned. From Fig. 4, it\nis also evident that the cleaner is the sample, the easier\nit is to depin the spins by stretching, as smaller Aimpre-\nquires smaller threshold value of ∆ θ. We conclude that\nthephasepinning impuritiesdonot hampertheproposed\nelectrically driven multiferroic magnonics as long as the\ndirtiness of the sample is limited, the winding speed of\nthe boundary spin is sufficient, and the sample size is\nshort enough.\n[1] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[2] A. Manchon and S. Zhang, Phys. Rev. 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B 19, 3970 (1979)."    },    {        "title": "2204.09923v1.Transport_theory_for_topological_Josephson_junctions_with_a_Majorana_qubit.pdf",        "content": "Transport theory for topological Josephson junctions with a Majorana qubit\nZhi Wang,1,\u0003Jia-Jin Feng,2,\u0003Zhao Huang,3,yand Qian Niu4\n1School of Physics, Sun Yat-sen University, Guangzhou 510275, China\n2International Center for Quantum Materials, Peking University, Beijing 100871, China\n3Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA\n4School of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China\nWe construct a semiclassical theory for the transport of topological junctions starting from a microscopic\nHamiltonian that comprehensively includes the interplay among the Majorana qubit, the Josephson phase, and\nthedissipationprocess. Withthepathintegralapproach,wederiveasetofsemiclassicalequationsofmotionthat\ncan be used to calculate the time evolution of the Josephson phase and the Majorana qubit. In the equations we\nrevealrichdynamicalphenomenasuchasthequbitinducedchargepumping,theeffectivespin-orbittorque,and\nthe Gilbert damping. We demonstrate the influence of these dynamical phenomena on the transport signatures\nof the junction. We apply the theory to study the Shapiro steps of the junction, and find the suppression of the\nfirst Shapiro step due to the dynamical feedback of the Majorana qubit.\nIntroduction. –Josephsonphysicsreceivesrevivinginterests\nduetotherapidprogressofsuperconductingquantumcompu-\ntationinrecentyears[1,2]. Thedemandfortheminificationof\nthe superconducting quantum circuits pushes the limits of the\nsizeofJosephsonjunctions[3,4]. Forjunctionsthataresmall\nenough,asingleembeddedqubitmaysignificantlymodifythe\ntransport signatures[5]. While this effect has been discussed\nin a number of systems with various models[6–8], a compre-\nhensive theory that takes account the qubit dynamics is still\nabsent.\nThis issue is particularly relevant to topological Josephson\njunctionswithMajoranazeromodes[9–31]. ThetwoMajorana\nzero modes in the junction construct a Majorana qubit which\nresults in a 4𝜋-periodic Josephson current[32, 33]. Previous\ntheoretical studies take a variety of phenomenological mod-\nels which are different extensions of the standard resistively\nshunted junction model for conventional junctions [6, 7, 34–\n37]. However, these phenomenological models still have dif-\nficultiesinexplainingtheexperimentalreportedtransportfea-\ntures such as the suppression of the first Shapiro step[38–45].\nIt is highly desirable to construct a microscopic theory to ex-\namine the validity of the phenomenological models and to\nunderstand the experimental results.\nInthiswork,wedevelopasemiclassicaltheoryforstudying\nthe transport properties of the topological junctions. Our the-\nory starts from a microscopic Hamiltonian that characterizes\nthe coupling between the Josephson junction and the Majo-\nranaqubit. Wetakeapathintegralapproachtoincorporatethe\ndissipation process that is essential for studying the transport\nproperties,andderivethesemiclassicalequationsofmotionfor\ntheJosephsonphaseandtheMajoranaqubit. Intheequations\nof motion, we identify the effective spin-orbit torque and the\nGilbert damping in the qubit dynamics, and reveal the charge\npumping driven by the qubit rotation in the dynamics of the\nJosephson phase. Solving the equations of motion, we ob-\ntainthetimeevolutionoftheJosephsonphasewhichprovides\ntransport and spectroscopic signatures for the junction.\nAs an application of this theory, we calculate the Shapiro\nsteps of the topological junction. We find that the first step\nis strongly suppressed while higher odd-number steps are ro-bustlyvisibleforarangeofjunctionparameters. Weshowthat\nthis bizarre behavior is due to the feedback of the Majorana\nqubitdynamicstothetransportofthejunction. Atthevoltage\nof the first step, the Majorana qubit evolves to a stable state\nwhich supports a finite 4𝜋-period Josephson current, and this\n4𝜋-periodicity in the Josephson phase dynamics suppresses\nthefirstShapirostep. Atvoltagesofhigherodd-numbersteps,\nhowever, the Majorana qubit evolves to a stable state which\ncontributes a vanishing 4𝜋-period Josephson current, and the\nShapiro steps are naturally intact. Our theory provides an\nintrinsic mechanism for the reported Shapiro step missing in\ntopological junctions.\nMicroscopic Hamiltonian and equations of motion. –The\nlow-energy effective Hamiltonian for the junction with a Ma-\nTSC TSC\nMajorana qubit(a)\n(b)\n0 1 2-101(c)\n0 1 2-101\nFIG. 1. (a) Schematic illustration of the two tunneling processes\nthrough the Majorana qubit: The qubit assisted half-pair tunneling\nthat leads to the fractional Josephson effect, and the qubit rotation\ninduced charge pumping. (b) The energy levels of the Hamiltonian\nh\u0001ˆ𝜎. Landau-Zener transitions happen at the anti-crossing points\nat𝜃=¹2𝑛¸1º𝜋. (c) The energy levels of the Hamiltonian ¹h¸\n¤𝜃B𝑓º\u0001ˆ𝜎for high voltage of ¤𝜃𝐵𝑓\u001d𝐸M, where B𝑓=𝐵𝑓ˆ𝑥. The\nLandau-Zener transition at the anti-crossing points are significantly\nsuppressed and the qubit dynamics would follow one of the levels.arXiv:2204.09923v1  [cond-mat.supr-con]  21 Apr 20222\njorana qubit can be written as[46],\nHJ=ˆ𝑝2\n𝜃\n𝐶0\u0000𝐸Jcosˆ𝜃\u0000𝐼exˆ𝜃\u0000𝐸M𝜎𝑧cosˆ𝜃\n2¸𝐸0\nM𝜎𝑥(1)\nwhere ˆ𝜃is the Josephson phase with canonical momentum\nˆ𝑝𝜃=2𝑒𝜕𝜃,𝐸Jis the Josephson energy, 𝐶0=2𝐶~2¹2𝑒º2\nis a dimensionless constant determined by the effective ca-\npacitance𝐶of the junction, 𝐼exrepresents the experimentally\ncontrollable external current injected into the junction, 𝜎𝑥𝑧\nare Pauli matrices which represent the pseudo-spin direction\nfor the Majorana qubit, 𝐸Mand𝐸0\nMrepresent the energies\nof the Majorana qubit from various couplings between Ma-\njorana zero modes. The first three terms of the Hamiltonian\nhave been widely adopted for studying conventional Joseph-\nson junctions[5], while the last two terms come from the\nMajorana zero modes[32, 47] and can be derived from the\nBogoliubov- de Gennes Hamiltonian of a topological Joseph-\nson junction[48, 49].\nThis Hamiltonian can be understood as describing a spin-\none-half particlewith a mass of 𝐶0, moving under a po-\ntential energy 𝑈𝑝=\u0000𝐸Jcos𝜃\u0000𝐼ex𝜃, and a Zeeman en-\nergy𝑈𝑧=h\u0001ˆ𝜎where the direction of the Zeeman field\nh=\u0000𝐸0\nM0\u0000𝐸Mcos𝜃\n2\u0001varies along the path of the motion.\nThepotentialenergyisidenticaltothetilted-washboardpoten-\ntialthatwastakeninstudyingconventionaljunctions[5],while\nthe unique Zeeman energy comes from the coupling between\nthe Josephson phase and the Majorana qubit.\nThe time-evolution of the Josephson phase determines the\ntransport properties of the junction through the ac Josephson\nrelation[50]. To derive the equation of motion for this time-\nevolution, we rewrite the Hamiltonian of Eq. (1) into an\naction[46, 51, 52],\n𝑆J=∫\n𝑑𝑡\u0012𝐶0\n2¤𝜃2¸𝐸Jcos𝜃¸𝐼ex𝜃¸A𝑠\u0001¤s¸h\u0001s\u0013\n(2)\nwhere s=𝜓yˆ𝜎𝜓=¹sin𝜑sin𝜙sin𝜑cos𝜙cos𝜑ºrepre-\nsents the psuedo-spin state on the Bloch sphere with 𝜓=\n¹𝑒\u0000𝑖𝜙cos𝜑\n2sin𝜑\n2ºthe spinor wave function of the qubit,\nA𝑠=ˆe𝜙¹1\u0000cos𝜑ºsin𝜑represents the Berry connection\non the Bloch sphere[52], which provides a Berry curvature\nofr\u0002A𝑠=s. The extreme action path of Eq. (2) gives\nthesemiclassicalequationsofmotionfortheJosephsonphase\n𝐶0¥𝜃¸𝐼𝑐1sin𝜃¸𝐼𝑐2𝑠𝑧sin𝜃\n2\u0000𝐼ex=0, and the pseudo spin\n¤s=h\u0002s, where𝐼𝑐1=2𝑒𝐸J~is the supercurrent from the\nCooper pair tunneling and 𝐼𝑐2=𝑒𝐸M~is the supercurrent\nfrom the half-pair tunneling through the Majorana qubit. The\nequations of motion explicitly demonstrate the coupling be-\ntween the Josephson phase and the Majorana qubit through\nthe𝑠𝑧dependent Zeeman term in the first equation and the\n𝜃dependent effective magnetic field in the second equation.\nHowever,theseequationsareinadequateforstudyingthetrans-\nportpropertiesofthejunction. Themissingpieceisthedissi-\npation process.\nTo include the dissipation into the equations of motion, we\nfollowtheCaldeira-Leggettapproachandintroduceathermalbath of harmonic modes to characterize the environment[53,\n54]. Theenvironmentaldegreesoffreedomandtheircoupling\nwith the junction can be described with the action[46],\n𝑆en=∑︁\n𝑖∫\n𝑑𝑡\u00121\n2\u0010\n¤ℎ2\n𝑖\u0000Ω2\n𝑖ℎ2\n𝑖\u0011\n¸ℎ𝑖¹𝑔𝑖𝜃¸B𝑖\u0001sº\u0013\n(3)\nwhereℎ𝑖are the coordinates of the environmental modes and\nΩ𝑖are their oscillating energies, 𝑔𝑖represents the minimal\ncouplingbetweentheenvironmentalmodesandtheJosephson\nphase[54], and B𝑖represents the minimal coupling between\nthe environmental modes and the qubit[55]. The details of\n𝑔𝑖andB𝑖are determined by the coupling between each envi-\nronmental mode and the junction. For topological junctions\ndescribed by Eq. (1), the environmental modes that modulate\nthe tunneling barrier of the junction can be understood as a\nfluctuation on the amplitude of 𝐸Mand we have B𝑖=𝐵𝑖ˆ𝑧,\nwhile the environmental modes that modulate the coupling\nbetween Majorana zero modes in one side of the junction can\nbeunderstoodasafluctuationon 𝐸0\nMandwehave B𝑖=𝐵𝑖ˆ𝑥. In\ngeneral, the environment modes may originate from multiple\nsources, such as electromagnetic perturbations, lattice vibra-\ntions, as well as thermally activated electrons. These modes\ncould have much more complex couplings with the junction,\nwhile the Caldeira-Leggett model is the minimal model for\ndescribing the dissipation processes.\nThe dissipated evolutionof the junction can beobtained by\nintegratingouttheenvironmentaldegreesoffreedom. Thisis\nachievable since the environment is modeled with harmonic\nmodes. After the integration, we arrive at an effective action\nfor the junction variables[46],\n𝑆eff=∫\n𝑑𝑡\u0012𝐶0\n2¤𝜃2¸𝐸Jcos𝜃¸𝐼ex𝜃¸A𝑠\u0001¤s¸h\u0001s\u0013\n(4)\n¸1\n4∫\n𝑑𝑡𝑑𝑡0»𝜂𝛼¹𝑡º\u0000𝜂𝛼¹𝑡0º¼𝐺𝛼𝛽¹𝑡𝑡0º\u0002\n𝜂𝛽¹𝑡º\u0000𝜂𝛽¹𝑡0º\u0003\n\nwhere𝜂𝛼=¹𝜃𝑠𝑥𝑠𝑦𝑠𝑧ºrepresents the junction degree of\nfreedom,𝐺𝛼𝛽¹𝑡𝑡0º=\u0000𝑖˜𝑀\n𝜋1\nj𝑡\u0000𝑡0j2is the averaged Green func-\ntionfromtheenvironmentalmodes,with ˜𝑀theaveragedcou-\nplingmatrix[46]. Theleastactionpathforthiseffectiveaction\nprovides the full semiclassical equations of motion for the\njunction variables,\n𝐼ex=𝐶0¥𝜃¸¤𝜃\n𝑅¸𝐼𝑐1sin𝜃¸𝐼𝑐2𝑠𝑧sin𝜃\n2¸B𝑓\u0001¤s(5a)\n¤s=h\u0002s¸¤𝜃B𝑓\u0002s¸¹˜𝛾\u0001¤sº\u0002s (5b)\nwhere𝑅=1Í\n𝑖𝑔2\n𝑖is the effective resistance of the junction\nwhichcomesfromthecouplingbetweentheenvironmentand\nthe Josephson phase, B𝑓=Í\n𝑖𝑔𝑖B𝑖is the environment medi-\natedcouplingfieldbetweentheJosephsonphaseandthequbit,\nand˜𝛾𝛼𝛽=Í\n𝑖𝐵𝑖𝛼𝐵𝑖𝛽representstheenvironmentinduceddis-\nsipation to the qubit.\nThe equation (5) is the central result of this work. Solving\nthese two self-consistent equations we can obtain 𝜃¹𝑡ºwhich3\ndetermines the dc and ac voltage of the junction through the\nJosephson relation 𝑉¹𝑡º=~¤𝜃¹𝑡º2𝑒. Starting from a mi-\ncroscopic Hamiltonian, we provide a framework to study the\ntransportpropertiesofaJosephsonjunctionwithanembedded\nMajorana qubit.\nPhysical interpretation of the equations of motion. – Let us\nsetup a physicalpicture for interpreting the termsin the equa-\ntions of motion, particularly those terms coming from the\nembedded Majorana qubit. The equation (5a) is a current\nconservationequationwhichstatesthattheexternallyinjected\ncurrent𝐼exequals the current flowing through all the physical\nchannels in the junction. If the qubit is completely ignored,\nthenthelasttwotermsontherightsideoftheEq. (5a)should\nbe dropped out and the equation becomes a self-consistent\nequation for the Josephson phase. This is exactly the resis-\ntively and capacitively shunted junction model that has been\nwidely used for studying bulk Josephson junctions[5].\nThe qubit provides two additional terms in Eq. (5a) as\nillustratedinFig. 1a. Thefirsttermisthe 4𝜋-periodicJoseph-\nson current 𝐼𝑐2𝑠𝑧sin𝜃\n2, which is linearly dependent on the\nz-component of the pseudo spin. This is the extensively dis-\ncussed fractional Josephson effect[32, 48, 56], which comes\nfromthequbitassistedhalf-pairtunnelinginthejunction. The\nother term is B𝑓\u0001¤swhich is non-vanishing only when the\npseudo spin rotates. Since the pseudo-spin state of the Ma-\njorana qubit is defined by the parity of the superconducting\nground state, this current represents the pumped current by\nthe parity flipping of the Majorana qubit. This qubit pumping\nhasneverbeenrevealedinpreviousmodels,andonlybecomes\napparent from the effective action of the microscopic theory.\nNow we take a closer look at the Eq. (5b). In the ab-\nsence of the environment, only the first term on the right\nside of equation survives. The residing equation, ¤s=h\u0002s,\ndescribes a qubit procession where the direction of the pro-\ncession hoscillates with 𝜃. When the oscillating component\nℎ𝑧is much larger than the stable component ℎ𝑥, the qubit\nwould evolve under an oscillating energy spectrum shown in\nFig. 1b. When 𝜃moves through the anti-crossing points\n𝜃=¹2𝑛¸1º𝜋,thequbitexperiencesLandau-Zenertransitions,\nandmultiplecoherentLandau-ZenertransitionsexhibitStück-\nelburg interference[57]. These effects have been thoroughly\nanalyzed in previous phenomenological models[6, 58].\nThe second term on the right side of Eq. (5b) is a unique\ndiscoveryofourtheory. Itresemblesaspin-orbittorquewhich\nlinearly depends on the velocity of the Josephson phase. This\nspin-orbittorquedominatesthequbitdynamicsathighvoltage\nof¤𝜃\u001d𝐸0\nM~,causingasignificantsuppressionoftheenergy\ncrossing and the Landau-Zener transition, as shown in Fig.\n1c. For this reason, the transport and spectroscopic signals\nof the junction are expected to exhibit qualitatively different\nbehaviors for different voltage regimes. This is useful for\nunderstandingthevoltage-dependentsignaturesthathavebeen\nwidelyreportedintheI-VcharacteristicscurvesandJosephson\nradiations of Josephson junctions constructed by topological\nsystems[10, 13, 19].\nThethirdtermontherightsideofEq. (5b)istheanisotropic\n01234\n104-101(a)\n01234\n104-101(b)FIG. 2. Typical time evolution of the Majorana qubit for the low\nvoltage regime (a) and the high voltage regime (b). The damped\noscillation are combined effect of the Landau-Zener-Stückelburg in-\nterference and the Gilbert damping. The two different stable values\nof the𝑠𝑧represent the different fixed points in the dynamics of the\nMajorana qubit.\nGilbertdampingwhichdeterminesthedissipationofthequbit\nfrom the coupling to the environment. For isotropic case\nwhere the matrix ˜𝛾becomes a number, this term turns into\nthe standard Gilbert damping which appeared in the Landau-\nLifshiz-Gilbert equation[59]. While the Gilbert damping has\nbeenwidelytakentostudythedynamicsofthemagnetization,\nourworkprovidesaderivationforitsmicroscopicorigininthe\nMajorana qubit. This damping process influences the dynam-\nics of the qubit and thereby modifies the transport properties\nof the junction.\nFinally, we hope to point out that if the environment me-\ndiated coupling B𝑓and the Gilbert damping ˜𝛾are ignored,\nthen Eq. (5) will reduce to the phenomenological quantum\nresistively and capacitively shunted junction model that has\nbeen taken to study the I-V characteristics and the Josephson\nradiationsofthetopologicaljunction[6,58]. Ourmicroscopic\ntheoryclarifiesthevalidityandlimitsofthephenomenological\nmodel.\nFixed point analysis. – The Eq. (5) are complicate nonlin-\near equations for which obtaining analytical solutions is im-\npossible. However, the fixed points of the equations can be\nanalyticalcalculatedwiththemethodofaveraging,whichisa\nmethod to decouple the nonlinear equations with the division\nofthedynamicalvariablestothe\"fastvariables\"andthe\"slow\nvariables\" based on their time scales[60]. In Eq. (5) we treat\nthe psuedo spin sas the slow variable since it has a larger\ntime scale. We take it as constant to solve the Eq. (5a) for the\nfastvariable 𝜃¹𝑡º,andthesolutionprovidesthetime-averaged\nJosephsonenergy∫\n𝑑𝑡𝐸 Mcos𝜃¹𝑡º2\u0019𝛼𝑠𝑧𝐸M. Pluggingthis\ninto the Eq. (5b), we obtain an approximated self-consistent\nequation for s, and the fixed points of this equation can be\ndetermined analytically. There are two sets of fixed points.\nThe first is the trivial fixed points at s0=\u0006¹100ºwhich are\nstable fixed points for all parameters. If the system evolves\ntowards these fixed points, the 4𝜋-periodic Josephson current\nin Eq. (5a) vanishes and all experimental 4𝜋-periodic signa-\ntureswoulddisappear. Wealsofindanothersetoffixedpoints4\n0123456789101112131401234567(a)\n0500010000-101(c)\n05000 10000-101(b)\nFIG. 3. (a) The Shapiro steps for the topological Josephson junction\nsimulated with Eq. (5). The first Shapiro step is strongly suppressed\nwhile all other steps are clearly visible. (b) The time evolution of 𝑠𝑧\nfor the voltage around the first Shapiro step. The pseudo spin oscil-\nlates with a non-zero averaging value, and the resulted 4𝜋-periodic\nsupercurrentstronglysuppressesthefirststep. (c)Thetimeevolution\nof𝑠𝑧for the voltage around the third Shapiro step. The pseudo spin\ngoes to the fixed point of 𝑠𝑧\u00190which effectively shuts down the\n4𝜋-periodic channel for the Josephson current.\nat[46]\ns1=\u0006\u0012\n𝐸0\nM¸𝑉0𝐵𝑓𝑅0√︃\n1\u0000¹𝐸0\nM¸𝑉0𝐵𝑓𝑅º2\u0013\n(6)\nwhicharestablefixedpointsonlywhentheinjectedcurrentis\nsmall so that the dc voltage is smaller than a critical value of\n𝑉𝑐=j𝐸M𝛼\u0000𝐸0\nMj𝑅𝐵𝑓. The existence of these fixed points\nis voltage-dependent, which is qualitatively different from the\ntrivial fixed points.\nTheseanalyticalresultsforfixedpointsprovideinsightinto\nthe experimentally reported voltage-dependent behaviors of\ntopological junctions[10, 13]. From the fixed point analysis,\nwe find two different voltage regimes. At dc voltage below\n𝑉𝑐, there are two sets of fixed points, and the system has a\nchance of evolving to either of them. If the system evolves to\nthe fixed point s1as shown in Fig. 2a, the final stable state\nwould have a non-vanishing 𝑠𝑧and therefore a non-vanishing\n4𝜋-periodic Josephson current shows up in the equation for\nthe Josephson phase. In this voltage regime, we should ex-\npect transport signatures for 4𝜋-periodicity. However, for the\nvoltage above 𝑉𝑐, there exits only the trivial fixed points at\ns0. When the system evolves towards it as shown in Fig. 2b,\nthe final stable state would have a vanishing 𝑠𝑧, and the 4𝜋-\nperiodic Josephson current vanishes. In this voltage regime,\nall the transport signatures for the 4𝜋-periodicity should dis-\nappear. Basedonthesefixedpointanalysis,wepredictthatthe\ntransportoftopologicaljunctionswouldexhibitnontrivial 4𝜋-\nperiodicsignaturesonlyatlowvoltage,whileathighvoltages\nit would look quite similar to the trivial junctions.Shapirosteps. –TheShapirostepsaretheplateausoftheI-V\ncurve at voltages 𝑉𝑛=𝑛~𝜔2𝑒under an injected ac current\nwith frequency 𝜔. It is a powerful tool for probing the dy-\nnamics of Josephson junctions since it reflects the resonance\nbetweenthedcandtheacJosephsonrelation. Fortopological\njunctions,itwasanticipatedthattheoddnumberShapirosteps\nwith𝑛=135should be suppressed by the 4𝜋-periodic\nsupercurrent. The experimental results, however, often show\nstrong suppression of low order odd number steps such as the\nonewith𝑛=1,whileotheroddnumberShapirostepsathigher\nvoltagesarerobust. Sincetheunderstandingoftheexperimen-\ntal results are crucial for detecting Majorana zero modes, it is\ntimely to implement the Eq. (5) to calculate the Shapiro steps\nof Majorana Josephson junctions.\nWe consider an injected current of 𝐼ex¹𝑡º=𝐼¸𝐼0cos𝜔𝑡\nand calculate the I-V curve of the junction, with the results\nfor a typical junction parameter shown in Fig. 3a. We find\nShapiro steps at 𝑉=𝑛~𝜔2𝑒, where𝑛labels the number of\nthestep. Intriguingly,itisclearthatthefirstShapirostepwith\n𝑛=1is strongly suppressed, while all other steps are clearly\nvisible. At first glance, the suppression of only one Shapiro\nstep seems mysterious. One would expect a suppression of\nall odd-number steps if the 4𝜋-periodic supercurrent carried\nby the Majorana qubit is significant, or no suppression to any\nof the steps if the 4𝜋-periodic supercurrent is irrelevant. For\nthis phenomenon, our theory provides a possible mechanism:\nthe feedback from the dynamics of the Majorana qubit. As\nwe have shown in the analytical results, the Majorana qubit\nevolves to different stable states for different voltages. We\nexamine the qubit dynamics at the voltages for the first step\nand the third step. As shown in Fig. 3b, at the voltage where\nthe first step should appear, the Majorana qubit evolves to the\nstable state with a finite 𝑠𝑧. Then the 4𝜋-periodic Josephson\ncurrent will dominate and the Shapiro step is suppressed. For\nthe higher voltage of the third step, however, the Majorana\nqubitevolvestoastablestatewith 𝑠𝑧\u00190,asshowninFig. 3c.\nThenthe 4𝜋-periodicsupercurrentisblocked,andthejunction\nwould behaves similar to a conventional junction presenting\nShapiro steps. This feed back of the qubit dynamics provides\na simple mechanism for the suppression of the first Shapiro\nstep, and gives a possible explanation to one of the puzzles in\nthe experimental findings of topological superconductors.\nWe emphasize that, while our theory is derived for topo-\nlogical junctions with Majorana qubit, it is actually valid for\nany junction with an embedded qubit that can be described\nby the low energy effective Hamiltonian Eq. (1). One such\nexample is the Josephson junction with quantum dots[8]. In\nthissense,ourcalculationofShapirostepsprovideasignalfor\nthefeedbackofembeddedqubit,insteadofauniquesignature\nof Majorana zero modes.\nConclusion. – In summary, we constructed a semiclassical\ntheory for the topological Josephson junctions with an em-\nbedded Majorana qubit. We revealed nontrivial qubit dynam-\nics such as the Landau-Zener transitions and the anisotropic\nGilbert damping. We found that the feedback of the qubit dy-\nnamicsstronglymodifiesthetransportfeaturesofthejunction.5\nWe applied the theory to study the Shapiro steps of the topo-\nlogicaljunctionsanddemonstratedthesuppressionofthefirst\nShapirostepwhichagreeswithrecentexperiments. Wereveal\nthat this phenomenon is due to the voltage-selective feedback\nfrom the dynamics of Majorana qubit.\nAcknowledgments.— We thank Zhongbo Yan, Peng Ye\nand Shuai Yin for valuable discussions. This work was\nsupported by NSFC (Grant No. 12174453), NKRDPC-\n2017YFA0206203, 2017YFA0303302, 2018YFA0305603,\nand Guangdong Basic and Applied Basic Research Founda-\ntion(GrantNo. 2019A1515011620). Z.H.issupportedbythe\nRobert A.Welch Foundation under Grant No. E-1146.\n\u0003These authors contributed equally to this work.\nyCorresponding author: huangzhaophysics@gmail.com\n[1] M. H. Devoret and R. J. Schoelkopf, Superconducting cir-\ncuits for quantum information: An outlook, Science 339, 1169\n(2013).\n[2] J.M.Martinis,M.H.Devoret,andJ.Clarke,Quantumjosephson\njunctioncircuitsandthedawnofartificialatoms,NaturePhysics\n16, 234 (2020).\n[3] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-\nJ. Wang, S. Gustavsson, and W. D. Oliver, Superconducting\nqubits: Current state of play, Annual Review of Condensed\nMatter Physics 11, 369 (2020).\n[4] I.Siddiqi,Engineeringhigh-coherencesuperconductingqubits,\nNature Reviews Materials 6, 875 (2021).\n[5] M. 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Malý \nFaculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3,  \n121 16 Prague 2, Czech Republic \n \nV. Novák, K. Olejník, M. Cukr, and T. Jungwirth \nInstitute of Physics ASCR  v.v.i., Cukrovarnická 10, 162 53 Prague, Czech Republic \n \n \nWe report dynamics of the transient polar Kerr rotation (KR) and of the transient \nreflectivity induced by femtosecond laser pulses in ferromagnetic (Ga,Mn)As with no \nexternal magnetic field applied. It is s hown that the measured KR signal consist of \nseveral different contributions, among which only the oscillatory signal is directly \nconnected with the ferromagnetic order in (Ga,Mn)As. The origin of the light-induced magnetization precession is discussed and the magnetization precession damping (Gilbert damping) is found to be strongly influenced by annealing of the sample. \n \n \n (Ga,Mn)As is the most intensively studied member of the family of diluted magnetic \nsemiconductors with carrier-mediated ferromagn etism [1]. The sensitiv ity of ferromagnetism \nto concentration of charge carriers opens up th e possibility of magneti zation manipulation on \nthe picosecond time scale using light pulses from ultrafast lasers [2]. Photoexcitation of a \nmagnetic system can strongly disturb the equili brium between the mobile  carriers (holes), \nlocalized spins (Mn ions), and the lattice. This in turn tr iggers a variety of dynamical \nprocesses whose characteristic time scales and st rengths can be investigated by the methods of \ntime-resolved laser spectroscopy [2]. In part icular, the magnetization reversal dynamics in \nvarious magnetic materials attracts a significant attention because it is directly related to the \nspeed of data storage in the magnetic reco rding [3]. The laser-induced precession of \nmagnetization in ferromagnetic (Ga,Mn)As has been recently re ported by two research groups \n[4-6] but the physical processes responsible fo r it are still not well unde rstood. In this paper \nwe report on simultaneous measurements of  the light-induced magnetization precession \ndynamics and of the dynamics of  photoinjected carriers.  \nThe experiments were performed on a 500 nm thick ferromagnetic Ga 1-xMn xAs film \nwith x = 0.06 grown by the low temperature molecular beam epitaxy (LT-MBE) on a \nGaAs(001) substrate. We studied both the as-grown sample, with the Curie temperature TC  ≈ \n60 K and the conductivity of 120 Ω-1cm-1, and the sample annealed at 200°C for 30 hours, \nwith TC  ≈ 90 K and the conductivity of 190 Ω-1cm-1; using the mobility vs. hole density \ndependence typical for GaMnAs [7] we can r oughly estimate their hole densities as 1.5 x 1020 \ncm-3 and 3.4 x 1020 cm-3, respectively. Magnetic properties of the samples were measured \nusing a superconducting quantum interference de vice (SQUID) with magnetic field of 20 Oe \napplied along different crysta llographic directions. The photo induced magnetization dynamics \nwas studied by the time-resolved Kerr rotation  (KR) technique [2] using a femtosecond \ntitanium sapphire laser (Tsunami, Spectra Physics) . Laser pulses, with th e time width of 80 fs \nand the repetition rate of 82 MHz, were tune d to 1.54 eV. The energy  fluence of the pump \npulses was typically 15 μJ.cm-2 and the probe pulses were always at least 10 times weaker. \nThe polarization of the pump pulses was either circular or linear, while the probe pulses were \n                                                 \na) Electronic mail: nemec@karlov.mff.cuni.cz \n 1linearly polarized (typically along the [010] crystallographic direction in the sample, but \nsimilar results were obtained also for other orientations). The rotation angle of the \npolarization plane of the reflected probe pulses was obtained by taking the difference  of \nsignals measured by detectors in an optical br idge detection system [2]. Simultaneously, we \nmeasured also the sum of signals from the detectors, which corresponded to a probe intensity \nchange due to the pump induced modification of  the sample reflectivity. The experiment was \nperformed with no external magnetic field applie d. However, the sample was cooled in some \ncases with no external magnetic field applied or  alternatively with a ma gnetic field of 170 Oe \napplied along the [-110] direction. \n \n \n \nFig. 1. Dynamics of photoinduced Kerr rotation angle (KR) measured for the as-grown sample at 10 K. (a) KR \nmeasured for σ + and σ - circularly polarized (CP) pump  pulses; (b) KR measured for p and s linearly polarized \n(LP) pump pulses. Polarization-independent part (c) (polar ization-dependent part (d)) of KR signal, which was \ncomputed from the measured traces as an average of the signals (a half of  the difference between the signals) \ndetected for pump pulses with the opposite CP (LP). Inset: Fourier transform of the oscillations. No external \nmagnetic field was applied during the sample cooling. \n \n In Fig. 1 we show typical te mporal traces of the transien t angles of KR measured for \nthe as-grown sample at 10 K. The KR signal wa s dependent on the light polarization but there \nwere certain features presen t for both the circular (Fig. 1 (a)) and linear (Fig. 1 (b)) \npolarizations. In Fig. 1 (c) we show the polari zation-independent part of the measured KR \nsignal, which was the same for circular and line ar polarization of pump pulses. On the other \nhand, the amplitude of the polarization-dependent part of the signal (Fig. 1 (d)) was larger for \nthe circular polarization. The interpretation of the polarization-dependent part of the signal is \nsignificantly complicated by the fact that the circularly polarized light generates spin-\npolarized carriers (electrons in particular), whose contribution to the measured KR signal can \neven exceed that of ferromagnetic ally coupled Mn spins [8]. In  the following we concentrate \non the polarization-independent part of the KR signal (Fig. 1 (c)). This signal can be fitted \nwell (see Fig. 2) by an exponentially damped sine harmonic oscillation superimposed on a \npulse-like function: \n \n() ( ) ( ) () [ ]()2 1 / exp / exp 1 sin / exp τ τ ϕωτ t t B t t A t KRD − −−+ + − = .    (1) \n \n 2The oscillatory part of the KR signa l is characterized by the amplitude (A ), damping time ( τD), \nangular frequency ( fπω2= ), and phase ( ϕ). The pulse-like part of the KR signal is \ndescribed by the amplitude ( B), rise time ( τ1), and decay time ( τ2). In the inset of Fig. 2 we \nshow the dynamics of the sample reflectivity change ΔR/R. This signal monitored the change \nof the complex index of refraction of the sa mple due to carriers photoinjected by the pump \npulse. From the dynamics of ΔR/R we can conclude that the population of photogenerated free \ncarriers (electrons in par ticular [9]) decays within ≈ 50 ps after the photoinjection. This rather \nshort lifetime of free electrons is similar to th at reported for the low temperature grown GaAs \n(LT-GaAs), which is generally interpreted as  a consequence of a high concentration of \nnonradiative recombination centers induced by th e low temperature growth mode of the MBE \n[9]. It is also clearly apparent from the inset of Fig. 2 that the KR data can be fitted well by \nEq. (1) only for time delays larger than ≈ 50 ps (i.e., just after th e population of photoinjected \nfree electrons nonradiatively decayed). We will come back to this point later. \n \n \n \nFig. 2. The fitting procedure applied to the polarization-independent part of KR signal. (a) The measured data \nfrom Fig. 1 (c) (points) are fitted (solid line) by a sum of the exponentially damped sine harmonic oscillation \n(solid line in part (b)) and the pulse-like KR signal (dashe d line in part (b)). Inset: Dynamics of the reflectivity \nchange (thick solid line) and the detail of the fitted KR signal. \n \n In Fig. 3(a) we show the intensity dependence of A and B, and in Fig. 3(b) of ω and τD  \nmeasured at 10 K. For the increasing inte nsity of pump pulses  the magnitudes of A and B \nwere increasing, ω was decreasing and the values of τD were not changing significantly. The \napplication of magnetic field applied along the [-110] directi on during the sample cooling \nmodified the value of ω. For 10 K (and pump intensity I0) the frequency decreased from 24.5 \nto 20 GHz (open and solid point in Fig. 3 (d ), respectively). The measured temperature \ndependence of A and B (Fig. 3 (c)) revealed that the oscillatory signal vanished above TC, \nwhile a certain fraction of  the pulse-like KR signal persisted even above TC. This shows that \nonly the oscillatory part of the KR si gnal was directly c onnected with the ferromagnetic  order \nin (Ga,Mn)As. (It is worth noti ng that also the polarization-de pendent part of the KR signal \nwas non-zero even above TC.) The frequency of oscillations  was decreasing with the sample \ntemperature (Fig. 3 (d)), but the values of τD were not changing sign ificantly (not shown \nhere). \n 3 \n \nFig. 3. Intensity dependence of ⎪A⎪and ⎪B⎪ (a), ω and τD (b) measured at 10 K; I0 = 15 μJ.cm-2, no external \nmagnetic field was applied during the sample cooling. (c), (d) Temperature dependence of ⎪A⎪, ⎪B⎪ and ω \n(points) measured at pump intensity I0. The open point in (d) was obtained for the sample cooled with no \nexternal magnetic field applied and the data in (c) and the solid points in (d) were obtained for the sample cooled \nwith magnetic field applied along the [-110]. The lines in (d) are the temperature dependence of the sample \nmagnetization projections to different crystallographic directions measured by SQUID. \n \n The photoinduced magnetization precession was reported by A. Oiwa  et al. , who \nattributed it to the precession of ferromagnetically coupled Mn spins induced by a change in \nmagnetic anisotropy initiated by an increase in hole concentration [4]. It was also shown that \nthe photoinduced magnetization precession a nd the ferromagnetic resonance (FMR) can \nprovide similar information [4]. Magnetic an isotropy in (Ga,Mn)As is influenced by the \nintrinsic cubic anisotropy, which is arising from its zinc-blende symmetry, and by the uniaxial anisotropy, which is a result of a strain induced  by different lattice constants of GaMnAs and \nthe substrate. For the standard stressed GaMnAs  films with  Mn content above 2% grown on \nGaAs substrates the magnetic easy axes are in -plain. Consequently, the measured polar Kerr \nrotation is not sensitive to the steady state magn etization of the sample, but only to the light-\ninduced transient out-of-plane magnetization due to the polar Kerr effect [2]. In our \nexperiment, the pump pulses with a fluence I\n0 = 15 μJ.cm-2 photoinjected electron-hole pairs \nwith an estimated concentration Δp = Δn ≈ 8 x 1017 cm-3. This corresponded to Δp/p ≈ 0.5% \nand such a small increase in the hole concentrati on is highly improbable to lead to any sizable \nchange of the sample anisotropy [1]. Another hypothesis about the origin  of the light-induced \nmagnetization precession was reported recently by J. Qi et al. [6]. The authors suggested that \nnot only the transient increase in local hole concentration Δp but also the local temperature \nincrease ΔT contributes to the change of anisotr opy constants.  This modification of the \nsample anisotropy changes in turn the direction of the in-plane magnetic easy axis and, \nconsequently, triggers a precessional motion of  the magnetization around the altered magnetic \nanisotropy field . The magnitude of decreases as T (the sample temperature) or ΔT \nincreases, primarily due to the decrease in the cubic anisotropy constant KMn\nanisHMn\nanisH\n1c [6]. Our samples \nexhibit in-plane easy axis behavior typica l for stressed GaMnAs layers grown on GaAs \nsubstrates. To characterize their in-plane anis otropy we measured the temperature dependent \nmagnetization projections to [110] , [010], and [-110] crystallographic directions – the results \nare shown in Fig. 3 (d) and in inset of Fig. 4 for the as-grown and the annealed sample, \nrespectively. At low temperatures the cubi c anisotropy dominates (as indicated by the \n 4maximal projection measured along the [010] di rection) but the uniaxial in-plane component \nis not negligible and the sample magnetization is slightly tilted from the [010] direction \ntowards the [-110] direction. Both samples exhibit rotation of magnetizat ion direction in the \ntemperature region 10-25K, which is in agreement with the expected fast weakening of the \ncubic component with an increasing temperatur e. In our experiment, the excitation fluence I0 \nled to ΔT ≈ 10 K (as estimated from the GaAs specifi c heat of 1 mJ/g/K [6]) that can be \nsufficient for a change of the easy axis positi on. This temperature-ba sed hypothesis about the \norigin of magnetization precession is supported also by our observation that the oscillations \nwere not fully developed immediately after the photoinj ection of carriers but only after ≈ 50 \nps when phonons were emitted by the nonradiative  decay of the population of free electrons \n(see inset in Fig. 2). We also point out that the measured precession frequency ω and the \nsample magnetization M (measured by SQUID) had very similar temperature dependence (see \nFig. 3(d)).  \n \n \n \nFig. 4. Polarization-independent part of KR signal measured for the annealed sample at 10 K; I0 = 15 μJ.cm-2, the \nsample was cooled with magnetic field applied along the [-110]. Inset: Temperature dependence of the sample \nmagnetization projections to different crystallographic directions measured by SQUID. \n \n An example of the results measured for the annealed sample is shown in Fig. 4. The \nanalysis of the data revealed that at simila r conditions the precession frequency was slightly \nhigher in the annealed sample (20 GHz and 24 GHz for the as-grown and the annealed \nsample, respectively). However, a major effect of the sample annealing was on the oscillation \ndamping time τD, which increased from 0.4 ns to  1.1 ns. This prolongation of τD can be \nattributed to the improved quality of the ann ealed sample, which is indicated by the higher \nvalue of TC and by the more Brillouin-like temperature dependence of the magnetization (cf. \nFig. 3 (d) and inset in Fig. 4). The damping of  oscillations is connected with the precession \ndamping in the Landau-Lifshitz-Gilbert equation [1]. The exact determination of the intrinsic \nGilbert damping coefficient α from the measured data is not straightforward because it is \ndifficult to decouple the contribution due to the inhomogeneous broadening [10]. In Ref. 6 the \nvalues of α from 0.12 to 0.21 were deduced for the as -grown sample from the analysis of the \noscillatory KR signal. The time-domain KR should provide similar information as the \nfrequency-domain based FMR, where the relaxa tion rate of the magnetization is connected \nwith the peak-to-peak ferromagnetic resonance linewidth ΔHpp [10]. Indeed both methods \nshowed that the relaxation rate of the magnetization is consider ably slower in the annealed \nsamples (as indicated by the prolongation of τD in our experiment and by the reduction of \nΔHpp in FMR [10]).  \n 5 In conclusion, we studied the transient Ke rr rotation (KR) and the reflectivity change \ninduced by laser pulses in (Ga ,Mn)As with no external magnetic field applied. We revealed \nthat the measured KR signals consisted of seve ral different contributions  and we showed that \nonly the oscillatory KR signal was directly connected with the fe rromagnetic order in \n(Ga,Mn)As. Our data indicated that the phonons  emitted by photoinjected  carriers during their \nnonradiative recombination in (Ga,Mn)As can be re sponsible for the magnetic anisotropy \nchange that was triggering the magnetization precession. We also observed that the precession \ndamping was strongly suppressed in the ann ealed sample, which reflected its improved \nmagnetic properties. This work was supported by Ministry of Education of the Czech Republic in the framework of the rese arch centre LC510, the research plans \nMSM0021620834 and AV0Z1010052, by the Grant Agen cy of the Charles University in \nPrague under Grant No. 252445, and by the Grant Agency of Academy of Sciences of the Czech Republic Grants FON/06/E 001, FON/06/E002, and KAN400100652. \n \n \nReferences \n \n[1] T. Jungwirth, J. Sinova, J. Maše k, A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). \n[2] J. Wang, Ch. Sun, Y. Hashimoto, J. Kono, G.A. Khodaparast, L. Cywinski, L.J. Sham, G.D. Sanders, Ch.J. Stanton, H. Munekata, J. Phys.: Condens. Matter \n18, R501 (2006). \n[3] A.V. Kimel, A. Kirilyuk, F.  Hansteen, R.V. Pisarev, T. Ra sing, J. Phys.: Condens. Matter \n19, 043201 (2007). \n[4] A. Oiwa, H. Takechi, H.  Munekata, J. Supercond. 18, 9 (2005). \n[5] H. Takechi, A. Oiwa, K. Nomura, T. Kondo, H. Munekata, phys. stat. sol. (c) 3, 4267 \n(2006). \n[6] J. Qi, Y. Xu, N.H. Tolk, X. Liu, J.K.  Furdyna, I.E. Perakis, Appl. Phys. Lett. 91, 112506 \n(2007). \n[7] T. Jungwirth et al., Phys. Rev. B 76, 125206 (2007). \n[8] A.V. Kimel, G.V. Astakhov,  G.M. Schott, A. Kirilyuk, D. R. Yakovlev, G. Karczewski, \nW. Ossau, G. Schmidt, L.W. Molenkamp, Th. Rasing, Phys. Rev. Lett. 92, 237203 (2004). \n[9] M. Stellmacher, J. Nagle, J.F. Lampin, P. Santoro, J. Van eecloo, A. Alexandrou, J. Appl. \nPhys. 88, 6026 (2000). \n[10] X. Liu, J.K. Furdyna, J. Phys.: Condens. Matter 18, R245 (2006). \n 6"    },    {        "title": "1012.5473v1.Screw_pitch_effect_and_velocity_oscillation_of_domain_wall_in_ferromagnetic_nanowire_driven_by_spin_polarized_current.pdf",        "content": "arXiv:1012.5473v1  [cond-mat.other]  25 Dec 2010Screw-pitch effect and velocity oscillation of domain-wall in ferromagnetic nanowire\ndriven by spin-polarized current\nZai-Dong Li1,2,3, Qiu-Yan Li1, X. R. Wang3, W. M. Liu4, J. Q. Liang5, and Guangsheng Fu2\n1Department of Applied Physics, Hebei University of Technol ogy, Tianjin 300401, China\n2School of Information Engineering, Hebei University of Tec hnology, Tianjin, 300401, China\n3Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China.\n4Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100080, China\n5Institute of Theoretical Physics and Department of Physics , Shanxi University, Taiyuan 030006, China\nWe investigate the dynamics of domain wall in ferromagnetic nanowire with spin-transfer torque.\nThe critical current condition is obtained analytically. B elow the critical current, we get the static\ndomain wall solution which shows that the spin-polarized cu rrent can’t drive domain wall moving\ncontinuously. In this case, the spin-transfer torque plays both the anti-precession and anti-damping\nroles, which counteracts not only the spin-precession driv en by the effective field but also Gilbert\ndamping to the moment. Above the critical value, the dynamic s of domain wall exhibits the novel\nscrew-pitch effect characterized by the temporal oscillati on of domain wall velocity and width,\nrespectively. Both the theoretical analysis and numerical simulation demonstrate that this novel\nphenomenon arise from the conjunctive action of Gilbert-da mping and spin-transfer torque. We\nalso find that the roles of spin-transfer torque are entirely contrary for the cases of below and above\nthe critical current.\nPACS numbers: 75.75.+a, 75.60.Ch, 75.40.Gb\nKeywords: Screw-Pitch effect, Velocity Oscillation of Doma in-Wall, Spin-Polarized Current.\nA magnetic domain wall (DW) is a spatially local-\nized configuration of magnetization in ferromagnet, in\nwhich the direction of magnetic moments inverses gradu-\nally. Whenaspin-polarizedelectriccurrentflowsthrough\nDW, the spin-polarization of conduction electrons can\ntransfer spin momentum to the local magnetization,\nthereby applying spin-transfer torque which can manip-\nulate the magnetic DW without the applied magnetic\nfield. This spin-transfer effect was theoretically proposed\nby Slonczewski [1] and Berger [2], and subsequently veri-\nfied experimentally [3]. As a theoretical model the mod-\nified Landau-Lifshitz-Gilbert (LLG) equation [4–6] with\nspin-transfertorquewasderivedto describesuchcurrent-\ninduced magnetization dynamics in a fully polarized fer-\nromagnet. With these novel forms of spin torque many\ninteresting phenomena have been studied, such as spin\nwave excitation [4, 7, 8] and instability [4, 9], magneti-\nzation switching and reversal [10–13], and magnetic soli-\ntons [14, 15]. For the smooth DW, this spin torque can\ndisplace DW opposite to the current direction which has\nbeenconfirmedexperimentallyinmagneticthinfilmsand\nmagnetic wires [16–20].\nWith the remarkable experimental success measur-\ning the motion of DW under the influence of current\npulse,considerableprogresshasbeenmadetounderstand\nthe current-induced DW motion in magnetic nanowire\n[5, 6, 17–20]. These studies haveimproved the pioneering\nwork of current-driven DW motion by Berger [21]. Al-\nthough both the theory and the quasi-static experiments\nhave indicated that the spin-polarized current can cause\nDW motion, the current-driven DW dynamics is not well\nunderstood. The dynamics of magnetization described\nby the LLG equation admits the static solutions for DWmotion. In the presence of spin torque and the exter-\nnal magnetic field, it is difficult to derive the dynamic\nsolutions. A circumvented approach is Walker solution\nanalysis [22] for the moving DW in response to a steady\nmagnetic field smaller than some critical value. How-\never, this approximation applying to DW motion driven\nby the electric current is unclear, and its reliability has\nto be verified theoretically and numerically.\nInthispaper,wereportanalyticallythecriticalcurrent\ncondition for anisotropic ferromagnetic nanowire driven\nonly by spin-transfer torque. Below the critical cur-\nrent, the ferromagnetic nanowire admits only the final\nstatic DW solution which implies that the spin-polarized\ncurrent can’t drive DW moving continuously. When\nthe spin-polarized current exceeds the critical value, the\ndynamics of DW exhibits the novel Screw-pitch effect\nwith the periodic temporal oscillation of DW velocity and\nwidth. A detail theoretical analysis and numerical sim-\nulation demonstrate that this novel phenomenon arises\nfrom the natural conjunction action of Gilbert-damping\nand spin-transfer torque. We also observe that the spin-\ntransfer torque plays the entirely opposite roles in the\nabove two cases. At last, our theoretical prediction can\nbe confirmed by the numerical simulation in terms of\nRKMK method [23].\nWe consider an infinite long uniaxial anisotropic fer-\nromagnetic nanowire, where the electronic current flows\nalong the long length of the wire defined as xdirection\nwhichis alsothe easyaxisofanisotropyferromagnet. For\nconvenience the magnetization is assumed to be nonuni-\nform only in the direction of current. Since the magneti-\nzation varies slowly in space, it is reasonable to take the\nadiabatic limit. Then the dynamics of the localized mag-2\nnetization can be described by the modified LLG equa-\ntion with spin-transfer torque\n∂M\n∂t=−γM×Heff+α\nMsM×∂M\n∂t+bJ∂M\n∂x,(1)\nwhereM≡M(x,t) is the localized magnetization, γ\nis the gyromagnetic ratio, αis the damping parameter,\nandHeffrepresents the effective magnetic field. The\nlast term of Eq. (1) denotes the spin-transfer torque,\nwherebJ=PjeµB/(eMs),Pis the spin polarization of\nthe current, jeis the electric current density and flows\nalong the xdirection, µBis the Bohr magneton, eis\nthe magnitude of electron charge, and Msis the sat-\nuration magnetization. For the uniaxial ferromagnetic\nnanowire the effective field can be written as Heff=/parenleftbig\n2A/M2\ns/parenrightbig\n∂2M/∂x2+HxMx/Msex−4πMzez, whereAis\nthe exchange constant, Hxis the anisotropy field, and ei,\ni=x,y,z,is the unit vector, respectively. Introducing\nthe normalized magnetization, i.e., m=M/Ms, Eq. (1)\ncan be simplified as the dimensionless form\nα1∂m\n∂t=−m×heff−αm×(m×heff)\n+αb1m×∂m\n∂x+b1∂m\n∂x, (2)\nwhereα1=/parenleftbig\n1+α2/parenrightbig\nandb1=bJt0/l0. The time\ntand space coordinate xhave been rescaled by the\ncharacteristic time t0= 1/(16πγMs) and length l0=/radicalbig\nA/(8πM2s), respectively. The dimensionless effective\nfield becomes heff=∂2m/∂x2+C1mxex−C2mzez, with\nC1=Hx/(16πMs) andC2= 0.25.\nIn the following, we seek for the exact DW solutions of\nEq. (2), and then study the dynamics of magnetization\ndriven by spin-transfer torque. To this purpose we make\nthe ansatz\nmx= tanhΘ 1,my=sinφ\ncoshΘ 1,mz=cosφ\ncoshΘ 1,(3)\nwhere Θ 1=k1x−ω1t, with the temporal and spatial\nindependent parameters φ,k1, andω1to be determined,\nrespectively. Substituting Eq. (3) into Eq. (2) we have\nk2\n1=C1+C2cos2φ, (4)\n−ω1/parenleftbig\n1+α2/parenrightbig\n=b1k1+C2sinφcosφ,(5)\nαb1k1cosφ=α/parenleftbig\nC1−k2\n1/parenrightbig\nsinφ, (6)\nαb1k1sinφ=−αC2sin2φcosφ. (7)\nFrom the above equations we can get three cases of DW\nsolutions for Eq. (2). Firstly, in the absence of damping\nEqs. (4) to (7) admit the solution\nk1=±/radicalbig\nC1+C2cos2φ,ω1=−b1k1−C2\n2sin2φ,(8)\nwith the arbitrary angle φ. This solution show that the\nspin-transfer torque contributes a dimensionless veloc-\nity−b1only without damping. The velocity of DW isformed by two parts, i.e., v=−(C2sin2φ)/(2k1)−b1,\nwhich can be affected by adjusting the angle φand the\nspin-transfer torque. Secondly, in the absence of spin\ntorque, we have the solution of Eqs. (4) to (7) as ω1= 0,\nφ=±π/2,k1=±√C1, i.e., the static DW solution. In\nterms of RKMK method [23] we perform direct numeri-\ncal simulation for Eq. (2) with various initial condition,\nand all numerical results show that the damping drives\nthe change of φwhich in turn affects the DW velocity\nand width defined by 1 /|k1|. At last φ=±π/2,ω1= 0,\ni.e., the DW loses moving, and the DW width attains\nits maximum value√C1, which confirms the Walker’s\nanalysis [22] that the damping prevents DW from mov-\ning without the external magnetic field or spin-transfer\ntorque. However, as shown later, the presence of damp-\ning is prerequisite for the novel Screw-pitch property of\nDW driven by spin-transfer torque. At last, we consider\nthe case of the presence of damping and spin-transfer\ntorque. Solving Eqs. (4) to (7) we have\nk1=±1\n2(B1−/radicalbig\nB2),ω1= 0,sin2φ=−2b1k1\nC2,(9)\nwhereB1= 2C1+C2−b2\n1,B2=/parenleftbig\nC2−b2\n1/parenrightbig2−4C1b2\n1.\nIt is clear that Eq. (9) implies the critical spin-\npolarized current condition, namely\nbJ≤(/radicalbig\nC1+C2−/radicalbig\nC1)l0/t0,\nwhich is determined by the character velocity l0/t0,\nthe anisotropic parameter C1, and the demagnetiza-\ntion parameter C2. Below the critical current, i.e.,\nb2\n1≤(√C1+C2−√C1)2, the DW width falls into the\nrange that 1 //radicalbig\nC2\n1+C1C2≤1/|k1| ≤1/C1. From\nEq. (9) we get four solutions of φ, i.e.,φ=±π/2 +\n1/2arcsin(2 b1k1/C2) fork1>0 andφ=±π/2−\n1/2arcsin(2 b1|k1|/C2) fork1<0. In fact, the signs\n“+” and “ −” in Eq. (9) denotes kink and anti-kink so-\nlution, respectively, and the corresponding solution in\nEq. (3) represents the static tail-to-tail or head-to-head\nN´ eel DW, respectively. This result shows that below the\ncritical current, the final equilibrium DW solution must\nbe realized by the condition that m×heff=b1∂m/∂x.\nIt clearly demonstrates that the spin-transfer torque has\ntwo interesting effects. One is that the term b1∂m/∂x\nin Eq. (2) plays the anti-precession role counteracting\nthe precession driven by the effective field heff. How-\never, the third term in the right hand of Eq. (2), namely\nαb1m×∂m/∂x, has the anti-damping effect counteract-\ning the damping term −αm×(m×heff). It is to say\nthat below the critical value, the spin-polarized current\ncan’t drive DW moving continuously without the applied\nexternal magnetic field.\nWhen the spin-polarized current exceed the critical\nvalue, the dynamics of DW possesses two novel prop-\nerties as shown in the following section. Above the\ncritical current, the precession term −m×heffcan’t be\ncounteracted by spin-transfer torque, and the static DW3\nsolution of Eq. (2) doesn’t exist. Because the mag-\nnetization magnitude is constant, i.e., m2= 1, so we\nhavem·∂m/∂x= 0 which shows that the direction of\n∂m/∂xis always perpendicular to the direction of m, or\n∂m/∂x= 0. It is well known that a magnetic DW sep-\narates two opposite domains by minimizing the energy.\nIn the magnetic DW the direction of magnetic moments\ngradually changes, i.e., ∂m/∂x/negationslash= 0,so the direction of\n∂m/∂xshould adopt the former case. Out of region of\nDW the normalized magnetization will site at the easy\naxis, i.e., mx= 1(or−1), in which ∂m/∂x= 0.\nWiththeaboveconsiderationwemakeadetailanalysis\nfor Eq. (2). As a characteristic view we mainly consider\nthe DW center, defined by mx= 0. The magnetic mo-\nment must be in the y-zplane, while the direction of\n∂m/∂xshould lay in x-axis (+x-axis for k1>0, and\n−x-axis for k1<0). In order to satisfy Eq. (2) the mag-\nnetic moment in DW center should include both the pre-\ncession around the effective spin-torque field αb1∂m/∂x\nand the tendency along the direction of ∂m/∂xcontinu-\nouslyfromthelasttwotermsintherighthandofEq. (2).\nThe formerprecessionmotion implies that the parameter\nφwill rotate around x-axis continuously, while the lat-\nter tendency forces the DW center moving toward to the\nopposite direction of the current, i.e., −x-axis direction,\nconfirming the experiment [16–20] in magnetic thin films\nand magnetic wires. Combining the above two effects we\nfind that this rotating and moving phenomenon is very\nsimilar to Screw-pitch effect. The continuous rotation of\nmagnetic moment in DW center, i.e., the periodic change\nofφ, can result in the periodic oscillation of DW veloc-\nity and width from Eq. (8) under the action of the first\ntwo terms in the right hand of Eq. (2). It is interesting\nto emphasize that when the current exceeds the critical\nvalue, the term αb1m×∂m/∂xplays the role to induce\nthe precession, while the term b1∂m/∂xhas the effect\nof damping, which is even entirely contrary to the case\nbelow the critical current as mentioned before. Combin-\ning the above discussion we conclude that the motion of\nmagnetic moment in the DW center will not stop, except\nit falls into the easy axis, i.e., out of the range of DW. In\nfact, all the magnetic moments in DW can be analyzed\nin detail with the above similar procedure.\nNow it is clear for the dynamics of DW driven only\nby spin-transfer torque. Coming back to Eq. (2) we can\nsee that this novel Screw-pitch effect with the periodicoscillation of DW velocity and width occurs even at the\nconjunct action of the damping and spin-transfer torque.\nTo confirm our theoretical prediction we perform direct\nnumerical simulation for Eq. (2) with an arbitrary initial\ncondition by means of RKMK method [23] with the cur-\nrent exceeding the critical value. In figure 1(a) to 1(c)\nwe plot the time-evolution of the normalized magnetiza-\ntionm, while the displacement of DW center is shown in\nfigure 1(d). The result in figure 1 confirms entirely our\ntheoretical analysis above. The evolution of cos φand\nthe DW velocity and width are shown in figure 2. From\nfigure 2 we can see that the periodic change of cos φleads\nto the periodic temporal oscillation of DW velocity and\nwidth. From Eq. (8) and the third term of Eq. (2)\nwe can infer that cos φpossesses of the uneven change\nas shown in figure 2(a), i.e., the time corresponding to\n0< φ+nπ≤π/2 is shorter than that corresponding to\nπ/2< φ+nπ≤π,n= 1,2..., in each period, and the\nDW velocity has the same character. It leads to the DW\ndisplacement firstly increases rapidly, and then slowly as\nshown in figure 1(d). This phenomenon clarifies clearly\nthe presence of Screw-pitch effect . The DW velocity os-\ncillation driven by the external magnetic field has been\nobserved experimentally [24]. Our theoretical prediction\nfor the range of DW velocity oscillation driven by the\nabove critical current could be observed experimentally.\nIn summary, the dynamics of DW in ferromagnetic\nnanowire driven only by spin-transfer torque is theoret-\nically investigated. We obtain an analytical critical cur-\nrent condition, below which the spin-polarized current\ncan’t drive DW moving continuously and the final DW\nsolution is static. An external magnetic field should be\napplied in order to drive DW motion. We also find that\nthe spin-transfer torque counteracts both the precession\ndrivenbytheeffectivefieldandtheGilbertdampingterm\ndifferent from the common understanding. When the\nspin current exceeds the critical value, the conjunctive\naction of Gilbert-damping and spin-transfer torque leads\nnaturally the novel screw-pitch effect characterized by\nthe temporal oscillation of DW velocity and width.\nThis work was supported by the Hundred Innovation\nTalents Supporting Project of Hebei Province of China,\nthe NSF of China under grants Nos 10874038, 10775091,\n90406017, and 60525417, NKBRSFC under grant No\n2006CB921400, and RGC/CERG grant No 603007.\n[1] Slonczewski J C, 1996 J. Magn. Magn. Mater. 159 L1\n[2] Berger L, 1996 Phys. Rev. B 54 9353\n[3] Katine J A, Albert F J, Buhrman R A Myers E B, and\nRalph D C 2000 Phys. Rev. Lett. 84 3149\n[4] Bazaliy Y B, Jones B A, and Zhang Shou-Cheng, 1998\nPhys. Rev. B 57 R3213\nSlonczewski J C, 1999 J. Magn. Magn. Mater. 195 L261\n[5] Tatara G, Kohno H, 2004 Phys. Rev. Lett. 92 086601\nHo J, Khanna F C, and Choi B C, 2004 Phys. Rev. Lett.92 097601\n[6] Li Z and Zhang S, 2004 Phys. Rev. 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Wade T, Ansermet J Ph, 2002 J. Appl. Phys.\n91 6806\nYamanouchi Y, Chiba D, Matsukura F, and Ohno H,\n2004 Nature (London) 428 539\nKrivorotov I N, Emley N C, Sankey J C, Kiselev S I,\nRalph D C, and Buhrman R A, 2005 Science 307 228\n[12] Chen T Y, Ji Y, Chien C L, and Stiles M D, 2004 Phys.\nRev. Lett. 93 026601\nMyers E B, Ralph D C, Katine J A, Louie R N, Buhrman\nR A, 1999 Science 285 867\n[13] Jiang Y, Nozaki T, Abe S, Ochiai T, Hirohata A, Tezuka\nN, and Inomata K, 2004 Nature Materials 3 361\nJiang Y, Yu G H, Wang Y B, Teng J, Ochiai T, Tezuka\nN, and Inomata K, 2005 Appl. Phys. Lett. 86 192515\nJiang Y, Tezuka N, Inomata K, 2006 Appl. Phys. Lett.\n89 122514\n[14] Li Zai-Dong, Liang J Q, Li L, Liu W M 2004 Phys. Rev.\nE 69 066611\nLi Zai-Dong, Li Qiu-Yan Li L, Liu W M, 2007 Phys. Rev.\nE 76 026605\n[15] He P B, Xie X C, Liu W M, 2005 Phys. Rev. B 72 064410\nHe P B, Liu W M, 2005 Phys. Rev. B 72 172411\n[16] Koo H, Krafft C, Gomez R D, 2002 Appl. Phys. Lett. 81\n862\nGrollier J, Boulenc, P, Cros V, Hamzic A, Vaures A, Fert\nA, and Faini G, 2003 Appl. Phys. Lett. 83 509\nTsoi M, Fontana R E, Parkin S S 2003 Appl. Phys. Lett.\n83 2617[17] Yamaguchi A, Ono T, Nasu S, Miyake K, Mibu K, Shinjo\nT, 2004 Phys. Rev. Lett. 92 077205\n[18] Saitoh E, Miyajima H, Yamaoka T and Tatara G, 2004\nNature 432 203\n[19] Lim C K, Devolder T, Chappert C, Grollier J, Cros V,\nVaures A, Fert A, and Faini G, 2004 Appl. Phys. Lett.\n84 2820\nTatara G, Saitoh E, Ichimura M and Kohno H, 2005\nAppl. Phys. Lett. 86 232504\n[20] Ohe J, Kramer B, 2006 Phys. Rev. Lett. 96 027204\nBeach G S D, Knutson C, Nistor C, Tsoi M, and Erskine\nJ L, 2006 Phys. Rev. Lett. 97 057203\nDugaev V K, Vieira V R, Sacramento P D, Barna J,\nAra´ ujo M A N, and J. Berakdar J, 2006 Phys. Rev. B\n74 054403\n[21] Berger L, 1978 J. Appl. Phys. 49 2156\nBerger L, 1984 J. Appl. Phys. 55 1954\nBerger L, 1992 J. Appl. Phys. 71 2721\nSalhi E and Berger L, 1993 J. Appl. Phys. 73 6405\n[22] Thiele A A, 1973 Phys. Rev. B 7 391\nSchryer N L, and Walker L R, 1974 J. Appl. Phys. 45\n5406\n[23] Munthe-Kaas H, 1995 BIT. 35 572\nEngo K, 2000 BIT. 40 41\n[24] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine\nJ L, 2005 Nat. Mater. 4 741\nYang J, Nistor C, Beach G S D, and Erskine J L, 2008\nPhys. Rev. B 77 014413\nFigure Captions\nFig. 1. The dynamics of DW above the critical\ncurrent. (a)-(c) Evolution of the normalized magneti-\nzationm. (d) The displacement of DW driven only\nby spin-transfer torque. The parameters are α= 0.2,\nC1= 0.05,C2= 0.25,bJ= 0.6, and the initial angle\nφ= 0.01π.\nFig. 2. (a) The evolution of cos φand the periodic\noscillation of DW velocity. (b) The periodic temporal\noscillation of DW width. The parameters are same as in\nfigure 1.0 20 40 60 80 100 120 140 160 180 200-1-0.8-0.6-0.4-0.200.20.40.60.81\nTimeEvolution of cos( I) and velocity of DW center(a)Velocity of DW center\ncos(I) of DW center0 20 40 60 80 100 120 140 160 180 20012345\nTimeEvolution of DW width(b)"    },    {        "title": "0705.1432v3.Effective_temperature_and_Gilbert_damping_of_a_current_driven_localized_spin.pdf",        "content": "arXiv:0705.1432v3  [cond-mat.mes-hall]  4 Feb 2008Effective temperature and Gilbert damping of a current-driv en localized spin\nAlvaro S. N´ u˜ nez∗\nDepartamento de F´ ısica, Facultad de Ciencias Fisicas y Mat ematicas,\nUniversidad de Chile, Casilla 487-3, Codigo postal 837-041 5, Santiago, Chile\nR.A. Duine†\nInstitute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n(Dated: October 31, 2018)\nStarting from a model that consists of a semiclassical spin c oupled to two leads we present a\nmicroscopic derivation of the Langevin equation for the dir ection of the spin. For slowly-changing\ndirection it takes on the form of the stochastic Landau-Lifs chitz-Gilbert equation. We give ex-\npressions for the Gilbert damping parameter and the strengt h of the fluctuations, including their\nbias-voltage dependence. At nonzero bias-voltage the fluct uations and damping are not related by\nthe fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible\nto introduce a voltage-dependent effective temperature tha t characterizes the fluctuations in the\ndirection of the spin, and its transport-steady-state prob ability distribution function.\nPACS numbers: 72.25.Pn, 72.15.Gd\nI. INTRODUCTION\nOne of the major challenges in the theoretical descrip-\ntion of various spintronics phenomena1, such as current-\ninduced magnetization reversal2,3,4,5and domain-wall\nmotion6,7,8,9,10,11,12, is their inherent nonequilibrium\ncharacter. In addition to the dynamics of the collective\ndegreeoffreedom, themagnetization, thenonequilibrium\nbehavior manifests itself in the quasi-particle degrees of\nfreedomthataredrivenoutofequilibriumbythe nonzero\nbias voltage. Due to this, the fluctuation-dissipation\ntheorem13,14cannot be applied to the quasi-particles.\nThis, in part, has led to controversysurrounding the the-\nory of current-induced domain wall motion15,16.\nEffective equations of motion for order-parameter\ndynamics that do obey the equilibrium fluctuation-\ndissipation theorem often take the form of Langevin\nequations, or their corresponding Fokker-Planck\nequations13,14,17. In the context of spintronics the rele-\nvant equation is the stochastic Landau-Lifschitz-Gilbert\nequationforthe magnetizationdirection18,19,20,21,22,23,24.\nIn this paper we derive the generalization of this equa-\ntion to the nonzero-current situation, for a simple\nmicroscopic model consisting of a single spin coupled\nto two leads via an onsite Kondo coupling. This model\nis intended as a toy-model for a magnetic impurity\nin a tunnel junction25,26,27. Alternatively, one may\nthink of a nanomagnet consisting of a collection of\nspins that are locked by strong exchange coupling.\nThe use of this simple model is primarily motivated\nby the fact that it enables us to obtain analytical\nresults. Because the microscopic starting point for\ndiscussing more realistic situations has a similar form,\nhowever, we believe that our main results apply quali-\ntatively to more complicated situations as well. Similar\nmodels have been used previously to explicitly study\nthe violation of the fluctuation-dissipation relation28, 1 1.1 1.2 1.3\n 0 0.1 0.2 0.3 0.4 0.5α/α0\n|e|V/µ\n 0 0.5 1 1.5 2 2.5 3\n 0  2  4  6  8  10Teff/T\n|e|V/(kB T)\nFIG. 1: Effective temperature as a function of bias voltage.\nThedashedlineshows thelarge bias-voltage asymptoticres ult\nkBTeff≃ |e|V/4 +kBT/2. The inset shows the bias-voltage\ndependence of the Gilbert damping parameter normalized to\nthe zero-bias result.\nand the voltage-dependence of the Gilbert damping\nparameter27. Starting from this model, we derive an\neffective stochastic equation for the dynamics of the\nspin direction using the functional-integral description\nof the Keldysh-Kadanoff-Baymnonequilibrium theory29.\n(For similar approaches to spin and magnetization\ndynamics, see also the work by Rebei and Simionato30,\nNussinov et al.31and Duine et al.32.) This formalism\nleads in a natural way to the path-integral formulation\nof stochastic differential equations33,34. One of the\nattractive features of this formalism is that dissipation\nand fluctuations enter the theory separately. This allows\nus to calculate the strength of the fluctuations even\nwhen the fluctuation-dissipation theorem is not valid.\nWe find that the dynamics of the direction of the spin\nis described by a Langevin equation with a damping ker-2\n,T\nSµL,T µR\nFIG. 2: Model system of a spin Sconnected to two tight-\nbinding model half-infinite leads. The chemical potential o f\nthe left lead is µLand different from the chemical potential\nof the right lead µR. The temperature Tof both leads is for\nsimplicity taken to be equal.\nnel and a stochastic magnetic field. We give explicit\nexpressions for the damping kernel and the correlation\nfunction of the stochastic magnetic field that are valid\nin the entire frequency domain. In general, they are not\nrelated by the fluctuation-dissipation theorem. In the\nlow-frequency limit the Langevin equation takes on the\nform ofthe stochasticLandau-Lifschitz-Gilbertequation.\nMoreover, in that limit it is always possible to introduce\nan effective temperature that characterizes the fluctua-\ntions and the equilibrium probability distribution for the\nspin direction. In Fig. 1 we present our main results,\nnamely the bias-voltage dependence of the effective tem-\nperature and the Gilbert damping parameter. We find\nthat the Gilbert damping constant initially varies lin-\nearly with the bias voltage, in agreement with the re-\nsult of Katsura et al.27. The voltage-dependence of the\nGilbert damping parameter is determined by the den-\nsity of states evaluated at an energy equal to the sum\nof the Fermi energy and the bias voltage. The effective\ntemperature is for small bias voltage equal to the actual\ntemperature, whereas for large bias voltage it is inde-\npendent of the temperature and proportional to the bias\nvoltage. This bias-dependence of the effective tempera-\nture is traced back to shot noise35.\nEffective temperatures for magnetization dynam-\nics have been introduced before on phenomenolog-\nical grounds in the context of thermally-assisted\ncurrent-driven magnetization reversal in magnetic\nnanopillars36,37,38. A current-dependent effective tem-\nperature enters in the theoretical description of these\nsystems because the current effectively lowers the energy\nbarrier thermal fluctuations have to overcome. In addi-\ntion to this effect, the presence of nonzero current alters\nthe magnetization noise due to spin current shot noise35.\nCovington et al.39interpret their experiment in terms of\ncurrent-dependent noise although this interpretation is\nstill under debate30. Foroset al.35also predict, using a\ndifferent model and different methods, a crossover from\nthermal to shot-noise dominated magnetization noise for\nincreasing bias voltage. Our main result in Fig. 1 is an\nexplicit example of this crossover for a specific model.\nThe remainderofthe paperis organizedasfollows. We\nstart in Sec. II by deriving the general Langevin equation\nforthe dynamicsofthe magneticimpurity coupledtotwo\nleads. In Sec. III and IV we discuss the low-frequency\nlimit in the absence and presence of a current, respec-\ntively. We end in Sec. V with our conclusions.II. DERIVATION OF THE LANGEVIN\nEQUATION\nWe use a model that consists of a spin Son a site\nthat is coupled via hopping to two semi-infinite leads, as\nshown in Fig. 2. The full probability distribution for the\ndirection ˆΩ of the spin on the unit sphere is written as a\ncoherent-state path integral over all electron Grassmann\nfield evolutions ψ∗(t) andψ(t), and unit-sphere paths\nS(t), that evolve from −∞totand back on the so-called\nKeldysh contour Ct. It is given by29\nP[ˆΩ,t] =/integraldisplay\nS(t)=ˆΩd[S]δ/bracketleftBig\n|S|2−1/bracketrightBig\nd[ψ∗]d[ψ]\n×exp/braceleftbiggi\n/planckover2pi1S[ψ∗,ψ,S]/bracerightbigg\n, (1)\nwhere the delta functional enforces the length constraint\nof the spin. In the above functional integral an inte-\ngration over boundary conditions at t=−∞, weighted\nby an appropriate initial density matrix, is implicitly in-\ncluded in the measure. We have not included boundary\nconditions on the electron fields, because, as we shall see,\nthe electron correlation functions that enter the theory\nafter integrating out the electrons are in practice conve-\nniently determined assumingthat the electronsareeither\nin equilibrium or in the transport steady state.\nThe action S[ψ∗,ψ,S] is the sum of four parts,\nS[ψ∗,ψ,S] =SL/bracketleftBig/parenleftbig\nψL/parenrightbig∗,ψL/bracketrightBig\n+SR/bracketleftBig/parenleftbig\nψR/parenrightbig∗,ψR/bracketrightBig\n+SC/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ0,/parenleftbig\nψL/parenrightbig∗,ψL,/parenleftbig\nψR/parenrightbig∗,ψR/bracketrightBig\n+S0/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ0,S/bracketrightBig\n. (2)\nWe describe the leads using one-dimensional non-\ninteracting electron tight-binding models with the action\nSL/R/bracketleftBig/parenleftBig\nψL/R/parenrightBig∗\n,ψL/R/bracketrightBig\n=\n/integraldisplay\nCtdt′\n\n/summationdisplay\nj,σ/parenleftBig\nψL/R\nj,σ(t′)/parenrightBig∗\ni/planckover2pi1∂\n∂t′ψL/R\nj,σ(t′)\n+J/summationdisplay\n/an}bracketle{tj,j′/an}bracketri}ht;σ/parenleftBig\nψL/R\nj,σ(t′)/parenrightBig∗\nψL/R\nj′,σ(t′)\n\n, (3)\nwhere the sum in the second term of this action is\nover nearest neighbors only and proportional to the\nnearest-neighbor hopping amplitude Jin the two leads.\n(Throughout this paper the electron spin indices are de-\nnoted byσ,σ′∈ {↑,↓}, and the site indices by j,j′.) The\ncoupling between system and leads is determined by the\naction\nSC[/parenleftbig\nψ0/parenrightbig∗,ψ0,/parenleftbig\nψL/parenrightbig∗,ψL,/parenleftbig\nψR/parenrightbig∗,ψR] =/integraldisplay\nCtdt′JC/summationdisplay\nσ/bracketleftBig/parenleftbig\nψL\n∂L,σ(t′)/parenrightbig∗ψ0\nσ(t′)+/parenleftbig\nψ0\nσ(t′)/parenrightbig∗ψL\n∂L,σ(t′)/bracketrightBig\n+3\n/integraldisplay\nCtdt′JC/summationdisplay\nσ/bracketleftBig/parenleftbig\nψR\n∂R,σ(t′)/parenrightbig∗ψ0\nσ(t′)+/parenleftbig\nψ0\nσ(t′)/parenrightbig∗ψR\n∂R,σ(t′)/bracketrightBig\n,\n(4)\nwhere∂L and∂R denote the end sites of the semi-infinite\nleft and right lead, and the fields/parenleftbig\nψ0(t)/parenrightbig∗andψ0(t) de-\nscribe the electrons in the single-site system. The hop-\nping amplitude between the single-site system and the\nleads is denoted by JC. Finally, the action for the sys-\ntem reads\nS0/bracketleftBig/parenleftbig\nψ0/parenrightbig∗,ψ∗,S/bracketrightBig\n=/integraldisplay\nCtdt′\n/summationdisplay\nσ/parenleftbig\nψ0\nσ(t′)/parenrightbig∗i/planckover2pi1∂\n∂t′ψ0\nσ(t′)\n−/planckover2pi1SA(S(t′))·dS(t′)\ndt′+h·S(t′)\n+∆/summationdisplay\nσ,σ′/parenleftbig\nψ0\nσ(t′)/parenrightbig∗τσ,σ′·S(t′)ψ0\nσ′(t′)\n.(5)\nThe second term in this action is the usual Berry phase\nfor spin quantization40, withA(S) the vector potential\nof a magnetic monopole\nǫαβγ∂Aγ\n∂Sβ=Sα, (6)\nwhere a sum over repeated Greek indices α,β,γ∈\n{x,y,z}is implied throughout the paper, and ǫαβγis\nthe anti-symmetric Levi-Civita tensor. The third term in\nthe action in Eq. (5) describes the coupling of the spin to\nan external magnetic field, up to dimensionful prefactors\ngiven by h. (Note that hhas the dimensions of energy.)\nThe last term in the action models the s−dexchange\ncoupling of the spin with the spin of the conduction elec-\ntrons in the single-site system and is proportional to the\nexchange coupling constant ∆ >0. The spin of the con-\nduction electronsisrepresentedbythe vectorofthe Pauli\nmatrices that is denoted by τ.\nNext, we proceed to integrate out the electrons using\nsecond-order perturbation theory in ∆. This results in\nan effective action for the spin given by\nSeff[S] =/integraldisplay\nCtdt′/bracketleftbigg\nS/planckover2pi1A(S(t′))·dS(t′)\ndt′+h·S(t′)\n−∆2/integraldisplay\nCtdt′′Π(t′,t′′)S(t′)·S(t′′)/bracketrightbigg\n. (7)\nThis perturbation theory is valid as long as the electron\nband width is much larger than the exchange interac-\ntion with the spin, i.e., J,JC≫∆. The Keldysh quasi-\nparticleresponsefunctionisgivenintermsoftheKeldysh\nGreen’s functions by\nΠ(t,t′) =−i\n/planckover2pi1G(t,t′)G(t′,t), (8)\nwhere the Keldysh Green’s function is defined by\niG(t,t′) =/angbracketleftBig\nψ0\n↑(t)/parenleftbig\nψ0\n↑(t′)/parenrightbig∗/angbracketrightBig\n=/angbracketleftBig\nψ0\n↓(t)/parenleftbig\nψ0\n↓(t′)/parenrightbig∗/angbracketrightBig\n.(9)We willgiveexplicit expressionsforthe responsefunction\nand the Green’s function later on. For now, we will only\nmake use of the fact that a general function A(t,t′) with\nits argumentson the Keldysh contour is decomposed into\nits analytic pieces by means of\nA(t,t′) =θ(t,t′)A>(t,t′)+θ(t′,t)A<(t,t′),(10)\nwhereθ(t,t′) is the Heaviside step function on the\nKeldysh contour. There can be also a singular piece\nAδδ(t,t′), but suchageneraldecompositionisnot needed\nhere. Also needed are the advanced and retarded com-\nponents, denoted respectively by the superscript ( −) and\n(+), and defined by\nA(±)(t,t′)≡ ±θ(±(t−t′))/bracketleftbig\nA>(t,t′)−A<(t,t′)/bracketrightbig\n,(11)\nand, finally, the Keldysh component\nAK(t,t′)≡A>(t,t′)+A<(t,t′), (12)\nwhich, as we shall see, determines the strength of the\nfluctuations.\nNext we write the forward and backward paths of the\nspin on the Keldysh contour, denoted respectively by\nS(t+) andS(t−), as a classical path Ω(t) plus fluctua-\ntionsδΩ(t), by means of\nS(t±) =Ω(t)±δΩ(t)\n2. (13)\nMoreover,it turns out to be convenient to write the delta\nfunctional, which implements the length constraintofthe\nspin, as a path integral over a Lagrange multiplier Λ( t)\ndefined on the Keldysh contour. Hence we have for the\nprobability distribution in first instance that\nP[ˆΩ,t] =/integraldisplay\nS(t)=ˆΩd[S]d[Λ]exp/braceleftbiggi\n/planckover2pi1Seff[S]+i\n/planckover2pi1SΛ[S,Λ]/bracerightbigg\n,\n(14)\nwith\nSΛ[S,Λ] =/integraldisplay\nCtdt′Λ(t′)/bracketleftBig\n|S(t′)|2−1/bracketrightBig\n.(15)\nWe then also have to split the Lagrange multiplier into\nclassical and fluctuating parts according to\nΛ(t±) =λ(t)±δλ(t)\n2. (16)\nNote that the coordinate transformations in\nEqs. (13) and (16) have a Jacobian of one. Before\nwe proceed, we note that in principle we are required\nto expand the action up to all orders in δΩ. Also note\nthat for some forward and backward paths S(t+) and\nS(t−) on the unit sphere the classical path Ωis not\nnecessarily on the unit sphere. In order to circumvent\nthese problems we note that the Berry phase term in\nEq. (5) is proportional to the area on the unit sphere\nenclosed by the forward and backward paths. Hence, in4\nthe semi-classical limit S→ ∞27,40paths whose forward\nand backward components differ substantially will be\nsuppressed in the path integral. Therefore, we take this\nlimit from now on which allows us to expand the action\nin terms of fluctuations δΩ(t) up to quadratic order. We\nwill see that the classical path Ω(t) is now on the unit\nsphere. We note that this semi-classical approximation\nis not related to the second-order perturbation theory\nused to derive the effective action.\nSplitting the paths in classical and fluctuation parts\ngives for the probability distribution\nP[ˆΩ,t] =/integraldisplay\nΩ(t)=ˆΩd[Ω]d[δΩ]d[λ]d[δλ]exp/braceleftbiggi\n/planckover2pi1S[Ω,δΩ,λ,δλ]/bracerightbigg\n,\n(17)\nwith the action, that is now projected on the real-time\naxis,\nS[Ω,δΩ,λ,δλ] =/integraldisplay\ndt/braceleftbigg\n/planckover2pi1SǫαβγδΩβ(t)dΩα(t)\ndtΩγ(t)\n+δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)\n+δλ(t)/bracketleftbig\n|Ω(t)|2−1+|δΩ(t)|2/4/bracketrightbig/bracerightbigg\n−∆2/integraldisplay\ndt/integraldisplay\ndt′/braceleftBig\nδΩα(t)/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\nΩα(t′)/bracerightBig\n−∆2\n2/integraldisplay\ndt/integraldisplay\ndt′/bracketleftbig\nδΩα(t)ΠK(t,t′)δΩα(t′)/bracketrightbig\n. (18)\nFrom this action we observe that the integration over\nδλ(t) immediately leads to the constraint\n|Ω(t)|2= 1−|δΩ(t)|2\n4, (19)\nas expected. Implementing this constraint leads to terms\nof order O(δΩ3) or higher in the above action which we\nare allowed to neglect because of the semi-classical limit.\nFrom now on we can therefore take the path integration\noverΩ(t) on the unit sphere.\nThephysicalmeaningofthetermslinearandquadratic\ninδΩ(t) becomes clear after a so-called Hubbard-\nStratonovich transformation which amounts to rewrit-\ning the action that is quadratic in the fluctuations as\na path integral over an auxiliary field η(t). Performing\nthis transformation leads to\nP[ˆΩ,t] =/integraldisplay\nΩ(t)=ˆΩd[Ω]d[δΩ]d[η]d[λ]\n×exp/braceleftbiggi\n/planckover2pi1S[Ω,δΩ,λ,η]/bracerightbigg\n,(20)\nwhere the path integration over Ωis now on the unit\nsphere. The action that weighs these paths is given by\nS[Ω,δΩ,λ,η] =/integraldisplay\ndt/bracketleftbigg\n/planckover2pi1SǫαβγδΩβ(t)dΩα(t)\ndtΩγ(t)\n+δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)+δΩα(t)ηα(t)/bracketrightbigg−∆2/integraldisplay\ndt/integraldisplay\ndt′/braceleftBig\nδΩα(t)/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\nΩα(t′)/bracerightBig\n+1\n2∆2/integraldisplay\ndt/integraldisplay\ndt′/bracketleftBig\nηα(t)/parenleftbig\nΠK/parenrightbig−1(t,t′)ηα(t′)/bracketrightBig\n.(21)\nNote that the inverse in the last term is defined as/integraltext\ndt′′ΠK(t,t′′)/parenleftbig\nΠK/parenrightbig−1(t′′,t′) =δ(t−t′).\nPerforming now the path integral over δΩ(t), we ob-\nserve that the spin direction Ω(t) is constraint to obey\nthe Langevin equation\n/planckover2pi1SǫαβγdΩβ(t)\ndtΩγ(t) =hα+2λ(t)Ωα(t)\n+ηα(t)+/integraldisplay∞\n−∞dt′K(t,t′)Ωα(t′),(22)\nwith the so-called damping or friction kernel given by\nK(t,t′) =−∆2/bracketleftBig\nΠ(−)(t′,t)+Π(+)(t,t′)/bracketrightBig\n.(23)\nNote that the Heaviside step functions in Eq. (11) appear\nprecisely such that the Langevin equation is causal. The\nstochastic magnetic field is seen from Eq. (21) to have\nthe correlations\n∝an}bracketle{tηα(t)∝an}bracketri}ht= 0 ;\n∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht=iδαβ/planckover2pi1∆2ΠK(t,t′).(24)\nUsing the fact that Ω(t) is a unit vector within our semi-\nclassical approximation, the Langevin equation for the\ndirection of the spin ˆΩ(t) is written as\n/planckover2pi1SdˆΩ(t)\ndt=ˆΩ(t)×/bracketleftbigg\nh+η(t)+/integraldisplay∞\n−∞dt′K(t,t′)ˆΩ(t′)/bracketrightbigg\n,\n(25)\nwhich has the form of a Landau-Lifschitz equation with\na stochastic magnetic field and a damping kernel. In\nthe next sections we will see that for slowly-varying spin\ndirection we get the usual form of the Gilbert damping\nterm.\nSo far, we have not given explicit expressions for the\nresponsefunctionsΠ(±),K(t,t′). Todeterminethesefunc-\ntions, we assume that the left and right leads are in\nthermal equilibrium at chemical potentials µLandµR,\nrespectively. Although not necessary for our theoretical\napproachwe assume, for simplicity, that the temperature\nTof the two leads is the same. The Green’s functions for\nthe system are then given by41,42\n−iG<(ǫ) =A(ǫ)\n2/summationdisplay\nk∈{L,R}N(ǫ−µk) ;\niG>(ǫ) =A(ǫ)\n2/summationdisplay\nk∈{L,R}[1−N(ǫ−µk)] ;\nG≶,K(t−t′) =/integraldisplaydǫ\n(2π)e−iǫ(t−t′)//planckover2pi1G≶,K(ǫ),(26)\nwithN(ǫ) ={exp[ǫ/(kBT)]+1}−1the Fermi-Dirac dis-\ntribution function with kBBoltzmann’s constant, and\nA(ǫ) =i/bracketleftBig\nG(+)(ǫ)−G(−)(ǫ)/bracketrightBig\n, (27)5\nthe spectral function. Note that Eq. (26) has a particu-\nlarlysimpleformbecausewearedealingwithasingle-site\nsystem. The retarded and advanced Green’s functions\nare determined by\n/bracketleftBig\nǫ±−2/planckover2pi1Σ(±)(ǫ)/bracketrightBig\nG(±)(ǫ) = 1, (28)\nwithǫ±=ǫ±i0, and the retarded self-energy due to one\nlead follows, for a one-dimensional tight-binding model,\nas\n/planckover2pi1Σ(+)(ǫ) =−J2\nC\nJeik(ǫ)a, (29)\nwithk(ǫ) = arccos[ −ǫ/(2J)]/athe wave vector in the\nleads at energy ǫ, andathe lattice constant. The ad-\nvanced self-energy due to one lead is given by the com-\nplex conjugate of the retarded one.\nBefore proceeding we give a brief physical description\nof the above results. (More details can be found in\nRefs. [41] and [42].) They arise by adiabatically elim-\ninating (“integrating out”) the leads from the system,\nassuming that they are in equilibrium at their respective\nchemical potentials. This procedure reduces the problem\nto a single-site one, with self-energy corrections for the\non-site electron that describe the broadening of the on-\nsite spectral function from a delta function at the (bare)\non-site energy to the spectral function in Eq. (27). More-\nover, the self-energy corrections also describe the non-\nequilibrium occupation of the single site via Eq. (26)\nFor the transport steady-state we have that\nΠ(±),K(t,t′) depends only on the difference of the time\narguments. Using Eq. (8) and Eqs. (10), (11), and (12)\nwe find that the Fourier transforms are given by\nΠ(±)(ǫ)≡/integraldisplay\nd(t−t′)eiǫ(t−t′)//planckover2pi1Π(±)(t,t′)\n=/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)1\nǫ±+ǫ′−ǫ′′\n×/bracketleftbig\nG<(ǫ′)G>(ǫ′′)−G>(ǫ′)G<(ǫ′′)/bracketrightbig\n,(30)\nand\nΠK(ǫ) =−2πi/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)δ(ǫ+ǫ′−ǫ′′)\n×/bracketleftbig\nG>(ǫ′)G<(ǫ′′)+G<(ǫ′)G>(ǫ′′)/bracketrightbig\n.(31)\nIn the next two sections we determine the spin dynamics\nin the low-frequency limit, using these expressions to-\ngether with the expressions for G≶(ǫ). We consider first\nthe equilibrium case.\nIII. EQUILIBRIUM SITUATION\nIn equilibrium the chemical potentials of the two leads\nare equal so that we have µL=µR≡µ. Combining re-\nsults from the previous section, we find for the retardedand advanced response functions (the subscript “0” de-\nnotes equilibrium quantities) that\nΠ(±)\n0(ǫ) =/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)A(ǫ′)A(ǫ′′)\n×[N(ǫ′−µ)−N(ǫ′′−µ)]\nǫ±+ǫ′−ǫ′′.(32)\nThe Keldysh component of the response function is in\nequilibrium given by\nΠK\n0(ǫ) =−2πi/integraldisplaydǫ′\n(2π)/integraldisplaydǫ′′\n(2π)A(ǫ′)A(ǫ′′)δ(ǫ−ǫ′+ǫ′′)\n{[1−N(ǫ′−µ)]N(ǫ′′−µ)+N(ǫ′−µ)[1−N(ǫ′′−µ)]}.(33)\nThe imaginary part of the retarded and advanced re-\nsponse functions are related to the Keldysh component\nby means of\nΠK\n0(ǫ) =±2i[2NB(ǫ)+1]Im/bracketleftBig\nΠ(±)\n0(ǫ)/bracketrightBig\n,(34)\nwithNB(ǫ) ={exp[ǫ/(kBT)]−1}−1the Bose distribu-\ntion function. This is, in fact, the fluctuation-dissipation\ntheorem which relates the dissipation, determined as we\nshall see by the imaginary part of the retarded and\nadvanced components of the response function, to the\nstrength of the fluctuations, determined by the Keldysh\ncomponent.\nFor low energies, corresponding to slow dynamics, we\nhave that\nΠ(±)\n0(ǫ)≃Π(±)\n0(0)∓i\n4πA2(µ)ǫ . (35)\nWith this result the damping term in the Langevin equa-\ntion in Eq. (25) becomes\n/integraldisplay∞\n−∞dt′K(t,t′)ˆΩ(t′) =−/planckover2pi1∆2A2(µ)\n2πdˆΩ(t)\ndt,(36)\nwhere we have not included the energy-independent part\nofEq. (35) because it does not contribute to the equation\nof motion for ˆΩ(t). In the low-energy limit the Keldysh\ncomponent of the response function is given by\nΠK\n0(ǫ) =A2(µ)\niπkBT . (37)\nPutting all these results together we find that the dy-\nnamics of the spin direction is, as long as the two leads\nare in equilibrium at the same temperature and chemical\npotential,determinedbythestochasticLandau-Lifschitz-\nGilbert equation\n/planckover2pi1SdˆΩ(t)\ndt=ˆΩ(t)×[h+η(t)]−/planckover2pi1α0ˆΩ×dˆΩ(t)\ndt,(38)\nwith the equilibrium Gilbert damping parameter\nα0=∆2A2(µ)\n2π. (39)6\nUsing Eqs. (24), (37), and (39) we find that the strength\nof the Gaussian stochastic magnetic field is determined\nby\n∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht= 2α0/planckover2pi1kBTδ(t−t′)δαβ.(40)\nNote that these delta-function type noise correlations\nare derived by approximating the time dependence of\nΠK(t,t′) by a delta function in the difference of the time\nvariables. This means that the noisy magnetic field η(t)\ncorresponds to a Stratonovich stochastic process13,14,17.\nThe stationary probability distribution function gen-\nerated by the Langevin equation in Eqs. (38) and (40) is\ngiven by the Boltzmann distribution18,19,20,21,22,23,24\nP[ˆΩ,t→ ∞]∝exp/braceleftBigg\n−E(ˆΩ)\nkBT/bracerightBigg\n, (41)\nwith\nE[ˆΩ] =−h·ˆΩ, (42)\nthe energy of the spin in the external field. It turns\nout that Eq. (41) holds for any effective field h=\n−∂E[ˆΩ]/∂ˆΩ, and in particular for the case that E[ˆΩ] is\nquadratic in the components of ˆΩ as is often used to\nmodel magnetic anisotropy.\nIt is important to realize that the equilibrium prob-\nability distribution has precisely this form because of\nthe fluctuation-dissipation theorem, which ensures that\ndissipation and fluctuations cooperate to achieve ther-\nmal equilibrium13,14. Finally, it should be noted that\nthis derivation of the stochastic Landau-Lifschitz-Gilbert\nequation from a microscopic starting point circumvents\nconcerns regarding the phenomenological form of damp-\ning and fluctuation-dissipation theorem, which is subject\nof considerable debate22,23.\nIV. NONZERO BIAS VOLTAGE\nIn this section we consider the situation that the chem-\nical potential of the left lead is given by µL=µ+|e|V,\nwith|e|V >0 the bias voltage in units of energy, and\nµ=µRthe chemical potential of the right lead. Using\nthe general expressions given for the response functions\nderived in Sec. II, it is easy to see that the imaginary\npart of the retarded and advanced components of the\nresponse functions are no longer related to the Keldysh\ncomponent by means of the fluctuation-dissipation theo-\nrem in Eq. (34). See also the work by Mitra and Millis28\nfor a discussion of this point. As in the previous section,\nwe proceed to determine the low-frequency behavior of\nthe response functions.\nUsing Eqs. (26), (27), and (30) we find that the re-\ntarded and advanced components of the response func-\ntion are given by\nΠ(±)(ǫ) =∓i\n8π/bracketleftbig\nA2(µ+|e|V)+A2(µ)/bracketrightbig\nǫ .(43)In this expression we have omitted the energy-\nindependent part andthe contribution followingfrom the\nprincipal-value part of the energy integral because, as we\nhave seen previously, these do not contribute to the final\nequation of motion for the direction of the spin. Follow-\ning the same steps as in the previous section, we find\nthat the damping kernel in the general Langevin equa-\ntion in Eq. (25) reduces to a Gilbert damping term with\na voltage-dependent damping parameter given by\nα(V) =∆2\n4π/bracketleftbig\nA2(µ+|e|V)+A2(µ)/bracketrightbig\n≃α0/bracketleftbigg\n1+O/parenleftbigg|e|V\nµ/parenrightbigg/bracketrightbigg\n. (44)\nThis result is physically understood by noting that the\nGilbert damping is determined by the dissipative part of\nthe response function Π(+)(ǫ). In this simple model, this\ndissipative part gets contributions from processes that\ncorrespond to an electron leaving or entering the system,\nto or from the leads, respectively. The dissipative part\nis in general proportional to the density of states at the\nFermi energy. Since the Fermi energy of left and right\nlead is equal to µ+|e|Vandµ, respectively, the Gilbert\ndamping has two respective contributions corresponding\nto the two terms in Eq. (44).\nNote that the result that the Gilbert damping param-\neter initially varies linearly with the voltage is in agree-\nment with the results of Katsura et al.27, although these\nauthorsconsideraslightlydifferentmodel. Inthe insetof\nFig. 1 we show the Gilbert damping parameter as a func-\ntion of voltage. The parameters taken are ∆ /J= 0.1,\nJC=J,µ/J= 1 andµ/(kBT) = 100.\nAlthoughwe cannolongermakeuseofthefluctuation-\ndissipation theorem, we are nevertheless able to deter-\nmine the fluctuations by calculating the low-energy be-\nhavioroftheKeldyshcomponentoftheresponsefunction\nin the nonzero-voltage situation. It is given by\nΠK(ǫ) =−i\n2/integraldisplaydǫ′\n(2π)A2(ǫ′){[N(µL−ǫ′)+N(µR−ǫ′)]\n×[N(ǫ′−µL)+N(ǫ′−µR)]}. (45)\nWe define an effective temperature by means of\nkBTeff(T,V)≡iΠK(ǫ)∆2\n2α(V). (46)\nThis definition is motivated by the fact that, as we\nmention below, the spin direction obeys the stochas-\ntic Landau-Lifschitz-Gilbert equation with voltage-\ndependentdampingandfluctuationscharacterizedbythe\nabove effective temperature43. From the expression for\nα(V) and ΠK(ǫ) we see that in the limit of zero bias\nvoltage we recover the equilibrium result Teff=T. In\nthe situation that |e|Vis substantially larger than kBT,\nwhichis usuallyapproachedin experiments, wehavethat\nkBTeff(T,V)≃|e|V\n4+kBT\n2, (47)7\nwhich in the limit that |e|V≫kBTbecomes indepen-\ndent of the actual temperature of the leads. In Fig. 1\nthe effective temperature as a function of bias voltage is\nshown, using the expression for ΠK(ǫ) given in Eq. (45).\nThe parameters are the same as before, i.e., ∆ /J= 0.1,\nJC=J,µ/J= 1 andµ/(kBT) = 100. Clearly the ef-\nfective temperature changes from Teff=Tat zero bias\nvoltagetotheasymptoticexpressioninEq.(47)shownby\nthe dashed line in Fig. 1. The crossover between actual\ntemperatureandvoltageasameasureforthefluctuations\nis reminiscent of the theory of shot noise in mesoscopic\nconductors44. This is not surprising, since in the single-\nsite model we use the noise in the equation of motion ul-\ntimately arises because of fluctuations in the number of\nelectronsin thesingle-sitesystem, andisthereforeclosely\nrelated to shot noise in the current through the system.\nForoset al.35calculate the magnetization noise arising\nfrom spin currentshot noisein the limit that |e|V≫kBT\nand|e|V≪kBT. In these limits our results are similar\nto theirs.\nWith the above definition of the effective temperature\nwefind that in the nonzerobiasvoltagesituationthe spin\ndirection obeys the stochastic Landau-Lifschitz-Gilbert\nequation, identical in form to the equilibrium case in\nEqs. (38) and (40), with the Gilbert damping parame-\nter and temperature replaced according to\nα0→α(V) ;\nT→Teff(T,V). (48)\nMoreover, the transport-steady-state probability distri-\nbution for the direction of the spinis a Boltzmann distri-\nbution with the effective temperature characterizing the\nfluctuations.\nV. DISCUSSION AND CONCLUSIONS\nWe have presented a microscopic derivation of the\nstochastic Landau-Lifschitz-Gilbert equation for a semi-\nclassical single spin under bias. We found that the\nGilbert damping parameter is voltage dependent and to\nlowest order acquires a correction linear in the bias volt-\nage, in agreement with a previous study for a slightly\ndifferent model27. In addition, we have calculated the\nstrength of the fluctuations directly without using the\nfluctuation-dissipation theorem and found that, in the\nlow-frequency regime, the fluctuations are characterized\nby a voltage and temperature dependent effective tem-\nperature.\nTo arrive at these results we have performed a low\nfrequency expansion of the various correlation functions\nthat enter the theory. Such an approximation is valid as\nlong as the dynamics is much slower than the times set\nby the other energy scales in the system such as temper-\nature and the Fermi energy. Moreover, in order for the\nleads to remain in equilibrium as the spin changes direc-\ntion, the processes in the leads that lead to equilibrationhave to be much faster than the precession period of the\nmagnetizationspin. Both these criteria are satisfied in\nexperiments with magnetic materials. In principle how-\never, the full Langevinequationderivedin Sec. II alsode-\nscribes dynamics beyond this low-frequency approxima-\ntion. The introduction of the effective temperature relies\non the low-frequency approximation though, and for ar-\nbitrary frequencies such a temperature can no longer be\nuniquely defined28.\nAn effective temperature for magnetization dynam-\nics has been introduced before on phenomenological\ngrounds36,37,38. Interestingly, the phenomenological ex-\npression of Urazhdin et al.36, found by experimentally\nstudying thermal activation of current-driven magneti-\nzation reversal in magnetic trilayers, has the same form\nas our expression for the effective temperature in the\nlarge bias-voltage limit [Eq. (47)] that we derived micro-\nscopically. Zhang and Li37, and Apalkov and Visscher38,\nhave, on phenomenological grounds, also introduced an\neffective temperature to study thermally-assisted spin-\ntransfer-torque-induced magnetization switching. In\ntheir formulation, however, the effective temperature is\nproportional to the real temperature because the current\neffectively modifies the energy barrier for magnetization\nreversal.\nForoset al.35consider spin current shot noise in the\nlargebias-voltagelimit andfind forsufficiently largevolt-\nage that the magnetization noise is dominated by shot\nnoise. Moreover, they also consider the low bias-voltage\nlimit and predict a crossover for thermal to shot-noise\ndominated magnetization fluctuations. Our main result\nin Fig. 1 provides an explicit example of this crossover\nfor a simple model system obtained by methods that are\neasily generalized to more complicated models. In the\nexperiments of Krivorotov et al.45the temperature de-\npendence of the dwell time of parallel and anti-parallel\nstates of a current-driven spin valve was measured. At\nlow temperatures kBT/lessorsimilar|e|Vthe dwell times are no\nlonger well-described by a constant temperature, which\ncould be a signature of the crossover from thermal noise\nto spin current shot noise. However, Krivorotov et al.\ninterpret this effect as due to ohmic heating, which is\nnot taken into account in the model presented in this\npaper, nor in the work by Foros et al.35. Moreover, in\nrealistic materials phonons provide an additional heat\nbath for the magnetization, with an effective tempera-\nture that may depend in a completely different manner\non the bias voltage than the electron heat-bath effec-\ntive temperature. Nonetheless, we believe that spin cur-\nrent shot noise may be observable in future experiments\nand that it may become important for applications as\ntechnological progress enables further miniaturization of\nmagnetic materials. Moreover, the formalism presented\nhereisanimportantstepinunderstandingmagnetization\nnoise from a microscopic viewpoint as its generalization\nto more complicated models is in principle straightfor-\nward. Possible interesting generalizations include mak-\ning one of the leads ferromagnetic (see also Ref. [46]).8\nSince spin transfer torques will occur on the single spin\nas a spin-polarized current from the lead interacts with\nthe single-spin system, the resulting model would be a\ntoy model for microscopically studying the attenuation\nof spin transfer torques and current-driven magnetiza-\ntion reversal by shot noise. Another simple and use-\nful generalization would be enlarging the system to in-\nclude more than one spin. The formalism presented here\nwould allow for a straightforward microscopic calcula-\ntion of Gilbert damping and adiabatic and nonadiabatic\nspin transfer torques which are currently attracting a lot\nof interest in the context of current-driven domain wall\nmotion6,7,8,9,10,11,12. The application of our theory in thepresentpaperis, in additiontoitsintrinsicphysicalinter-\nest, chosen mainly because of the feasibility of analytical\nresults. Theapplicationsmentionedabovearemorecom-\nplicated and analytical results may be no longer obtain-\nable. 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Buhrman,\nPhys. Rev. Lett. 93, 166603 (2004) .\n46New. J. Phys. 10, 013017 (2008)."    },    {        "title": "1304.7295v2.Landau_Lifshitz_theory_of_the_longitudinal_spin_Seebeck_effect.pdf",        "content": "Landau-Lifshitz theory of the longitudinal spin Seebeck e\u000bect\nSilas Ho\u000bman, Koji Sato, and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nThermal-bias-induced spin angular momentum transfer between a paramagnetic metal and ferro-\nmagnetic insulator is studied theoretically based on the stochastic Landau-Lifshitz-Gilbert (LLG)\nphenomenology. Magnons in the ferromagnet establish a nonequilibrium steady state by equilibrat-\ning with phonons via bulk Gilbert damping and electrons in the paramagnet via spin pumping,\naccording to the \ructuation-dissipation theorem. Subthermal magnons and the associated spin cur-\nrents are treated classically, while the appropriate quantum crossover is imposed on high-frequency\nmagnetic \ructuations. We identify several length scales in the ferromagnet, which govern qualitative\nchanges in the dependence of the thermally-induced spin current on the magnetic \flm thickness.\nPACS numbers: 85.75.-d,72.25.Mk,73.50.Lw,75.30.Ds\nI. INTRODUCTION\nOver the past three decades, spintronics has evolved\nfrom a focus on equilibrium phenomena in magnetic het-\nerostructures, such as giant magnetoresistance1and in-\nterlayer exchange interactions,2to dynamic processes,\nsuch as spin-transfer torque3,4and spin pumping,5,6and,\nmore recently, nonequilibrium thermodynamics, heralded\nby the spin Seebeck e\u000bect7{9and thermally-induced\nmotion of domain walls.10,11From a practical stand-\npoint, magnetic nanostructures are useful for \feld sens-\ning and nonvolatile information storage,12where magne-\ntoresistance is paramount for the readout, while current-\ninduced spin torques are useful for fast and scalable bit\nswitching.13\nOne rapidly-developing avenue of research concerns\nout-of-equilibrium spin phenomena in insulating systems,\nwhere spin is carried by collective excitations, such as\nspin waves (magnons), rather than electronic quasipar-\nticles. To this end, spin waves in the ferrimagnetic in-\nsulator yttrium iron garnet (YIG) appear particularly\npromising as they su\u000ber from a remarkably low Gilbert\ndamping (at microwave frequencies), \u000b\u001810\u00004, and\nthe host material has Curie temperature of \u0018500 K,\nthus remaining magnetic at room temperature.14Spin\nwaves in YIG have recently been shown to undergo room-\ntemperature Bose-Einstein condensation under nonlinear\nmicrowave pumping,15exhibit large spin pumping into\nadjacent conductors,16{18manifest the longitudinal spin\nSeebeck e\u000bect,19and e\u000eciently move domain walls un-\nder small thermal gradients.11These phenomena hold\npromise for integrated circuits based on nonvolatile mag-\nnetic elements20with essentially no Ohmic losses and\nthus very low dissipation.\nFurthermore, thermal control of magnetic dynamics\nand spin currents21provides an attractive alternative\nto voltage control, especially since magnons, which are\nneutral objects, can respond more directly to tempera-\nture gradients. The spin Seebeck e\u000bect, i.e., the gen-\neration of thermal spin current between magnetic insu-\nlators and normal metals, is the basic phenomenon of\ncentral interest in this context. The purpose of this pa-per is to develop a systematic semi-phenomenological ap-\nproach to this problem, based on the Landau-Lifshitz-\nGilbert (LLG) theory of ferromagnetic dynamics,22de-\nparting from the spin-pumping6perspective on the inter-\naction between electrons and magnons at ferromagnetic-\ninsulatorjnormal-metal interfaces put forward in Ref. 8.\nIn Sec. VIII, we comment on how the theory could be ex-\npanded to account for magnon and phonon kinetics when\nthe standard LLG phenomenology fails.\nII. FERROMAGNETIC BULK DYNAMICS\nIn the ferromagnetic bulk, away from the Curie\ntemperature, magnetic dynamics are described by the\nstochastic LLG equation22\n@tm=\u0000\rm\u0002(He\u000b+hl) +\u000bm\u0002@tm; (1)\nwhere m=M=Msis the unit-vector magnetization di-\nrection (Ms=jMjbeing the saturated magnetization\nmagnitude), \r(minus) the gyromagnetic ratio ( \r > 0\nfor free electrons), \u000bdimensionless Gilbert damping con-\nstant,\nHe\u000b\u0011\u0000\u000eMF=Haz+Axr2m+Hr (2)\nthe e\u000bective \feld (consisting of applied \feld Hain the\nzdirection, exchange \feld /Ax, and relativistic cor-\nrections Hrthat include dipolar interactions and crys-\ntalline anisotropies), and hlrandom Langevin \feld with\ncorrelator23\nhhl;i(r;t)hl;j(r0;t0)i=2\u000b\n\rMskBT(r)\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);\n(3)\nin accordance with the \ructuation-dissipation theorem.\nWe are interested at intermediate temperatures: much\nlower than the Curie temperature, such that the Landau-\nLifshitz phenomenology based on the directional magne-\ntization dynamics [SO(3) nonlinear \u001bmodel] is appro-\npriate, while not too low such that the classical theory\ncan be used as a starting point. We will, furthermore, ne-\nglectHrin Eq. (2), for simplicity, which is justi\fed whenarXiv:1304.7295v2  [cond-mat.mes-hall]  12 Aug 20132\nxyz0dT1T2With a little effort the layout of this diagram can actually be improvedby enlarging the inner box, see page 29 below.Here is the resonants-channel contribution toe+e−→4f. (From nowon, we do no longer display the\\begin{fmfgraph}(40,25)\\fmfpen{thick}...\\end{fmfgraph}environment surrounding all pictures.)\\fmfleftn{i}{2}\\fmfrightn{o}{4}\\fmf{fermion}{i1,v1,i2}\\fmf{photon}{v1,v2}\\fmfblob{.15w}{v2}\\fmf{photon}{v2,v3}\\fmf{fermion}{o1,v3,o2}\\fmf{photon}{v2,v4}\\fmf{fermion}{o4,v4,o3}\ne−e+µ+νµs¯cAnd the resonantt-channel contribution:\\fmfleftn{i}{2}\\fmfrightn{o}{4}\\fmf{fermion}{i1,v1,v2,i2}\\fmf{photon}{v1,v3}\\fmf{fermion}{o1,v3,o2}\\fmf{photon}{v2,v4}\\fmf{fermion}{o4,v4,o3}\ne−e+µ+νµs¯cTwo point loop diagrams pose another set of problems. We must havea way of specifying that one or more of the lines connecting the twovertices arenotconnected by a straight line. The optionsleft,rightandstraightoffer the possibility to connect two vertices by a semicircledetour, either on the left or on the right. Since by default all lines con-tribute to the tension between two vertices, thetensionoption allows usto reduce this tension. The next examples shows both options in action.The lower fermion line is given an tension of 1/3 to make is symmetricalwith the upper line with consists of three parts. The loop photon is usinga detour on the right and does not contribute any tension.\\fmfleft{i1,i2}\\fmfright{o1}\\fmf{fermion,tension=1/3}{i1,v1}\\fmf{plain}{v1,v2}\\fmf{fermion}{v2,v3}\\fmf{photon,right,tension=0}{v2,v3}21nHa⊗N1N2Fjs1js2\nFIG. 1. Schematic of an N 1jFjN2sandwich structure studied\nin this paper. The normal-metal layer N 1is treated as a poor\nspin sink, which blocks spin current, js1\u00190. The normal-\nmetal layer N 2, on the other hand, is a perfect spin sink, thus\nestablishing a thermal contact between its itinerant electrons\nand magnons in the ferromagnetic insulator (F), which results\nin spin current js2\u0011js. We assume the phonons in the F layer\nfollow a linear temperature pro\fle from T1atx= 0 (N 1jF\ninterface) to T2atx=d(FjN2interface), corresponding to\nelectron temperatures in N 1and N 2, respectively.\nkBT\u001d~\rMs. The Langevin correlator (3) is white at\nfrequencies !\u001ckBT=~, corresponding to classical be-\nhavior. In Sec. VI, we will adapt our theory to account\nfor quantum \ructuations at !&kBT=~, by matching\nwith the fully quantum treatment of Ref. 24.\nIn order to streamline discussion of the spin transfer,\nlet us switch from the magnetization to the spin density:\ns\u0011sn=\u0000Ms\n\rm; (4)\nwheres=Ms=\ris the saturated spin density and n=\n\u0000mits direction. The LLG equation then becomes\ns(1 +\u000bn\u0002)@tn+n\u0002(Hz+h) +@ijs;i= 0;(5)\nwhere\njs;i=\u0000An\u0002@in (6)\nis identi\fed as the magnetic spin current and\nhhi(r;t)hj(r0;t0)i= 2\u000bskBT(r)\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);(7)\nwhereiandjstand for the Cartesian coordinates. Here,\nH\u0011MsHaandA\u0011MsAx. In equilibrium, n=\u0000z,\nassuming the applied \feld Ha>0.\nIn this paper, we focus on the trilayer heterostructure\ndepicted in Fig. 1. The temperature T(r) entering Eq. (3)\nis taken to correspond to the x-dependent phonon tem-\nperature inside of the ferromagnetic \flm,\nT(x) =T1+x\nd(T2\u0000T1); (8)\nassuming Gilbert damping stems from the local magnon-\nphonon scattering. (We will revisit this assumption in\nSec. VIII.)III. BOUNDARY CONDITIONS\nThe boundary conditions for a ferromagnet sand-\nwiched between two normal metals need to be simi-\nlarly constructed to account both for deterministic6and\nstochastic25spin-transfer torques. We will start with\nthe former and then include the latter according to the\n\ructuation-dissipation theorem.\nWe assume the spin current is blocked by the N 1layer\natx= 0, due to its weak spin-relaxation rate:\njs;x= 0 (x= 0): (9)\nIn other words, the spin current pumped across the F jN1\ninterface is balanced by an equal back\row.6For our pur-\nposes, N 1can thus be replaced by an insulator, as long\nas it makes a good thermal contact with phonons in the\nferromagnet. A net spin current across the F jN2inter-\nface, on the other hand, is allowed, if we treat N 2as a\nperfect spin sink:6\njs;x=~g\"#\n4\u0019n\u0002dn\ndt(x=d); (10)\nwhereg\"#is the real part of the dimensionless interfa-\ncial spin-mixing conductance (per unit area).26We disre-\ngard the imaginary part of the spin-mixing conductance,\nsince it governs the typically smaller6nondissipative spin-\ncurrent component /dn=dt, which vanishes over a cycle\nof precession. A speci\fc realization for such an N 1jFjN2\ntrilayer could be provided by the Cu jYIGjPt combina-\ntion, where Cu (Pt) is a light (heavy) element with weak\n(strong) spin-orbit interaction. (We will generalize our\n\fndings to arbitrary N 1jFjN2trilayers, such as symmetric\nPtjYIGjPt type structures or general asymmetric struc-\ntures, in Sec. VII.)\nFor the two N 1jFjN2interfaces, we correspondingly\nhave the following (deterministic) boundary conditions\nfor magnetic dynamics (as T!0):\n8\n<\n:@xn= 0; x = 0\nA@xn+~g\"#\n4\u0019@tn= 0; x=d; (11)\nre\recting continuity of spin current, which is given by\nEq. (6) inside the ferromagnet and Eqs. (9) and (10) in\nN1and N 2, respectively, across the corresponding inter-\nfaces. Since spin pumping (10) a\u000bects magnetic dynamics\nsimilarly to Gilbert damping,6it is accompanied with a\nsimilar stochastic term.25The latter can be accounted\nfor by modifying the boundary condition at x=d:\nA@xn+~g\"#\n4\u0019@tn+h0= 0; (12)\nwhere\n\nh0\ni(\u001a;t)h0\nj(\u001a0;t0)\u000b\n=~g\"#\n2\u0019kBT2\u000eij\u000e(\u001a\u0000\u001a0)\u000e(t\u0000t0) (13)3\nand\u001a= (y;z) is the two-dimensional position along the\ninterface at x=d. The Langevin correlator strength is\nproportional to the electron temperature T2at the FjN2\ninterface, since the noise originates in the thermal \ruc-\ntuations of electronic spin currents in N 2.\nThe spin Seebeck e\u000bect is embodied in the thermal-\naveraged spin current \rowing through the F jN2\ninterface:8\njs;x=\u0000An\u0002@xn=n\u0002\u0012~g\"#\n4\u0019@tn+h0\u0013\n: (14)\nSince our system is axially symmetric with respect to the\nzaxis, it is convenient to switch to complex notation:\nn\u0011nx\u0000iny. Thermal spin-current density, hjs;xi=jsz,\ncan thus be written for small-angle dynamics (relevant\nat temperatures well below the Curie temperature) as\njs=AImhn\u0003@xnijx=d: (15)\nExploiting, furthermore, translational invariance in the\nyzplane, we \fnd in the steady state:\njs=AImZd2qd!\n(2\u0019)3hn(q;!)\u0003@xn(q0;!0)i\n(2\u0019)3\u000e(q\u0000q0)\u000e(!\u0000!0);(16)\nwhere\nn(q;!) =Z\nd2\u001adtei(!t\u0000q\u0001\u001a)n(\u001a;d;t) (17)\nis the Fourier transform over \u001aand timet. The delta\nfunctions in the denominator of Eq. (16) cancel delta\nfunctions that factor out of the numerator when evaluat-\ning the averageh:::i(with the remaining integrand inde-\npendent of q0and!0). Similarly transforming Langevin\ncorrelators, Eqs. (7) and (13), we have:\nhh(x;q;!)\u0003h(x0;q0;!0)i=4(2\u0019)3\u000bskBT(x)\n\u0002\u000e(x\u0000x0)\u000e(q\u0000q0)\u000e(!\u0000!0);(18)\nfor the bulk and\nhh0(q;!)\u0003h0(q0;!0)i= 4(2\u0019)3\u000b0skBT2\u000e(q\u0000q0)\u000e(!\u0000!0)\n(19)\nfor the FjN2interface, de\fning\n\u000b0\u0011~g\"#\n4\u0019s; (20)\nwhich has dimensions of length. \u000b0=dis the enhanced\nGilbert damping for a monodomain precession of the fer-\nromagnetic \flm.6\nIV. SPIN SEEBECK COEFFICIENT\nWe now have all the necessary ingredients in order\nto evaluate the (longitudinal) spin Seebeck coe\u000ecient\n(which has units of inverse length squared)27\nS\u0011js\nkB(T1\u0000T2)(21)of the N 1jFjN2structure shown in Fig. 1. To simplify\nour subsequent analysis, let us optimize the notation, as\nfollows. The stochastic LLG equation (5) in the \flm bulk\nis written as\nA(@2\nx\u0000\u00142)n(x;q;!) =h(x;q;!); (22)\nafter linearizing transverse dynamics and Fourier trans-\nforming it in the yzplane and time. Here,\n\u00142\u0011q2+H\u0000(1 +i\u000b)s!\nA: (23)\nThe stochastic boundary condition at x=d, Eq. (12), in\nthis notation is\nA(@x\u0000\u00140)n(x;q;!) =\u0000h0(q;!) (x=d); (24)\nwhile@xn= 0 atx= 0. Here,\n\u00140\u0011i\u000b0s!\nA: (25)\nEqs. (22)-(25) now form a closed system of inhomo-\ngeneous linear di\u000berential equations, with source terms\ngiven by stochastic \felds handh0. These are straight-\nforward to solve for nusing Green's functions. Substitut-\ning the solution for ninto Eq. (16), we \fnd, after some\nalgebra, the spin Seebeck coe\u000ecient (21):\nS=\u000b\u000b0s2\n2\u00193A2dZ1\n\u00001d2qZ1\n\u00001d!!\n\u0002Zd\n0dxx\f\f\f\fcosh[\u0014(x\u0000d)]\n\u0014sinh(\u0014d)\u0000\u00140cosh(\u0014d)\f\f\f\f2\n:(26)\nIntegrating over the longitudinal coordinate x, this \fnally\nbecomes\nS=\u000b\u000b0s2\n8\u00193A2dZ1\n\u00001d2qZ1\n\u00001d!!\nj\u0014sinh(\u0014d)\u0000\u00140cosh(\u0014d)j2\n\u0002\u0014sin2(\u0014id)\n\u00142\ni+sinh2(\u0014rd)\n\u00142r\u0015\n; (27)\nwhere\u0014r(\u0014i) is the real (imaginary) part of \u0014. Eq. (27)\nis our central results and the main departure point for\nthe subsequent analysis.\nLet us, for convenience, de\fne the following length\nscales:\n\u0018\u0011r\nA\nH(28)\nis the magnetic exchange length,\nl0\u0011\u000b0\n\u000b(29)\nis the spin-pumping length (i.e., the F thickness at which\nthe monodomain Gilbert damping enhancement due to\nspin pumping6equals the intrinsic damping),\n\u0015\u0011r\n~A\nskBT(30)4\nis the thermal de Broglie wavelength (in the absence of\napplied \feld), where Tis the ambient temperature, and\nl\u0011\u0015\n\u000b(31)\nis the decay length for thermal magnons in the bulk. In\nthis notation,\n\u00142=q2+1\n\u00182\u00001 +i\u000b\n\u00152~!\nkBT: (32)\nV. QUASIPARTICLE APPROXIMATION\nIn the following, we are primarily interested in the\nthickness,d, dependence of the spin Seebeck coe\u000ecient,\nS, assuming the the length-scale hierarchy \u0015\u001cl0\u001cl.\nIn YIG, for example, taking144\u0019Ms\u00182 kG,A\u0018\n1=2\u000210\u00006erg/cm, and \u000b\u001810\u00004, we \fnd the follow-\ning lengths: (1) \u0015.1 nm, at room temperature, (2)\nl0\u0018100 nm, taking g\"#\u00181014cm\u00002from Ref. 17 and\nproportionately larger l0withg\"#\u00185\u00021014cm\u00002from\nRef. 18 (g\"#is very sensitive to the preparation and qual-\nity of the YIGjmetal interfaces), (3) l.10\u0016m, and (4)\n\u0018\u001810 nm at 1 kG (corresponding to typical magneto-\nstatic \felds).\nWe start by performing integration over frequency !\nin Eq. (27) in the limit of low damping (both intrinsic\nand spin pumping). In this case, the integrand is peaked\nat sinh(\u0014d)\u00190, corresponding to\n!n(q) =A\ns\u0012\nq2+n2\u00192\nd2+1\n\u00182\u0013\n; (33)\nwheren= 0;1;2;3;::: labels magnon subbands [not to\nbe confused with unit vector nintroduced in Eq. (4)].\nThese resonances are well separated when their width is\nmuch smaller than their spacing, allowing for a quasipar-\nticle treatment of the energy integral. For the bulk damp-\ning, this condition is \u000b\u001c1=dp\n1=\u00152\u00001=\u00182, for thermal\nmagnons. Additionally, the occupation of magnons is ex-\nponentially suppressed when the temperature is smaller\nthan the gap, i.e., kBT=~.!0\u0011H=s. Therefore, we are\ninterested in the opposite regime, when the thermal de\nBroglie wavelength is smaller than the magnetic exchange\nlength,\u0015 < \u0018 . Thus in the regime d\u001cl, these reso-\nnances are well-de\fned quasiparticle peaks corresponding\nto monodomain precession ( n= 0) and standing waves\n(n>0) along the longitudinal direction. See Fig. 2. This\nallows us to evaluate the spin Seebeck coe\u000ecient by sum-\nming the contributions from individual magnon modes,\nwhend\u001cl.\nExpanding around !0, the contribution from the lowest\nenergy resonance is\nS0=\u000b\u000b0\n4\u00193dZ1\n\u00001d2qZ1\n\u00001d!!\n(!\u0000!0)2+ (\u000b+\u000b0=d)2!2;\n(34)\n1.01.21.41.61.82.011.52!/!0(q)FIG. 2. Integrand of the spin Seebeck coe\u000ecient, Eq. (27),\nin arbitrary units, illustrating the \frst four quasiparticle res-\nonances,n= 0;1;2;3, according to Eq. (33), for a \fxed q.\nWe set\u000b= 10\u00004,\u000b0= 10\u00002nm,q2+\u0018\u00002= 1 nm\u00002, and\nd= 10 nm in the main plot. The inset shows the essentially\ncontinuum spectrum when d= 100\u0016m, keeping other param-\neters unmodi\fed.\nwhich can be readily integrated over frequency:\nS0=\u000b\u000b0\n4\u00192dZ1\n\u00001d2q\n(\u000b+\u000b0=d) [1 + (\u000b+\u000b0=d)2]\n\u0019\u000b0=d\n1 +l0=dZ1\n\u00001d2q\n(2\u0019)2; (35)\nwhere we have assumed small damping, \u000b+\u000b0=d\u001c1.\nSimilarly, we \fnd for the n>0 subbands:28\nSn\u00192\u000b0=d\n1 + 2l0=dZ1\n\u00001d2q\n(2\u0019)2: (36)\nThe total Seebeck coe\u000ecient in the quasiparticle approx-\nimation is thus\nS=S0+X\nn>0Sn: (37)\nThe wave vector qin integrals (35) and (36) should be\nbounded by requiring that ~!n(q)<kBT, at which point\nour classical treatment breaks down, as discussed in the\nnext section. The spurious ultraviolet divergence is elim-\ninated by cutting o\u000b at a total three-dimensional wave\nnumber,qc=p\n1=\u00152\u00001=\u00182, corresponding to energy\nkBT. While a more careful quantum treatment for large\nwave numbers is constructed in the next section, we ex-\npect a crude cuto\u000b to adequately capture the behavior of\nthe Seebeck coe\u000ecient, up to an overall factor of order\none.\nAllow us to momentarily focus on the regime when d\u001d\n\u0015, where the quasiparticle peaks are dense (recovering\nthree-dimensional behavior), so that\nS\u0019X\nn>0Sn\u00192\u000b0\n1 + 2l0=dZ\nqcd3q\n(2\u0019)3=\u000b0\n1 + 2l0=dq3\nc\n3\u00192(38)5\nand, therefore,\nS=\u00121\n\u00152\u00001\n\u00182\u00133=2\n\u00028\n>><\n>>:\u000bd\n6\u00192; d\u001cl0\n\u000b0\n3\u00192; d\u001dl0: (39)\nThat is, when the thickness of the ferromagnet is much\nsmaller or larger than spin-pumping length, the spin cur-\nrent scales linearly with dor isdindependent, respec-\ntively.\nIf the quasiparticle peaks are not dense, d\u0018\u0015, \fnite-\nsize e\u000bects are important, re\recting individual magnon\nsubbands. In the extreme low-temperature case when\nd\u001c\u0015, only monodomain precession along the longi-\ntudinal direction contributes to the Seebeck coe\u000ecient:\nS\u0019S0. If the transverse dimensions are also much\nsmaller than \u0015, the full volume of the (nano)magnet un-\ndergoes stochastic monodomain precession, and\nS=\u000b0\nV1\n1 +l0=d; (40)\nwhere we have retained only one mode associated with\nthe transverse momentum in Eq. (35). Vhere is the vol-\nume of the F layer. This coincides with the spin Seebeck\ncoe\u000ecient for a monodomain obtained in Ref. 8 [de\fning\n(T1+T2)=2!TF,T2!TN, and\u000b0=d!\u000b0, to match\ntheir notation].\nFinally, for largest thicknesses d\u001dl, the quasiparticles\nare no longer well-de\fned (see inset of Fig. 2) and the\nabove analysis cannot be applied. Because the thickness\nis beyond the magnon propagation length, only magnons\nwithin a distance lfrom the FjN2interface contribute to\nthe spin current, which should, therefore, be independent\nof thickness, d, for a \fxed thermal gradient, ( T1\u0000T2)=d.\nSince, in this regime, the magnon propagation length is\nthe largest length scale in the problem, we can send d!\n1in Eq. (27), which gives\nS=\u000b\u000b0s2\n8\u00193A2dZ1\n\u00001d2qZ1\n\u00001d!!\n\u00142rj\u0014\u0000\u00140j2: (41)\nThe integrand, which can be evaluated numerically, is\nindependent of thickness, and, therefore, S/1=d, as\nexpected.\nVI. QUANTUM CROSSOVER\nOur classical Langevin theory needs to be appropri-\nately modi\fed when approaching magnon frequencies of\n~!\u0018kBT. On the one hand, this is an important limit,\nas the spin transport is dominated by thermal magnons\nin our model. On the other hand, the classical theory\nis inadequate for the treatment of quantum \ructuations\nthat dominate at high (on the scale of the ambient tem-\nperature) frequencies.To this end, we use a quantum-mechanical result24for\nthe thermal spin current, which is exact for a tunneling\nspin-exchange Hamiltonian at an F jN interface:29\njs=\u00004kB\u000eT\u000b0Z1\n\u000f0d\u000f\u000fD (\u000f)\f2@nBE(\f\u000f)\n@\f; (42)\nwhereD(\u000f) is the magnon density of states, nBE(x)\u0011\n(ex\u00001)\u00001is the Bose-Einstein distribution, \f\u00111=kBT,\nand\u000eTis the temperature drop across the F jN inter-\nface (assuming magnons are equilibrated to a uniform\ntemperature T, such that \u0015\u001cd). This limit can be di-\nrectly compared to our Eqs. (21) and (27), by \frst send-\ning\u000b0!0 in the integrand (thus reproducing the weak\nFjN contact and allowing for the magnons in F to equili-\nbrate with phonons) and then sending \u000b!0 [such that\nthe magnon spectral properties are una\u000bected by Gilbert\ndamping, as assumed in the derivation of Eq. (42)]. The\nmagnons are correspondingly equilibrated to the average\nphonon temperature ( T1+T2)=2, such that we identify\n\u000eT= (T1\u0000T2)=2, in the present notation. According to\nEq. (38), our semiclassical spin current becomes in this\nlimit\njs!2kB(T1\u0000T2)\u000b0Zd3q\n(2\u0019)3: (43)\nThis agrees with Eq. (42) in the limit \u000f\u001ckBT, where\n\u0000Z1\n\u000f0d\u000f\u000fD (\u000f)\f2@nBE(\f\u000f)\n@\f!Z1\n\u000f0d\u000fD(\u000f)\u0011Zd3q\n(2\u0019)3:\n(44)\nWe conclude that the classical-to-quantum crossover\ncan be accounted for by inserting the factor\n\u0000\u000f\f2@nBE(\f\u000f)\n@\f=\u0014\f~!=2\nsinh(\f~!=2)\u00152\n\u0011F(\f~!) (45)\nin the energy integrand of Eq. (27), which e\u000bectively cuts\no\u000b the contribution from magnons with energy \u000f\u0011~!\u001d\nkBT.\nVII. RESULTS\nWe summarize the spin Seebeck coe\u000ecient dependence\non the ferromagnetic layer thickness, when \u0015\u001cl0\u001cl:\nS(d)\u00181\n2\u0019\u001528\n><\n>:\u000b; d\u001c\u0015\nd=l; \u0015\u001cd\u001cl0\nl0=l; l0\u001cd\u001cl\nl0=d; l\u001cd; (46)\nassuming\u0015\u001c\u0018(or else the magnon transport is ex-\nponentially frozen out). The four regimes correspond\nrespectively to the following physical situations: (1)\nOnly the lowest magnon subband is thermally active\n(d\u001c\u0015), (2) quasi-3D subband structure is activated, but\ndamping is still dominated by interfacial spin pumping6\n0.01110010410610-510-410-310-210-1\u0000l0l\nd/\u0000S\u00002\nFIG. 3. Plot of the spin Seebeck coe\u000ecient, Eq. (27), as a\nfunction of ferromagnet thickness dfor\u0015= 1 nm,\u0018= 10 nm,\nl0= 100 nm, and l= 10\u0016m. We use Eq. (37) (solid curve\non the left) and Eq. (41) (solid line on the right), which are\nvalid when d.landd&l, respectively, when N 1is a poor\nspin sink and N 2a perfect spin sink. To account for the\nclassical-to-quantum crossover, we have inserted factor (45)\nin the integrands of Eqs. (35) and (36) [with !!!n(q)] and\nEq. (41). The dotted curve shows the enhanced spin Seebeck\ncoe\u000ecient when also N 1is a perfect spin sink (which increases\nthe e\u000bective thermal bias between magnons and electrons at\nthe FjN2interface).\n(\u0015\u001cd\u001cl0), (3) bulk damping overtakes spin pumping,\nbut the magnetic \flm is still thinner than the thermal\nmagnon decay length, such that magnons probe the full\n\flm width ( l0\u001cd\u001cl), and (4) bulk regime is \fnally\nestablished when the \flm is thicker than the magnon de-\ncay length ( l\u001cd). To illustrate these crossovers, we plot\nthe spin Seebeck coe\u000ecient as a function of din Fig. 3,\nusing lengths characteristic of YIG, which are consistent\nwith the above length-scale hierarchy. Notice that even\nthough\u0018determines the magnon energy gap and the as-\nsociated Ginzburg-Landau correlation length in the clas-\nsical theory, it does not govern any prominent crossover\nin the function S(d).\nWe conclude that S(d) has nonmonotonic thickness de-\npendence, with the maximum value\nS(max)\u0018\u000b0\n2\u0019\u00153; (47)\nattained at l0.d.l, i.e., below the magnon decay\nlength but above l0, such that magnons equilibrate fully\nto the average phonon temperature \u0016T= (T1+T2)=2\n(d&l0) but still remain coherent on the scale of d\n(d.l). This agrees with the result obtained in Ref. 30.\nS(max)is proportional to the spin-mixing conductance\n[see Eq. (20)] but is independent of the bulk Gilbert\ndamping (as the magnon quasiparticle structure is still\nwell resolved). According to Eq. (21), S(max)determines\nthe largest spin current emitted thermally by a \flm of\nmagnetic insulator, as a function of d, when subjectedto a certain temperature di\u000berence (for example, in a\nwedged magnetic insulator coated by metallic contacts).\nIf, on the other hand, a well-de\fned temperature gradi-\nentis supplied (corresponding, for example, to a certain\nphonon-dominated heat-\rux density), while thickness d\nis varied, the spin-current density js/Sdincreases with\ndsaturating at d&l(the magnon decay length):\nj(max)\ns\f\f\f\n\fxed@xT\u0018l0\n2\u0019\u00152kB@xT; (48)\nwhich corresponds to the bulk regime. j(max)\ns vanishes\nwhen\u000b0!0 (no spin pumping) or \u000b!1 (no magnetic\ndynamics).\nIt is interesting to ask how the above results would\nmodify if both N 1and N 2in our model (see Fig. 1) were\nperfect spin sinks. For an inversion-symmetric struc-\nture (e.g., PtjYIGjPt), spin currents at the two inter-\nfaces must be equal, js1=js2\u0011js. Whend\u001dl, we\nshould recover the bulk limit (41), since the magnons de-\ncay before traversing the full width of the \flm (and thus\nthe spin current at one interface should not be sensitive\nto the boundary condition at the other). When d\u001cl,\nhowever,S0andSnentering Eq. (37) need to be modi-\n\fed. To that end, we notice that the factor (1+ l0=d)\u00001in\nEq. (35) re\rects the di\u000berence between Tm;0, the e\u000bective\ntemperature of the magnons, and T2, the temperature of\nthe electrons in N 2:8\nTm;0\u0000T2=\u000b\u0016T+ (\u000b0=d)T2\n\u000b+\u000b0=d=T1\u0000T2\n21\n1 +l0=d:(49)\nSimilarly for the n>0 subbands, the factor (1+2 l0=d)\u00001\nin Eq. (36) stems from the e\u000bective magnon-electron tem-\nperature di\u000berence across the F jN2interface of\nTm;n\u0000T2=T1\u0000T2\n21\n1 + 2l0=d: (50)\nWhen the bulk damping dominates over the interfa-\ncial spin pumping, i.e., l0\u001cd(while still d\u001cl),\nTm;n!(T1+T2)=2, while in the opposite limit, i.e.,\nd\u001cl0,Tm;n!T2. The magnon temperature thus\nbecomes strongly skewed toward the F jN2interface for\nthinner ferromagnetic \flms, when N 1is a poor spin sink\n(with electrons and magnons, therefore, being essentially\ndecoupled at the N 1jF interface, for our purposes). In\nthe case when both N 1and N 2are perfect spin sinks, on\nthe other hand, the e\u000bective magnon-electron tempera-\nture di\u000berence driving spin current is given simply by\n\u0016T\u0000T2= (T1\u0000T2)=2 for all subbands (when d\u001cl). We\naccount for this increased thermal gradient by dropping\nthe factor (1 + l0=d)\u00001on the right-hand side of Eq. (35)\nand likewise (1+2 l0=d)\u00001in Eq. (36). The corresponding\nenhancement of the spin Seebeck coe\u000ecient reverses the\ntrend in Eq. (46) at d.l0to give\nS(d)\u00181\n2\u0019\u001528\n<\n:\u000b0=d; d\u001c\u0015\nl0=l; \u0015\u001cd\u001cl\nl0=d; l\u001cd; (51)7\n0.01110010410610-510-410-310-210-1\u0000l0l\nd/\u0000S\u00002\nFIG. 4. Plot of the spin Seebeck coe\u000ecient using magnetic\nlength scales as in Fig. (3) but calculated for a non-inversion-\nsymmetric N 1jFjN2structure with spin-pumping parameters\n\u000b0\n1=\u000b0=10,\u000b0\n2=\u000b0, respectively, at the two interfaces (upper\ntrace) and\u000b0\n1=\u000b0,\u000b0\n2=\u000b0=10 (lower trace). Note that the\nformer case is intermediate between the two curves plotted in\nFig. 3 (where the solid curve corresponds to \u000b0\n1= 0,\u000b0\n2=\u000b0\nand the dotted curve to \u000b0\n1=\u000b0\n2=\u000b0).\nwhich is now monotonically decreasing with d, as plotted\nby the dotted line in Fig. 3.\nWhen the structure N 1jFjN2is not mirror symmetric\n(either because the spin-mixing conductances or the spin-\nsink characteristics are di\u000berent), which we characterize\nby di\u000berent \u000b0\n1and\u000b0\n2spin-pumping parameters at the\ntwo interfaces, we can repeat the above analysis for d\u001cl,\n\fnding\nTm;0\u0000T2=\u000b\u0016T+ (\u000b0\n1=d)T1+ (\u000b0\n2=d)T2\n\u000b+\u000b0\n1=d+\u000b0\n2=d\n=T1\u0000T2\n21 + 2l0\n1=d\n1 + (l0\n1+l0\n2)=d(52)\nand\nTm;n\u0000T2=T1\u0000T2\n21 + 4l0\n1=d\n1 + 2(l0\n1+l0\n2)=d; (53)\nwhere we de\fned the spin-pumping lengths l0\ni\u0011\u000b0\ni=\u000bas-\nsociated with the left, i= 1, and right, i= 2, interfaces.\nWe thus generalize the spin Seebeck contributions (at the\nFjN2interface) from di\u000berent magnon subbands to\nS0\u0019\u000b0\n2\nd1 + 2l0\n1=d\n1 + (l0\n1+l0\n2)=dZ1\n\u00001d2q\n(2\u0019)2F[\f~!0(q)] (54)\nand\nSn>0\u00192\u000b0\n2\nd1 + 4l0\n1=d\n1 + 2(l0\n1+l0\n2)=dZ1\n\u00001d2q\n(2\u0019)2F[\f~!n(q)]\n(55)\nin lieu of Eqs. (35) and (36), respectively. The asymptotic\nand crossover trends are now given by (assuming \u0015\u001cl0\n1;l0\n2\u001cl):\nS(d)\u00181\n2\u0019\u001528\n>><\n>>:2=(\u000b0\n1\u00001+\u000b0\n2\u00001)d; d\u001c\u0015\n2=(l0\n1\u00001+l0\n2\u00001)l; \u0015\u001cd\u001cl0\n1\nl0\n2=l; l0\n1;l0\n2\u001cd\u001cl\nl0\n2=d; l \u001cd;\n(56)\nwhich coincides with Eq. (51) when \u000b0\n1=\u000b0\n2but has an\nadditional shoulder-like feature at d\u0018(l0\n1+l0\n2)=2 . This\nfeature makes S(d) nonmonotonic when \u000b0\n1< \u000b0\n2. In\nFig. 4, we plot the spin Seebeck coe\u000ecient for two asym-\nmetric cases: (1) \u000b0\n1=\u000b0=10,\u000b0\n2=\u000b0and (2)\u000b0\n1=\u000b0,\n\u000b0\n2=\u000b0=10 (physically corresponding to spin currents on\ntwo sides of a non-inversion-symmetric N 1jFjN2struc-\nture, such as, PdjYIGjPt).\nVIII. DISCUSSION\nOur theory provides a minimalistic application of the\nLLG phenomenology to the problem of the spin See-\nbeck e\u000bect, yet disregards magnon-magnon interactions.\nThese become important at high temperatures, especially\napproaching the Curie temperature. Magnon-phonon in-\nteractions are included only insofar as a contribution to\nthe total Gilbert damping. Elastic magnon scattering on\nimpurities, which would manifest as an inhomogeneous\nbroadening of ferromagnetic-resonance linewidth, may\nbe an important impediment to the thermally-induced\nspin currents in disordered \flms. The bulk limit of spin\ncurrent, Eq. (48), is reduced by disorder, as well as the\nmagnon decay length describing the crossover to the bulk\nregime. When the magnon mean free path l\u0003is shorter\nthan our Gilbert damping decay length l, in particular,\nwe expect the e\u000bective decay length to be le\u000b\u0018p\nll\u0003(the\nspin di\u000busion length) and j(max)\ns in Eq. (48) to be reduced\nby a factor of l=le\u000b\u0018p\nl=l\u0003(assuming that l0\u001cle\u000b,\nsuch that our length-scale hierarchy is unchanged). The\nSeebeck coe\u000ecient behavior (46) [as well as Eq. (47)],\nhowever, remain essentially intact up to the thickness\nd\u0018le\u000b.\nFinally, we want to comment on a possibility of nonlo-\ncal magnetic relaxation. In this paper, we have assumed\nthat Gilbert damping is a local and isotropic tensor.\nThe locality would be a reasonable approximation if the\ndamping bottleneck was due to some local dynamic de-\nfects. In the case of YIG,31which is known for its highly\ncoherent elastic properties, nonlocality of the bulk mag-\nnetic relaxation could signi\fcantly modify our \fndings.\nFirst of all, this could introduce new phonon-dependent\nlength scales into the problem, which would show in the\nS(d) dependence. Standard long-wavelength ferromag-\nnetic resonance on thick \flms would reveal damping \u000b\nthat could be very di\u000berent from that of short-wavelength\nthermal magnons relevant here. An e\u000bective damping\nparameter ~ \u000bof thermal magnons (which may itself be\nthickness dependent) would result in the bulk crossover\nthickness of ~l\u0018\u0015=~\u000b6=l. Whend.~l, we may still invoke8\nthe \frst three regimes of our \fndings, Eq. (46) (with \u000b\nandlcorresponding to the thermal-magnon inverse qual-\nity factor and decay length, respectively, due to magnon-\nphonon scattering), which should, furthermore, be indif-\nferent to the fact that the local temperature, Eq. (8), is\nnot well de\fned for highly-coherent phonons. The rea-\nson for this is that, in these regimes, when dis below the\nmagnon decay length ~l, only the average phonon tem-\nperature \u0016Tis relevant for our theory. 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W. Bauer, K. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 82, 099904(E) (2010).\n31V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep.\n229, 81 (1993)."    },    {        "title": "1605.08698v1.A_reduced_model_for_precessional_switching_of_thin_film_nanomagnets_under_the_influence_of_spin_torque.pdf",        "content": "A reduced model for precessional switching of thin-\flm nanomagnets under the\nin\ruence of spin-torque\nRoss G. Lund1, Gabriel D. Chaves-O'Flynn2, Andrew D. Kent2, Cyrill B. Muratov1\n1Department of Mathematical Sciences, New Jersey Institute of Technology , University Heights, Newark, NJ 07102, USA\n2Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA\n(Dated: May 25, 2022)\nWe study the magnetization dynamics of thin-\flm magnetic elements with in-plane magnetization\nsubject to a spin-current \rowing perpendicular to the \flm plane. We derive a reduced partial\ndi\u000berential equation for the in-plane magnetization angle in a weakly damped regime. We then apply\nthis model to study the experimentally relevant problem of switching of an elliptical element when\nthe spin-polarization has a component perpendicular to the \flm plane, restricting the reduced model\nto a macrospin approximation. The macrospin ordinary di\u000berential equation is treated analytically\nas a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand\ntransitions in solution behaviors in terms of a discrete dynamical system. The predictions of our\nreduced model are compared to those of the full Landau{Lifshitz{Gilbert{Slonczewski equation for\na macrospin.\nI. INTRODUCTION\nMagnetization dynamics in the presence of spin-\ntransfer torques is a very active area of research with ap-\nplications to magnetic memory devices and oscillators1{3.\nSome basic questions relate to the types of magnetiza-\ntion dynamics that can be excited and the time scales on\nwhich the dynamics occurs. Many of the experimental\nstudies of spin-transfer torques are on thin \flm magnetic\nelements patterned into asymmetric shapes (e.g. an el-\nlipse) in which the demagnetizing \feld strongly con\fnes\nthe magnetization to the \flm plane. Analytic models\nthat capture the resulting nearly in-plane magnetization\ndynamics (see e.g.4{8) can lead to new insights and guide\nexperimental studies and device design. A macrospin\nmodel that treats the entire magnetization of the ele-\nment as a single vector of \fxed length is a starting point\nfor most analyses.\nThe focus of this paper is on a thin-\flm magnetic el-\nement excited by a spin-polarized current that has an\nout-of-plane component. This out-of-plane component\nof spin-polarization can lead to magnetization precession\nabout the \flm normal or magnetization reversal. The for-\nmer dynamics would be desired for a spin-transfer torque\noscillator, while the latter dynamics would be essential in\na magnetic memory device. A device in which a perpen-\ndicular component of spin-polarization is applied to an\nin-plane magnetized element was proposed in Ref. [9] and\nhas been studied experimentally10{12. There have also\nbeen a number of models that have considered the in\ru-\nence of thermal noise on the resulting dynamics, e.g., on\nthe rate of switching and the dephasing of the oscillator\nmotion13{15.\nHere we consider a weakly damped asymptotic regime\nof the Landau{Lifshitz{Gilbert{Slonczewski (LLGS)\nequation for a thin-\flm ferromagnet, in which the oscil-\nlatory nature of the in-plane dynamics is highlighted. In\nthis regime, we derive a reduced partial di\u000berential equa-\ntion (PDE) for the in-plane magnetization dynamics un-\nder applied spin-torque, which is a generalization of theunderdamped wave-like model due to Capella, Melcher\nand Otto8. We then analyze the solutions of this equa-\ntion under the macrospin (spatially uniform) approxima-\ntion, and discuss the predictions of such a model in the\ncontext of previous numerical studies of the full LLGS\nequation16.\nThe rest of this article is organized as follows. In Sec.\nII, we perform an asymptotic derivation of the reduced\nunderdamped equation for the in-plane magnetization\ndynamics in a thin-\flm element of arbitrary cross sec-\ntion, by \frst making a thin-\flm approximation to the\nLLGS equation, then a weak-damping approximation. In\nSec. III, we then further reduce to a macrospin ordinary\ndi\u000berential equation (ODE) by spatial averaging of the\nunderdamped PDE, and restrict to the particular case of\na soft elliptical element. A brief parametric study of the\nODE solutions is then presented, varying the spin-current\nparameters. In Sec. IV, we make an analytical study of\nthe macrospin equation using an orbit-averaging method\nto reduce to a discrete dynamical system, and compare\nits predictions to the full ODE solutions. In Sec. V, we\nseek to understand transitions between the di\u000berent so-\nlution trajectories (and thus predict current-parameter\nvalues when the system will either switch or precess) by\nstudying the discrete dynamical system derived in Sec.\nIV. Finally, we summarize our \fndings in Sec. VI.\nII. REDUCED MODEL\nWe consider a domain \n \u001aR3occupied by a ferromag-\nnetic \flm with cross-section D\u001aR2and thickness d, i.e.,\n\n =D\u0002(0;d). Under the in\ruence of a spin-polarized\nelectric current applied perpendicular to the \flm plane,\nthe magnetization vector m=m(r;t), withjmj= 1 in \nand 0 outside, satis\fes the LLGS equation (in SI units)\n@m\n@t=\u0000\r\u00160m\u0002He\u000b+\u000bm\u0002@m\n@t+\u001cSTT (1)arXiv:1605.08698v1  [cond-mat.mes-hall]  27 May 20162\nin \n, with@m=@n= (n\u0001r)m= 0 on@\n, where nis the\noutward unit normal to @\n. In the above, \u000b > 0 is the\nGilbert damping parameter, \ris the gyromagnetic ratio,\n\u00160is the permeability of free space, He\u000b=\u00001\n\u00160Ms\u000eE\n\u000emis\nthe e\u000bective magnetic \feld,\nE(m) =Z\n\n\u0010\nAjrmj2+K\b(m)\u0000\u00160MsHext\u0001m)\u0011\nd3r\n+\u00160M2\nsZ\nR3Z\nR3r\u0001m(r)r\u0001m(r0)\n8\u0019jr\u0000r0jd3rd3r0(2)\nis the micromagnetic energy with exchange constant A,\nanisotropy constant K, crystalline anisotropy function\n\b, external magnetic \feld Hext, and saturation magne-\ntizationMs. Additionally, the Slonczewski spin-transfer\ntorque\u001cSTTis given by\n\u001cSTT=\u0000\u0011\r~j\n2deMsm\u0002m\u0002p; (3)\nwherejis the density of current passing perpendicularly\nthrough the \flm, eis the elementary charge (positive),\npis the spin-polarization direction, and \u00112(0;1] is the\nspin-polarization e\u000eciency.\nWe now seek to nondimensionalize the above system.\nLet\n`=s\n2A\n\u00160M2s; Q =2K\n\u00160M2s;hext=Hext\nMs:(4)\nWe then rescale space and time as\nr!`r; t!t\n\r\u00160Ms; (5)\nobtaining the nondimensional form\n@m\n@t=\u0000m\u0002he\u000b+\u000bm\u0002@m\n@t\u0000\fm\u0002m\u0002p;(6)\nwhere he\u000b=He\u000b=Ms, and\n\f=\u0011~j\n2de\u00160M2s(7)\nis the dimensionless spin-torque strength.\nSince we are interested in thin \flms, we now assume\nthatmis independent of the \flm thickness. Then, after\nrescaling\nE!\u00160M2\nsd`2E; (8)\nwe have he\u000b'\u0000\u000eE\n\u000em, whereEis given by a local energy\nfunctional de\fned on the (rescaled) two-dimensional do-\nmainD(see, e.g., Ref. [17]):\nE(m)'1\n2Z\nD\u0000\njrmj2+Q\b(m)\u00002hext\u0001m\u0001\nd2r\n+1\n2Z\nDm2\n?d2r+1\n4\u0019\u000ejln\u0015jZ\n@D(m\u0001n)2ds;(9)in which now m:D!S2,m?is its out-of-plane com-\nponent,\u000e=d=`is the dimensionless \flm thickness, and\n\u0015=d=L\u001c1 (whereLis the lateral size of the \flm) is the\n\flm's aspect ratio. The e\u000bective \feld is given explicitly\nby\nhe\u000b= \u0001m\u0000Q\n2rm\b(m)\u0000m?ez+hext; (10)\nandmsatis\fes equation (6) in Dwith the boundary\ncondition\n@m\n@n=\u00001\n2\u0019\u000ejln\u0015j(m\u0001n)(n\u0000(m\u0001n)m) (11)\non@D.\nWe now parametrize min terms of spherical angles as\nm= (\u0000sin\u0012cos\u001e;cos\u0012cos\u001e;sin\u001e); (12)\nand the current polarization direction pin terms of an\nin-plane angle  and its out-of-plane component p?as\np=1p\n1 +p2\n?(\u0000sin ;cos ;p?): (13)\nWriting\f\u0003=\f=p\n1 +p2\n?, after some algebra, one may\nthen write equation (6) as the system\n@\u001e\n@t=\u00001\ncos\u001ehe\u000b\u0001m\u0012+\u000bcos\u001e@\u0012\n@t\n+\f\u0003(p?cos\u001e\u0000sin\u001ecos(\u0012\u0000 ));(14)\n\u0000cos\u001e@\u0012\n@t=\u0000he\u000b\u0001m\u001e+\u000b@\u001e\n@t+\f\u0003sin(\u0012\u0000 );(15)\nwhere m\u0012=@m=@\u0012andm\u001e=@m=@\u001eformgiven by\n(12). Again, since we are working in a soft thin \flm, we\nassume\u001e\u001c1 and that the out-of-plane component of\nthe e\u000bective \feld in equation (10) is dominated by the\ntermhe\u000b\u0001ez'\u0000m?=\u0000sin\u001e. Note that this assumes\nthat the crystalline anisotropy and external \feld terms\nin the out-of-plane directions are relatively small, so we\nassume the external \feld is only in plane, though it is still\npossible to include a perpendicular anisotropy simply by\nrenormalizing the constant in front of the m?term in\nhe\u000b. We then linearize the above system in \u001e, yielding\n@\u001e\n@t=\u000eE\n\u000e\u0012+\u000b@\u0012\n@t+\f\u0003(p?\u0000\u001ecos(\u0012\u0000 )); (16)\n\u0000@\u0012\n@t=\u001e+\f\u0003sin(\u0012\u0000 )\n+\u001e(\u0000hxsin\u0012+hycos\u0012) +\u000b@\u001e\n@t:(17)\nwherehx=he\u000b\u0001exandhy=he\u000b\u0001ey, andE(\u0012) isE(m)\nevaluated at \u001e= 0.3\nWe now note that the last two terms in (17) are neg-\nligible relative to \u001ewheneverjhxj;jhyjand\u000bare small,\nwhich is true of typical clean thin-\flm samples of su\u000e-\nciently large lateral extent. Neglecting these terms, one\nhas\n@\u001e\n@t=\u000eE\n\u000e\u0012+\u000b@\u0012\n@t+\f\u0003(p?\u0000\u001ecos(\u0012\u0000 )); (18)\n\u0000@\u0012\n@t=\f\u0003sin(\u0012\u0000 ) +\u001e: (19)\nThen, di\u000berentiating (19) with respect to tand using the\nresult along with (19) to eliminate \u001eand@\u001e\n@tfrom (18),\nwe \fnd a second-order in time equation for \u0012:\n0 =@2\u0012\n@t2+@\u0012\n@t(\u000b+ 2\f\u0003cos(\u0012\u0000 )) +\u000eE\n\u000e\u0012\n+\f\u0003p?+\f2\n\u0003sin(\u0012\u0000 ) cos(\u0012\u0000 );(20)\nwhere, explicitly, one has\n\u000eE\n\u000e\u0012=\u0000\u0001\u0012+Q\n2~\b0(\u0012) +hext\u0001(cos\u0012;sin\u0012); (21)\nand~\b(\u0012) = \b( m(\u0012)). In turn, from the boundary condi-\ntion on min (11), we can derive the boundary condition\nfor\u0012as\nn\u0001r\u0012=1\n2\u0019\u000ejln\u0015jsin(\u0012\u0000') cos(\u0012\u0000'); (22)\nwhere'is the angle parametrizing the normal nto@D\nvian= (\u0000sin';cos').\nThe model comprised of (20){(22) is a damped-driven\nwave-like PDE for \u0012, which coincides with the reduced\nmodel of Ref. [8] for vanishing spin-current density in\nan in\fnite sample. This constitutes our reduced PDE\nmodel for magnetization dynamics in thin-\flm elements\nunder the in\ruence of out-of-plane spin currents. It is\neasy to see that all of the terms in (20) balance when the\nparameters are chosen so as to satisfy\n\f\u0003\u0018p?\u0018\u000b\u0018Q1=2\u0018jhextj1=2\u0018`\nL\u0018\u000ejln\u0015j:(23)\nThis shows that it should be possible to rigorously obtain\nthe reduced model in (20){(22) in the asymptotic limit\nofL!1 and\u000b;\f\u0003;p?;Q;jhextj;\u000e!0 jointly, so that\n(23) holds.\nIII. MACROSPIN SWITCHING\nIn this section we study the behavior of the reduced\nmodel (20){(22) in the approximation that the magneti-\nzation is spatially uniform on an elliptical domain, and\ncompare the solution phenomenology to that found by\nsimulating the LLGS equation in the same physical situ-\nation, as studied in Ref. [16].A. Derivation of macrospin model\nIntegrating equation (20) over the domain Dand using\nthe boundary condition (22), we have\nZ\nD\u0012@2\u0012\n@t2+@\u0012\n@t(\u000b+ 2\f\u0003cos(\u0012\u0000 ))\n+\f\u0003p?+\f2\n\u0003sin(\u0012\u0000 ) cos(\u0012\u0000 )\n+Q\n2~\b0(\u0012) +hext\u0001(cos\u0012;sin\u0012)\u0013\nd2r\n=1\n2\u0019\u000ejln\u0015jZ\n@Dsin(\u0012\u0000') cos(\u0012\u0000') ds:(24)\nAssume now that \u0012does not vary appreciably across the\ndomainD, which makes sense in magnetic elements that\nare not too large. This allows us to replace \u0012(r;t) by\nits spatial average \u0016\u0012(t) =1\njDjR\nD\u0012(r;t) d2r, wherejDj\nstands for the area of Din the units of `2. Denoting\ntime derivatives by overdots, and omitting the bar on \u0016\u0012\nfor notational simplicity, this spatial averaging leads to\nthe following ODE for \u0012(t):\n\u0012+_\u0012(\u000b+ 2\f\u0003cos(\u0012\u0000 )) +\f2\n\u0003sin(\u0012\u0000 ) cos(\u0012\u0000 )\n+\f\u0003p?+Q\n2~\b0(\u0012) +hext\u0001(cos\u0012;sin\u0012)\n=\u000ejln\u0015j\n4\u0019jDjsin 2\u0012Z\n@Dcos(2') ds\n\u0000\u000ejln\u0015j\n4\u0019jDjcos 2\u0012Z\n@Dsin(2') ds:(25)\nNext, we consider a particular physical situation in\nwhich to study the macrospin equation, motivated by\nprevious work10,11. As in Refs. [14{16], we consider an\nelliptical thin-\flm element (recall that lengths are now\nmeasured in the units of `):\nD=\u001a\n(x;y) :x2\na2+y2\nb2<1\u001b\n; (26)\nwith no in-plane crystalline anisotropy, Q= 0, and no\nexternal \feld, hext= 0. We take the long axis of the\nellipse to be aligned with the ey-direction, i.e. b > a ,\nwith the in-plane component of current polarization also\naligned along this direction, i.e., taking  = 0. One can\nthen compute the integral over the boundary in equation\n(25) explicitly, leading to the equation\n\u0012+_\u0012(\u000b+\f\u0003cos\u0012) + \u0003 sin\u0012cos\u0012\n+\f2\n\u0003sin\u0012cos\u0012+\f\u0003p?= 0;(27)\nwhere we introduced the geometric parameter 0 <\u0003\u001c1\nobtained by an explicit integration:\n\u0003 =\u000ejln\u0015j\n2\u00192abZ2\u0019\n0b2cos2\u001c\u0000a2sin2\u001cp\nb2cos2\u001c+a2sin2\u001cd\u001c: (28)4\n(d)\n(c)\n(a)\n(b)\nFIG. 1: Solutions of macrospin equation (30) for \u000b= 0:01, \u0003 = 0:1. In (a),p?= 0:2,\u001b= 0:03: decaying solution; in (b),\np?= 0:2,\u001b= 0:06: limit cycle solution (the initial conditions in (a) and (b) are \u0012(0) = 3:5, to better visualize the behavior).\nIn (c),p?= 0:3,\u001b= 0:08: switching solution; in (d), p?= 0:6,\u001b= 0:1: precessing solution.\nThis may be computed in terms of elliptic integrals,\nthough the expression is cumbersome so we omit it here.\nImportantly, up to a factor depending only on the eccen-\ntricity the value of \u0003 is given by\n\u0003\u0018d\nLlnL\nd: (29)\nFor example, for an elliptical nanomagnet with dimen-\nsions 100\u000230\u00022:5 nm (similar to those considered in\nRef. [16]), this yields \u0003 '0:1.\nIt is convenient to rescale time byp\n\u0003 and divide\nthrough by \u0003, yielding\n\u0012+1p\n\u0003_\u0012(\u000b+ 2\u001b\u0003 cos\u0012) + sin\u0012cos\u0012\n+\u001bp?+\u001b2\u0003 sin\u0012cos\u0012= 0;(30)\nwhere we introduced \u001b=\f\u0003=\u0003. We then apply this\nODE to model the problem of switching of the thin-\flm\nelements, taking the initial in-plane magnetization direc-tion to be static and aligned along the easy axis, an-\ntiparallel to the in-plane component of the spin-current\npolarization. Thus, we take\n\u0012(0) =\u0019; _\u0012(0) = 0; (31)\nand study the resulting initial value problem.\nB. Solution phenomenology\nLet us brie\ry investigate the solution phenomenology\nas the dimensionless spin-current parameters \u001bandp?\nare varied, with the material parameters, \u000band \u0003, \fxed.\nWe take all parameters to be constant in time for simplic-\nity. We \fnd, by numerical integration, 4 types of solution\nto the initial value problem de\fned above. The sample\nsolution curves are displayed in Fig. 1 below. The \frst\n(panel (a)) occurs for small values of \u001b, and consists sim-\nply of oscillations of \u0012around a \fxed point close to the\nlong axis of the ellipse, which decay in amplitude towards\nthe \fxed point, without switching.5\nSecondly (panel (b)), still below the switching thresh-\nold, the same oscillations about the \fxed point can reach\na \fnite \fxed amplitude and persist without switching.\nThis behavior corresponds to the onset of relatively small\namplitude limit-cycle oscillations around the \fxed point.\nThirdly (panel (c)), increasing either \u001b;p?or both, we\nobtain switching solutions. These have initial oscillations\nin\u0012about the \fxed point near \u0019, which increase in ampli-\ntude, and eventually cross the short axis of the ellipse at\n\u0012=\u0019=2. Then\u0012oscillates about the \fxed point near 0,\nand the oscillations decay in amplitude toward the \fxed\npoint.\nFinally (panel (d)), further increasing \u001bandp?we\nobtain precessing solutions. Here, the initial oscillations\nabout the \fxed point near \u0019quickly grow to cross \u0019=2,\nafter which \u0012continues to decrease for all t, the magne-\ntization making full precessions around the out-of-plane\naxis.\nIV. HALF-PERIOD ORBIT-AVERAGING\nAPPROACH\nWe now seek to gain some analytical insight into the\ntransitions between the solution types discussed above.\nWe do this by averaging over half-periods of the oscil-\nlations observed in the solutions to generate a discrete\ndynamical system which describes the evolution of the\nenergy of a solution \u0012(t) on half-period time intervals.\nFirstly, we observe that in the relevant parameter\nregimes the reduced equation (30) can be seen as a weakly\nperturbed Hamiltonian system. We consider both \u000band\n\u0003 small, with \u000b.p\n\u0003, and assume \u001b\u0018\u000b=\u0003 and\n\u001bp?.1. The arguments below can be rigorously jus-\nti\fed by considering, for example, the limit \u0003 !0 while\nassuming that \u000b=O(\u0003) and that the values of \u001band\np?are \fxed. This limit may be achieved in the origi-\nnal model by sending jointly d!0 andL!1 , while\nkeeping17\nLd\n`2lnL\nd.1: (32)\nThe last condition ensures the consistency of the assump-\ntion that\u0012does not vary appreciably throughout D.\nIntroducing !(t) =_\u0012(t), (30) can be written to leading\norder as\n_\u0012=@H\n@!;_!=\u0000@H\n@\u0012; (33)\nwhere we introduced\nH=1\n2!2+V(\u0012); V (\u0012) =1\n2sin2\u0012+\u001bp?\u0012: (34)\nAt the next order, the e\u000bects of \fnite \u000band \u0003 appear\nin the \frst-derivative term in (30), while the other forc-\ning term is still higher order. The behavior of (30) is\ntherefore that of a weakly damped Hamiltonian system\nwith Hamiltonian H, with the e\u000bects of \u000band\u001bservingto slowly change the value of Has the system evolves.\nThus, we now employ the technique of orbit-averaging to\nreduce the problem further to the discrete dynamics of\nH(t), where the discrete time-steps are equal (to the lead-\ning order) to half-periods of the underlying Hamiltonian\ndynamics (which thus vary with H).\nLet us \frst compute the continuous-in-time dynamics\nofH. From (34),\n_H=!( _!+V0(\u0012)); (35)\nwhich vanishes to leading order. At the next order, from\n(30), one has\n_H=\u0000!2\np\n\u0003(\u000b+ 2\u001b\u0003 cos\u0012): (36)\nWe now seek to average this dynamics over the Hamil-\ntonian orbits. The general nature of the Hamiltonian\norbits is either oscillations around a local minimum of\nV(\u0012) (limit cycles) or persistent precessions. If the lo-\ncal minimum of Vis close to an even multiple of \u0019,H\ncannot increase, while if it is close to an odd multiple\nthenHcan increase if \u001bis large enough. The switching\nprocess involves moving from the oscillatory orbits close\nto one of these odd minima, up the energy landscape,\nthen jumping to oscillatory orbits around the neighbor-\ning even minimum, and decreasing in energy towards the\nnew local \fxed point.\nWe focus \frst on the oscillatory orbits. We may de\fne\ntheir half-periods as\nT(H) =Z\u0012\u0003\n+\n\u0012\u0003\n\u0000d\u0012\n_\u0012; (37)\nwhere\u0012\u0003\n\u0000and\u0012\u0003\n+are the roots of the equation V(\u0012) =\nHto the left and right of the local minimum of V(\u0012)\nabout which \u0012(t) oscillates. To compute this integral, we\nassume that \u0012(t) follows the Hamiltonian trajectory:\n_\u0012=\u0006p\n2(H\u0000V(\u0012)): (38)\nWe then de\fne the half-period average of a function\nf(\u0012(t)) as\nhfi=1\nT(H)Z\u0012\u0003\n+\n\u0012\u0003\n\u0000f(\u0012) d\u0012p\n2(H\u0000V(\u0012)); (39)\nwhich agrees with the time average over half-period to\nthe leading order. Note that this formula applies irre-\nspectively of whether the trajectory connects \u0012\u0003\n\u0000to\u0012\u0003\n+\nor\u0012\u0003\n+to\u0012\u0003\n\u0000. Applying this averaging to _H, we then have\nD\n_HE\n=\u00001\nT(H)Z\u0012\u0003\n+\n\u0012\u0003\n\u0000\u001f(\u0012;H) d\u0012; (40)\nwhere we de\fned\n\u001f(\u0012;H) =(\u000b+ 2\u001b\u0003 cos\u0012)p\n2(H\u0000V(\u0012))p\n\u0003: (41)6\nIf the value ofHis such that either of the roots \u0012\u0003\n\u0006no\nlonger exist, this indicates that the system is now on a\nprecessional trajectory. In order to account for this, we\ncan de\fne the period on a precessional trajectory instead\nas\nT(H) =Z\u0012C\n\u0012C\u0000\u0019d\u0012\n_\u0012; (42)\nwhere\u0012Cis a local maximum of V(\u0012). On the preces-\nsional trajectories, we then have\nD\n_HE\n=\u00001\nT(H)Z\u0012C\n\u0012C\u0000\u0019\u001f(\u0012;H) d\u0012: (43)\nIn order to approximate the ODE solutions, we now\ndecompose the dynamics of Hinto half-period time in-\ntervals. We thus take, at the n'th timestep,Hn=H(tn),\ntn+1=tn+T(Hn) and\nHn+1=Hn\u0000Z\u0012\u0003\n+(Hn)\n\u0012\u0003\n\u0000(Hn)\u001f(\u0012;Hn) d\u0012; (44)\nifHncorresponds to a limit cycle trajectory. The same\ndiscrete map applies to precessional trajectories, but with\nthe integration limits replaced with \u0012C\u0000\u0019and\u0012C, re-\nspectively.\nA. Modelling switching with discrete map\nIn order to model switching starting from inside a well\nofV(\u0012), we can iterate the discrete map above, starting\nfrom an initial energy H0. We chooseH0by choosing a\nstatic initial condition \u0012(0) =\u00120close to an odd multiple\nof\u0019(let us assume without loss of generality that we are\nclose to\u0019), and computing H0=V(\u00120).\nOn the oscillatory trajectories, the discrete map then\npredicts the maximum amplitudes of oscillation ( \u0012\u0003\n\u0006(Hn))\nat each timestep, by locally solving Hn=V(\u0012) for each\nn. After some number of iterations, the trajectory will\nescape the local potential well, and one or both roots of\nHn=V(\u0012) will not exist. Due to the positive average\nslope ofV(\u0012) the most likely direction for a trajectory to\nescape the potential well is _\u0012<0 (`downhill'). Assuming\nthis to be the case, at some timestep tN, it will occur that\nthe equationHN=V(\u0012) has only one root \u0012=\u0012\u0003\n+>\u0019,\nimplying that the trajectory has escaped the potential\nwell, and will proceed on a precessional trajectory in a\nnegative direction past \u0012=\u0019=2 towards\u0012= 0.\nTo distinguish whether a trajectory results in switching\nor precession, we then perform a single half-period step\non the precessional orbit from \u0012Cto\u0012C\u0000\u0019, and check\nwhetherH< V (\u0012C\u0000\u0019): if this is the case, the tra-\njectory moves back to the oscillatory orbits around the\nwell close to \u0012= 0, and decreases in energy towards the\n\fxed point near \u0012= 0, representing switching. If how-\neverH> V(\u0012C\u0000\u0019) after the precessional half-period,\nthe solution will continue to precess.In Fig. 2 below, we display the result of such an iter-\nated application of the discrete map, for the same param-\neters as the switching solution given in Fig. 1(c). In Fig.\n2(a), the continuous curve represents the solution to (30),\nand the points are the predicted peaks of the oscillations,\nfrom the discrete map (44). Fig. 2(b) shows the energy\nof the same solution as a function of \u0012. Again the blue\ncurve givesH(t) for the ODE solution, the green points\nare the prediction of the iterated discrete map, and the\nred curve is V(\u0012). The discrete map predicts the switch-\ning behavior quite well, only su\u000bering some error near\nthe switching event, when the change of His signi\fcant\non a single period.\nB. Modelling precession\nHere we apply the discrete map to a precessional\nsolution|one in which the trajectory, once it escapes\nthe potential well near \u0019, does not get trapped in the\nnext well, and continues to rotate. Fig. 3(a) below dis-\nplays such a solution \u0012(t) and its discrete approximation,\nand Fig. 3(b) displays the energy of the same solution.\nAgain, the prediction of the discrete map is excellent.\nV. TRANSITIONS IN TRAJECTORIES\nIn this section we seek to understand the transi-\ntions between the trapping, switching, and precessional\nregimes as the current parameters \u001bandp?are varied.\nA. Escape Transition\nFirstly, let us consider the transition from states which\nare trapped in a single potential well, such as those in\nFigs. 1(a,b), to states which can escape and either switch\nor precess. E\u000bectively, the absolute threshold for this\ntransition is for the value of Hto be able to increase for\nsome value \u0012close to the minimum of V(\u0012) near\u0019. Thus,\nwe consider the equation of motion (36) for H, and wish\nto \fnd parameter values such that _H>0 for some\u0012near\n\u0019. This requires that\n!2\np\n\u0003(\u000b+ 2\u001b\u0003 cos\u0012)<0: (45)\nAssuming that !6= 0, we can see that the optimal value\nof\u0012to hope to satisfy this condition is \u0012=\u0019, yield-\ning a theoretical minimum \u001b=\u001bsfor the dimensionless\ncurrent density for motion to be possible, with\n\u001bs=\u000b\n2\u0003: (46)\nThis is similar to the critical switching currents derived\nin previous work14. We then require \u001b>\u001bsfor the possi-\nbility of switching or precession. Note that this estimate\nis independent of the value of p?.7\n(b) (a)\nFIG. 2: Switching solution (blue line) and its discrete approximation (green circles). Parameters: \u000b= 0:01, \u0003 = 0:1,p?= 0:3,\n\u001b= 0:08. Panel (a) shows the solution \u0012(t), and panel (b) shows the trajectory for this solution in the H\u0000\u0012plane. The red\nline in (b) shows V(\u0012).\nB. Switching{Precessing Transition\nWe now consider the transition from switching to pre-\ncessional states. This is rather sensitive and there is not\nin general a sharp transition from switching to precession.\nIt is due to the fact that for certain parameters, the path\nthat the trajectory takes once it escapes the potential\nwell depends on how much energy it has as it does so. In\nfact, for a \fxed \u000b;\u0003, and values of \u001b > \u001bswe can sep-\narate the (\u001b;p?)-parameter space into three regions: (i)\nafter escaping the initial well, the trajectory always falls\ninto the next well, and thus switches; (ii) after escaping,\nthe trajectory may either switch or precess depending on\nits energy as it does so (and thus depending on its initial\ncondition); (iii) after escaping, the trajectory completely\npasses the next well, and thus begins to precess.\nWe can determine in which region of the parameter\nspace a given point ( \u001b;p?) lies by studying the discrete\nmap (44) close to the peaks of V(\u0012). Assume that the\ntrajectory begins at \u0012(0) =\u0019, and is thus initially in\nthe potential well spanning the interval \u0019=2\u0014\u0012\u00143\u0019=2.\nDenote by\u0012Cthe point close to \u0012=\u0019=2 at which V(\u0012)\nhas a local maximum. It is simple to compute\n\u0012C=\u0019\n2+1\n2sin\u00001(2\u001bp?): (47)\nMoreover, it is easy to see that all other local maxima of\nV(\u0012) are given by \u0012=\u0012C+k\u0019, fork2Z.\nWe now consider trajectories which escape the initial\nwell by crossing \u0012C. These trajectories have, for some\nvalue of the timestep nwhile still con\fned in the initial\nwell, an energy value Hnin the range\nHtrap<Hn<V(\u0012C+\u0019); (48)\nwhere we de\fneHtrapto be the value of Hnsuch that\nthe discrete map (44) gives Hn+1=V(\u0012C). We thushaveHn+1> V (\u0012C). In order to check whether the\ntrajectory switches or precesses, we then compute Hn+2\nand compare it to V(\u0012C\u0000\u0019). We may then classify the\ntrajectories as switching if Hn+2\u0000V(\u0012C\u0000\u0019)<0, and\nprecessional ifHn+2\u0000V(\u0012C\u0000\u0019)>0.\nFigure 4 displays a plot of Hn\u0000V(\u0012C+\u0019) against\nHn+2\u0000V(\u0012C\u0000\u0019). The blue line shows the result of\napplying the discrete map, while the red line is the iden-\ntity line. Values of Hn\u0000V(\u0012C+\u0019) which are inside the\nrange speci\fed in (48) are thus on the negative x-axis\nhere. We can classify switching trajectories as those for\nwhich the blue line lies below the x-axis, and precessing\ntrajectories as those which lie above. In Fig. 4, the pa-\nrameters are such that both of these trajectory types are\npossible, depending on the initial value of Hn, and thus\nthis set of parameters are in region (ii) of the parameter\nspace. We note that, since the curve of blue points and\nthe identity line intersect for some large enough value of\nH, this \fgure implies that if the trajectory has enough\nenergy to begin precessing, then after several precessions\nthe trajectory will converge to one which conserves en-\nergy on average over a precessional period (indicated by\nthe arrows). In region (i) of the parameter space, the\nportion of the blue line for Hn\u0000V(\u0012C+\u0019)<0 would\nhaveHn+2\u0000V(\u0012C\u0000\u0019)<0, while in region (iii), they\nwould all haveHn+2\u0000V(\u0012C\u0000\u0019)>0.\nWe can classify the parameter regimes for which\nswitching in the opposite direction (i.e. \u0012switches from\n\u0019to 2\u0019) is possible in a similar way. It is not possible\nto have a precessional trajectory moving in this direction\n(_\u0012>0), though.\nWe may then predict, for a given point ( \u001b;p?) in pa-\nrameter space, by computing relations similar to that in\nFig. 4, which region that point is in, and thus generate\na theoretical phase diagram.\nIn Fig. 5 below, we display the phase diagram in the8\n(a) (b)\nFIG. 3: Precessing solution (blue line) and its discrete approximation (green circles). Parameters: \u000b= 0:01, \u0003 = 0:1,p?= 0:6,\n\u001b= 0:1. Panel (a) shows the solution \u0012(t), and panel (b) shows the trajectory for this solution in the H\u0000\u0012plane. The red\nline in (b) shows V(\u0012).\n(\u001b;p?)-parameter space, showing the end results of solv-\ning the ODE (30) as a background color, together with\npredictions of the bounding curves of the three regions\nof the space, made using the procedure described above.\nThe predictions of the discrete map, while not perfect,\nare quite good, and provide useful estimates on the dif-\nferent regions of parameter space. In particular, we note\nthat the region where downhill switching reliably occurs\n(the portion of region (i) above the dashed black line) is\nestimated quite well. We would also note that we would\nexpect the predictions of the discrete map to improve if\nthe values of \u0003 and \u000bwere decreased.\nVI. DISCUSSION\nWe have derived an underdamped PDE model for mag-\nnetization dynamics in thin \flms subject to perpendic-\nular applied spin-polarized currents, valid in the asymp-\ntotic regime of small \u000band \u0003, corresponding to weak\ndamping and strong penalty for out-of-plane magnetiza-\ntions. We have examined the predictions of this model\napplied to the case of an elliptical \flm under a macrospin\napproximation by using an orbit-averaging approach. We\nfound that they qualitatively agree quite well with pre-\nvious simulations using full LLGS dynamics16.\nThe bene\fts of our reduced model are that they should\nfaithfully reproduce the oscillatory nature of the in-\nplane magnetization dynamics, reducing computational\nexpense compared to full micromagnetic simulations. In\nparticular, in su\u000eciently small and thin magnetic ele-\nments the problem further reduces to a single second-\norder scalar equation.\nThe orbit-averaging approach taken here enables the\ninvestigation of the transition from switching to preces-\nsion via a simple discrete dynamical system. The regionsin parameter space where either switching or precession\nare predicted, as well as an intermediate region where\nthe end result depends sensitively on initial conditions.\nIt may be possible to further probe this region by includ-\ning either spatial variations in the magnetization (which,\nin an earlier study16were observed to simply `slow down'\nthe dynamics and increase the size of the switching re-\ngion), or by including thermal noise, which could result\nin instead a phase diagram predicting switching proba-\nbilities at a given temperature, or both.\n−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−0.05−0.04−0.03−0.02−0.0100.010.020.030.040.05\nHn−V(θC+π)Hn+2−V(θC−π)Switch Precess\nFIG. 4: Precession vs switching prediction from the discrete\nmap. Parameters: \u000b= 0:01, \u0003 = 0:1,p?= 0:35,\u001b= 0:08.\nValues ofHn\u0000V(\u0012C+\u0019) to the left of the dashed line switch\nafter the next period, the trajectory becoming trapped in the\nwell around \u0012= 0. Values to the right begin to precess, and\nconverge to a precessional \fxed point of the discrete map.9\nσp⊥\n0 0.05 0.1 0.15 0.2 0.25 0.300.10.20.30.40.50.60.70.80.91\n(i)(ii)(iii)\nFIG. 5: Macrospin solution phase diagram: \u000b= 0:01;\u0003 = 0:1.\nThe background color indicates the result of solving the ODE\n(30) with initial condition (31): the dark region to the left of\nthe \fgure indicates solutions which do not escape their initial\npotential well, and the vertical dashed white line shows the\ncomputed value of the minimum current required to escape,\n\u001bs=\u000b=(2\u0003). The black band represents solutions which\ndecay, like in Fig. 1(a), while the dark grey band represents\nsolutions like in Fig. 1(b). In the rest of the \fgure, the\ngreen points indicate switching in the negative direction like\nin Fig. 1(c), grey indicate switching in the positive direction,\nand white indicates precession like in Fig. 1(d). The solid\nblack curves are the predictions of boundaries of the regions\n(as indicated in the \fgure) by using the discrete map, and\nthe dashed line is the prediction of the boundary below which\nswitching in the positive direction is possible.ACKNOWLEDGMENTS\nResearch at NJIT was supported in part by NSF via\nGrant No. DMS-1313687. Research at NYU was sup-\nported in part by NSF via Grant No. DMR-1309202.\n1S. D. Bader and S. S. P. Parkin, Annu. Rev. Condens.\nMatter Phys. 1, 71 (2010).\n2A. Brataas and A. D. Kent and H. Ohno, Nature Mat. 11,\n372 (2012).\n3A. D. Kent and D. C. Worledge, Nature Nanotechnol. 10,\n187 (2015).\n4C. J. Garc\u0013 \u0010a-Cervera and W. E, J. Appl. Phys. 90, 370\n(2001).\n5A. DeSimone, R. V. Kohn, S. M uller and F. Otto Comm.\nPure Appl. Math. 55, 1408 (2002).\n6R. V. Kohn and V. V. Slastikov, Proc. R. Soc. Lond. Ser.\nA461, 143 (2005).\n7C. B. Muratov and V. V. Osipov, J. Comput. Phys. 216,\n637 (2006).\n8A. Capella, C. Melcher and F. Otto, Nonlinearity 20, 2519\n(2007).\n9A. D. Kent, B. Ozyilmaz and E. del Barco Appl. Phys.\nLett.84, 3897 (2004).10H. Liu, D. Bedau, D. Backes, J. A. Katine, J. Langer and\nA. D. Kent Appl. Phys. Lett. 97, 242510 (2010).\n11H. Liu, D. Bedau, D. Backes, J. A. Katine, and A. D. Kent,\nAppl. Phys. Lett. 101, 032403 (2012).\n12L. Ye, G. Wolf, D. Pinna, G. D. Chaves-O'Flynn and A.\nD. Kent, J. Appl. Phys. 117, 193902 (2015).\n13K. Newhall and E. Vanden-Eijnden, J. Appl. Phys. 113,\n184105 (2013).\n14D. Pinna, A. D. Kent and D. L. Stein Phys. Rev. B 88,\n104405 (2013).\n15D. Pinna, D. L. Stein and A. D. Kent Phys. Rev. B 90,\n174405 (2014).\n16G. D. Chaves-O'Flynn, G. Wolf, D. Pinna and A. D. Kent,\nJ. Appl. Phys. 117, 17D705 (2015).\n17R. V. Kohn and V. V. Slastikov, Arch. Rat. Mech. Anal.\n178, 227 (2005)."    },    {        "title": "0804.0820v2.Inhomogeneous_Gilbert_damping_from_impurities_and_electron_electron_interactions.pdf",        "content": "arXiv:0804.0820v2  [cond-mat.mes-hall]  9 Aug 2008Inhomogeneous Gilbert damping from impurities and electro n-electron interactions\nE. M. Hankiewicz,1,2,∗G. Vignale,2and Y. Tserkovnyak3\n1Department of Physics, Fordham University, Bronx, New York 10458, USA\n2Department of Physics and Astronomy, University of Missour i, Columbia, Missouri 65211, USA\n3Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n(Dated: October 30, 2018)\nWe present a unified theory of magnetic damping in itinerant e lectron ferromagnets at order q2\nincluding electron-electron interactions and disorder sc attering. We show that the Gilbert damping\ncoefficient can be expressed in terms of the spin conductivity , leading to a Matthiessen-type formula\nin which disorder and interaction contributions are additi ve. Inaweak ferromagnet regime, electron-\nelectron interactions lead to a strong enhancement of the Gi lbert damping.\nPACS numbers: 76.50.+g,75.45.+j,75.30.Ds\nIntroduction – In spite of much effort, a complete\ntheoretical description of the damping of ferromagnetic\nspin waves in itinerant electron ferromagnets is not yet\navailable.1Recent measurements of the dispersion and\ndamping of spin-wave excitations driven by a direct spin-\npolarized current prove that the theoretical picture is in-\ncomplete, particularly when it comes to calculating the\nlinewidth of these excitations.2One of the most impor-\ntant parameters of the theory is the so-called Gilbert\ndamping parameter α,3which controls the damping rate\nand thermal noise and is often assumed to be indepen-\ndent of the wave vector of the excitations. This assump-\ntion is justified for excitations of very long wavelength\n(e.g., a homogeneous precession of the magnetization),\nwhereαcanoriginateinarelativelyweakspin-orbit(SO)\ninteraction4. But it becomes dubious as the wave vector\nqof the excitations grows. Indeed, both electron-electron\n(e-e) and electron-impurity interactions can cause an in-\nhomogeneous magnetization to decay into spin-flipped\nelectron-hole pairs, giving rise to a q2contribution to the\nGilbert damping. In practice, the presence of this contri-\nbution means that the Landau-Lifshitz-Gilbert equation\ncontains a term proportional to −m×∇2∂tm(wherem\nis the magnetization) and requires neither spin-orbit nor\nmagnetic disorder scattering. By contrast, the homoge-\nneous damping term is of the form m×∂tmand vanishes\nin the absence of SO or magnetic disorder scattering.\nThe influence of disorder on the linewidth of spin\nwaves in itinerant electron ferromagnets was discussed in\nRefs. 5,6,7, and the role of e-e interactions in spin-wave\ndamping was studied in Refs. 8,9 for spin-polarized liq-\nuid He3and in Refs. 10,11fortwo-and three-dimensional\nelectron liquids, respectively. In this paper, we present\na unified semiphenomenological approach, which enables\nus to calculate on equal footing the contributions of dis-\norder and e-e interactions to the Gilbert damping pa-\nrameter to order q2. The main idea is to apply to the\ntransverse spin fluctuations of a ferromagnet the method\nfirst introduced by Mermin12for treating the effect of\ndisorder on the dynamics of charge density fluctuations\nin metals.13Following this approach, we will show that\ntheq2contribution to the damping in itinerant electron\nferromagnets can be expressed in terms of the transversespin conductivity, which in turn separates into a sum of\ndisorder and e-e terms.\nA major technical advantage of this approach is that\nthe ladder vertex corrections to the transverse spin-\nconductivity vanish in the absence of SO interactions,\nmaking the diagrammatic calculation of this quantity a\nstraightforwardtask. Thusweareabletoprovideexplicit\nanalytic expressions for the disorder and interaction con-\ntribution to the q2Gilbert damping to the lowest order\nin the strength of the interactions. Our paper connects\nand unifies different approaches and gives a rather com-\nplete and simple theory of q2damping. In particular, we\nfind that for weak metallic ferromagnets the q2damping\ncan be strongly enhanced by e-e interactions, resulting in\na value comparable to or larger than typical in the case\nof homogeneous damping. Therefore, we believe that the\ninclusionofadampingtermproportionalto q2inthephe-\nnomenologicalLandau-Lifshitzequationofmotionforthe\nmagnetization14is a potentially important modification\nof the theory in strongly inhomogeneous situations, such\nas current-driven nanomagnets2and the ferromagnetic\ndomain-wall motion15.17\nPhenomenological approach – In Ref. 12, Mermin con-\nstructed the density-density response function of an elec-\ntron gas in the presence of impurities through the use\nof a local drift-diffusion equation, whereby the gradient\nof the external potential is cancelled, in equilibrium, by\nan opposite gradient of the local chemical potential. In\ndiagrammatic language, the effect of the local chemical\npotential corresponds to the inclusion of the vertex cor-\nrection in the calculation of the density-density response\nfunction. Here, we use a similar approach to obtain the\ntransverse spin susceptibility of an itinerant electron fer-\nromagnet, modeled as an electron gas whose equilibrium\nmagnetization is along the zaxis.\nBefore proceeding we need to clarify a delicate point.\nThe homogeneous electron gas is not spontaneously fer-\nromagnetic at the densities that are relevant for ordinary\nmagneticsystems.13Inordertoproducethe desired equi-\nlibrium magnetization, we must therefore impose a static\nfictitious field B0. Physically, B0is the “exchange” field\nBexplus any external/applied magnetic field Bapp\n0which\nmaybeadditionallypresent. Therefore,inordertocalcu-2\nlate the transverse spin susceptibility we must take into\naccount the fact that the exchange field associated with\na uniform magnetization is parallel to the magnetization\nand changes direction when the latter does. As a result,\nthe actual susceptibility χab(q,ω) differs from the sus-\nceptibility calculated at constant B0, which we denote\nby ˜χab(q,ω), according to the well-known relation:11\nχ−1\nab(q,ω) = ˜χ−1\nab(q,ω)−ωex\nM0δab. (1)\nHere,M0is the equilibrium magnetization (assumed to\npoint along the zaxis) and ωex=γBex(whereγis the\ngyromagnetic ratio) is the precession frequency associ-\nated with the exchange field. δabis the Kronecker delta.\nThe indices aandbdenote directions ( xory) perpen-\ndicular to the equilibrium magnetization and qandω\nare the wave vector and the frequency of the external\nperturbation. Here we focus solely on the calculation of\nthe response function ˜ χbecause term ωexδab/M0does\nnot contribute to Gilbert damping. We do not include\nthe effects of exchange and external fields on the orbital\nmotion of the electrons.\nThe generalized continuity equation for the Fourier\ncomponent of the transverse spin density Main the di-\nrectiona(xory) at wave vector qand frequency ωis\n−iωMa(q,ω) =−iγq·ja(q,ω)−ω0ǫabMb(q,ω)\n+γM0ǫabBapp\nb(q,ω), (2)\nwhereBapp\na(q,ω)isthetransverseexternalmagneticfield\ndriving the magnetization and ω0is the precessional fre-\nquency associated with a static magnetic field B0(in-\ncluding exchange contribution) in the zdirection. jais\ntheath component of the transverse spin-current density\ntensor and we put /planckover2pi1= 1 throughout. The transverse\nLevi-Civita tensor ǫabhas components ǫxx=ǫyy= 0,\nǫxy=−ǫyx= 1, and the summation over repeated in-\ndices is always implied.\nThe transverse spin current is proportional to the gra-\ndient of the effective magnetic field, which plays the role\nanalogousto the electrochemicalpotential, and the equa-\ntion that expressesthis proportionalityis the analogueof\nthe drift-diffusion equation of the ordinary charge trans-\nport theory:\nja(q,ω) =iqσ⊥/bracketleftbigg\nγBapp\na(q,ω)−Ma(q,ω)\n˜χ⊥/bracketrightbigg\n,(3)\nwhereσ⊥(=σxxorσyy) is the transverse dc (i.e., ω= 0)\nspin-conductivity and ˜ χ⊥=M0/ω0is the static trans-\nverse spin susceptibility in the q→0 limit.18Just as in\nthe ordinary drift-diffusion theory, the first term on the\nright-hand side of Eq. (3) is a “drift current,” and the\nsecond is a “diffusion current,” with the two canceling\nout exactly in the static limit (for q→0), due to the\nrelationMa(0,0) =γ˜χ⊥Bapp\na(0,0). Combining Eqs. (2)\nand (3) gives the following equation for the transversemagnetization dynamics:\n/parenleftbigg\n−iωδab+γσ⊥q2\n˜χ⊥δab+ω0ǫab/parenrightbigg\nMb=\n/parenleftbig\nM0ǫab+γσ⊥q2δab/parenrightbig\nγBapp\nb,(4)\nwhich is most easily solved by transforming to the\ncircularly-polarized components M±=Mx±iMy, in\nwhich the Levi-Civita tensor becomes diagonal, with\neigenvalues ±i. Solving in the “+” channel, we get\nM+=γ˜χ+−Bapp\n+=M0−iγσ⊥q2\nω0−ω−iγσ⊥q2ω0/M0γBapp\n+,\n(5)\nfrom which we obtain to the leading order in ωandq2\n˜χ+−(q,ω)≃M0\nω0/parenleftbigg\n1+ω\nω0/parenrightbigg\n+iωγσ⊥q2\nω2\n0.(6)\nThe higher-orderterms in this expansion cannot be legit-\nimately retained within the accuracy of the present ap-\nproximation. We also disregard the q2correction to the\nstatic susceptibility, since in making the Mermin ansatz\n(3) we are omitting the equilibrium spin currents respon-\nsible for the latter. Eq. (6), however, is perfectly ade-\nquate for our purpose, since it allows us to identify the\nq2contribution to the Gilbert damping:\nα=ω2\n0\nM0lim\nω→0ℑm˜χ+−(q,ω)\nω=γσ⊥q2\nM0.(7)\nTherefore, the Gilbert damping can be calculated from\nthe dc transverse spin conductivity σ⊥, which in turn\ncan be computed from the zero-frequency limit of the\ntransverse spin-current—spin-current response function:\nσ⊥=−1\nm2∗Vlim\nω→0ℑm/angb∇acketleft/angb∇acketleft/summationtextN\ni=1ˆSiaˆpia;/summationtextN\ni=1ˆSiaˆpia/angb∇acket∇ight/angb∇acket∇ightω\nω,(8)\nwhereˆSiaisthexorycomponentofspinoperatorforthe\nith electron, ˆ piais the corresponding component of the\nmomentum operator, m∗is the effective electron mass, V\nisthe systemvolume, Nisthe totalelectronnumber, and\n/angb∇acketleft/angb∇acketleftˆA;ˆB/angb∇acket∇ight/angb∇acket∇ightωrepresents the retarded linear response func-\ntion for the expectation value of an observable ˆAunder\nthe action of a field that couples linearly to an observable\nˆB. Both disorder and e-e interaction contributions can\nbe systematically included in the calculation of the spin-\ncurrent—spin-current response function. In the absence\nof spin-orbit and e-e interactions, the ladder vertex cor-\nrections to the conductivity are absent and calculation\nofσ⊥reduces to the calculation of a single bubble with\nGreen’s functions\nG↑,↓(p,ω) =1\nω−εp+εF±ω0/2+i/2τ↑,↓,(9)\nwhere the scattering time τsin general depends on the\nspin band index s=↑,↓. In the Born approximation,3\nthe scattering rate is proportional to the electron den-\nsity of states, and we can write τ↑,↓=τν/ν↑,↓, whereνs\nis the spin- sdensity of states and ν= (ν↑+ν↓)/2.τ\nparametrizes the strength of the disorder scattering. A\nstandard calculation then leads to the following result:\nσdis\n⊥=υ2\nF↑+υ2\nF↓\n6(ν−1\n↓+ν−1\n↑)1\nω2\n0τ. (10)\nThis, inserted in Eq. (7), gives a Gilbert damping pa-\nrameter in full agreement with what we have also calcu-\nlated from a direct diagrammatic evaluation of the trans-\nverse spin susceptibility, i.e., spin-density—spin-density\ncorrelation function. From now on, we shall simplify the\nnotation by introducing a transversespin relaxation time\n1\nτdis\n⊥=4(EF↑+EF↓)\n3n(ν−1\n↓+ν−1\n↑)1\nτ, (11)\nwhereEFs=m∗υ2\nFs/2istheFermienergyforspin- selec-\ntrons and nis the total electron density. In this notation,\nthe dc transverse spin-conductivity takes the form\nσdis\n⊥=n\n4m∗ω2\n01\nτdis\n⊥. (12)\nElectron-electron interactions – One of the attractive fea-\ntures of the approach based on Eq. (8) is the ease with\nwhich e-e interactions can be included. In the weak cou-\npling limit, the contributions of disorder and e-e inter-\nactions to the transverse spin conductivity are simply\nadditive. We can see this by using twice the equation of\nmotion for the spin-current—spin-current response func-\ntion. This leads to an expression for the transverse\nspin-conductivity (8) in terms of the low-frequency spin-\nforce—spin-force response function:\nσ⊥=−1\nm2∗ω2\n0Vlim\nω→0ℑm/angb∇acketleft/angb∇acketleft/summationtext\niˆSiaˆFia;/summationtext\niˆSiaˆFia/angb∇acket∇ight/angb∇acket∇ightω\nω.(13)\nHere,ˆFia=˙ˆpiais the time derivative of the momentum\noperator, i.e., the operator of the force on the ith elec-\ntron. The total force is the sum of electron-impurity and\ne-e interaction forces. Each of them, separately, gives a\ncontribution of order |vei|2and|vee|2, whereveiandvee\nare matrix elements of the electron-impurity and e-e in-\nteractions, respectively, while cross terms are of higher\norder, e.g., vee|vei|2. Thus, the two interactions give ad-\nditive contributions to the conductivity. In Ref.16, a phe-\nnomenological equation of motion was used to find the\nspin current in a system with disorder and longitudinal\nspin-Coulomb drag coefficient. We can use a similar ap-\nproach to obtain transversespin currents with transverse\nspin-Coulomb drag coefficient 1 /τee\n⊥. In the circularly-\npolarized basis,\ni(ω∓ω0)j±=−nE\n4m∗+j±\nτdis\n⊥+j±\nτee\n⊥,(14)and correspondingly the spin-conductivities are\nσ±=n\n4m∗1\n−(ω∓ω0)i+1/τdis\n⊥+1/τee\n⊥.(15)\nIn the dc limit, this gives\nσ⊥(0) =σ++σ−\n2=n\n4m∗1/τdis\n⊥+1/τee\n⊥\nω2\n0+/parenleftbig\n1/τdis\n⊥+1/τee\n⊥/parenrightbig2.(16)\nUsing Eq. (16), an identification of the e-e contribution is\npossible in a perturbative regime where 1 /τee\n⊥,1/τdis\n⊥≪\nω0, leading to the following formula:\nσ⊥=n\n4m∗ω2\n0/parenleftbigg1\nτdis\n⊥+1\nτee\n⊥/parenrightbigg\n. (17)\nComparison with Eq. (13) enables us to immediately\nidentify the microscopic expressions for the two scatter-\ning rates. For the disorder contribution, we recover what\nwe already knew, i.e., Eq. (11). For the e-e interaction\ncontribution, we obtain\n1\nτee\n⊥=−4\nnm∗Vlim\nω→0ℑm/angb∇acketleft/angb∇acketleft/summationtext\niˆSiaˆFC\nia;/summationtext\niˆSiaˆFC\nia/angb∇acket∇ight/angb∇acket∇ightω\nω,(18)\nwhereFCis just the Coulomb force, and the force-force\ncorrelation function is evaluated in the absence of disor-\nder. The correlation function in Eq. (18) is proportional\nto the function F+−(ω) which appeared in Ref. 11 [Eqs.\n(18) and (19)] in a direct calculation of the transverse\nspin susceptibility. Making use of the analytic result for\nℑmF+−(ω)presentedinEq. (21)ofthatpaperweobtain\n1\nτee\n⊥= Γ(p)8α0\n27T2r4\nsm∗a2\n∗k2\nB\n(1+p)1/3, (19)\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s49/s48/s45/s54/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49\n/s112/s61/s48/s46/s57/s57/s40/s110/s111/s32/s101/s45/s101/s32/s105/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s41\n/s112/s61/s48/s46/s53/s112/s61/s48/s46/s49/s112/s61/s48/s46/s49\n/s32/s32\n/s49/s47 /s32/s91/s49/s47/s110/s115/s93\nFIG. 1: (Color online) The Gilbert damping αas a function\nof the disorder scattering rate 1 /τ. Red (solid) line shows the\nGilbertdampingfor polarization p= 0.1inthepresenceofthe\ne-e and disorder scattering, while dashed line does not incl ude\nthee-escattering. Blue(dotted)andblack(dash-dotted)l ines\nshow Gilbert damping for p= 0.5 andp= 0.99, respectively.\nWe took q= 0.1kF,T= 54K,ω0=EF[(1+p)2/3−(1−p)2/3],\nM0=γpn/2,m∗=me,n= 1.4×1021cm−3,rs= 5,a∗= 2a04\nwhereTis the temperature, p= (n↑−n↑)/nis the degree\nof spin polarization, a∗is the effective Bohr radius, rsis\nthe dimensionless Wigner-Seitz radius, α0= (4/9π)1/3\nand Γ(p) – a dimensionless function of the polarization\np– is defined by Eq. (23) of Ref. 11. This result is valid\nto second order in the Coulomb interaction. Collecting\nour results, we finally obtain a full expression for the q2\nGilbert damping parameter:\nα=γnq2\n4m∗M01/τdis\n⊥+1/τee\n⊥\nω2\n0+/parenleftbig\n1/τdis\n⊥+1/τee\n⊥/parenrightbig2.(20)\nOne of the salient features of Eq. (20) is that it scales\nas the total scattering ratein the weak disorder and\ne-e interactions limit, while it scales as the scattering\ntimein the opposite limit. The approximate formula\nfor the Gilbert damping in the more interesting weak-\nscattering/strong-ferromagnet regime is\nα=γnq2\n4m∗ω2\n0M0/parenleftbigg1\nτdis\n⊥+1\nτee\n⊥/parenrightbigg\n, (21)\nwhile in the opposite limit, i.e. for ω0≪1/τdis\n⊥,1/τee\n⊥:\nα=γnq2\n4m∗M0/parenleftbigg1\nτdis\n⊥+1\nτee\n⊥/parenrightbigg−1\n. (22)\nOur Eq. (20) agrees with the result of Singh and\nTeˇ sanovi´ c6on the spin-wave linewidth as a function of\nthe disorder strength and ω0. However, Eq. (20) also\ndescribes the influence of e-e correlations on the Gilbert\ndamping. A comparison of the scattering rates originat-\ning from disorder and e-e interactions shows that the lat-\nter is important and can be comparable or even greater\nthan the disorder contribution for high-mobility and/or\nlow density 3D metallic samples. Fig. 1 shows the be-\nhavior of the Gilbert damping as a function of the dis-\norder scattering rate. One can see that the e-e scatter-\ning strongly enhances the Gilbert damping for small po-\nlarizations/weak ferromagnets, see the red (solid) line.\nThis stems from the fact that 1 /τdis\n⊥is proportional to\n1/τand independent of polarization for small polar-\nizations, while 1 /τee\n⊥is enhanced by a large prefactorΓ(p) = 2λ/(1−λ2) + (1/2)ln[(1 + λ)/(1−λ)], where\nλ= (1−p)1/3/(1+p)1/3. On the other hand, for strong\npolarizations(dotted anddash-dottedlinesinFig.1), the\ndisorder dominates in a broad range of 1 /τand the inho-\nmogenous contribution to the Gilbert damping is rather\nsmall. Finally, we note that our calculation of the e-e in-\nteractioncontributiontothe Gilbertdampingisvalidun-\nder the assumption of /planckover2pi1ω≪kBT(which is certainly the\ncase ifω= 0). More generally, as follows from Eqs. (21)\nand (22) of Ref. 11, a finite frequency ωcan be included\nthrough the replacement (2 πkBT)2→(2πkBT)2+(/planckover2pi1ω)2\nin Eq. (19). Thus 1 /τee\n⊥is proportional to the scattering\nrateofquasiparticlesnearthe Fermi level, andour damp-\ning constant in the clean limit becomes qualitatively sim-\nilar to the damping parameter obtained by Mineev9for\nωcorresponding to the spin-wave resonance condition in\nsome external magnetic field (which in practice is much\nsmaller than the ferromagnetic exchange splitting ω0).\nSummary – We have presented a unified theory of the\nGilbert damping in itinerant electron ferromagnets at\nthe order q2, including e-e interactions and disorder on\nequal footing. For the inhomogeneous dynamics ( q/negationslash= 0),\nthese processes add to a q= 0 damping contribution\nthat is governed by magnetic disorder and/or spin-orbit\ninteractions. We have shown that the calculation of the\nGilbertdampingcanbe formulatedinthe languageofthe\nspin conductivity, which takes an intuitive Matthiessen\nform with the disorder and interaction contributions be-\ning simply additive. It is still a common practice, e.g., in\nthe micromagnetic calculations of spin-wave dispersions\nand linewidths, to use a Gilbert damping parameter in-\ndependent of q. However, such calculations are often at\nodds with experiments on the quantitative side, particu-\nlarly where the linewidth is concerned.2We suggest that\nthe inclusion of the q2damping (as well as the associ-\nated magnetic noise) may help in reconciling theoretical\ncalculations with experiments.\nAcknowledgements – This work was supported in part\nby NSF Grants Nos. DMR-0313681 and DMR-0705460\nas well as Fordham Research Grant. Y. T. thanks A.\nBrataas and G. E. W. Bauer for useful discussions.\n∗Electronic address: hankiewicz@fordham.edu\n1Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n2I. N. Krivorotov et al., Phys. Rev. B 76, 024418 (2007).\n3T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n4E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B75, 174434 (2007).\n5A. Singh, Phys. Rev. B 39, 505 (1989).\n6A. Singh and Z. Tesanovic, Phys. Rev. B 39, 7284 (1989).\n7V.L.SafonovandH.N.Bertram, Phys.Rev.B 61, R14893\n(2000).\n8V. P. Silin, Sov. Phys. JETP 6, 945 (1958).9V. P. Mineev, Phys. Rev. B 69, 144429 (2004).\n10Y. Takahashi, K. Shizume, and N. Masuhara, Phys. Rev.\nB60, 4856 (1999).\n11Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 56404 (2002).\n12N. D. Mermin, Phys. Rev. B 1, 2362 (1970).\n13G. F. Giuliani and G. Vignale, Quantum Theory of the\nElectron Liquid (Cambridge University Press, UK, 2005).\n14E.M.Lifshitz andL.P.Pitaevskii, Statistical Physics, Part\n2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n15Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn.\nMagn. Mater. 320, 1282 (2008), and reference therein.5\n16I. D’Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000).\n17In ferromagnets whose nonuniformities are beyond the\nlinearized spin waves, there is a nonlinear q2contribu-\ntion to damping, (see J. Foros and A. Brataas and Y.\nTserkovnyak, and G. E. W. Bauer, arXiv:0803.2175) which\nhas a different physical origin, related to the longitudinalspin-current fluctuations.\n18Although both σ⊥and ˜χ⊥are in principle tensors in trans-\nverse spin space, they are proportional to δabin axially-\nsymmetric systems—hence we use scalar notation."    },    {        "title": "1104.3002v1.Lagrangian_approach_and_dissipative_magnetic_systems.pdf",        "content": "arXiv:1104.3002v1  [cond-mat.stat-mech]  15 Apr 2011Lagrangian approach and dissipative magnetic systems\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗\n(Dated: September 18, 2018)\nAbstract\nA Lagrangian is introduced which includes the coupling betw een magnetic moments mand the\ndegrees of freedom σof a reservoir. In case the system-reservoir coupling break s the time reversal\nsymmetry the magnetic moments perform a damped precession a round an effective field which is\nself-organized by the mutual interaction of the moments. Th e resulting evolution equation has the\nform of the Landau-Lifshitz-Gilbert equation. In case the b ath variables are constant vector fields\nthe moments mfulfill the reversible Landau-Lifshitzequation. Applying Noether’s theorem we find\nconserved quantities under rotation in space and within the configuration space of the moments.\nPACS numbers: 75.78.-n, 11.10.Ef, 75.10.Hk\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nThe dynamics of magnetic systems is described in a wide range of time a nd length scales\nfrom a quantum approach up to a macroscopic thermodynamic acce ss. On a coarse-grained\nmesoscopic level the relevant electronic degrees of freedom are g rouped into effective mag-\nnetic moments. As the consequence the magnetization is characte rized by a spatiotemporal\nvector field m(r,t). Owing to the mutual interaction between the magnetic moments t hey\nperform a precession motion around a local effective field giving rise t o the propagation of\nspin-wave excitations. Due to a system-reservoir coupling the pre cession of the moments\nshould be a damped one. To analyze this situation one has to specify t he coupling between\nthe system and the bath. The most popular approach to incorpora te dissipation is the em-\nbedding of the relevant system into a quantum-statistical environ ment which is assumed to\nremain in thermal equilibrium. The reservoir is often represented by harmonic oscillators or\nspin moments which are analyzed by path integral techniques [1–3]. A specification of the\npath integral approach for spin systems can be found in [4–7]. A mor e generic description\nconcerning dissipative semiclassical dynamics is presented in [8]. Altho ugh the application\nof path integrals can be considered as an intuitive formalism analytica l calculations are often\nimpossible and numerical schemes are necessary. In the present p aper we propose an alter-\nnative way to include dissipative effects for mesoscopic magnetic sys tems. On this level the\nanalysis of magnetodynamics is performed properly by applying the L andau-Lifshitz-Gilbert\nequation designated as LLG [9, 10]. A comprehensive survey of magn etization dynamics is\ngiven in [11]. Our investigation can be grouped in the effort to underst and dissipative mech-\nanisms in magnets. So, a non-linear dissipative model for magnetic sy stems was discussed in\n[12]. On the relation between fluctuation-dissipation theorems and d amping terms like that\none occurring in the LLG was reported in [13]. The dynamical respons e of ferromagnetic\nshape memory alloy actuators can be modeled by means of a dissipativ e Euler-Lagrange\nequation as performed in [14]. Likewise, the pinning of magnetic domain walls in multifer-\nroics is discussed in terms of the EL equations in [15]. An alternative a nsatz is introduced in\n[16], where a Lagrangian density is obtained based on a projection on to the complex plane.\nThe procedure gives rise to a dynamical equation which is equivalent t o the Landau-Lifshitz\nequation. Different to the mentioned approaches the present pap er is aimed to derive an\nequation of motion for a magnetic system on a mesoscopic scale unde r the influence of a\n2bath which likewise consists of mesoscopic moments. Following this idea we propose a La-\ngrangian comprising both fields, m(r,t) as the system variables and σas the bath variables.\nThe bath becomes dynamically active by the coupling to the system. I n case the coupling\nbetween system and reservoir breaks the time reversal symmetr y the motion of the moments\nm(r,t) is damped. The Lagrangian is modified in such a manner that dissipatio n can occur.\nII. THE LAGRANGIAN\nAs indicated we are interested to construct a Lagrangian describin g the motion of a magne-\ntization vector field within a bath of spins. This reservoir should influe nce the measurable\nmagnetization due to the mutual interaction. Let us formulate the general assumptions for\nthe underlying model. The magnetic order is originated by single magne tic atoms which\noccupy equivalent crystal positions. Here we refer to a continuou s description in terms of\na field vector denoted as m(r,t). Because the ferromagnet is considered below the Curie\ntemperature a sufficient number of microscopic spins preferring a p arallel alignment are in-\ncluded in m, i.e. the effective magnetic moment is given by m(r,t) =/summationtext\niµiwhere the\nsum is extended over all microscopic moments within a small volume aro und the spatial\ncoordinate rat timet. As each axial vector the moment fulfills m(−t) =−m(t). The bath\nin which the moments are embedded consists likewise of mesoscopic sp ins. They are denoted\nasσand are also composed of microscopic moments ηi. This bath moments which play\nthe role of ’virtual’ moments are also axial vectors changing their sig n by time inversion.\nA further new aspect is that the coupling between the real and the virtual moments is not\nassumed to be weak. As the result the complete system consists of two subsystems. One of\nthem abbreviated as L1 is occupied exclusively by the real spins with t he moments mand\nthe other one denoted as L2 is occupied by the bath spins σ. The situation is illustrated in\nFIG. 1. Now let us introduce the action\nS[{qα}] =/integraldisplay\ndt/integraldisplay\nd3xL[{qα}], (1)\nwhere the set {qα}consists of the set of both moments σandm. The Lagrange density\ncomprises three terms\nL[m,˙m,∇m;σ,∇σ] =L(m)[m,˙m,∇m]+L(σ)[∇σ]+L(mσ)[σ,˙m], (2)\n3µi−1 µi µi+1 ηj−k ηj ηj+1 ηj+1+kJµηJηηJµµ\nFIG. 1. (Color online) Schematic illustration of the basic m odel. The red spins represent the\nmagnetic moments µiand refer to the lattice L1 introduced in the text. The green s pin vectors\nηibuild the bath lattice L2. Interactions are possible betwee n theµiandηj,µiandµjandηi\nandηj. The respective coupling strengths correspond to the coupl ing parameters in Eqs. (3)-(5)\nas follows: Jµη↔J(mσ),Jµµ↔J(m)andJηη↔J(σ).\nwhereL(m)indicates the Lagrangian of the magnetic system, L(σ)represents the reservoir\nand the interaction term is denoted as L(mσ). To be more specific the magnetic moments of\nthe system interact via exchange coupling defined by the Lagrangia n\nL(m)=1\n2J(m)\nαβ∂mν\n∂xα∂mν\n∂xβ+Aν(m) ˙mν, (3)\nwhereJ(m)\nαβisthecouplingparameter, diagonalintheisotropiccase. Thefirstt ermrepresents\nthe energy density of the magnetic system. Because we are not co nsidering the acceleration\nof magnetic moments a term of the order ˙m2is missing. Moreover, the magnetic moments\nperformaprecessionaroundaneffective magneticfield, whichisself -organizedbythemutual\ninteraction. Thereforethevectorpotential Adependsonthemoments, i. e. A=A(m(r,t)).\nThe coupling has the same form as the minimal coupling in electrodynam ics. The bath\nLagrangian is defined in a similar manner as\nL(σ)=1\n2J(σ)\nαβ∂σν\n∂xα∂σν\n∂xβ, (4)\nwith the coupling constant J(σ)\nαβ. Eventually, the interacting part between system and bath\nis written as\nL(mσ)=J(mσ)\nαβ∂mν\n∂xα∂σν\n∂xβ+Bν(σ) ˙mν, (5)\n4with the coupling strength J(mσ)\nαβ. The second term is constructed in the same manner as\nin Eq. (3), where the potential B(σ) will be specified below, see Eq. (8). The dynamics\nof the bath variable σremains unspecified for the present, i.e. the Lagrangian does not\ninclude a term of the form ∝˙σ. Owing to the constraint, introduced in the next section,\nthe dynamically passive bath is sensitive to a change of the system va riablesmin such a\nmanner that small variations of the system variables mare related to small variations of\nσ. This procedure leads to a coupling between bath and system so tha t the time reversal\nsymmetry is broken.\nIII. RELATION TO THE LANDAU-LIFSHITZ-GILBERT EQUATION\nIn this section we find the equation of motion for the magnetization m(r,t) from Eq. (2)\ncombined with Eqs. (3)-(5). Using the principle of least action it follow s\n/bracketleftbigg∂L\n∂σβ−∂\n∂xα∂L\n∂/parenleftBig\n∂σβ\n∂xα/parenrightBig/bracketrightbigg\nδσβ+/bracketleftbigg∂L\n∂mβ−∂\n∂t∂L\n∂˙mβ−∂\n∂xα∂L\n∂/parenleftBig\n∂mβ\n∂xα/parenrightBig/bracketrightbigg\nδmβ= 0,(6)\nwhereδmβandδσβare the small variations which drive the value for the action out of th e\nstationary state. In general, one derives a system of coupled par tial differential equations.\nHowever, to proceed further let us impose a constraint on the sys tem. A small variation of\nσβshould be related to a small variation of mβ. Thus, we make the ansatz\nδσβ=−κδmβ,withκ= const>0. (7)\nNotice that this condition should be valid only locally but not globally. Ins ofar Eq. (7) is\ncomparable to an anholonom condition in mechanics. Moreover relatio n (7) is in accordance\nwith thebehavior of themoments mandσunder timeinversion. Physically the last relation\nmeans that the bath reacts to a change of the system only tempor arily. Because the system-\nreservoir coupling should typically break the time reversal symmetr y the expansion of the\nfunctionBν(σ) in terms of σincludes only odd terms. In lowest order we get from Eq. (5)\nBν(σ) =−cσν,withc= const. (8)\nDue to Eqs. (7) and (8) the second term in Eq. (5) is of the form ∝σ·˙m. Such a term is not\ninvariant under time reversal symmetry t→ −t. As demonstrated below the broken time\n5inversion invariance gives rise to damping effects. Inserting Eqs. (7 ) and (8) into Eq. (2)\nand performing the variation according to Eq. (6) we get\n0 =/parenleftbigg∂Aν\n∂mβ−∂Aβ\n∂mν/parenrightbigg\n˙mν+c˙σβ+κc˙mβ\n−[J(m)−κJ(mσ)]∇2mβ−[J(mσ)−κJ(σ)]∇2σβ.(9)\nHere we have assumed for simplicity that all coupling tensors Jare diagonal: Jαβ=Jδαβ.\nThe first term on the right hand side in Eq. (9) reminds of the field str ength tensor in\nelectrodynamics [17]. Thus, we rewrite\n/parenleftbigg∂Aν\n∂mβ−∂Aβ\n∂mν/parenrightbigg\n˙mν≡Fβν˙mν=/bracketleftBig\n˙m×(∇m×A(m))/bracketrightBig\nβ. (10)\nAs mentioned above the vector function A(m) is regarded as vector potential which depends\non space-time coordinates via the magnetic moment m(r,t). In vector notation the last\nequation reads\n˙m×(∇m×A) = [J(m)−κJ(mσ)]∇2m−κc˙m−c˙σ+[Jmσ−κJ(σ)]∇2σ.(11)\nIf one is interested in weak excited states of a ferromagnet it is rea sonable to assume that\nthe direction of the magnetization in space changes slowly while its abs olute value is fixed,\nthat ism2= 1. Without loss of generality we have set the amplitude of mto unity. In order\nto proceed it is necessary to specify the condition which should be fu lfilled by the function\nA(m). Having in mind the LLG then we make the ansatz ∇m×A(m) =gm,g= const.\nBased on these assumptions we get from Eq. (11)\n∂m\n∂t=1\ng/parenleftBigg\nm×Heff/parenrightBigg\n−κc\ng/parenleftBigg\nm×∂m\n∂t/parenrightBigg\n. (12)\nHere the effective field is given by the expression\nHeff=/parenleftBig\nJ(m)−κJ(mσ)/parenrightBig\n∇2m−c∂σ\n∂t+/parenleftBig\nJ(mσ)−κJ(σ)+/parenrightBig\n∇2σ. (13)\nEq. (12) is nothing else than the Gilbert equation [10] by relating the prefactors as follows\nγ=−1\ng, α=−κc\ng=κcγ, (14)\nwhereγandαare the gyromagnetic ratio and the Gilbert damping parameter, res pectively.\nSince bothparameters arepositive quantities itfollows that g<0aswell as κc>0. Further,\n6Eq. (12) can be converted into the form of the equivalent and widely used Landau-Lifshitz-\nGilbert equation which reads\n∂m\n∂t=−γ\n(1+α)2(m×Heff)−αγ\n(1+α2)/bracketleftBig\nm×(m×Heff)/bracketrightBig\n, (15)\nBoth quantities γandαare still related to the model parameters by the expressions in\nEq. (14) whereas the the effective field Heffis given by Eq. (13). Now we want to analyze\nthis expression and in particular, to assign a physical meaning to the more or less ad hoc\nintroduced quantity σ. In doing so one can distinguish four different cases:\n(i) The bath is not included which corresponds formally to σis a constant vector depend-\ning neither on coordinates nor on time. Then obviously all derivatives with respect to the\ncoordinates and the time of σdisappear in Eq. (13) and consequently, the set of {qα}in\nEq. (1) does not include σ. From here we conclude that the variation fulfills δσ= 0 in\nEq. (6) which can be easily realized setting κ= 0, cf. Eq. (7). Thus, the effective field\nin Eq. (13) comprises the pure exchange interaction J(m)between the magnetic moments\nand the damping term in Eq. (12) is absent due to α= 0 in Eq. (14). A constant bath\nfieldσlead to the Landau-Lifshitz equation in the exchange interaction ap proach without\ndamping, compare [18]. It describes the precession of magnetic mom ents of an effective field\nwhich is self-organized by the mutual interaction of the moments.\n(ii)σ=σ(t) depends only on the time and not on the spatial coordinates. Rega rding\nEq. (13) the effective field is modified by two additional contributions , namely one propor-\ntional to ∇2m, originated in the exchange interaction of the magnetic moments, a nd the\nother one ∝˙σ. The latter one could be associated with an external time dependen t field\nor, ifσpoints into a fixed direction, gives rise to magnetic anisotropy. In th at case the\nanisotropy axis is spatially constant but the amount of the anisotro py is changing in time.\nSuch a situation could be realized for instance when the ferromagne tic sample is excited by\nthe irradiation with electromagnetic waves. As already mentioned th e exchange coupling\nJ(m)is supplemented by a term −κJ(mσ). In this manner the exchange interaction is influ-\nenced by the coupling between mandσalthough the spatial dependence of σis not taken\ninto account explicitly.\n(iii)σ=σ(r) depends only on the spatial coordinates and not on the time. In th is case we\nfirst recognize that the coupling strength J(m)in the term ∝ ∇2mis influenced in the same\nmanner as in case when σ=σ(t), see the previous point. Different to the former cases the\n7expression ∝ ∇2σbecomes important for the effective field in Eq. (13). The appearan ce of\nthis term suggests that spatial inhomogeneities of the surroundin gs of the magnetic system\nrepresented by mhave to be incorporated into the effective field. It seems to be reas onable\nthat the origin of this term is an inherent one and should not be led bac k to external fields.\nAs possible sources we have in mind local varying fields like inner and out er demagnetization\nfields as well as accessible fields created for instance by different loc al temperatures.\n(iv)σ=σ(r,t) isthemost general case. Then external aswell asinternal fields arecaptured\nin the model. Thus, the effective field in Eq. (13) can be rewritten as\nHeff(r,t) =Hexch(r)+h(r,t), (16)\nwhereHeffconsists of two parts. The term Hexch= (J(m)−κJ(mσ))∇2mis due to the\nexchange interaction between the magnetic moments whereas h(r,t) represents other possi-\nble influences as discussed under the points (ii) and (iii). The function his related to the\nquantity σby\nh(r,t) =−c∂σ(r,t)\n∂t+/bracketleftbig\nJ(mσ)−κJ(σ)/bracketrightbig\n∇2σ(r,t). (17)\nRemark that the formerly introduced quantity σis related to the physically relevant effec-\ntive field by the first derivation with respect to the time and the seco nd derivation with\nrespect to the spacial coordinates via Eq. (17). This equation is an inhomogeneous diffusion\nequation which can be generally solved by means of the expansion into Fourier series and the\nassumption of accurate initial and boundary conditions which depen d on the actual physical\nproblem.\nIV. SYMMETRY AND CONSERVATION\nAfter regarding the special example of the LLG we proceed with the investigation of more\ngeneral aspects. The Lagrangian density allows to discuss the beh avior under space-time\ndependent group transformation. For this purpose we apply Noet her’s theorem [19] to our\nmodel. To be more precise we consider the conservation equation [20 ]\n∂\n∂Xα/bracketleftbigg/parenleftBig\nLδαβ−∂L\n∂(∂αΨγ)∂βΨγ/parenrightBig\n∆Xβ+∂L\n∂(∂αΨγ)∆Ψγ/bracketrightbigg\n= 0. (18)\nHere, the expression in the square brackets are the components of the Noether current Iα.\nThe term∂/∂Xαin front ofIαshould be interpreted as an implicit derivative with respect\n8to time and three spatial coordinates. The symmetry operations ∆ Xαand ∆Ψ αwill be\nspecified below. With regard to the Lagrangian in Eq. (2) we introduc e the components\nΨα= (mx,my,mz,σx,σy,σz) and their partial derivatives with respect to the independent\nvariables∂βΨα=∂Ψα/∂Xβ. Since we examine an Euclidean field theory a distinction\nbetween upper and lower indices is not necessary. Eq. (18) can be r ewritten by using\nEq. (6). This yields\n∂\n∂tL∆t+∂\n∂xαL∆xα+∂L\n∂Ψα/parenleftbig\n∆Ψα−∂\n∂tΨα∆t−∂\n∂xβΨα∆xβ/parenrightbig\n= 0. (19)\nIn this equation we distinguish between the time and space variables tandxαexplicitly.\nEq. (19) is the basis for the application of the following symmetry ope rations. Now we study\nthe rotation around a certain axis as a relevant one. Here we select for instance the z-axis.\nPerforming a rotation in coordinate space with the infinitesimal angle ∆Θ the change of the\nxandy-coordinates obeys\n∆t= 0,∆xα= ∆Rαβxβ,∆R=\n0 ∆Θ\n−∆Θ 0\n. (20)\nIn the same manner one can perform the rotation in the configurat ion space of the moments\nmandσsymbolized by the before introduced vector Ψ α={mx,my,mz,σx,σy,σz}. The\ntransformation reads ∆Ψ α= ∆Sαβ(∆Φ)Ψ β, where the rotation matrix is a 6 ×6-matrix\ndetermined by the rotation angle ∆Φ. Because both rotations in coo rdinate space and\nconfiguration space, respectively, are in general independent fr om each other we find two\nconserved quantities. Using Eq. (19) it results\nˆDzL= 0,ˆΓzL= 0. (21)\nHere the two operators ˆDzandˆΓzare expressed by\nˆDz=ˆLz−/parenleftBig\nˆLzψα/parenrightBig∂\n∂Ψα,\nˆΓz=ˆS(m)\nz+ˆS(σ)\nz.(22)\nThe quantity ˆLzis the generator of an infinitesimal rotation around the z-axis in the coor-\ndinate space\nˆLz=y∂\n∂x−x∂\n∂y, (23)\n9and therefore, it is identical with the angular momentum operator. The other quantities\nˆS(m)\nzandˆS(σ)\nzare the corresponding generators in the configuration space of t he moments.\nThey are defined as\nˆS(m)\nz=my∂\n∂mx−mx∂\n∂my,\nˆS(σ)\nz=σy∂\n∂σx−σx∂\n∂σy.(24)\nThese operators reflect the invariance of the total magnetic mom entm+σunder rotation.\nMoreover the system is invariant under the combined transformat ion expressed by ˆDzand\nˆΓz, where ˆDzoffers due to the coupling between system and bath variables as well as the\nbreaking of time reversal invariance a coupling between magnetic mo ments and the angular\nmomentum.\nV. CONCLUSION\nIn this paper we have presented an approach for a mesoscopic mag netic system with dissi-\npation. The Lagrangian consists of two interacting subsystems ch aracterized by the active\nmagnetic moments of the system mand the dynamically inactive moments of the bath\ndenoted as σ. Both systems are in contact so that a small local alteration of the system\nvariables mis related as well to a small change of the bath variables σand vice versa.\nDue to this constraint we are able to describe the system by a commo n Lagrangian which\nincorporates both degrees of freedom and their coupling. In case the bath variables are\nconstant then the coupling between both systems is absent and th e whole system decays\ninto two independent subsystems. The magnetic moments mperform a precession around\nan effective field which is self-organized by the mutual interaction of the moments. If the\ncoupling between both subsystems breaks the time reversal symm etry the related evolution\nequation of the moments mis associated with the Landau-Lifshitz-Gilbert equation which\ndescribes both the precession of magnetic moments as well as their damping. It turned\nout that the bath variable σcan be linked to the effective magnetic field which drives the\nmotion of the magnetic moments. As consequence the motion of the moments is influenced\nby the additional bath degrees of freedom. This influence is formula ted mathematically and\nis described by an inhomogeneous diffusion equation. Finally, we have f ound conservation\nlaws by means of symmetry considerations based on Noether’s theo rem. Aside from the\n10expected symmetry transformation in the coordinate space and t he configuration space of\nthe moments, the analysis offers in a non-relativistic Euclidean field th eory an unexpected\ncoupling between both. This point deserves further consideration . Our approach could be\nalso considered as starting point for a further analysis in magnetic a nd multiferroic systems.\nEspecially, we are interested in more refined models which include for in stance higher order\ncouplings or anisotropy in the Lagrangian. In multiferroic systems o ne could study the case\nthat the magnetic and the polar subsystem have their own reservo irs.\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which\nis supported by the Saxony-Anhalt State, Germany.\n11[1] R. Feynman, A. Hibbs, and D. Styer, Quantum Mechanics and Path Integrals: Emended\nEdition(Dover Publications, 2010).\n[2] H. Kleinert, Path integrals in quantum mechanics, statistics, polymer p hysics, and financial\nmarkets (World Scientific, 2009).\n[3] U. Weiss, Quantum dissipative systems , Series in modern condensed matter physics (World\nScientific, 1999).\n[4] L. Schulman, Phys. Rev. 176, 1558 (1968).\n[5] D. C. Cabra, A. Dobry, A. Greco, and G. L. Rossini, J. Phys. A30, 2699 (1997).\n[6] V. V. Smirnov, J. Phys. A 32, 1285 (1999).\n[7] H. Grinberg, Phys. Lett. A 311, 133 (2003).\n[8] W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold ,\nPhys. Rev. Lett. 100, 230402 (2008).\n[9] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[10] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[11] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halp erin, Rev. Mod. Phys. 77, 1375\n(2005).\n[12] P. Durand and I. Paidarov ˜A¡, EPL 89, 67004 (2010).\n[13] V. L. Safonov and H. N. Bertram, Phys. Rev. B 71, 224402 (2005).\n[14] P. Weetman and G. Akhras, J. Appl. Phys. 105, 023917 (2009).\n[15] Z. V. Gareeva and A. K. Zvezdin, EPL 91, 47006 (2010).\n[16] I. V. Ovchinnikov and K. L. Wang, Phys. Rev. B 82, 024410 (2010).\n[17] J. Jackson, Classical electrodynamics (Wiley, 1999).\n[18] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Statistical Physics Part 2: Theory of the Con-\ndensed State (Pergamon Press, Oxford, 1980).\n[19] E. Noether, Nachr. Ges. Wiss. G¨ ottingen , 235 (1918).\n[20] E. L. Hill, Rev. Mod. Phys. 23, 253 (1951).\n12"    },    {        "title": "0810.2870v1.Interaction_of_reed_and_acoustic_resonator_in_clarinetlike_systems.pdf",        "content": "arXiv:0810.2870v1  [physics.class-ph]  16 Oct 2008\n/C1/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D3/CU /D6/CT/CT/CS /CP/D2/CS /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CX/D2 \r/D0/CP/D6/CX/D2/CT/D8/B9/D0/CX/CZ /CT /D7/DD/D7/D8/CT/D1/D7/BY /CP/CQ /D6/CX\r/CT /CB/CX/D0/DA/CP/B8 /C2/CT/CP/D2 /C3/CT/D6/CV/D3/D1/CP /D6/CS/B8 /CP/D2/CS /BV/CW/D6/CX/D7/D8/D3/D4/CW/CT /CE /CT/D6/CV/CT/DE\n/CP/B5/C4 /CP/CQ /D3/D6 /CP/D8/D3/CX/D6 /CT /CS/CT /C5/GH \r /CP/D2/CX/D5/D9/CT /CT/D8 /CS/B3/BT \r /D3/D9/D7/D8/CX/D5/D9/CT /CD/C8/CA /BV/C6/CA/CB /BJ/BC/BH/BD/B8 /BD/BF/BG/BC/BE /C5/CP/D6/D7/CT/CX/D0 /D0/CT \r /CT /CS/CT/DC /BE/BC/B8 /BY /D6 /CP/D2\r /CT/C2/D3 /GJ/D0 /BZ/CX/D0/CQ /CT/D6/D8\n/CQ/B5/C4 /CP/CQ /D3/D6 /CP/D8/D3/CX/D6 /CT /CS/B3/BT \r /D3/D9/D7/D8/CX/D5/D9/CT /CS/CT /D0/B3/CD/D2/CX/DA/CT/D6/D7/CX/D8/GH /CS/D9 /C5/CP/CX/D2/CT /CD/C5/CA 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/CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BE/D7/D3/D0/D9/D8/CX/D3/D2/D7 /CW/CP /DA/CX/D2/CV /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT exp(jωt) /CP/D6/CT /D7/D3/D9/CV/CW /D8/BA/BV/CP/D2\r/CT/D0/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8 /D3/CUω \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP/D2/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D7 /D2/CT/CX/D8/CW/CT/D6 /CS/CP/D1/D4 /CT/CS /D2/D3/D6 /CP/D1/D4/D0/CX/AS/CT/CS/BM /CX/D8 \r /CW/CP/D6/B9/CP\r/D8/CT/D6/CX/DE/CT/D7 /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3/CU /D7/D8/CP/D8/CX\r /D6/CT/CV/CX/D1/CT/BA /BT /D8/B9/D8/CT/D2 /D8/CX/D3/D2 /CX/D7 /CS/D6/CP /DB/D2 /D8/D3 /D8/CW/CT /CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CX/D7 /D5/D9/CP/D2 /D8/CX/D8 /DD /D1/CP /DD /CS/CX/AR/CT/D6/CU/D6/D3/D1 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CS/CT/D4 /CT/D2/CS/CX/D2/CV /D3/D2 /D8/CW/CT /D2/CP/D8/D9/D6/CT/D3/CU /D8/CW/CT /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2/B8 /D7/D8/D9/CS/CX/CT/CS /CX/D2 /CB/CT\r/BA /C1/CE/BA/BU/BA /BT/D7 /CP /D0/CP/D2/CV/D9/CP/CV/CT/CP/CQ/D9/D7/CT /D3/D7\r/CX/D0 /D0/CP/D8/CX/D3/D2 /D8/CW/D6 /CT/D7/CW/D3/D0/CS /CX/D7 /D3/CU/D8/CT/D2 /D9/D7/CT/CS /CX/D2/D7/D8/CT/CP/CS /D3/CU /CX/D2/D7/D8/CP/B9/CQ/CX/D0/CX/D8/DD /D8/CW/D6 /CT/D7/CW/D3/D0/CS /BA/BT/D7/D7/D9/D1/CX/D2/CV /D7/D1/CP/D0/D0 /DA/CX/CQ/D6/CP/D8/CX/D3/D2/D7 /CP/D6/D3/D9/D2/CS /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT/B4/D1/CT/CP/D2 /DA /CP/D0/D9/CT/D7 /D3/CUy /CP/D2/CSp /CP/D6/CTy0−Pm/K /CP/D2/CS0 /B8 /D6/CT/D7/D4 /CT\r/B9/D8/CX/DA /CT/D0/DD/B5/B8 /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /B4/BG/B5 /CX/D7 /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/BA /BW/CX/B9/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9\r/CT/CS /CW/CT/D6/CT/BMθ /B8Ye\n/B8 /CP/D2/CSD/CP/D6/CT /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /CX/D2/D4/D9/D8 /CP/CS/D1/CX/D8/D8/CP/D2\r/CT /CP/D2/CS/D8/CW/CT /D6/CT/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /CU/D9/D2\r/D8/CX/D3/D2/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA\nθ=ω\nωr,Ye(θ) =Zc\nZe(θ)\n/CP/D2/CSD(θ) =1\n1+jqrθ−θ2. /B4/BH/B5/CC/CW/CT/D6/CT /CP/D6/CT /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 \r/D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BM\nγ=Pm\nKy0\n/CP/D2/CSζ=ZcW/radicalbigg2y0\nKρ. /B4/BI/B5\nγ /CX/D7 /D8/CW/CT /D6/CP/D8/CX/D3 /CQ /CT/D8 /DB /CT/CT/D2 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT/D6/CT/D5/D9/CX/D6/CT/CS /D8/D3 \r/D3/D1/D4/D0/CT/D8/CT/D0/DD \r/D0/D3/D7/CT /D8/CW/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /CX/D2 /D7/D8/CP/D8/CX\r/D6/CT/CV/CX/D1/CT/B8 /DB/CW/CX/D0/CTζ /D1/CP/CX/D2/D0/DD /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D1/D3/D9/D8/CW/D4/CX/CT\r/CT \r/D3/D2/D7/D8/D6/D9\r/B9/D8/CX/D3/D2 /CP/D2/CS /D0/CX/D4 /D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS /CP/D2/CS /CX/D7 /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT /D1/CP/DC/B9/CX/D1 /D9/D1 /AT/D3 /DB /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /B4ζ /CT/D5/D9/CP/D0/D7 /D5/D9/CP/D2 /D8/CX/D8 /DD\n2β /CX/D2 /CA/CT/CU/BA /BE/B5/BA/C4/CX/D2/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BG/B5 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /D7/D3/B9\r/CP/D0/D0/CT/CS \r /CW/CP/D6/CP\r/B9/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2/BM\nYe(θ) =ζ√γ/braceleftbigg\nD(θ)−1−γ\n2γ/bracerightbigg\n, /B4/BJ/B5/DB/CW/CX\r /CW \r/CP/D2 /CQ /CT /D7/D4/D0/CX/D8 /CX/D2 /D8/D3 /D6/CT/CP/D0 /CP/D2/CS /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8/D7/BM/C1/D1(Ye(θ)) =ζ√γ /C1/D1(D(θ)), /B4/BK/B5/CA/CT(Ye(θ)) =ζ√γ/parenleftbigg/CA/CT(D(θ))−1−γ\n2γ/parenrightbigg\n. /B4/BL/B5/BT /D8 /D0/CP/D7/D8/B8 /CP /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D0/CT/D2/CV/D8/CWkrL=ωrL/c /CX/D7 /CX/D2 /D8/D6/D3/B9/CS/D9\r/CT/CS/BA/BV/BA /C6/D9/D1/CT/D6/CX\r/CP/D0 /D8/CT\r/CW/D2/CX/D5/D9/CT/D7/CC/CW/CT /D9/D2/CZ/D2/D3 /DB/D2/D7 θ, γ∈R+/D7/CP/D8/CX/D7/CU/DD/CX/D2/CV /BX/D5/BA /B4/BJ/B5 /CP/D6/CT /D2 /D9/D1/CT/D6/B9/CX\r/CP/D0/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/D3/D6 /CP /D6/CP/D2/CV/CT /D3/CU /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7/B8 /D4/CP/D6/CP/D1/CT/B9/D8/CT/D6/D7(qr,ζ,ωr) /CQ /CT/CX/D2/CV /D7/CT/D8/BA /CC/CW/CT/DD \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CU/D6/CT/D5/D9/CT/D2\r/DD/CP/D2/CS /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D8 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX\r/D6/CT/CV/CX/D1/CT/BA /CF/CW/CT/D2 /DA /CP/D6/CX/D3/D9/D7 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CT/DC/CX/D7/D8 /CU/D3/D6 /CP /CV/CX/DA /CT/D2 \r/D3/D2/B9/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/DB/CX/D8/CW /D8/CW/CT /D7/CT/DA /CT/D6/CP/D0 /CQ /D3/D6/CT /D6/CT/D7/D3/D2/CP/D2\r/CT/D7/B8 /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3/CQ/D7/CT/D6/DA /CT/CS/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /CQ /DD /CX/D2\r/D6/CT/CP/D7/CX/D2/CV /D8/CW/CT /CQ/D0/D3 /DB/CX/D2/CV /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D8/CW/CT/D3/D2/CT /CW/CP /DA/CX/D2/CV /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /DA /CP/D0/D9/CT /D3/CUγ /BA/CC/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D8/D6/CP/D2/D7\r/CT/D2/CS/CT/D2 /D8/CP/D0 /CP/D2/CS /D1/CP /DD/CW/CP /DA /CT /CP/D2 /CX/D2/AS/D2/CX/D8/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA /CI/CT/D6/D3 /AS/D2/CS/CX/D2/CV /CX/D7 /CS/D3/D2/CT/D9/D7/CX/D2/CV /D8/CW/CT /C8 /D3 /DB /CT/D0/D0 /CW /DD/CQ/D6/CX/CS /D1/CT/D8/CW/D3 /CS\n/BF/BD/B8 /DB/CW/CX\r /CW \r/D3/D1 /CQ/CX/D2/CT/D7 /D8/CW/CT/CP/CS/DA /CP/D2 /D8/CP/CV/CT/D7 /D3/CU /CQ /D3/D8/CW /C6/CT/DB/D8/D3/D2 /D1/CT/D8/CW/D3 /CS /CP/D2/CS /D7\r/CP/D0/CT/CS /CV/D6/CP/CS/CX/CT/D2 /D8/D3/D2/CT/BA /BT \r/D3/D2 /D8/CX/D2 /D9/CP/D8/CX/D3/D2 /D8/CT\r /CW/D2/CX/D5/D9/CT /CX/D7 /CP/CS/D3/D4/D8/CT/CS /D8/D3 /D4/D6/D3 /DA/CX/CS/CT /CP/D2\n/CX/D2/CX/D8/CX/CP/D0 /DA /CP/D0/D9/CT /D8/D3 /D8/CW/CT /CP/D0/CV/D3/D6/CX/D8/CW/D1/BM /D8/CW/CT /AS/D6/D7/D8 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/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CP/D2/CS /D4/D6/CT/D7/D7/D9/D6/CT/CU/D3/D6 /CP /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/BMqr= 0.008 /B8fr= 700 Hz /B8β=\n0.05 /BA /CA/CT/D7/D9/D0/D8/D7 /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7/B5 /CP/D2/CS /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /B4/D7/D5/D9/CP/D6/CT/D7/B5 /CU/D6/D3/D1/CA/CT/CU/BA /BE/BN /D3/D9/D6 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/B5/BA\r/CP/D2 /CQ /CT /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8/BA /C1/D2 /CP /DA /CP/D0/CX/CS/CP/D8/CX/D3/D2 /D4/CW/CP/D7/CT /D3/CU /D3/D9/D6 /D2 /D9/D1/CT/D6/B9/CX\r/CP/D0 /CP/D0/CV/D3/D6/CX/D8/CW/D1/D7/B8 /D3/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/CT/CT /BY/CX/CV/BA /BD /CP/D2/CS /BE/B5 /DB /CT/D6/CT \r/D3/D1/B9/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT/CX/D6 /D3/D2/CT/D7 /B4/D8/CW/CT /CS/CP/D8/CP /CQ /CT/CX/D2/CV /CT/DC/D8/D6/CP\r/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT/CX/D6/CP/D6/D8/CX\r/D0/CT/B5/B8 /D7/D3/D0/DA/CX/D2/CV /CT/DC/CP\r/D8/D0/DD /D8/CW/CT /D7/CP/D1/CT /CT/D5/D9/CP/D8/CX/D3/D2/BA /BW/CX/AR/CT/D6/CT/D2\r/CT/D7/CP/D4/D4 /CT/CP/D6 /CQ /CT/D8 /DB /CT/CT/D2 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 \r/D3/D2\r/CT/D6/D2/CX/D2/CV /D8/CW/D6/CT/D7/CW/D3/D0/CS/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7/B8 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /DB/CW/CT/D2 /D8/CW/CT /D0/CT/D2/CV/D8/CW/CS/CT\r/D6/CT/CP/D7/CT/D7/BA /C1/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8 /D1/D3 /CS/CT/D0 /D0/CX/D1/CX/D8/D7 /D8/CW/CT/CP/D9/D8/CW/D3/D6/D7 /CS/D3 /D2/D3/D8 /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8/BM /D0/CX/D2/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /AT/D3 /DB/D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /CX/D7 /DA /CP/D0/CX/CS /D3/D2/D0/DD /DB/CW/CX/D0/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /CX/D7 /D2/D3/D8 \r/D0/D3/D7/CT/CS/CP/D8 /D6/CT/D7/D8/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2Pm< Ky 0\n/BA /CD/D7/CX/D2/CV /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CU/D3/D6/D1/CU/D3/D6 /CW/CX/CV/CW/CT/D6 /DA /CP/D0/D9/CT/D7 /D3/CUPm\n/DB /D3/D9/D0/CS /CQ /CT /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7/B8 /CT/DA /CT/D2 /CU/D3/D6/CU/D6/CT/CT /D6/CT/CT/CS /CP/CT/D6/D3/D4/CW/D3/D2/CT/D7/BM /D8/CW/CT /D3/D4 /CT/D2/CX/D2/CV /CU/D9/D2\r/D8/CX/D3/D2 /B4/D0/CX/D2/CZ /CT/CS /D8/D3/D8/CW/CT /D6/CT/CT/CS /CS/CX/D7/D4/D0/CP\r/CT/D1/CT/D2 /D8/B5 /D8/CP/CZ/CX/D2/CV /D4/CP/D6/D8 /CX/D2 /AT/D3 /DB \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2\r/CP/D2 /D2/CT/DA /CT/D6 /CQ /CT /D2/CT/CV/CP/D8/CX/DA /CT/BA /BY /D3/D6 /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /CU/D3/D6 /DB/CW/CX\r /CW /D6/CT/CT/CS/CQ /CT/CP/D8/D7 /CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D1/D3/D9/D8/CW/D4/CX/CT\r/CT/B8 /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /CX/D7 \r/D3/D1/D4/D0/CT/D8/CT/D0/DD\r/D0/D3/D7/CT/CS /CP/D2/CS /D8/CW/CT/D2 /D7/D9/D7/D8/CP/CX/D2/CT/CS /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 \r/CP/D2/D2/D3/D8 /D3 \r\r/D9/D6 /CU/D3/D6/CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7 /DB/CW/CT/D6/CT /CF/BU /D8/CW/CT/D3/D6/DD /D4/D6/CT/CS/CX\r/D8/D7 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS /CP/CQ /D3 /DA /CT /D7/D8/CP/D8/CX\r /CQ /CT/CP/D8/CX/D2/CV /D6/CT/CT/CS /D4/D6/CT/D7/D7/D9/D6/CT /CQ /DD /CT/DC/D8/CT/D2/CS/CX/D2/CV /D0/CX/D2/B9\n/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /CQ /CT/DD /D3/D2/CS /D1/D3 /CS/CT/D0 /D0/CX/D1/CX/D8/D7/BA/CC/CW/CT /CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /D6/CT/B9/D1/CP/CX/D2/D7 /D0/D3 /DB /CT/D6 /D8/CW/CP/D2 /BD /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /CP /D1/CP/DC/CX/D1 /D9/D1/D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2\r/DD /CU/D3/D6 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/BA /CF /CT /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8/D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1 /DA /CP/D0/D9/CTθ/D1/CP/DC\n/CX/D7 /CP/D7/D7/D3 \r/CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1/DA /CP/D0/D9/CT /D3/CUγ /B4/CX/BA/CT/BA/B8/BD/B5/BA /C6/D3 /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D4/D6/D3 /D3/CU /CX/D7 /CV/CX/DA /CT/D2 /CW/CT/D6/CT/B8 /CQ/D9/D8/D8/CW/CX/D7 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 /CP/D0/D0 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/B9/D8/CP/CX/D2/CT/CS/BA /CC/CW/CT/D2 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 θ/D1/CP/DC\n/CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nYe(θ/D1/CP/DC) =ζD(θ/D1/CP/DC), /B4/BD/BC/B5/DB/CW/CX\r /CW /D0/CT/CP/CS/D7 /D8/D3 /D6/CT/D0/CP/D8/CX/D3/D2\n1−θ2/D1/CP/DC=ζ /CA/CT(Ze(θ/D1/CP/DC))>0 /B4/BD/BD/B5/CP/D7 /D8/CW/CT /CQ /D3/D6/CT /CX/D7 /CP /D4/CP/D7/D7/CX/DA /CT /D7/DD/D7/D8/CT/D1/BA /BT/D7 /CP \r/D3/D2\r/D0/D9/D7/CX/D3/D2/B8 /CX/D8 /CX/D7 /D2/D3/D8/D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D4/D0/CP /DD /D7/CW/CP/D6/D4 /CT/D6 /D8/CW/CP/D2 /CP /CU/D6/CT/D5/D9/CT/D2\r/DD /D7/D0/CX/CV/CW /D8/D0/DD /AT/CP/D8/B9/D8/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D6/CT/CT/CS/B9/D0/CX/D4/B9/D1/D3/D9/D8/CW/D4/CX/CT\r/CT /D7/DD/D7/D8/CT/D1 /D6/CT/D7/D3/D2/CP/D2\r/CT /CU/D6/CT/B9/D5/D9/CT/D2\r/DD /B8 /D8/CW/CT /CS/CT/DA/CX/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CX/D7 /D0/CP/D8/D8/CT/D6 /CQ /CT/CX/D2/CV /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT/D0/D3/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /CQ /D3/D6/CT/BA/BY/BA /C5/CX/D2/CX/D1/D9/D1 /D4 /D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/BM /C1/D1/D4 /D6/D3/DA/CT/CS /D4/D0/CP /DD /CP/CQ/CX/D0/CX/D8 /DD /CU/D3 /D6/CX/D2/D8/CT/D6/CP\r/D8/CX/D2/CV /D6/CT/D7/D3/D2/CP/D2\r/CT/D7/BY /D3/D6 /CT/CP\r /CW /DA /CP/D0/D9/CT /D3/CUqr\n/B8 /D8/CW/CT/D6/CT /CT/DC/CX/D7/D8/D7 /D3/D2/CT /D3/D6 /D1/D3/D6/CT /D6/CP/D2/CV/CT/D7 /D3/CU/CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7 /DB/CW/CT/D6/CT /D4/D0/CP /DD /CP/CQ/CX/D0/CX/D8 /DD /CX/D7 /CV/D6/CT/CP/D8/D0/DD /CX/D1/D4/D6/D3 /DA /CT/CS/BA /C1/D2/B9/CS/CT/CT/CS/B8 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS \r/D9/D6/DA /CT/D7 /D7/CW/D3 /DB /CP /D1/CX/D2/CX/D1 /D9/D1 /CU/D3/D6 /CP\r/CT/D6/D8/CP/CX/D2 /DA /CP/D0/D9/CT /D3/CUkrL /B8 /CS/CT/D2/D3/D8/CX/D2/CV /CP/D2 /CX/D2\r/D6/CT/CP/D7/CT/CS /CT/CP/D7/CX/D2/CT/D7/D7 /D8/D3/D4/D6/D3 /CS/D9\r/CT /D8/CW/CT /D2/D3/D8/CT \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CX/D7 /D0/CT/D2/CV/D8/CW/BA /BT/D7/D7/D3 \r/CX/B9/CP/D8/CX/D2/CV /CP \r/D0/CP/D6/CX/D2/CT/D8 /D1/D3/D9/D8/CW/D4/CX/CT\r/CT /DB/CX/D8/CW /CP /D8/D6/D3/D1 /CQ /D3/D2/CT /D7/D0/CX/CS/CT/B8 /CX/D2/B9/CU/D3/D6/D1/CP/D0 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 \r/D3/D2/AS/D6/D1 /D8/CW/CP/D8 /CX/D8 /CX/D7 /CT/CP/D7/CX/CT/D6 /D8/D3 /D4/D6/D3 /CS/D9\r/CT/D7/D3/D1/CT /D2/D3/D8/CT/D7 /D8/CW/CP/D2 /D3/D8/CW/CT/D6 /D3/D2/CT/D7/BA /BT/D2/CP/D0/DD/D8/CX\r/CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /CT/DC/B9/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CW/CP /DA /CT /CQ /CT/CT/D2 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA /CD/D2/B9/CS/CT/D6 /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CX/D7 /D1/CX/D2/CX/D1/CP/D0 /DA /CP/D0/D9/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS/CU/D3/D6 /CP/D2 /CT/D1/CT/D6/CV/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2\r/DD /D0/D3 \r/CP/D8/CT/CS \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/B9/D2/CP/D2\r/CT/B8 /CP/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /CX/D7 /D1/CP/CX/D2/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D6/CT/CT/CS /CS/CP/D1/D4/B9/CX/D2/CV/B8 /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 \r/CP/D2 /CQ /CT /CX/CV/D2/D3/D6/CT/CS /CA/CT(Ye(ω)) = 0 /B8 /BX/D5/BA /B4/BL/B5/D0/CT/CP/CS/CX/D2/CV /D8/CW /D9/D7 /D8/D3/BM\nγ=1\n1+2 /CA/CT(D(θ)). /B4/BD/BE/B5/C1/D2 /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS /D1/D3 /CS/CT/D0 /B4D(θ) = 1 /B5/B8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /D4/D6/CT/D7/B9/D7/D9/D6/CT /CX/D7 /CT/D5/D9/CP/D0 /D8/D31/3 /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CU/D6/CT/B9/D5/D9/CT/D2\r/CX/CT/D7 /CU/D3/D6 /DB/CW/CX\r /CW /D8/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8 /D3/CU /D8/CW/CT /CQ /D3/D6/CT /CX/D2/D4/D9/D8/CX/D1/D4 /CT/CS/CP/D2\r/CT /DA /CP/D2/CX/D7/CW/CT/D7/B8 /DB/CW/CX\r /CW /CX/D7 \r/D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D6/CT/D7/D9/D0/D8/D7 /CP/D0/B9/D6/CT/CP/CS/DD /D4/D9/CQ/D0/CX/D7/CW/CT/CS\n/BF/BG/BA /CC/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3 \r/B9\r/D9/D6/D7 /CP/D8 /CP /D1/CP/DC/CX/D1 /D9/D1 /D3/CU /CA/CT(D(θ)) /BM/CA/CT(D(θ)) =1−θ2\n(1−θ2)2+(qrθ)2, /B4/BD/BF/B5/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6θ=√1−qr\n/B4/DB/CW/CX\r /CW /CX/D7 \r/D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 θ≃1 /B5/B8 /D8/CW /D9/D7/B8\nγ0=qr(2−qr)\n2+qr(2−qr), /B4/BD/BG/B5/D2/CT/CP/D6/D0/DD /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3qr\n/CU/D3/D6 /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/D7/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /CT/DA /CP/D0/D9/CP/D8/CT /D8/CW/CT /CT/AR/CT\r/D8 /D3/CU /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /D3/D2 /D8/CW/CX/D7/D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CP /D3/D2/CT/B9/D1/D3 /CS/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /DB/CX/D8/CW/D0/D3/D7/D7/CT/D7 /CX/D7 /D2/D3 /DB \r/D3/D2/D7/CX/CS/CT/D6/CT/CS/BM\nYe(ω) =Yn/parenleftbigg\n1+jQn/parenleftbiggω\nωn−ωn\nω/parenrightbigg/parenrightbigg\n, /B4/BD/BH/B5/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BG/DB/CW/CT/D6/CTYn\n/CX/D7 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D8/CW/CT /CP/CS/D1/CX/D8/D8/CP/D2\r/CT/CP/D2/CSQn\n/CX/D7 /D8/CW/CT /D5/D9/CP/D0/CX/D8 /DD /CU/CP\r/D8/D3/D6/BN /BX/D5/D7/BA /B4/BK/B5 /CP/D2/CS /B4/BL/B5 /CQ /CT\r/D3/D1/CT\nYn+ζ1−γ\n2√γ=ζ√γ1−θ2\n(1−θ2)2+(qrθ)2, /B4/BD/BI/B5\nYnQn/parenleftbiggθ\nθn−θn\nθ/parenrightbigg\n=−ζ√γqrθ\n(1−θ2)2+(qrθ)2. /B4/BD/BJ/B5/C1/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CUYn\n/DB/CX/D8/CW /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CQ /D3/D6/CT/B8/D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BD/BI /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3krL /D0/CT/CP/CS/D7 /D8/D3/CP /D1/CX/D2/CX/D1 /D9/D1 /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CU/D9/D2\r/D8/CX/D3/D2 γ=f(krL) /CU/D3/D6θ2/D1/CX/D2=\n1−qr\n/CP/D2/CSγ/D1/CX/D2\n/D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /CT/D5/D9/CP/D8/CX/D3/D2\nYn\nζ√γ/D1/CX/D2+1−γ/D1/CX/D2\n2γ/D1/CX/D2=1\nqr(2−qr), /B4/BD/BK/B5/DB/CW/CX\r /CW /D8/CW/CT /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nγmin≃γ0/parenleftbigg\n1+2Yn\nζ√γ0/parenrightbigg/B4/BD/BL/B5/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6\nωn=ωr/parenleftbigg\n1−qr\n2+1\n2Qn+ζ\n2YnQn√qr/parenrightbigg\n. /B4/BE/BC/B5/BY /D3/D6 /CP/D2 /D3/D4 /CT/D2/BB\r/D0/D3/D7/CT/CS \r/DD/D0/CX/D2/CS/CT/D6 ωn= (2n−1)πc/2L /B8 /D8/CW/CT/D6/CT/D7/D9/D0/D8 /CX/D7\n(krL)min≃(2n−1)π\n2/parenleftbigg\n1+qr\n2−1\n2Qn−ζ\n2YnQn√qr/parenrightbigg\n./B4/BE/BD/B5/CC /DD/D4/CX\r/CP/D0 /DA /CP/D0/D9/CT/D7Yn= 1/25 /B8ζ= 0.4 /B8 /CP/D2/CSqr= 0.4 /D0/CT/CP/CS /D8/D3/CP/D2 /CX/D2\r/D6/CT/CP/D7/CT /CX/D2γmin\n/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3γ0\n/D3/CU /CP/CQ /D3/D9/D88% /B8 \r/D3/D2/AS/D6/D1/CX/D2/CV/D8/CW/CT /D4/D6/CT/D4 /D3/D2/CS/CT/D6/CP/D2 /D8 /CT/AR/CT\r/D8 /D3/CU /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1/D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /CW/D3 /DB \r/D3/D9/D4/D0/CX/D2/CV /CP\r/D3/D9/D7/D8/CX\r/CP/D0 /CP/D2/CS/D1/CT\r /CW/CP/D2/CX\r/CP/D0 /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 \r/D3/D9/D0/CS /D6/CT/CS/D9\r/CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D3 /DA /CT/D6 /CP /DB/CX/CS/CT/D6 /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /D6/CP/D2/CV/CT/B8 /D8/CW/CT /D2/CT/CX/CV/CW /CQ /D3/D6/CW/D3 /D3 /CS /D3/CU /D8/CW/CT/D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D1/CX/D2/CX/D1 /D9/D1 /CW/CP/D7 /CQ /CT/CT/D2 /D7/D8/D9/CS/CX/CT/CS/BA /CD/D7/CX/D2/CV/CP/CV/CP/CX/D2 /CP /D7/CX/D2/CV/D0/CT /CP\r/D3/D9/D7/D8/CX\r/CP/D0 /D1/D3 /CS/CT /CU/D3/D6 /D8/CW/CT \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/B8 /CS/CT/D6/CX/DA /CP/B9/D8/CX/D3/D2 /D3/CU /CP /D4/CP/D6/CP/CQ /D3/D0/CX\r /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /DB /CP/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CF /D6/CX/D8/CX/D2/CV\nγ=γ/D1/CX/D2(1+ε2) /B8θ2=θ2/D1/CX/D2+δ /CP/D2/CSωn= (ωn)/D1/CX/D2(1+ν) /B8/DB/CX/D8/CWε /B8δ /B8 /CP/D2/CSν /D7/D1/CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /D8/CW/CT /CC /CP /DD/D0/D3/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D3/CU /BX/D5/D7/BA /B4/BD/BI /B5 /CP/D2/CS /B4/BD/BJ /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT/D7/CT /DA /CP/D0/D9/CT/D7 /D0/CT/CP/CS/D7/D8/D3 /D8/CW/CT /D2/CT/DC/D8 /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/D7/BM\nε2∼δ2/(2q2\nr), /B4/BE/BE/B5\n2q2\nrYnQn(δ−2ν) =−δζ√qr. /B4/BE/BF/B5/BY/CX/D2/CP/D0/D0/DD /B8 /D2/CT/CP/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CS/CT/D4 /CT/D2/B9/CS/CT/D2\r/CT /D3/D2 /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nγ=γ/D1/CX/D2\n1+2qr/parenleftbigg\nq3/2\nr+ζ\n2YnQn/parenrightbigg2/parenleftbiggkrL−krL/D1/CX/D2\nkrL/D1/CX/D2/parenrightbigg2\n./B4/BE/BG/B5/C1/D2 /CP /AS/D6/D7/D8 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CP/D4 /CT/D6/D8/D9/D6/CT /D3/CU /D8/CW/CT /CP/D4/D4/D6/D3 /DC/CX/B9/D1/CP/D8/CT/CS /D4/CP/D6/CP/CQ /D3/D0/CP/B8 /D8/CW /D9/D7 /D8/CW/CT /DB/CX/CS/D8/CW /D3/CU /D8/CW/CT /D6/CP/D2/CV/CT /CU/D3/D6 /DB/CW/CX\r /CW/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D7 /D0/D3 /DB /CT/D6/CT/CS/B8 /CX/D7 /D1/CP/CX/D2/D0/DD \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD\n/D8/CW/CT /D1 /D9/D7/CX\r/CX/CP/D2 /CT/D1 /CQ /D3/D9\r /CW /D9/D6/CT/B8 /CX/BA/CT/BA/B8 /CQ /DD /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /CP/D2/CS /D0/CX/D4/D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8/B8 /D8/CW/CP/D2/CZ/D7 /D8/D3 /CX/D8/D7 /CT/D1/B9/CQ /D3/D9\r /CW /D9/D6/CT/B8 /D8/CW/CT /D4/D0/CP /DD /CT/D6 \r/CP/D2 /CT/DC/D4 /CT\r/D8 /CP/D2 /CT/CP/D7/CX/CT/D6 /D4/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU/D8/D3/D2/CT/D7 /CU/D3/D6 \r/CT/D6/D8/CP/CX/D2 /D2/D3/D8/CT/D7/BA/BY /D3/D6 /CP /D0/D3/D7/D7/DD \r/DD/D0/CX/D2/CS/D6/CX\r/CP/D0 /D3/D4 /CT/D2/BB\r/D0/D3/D7/CT/CS /CQ /D3/D6/CT/B8 /D1/D3 /CS/CP/D0 /CT/DC/B9/D4/CP/D2/D7/CX/D3/D2 /D3/CU /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2\r/CT /CV/CX/DA /CT/D7YnQn=ωnL/2c=\n(2n−1)π/4 /D7/D3 /D8/CW/CP/D8 /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /CS/D3 /D2/D3/D8 /D7/CT/CT/D1 /D8/D3 /CW/CP /DA /CT /CP/CV/D6/CT/CP/D8 /CX/D2/AT/D9/CT/D2\r/CT /D3/D2 /D4/D0/CP /DD/CX/D2/CV /CU/CP\r/CX/D0/CX/D8 /DD /B8 /CP/D8 /D0/CT/CP/D7/D8 /DB/CW/CT/D2 \r/D3/D2/D7/CX/CS/B9/CT/D6/CX/D2/CV /D1/CX/D2/CX/D1/CP/D0 /CQ/D0/D3 /DB/CX/D2/CV /D4/D6/CT/D7/D7/D9/D6/CT γmin\n/B4Qn\n/CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6/CP/D0/D3/D2/CT /CX/D2 /AS/D6/D7/D8 /D3/D6/CS/CT/D6 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/B5/BA /C7/D2 /D8/CW/CT \r/D3/D2 /D8/D6/CP/D6/DD /B8 /D8/CW/CT/DD/CP/D6/CT /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CU/D3/D6 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D8/CW/CT /CT/DC/D8/CX/D2\r/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2\n/BF/BH/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2 /D8/CW/CT /D6/CT/CT/CS /CX/D7 /CW/CT/D0/CS /D1/D3/D8/CX/D3/D2/D0/CT/D7/D7/D0/DD/CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D0/CP /DD /BA/C1 /C1 /C1/BA /C5/C7/BW/BX/C4 /C1/C5/C8/CA/C7 /CE/BX/C5/BX/C6/CC/CB/C4/CP/D7/D8 /CU/D3/D9/D6 /CS/CT\r/CP/CS/CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /CU/D6/D9/CX/D8/CU/D9/D0 /CX/D2 /D4/CW /DD/D7/CX\r/CP/D0 /D1/D3 /CS/B9/CT/D0/CX/D2/CV /D3/CU /D1 /D9/D7/CX\r/CP/D0 /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7/B8 /CT/D7/D4 /CT\r/CX/CP/D0/D0/DD /CU/D3/D6 /D7/CX/D2/CV/D0/CT /D6/CT/CT/CS/CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7/BA /C8/CX/D4 /CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /D8/CW/CT /CU/D3 \r/D9/D7 /D3/CU /CP /CV/D6/CT/CP/D8 /D2 /D9/D1/B9/CQ /CT/D6 /D3/CU /D7/D8/D9/CS/CX/CT/D7 /D7/CX/D2\r/CT /BU/CT/D2/CP/CS/CT\n/BF/BI/B8 /CP/D7 /DB /CT/D0/D0 /CP/D7 /D8/CW/CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU/D4 /CT\r/D9/D0/CX/CP/D6/CX/D8/CX/CT/D7 /D3/CU /D8/CW/CT /AT/D3 /DB /B4/BU/CP\r /CZ/D9/D7\n/BF/B8 /C0/CX/D6/D7\r /CW /CQ /CT/D6/CV\n/BE/BI/B8 /CP/D2/CS /BW/CP/D0/B9/D1/D3/D2 /D8 /CT/D8 /CP/D0/BA\n/BE/BJ/B5/BA /CC/CW/CT /CP/CX/D1 /CW/CT/D6/CT /CX/D7 /D8/D3 /D8/D6/DD /D8/D3 /D6/CT/CS/D9\r/CT /CS/CX/D7\r/D6/CT/D4/CP/D2/B9\r/CX/CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CP/D2/CS /D8/CW/CT/D3/D6/DD /B8 /CQ/CP/D7/CT/CS /D3/D2 /D7/D3/D1/CT/D3/CU /D8/CW/D3/D7/CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /DB/CW/CX\r /CW /D0/D3 /D3/CZ /D6/CT/D0/CT/DA /CP/D2 /D8 /D8/D3 /D8/CW/CT /D7/D8/D9/CS/DD /D3/CU/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /C1/D2 /CP /D6/CT/CP/D0 \r/D0/CP/D6/CX/D2/CT/D8/B9/D4/D0/CP /DD /CT/D6 /D7/DD/D7/D8/CT/D1/B8 /D8/CW/CT/DA /D3 \r/CP/D0 /D8/D6/CP\r/D8 /D1/CP /DD /CW/CP /DA /CT /CP/D2 /CT/AR/CT\r/D8 /D3/D2 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BF/BJ /B5/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /D9/D7/CT /D3/CU /D4 /D3/D6/D3/D9/D7 /D1/CP/D8/CT/D6/CX/CP/D0/CX/D2 /D8/CW/CT /CF/BU /CP/D4/D4/CP/D6/CP/D8/D9/D7 /D7/D9/CV/CV/CT/D7/D8/D7 /D8/CW/CP/D8 /CP\r/D3/D9/D7/D8/CX\r /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/CQ /CT/D8 /DB /CT/CT/D2 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/AS\r/D8/CX/D8/CX/D3/D9/D7 /DA /D3/D0/D9/D1/CT /DB/CW/CT/D6/CT /D8/CW/CT/D6/CT/CT/CS /D7/DB/CX/D2/CV/D7/BA /BW/CP/D0/D1/D3/D2 /D8 /CT/D8 /CP/D0/BA\n/BD/BF/D6/CT/D4 /D3/D6/D8/CT/CS /D8 /DD/D4/CX\r/CP/D0 /DA /CP/D0/D9/CT/D7/D3/CU10mm /CU/D3/D6 /CP \r/D0/CP/D6/CX/D2/CT/D8/BA /C6/CT/CS/CT/D6/DA /CT/CT/D2\n/BG/BD/D0/CX/D2/CZ /CT/CS∆l /D8/D3 /D6/CT/CT/CS/D7/D8/D6/CT/D2/CV/D8/CW /B4/D3/D6 /CW/CP/D6 /CS/D2/CT/D7/D7 /B5/BM∆l /D1/CP /DD /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/D0/DD /DA /CP/D6/DD /CU/D6/D3/D1\n6mm /B4/D7/D8/D6/D3/D2/CV /D6/CT/CT/CS/D7/B5 /D8/D39mm /B4/D7/D3/CU/D8/CT/D6 /D6/CT/CT/CS/D7/B5/BA /CC/CW/CT/D7/CT /DA /CP/D0/D9/CT/D7/CQ /CT/CX/D2/CV /D7/D1/CP/D0/D0 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 \r/D0/CP/D6/CX/D2/CT/D8 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/B8 /D8/CW/CT /D6/CT/CT/CS/D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB \r/CP/D2 /CQ /CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/CW/D6/D3/D9/CV/CW /CP /D1/CT/D6/CT/D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2 /CX/D2 \r/D3/D1/D1/D3/D2 /DB /D3/D6/CZ/B8 /CQ/D9/D8 /CX/D8/D7 /CX/D2/AT/D9/CT/D2\r/CT /D3/D2/D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CP/D2/CS /D6/CT/CT/CS /CX/D7/D2/D3/D8 /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /D3/D2 /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /CP/D7 /CX/D8 /CX/D7 /D7/D8/D9/CS/CX/CT/CS/D2/D3 /DB/BA/C1/D2 /CP /AS/D6/D7/D8 /D7/D8/CT/D4/B8 /D8/CW/CX/D7 /CT/AR/CT\r/D8 /CX/D7 \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /B8 /CP/D0/D0/D0/D3/D7/D7/CT/D7 /CQ /CT/CX/D2/CV /CX/CV/D2/D3/D6/CT/CS /B4qr= 0 /CP/D2/CSη= 0 /B5/BA /BX/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BJ /B5\r/D3/D9/D4/D0/CT/CS /D8/D3 /BX/D5/BA /B4/BJ /B5 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D7/DD/D7/D8/CT/D1/BM\n− /C1/D1(Ye) =kr∆lθ\n1−θ2, /B4/BE/BK/B5\n1\n1−θ2−1−γ\n2γ= 0⇔γ=1−θ2\n3−θ2. /B4/BE/BL/B5/BT/D7 /D7/CT/CT/D2 /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /B8 /D7/CT/DA /CT/D6/CP/D0 /CU/D6/CT/D5/D9/CT/D2\r/DD /D7/D3/D0/D9/D8/CX/D3/D2/D7 θ /CT/DC/CX/D7/D8 /CU/D3/D6/CS/CX/AR/CT/D6/CT/D2 /D8 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D7 γ /BA /BX/DC/CP/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BE/BL /B5/CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 γ=f(θ) /B4/CU/D3/D6θ <1 /B5 /D6/CT/DA /CT/CP/D0/D7 /D8/CW/CP/D8 /D8/CW/CT /D7/D3/B9/D0/D9/D8/CX/D3/D2 /CW/CP /DA/CX/D2/CV /D8/CW/CT /D0/D3 /DB /CT/D7/D8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D7 /D8/CW/CT /D3/D2/CT /CQ /CT/CX/D2/CV /D8/CW/CT\r/D0/D3/D7/CT/D7/D8 /D8/D3 /D8/CW/CT /D6/CT/CT/CS /CU/D6/CT/D5/D9/CT/D2\r/DD /BA/BT/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D7 \r/CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS /CX/D2 /D7/D3/D1/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7/BA/CF/CW/CT/D2 /D4/D0/CP /DD/CX/D2/CV \r/D0/D3/D7/CT /D8/D3 /CP /CQ /D3/D6/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /CU/D6/CT/D5/D9/CT/D2\r/DD θn≪/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BI/BY/C1/BZ/BA /BG/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CT/AR/CT\r/D8 /CP/D2/CS/D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /CT/AR/CT\r/D8 /B4∆l= 12mm /CP/D2/CS /D3/D8/CW/CT/D6 \r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7/CX/D2 /BY/CX/CV/BA /BD/B5/BM /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /D3/D2/D0/DD /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7/B5/B8/D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2/D0/DD /B4/CS/CP/D7/CW/B9/CS/D3/D8/D8/CT/CS /D0/CX/D2/CT/D7/B5/B8 /CP/D2/CS /CQ /D3/D8/CW /CT/AR/CT\r/D8/D7 /B4/D4/D0/CP/CX/D2/D0/CX/D2/CT/D7/B5/BA /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT\r/CP/D0/D0/CT/CS /B4/D7/D5/D9/CP/D6/CT/D7/B5/BA\n1 /B8 /D8/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D8/CT/D6/D1 /CX/D2 /BX/D5/BA /B4/BE/BK /B5 /CX/D7 /D7/D1/CP/D0/D0/B8 /D7/D3 /D8/CW/CP/D8/B8 /DB/CX/D8/CW\nYe=−jcot(θkrL) /B8 /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CP\r/D8/D7/D1/CT/D6/CT/D0/DD /CP/D7 /CP /D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2/BM\n∆l1\n1−θ2n\n/DB/CW/CT/D6/CTθn=(2n−1)π\n2krL. /B4/BF/BC/B5/CC/CW/CX/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CX/D7 /DA /CP/D0/CX/CS /DB/CW/CT/D2 /D8/CW/CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /CQ /D3/D6/CT/CU/D6/CT/D5/D9/CT/D2\r/DD θn\n/D6/CT/D1/CP/CX/D2/D7 /D7/D1/CP/D0/D0/CT/D6 /D8/CW/CP/D2 /D9/D2/CX/D8 /DD /BA /BT/D2 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2 \r/CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS /CU/D3/D6 /D8/CW/CT/CT/AR/CT\r/D8 /D3/CU /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CU/D6/D3/D1 /BX/D5/BA /B4/BK/B5/BM\n∆lq≃ζqr√\n3kr. /B4/BF/BD/B5/BT /D8/D6/CX/CP/D0 /CP/D2/CS /CT/D6/D6/D3/D6 /D4/D6/D3 \r/CT/CS/D9/D6/CT /CW/CP/D7 /CQ /CT/CT/D2 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /D8/D3 /CP/CS/CY/D9/D7/D8/D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /CE /CT/D6/DD /CU/CT/DB /CX/D8/CT/D6/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT /D2/CT/CT/CS/CT/CS/D8/D3 /CT/DC/CW/CX/CQ/CX/D8 /CP /DA /CP/D0/D9/CT /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /B4∆l≃\n12mm /B5 /CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /B4∆lq≃2mm /B5/CX/D2 /D8/CW/CT \r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/CU /BY/CX/CV/D7/BA /BD /CP/D2/CS /BG/BA /CF/CW/CT/D2 /CP\r/D3/D9/D7/D8/CX\r/CP/D0 /CP/D2/CS/D1/CT\r /CW/CP/D2/CX\r/CP/D0 /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /CP/D6/CT /DA /CT/D6/DD \r/D0/D3/D7/CT /B4θ= 1−ε /CP/D2/CSθn=\n1−εn\n/B5/B8 /CP /D7/CT\r/D3/D2/CS/B9/D3/D6/CS/CT/D6 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 \r/CP/D2 /CQ /CT /CS/CT/CS/D9\r/CT/CS/BM\nε=εn\n2/parenleftigg\n1+/radicaligg\n1+2∆l\nLε2n/parenrightigg\n, /B4/BF/BE/B5/D8/CW/CT /CP/D4/D4/CP/D6/CX/D8/CX/D3/D2 /D3/CU /CP /D7/D5/D9/CP/D6/CT /D6/D3 /D3/D8 /CQ /CT/CX/D2/CV /D8 /DD/D4/CX\r/CP/D0 /D3/CU /D1/D3 /CS/CT \r/D3/D9/B9/D4/D0/CX/D2/CV/B8 /D1/CP/CZ/CX/D2/CV /D8/CW/CT /CP\r /CW/CX/CT/DA /CT/D1/CT/D2 /D8 /D3/CU /CP/D2/CP/D0/DD/D8/CX\r/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7/CS/CXꜶ\r/D9/D0/D8/BA /CC/CW/CT/D2/B8 /DB/CW/CT/D2 /D8/CW/CT /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CS/CT\r/D6/CT/CP/D7/CT/D7 /CT/D2/D3/D9/CV/CW/D7/D3 /D8/CW/CP/D8 /D3/D2/CT /D3/CU /CX/D8/D7 /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /CX/D2\r/D6/CT/CP/D7/CT/D7 /CP/CQ /D3 /DA /CT /D8/CW/CT /D6/CT/CT/CS /D3/D2/CT/B4θn>1 /B5/B8 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D4/D4/D6/D3/CP\r /CW/CT/D7 /D8/CW/CT /D6/CT/CT/CS/D3/D2/CT /D9/D2 /D8/CX/D0 /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT\r/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CS/CX/D7/CP/D4/D4 /CT/CP/D6/D7 /CU/D3/D6krL=nπ/B4/D7/CT/CT /BY/CX/CV/BA /BH /B5/BA /C6/CT/CP/D6 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/B8 /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /CP/D4/D4/D6/D3 /DC/B9/CX/D1/CP/D8/CX/D3/D2/D7 \r/CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS/BM\nθ≃1−1\n2kr∆ltan(krL), /B4/BF/BF/B5\nγ≃1\n2kr∆ltan(krL). /B4/BF/BG/B51rL k2/ π=θ1rL k2/ π2 =θ13rL k2/ π3 =θ\nθ0r)2θ−1 (/ l ∆ k θ r)L k θ(t o c/BY/C1/BZ/BA /BH/BA /BZ/D6/CP/D4/CW/CX\r/CP/D0 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BE/BK /B5 /CV/CX/DA/CX/D2/CV /D3/D7\r/CX/D0/D0/CP/B9/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D8 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BM /D0/CT/CU/D8/B9/CW/CP/D2/CS /D8/CT/D6/D1− /C1/D1(Y) /B4/D8/CW/CX\r /CZ/D0/CX/D2/CT/D7/B5/B8 /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D8/CT/D6/D1kr∆lθ/(1−θ2) /B4/D8/CW/CX/D2 /D0/CX/D2/CT/B5/B8 /CP/D2/CS /D7/D3/D0/D9/B9/D8/CX/D3/D2/D7 /B4/D1/CP/D6/CZ /CT/D6/D7/B5/BA/CC/CW/CT/D7/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CP/D6/CT /DA /CP/D0/CX/CS /CX/CUtan(krL)/greaterorsimilar0 /B8 /CX/BA/CT/BA/B8krL/greaterorsimilar\nnπ /BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /BX/D5/BA /B4/BE/BL /B5/B8 /DB/CW/CT/D2 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D4/B9/D4/D6/D3/CP\r /CW/CT/D7 fr\n/B8 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CS/CT\r/D6/CT/CP/D7/CT/D7 /D8/D3 /DE/CT/D6/D3 \r/D3/D2/B9/D8/D6/CP/D6/DD /D8/D3 /DB/CW/CP/D8 /CW/CP/D4/D4 /CT/D2/D7 /DB/CW/CT/D2 \r/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT /D6/CT/CT/CS /CS/CP/D1/D4/B9/CX/D2/CV /CT/AR/CT\r/D8/BA/BY/CX/CV/D9/D6/CT/D7 /BG /CP/D2/CS /BI /D7/CW/D3 /DB /CP /D2 /D9/D1/CT/D6/CX\r/CP/D0 \r/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT/D6/CT/D7/D4 /CT\r/D8/CX/DA /CT /CT/AR/CT\r/D8/D7 /D3/CU /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CP/D2/CS /D6/CT/CT/CS/CS/CP/D1/D4/CX/D2/CV/BA /CC/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /DB/CX/D8/CW∆l=\n12mm /CX/D2 /BY/CX/CV/BA /BG /CP/D2/CS5mm /CX/D2 /BY/CX/CV/BA /BI /CP/CS/CY/D9/D7/D8/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD/CS/CT/DA/CX/CP/D8/CX/D3/D2 /CU/D3/D6 /CQ /D3/D8/CW /CW/CT/CP /DA/CX/D0/DD /CP/D2/CS /D7/D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/D7/B8/CT/DA /CT/D2 /DB/CW/CT/D2 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/B8 /CP/D2/CS /CX/D7 /D4/D6/CT/D4 /D3/D2/B9/CS/CT/D6/CP/D2 /D8 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /CT/AR/CT\r/D8/BA /BT/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU/CP /D1/CX/D7/D8/D9/D2/CT/CS /D0/CT/D2/CV/D8/CW \r/D3/D6/D6/CT\r/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B4∆l= 12mm /B5/CX/D7 /CT/DC/CW/CX/CQ/CX/D8/CT/CS /CX/D2 /BY/CX/CV/BA /BI /B8 /D6/CT/DA /CT/CP/D0/CX/D2/CV /CP /D1/CX/D7/D1/CP/D8\r /CW /D3/CU /D3/D7\r/CX/D0/D0/CP/B9/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /CC/CW/CX/D7 /D0/CP/D8/D8/CT/D6 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /D1/CP/CX/D2/D0/DD \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS/CQ /DD /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CT/AR/CT\r/D8 /DB/CW/CT/D2 /CP/D4/D4/D6/D3/CP\r /CW/B9/CX/D2/CV /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /BV/D0/CP/D7/D7/CX\r/CP/D0 /CP/D4/D4/D6/D3/CP\r /CW/CT/D7/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BG/BD /B5 /CU/D3/D6 /D8/CW/CT \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /D4/D0/CP /DD/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2/B9\r/CX/CT/D7/B8 /CX/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT /AT/D3 /DB /CS/D9/CT /D8/D3 /D4/D6/CT/D7/D7/D9/D6/CT /CS/D6/D3/D4/B8 /CP/D2/CS /D7/CT/CP/D6\r /CW/B9/CX/D2/CV /CU/D3/D6 /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /D3/CU /D8/CW/CT /D4/CP/D7/D7/CX/DA /CT /D7/DD/D7/D8/CT/D1 /CX/D2\r/D0/D9/CS/CX/D2/CV/D8/CW/CT /CQ /D3/D6/CT /CP/D2/CS /D8/CW/CT /D6/CT/CT/CS /D3/D2/D0/DD /CP/D6/CT /D8/CW /D9/D7 /CY/D9/D7/D8/CX/AS/CT/CS/BA /C1/D8 \r/CP/D2 /CQ /CT/D2/D3/D8/CX\r/CT/CS /D8/CW/CP/D8 /BX/D5/BA /B4/BE/BK/B5 /DB /CP/D7 /CP/D0/D6/CT/CP/CS/DD /CV/CX/DA /CT/D2 /CQ /DD /CF /CT/CQ /CT/D6\n/BG/BF/CX/D2/D8/CW/CT /CT/CP/D6/D0/DD /BD/BL/D8/CW \r/CT/D2 /D8/D9/D6/DD /B4/D7/CT/CT /D4/CP/CV/CT /BE/BD/BI/B5/B8 /CP/D7/D7/D9/D1/CX/D2/CV /CP /D6/CT/CT/CS/CP/D6/CT/CP /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT \r/DD/D0/CX/D2/CS/D6/CX\r/CP/D0 /D8/D9/CQ /CT /D7/CT\r/D8/CX/D3/D2/BA /CC/CW/CX/D7 /D8/CW/CT/D3/D6/DD /B8/D9/D7/CT/CS /CQ /DD /D7/CT/DA /CT/D6/CP/D0 /CP/D9/D8/CW/D3/D6/D7 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BF/BE /B5/B8 /DB /CP/D7 /CS/CX/D7\r/D9/D7/D7/CT/CS/CQ /DD /DA /D3/D2 /C0/CT/D0/D1/CW/D3/D0/D8/DE\n/BF/BF/CP/D2/CS /BU/D3/D9/CP/D7/D7/CT\n/BG/BG/B8 /CT/D7/D4 /CT\r/CX/CP/D0/D0/DD \r/D3/D2\r/CT/D6/D2/CX/D2/CV/D8/CW/CT /D0/CP\r /CZ /D3/CU /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D4/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU /D7/CT/D0/CU/B9/D7/D9/D7/D8/CP/CX/D2/CT/CS/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7/BA /C7/D2 /D8/CW/CT \r/D3/D2 /D8/D6/CP/D6/DD /B8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /D4/D6/CT/D7/D7/D9/D6/CT \r/D9/D6/DA /CT/D7/CT/DC/CW/CX/CQ/CX/D8 /D8/CW/CP/D8 /CQ /D3/D8/CW /CS/CP/D1/D4/CX/D2/CV /CP/D2/CS /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /AT/D3 /DB /CW/CP /DA /CT /CX/D2/B9/AT/D9/CT/D2\r/CT /D3/D2 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D6/CT/D5/D9/CX/D6/CT/CS /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /D8/D3 /D3/D7\r/CX/D0/D0/CP/D8/CT/BA/CB/D3 /CP \r/D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/CW/CT/D2/D3/D1/CT/D2/CP /CW/CP/D7 /D8/D3 /CQ /CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3/CP\r\r/D3/D9/D2 /D8/B8 /D2/D3/D2/CT /D3/CU /D8/CW/CT/D1 /CQ /CT/CX/D2/CV /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /CX/D2 /D8/CW/CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS/CS/D3/D1/CP/CX/D2/BA/C1/CE/BA /BZ/C7/C1/C6/BZ /BU/BX/CH/C7/C6/BW /C1/C6/CB/CC /BT/BU/C1/C4/C1/CC/CH /CC/C0/CA/BX/CB/C0/C7/C4/BW/BT/BA /C4/CX/D2/CT/CP /D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CP/D0/DD/D7/CX/D7 /DB/CX/D8/CW /D1/D3 /CS/CP/D0 /CS/CT\r/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/BT/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS \r/CP/D2 /CQ /CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS/D9/D7/CX/D2/CV \r/D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2\r/DD /CU/D3/D6/D1/CP/D0/CX/D7/D1/BA /BY /D3/D6 /CP /CV/CX/DA /CT/D2 \r/D3/D2/AS/CV/B9/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1 /CQ /D3/D6 /CT/B9/D6 /CT /CT /CS/B9/D1/D9/D7/CX\r/CX/CP/D2 /B4L /B8S /B8/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BJ/BY/C1/BZ/BA /BI/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS /AT/D3 /DB /CT/AR/CT\r/D8 /CP/D2/CS /D6/CT/CT/CS/CS/CP/D1/D4/CX/D2/CV /CT/AR/CT\r/D8 /B4\r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7 /CX/D2 /BY/CX/CV/BA /BE/B5/BM /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9\r/CT/CS/AT/D3 /DB /D3/D2/D0/DD /B4∆l= 5mm /BM /D4/D0/CP/CX/D2 /D8/CW/CX/D2 /D0/CX/D2/CT/D7/B8∆l= 12mm /BM /CS/D3/D8/D8/CT/CS/D0/CX/D2/CT /CX/D2 /D9/D4/D4 /CT/D6 /CV/D6/CP/D4/CW /D3/D2/D0/DD/B5/B8 /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2/D0/DD /B4/CS/CP/D7/CW/CT/CS /D8/CW/CX/D2/D0/CX/D2/CT/D7/B5/B8 /CP/D2/CS /CQ /D3/D8/CW /CT/AR/CT\r/D8/D7 /B4/D4/D0/CP/CX/D2 /D8/CW/CX\r /CZ /D0/CX/D2/CT/D7/B5/BA /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT/CP\r/CP/D0/D0/CT/CS /B4/D7/D5/D9/CP/D6/CT/D7/B5/BA\nωr\n/B8qr\n/B8γ /B8 /CP/D2/CSζ /CQ /CT/CX/D2/CV /D7/CT/D8/B5/B8 /CX/D8/D7 \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7\nsn=jωn−αn\n\r/CP/D2 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/BA /CC/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8/D3/CUsn\n\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /D8/CW/CT /D6/CT/CP/D0 /D4/CP/D6/D8αn\n/CQ /CT/B9/CX/D2/CV /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /D3/CU /D8/CW/CX/D7 /D1/D3 /CS/CT/B8 /CU/D3/D6 /D8/CW/CT \r/D3/D9/D4/D0/CT/CS /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/D7/DD/D7/D8/CT/D1 \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D7/D8/CP/D8/CX\r /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT/BA /BV/D0/CP/D7/D7/CX\r/CP/D0/D0/DD /B8/DB/CW/CT/D2 /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /CQ /CT/D0/D3 /DB /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS/B8 /CP/D0/D0 /CT/CX/CV/CT/D2/CS/CP/D1/D4/CX/D2/CV/D7 αn\n/CP/D6/CT /D4 /D3/D7/CX/D8/CX/DA /CT/B8 /D8/CW/CT /D7/D8/CP/D8/CX\r /D6/CT/CV/CX/D1/CT/CQ /CT/CX/D2/CV /D7/D8/CP/CQ/D0/CT/BA /BT/D2 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D1/CP /DD /CP/D4/D4 /CT/CP/D6 /DB/CW/CT/D2 /CP/D8 /D0/CT/CP/D7/D8/D3/D2/CT /D1/D3 /CS/CT /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1 /CQ /CT\r/D3/D1/CT/D7 /D9/D2/D7/D8/CP/CQ/D0/CT/B8 /CX/BA/CT/BA/B8/DB/CW/CT/D2 /CP/D8 /D0/CT/CP/D7/D8 /D3/D2/CT /D3/CU /D8/CW/CTαn\n/CQ /CT\r/D3/D1/CT/D7 /D2/CT/CV/CP/D8/CX/DA /CT/BA /C4/D3 /D3/CZ/CX/D2/CV/CU/D3/D6 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS \r/CP/D2 /CQ /CT /CS/D3/D2/CT /CQ /DD /DA /CP/D6/DD/CX/D2/CV /CP /CQ/CX/CU/D9/D6\r/CP/B9/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B4/CT/CX/D8/CW/CT/D6L /B8γ /B8 /D3/D6ζ /B5 /CP/D2/CS /CT/DC/CP/D1/CX/D2/CX/D2/CV /D8/CW/CT /D6/CT/CP/D0/D4/CP/D6/D8 /D3/CU \r/D3/D1/D4/D9/D8/CT/CS /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7/BA /CC /DB /D3 /CT/DC/CP/D1/D4/D0/CT/D7 /CP/D6/CT/D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/D7/BA /BJ /CP/D2/CS /BK /BA /C1/D8 /CX/D7 /D2/D3/D8/CX\r/CT/CP/CQ/D0/CT /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2/B9\r/CX/CT/D7 /CT/DA /D3/D0/DA /CT /D3/D2/D0/DD /D7/D0/CX/CV/CW /D8/D0/DD /DB/CX/D8/CW /D8/CW/CT /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 γ/CP/D2/CS /CP/D6/CT \r/D0/D3/D7/CT /D8/D3 /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /D3/CU /CT/CX/D8/CW/CT/D6 /D8/CW/CT /CQ /D3/D6/CT /D3/D6 /D8/CW/CT/D6/CT/CT/CS /B4 /C1/D1(jω/ωr)≃1 /B5/BA /BY /D3/D6 /D8/CW/CT /AS/D6/D7/D8 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D7/D8/CP/D8/CX\r/D6/CT/CV/CX/D1/CT /CQ /CT\r/D3/D1/CT/D7 /D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6γ≃0.28 /CP/D2/CS /CP /CU/D6/CT/D5/D9/CT/D2\r/DD/D2/CT/CP/D6 /D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D2/CP/D2\r/CT /D3/CU /D8/CW/CT /CQ /D3/D6/CT /B4/CS/D3/D8/B9/CS/CP/D7/CW/CT/CS \r/D9/D6/DA /CT/D7/B5/BA/C7/D8/CW/CT/D6 /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /B4/D8/CW/CT/CX/D6 /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CQ /CT/CX/D2/CV /D3 /CS/CS/D1 /D9/D0/D8/CX/D4/D0/CT/D7 /D3/CU /D8/CW/CT /AS/D6/D7/D8 /D3/D2/CT/B5 /CP/D2/CS /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /CW/CP /DA /CT /CW/CX/CV/CW/CT/D6/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /D6/CT/D1/CP/CX/D2 /CS/CP/D1/D4 /CT/CS /CU/D3/D6 /D8/CW/CX/D7 \r/D3/D2/AS/CV/D9/B9/D6/CP/D8/CX/D3/D2/BA /BY /D3/D6 /CP /D0/D3/D2/CV/CT/D6 /D8/D9/CQ /CT /B4/BY/CX/CV/BA /BK/B5/B8 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D4/D4 /CT/CP/D6/D7 /CU/D3/D6\nγ≃0.3 /CP/D8 /CP /CU/D6/CT/D5/D9/CT/D2\r/DD /D0/D3 \r/CP/D8/CT/CS /D2/CT/CP/D6 /D8/CW/CT /D8/CW/CX/D6/CS /D6/CT/D7/D3/D2/CP/D2\r/CT/D3/CU /D8/CW/CT /CQ /D3/D6/CT /CJ/CS/D3/D8/D8/CT/CS /D0/CX/D2/CT /CP/D8 /C1/D1(jω/ωr)≃0.8≃3×0.27 ℄/B8/D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D2/CP/D2\r/CT /CQ /CT\r/D3/D1/CX/D2/CV /D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6 /CP /D0/CP/D6/CV/CT/D6 /D1/D3/D9/D8/CW/D4/D6/CT/D7/D7/D9/D6/CT/BA /CC/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /B4/D7/D3/D0/CX/CS\r/D9/D6/DA /CT/D7/B5 /D7/D8/CX/D0/D0 /D6/CT/D1/CP/CX/D2/D7 /CS/CP/D1/D4 /CT/CS/BA/BV/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /D1/CP /DD /CQ /CT /D7/CX/D1/D4/D0/CX/AS/CT/CS /CP/D2/CS /CP\r\r/CT/D0/CT/D6/CP/D8/CT/CS /CQ /DD /D9/D7/B9/CX/D2/CV /CP /D1/D3 /CS/CP/D0 /CS/CT\r/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CQ /D3/D6/CT /CX/D1/D4 /CT/CS/CP/D2\r/CT Ze(ω) /BM/D8/CW/CX/D7 /CP/D0/D0/D3 /DB/D7 /CU/D3/D6 /D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /D8/D3 /CQ /CT /DB/D6/CX/D8/D8/CT/D2/CP/D7 /CP /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CUjω /B8 /CP/D2/CS /D3/D4/D8/CX/D1/CX/DE/CT/CS /CP/D0/CV/D3/B9/D6/CX/D8/CW/D1/D7 /CU/D3/D6 /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /D6/D3 /D3/D8 /AS/D2/CS/CX/D2/CV \r/CP/D2 /CQ /CT /D9/D7/CT/CS/BA /C5/D3 /CS/CP/D0/CT/DC/D4/CP/D2/D7/CX/D3/D2 \r/D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CTN /AS/D6/D7/D8 /CP\r/D3/D9/D7/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT/D7 /D3/CU /D8/CW/CT\n/BY/C1/BZ/BA /BJ/BA /BX/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CP/D7 /CP /CU/D9/D2\r/B9/D8/CX/D3/D2 /D3/CU /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /BAr= 7mm /B8ωr= 2π×1000rad/s /B8\nqr= 0.3 /B8L= 16cm /B4θ1= 0.53 /CP/D2/CSkrL= 2.96 /B5/B8 /CP/D2/CSζ= 0.2 /BA\n/BY/C1/BZ/BA /BK/BA /BX/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /CP/D7 /CP /CU/D9/D2\r/B9/D8/CX/D3/D2 /D3/CUγ /BAr= 7mm /B8ωr= 2π×1000rad/s /B8qr= 0.3 /B8\nL= 32cm /B4θ1= 0.26 /CP/D2/CSkrL= 5.91 /B5/B8 /CP/D2/CSζ= 0.2 /BA/CQ /D3/D6/CT/BM\nZe(ω)\nZc=jtan/parenleftbiggωL\nc−jα(ω)L/parenrightbigg/B4/BF/BH/B5\nZe(ω)\nZc≃2c\nLN/summationdisplay\nn=1jω\nω2n+jqnωωn+(jω)2, /B4/BF/BI/B5/DB/CW/CT/D6/CT /D1/D3 /CS/CP/D0 \r/D3 /CTꜶ\r/CX/CT/D2 /D8/D7 ωn\n/CP/D2/CSqn\n\r/CP/D2 /CQ /CT /CS/CT/CS/D9\r/CT/CS /CT/CX/B9/D8/CW/CT/D6 /CU/D6/D3/D1 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2\r/CT /D3/D6 /CU/D6/D3/D1 /CP/D2/CP/D0/DD/D8/CX\r/CP/D0/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CJ/BX/D5/BA /B4/BF/BH /B5℄/B8 /CP/D7/D7/D9/D1/CX/D2/CV α(ω) /D8/D3 /CQ /CT /CP /D7/D0/D3 /DB/D0/DD /DA /CP/D6/DD/B9/CX/D2/CV /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD /BA/BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CS/CX/D6/CT\r/D8 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D8/CW/D6/CT/D7/CW/D3/D0/CS γ/D8/CW\n/D9/D7/CX/D2/CV /CF/BU /D1/CT/D8/CW/D3 /CS /CP/D2/CS /CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /D9/D7/CX/D2/CV/D1/D3 /CS/CP/D0 /CS/CT\r/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /CP/D2/CS \r/D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 \r/D3/D1/B9/D4/D9/D8/CX/D2/CV /CW/CP/D7 /CQ /CT/CT/D2 /CS/D3/D2/CT/BA /CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT /CS/CXꜶ\r/D9/D0/D8/CX/CT/D7 /CU/D3/D6 /D8/CW/CT/AS/D6/D7/D8 /D1/CT/D8/CW/D3 /CS /CP/D6/CX/D7/CT /CS/D9/CT /D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7\r/CT/D2/CS/CT/D2 /D8/CP/D0 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/B9/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D8/CW/CT /D7/CT\r/D3/D2/CS /D3/D2/CT /D6/CT/D5/D9/CX/D6/CT/D7 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/CX/CV/CT/D2/B9/DA /CP/D0/D9/CT/D7 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7 /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /B8 /D9/D7/CX/D2/CV/CP/D2 /CX/D8/CT/D6/CP/D8/CX/DA /CT /D7/CT/CP/D6\r /CW /D3/CU /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /CC/CW/CT /D2 /D9/D1/B9/CQ /CT/D6 /D3/CU /D1/D3 /CS/CT/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8 /CW/CP/D7 /CQ /CT/CT/D2 \r /CW/D3/D7/CT/D2 /D7/D9\r /CW/D8/CW/CP/D8 /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2 /D3/CUγ/D8/CW\n/CX/D7 /D0/CT/D7/D7 /D8/CW/CP/D20.01 /B8 /DB/CW/CX\r /CW /CX/D7 /CP/D0/D7/D3 /D8/CW/CT/D8/D3/D0/CT/D6/CP/D2\r/CT /D9/D7/CT/CS /CU/D3/D6 /D8/CW/CT /CX/D8/CT/D6/CP/D8/CX/DA /CT /D7/CT/CP/D6\r /CW/BA /BT/D2 /CT/DC/CP/D1/D4/D0/CT /CX/D7/CV/CX/DA /CT/D2 /CX/D2 /CC /CP/CQ/D0/CT /C1 /CU/D3/D6 /CP /CW/CT/CP /DA/CX/D0/DD /CS/CP/D1/D4 /CT/CS /CP/D2/CS /D7/D8/D6/D3/D2/CV /D6/CT/CT/CS/BA/C1/D8 /D7/CW/D3 /DB/D7 /DA /CT/D6/DD /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /D1/CT/D8/CW/D3 /CS/D7/CU/D3/D6 /CQ /D3/D8/CWγ /CP/D2/CSθ /BA /CC/CW/CX/D7 /DA /CP/D0/CX/CS/CP/D8/CX/D3/D2 /CP/D0/D0/D3 /DB/D7 /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT\r/D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2\r/DD /CP/D4/D4/D6/D3/CP\r /CW/B8 /DB/CW/CX\r /CW /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /CP/D2 /CTꜶ\r/CX/CT/D2 /D8/CP/D0/CV/D3/D6/CX/D8/CW/D1 /D8/CW/CP/D8 \r/CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D1/D3/D6/CT \r/D3/D1/D4/D0/CT/DC /D6/CT/D7/B9/D3/D2/CP/D8/D3/D6/D7 /DB/CW/CT/D2/CT/DA /CT/D6 /CP /D1/D3 /CS/CP/D0 /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /CX/D7 /CP /DA /CP/CX/D0/CP/CQ/D0/CT/BA/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BKkrLθCFθWB∆θγCFγWB∆γ/BK/BA/BH /BC/BA/BD/BK/BH /BC/BA/BD/BK/BI /BC/BA/BH/B1 /BC/BA/BG/BF /BC/BA/BG/BF /BC/BA/BD/B1/BE /BC/BA/BJ/BG/BJ /BC/BA/BJ/BG/BJ /BC/BA/BD/B1 /BC/BA/BF/BC /BC/BA/BF/BC /BC/BA/BI/B1/BD /BD/BA/BC/BE/BE /BD/BA/BC/BE/BG /BC/BA/BE/B1 /BF/BA/BK/BI /BF/BA/BK/BE /BD/BA/BC/B1/BC/BA/BK/BD /BD/BA/BC/BF/BF /BD/BA/BC/BF/BG /BC/BA/BD/B1 /BK/BA/BJ/BJ /BK/BA/BJ/BD /BC/BA/BJ/B1/CC /BT/BU/C4/BX /C1/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/CU/D6/CT/D5/D9/CT/D2\r/DD \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D7/CX/D2/CV \r/D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2\r/DD /CU/D3/D6/D1/CP/D0/CX/D7/D1 /B4/CX/D2/B9/CS/CT/DC/CT/CS /CQ /DDCF /B5 /CP/D2/CS /CF/CX/D0/D7/D3/D2 /B2 /BU/CT/CP /DA /CT/D6/D7 /D1/CT/D8/CW/D3 /CS /B4WB /B5 /CU/D3/D6\nr= 7mm /B8ωr= 2π×750rad/s /B8qr= 0.4 /CP/D2/CSζ= 0.13 /BA/CC/CW/CT /DB/D6/CX/D8/CX/D2/CV /D3/CU /D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /CP /D7/CX/D2/B9/CV/D0/CT /CP\r/D3/D9/D7/D8/CX\r /D1/D3 /CS/CT /CT/DC/CW/CX/CQ/CX/D8/D7 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT \r/D3/D9/D4/D0/CT/CS/D3/D7\r/CX/D0/D0/CP/D8/D3/D6/D7/BM\n/bracketleftbigg\nω2\nn+jω/parenleftbigg\nqnωn+c\nLζ1−γ√γ/parenrightbigg\n−ω2/bracketrightbigg\n×/bracketleftbig\nω2\nr+jqrωωr−ω2/bracketrightbig\n=jω2c\nLω2\nr/parenleftbigg\nζ√γ+jω∆l\nc/parenrightbigg\n./B4/BF/BJ/B5/BV/D3/D9/D4/D0/CX/D2/CV /D6/CT/CP/D0/CX/DE/CT/CS /CQ /DD /D8/CW/CT /AT/D3 /DB /CX/D2 /D8/CW/CT /D6/CT/CT/CS \r /CW/CP/D2/D2/CT/D0 /D1/D3 /CS/B9/CX/AS/CT/D7 /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /D3/CU /D8/CW/CT /CP\r/D3/D9/D7/D8/CX\r /D1/D3 /CS/CT/BM /CX/D2 /CP/CS/CS/CX/D8/CX/D3/D2 /D8/D3/D8/CW/CT /D9/D7/D9/CP/D0 /D8/CT/D6/D1 /B4\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /DA/CX/D7\r/D3/D8/CW/CT/D6/D1/CP/D0 /D0/D3/D7/D7/CT/D7 /CP/D2/CS/CT/DA /CT/D2 /D8/D9/CP/D0/D0/DD /D6/CP/CS/CX/CP/D8/CX/D3/D2/B5/B8 /CS/CP/D1/D4/CX/D2/CV /CX/D7 /CX/D2\r/D6/CT/CP/D7/CT/CS /CQ /DD /CP /D5/D9/CP/D2 /D8/CX/D8 /DD/D6/CT/D0/CP/D8/CT/CS /D8/D3 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS/BA /CC/CW/CX/D7/D1/CP /DD /CQ /CT /D6/CT/CV/CP/D6/CS/CT/CS /D8/D3 /CP/D7 /CP /D6/CT/D7/CX/D7/D8/CX/DA /CT /CP\r/D3/D9/D7/D8/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /CP/D8 /D8/CW/CT/CQ /D3/D6/CT /CT/D2 /D8/D6/CP/D2\r/CT/BA/BT/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /CP /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS /D1/D3 /CS/CT/D0 /CX/D7 /D7/D8/CX/D0/D0 /D6/CT/D0/CT/DA /CP/D2 /D8 /CS/D9/D6/B9/CX/D2/CV /D8/CW/CT /CV/D6/D3 /DB/D8/CW /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /B4/CQ /CT/CU/D3/D6/CT /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D7/CP/D8/B9/D9/D6/CP/D8/CX/D3/D2 /D1/CT\r /CW/CP/D2/CX/D7/D1 /CP/D4/D4 /CT/CP/D6/D7/B5/B8 /D8/CW/CX/D7 /CP/D4/D4/D6/D3/CP\r /CW \r/CP/D2 /CQ /CT /CT/DC/B9/D8/CT/D2/CS/CT/CS /D8/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT \r/D3/D9/B9/D4/D0/CT/CS /D7/DD/D7/D8/CT/D1/BA /BV/CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CX/D2/CV /D8/CW/CT /CS/CT/CV/D6/CT/CT /D3/CU /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD/D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /DDσ= min nαn\n/B8 /D8/CW/CT /D7/D0/D3/D4 /CT /D3/CU /D8/CW/CT \r/D9/D6/DA /CT\nσ=f(γ) /CV/CX/DA /CT/D7 /CP/D2 /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/B9/CV/D6/CT/CT /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /DB/CW/CT/D2 /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D7/D0/CX/CV/CW /D8/D0/DD/CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /BT /CV/D6/CT/CP/D8 /D7/D0/D3/D4 /CT/DB /D3/D9/D0/CS \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CP /DA /CT/D6/DD /D9/D2/D7/D8/CP/CQ/D0/CT \r/D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS/CP /D5/D9/CX\r /CZ /CV/D6/D3 /DB/D8/CW /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT/CP/D7 /D2/CT/CP/D6/D0/DD \r/D3/D2/D7/D8/CP/D2 /D8\r/D9/D6/DA /CT /DB /D3/D9/D0/CS /D0/CT/CP/CS /D8/D3 /CP /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/AS\r/CP/D8/CX/D3/D2 \r/D3 /CTꜶ\r/CX/CT/D2 /D8 /CP/D2/CS/D7/D0/D3 /DB/D0/DD /D6/CX/D7/CX/D2/CV /DA/CX/CQ/D6/CP/D8/CX/D3/D2/D7 /CP/D2/CS /D8/CW/CT/D2 /D8/D3 /D0/D3/D2/CV/CT/D6 /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /CP/D8/B9/D8/CP\r /CZ /CQ /CT/CU/D3/D6/CT /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7/BA/C4/CX/D2/CZ/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT \r/D3/D1/D4/D9/D8/CT/CS /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 /D3/CU /D8/CW/CT \r/D3/D9/B9/D4/D0/CT/CS /D7/DD/D7/D8/CT/D1 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CW/CT/D6/CT /CP/D2/CS /D8/CW/CT /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /CQ /CT/CW/CP /DA/CX/D3/D6/CW/CP /DA /CT /D7/D8/CX/D0/D0 /D8/D3 /CQ /CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA/BU/BA /C5/D3/D9/D8/CW /D4 /D6/CT/D7/D7/D9/D6/CT /D6/CT/D5/D9/CX/D6/CT/CS /D8/D3 /D3/CQ/D8/CP/CX/D2 /CP /CV/CX/DA/CT/D2 /B4/D7/D1/CP/D0/D0/B5/CP/D1/D4/D0/CX/D8/D9/CS/CT/CC/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT\r/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4/CP/D4 /CT/D6 /CS/CT/CP/D0 /DB/CX/D8/CW /D8/CW/CT /D7/D8/CP/CQ/CX/D0/B9/CX/D8 /DD /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX\r /D6/CT/CV/CX/D1/CT/B8 /D0/D3 /D3/CZ/CX/D2/CV /CU/D3/D6 /D8/CW/CT \r/D3/D2/CS/CX/D8/CX/D3/D2 /D8/D3 /D1/CP/CZ /CT/CP /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CB/D3/D1/CT /CS/CT/DA /CT/D0/D3/D4/D1/CT/D2 /D8/D7 \r/D3/D2\r/CT/D6/D2/CX/D2/CV/D8/CW/CT /CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D2/CV /D6/CT/CV/CX/D1/CT /CP/CQ /D3 /DA /CT /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/CP/D6/CT /CS/CT/D6/CX/DA /CT/CS /D2/D3 /DB/BA /C6/CT/CX/D8/CW/CT/D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /D2/D3/D6 /D8/D3/D2/CT/CS/CT/DA/CX/CP/D8/CX/D3/D2 /CX/D7/D7/D9/CT /DB/CX/D0/D0 /CQ /CT /CS/CX/D7\r/D9/D7/D7/CT/CS /CW/CT/D6/CT/BA/BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA\n/BE/BC/D7/D9/CV/CV/CT/D7/D8/CT/CS /D8/CW/CT /CX/D2 /D8/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CX/D1/B9/CX/D8/CT/CS /BY /D3/D9/D6/CX/CT/D6 /D7/CT/D6/CX/CT/D7 /D3/CU /D4/D6/CT/D7/D7/D9/D6/CT /CX/D2 /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS \r/CP/D7/CT/BA\n/CC/CW/CT /D8/CT\r /CW/D2/CX/D5/D9/CT /CX/D7 /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX\r /CQ/CP/D0/CP/D2\r/CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D3/D7\r/CX/D0/B9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT/D7/BA /BV/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D2/CT /CW/CT/D6/CT/B9/CP/CU/D8/CT/D6 /CQ /DD /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8 /D8/CW/CT /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX\r/D7 /CX/D2 /D8/CW/CT/DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/B8 /DB/CW/CX\r /CW /CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6 /CX/D2 /D8/CW/CT/D1/CT/D2 /D8/CX/D3/D2/CT/CS /D4/CP/D4 /CT/D6/BA /BY /D3/D9/D6/CX/CT/D6 /D7/CT/D6/CX/CT/D7 /D3/CU /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /CS/CT/B9/D4 /CT/D2/CS/D7 /D3/D2 /BY /D3/D9/D6/CX/CT/D6 \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D7/CX/CV/D2/CP/D0Pm−p(t) /CP/D2/CS\ny(t) /BA /BT/D7/D7/D9/D1/CX/D2/CV /D7/D8/CT/CP/CS/DD /D7/D8/CP/D8/CT /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/B9/D5/D9/CT/D2\r/DDω /B8 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D7/CX/CV/D2/CP/D0/D7 /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7\np(t) =/summationdisplay\nn/negationslash=0pnenjωt, u(t) =u0+/summationdisplay\nn/negationslash=0Ynpnenjωt, /B4/BF/BK/B5\ny(t)\ny0= (1−γ)+/summationdisplay\nn/negationslash=0Dnpnenjωt, /B4/BF/BL/B5/DB/CW/CT/D6/CTYn=Ye(nω) /CP/D2/CSDn=D(nω) /CP/D6/CT /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CQ /D3/D6/CT /CP/CS/D1/CX/D8/D8/CP/D2\r/CT /CP/D2/CS /D6/CT/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /CU/D9/D2\r/D8/CX/D3/D2/CU/D3/D6 /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/D5/D9/CT/D2\r/DD nω /BA /CC/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/CX/D7 /D6/CT/DB/D6/CX/D8/D8/CT/D2 /CP/D7\nu2(t) =ζ2(y(t)/y0)2(γ−p(t)). /B4/BG/BC/B5/CB/D9/D7/D8/CP/CX/D2/CT/CS /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /D3/CU /DA /CT/D6/DD /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D6/CT /D7/D8/D9/CS/B9/CX/CT/CS/B8 /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8p1\n/CX/D7 /CP /D2/D3/D2 /DA /CP/D2/CX/D7/CW/CX/D2/CV \r/D3 /CTꜶ\r/CX/CT/D2 /D8 \r/D3/D2/B9/D7/CX/CS/CT/D6/CT/CS /CP/D7 /CP /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /C6/D3/D8/CP/D8/CX/D3/D2/D7 CEn\n/CP/D2/CS\nF(n\nm)\n/CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9\r/CT/CS/BM\nCEn=Yn/(ζ√γ)+1−γ\n2γ−Dn, /B4/BG/BD/B5\nF(m\nn)=F(n\nm)=DnDm−1−γ\nγ(Dn+Dm)−YnYm\nζ2γ/B4/BG/BE/B5/BV/CP/D2\r/CT/D0/D0/CP/D8/CX/D3/D2 /D3/CUCEn\n/CU/D3/D6 /CV/CX/DA /CT/D2ω /CP/D2/CSγ /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT\r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BJ/B5 /CX/D7 /D7/D3/D0/DA /CT/CS /CU/D3/D6nω /CP/D2/CSγ /BA /BX/DC/B9/D4/CP/D2/CS/CX/D2/CV /BX/D5/BA /B4/BG/BC /B5 /D0/CT/CP/CS/D7 /D8/D3\n0 =/bracketleftbiggu2\n0\nζ2γ−(1−γ)2/bracketrightbigg\n+2(1−γ)/summationdisplay\nn/negationslash=0/bracketleftbiggu0Yn\nζ2γ(1−γ)−Dn+1−γ\n2γ/bracketrightbigg\npnenjωt\n−/summationdisplay\nn,m/negationslash=0F(n\nm)pnpme(n+m)jωt\n+1\nγ/summationdisplay\nn,m,q/negationslash=0DnDmpnpmpqe(n+m+q)jωt./B4/BG/BF/B5/C1/D8 /CX/D7 /CW/CT/D6/CT /CP/D7/D7/D9/D1/CT/CS /D8/CW/CP/D8pn\n/CX/D7 /D3/CU /D3/D6/CS/CT/D6|n| /B4/DB/CX/D8/CWp−n=p∗\nn\n/B5/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BE/BC /B5/BA /CC/CW/CT \r/D3/D2 /D8/CX/D2 /D9/D3/D9/D7 \r/D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT/DA /D3/D0/D9/D1/CT /AT/D3 /DB /CX/D7 \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D4 /D8/D3 /D3/D6/CS/CT/D6 /BE/BM\nu2\n0\nζ2γ=(1−γ)2+/summationdisplay\nn/negationslash=0F“+n\n−n”|pn|2\n−1\nγ/summationdisplay\nn,m,n+m/negationslash=0DnDmpnpmp∗\nn+m\n≃(1−γ)2+2F“+1\n−1”|p1|2+o(p2\n1)\n/B4/BG/BG/B5/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BL/D7/D3 /D8/CW/CP/D8/B8 \r/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV u0\n/D8/D3 /CQ /CT /D6/CT/CP/D0\nu0≃ζ√γ(1−γ)\n1+|p1|2F“+1\n−1”\n(1−γ)2\n+o(p2\n1). /B4/BG/BH/B5/BY /D6/D3/D1 /BX/D5/BA /B4/BG/BF/B5/B8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD (Nω) /CX/D7 /CT/DC/D8/D6/CP\r/D8/CT/CS /CU/D3/D6\nN≥1 /BM\n0 = 2(1−γ)/bracketleftbigg\nDN−1−γ\n2γ−u0Yn\nζ2γ(1−γ)/bracketrightbigg\npN\n+/summationdisplay\nn/negationslash=0F(n\nN−n)pnpN−n\n−1\nγ/summationdisplay\nn,m/negationslash=0DnDmpnpmpN−n−m. /B4/BG/BI/B5/BY /D3/D6N≥2 /B8 /CC /CP /DD/D0/D3/D6 /D7/CT/D6/CX/CT/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D9/D4 /D8/D3 /D3/D6/CS/CT/D6N /CX/D7/CP/D4/D4/D0/CX/CT/CS/BM /CX/D2 /D8/CW/CT /AS/D6/D7/D8 /D7/D9/D1/B8 /D3/D2/D0/DD /D8/CT/D6/D1/D7 \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3\n0≤n≤N \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CT /CP/D8 /D3/D6/CS/CT/D6N /B8 /DB/CW/CX/D0/CT/B8 /CX/D2 /D8/CW/CT /D7/CT\r/D3/D2/CS/D3/D2/CT/B8 /D8/CW/CT /D8/CT/D6/D1/D7 /D8/D3 \r/D3/D2/D7/CX/CS/CT/D6 /CP/D6/CT /D8/CW/CT /D3/D2/CT/D7 /CU/D3/D6 /DB/CW/CX\r /CW0<\nn < N /CP/D2/CS0< m < N −n /BA /CC/CW/CT \r/D3/D1/D4 /D3/D2/CT/D2 /D8 pN\n\r/CP/D2 /CQ /CT/CS/CT/CS/D9\r/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D7/CT/D5/D9/CT/D2\r/CT (pn)0<n<N\n/BM\npN=1\n2(1−γ)CEN/bracketleftigg/summationdisplay\n0<n<NF(n\nN−n)pnpN−n\n−1\nγ/summationdisplay\n0<n<N\n0<m<N −nDnDmpnpmpN−n−m/bracketrightigg\n+o(pN\n1). /B4/BG/BJ/B5/BT/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT/B8 /CU/D3/D6N= 2 /B8 /D8/CW/CT /D7/CT\r/D3/D2/CS /D7/D9/D1 /CQ /CT/CX/D2/CV /CT/D1/D4/D8 /DD/BM\np2≃F(1\n1)p2\n1\n2(1−γ)CE2+o(p2\n1). /B4/BG/BK/B5/BT/D7 /CT/DC/D4 /CT\r/D8/CT/CS /CP/D2/CS /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D7/D3/B9\r/CP/D0/D0/CT/CS /AG/CF /D3/D6/B9/D1/CP/D2 /D6/D9/D0/CT/AH\n/BD/BK/B8 /CW/CX/CV/CW/CT/D6 \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D4/D4 /CT/CP/D6 /D8/D3 /CQ /CT /CW/CX/CV/CW/CT/D6 /D3/D6/B9/CS/CT/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BM /D3/D6/CS/CT/D6 /BE /CU/D3/D6p2\n/B8 /BF /CU/D3/D6p3\n/B8 /BG /CU/D3/D6p4\n/B8 /CT/D8\r/BA/BY /D3 \r/D9/D7 /CX/D7 /D2/D3 /DB /CV/CX/DA /CT/D2 /D8/D3 /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /BV/CP/D0\r/D9/B9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D2/CT /D9/D4 /D8/D3 /D3/D6/CS/CT/D6 /BF/BM\n0 =2(1−γ)/bracketleftbigg\nD1−1−γ\n2γ−u0Y1\nζ2γ(1−γ)/bracketrightbigg\np1\n+/summationdisplay\nn/negationslash=0,1F(n\n1−n)pnp1−n\n−1\nγ/summationdisplay\n1−n−m,n,m/negationslash=0DnDmpnpmp1−n−m\n≃2(1−γ)/bracketleftbigg\nD1−1−γ\n2γ−u0Y1\nζ2γ(1−γ)/bracketrightbigg\np1\n+2F(2\n−1)p2p∗\n1−1\nγD1p1|p1|2(D1+2D∗\n1)\n/B4/BG/BL/B5/CA/CT/D4/D0/CP\r/CX/D2/CV u0\n/CP/D2/CSp2\n/CV/CX/DA /CT/D7\n|p1|2≃2(1−γ)2CE1\nF(1\n1)F“+2\n−1”\nCE2−2Y1\nζ√γF“+1\n−1”−1−γ\nγ(D2\n1+2|D1|2)./B4/BH/BC/B5\n/CC/CW/CT /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /AS/D6/D7/D8 /CW/CP/D6/D1/D3/D2/CX\r /D1/CT/D8/CW/D3 /CS /CX/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT/CX/D2/AT/D9/CT/D2\r/CT /D3/CUpn≥2\n/D3/D2 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT |p1| /D0/CT/CP/CS/D7 /D8/D3\n|p1|2≃2(1−γ)2CE1\n−2Y1\nζ√γF“+1\n−1”−1−γ\nγ(D2\n1+2|D1|2). /B4/BH/BD/B5/C1/D2 /D8/CW/CT/CX/D6 /D4/CP/D4 /CT/D6/B8 /BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA\n/BE/BC/D7/D8/CP/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D4/D6/D3 \r/CT/CS/D9/D6/CT/D9/D7/CT/CS /CX/D2 /CP /AS/D6/D7/D8 /D7/CX/D1/D4/D0/CX/AS/CT/CS \r/CP/D7/CT \r/CP/D2 /CQ /CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /CP /D1/D3 /CS/CT/D0/CX/D2\r/D0/D9/CS/CX/D2/CV /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX\r/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /D8/CW/CP/D8Ze(ω) /CX/D7 /D6/CT/D4/D0/CP\r/CT/CS/CQ /DDZe(ω)D(ω) /BA /BV/D3/D2\r/D0/D9/D7/CX/D3/D2 /CX/D7 /D2/D3/D8 /D7/D3 /D7/D8/D6/CP/CX/CV/CW /D8/CU/D3/D6/DB /CP/D6/CS /CP/D7/D6/CT/CT/CS /CS/DD/D2/CP/D1/CX\r/D7 /CP/D0/D7/D3 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D7 /DB/CX/D8/CW /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/B9/D8/CX/D3/D2/D7/CW/CX/D4 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D2/CV /D8/D3 /CP /D1/D3/D6/CT \r/D3/D1/D4/D0/CT/DC /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6/D8/CW/CT /AS/D6/D7/D8 \r/D3/D1/D4 /D3/D2/CT/D2 /D8 /CP/D1/D4/D0/CX/D8/D9/CS/CT/BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT/CX/D6 /DB /D3/D6/CZ/B8\nDn\n/D8/CT/D6/D1/D7 /DB /D3/D9/D0/CS /D3/D2/D0/DD /D3 \r\r/D9/D6 /DB/CX/D8/CWYn\n/CX/D2 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CU\n|p1|2/DB/CW/CX\r /CW /CX/D7 /D2/D3/D8 /D8/CW/CT \r/CP/D7/CT/BA\n/BY/C1/BZ/BA /BL/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU|p1| \r/D3/D1/D4/D9/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX\r /CQ/CP/D0/B9/CP/D2\r/CT /CU/D3/D6 /D7/D1/CP/D0/D0 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7 /DA /CP/D0/D9/CT/D7 /D3/CUkrL /BM /CU/D6/D3/D1/D8/CW/CX/D2 /D8/D3 /D8/CW/CX\r /CZ /D0/CX/D2/CT/D7/B8krL= 9,5,4,3,2,1.8,1.5,1.25 /B4∆l= 0 /B8\nqr= 0.4 /B8 /CP/D2/CSfr= 1050Hz /B5/BA/C1/D2 /BY/CX/CV/BA /BL /CP/D6/CT /D7/CW/D3 /DB/D2 /D8/CW/CT /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CS/CX/CP/CV/D6/CP/D1/D7 /CU/D3/D6 /CP /CW/CT/CP /DA/B9/CX/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/BA /BY /D6/D3/D1 /D8/CW/CT /AT/CP/D8/D8/CT/D7/D8 /D8/D3/D2/CT /B4krL= 9.00 /B5 /D8/D3/D8/CW/CT /D7/CW/CP/D6/D4 /CT/D7/D8 \r/D3/D1/D4/D9/D8/CT/CS /B4krL= 1.25 /B5/B8 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW/DA /CT/D6/DD /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT/D7 /CP/D6/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /CP/D7 /D8/CW/CT /CS/CX/CP/CV/D6/CP/D1/D7 /D7/CW/D3 /DB/C0/D3/D4/CU /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2/D7 /D8/CW/CP/D8 /CP/D6/CT /D7/D9/D4 /CT/D6\r/D6/CX/D8/CX\r/CP/D0/B8 /CX/BA/CT/BA/B8 /CS/CX/D6/CT\r/D8/B8 /CU/D3/D6 /CP/D0/D0/D8/CW/CT \r/D3/D1/D4/D9/D8/CT/CS \r/CP/D7/CT/D7/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /CU/D3/D6 \r/CP/D7/CT/D7 /DB/CW/CT/D6/CT/D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /CX/D7 \r/D0/D3/D7/CT/CS /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D3/D2/CT /B4/BY/CX/CV/BA /BD/BC /B5/B8/CP /D7/D9/CQ \r/D6/CX/D8/CX\r/CP/D0 /C0/D3/D4/CU /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /D3 \r\r/D9/D6/D7 /CU/D3/D6 /CP /D6/CP/D2/CV/CT /D3/CU /CQ /D3/D6/CT/D0/CT/D2/CV/D8/CW/B8 /DB/CX/D8/CW /CP /DA /CT/D6/DD /D7/D1/CP/D0/D0 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D0/CX/D2/CZ /CT/CS /D8/D3/D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D3/CU /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2\r/CT /DB/CX/D8/CW /D8/CW/CT /D7/CT\r/D3/D2/CS /CQ /D3/D6/CT/D6/CT/D7/D3/D2/CP/D2\r/CT/BM /D8/CW/CT /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CX/D7 /D8/CW/CT/D6/CT /CX/D2 /DA /CT/D6/D7/CT/BA/C1/D2 /CP\r\r/D3/D6/CS/CP/D2\r/CT /DB/CX/D8/CW /BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA\n/BE/BC/B8 /CX/D8 /CP/D4/D4 /CT/CP/D6/D7 /D8/CW/CP/D8 /D8/CW/CT/CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CX/D7 /D2/D3/D8 /CP/D0/DB /CP /DD/D7 /CS/CX/D6/CT\r/D8/BA /CC/CW/CT/D6/CT /CT/DC/CX/D7/D8 \r/D3/D2/AS/CV/D9/D6/CP/B9/D8/CX/D3/D2/D7 /CU/D3/D6 /DB/CW/CX\r /CW \r/D3/D1/D4/D9/D8/CT/CS /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CS/CX/CP/CV/D6/CP/D1 /D7/CW/D3 /DB/D7 /CP/D7/D9/CQ \r/D6/CX/D8/CX\r/CP/D0 /D4/CX/D8\r /CW/CU/D3/D6/CZ/B8 /CX/BA/CT/BA/B8 /D7/D1/CP/D0/D0 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /DA /CP/D0/D9/CT/D7 /D3/CU/D4/D6/CT/D7/D7/D9/D6/CT /CQ /CT/D0/D3 /DB /D7/D8/CP/D8/CX\r /D6/CT/CV/CX/D1/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /CC/CW/CT/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BD/BC/BY/C1/BZ/BA /BD/BC/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU|p1| \r/D3/D1/D4/D9/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX\r/CQ/CP/D0/CP/D2\r/CT /CU/D3/D6 /D7/D1/CP/D0/D0 /D3/D7\r/CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /BD/BG /DA /CP/D0/D9/CT/D7 /D3/CUkrL /D6/CT/CV/D9/D0/CP/D6/D0/DD/D7/D4/CP\r/CT/CS /CQ /CT/D8 /DB /CT/CT/D2 3.6 /CP/D2/CS4.20 /DB/CX/D8/CW /CP /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS /B4∆l=\n0 /B8qr= 0.01 /B8 /CP/D2/CSfr= 1050Hz /B5/BA/CQ /D3/D9/D2/CS/CP/D6/DD /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 \r/CP/D7/CT/D7 /CX/D7 /D2/D3/D8 /D8/D6/CX/DA/CX/CP/D0 /D8/D3 /CT/DC/B9/D4/D0/D3/D6/CT /CP/D2/CP/D0/DD/D8/CX\r/CP/D0/D0/DD /B8 /D6/CT/D5/D9/CX/D6/CX/D2/CV /D8/CW/CT /D1/CP/D8/CW/CT/D1/CP/D8/CX\r/CP/D0 /D7/D8/D9/CS/DD /D3/CU/BX/D5/BA /B4/BH/BC/B5/BA 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Waals\nAntiferromagnets\nRoberto E. Troncoso,1Mike A. Lund,2, 1Arne Brataas,1and Akashdeep Kamra1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Engineering Sciences, University of Agder, 4879 Grimstad, Norway\nWe theoretically study spin pumping from a layered van der Waals antiferromagnet in its canted\nground state into an adjacent normal metal. We \fnd that the resulting dc spin pumping current\nbears contributions along all spin directions. Our analysis allows for detecting intra- and cross-\nsublattice spin-mixing conductances via measuring the two in-plane spin current components. We\nfurther show that sublattice symmetry-breaking Gilbert damping can be realized via interface en-\ngineering and induces a dissipative coupling between the optical and acoustic magnon modes. This\nrealizes magnon level attraction and exceptional points in the system. Furthermore, the dissipative\ncoupling and cross-sublattice spin pumping contrive to produce an unconventional spin current in\nthe out-of-plane direction. Our \fndings provide a route to extract the spin mixing conductance\nmatrix and uncovers the unique opportunities, such as level attraction, o\u000bered by van der Waals\nantiferromagnet-normal metal hybrids.\nIntroduction .{ The dawn of magnetic van der Waals\n(vdW) materials has renewed and invigorated interest\nin low-dimensional phenomena hosted by solid state sys-\ntems [1, 2]. These layered vdWs materials have been\nfound to host various forms of magnetic order [3{8] with\nantiferromagnets (AFs) taking a special place due to their\nvarious unique advantages [9{12]. Among these, a control\nover interfacial exchange coupling to an adjacent metal\nand canted or noncollinear ground state o\u000bers unprece-\ndented pathways to achieve intriguing physics and ap-\nplications [13{19]. The vdW magnets o\u000ber an e\u000bective\ncontrol over both - interface and ground state - due to\ntheir layered structure and relatively weak interlayer an-\ntiferromagnetic exchange [3, 4, 20].\nCapitalizing on these features, magnon-magnon cou-\npling resulting in level repulsion and hybridization has\nrecently been observed in the vdW AF CrCl 3[21{23].\nApproaching the challenge from a di\u000berent direction,\nsimilar magnonic hybridization via their mutual cou-\npling [24{26] has been discovered in carefully chosen plat-\nforms including compensated ferrimagnets [27] and syn-\nthetic AFs [28, 29]. These investigations have, in part,\nbeen driven by the desire to control magnonic systems\nfor quantum information applications [30] and were pre-\nceded by the realization of strong magnon-photon cou-\npling [31, 32]. The latter, being a relatively mature\n\feld, has started to explore dissipative magnon-photon\ncoupling [33, 34] resulting in intriguing phenomena that\nforay into the realm of non-Hermitian physics [35] pro-\nviding a powerful model platform. While not explored\nthus far, such phenomena can also result from dissi-\npative magnon-magnon coupling o\u000bering various advan-\ntages over the magnon-photon platform [36]. Demon-\nstrating these for a vdW AF interfaced with a heavy nor-\nmal metal (NM) forms a key contribution of this work.\nHeterostructures comprising a magnetic insulator in-\nterfaced with a thin NM layer have become basic build-ing blocks in an emerging spin-based paradigm for in-\nformation transport and processing [15, 37{45]. In such\nstructures, magnonic spin in the magnetic insulator can\nbe interfaced with the electronic spin in NM thereby\nallowing their integration with conventional electronics.\nFurthermore, spin generated in NM allows to control,\nand even negate [41], dissipation in the magnetic sys-\ntem via spin transfer torques [46]. Invigorated by re-\ncent breakthroughs, especially employing the magnonic\nspin in AFs [15, 40, 47, 48], an interface-engineering and\ncontrol of spin transfer from AF to NM via magnonic\nspin pumping [13, 14, 49{51] assumes a central role.\nWhile coherently driven spin pumping from an AF into\nNM has recently been observed in its collinear ground\nstate [47, 48], a noncollinear or canted AF should en-\nable unique and novel phenomena emerging from cross-\nsublattice spin pumping [14, 52, 53]. Besides providing\nthe much needed understanding of spin transfer across\nthe AF-NM interface, cross-sublattice pumping may also\no\u000ber a direct probe into the aforementioned dissipative\nand non-Hermitian magnon-magnon coupling phenom-\nena, as shown in this work.\nIn this Letter, we demonstrate heterostructures com-\nprising vdW AFs and a heavy NM to be a unique plat-\nform for observing intriguing phenomena emerging from\ncross-sublattice spin pumping and dissipative magnon-\nmagnon coupling. This niche is enabled by the layered\nstructure of vdW AFs resulting in the possibility of AF-\nNM interface engineering and canted AF ground states\nwith application of relatively small magnetic \felds. We\nshow that in such a non-collinear ground state, the AF\npumps spin into the NM along all three directions on\nexcitation via an rf magnetic \feld. A detection of the\ntwo in-plane spin components via inverse spin Hall e\u000bect\nallows to determine the complete spin mixing conduc-\ntance matrix of the interface. We further \fnd an un-\nconventional out-of-plane spin pumping component thatarXiv:2011.14314v1  [cond-mat.mes-hall]  29 Nov 20202\nx<latexit sha1_base64=\"E+xWb622b2P97o+CO1oWwc/7ors=\">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYNRo9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmhdlr1K+rFdK1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOjRjQQ=</latexit>\ny<latexit sha1_base64=\"0AzN9pSceEkVyYO5YY8D+4ugxy4=\">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoseiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ+6Lq1aqXzVqlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6lWNBQ==</latexit>\nz<latexit sha1_base64=\"mBNsSck29HYD+UA8I7CdsBnbA5A=\">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYNRo9ELx4hkUcCGzI79MLI7OxmZtYECV/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmhdlr1K+rFdK1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOvZjQY=</latexit>\n(a)\n<latexit sha1_base64=\"Ac+cs8EQeHY6Uohp+aTWQJMy8+k=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI97xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ai0ONUg==</latexit>\n(b)\n<latexit sha1_base64=\"zVpXaXYpCy/5Krr2sSqRENfT7FA=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqaFxWvWrm8r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjMiNUw==</latexit>\n(c)\n<latexit sha1_base64=\"0u7bBHPDKBCJbAK7avM20Uz0++4=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI77xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ajk2NVA==</latexit>\n(d)\n<latexit sha1_base64=\"+qTaDHOrJ165Z1nDA3KKJzLMLOY=\">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BItQL2VXKnosevFY0X5Au5RsNtuGZrNLkhXK0p/gxYMiXv1F3vw3pu0etPXBwOO9GWbm+Yng2jjONyqsrW9sbhW3Szu7e/sH5cOjto5TRVmLxiJWXZ9oJrhkLcONYN1EMRL5gnX88e3M7zwxpXksH80kYV5EhpKHnBJjpYdqcD4oV5yaMwdeJW5OKpCjOSh/9YOYphGThgqidc91EuNlRBlOBZuW+qlmCaFjMmQ9SyWJmPay+alTfGaVAIexsiUNnqu/JzISaT2JfNsZETPSy95M/M/rpSa89jIuk9QwSReLwlRgE+PZ3zjgilEjJpYQqri9FdMRUYQam07JhuAuv7xK2hc1t167vK9XGjd5HEU4gVOoggtX0IA7aEILKAzhGV7hDQn0gt7Rx6K1gPKZY/gD9PkDj9KNVQ==</latexit>\nFIG. 1. (a) Schematic setup for the measurement of pumped\nspin currents in the antiferromagnet (AF)-normal metal (NM)\nstructure. The in-plane spin-polarization of the spin current\ncan be detected by the measurements of inverse spin-Hall\nvoltagesVx\nISHE andVy\nISHE. Possible interface microstructures\nare schematically depicted in (b), (c), and (d). The cross-\nsublattice spin mixing conductance gABvanishes [is \fnite]\nfor the interface depicted in (c) [(b) and (d)].\nresults from a concerted e\u000bect of cross-sublattice spin\npumping and a dissipative coupling resulting from the\nsublattice-symmetry breaking at AF-NM interface. Fur-\nthermore, the ensuing dissipative coupling is found to re-\nsult in magnon-magnon level attraction and coalescence\nobservable via typical magnetic resonance experiments.\nThe system thus constitutes a novel and unique platform\nfor investigating this interplay between magnon level re-\npulsion, attraction, and non-Hermitian physics via in-situ\ndamping matrix engineering by, for example, spin trans-\nfer torques.\nModel .{ We treat the vdW material as a two-sublattice\nmagnet described by the magnetization \felds MAand\nMBthat correspond to the sublattices AandB. We con-\nsider magnetic free-energy density [21] F=\u00160HEMA\u0001\nMB=Ms+\u00160\u0000\nM2\nAz+M2\nBz\u0001\n=2\u0000\u00160H\u0001(MA+MB),\nwithMsthe saturation magnetization of each sublattice.\nThe inter-sublattice exchange coupling parametrized by\nHE>0 favors antiferromagnetic order. In addition, an\nexternal dc magnetic \feld His applied in-plane. The sec-\nond term represents the easy-plane anisotropy. Gilbert\ndamping is accounted for by the viscous Rayleigh dissi-\npation functional [52, 54, 55] via the symmetric matrix\n\u0011\u0010\u00100:R[_MA;_MB] =P\n\u0010\u00100R\nVdr\u0011\u0010\u00100_M\u0010\u0001_M\u00100=2, where\nf\u0010;\u00100g=fA;Bg. The ensuing magnetization dynam-\nics is described by the coupled Landau-Lifshitz-Gilbert\n(LLG) equations,\n_m\u0010=\u0000\u00160\rm\u0010\u0002he\u000b\n\u0010+\u000b\u0010\u00100m\u0010\u0002_m\u00100+\u001c\u0010; (1)\nin terms of the unit vectors m\u0010\u0011M\u0010=Ms. The e\u000bective\felds are given by he\u000b\n\u0010= (1=\u00160)\u0002@F=@M\u0010=H\u0000\nHE\u001bx\n\u0010\u00100m\u00100\u0000Ms(m\u0010\u0001^z)^z, with\u001bxthe Pauli matrix\nand\r > 0 is the gyromagnetic ratio magnitude. The\nGilbert damping parameters are de\fned through \u000b\u0010\u00100\u0011\n\rMs\u0011\u0010\u00100where, in particular, \u000bAB=\u000bBA\u0011\u000bod. Note\nthat sublattice asymmetry in our model is broken only\nby the Gilbert damping [52]. It results from the AF-\nNM interface and spin pumping-mediated losses [49, 52].\nThe magnetization dynamics may be excited by a time-\ndependent magnetic \feld h\u0011\u00160\rhRF(t) that produces\na torque\u001c\u0010=m\u0010\u0002h.\nMagnetization dynamics and magnon modes.{ We now\ninvestigate the magnon modes in the material when the\ntwo sublattice magnetizations are non-collinear in their\nequilibrium con\fguration [Fig. 1(a)]. In the presence of\nan in-plane external magnetic \feld H=H^y, the mag-\nnetic ground state becomes meq\n\u0010=\u0006cos\u001e^x+ sin\u001e^y,\nwhere the non-collinearity is captured by the \fnite an-\ngle\u001ethat satis\fes sin \u001e=H=2HE[see Fig. 1(a)]. The\nmagnetic ground state is invariant under a twofold rota-\ntional operation around the yaxis,C2y, in combination\nwith sublattice exchange A$B, i.e.C2ymeq\nA=meq\nB.\nLinearizing the LLG equations (1), considering m\u0010=\nmeq\n\u0010+\u000em\u0010ei!t, the coupled dynamical equations become,\ni!\u000em+=meq\nA\u0002(A+\u000em++i!\u0001\u0016\u000b\u000em\u0000) +\u001c+;(2a)\ni!\u000em\u0000=meq\nA\u0002(A\u0000\u000em\u0000+i!\u0001\u0016\u000b\u000em+) +\u001c\u0000;(2b)\nwith the two magnetization dynamics or magnon modes\ndescribed by the \felds \u000em\u0006=\u000emA\u0006C2y\u000emB, the\ntorques\u001c\u0006=\u001cA\u0006C2y\u001cBand the operator A\u0006=\n(\u00160\rHE+i!\u0016\u000b)\u0006(\u00160\rHE+i!\u000bod)C2y. Furthermore,\nwe have reformulated the Gilbert damping parameters\nas\u000bAA= \u0016\u000b+ \u0001\u0016\u000band\u000bBB= \u0016\u000b\u0000\u0001\u0016\u000b. Note that\nwhen sublattice symmetry is assumed, i.e., \u000bAA=\u000bBB,\nthe Eqs. (2a) and (2b) become decoupled since \u0001\u0016 \u000b=\n0. In the absence of dissipation, the magnon eigen-\nmodes are captured well by the \felds \u000em\u0006, with the\neigenfrequencies for the so-called optical and acoustic\nmagnon modes being !+=\u00160\rp\n2MsHEcos2\u001eand\n!\u0000=\u00160\rp\n2HE(Ms+ 2HE) sin\u001e, respectively. The\ntwo modes can be excited selectively by a careful choice\nof the rf-\feld h[21]. In general, the excitation of ( \u0006)-\nmodes one at a time, which demands \u001c\u0007= 0 for the\ntorque, imposes the conditions h=\u0006C2yh, respectively.\nSpin pumping .{ We now investigate spin transport\nacross the AF-NM interface resulting from the excitation\nof magnetization dynamics by rf magnetic \feld. The dc\nspin pumping current injected into the adjacent NM is\ngiven by [14]\ne\n~Is=X\n\u0010\u001002fA;Bgg\u0010\u00100hm\u0010\u0002_m\u00100i; (3)\nwhereh\u0001\u0001\u0001i stands for the time-average over the period\nof oscillation. 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The frequency and \feld dependence of the spin-pumping current is represented by the plot of Fj\n\u0006(!), withjthe\ndirections of polarization and damping parameters \u0016 \u000b= 0:05,\u000bod= 0:01 and \u0001\u0016\u000b= 0:005. At panels (a), (b) and (c), we have\ndepicted the spin current with spin-polarization along x,y- andz-direction, respectively, due to coherent excitation of optical\nmodes. Similarly, in panels (d), (e) and (f), we have plotted the polarization components of the spin current when acoustic\nmodes are excited. At the inset of each panel we show Fj\n\u0006, evaluated at the resonant frequencies !=!\u0006. In the inset of panels\n(a), (b), (d) and (e), the curves correspond to di\u000berent \u000bod, while at panels (c) and (f) \u0001\u0016 \u000bis modi\fed. Other parameters\nemployed at the plots were extracted for the vdW antiferromagnet CrCl 3[21].\ndescribe the intra-sublattice spin mixing conductance.\nAn asymmetric interfacial coupling [13{16], resulting in\ngAA6=gBB, occurs when the two magnetic sublattices\nare incommensurately exposed to the NM (see Fig. 1(c)\nfor an example in which gAA= 0). The o\u000b-diagonal\ncross-sublattice conductance satisfy gAB=gBAand are\nnonzero when both the sublattices are (partly) exposed to\nNM [Fig. 1(b) and (d)]. Such interfaces can be achieved\nwith layered magnets [5, 56] (Fig. 1), but are not possi-\nble with synthetic AFs [28]. Our goal here is to estab-\nlish the experimentally detectable spin pumping current\nas a direct probe of the 2 \u00022 spin mixing conductance\nmatrix [Eq. (3)]. In particular, we are interested in es-\ntablishing unique signatures of the cross-sublattice con-\nductancesgAB=gBAthat elude a direct experimental\nobservation thus far.\nIn typical experimental setups (Fig. 1), the spin\npumping current is detected via inverse spin-Hall e\u000bect\n(ISHE) [57{60]. In metals with strong spin-orbit cou-\npling, a nonequilibrium spin current jsinduces a trans-\nverse charge current jc=\u0012SH(2e=~)js\u0002~ \u001b, where~ \u001bde-\nnotes the spin polarization direction and \u0012SHis the spin\nHall angle [57]. Under open circuit conditions, the gen-\nerated charge current is countered by an induced inverse\nspin Hall voltage VISHE proportional to the charge, andthus spin, current. The voltage thus generated is propor-\ntional to the spin current injected into the NM [Eq. (3)]\nand is directly detected in experiments [61].\nIn order to relate spin mixing conductance matrix ele-\nments to experimental observables, we evaluate the spin\npumping current Eq. (3) in the limit \u000bAA\u0019\u000bBB, i.e.,\n\u0001\u0016\u000bsmall. In this perturbative regime, Eqs. (2a) and\n(2b) decouple and the system eigenmodes are the opti-\ncal and acoustic magnon modes. The corresponding spin\npumping currents are evaluated to be\nI\u0006\nz= (gAA\u0000gBB)Fx\n\u0006(!)^x+gABFz\n\u0006(!)^z\n+ (gAA+gBB\u00062gAB)Fy\n\u0006(!)^y:(4)\nThe\u0006index labels optical and acoustic modes, which\nare driven by the rf magnetic \feld with frequency !\nand amplitude h+;\u001e= 2hycos\u001eandh\u0000;\u001e= 2hxsin\u001e,\nrespectively. The expression in Eq. (4) represents a\npure spin current that \rows across the AF-NM inter-\nface (along zaxis). Its various components pertaining to\ndirections in the spin space are proportional to the func-\ntionsFj\n\u0006(!), as detailed in the Supplemental Material\nin [62] (see Eqs. (25)-(27)), which have been obtained to\nthe \frst order in \u0001\u0016 \u000b. The in-plane components Fx;y\n\u0006,\nsatisfying Fy\n\u0006(!) = tan\u001eFx\n\u0006(!), are independent of \u0001\u0016 \u000b.\nHowever, the out-of-plane component of the spin current4\n/Fz\n\u0006(!) scales linearly with \u0001\u0016 \u000b. The resulting \feld-\nand frequency-dependence of the spin current I\u0006\nzcompo-\nnents are plotted in Fig. 2 employing system parameters\nrelevant for the vdW AF CrCl 3[21]. The various spin\ncurrent components are displayed in panels (a), (b) and\n(c) of Fig. 2 for the optical mode, and in panels (d), (e),\nand (f) for the acoustic mode.\nIn contrast with the case of spin pumping via collinear\nmagnets, in which a dc spin current polarized along the\nequilibrium order is generated [47{49, 51, 60, 61], the\ncanted AF under consideration pumps spin with compo-\nnents along all three directions [Eq. (4)]. As detailed in\nthe Supplemental Material [62], Fy;z\n\u0006/sin\u001eimplying\nyandzcomponents of the spin pumping current vanish\nfor\u001e= 0. Our result thus reduces to the existing under-\nstanding of collinear AFs [14, 15]. These additional com-\nponents of the spin pumping current for the canted AF\ntogether with the existence of two independent magnon\nmodes constitute some of the unique features and op-\nportunities o\u000bered by this system. For example, detec-\ntion of the ISHE voltage in two orthogonal directions [as\ndepicted in Fig. 1(a)] enables determination of both in-\nplane spin current components. By detecting these while\nexciting the two magnon modes one at a time, we may\ndetermine the full spin mixing conductance matrix with\nthe cross-sublattice term given by\ngAB=gAA\u0000gBB\n4 tan\u001e\u0012I+\ns;y\nI+s;x\u0000I\u0000\ns;y\nI\u0000s;x\u0013\n; (5)\nwhich assumes the condition gAA6=gBB. This accom-\nplishes a key goal and constitutes a main result of this\npaper.\nFurthermore, existence of the spin current zcompo-\nnent [Eq. (4)] that we \fnd is unconventional and coun-\nterintuitive as the magnon spin is expected to lie in\nthe same plane as the equilibrium sublattice magnetiza-\ntions [24, 25]. However, this component is nonzero only\nwhen \u0001\u0016\u000b6= 0,gAA6=gBB,gAB6= 0, and\u001e6= 0 im-\nplying that it results from a complex interplay of the\nsublattice-symmetry breaking dissipative coupling and\na cross-sublattice interference e\u000bect. Such physics, es-\npecially dissipative coupling [33, 34], appears to go be-\nyond the magnon picture considered thus far and consti-\ntutes another key result of our work. While an out-of-\nplane spin component has not been measured in typical\nspin pumping experiments [57, 60, 61], the recent ther-\nmal drag-mediated detection of such an out-of-plane spin\ncomponent [63] provides one possible method for its di-\nrect observation.\nMagnon level attraction .{ We now discuss the magnon\neigenmodes which become dissipatively coupled [see\nEqs. (2a) and (2b)] due to the sublattice-symmetry\nbreaking Gilbert damping [52], i.e. nonzero \u0001\u0016 \u000b. While a\n\\reactive\" coupling between magnon modes has been ob-\nserved in various systems [21, 22, 27{29], dissipative cou-\npling remains less explored and invokes non-Hermitian\nµ0H[T]\n<latexit sha1_base64=\"BYO018/2BTGPnIVfz+lP5J4t3zQ=\">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSSi6LHopccK/YImhM120y7d3YTdSbGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvSjnT4LrfVmljc2t7p7xb2ds/ODyyj086OskUoW2S8ET1IqwpZ5K2gQGnvVRRLCJOu9H4Ye53J1RplsgWTFMaCDyULGYEg5FC2/ZFFrqNvg/0CfLWLAjtqltzF3DWiVeQKirQDO0vf5CQTFAJhGOt+56bQpBjBYxwOqv4maYpJmM8pH1DJRZUB/ni8plzYZSBEyfKlARnof6eyLHQeioi0ykwjPSqNxf/8/oZxHdBzmSaAZVkuSjOuAOJM4/BGTBFCfCpIZgoZm51yAgrTMCEVTEheKsvr5POVc27rt08Xlfr90UcZXSGztEl8tAtqqMGaqI2ImiCntErerNy68V6tz6WrSWrmDlFf2B9/gByiZON</latexit>\n(a)\n<latexit sha1_base64=\"Ac+cs8EQeHY6Uohp+aTWQJMy8+k=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI97xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ai0ONUg==</latexit>\n(b)\n<latexit sha1_base64=\"zVpXaXYpCy/5Krr2sSqRENfT7FA=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqaFxWvWrm8r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjMiNUw==</latexit>\n(c)\n<latexit sha1_base64=\"0u7bBHPDKBCJbAK7avM20Uz0++4=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI77xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ajk2NVA==</latexit>\n(d)\n<latexit sha1_base64=\"+qTaDHOrJ165Z1nDA3KKJzLMLOY=\">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BItQL2VXKnosevFY0X5Au5RsNtuGZrNLkhXK0p/gxYMiXv1F3vw3pu0etPXBwOO9GWbm+Yng2jjONyqsrW9sbhW3Szu7e/sH5cOjto5TRVmLxiJWXZ9oJrhkLcONYN1EMRL5gnX88e3M7zwxpXksH80kYV5EhpKHnBJjpYdqcD4oV5yaMwdeJW5OKpCjOSh/9YOYphGThgqidc91EuNlRBlOBZuW+qlmCaFjMmQ9SyWJmPay+alTfGaVAIexsiUNnqu/JzISaT2JfNsZETPSy95M/M/rpSa89jIuk9QwSReLwlRgE+PZ3zjgilEjJpYQqri9FdMRUYQam07JhuAuv7xK2hc1t167vK9XGjd5HEU4gVOoggtX0IA7aEILKAzhGV7hDQn0gt7Rx6K1gPKZY/gD9PkDj9KNVQ==</latexit>\nµ0H[T]\n<latexit sha1_base64=\"BYO018/2BTGPnIVfz+lP5J4t3zQ=\">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSSi6LHopccK/YImhM120y7d3YTdSbGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvSjnT4LrfVmljc2t7p7xb2ds/ODyyj086OskUoW2S8ET1IqwpZ5K2gQGnvVRRLCJOu9H4Ye53J1RplsgWTFMaCDyULGYEg5FC2/ZFFrqNvg/0CfLWLAjtqltzF3DWiVeQKirQDO0vf5CQTFAJhGOt+56bQpBjBYxwOqv4maYpJmM8pH1DJRZUB/ni8plzYZSBEyfKlARnof6eyLHQeioi0ykwjPSqNxf/8/oZxHdBzmSaAZVkuSjOuAOJM4/BGTBFCfCpIZgoZm51yAgrTMCEVTEheKsvr5POVc27rt08Xlfr90UcZXSGztEl8tAtqqMGaqI2ImiCntErerNy68V6tz6WrSWrmDlFf2B9/gByiZON</latexit>\nµ0H[T]\n<latexit sha1_base64=\"BYO018/2BTGPnIVfz+lP5J4t3zQ=\">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSSi6LHopccK/YImhM120y7d3YTdSbGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvSjnT4LrfVmljc2t7p7xb2ds/ODyyj086OskUoW2S8ET1IqwpZ5K2gQGnvVRRLCJOu9H4Ye53J1RplsgWTFMaCDyULGYEg5FC2/ZFFrqNvg/0CfLWLAjtqltzF3DWiVeQKirQDO0vf5CQTFAJhGOt+56bQpBjBYxwOqv4maYpJmM8pH1DJRZUB/ni8plzYZSBEyfKlARnof6eyLHQeioi0ykwjPSqNxf/8/oZxHdBzmSaAZVkuSjOuAOJM4/BGTBFCfCpIZgoZm51yAgrTMCEVTEheKsvr5POVc27rt08Xlfr90UcZXSGztEl8tAtqqMGaqI2ImiCntErerNy68V6tz6WrSWrmDlFf2B9/gByiZON</latexit>Im[\u0000+\u0000\u0000]<latexit sha1_base64=\"dsawMljXjT9cR+hs42pS7ayNkWE=\">AAACB3icbVBNS8NAEN3Ur1q/qh4FCRZBEEoiFT0Wveitgv2AJJbNdtss3WzC7kQsITcv/hUvHhTx6l/w5r9x0/agrQ8GHu/NMDPPjzlTYFnfRmFhcWl5pbhaWlvf2Nwqb++0VJRIQpsk4pHs+FhRzgRtAgNOO7GkOPQ5bfvDy9xv31OpWCRuYRRTL8QDwfqMYNBSt7zvAn2A9DrMHJcE7C49zrqpGwcsr8zrlitW1RrDnCf2lFTQFI1u+cvtRSQJqQDCsVKObcXgpVgCI5xmJTdRNMZkiAfU0VTgkCovHf+RmYda6Zn9SOoSYI7V3xMpDpUahb7uDDEEatbLxf88J4H+uZcyESdABZks6ifchMjMQzF7TFICfKQJJpLpW00SYIkJ6OhKOgR79uV50jqp2rXq6U2tUr+YxlFEe+gAHSEbnaE6ukIN1EQEPaJn9IrejCfjxXg3PiatBWM6s4v+wPj8Af2Wmgg=</latexit>\n0.120.130.140.155.86.26.6\n0.050.100.150.200246810\n0.050.100.150.200.00.51.01.52.0\n0.100.110.120.130.140.150.250.350.450.020.100.180.20.61.0FIG. 3. Eigenfrequencies of the coupled magnon modes are\nplotted as a function of the external dc magnetic \feld. The\nreal (!r) and imaginary ( !i) parts of the frequencies are dis-\nplayed in panels (a) and (c) respectively, for \u0016 \u000b= 0:1 and\n\u000bod= \u0001\u0016\u000b= 0:07. In panel (b), we zoom-in on the level\ncrossing of panel (a). The additional curves shown in pur-\nple and red correspond to the same \u0016 \u000band \u0001\u0016\u000bas (a), but\nwith\u000bod= 0. The level attraction is depicted in gray when\n\u0001\u0016\u000b= 0:07 and \u0016\u000b=\u000bod= 0. (d) Imaginary part of the\nmagnetic susceptibility Im[ \u001f+\n\u001e\u001e]vs.applied magnetic \feld for\nvarious values of \u0001\u0016 \u000b. The susceptibility corresponds to the\noptical mode and has been evaluated at !=!+.\nphysics [35, 64]. Solving Eqs. (2a) and (2b) without an\nexternal rf drive, we obtain the complex eigenmode fre-\nquencies!\u0006=!r\u0006+i!i\u0006.!rand!i, respectively, cap-\nture the energy and inverse lifetime of the magnon modes\nand have been plotted against the external dc magnetic\n\feld in Fig. 3 (a)-(c). Due to the dissipative nature of\nthe coupling, a magnon-magnon level attraction is ob-\nserved. The grey curve in Fig. 3 (b), corresponding to\n\u0016\u000b=\u000bod= 0 and \u0001\u0016 \u000b6= 0, depicts a perfect level coa-\nlescence or mode synchronization [33, 34]. Such values\nfor Gilbert damping matrix require a dc spin transfer\ntorque drive in the NM. An undriven system however\nimposes constraints \u000bAA;\u000bBB>0, i.e. \u0016\u000b > \u0001\u0016\u000band\n\u000bod\u0014p\u000bAA\u000bBB[52]. In this scenario, we \fnd a complex\ninterplay of repulsion and attraction between the two\nmodes. The resulting eigenfrequencies split slightly (see\nFig. 3(a) when \u0016 \u000b= 0:1,\u000bod= 0:07 and \u0001\u0016\u000b= 0:07) while\ncoalescing at a speci\fc point, the so-called exceptional\npoint [65]. Furthermore, Fig. 3(d) depicts the imaginary\npart of dynamic susceptibility, which is directly accessible\nin experiments [21, 27, 66]. A peak in this susceptibility\nprovides an additional experimental signature of the level\nattraction when \u0001\u0016 \u000bis su\u000eciently large. Thus, vdW AFs\nunder consideration constitute a rich platform for realiz-\ning non-Hermitian physics and magnon-magnon level at-\ntraction via AF-NM interface engineering and spin trans-\nfer torques exerted on AF by the NM.\nSummary .{ We have theoretically uncovered unique\nand intriguing cross-sublattice spin pumping and5\nmagnon-magnon level attraction e\u000bects in a model canted\nantiferromagnet. By providing guidance to experiments\nin extracting the interfacial spin mixing conductance ma-\ntrix and key signatures of level attraction, we hope to es-\ntablish van der Waals antiferromagnets interfaced with\na heavy metal layer as a fertile and convenient plat-\nform for realizing and investigating unconventional non-\nHermitian physics.\nNote added: During the manuscript preparation, we\nnoticed a recent related preprint [67] that studies level-\nrepulsion and hybridization of magnonic modes in bulk\nsymmetry-breaking synthetic antiferromagnets. It how-\never does not discuss spin pumping or the dissipative\nmagnon level attraction - the two key novelties of our\nwork.\nThis work was supported by the European Union's\nHorizon 2020 Research and Innovation Programme un-\nder Grant No. DLV-737038 \\TRANSPIRE,\" and the\nResearch Council of Norway through its Centres of Ex-\ncellence funding scheme, Project No. 262633, \\QuSpin\".\n[1] K. S. Burch, D. Mandrus, and J.-G. 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B 82, 214403 (2010).\n[62] SeeSee Supplemental Material at [link] for details of two-\nsubllatice magnetization dynamics and spin pumping..[63] C. O. Avci, E. Rosenberg, M. Huang, J. Bauer, C. A.\nRoss, and G. S. D. Beach, Phys. Rev. Lett. 124, 027701\n(2020).\n[64] B. Flebus, R. A. Duine, and H. M. Hurst, Phys. Rev. B\n102, 180408 (2020).\n[65] Y. Tserkovnyak, Phys. Rev. Research 2, 013031 (2020).\n[66] A. J. Berger, E. R. J. Edwards, H. T. Nembach, A. D.\nKarenowska, M. Weiler, and T. J. Silva, Phys. Rev. B\n97, 094407 (2018).\n[67] J. Lu, M. Li, and W. He, \\Symmetry breaking in-\nduced magnon-magnon coupling in synthetic antifer-\nromagnets,\" (2020), arXiv:2011.01583 [cond-mat.mes-\nhall].7\nSUPPLEMENTAL MATERIAL\nIn this Supplemental Material, we explicitly show the calculation of two-sublattice spin pumping in AF-NM struc-\ntures. The result is applied when the magnetization dynamics is coherently driven by a linearly polarized ac magnetic\n\feld. Basic details of magnon-magnon hybridization are also provided.\nMagnetization dynamics and magnon modes\nMagnetic \ructuations of the two-sublattice magnet results from the linearization of LLG equations. Representing\nthe \ructuations by m\u0010=meq\n\u0010+\u000em\u0010ei!t, with\u0010=fA;Bg, the resulting coupled dynamical equations becomes,\ni!\u000em+=meq\nA\u0002[(\u00160\rHE+i!\u0016\u000b)\u000em++ (\u00160\rHE+i!\u000bod)C2y\u000em+] +i!\u0001\u0016\u000bmeq\nA\u0002\u000em\u0000+\u001c+; (6a)\ni!\u000em\u0000=meq\nA\u0002[(\u00160\rHE+i!\u0016\u000b)\u000em\u0000\u0000(\u00160\rHE+i!\u000bod)C2y\u000em\u0000] +i!\u0001\u0016\u000bmeq\nA\u0002\u000em++\u001c\u0000; (6b)\nwith the torques \u001c\u0006=mA\u0002h\u0006. The Gilbert damping terms are represented by \u000bAB=\u000bBA=\u000bod,\u000bAA= \u0016\u000b+ \u0001\u0016\u000b\nand\u000bBB= \u0016\u000b\u0000\u0001\u0016\u000b. Note that when sublattice symmetry is restored, i.e., \u000bAA=\u000bBB, the modes \u000em+and\u000em\u0000\nbecome decoupled and characterize, optical and acoustic modes respectively. In a compact form the Eqs. (6a) and\n(6b) read,\n\u0012h+\nh\u0000\u0013\n=\u0012M+T\nT\u0003M\u0000\u0013\n|{z}\nM\u0012\n\u000em+\n\u000em\u0000\u0013\n(7)\nwith the ac \felds h+= (h+;\u001e;0)Tandh\u0000= (h\u0000;\u001e;h\u0000;\u0012)T, whereh+;\u001e= 2hycos\u001e,h\u0000;\u001e= 2hxsin\u001eandh\u0000;\u0012= 2hz.\nThe matrices M\u0006andTare de\fned as\nT=\u0012\u0000i!\u0001\u0016\u000b 0\n0i!\u0001\u0016\u000b\u0013\n; (8)\nM+=\u0012\u0000A\u0000i!(\u0016\u000b+\u000bodcos 2\u001e)i!\ni! B +i!(\u0016\u000b\u0000\u000bod)\u0013\n; (9)\nM\u0000=\u0012C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)\u0000i!\n\u0000i!\u0000D\u0000i!(\u0016\u000b+\u000bod)\u0013\n; (10)\nwhere we introduced the following constants A= 2\u00160\rHEcos2\u001eandB=\u00160\rMs,C= 2\u00160\rHEsin2\u001eandD=\n\u00160\r(Ms+ 2HE). It is worth noting that det [ M] = det [ M+] det\u0002\nM\u0000\u0000TM\u00001\n+T\u0003\n\u0019det [M+] det [M\u0000] in the limit of\nsmall \u0001\u0016\u000b. Beyond this approximation, the optical and acoustic magnonic modes are no longer decoupled. Instead,\nthese modes hybridize with \u0001\u0016 \u000bbeing the dissipative coupling that generates level attraction. The magnon-magnon\nhybrid eigenfrequencies were numerically calculated and depicted in Fig. 3.\nAssuming that \u0001\u0016 \u000bis small, we use the standard formula for the inverse of a block matrix to obtain the \felds \u000em\u0006,\n\u000em+=M\u00001\n+h+\u0000M\u00001\n+TM\u00001\n\u0000h\u0000; (11)\n\u000em\u0000=M\u00001\n\u0000h\u0000+M\u00001\n\u0000TM\u00001\n+h+: (12)\nThe \frst term at the right-hand side is the zero order correction in \u0001\u0016 \u000b, where M\u00001\n\u0006corresponds to the symmetric\ndynamic susceptibility matrix. The components are de\fned as\u0002\nM\u00001\n\u0006\u0003\n11=\u001f\u0006\n\u001e\u001e,\u0002\nM\u00001\n\u0006\u0003\n22=\u001f\u0006\n\u0012\u0012and\u0002\nM\u00001\n\u0006\u0003\n12=i\u001f\u0006\n\u001e\u0012,\nwhere\n\u001f+\n\u001e\u001e=B+i(\u0016\u000b\u0000\u000bod)!\n!2\u0000(B+i!(\u0016\u000b\u0000\u000bod)) (A+i!(\u0016\u000b+\u000bodcos 2\u001e)); (13)\n\u001f+\n\u001e\u0012=\u0000!\n!2\u0000(B+i!(\u0016\u000b\u0000\u000bod)) (A+i!(\u0016\u000b+\u000bodcos 2\u001e)); (14)8\nand\n\u001f\u0000\n\u001e\u001e=\u0000D\u0000i!(\u0016\u000b+\u000bod)\n!2\u0000(D+i!(\u0016\u000b+\u000bod)) (C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)); (15)\n\u001f\u0000\n\u0012\u0012=C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)\n!2\u0000(D+i!(\u0016\u000b+\u000bod)) (C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)); (16)\n\u001f\u0000\n\u001e\u0012=!\n!2\u0000(D+i!(\u0016\u000b+\u000bod)) (C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)): (17)\nThe second contribution in Eqs. (11) and (12) is linear in the damping \u0001\u0016 \u000b, with the 2\u00022 matrices,\nM\u00001\n+TM\u00001\n\u0000=!\u0001\u000b\ndet [M+] det [M\u0000]\u0012i\u0000\nBD+!2\u0001\n(B+C)!\n(A+D)!\u0000i\u0000\nAC+!2\u0001\u0013\n; (18)\nM\u00001\n\u0000TM\u00001\n+=!\u0001\u000b\ndet [M+] det [M\u0000]\u0012i\u0000\nBD+!2\u0001\n(A+D)!\n(B+C)!\u0000i\u0000\nAC+!2\u0001\u0013\n: (19)\nTwo-sublattice Spin Pumping\nIn this section we evaluate the spin pumping in the two-sublattice magnet. Final expressions for the injected spin\ncurrents are found in terms of the dynamical magnetic susceptibility. To start with, let us consider the spin pumping\ncurrent into the normal metal given by Eq. (3). The \ructuations are represented by m\u0010=meq\n\u0010+\u000eM\u0010, with the\nreal-valued \felds de\fned as \u000eMA=\u000eMA;\u0012^z+\u000eMA;\u001e(^z\u0002meq\nA) and\u000eMB=\u000eMB;\u0012^z+\u000eMB;\u001e(^z\u0002meq\nB). We \fnd\nthat the spin pumping current Isbecomes\n4\u0019\n~Is=gAAh\u000eMA\u0002\u000e_MAi+gAB\u0010\nh\u000eMA\u0002\u000e_MBi+h\u000eMB\u0002\u000e_MAi\u0011\n+gBBh\u000eMB\u0002\u000e_MBi: (20)\nNote that since a time-average is involved in the evaluation of previous equation, linear terms in the \ructuations do\nnot contribute. In order to relate each term in Eq. (20) with the dynamical susceptibility, Eqs. (13)-(17), we represent\n\u000eM\u0010by complex-valued \felds as \u000eM\u0010;\u0016= Re\u0002\n\u000em\u0010;\u0016ei!t\u0003\n, where\u0016=f\u0012;\u001eg. Next, we write the \felds \u000em\u0010;\u0016in the\neigenbasis by the following relations \u000emA=B;\u0012 =1\n2(\u000em\u0006;\u0012\u0006\u000em\u0007;\u0012) and\u000emA=B;\u001e =1\n2(\u000em\u0006;\u001e\u0006\u000em\u0007;\u001e). Therefore,\nwe obtain\nh\u000eMA\u0002\u000e_MAi=!\n4\u0000\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\n+ Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n+ Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n+;\u0012i\u0003\n+ Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\u0001\nmeq\nA;\n(21)\nh\u000eMB\u0002\u000e_MBi=!\n4\u0000\nIm\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n+ Im\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\n\u0000Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n+;\u0012i\u0003\n\u0000Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\u0001\nmeq\nB;\n(22)\nh\u000eMA\u0002\u000e_MBi+h\u000eMB\u0002\u000e_MAi=!\n2cos\u001e\u0000\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n\u0000Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n+;\u0012i\u0003\u0001^x\n+!\n2sin\u001e\u0000\nIm\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n\u0000Im\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\u0001^y+!\n2sin(2\u001e)Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003^z:(23)\nBelow, we present examples of spin pumping current, evaluating Eqs. (21-23), due to the coherent magnonic\nexcitation by linearly polarized external ac \felds.\nExciting the Optical and Acoustic Modes\nAccording to the selection rules described in the main text, excitation of the optical mode requires a \feld that\nsatisfyh\u0000=0andh+;\u001e6= 0. On the other hand, excitation of the acoustic mode requires h+= 0 andh\u00006= 0. From\nEqs. (11) and (12), we \fnd the \felds \u000em\u0006when the ac magnetic \feld is h+=hy^y(optical mode) and h\u0000=hx^x\n(acoustic mode). To determine the spin current, we evaluate each term given by the Eqs. (23)-(22). Up to linear\norder in the Gilbert damping di\u000berence \u0001\u0016 \u000b, we \fnd\ne\n~I\u0006\ns=!\n4Im\u0002\nh\u000em\u0006;\u001e\u000em\u0003\n\u0006;\u0012i\u0003\n[(gAA\u0000gBB) cos\u001e^x+ ((gAA+gBB)\u00072gAB) sin\u001e^y]\n+!\n2Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n\u0006gABsin(2\u001e)^z: (24)9\nThe functions Fj\n\u0006(!) introduced in the main text, are thus de\fned as\nFx\n\u0006(!) =!\n4Im\u0002\nh\u000em\u0006;\u001e\u000em\u0003\n\u0006;\u0012i\u0003\ncos\u001e; (25)\nFy\n\u0006(!) =!\n4Im\u0002\nh\u000em\u0006;\u001e\u000em\u0003\n\u0006;\u0012i\u0003\nsin\u001e; (26)\nFz\n\u0006(!) =!\n2Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n\u0006sin(2\u001e): (27)\nThe explicit expression for each contribution in Eq. (24) obeys\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\n=!Bh2\n+;\u001e\njdet [M+]j2; (28)\nIm\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n=!Dh2\n\u0000;\u001e\njdet [M\u0000]j2; (29)\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n+= \u0001\u000b!B\u0000\nBD+!2\u0001\u0000\n!2\n\u0000\u0000!2\u0001\nh2\n+;\u001e\njdet [M+]j2jdet [M\u0000]j2; (30)\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n\u0000= \u0001\u000b!D\u0000\nBD+!2\u0001\u0000\n!2\u0000!2\n+\u0001\nh2\n\u0000;\u001e\njdet [M+]j2jdet [M\u0000]j2: (31)"    },    {        "title": "1512.05408v2.Parity_time_symmetry_breaking_in_magnetic_systems.pdf",        "content": "Parity-time symmetry breaking in magnetic systems\nAlexey Galda1and Valerii M. Vinokur1\n1Materials Science Division, Argonne National Laboratory,\n9700 South Cass Avenue, Argonne, Illinois 60439, USA\n(Dated: July 1, 2016)\nThe understanding of out-of-equilibrium physics, especially dynamic instabilities and dynamic phase transi-\ntions, is one of the major challenges of contemporary science, spanning the broadest wealth of research areas that\nrange from quantum optics to living organisms. Focusing on nonequilibrium dynamics of an open dissipative\nspin system, we introduce a non-Hermitian Hamiltonian approach, in which non-Hermiticity reflects dissipation\nand deviation from equilibrium. The imaginary part of the proposed spin Hamiltonian describes the e \u000bects of\nGilbert damping and applied Slonczewski spin-transfer torque. In the classical limit, our approach reproduces\nLandau-Lifshitz-Gilbert-Slonczewski dynamics of a large macrospin. We reveal the spin-transfer torque-driven\nparity-time symmetry-breaking phase transition corresponding to a transition from precessional to exponen-\ntially damped spin dynamics. Micromagnetic simulations for nanoscale ferromagnetic disks demonstrate the\npredicted e \u000bect. Our findings can pave the way to a general quantitative description of out-of-equilibrium phase\ntransitions driven by spontaneous parity-time symmetry breaking.\nINTRODUCTION\nA seminal idea of parity-time ( PT)-symmetric quantum\nmechanics [1,2], that has stated that the condition of Hermitic-\nity in standard quantum mechanics required for physical ob-\nservables and energy spectrum to be real can be replaced by\na less restrictive requirement of invariance under combined\nparity and time-reversal symmetry, triggered an explosive de-\nvelopment of a new branch of science. The interpretation of\nPTsymmetry as “balanced loss and gain” [3] connected PT\nsymmetry breaking to transitions between stationary and non-\nstationary dynamics and established its importance to under-\nstanding of the applied field-driven instabilities. Experiments\non a diverse variety of strongly correlated systems and phe-\nnomena including optics and photonics [4–10], superconduc-\ntivity [11–13], Bose-Einstein condensates [14], nuclear mag-\nnetic resonance quantum systems [15], and coupled electronic\nand mechanical oscillators [16–18] revealed PT symmetry-\nbreaking transitions driven by applied fields. These observa-\ntions stimulated theoretical focus on far-from-equilibrium in-\nstabilities of many-body systems [12,13,19] that are yet not\nthoroughly understood.\nHere we demonstrate that the non-Hermitian extension\nof classical Hamiltonian formalism provides quantitative de-\nscription of dissipative dynamics and dynamic phase transi-\ntions in out-of-equilibrium systems. Focusing on the case of\nspin systems, we consider the zero-temperature spin dynamics\nunder the action of basic nonconservative forces: phenomeno-\nlogical Gilbert damping [20] and Slonczewski spin-transfer\ntorque [21] (STT). The latter serves as the most versatile way\nof directly manipulating magnetic textures by external cur-\nrents. We propose a general complex spin Hamiltonian, in\nwhich Slonczewski STT emerges from an imaginary magnetic\nfield. ThePT-symmetric version of the Hamiltonian is shown\nto exhibit a phase transition associated with inability of the\nsystem to sustain the balance between loss and gain above a\ncertain threshold of external nonconservative field.\nIn the classical limit of a large spin, our formalism repro-\nduces the standard Landau-Lifshitz-Gilbert-Slonczewski [20–22] (LLGS) equation of spin dynamics and predicts the PT\nsymmetry-breaking phase transition between stationary (con-\nservative) and dissipative (nonconservative) spin dynamics. In\nthis Letter we focus on a single spin, yet our theory can be ex-\ntended to coupled spin systems in higher dimensions. More-\nover, as spin physics maps onto a wealth of strongly corre-\nlated systems and phenomena ranging from superconductiv-\nity to cold-atom and two-level systems, our results provide\nquantitative perspectives on the nature of phase transitions as-\nsociated withPTsymmetry breaking in a broad class of far-\nfrom-equilibrium systems.\nWe introduce the non-Hermitian Hamiltonian for a single\nspin operator ˆS:\nˆH=E\u0000ˆS\u0001+ij\u0001ˆS\n1\u0000i\u000b; (1)\nwhere E\u0000ˆS\u0001denotes the standard Hermitian spin Hamil-\ntonian determined by the applied magnetic field Hand\nmagnetic anisotropy constants kiin the x;y;zdirections:\nE\u0000ˆS\u0001=P\nikiˆS2\ni+\rH\u0001ˆS. A schematic system setup is shown\nin Fig. 1. The phenomenological constant \u000b>0 in Eq. (1)\ndescribes damping; the imaginary field ijis responsible for\nthe applied Slonczewski STT, with jS=ep(~=2e)\u0011Jbeing\nthe spin-angular momentum deposited per second in the di-\nrection epwith spin polarization \u0011=(J\"\u0000J#)=(J\"+J#) of the\nincident current J; and\r=g\u0016B=~is the absolute value of the\ngyromagnetic ratio; g'2,\u0016Bis the Bohr magneton, ~is the\nPlanck’s constant, and eis the elementary charge. We conjec-\nture that Eq. (1) serves as a fundamental generalization of the\nHamiltonian description of both quantum and classical spin\nsystems, which constitutes one of our core results. This form\nof the Hamiltonian proves extremely useful for the general un-\nderstanding of STT-driven dissipative spin dynamics. In this\nwork we focus primarily on the classical limit of spin dynam-\nics, while the semiclassical limit of finite spin will be consid-\nered elsewhere.\nSpin dynamics in the classical limit is conveniently\nobtained by studying expectation value of the Hamilto-\nnian (1) with respect to SU(2) spin-coherent states [23,24]:arXiv:1512.05408v2  [cond-mat.other]  30 Jun 20162\nFIG. 1. Schematic representation of the system setup. Ferromagnetic\ncylinder (blue) is placed in magnetic field Happlied along the xaxis,\nand STT-inducing electric current Jis polarized in the direction ep\nalong theyaxis. Spin-polarized current passes through a nonmag-\nnetic metallic spacer and induces torque (Slonczewski STT, shown\nby the small red arrow) on the total spin S.\njzi=ezˆS+jS;\u0000Si, where ˆS\u0006\u0011ˆSx\u0006iˆSy, and z2Cis the stan-\ndard stereographic projection of the spin direction on a unit\nsphere, z=(sx+isy)=(1\u0000sz), with si\u0011Si=S. Note that such\nparametrization of the phase space for a classical single-spin\nsystem (i.e., in the limit S!1 ) guarantees the invariance of\nthe traditional equation of motion [24] under generalization to\nnon-Hermitian Hamiltonians (see Appendix A):\n˙z=i(1+¯zz)2\n2S@H\n@¯z; (2)\nwhere zand ¯zform a complex conjugate pair of stereographic\nprojection coordinates, and\nH(z;¯z)=hzjˆHjzi\nhzjzi(3)\nis the expectation value of the Hamiltonian (1) in spin-\ncoherent states (for a detailed review see, e.g., Ref. [25]). In\nthis formulation, the eigenstates of ˆHcorrespond to the fixed\npoints ziof the equation of motion for H, while the eigen-\nvalues (i.e., energy values) are equal to Hevaluated at the\ncorresponding fixed points, Ei=H(zi;¯zi).\nAssuming a constant magnitude of the total spin, ˙S=0,\nEq. (2) reduces to the following equation of spin dynamics in\nthe classical limit:\n˙S=rS(ReH)\u0002S+1\nS\u0002rS(ImH)\u0002S\u0003\u0002S: (4)\nHere we refer to the real and imaginary parts of the Hamil-\ntonian functionHwritten in the spin Srepresentation. For\nthe non-Hermitian Hamiltonian (1), Eq. (4) reproduces theLLGS equation describing dissipative STT-driven dynamics\nof a macrospin:\n\u0010\n1+\u000b2\u0011˙S=\rHe\u000b\u0002S+\u000b\r\nS[\rHe\u000b\u0002S]\u0002S+1\nSS\u0002[S\u0002j]\n+\u000bS\u0002j; (5)\n\rHe\u000b=rSE(S): (6)\nThe first two terms in Eq. (5) describe the standard Landau-\nLifshitz torque and dissipation, while the last two are respon-\nsible for the dissipative (antidamping) and conservative (e \u000bec-\ntive field) Slonczewski STT contributions, correspondingly,\nboth of which appear naturally from the imaginary magnetic\nfield term in the Hamiltonian (1).\nPT-SYMMETRIC HAMILTONIAN\nSlonczewski STT turns the total spin-angular momentum,\nS, in the direction of spin-current polarization, ep, without\nchanging its magnitude. On the S-sphere this can be repre-\nsented by a vector field converging in the direction of epand\noriginating from the antipodal point. It is the imaginary mag-\nnetic field ijthat produces exactly the same e \u000bect on spin dy-\nnamics, according to Eq. (2). The action of STT is invari-\nant under the simultaneous operations of time reversal and\nreflection with respect to the direction ep, which is the un-\nderlying reason behind the inherent PTsymmetry of certain\nSTT-driven magnetic systems, including the one considered\nbelow.\nBefore turning to the PT-symmetric form of Hamilto-\nnian (1), we note that PT-symmetric systems play an im-\nportant role in the studies of nonequilibrium phenomena\nand provide a unique nonperturbative tool for examining the\nphase transition between stationary and nonstationary out-\nof-equilibrium dynamics. We show that despite being non-\nHermitian, such systems can exhibit both of the above types\nof behavior, depending on the magnitude of the external non-\nconservative force. In the parametric regime of unbroken\nPTsymmetry, systems exhibit physical properties seemingly\nequivalent [26] to those of Hermitian systems: real energy\nspectrum, existence of integrals of motion (see Appendix C),\nand, notably, the validity of the quantum Jarzynski equal-\nity [27]. However, in the regime of brokenPT symme-\ntry, one observes complex energy spectrum and nonconser-\nvative dynamics. Therefore, the true transition between sta-\ntionary and nonstationary dynamics can be identified as the\nPTsymmetry-breaking phase transition.\nSpin systems are generally subject to various non-\nlinear magnetic fields including ones originating from\nshape, exchange, and magnetocrystalline and magnetoelastic\nanisotropies. Restricting ourselves for simplicity to a second-\norder anisotropy term, we arrive at the following Hamiltonian\nfor a nonlinear magnetic system with uniaxial anisotropy and\napplied Slonczewski STT:\nˆHPT=\rH0\u0010\nkzˆS2\nz+hxˆSx+i\fˆSy\u0011\n; (7)3\nFIG. 2. Real (a) and imaginary (b) parts of energy spectrum of the Hamiltonian (7) as a function of the STT parameter \fforhx=1 and\nD=20. Blue and red lines correspond to the eigenvalues E1;2andE3\u00006, respectively. The first PTsymmetry-breaking transition occurs at\nj\fj=\f1\u00194:5.\nwhere the applied magnetic field hxis measured in units of\nsome characteristic magnetic field H0, and\fis a dimension-\nless STT parameter determining the relative to Samount of\nangular momentum transfered in time \u001c\u0011(\rH0)\u00001(charac-\nteristic timescale of the dynamics, used as a unit of dimen-\nsionless time in what follows). The Hamiltonian (7) modeling\nthe dynamics of the free magnetic layer in a typical nanopil-\nlar device with fixed polarizer layer (see Fig. 1) is PTsym-\nmetric: It is invariant under simultaneous action of parity and\ntime-reversal operators ( y!\u0000y,t!\u0000t,i!\u0000i). Because\nthe Hamiltonian ˆHPTcommutes with an antilinear operator\nPT, its eigenvalues are guaranteed to appear in complex con-\njugate pairs. Notice that PT-symmetric Hamiltonian (7) does\nnot contain damping, which is assumed to be negligibly small,\nas is the case in many experimental systems.\nCLASSICAL SPIN SYSTEM\nIn order to best illustrate the mechanism of PT symme-\ntry breaking, we focus on the classical limit, S!1 and\nkzS!D=2, where Dis the dimensionless uniaxial anisotropy\nconstant. Formula (2) then yields the following equation of\nmotion for the Hamiltonian (7):\n˙z(t)=\u0000i(hx+\f)\n2 \nz2\u0000hx\u0000\f\nhx+\f!\n\u0000i D z1\u0000jzj2\n1+jzj2; (8)\nwith up to six fixed points zk,k=1;:::; 6.\nShown in Fig. 2 are the real and imaginary parts of the en-\nergy spectrum E1\u00006as functions of the STT amplitude \f. It\nreveals that in a system with strong anisotropy, D\u001d1,PT\nsymmetry breaking occurs in three separate transitions, with\nthe first one atj\fj=\f1=jhxjq\u00021+p\n1+(2D=jhxj)2\u0003=2,\nwhich corresponds to the smallest amplitude of STT at which\nIm(E),0. Therefore,PTsymmetry is not broken in the en-\ntire phase space of initial spin directions simultaneously, at\nvariance to the linear spin system with D=0 (see AppendixB). Instead, the regions of broken and unbroken PTsymme-\ntry may coexist in the phase diagram of a nonlinear spin sys-\ntem.\nIn what follows we consider a system described by\nthe Hamiltonian (7) with hx=1 and D=20. For all\nj\fj<\f 1\u00194:5,PT symmetry is unbroken and the character\nof spin (magnetization) dynamics is oscillatory in the entire\nphase diagram, i.e., for all possible initial conditions z. At\nj\fj=\f1the phase transition (first of the three, see Fig. 2) oc-\ncurs sharply in a wide region around the easy plane, jzj=1,\ni.e. near the equator of the unit S-sphere, shown in gray in\nFigs. 3(a) and 3(b) in Cartesian and stereographic projection\ncoordinates, respectively. It this region the nature of spin dy-\nnamics becomes fundamentally di \u000berent—all spin trajectories\nfollow the lines connecting the fixed points z1andz2, where\nz1;2=\u0000\u0000Dhx\u0006ip\n\f4\u0000\f2h2x\u0000D2h2x\u0001=(hx+\f)\f, and no closed\ntrajectories are possible; see Fig. 3(b).\nAsj\fjis increased further, the region of broken PTsymme-\ntry expands until it eventually closes around the fixed point z5\nat\f2\u00199:3 (second bifurcation in Fig. 2) and, eventually, the\nlast region of unbroken PT symmetry near z3disappears at\n\f3\u001910:8. The second and third phase transitions are less rel-\nevant experimentally as they occur in the vicinity of the least\nfavorable spin directions (parallel and antiparallel to the hard\naxisz) and at considerably higher applied currents.\nThe predicted transition from precessional dynamics (un-\nbrokenPTsymmetry) to exponentially fast saturation in the\ndirection z1(hx;\f) for any initial spin position around the easy\nplane (brokenPT symmetry) occurs in the setup with mu-\ntually perpendicular applied magnetic field and Slonczewski\nSTT. Such a transition in nanoscale magnetic structures can\nbe used for STT- or magnetic-field-controlled magnetization\nswitching in spin valves and a variety of other experimental\nsystems. This e \u000bect can further be used for direct measure-\nments of the amplitude of the applied STT, which, unlike the\napplied current, can be hard to quantify experimentally.4\nFIG. 3. (a, b) Spin dynamics described by Eq. (8) with hx=1,\f=4:7, and D=20.PTsymmetry is broken in the shaded region around\nthe easy planejzj=1 (dashed line), encompassing two fixed points, z1;2(blue dots), appearing as source and sink nodes. The green line\ndepicts a typical nonoscillatory spin trajectory in the region of broken PTsymmetry. Red dots represent the fixed points z3\u00006. (c) Results of\nmicromagnetic simulations for \f\u0003as a function of stereographic projection of the initial spin direction z. In the blue region, 4 :6.\f\u0003.4:8, the\nPTsymmetry is broken at all j\fj<\f\u0003, and the spin takes under 0 :5 ns to saturate in the direction of z1, which is in full agreement with the\nanalytical result.\nNUMERICAL SIMULATIONS OF PTSYMMETRY\nBREAKING\nHere we present the results of numerical simulations con-\nfirming thePT symmetry-breaking phase transition in the\nclassical single-spin system (7) by modeling magnetization\ndynamics of a ferromagnetic disk 100 nm in diameter and\nd=5 nm thick, which is consistent with the anisotropy con-\nstant D=20 in Eq. (8). We used the following typical\npermalloy material parameters: damping constant \u000b=0:01,\nexchange constant Aex=13\u000210\u000012J/m and saturation mag-\nnetization Msat=800\u0002103A/m . The simulations were car-\nried out using the open-source GPU-accelerated micromag-\nnetic simulation program MuMax3 [28] based on the LLGS\nequation (5) discretized in space. We used a cubic discretiza-\ntion cell of 5 nm in size, which is smaller than the exchange\nlength in permalloy, lex=(2Aex=\u00160M2\nsat)1=2\u00195:7 nm.\nThe permalloy disk was simulated in an external magnetic\nfield applied along the xaxis, H0=400 Oe, which corre-\nsponds to the characteristic time \u001c\u00190:14 ns. The STT was\nproduced by applying electric current perpendicular to the\ndisk in the zdirection with spin polarization \u0011=0:7 along\nep=ˆy(see Fig. 1) and current density \fmeasured in dimen-\nsionless units of 2 eH0Msatd=\u0011~\u00190:7\u0002108A/cm2. While\nsuch current density is comparable to typical switching cur-\nrent densities in STT-RAM devices [29,30], its magnitude can\nbe optimized for various practical applications by changing\nH0and adjusting the size, shape, and material of the ferro-\nmagnetic element.\nFor all possible initial spin directions z, we calculated the\ncritical amplitude of the applied STT, \f\u0003, for which the char-\nacter of spin dynamics changes from oscillatory (at j\fj< \f\u0003)\nto exponential saturation. Shown in Fig. 3(c) is the color map\nof\f\u0003as a function of zin complex stereographic coordinates.\nThe region shown in the shades of blue corresponds to the\ninitial conditions z, for which the minimum values of \fthat\nwould guarantee saturation of spin dynamics in the directionofz1in under 0:5 ns are between 4.6 and 4.8. This is in full\nagreement with the region of broken PTsymmetry at \f=4:7\ncalculated analytically, i.e., the shaded area in Fig. 3(b) [the\noutline is repeated in Fig. 3(c) for comparison]. Outside of\nthis region, a considerably larger magnitude of the applied\nSTT is required to break PTsymmetry.\nThe agreement between theoretical results and micromag-\nnetic simulations is remarkable considering the non-zero\nGilbert damping parameter ( \u000b=0:01) and nonlinear e \u000bects\n(demagnetizing field, finite size and boundary e \u000bects, etc.)\ninherently present in the micromagnetic simulations but not\nincluded in the model Hamiltonian (7).\nCONCLUSION\nThe presented non-Hermitian Hamiltonian formulation of\ndissipative nonequilibrium spin dynamics generalizes the pre-\nvious result [31], where the classical Landau-Lifshitz equa-\ntion was derived from a non-Hermitian Hamilton operator,\nto open STT-driven spin systems. The introduction of Slon-\nczewski STT in the imaginary part of the Hamiltonian re-\nvealed the possibility of STT-driven PTsymmetry-breaking\nphase transition. Micromagnetic simulations confirm the\nPTsymmetry-breaking phenomenon in realistic mesoscopic\nmagnetic systems and its robustness against weak dissipation,\nindicating high potential for impacting spin-based informa-\ntion technology. The way STT enters the complex Hamilto-\nnian (1), i.e. as imaginary magnetic field, provides a unique\ntool for studying Lee-Yang zeros [32] in ferromagnetic Ising\nand Heisenberg models and, more generally, dynamics and\nthermodynamics in the complex plane of physical parame-\nters. We envision further realizations of the PT symmetry-\nbreaking phase transitions in diverse many-body condensed-\nmatter systems and the expansion of practical implementa-\ntions of thePT symmetry beyond the present realm of op-\ntics [33] and acoustics [34].5\nFIG. 4. Real (a) and imaginary (b) parts of energy spectrum of the linear Hamiltonian ˆH0PTas functions of \fforhx=1.PTsymmetry-\nbreaking transition occurs at j\fj=1.\nACKNOWLEDGEMENTS\nWe thank Alex Kamenev for critical reading of the\nmanuscript and valuable suggestions. This work was sup-\nported by the U.S. Department of Energy, O \u000ece of Science,\nBasic Energy Sciences, Materials Sciences and Engineering\nDivision.\nAPPENDIX A. GENERALIZED EQUATION OF MOTION\nFOR NON-HERMITIAN SPIN HAMILTONIANS IN THE\nCLASSICAL LIMIT\nThe remarkable simplicity of the equation of motion (2)\nfor an arbitrary non-Hermitian spin Hamiltonian function\nHstems from the choice of parametrization of the phase\nspace, i.e., the complex stereographic projection coordinates\nfz;¯zg. The extension of classical equations of motion to non-\nHermitian Hamiltonians in terms of canonical coordinates\nfq;pghas the following generalized form [35]:\n \nq\np!\n= \n\u00001rq;p(ReH)\u0000G\u00001rq;p(ImH); (A1)\nwhere \nandGare the symplectic structure and metric of\nthe underlying classical phase space, respectively, which must\nsatisfy the compatibility condition [36] written in the matrix\nform as\n\n\u00001=\u0010\n\u0000\u00001\n \u0000\u00001\u0011T: (A2)\nIn the stereographic projection coordinates, one obtains the\nfollowing symplectic structure and metric:\n\n =2\n\u00001+jzj2\u00012 \n0i\n\u0000i0!\n; G=2\n\u00001+jzj2\u00012 \n0 1\n1 0!\n:(A3)\nIt is the form of these matrices that leads to Eq. (2), where the\nreal and imaginary parts (as written in the Srepresentation)\nof the Hamiltonian combine naturally into a single complexfunctionH. Therefore, when written in stereographic projec-\ntion coordinates, the generalized classical equation of motion\nfor non-Hermitian Hamiltonians coincides with that for tradi-\ntional Hermitian Hamiltonians.\nAPPENDIX B.PTSYMMETRY BREAKING IN LINEAR\nSPIN SYSTEM\nIn the absence of magnetic anisotropy fields, the Hamilto-\nnian (1) from the main text becomes linear:\nˆH0= \rH+ij\n1\u0000i\u000b!\n\u0001ˆS; (B1)\nwith e \u000bects of applied magnetic field, damping and Slon-\nczewski STT contributions all incorporated in the complex\nmagnetic field (in parentheses). The PT-symmetric version\nof this Hamiltonian has mutually perpendicular real and imag-\ninary parts of the complex magnetic field:\nˆH0PT=\rH0\u0010\nhxˆSx+i\fˆSy\u0011\n: (B2)\nThe quantum spin-1\n2version of this Hamiltonian describes\na two-level quantum system with balanced loss and gain and\nis known [37,38] to exhibit PTsymmetry-breaking transition\nathx=\u0006\f. Whenjhxj>j\fj, the Hamiltonian ˆH0PThas real\neigenvalues, \u00151;2=\u0006p\nh2x\u0000\f2, while in the parametric region\njhxj<j\fjeigenvalues are imaginary, see Fig. 4.\nThe generalized Hamilton’s equation of motion (2) for this\nHamiltonian takes the form\n˙z(t)=f(t)=\u0000i(hx+\f)\n2 \nz2\u0000hx\u0000\f\nhx+\f!\n: (B3)\nIt has two fixed points: z1;2=\u0006p\n(hx\u0000\f)=(hx+\f), which\ncorrespond to stereographic coordinates of the spin equi-\nlibria directions on a unit Bloch sphere. The character\nof spin dynamics around the fixed points is fully deter-\nmined by the eigenvalues of the complex Jacobian matrix6\nFIG. 5. Spin trajectories for classical linear Hamiltonian H0PTin the regime of unbroken, (a) and (c), (at \f=0:8) and broken, (b) and (d), (at\n\f=1:2)PTsymmetry for hx=1.\nJC=@(Ref;Imf)=@(Rez;Imz)in their vicinity. The solu-\ntion of Eq. (B3) takes the form of a M ¨obius transformation:\nz(t)=z(0)+ihx+\fp\nh2x\u0000\f2tan p\nh2x\u0000\f2\n2t!\nihx\u0000\fp\nh2x\u0000\f2tan p\nh2x\u0000\f2\n2t!\nz(0)+1: (B4)\nThe parametric region jhxj>j\fjdefines the regime of un-\nbrokenPT symmetry with real Hamiltonian eigenvalues,\nE1;2=\u0006p\nh2x\u0000\f2. In the classical approximation, the spin\nperforms persistent oscillations along circular orbits about the\nfixed points z1;2situated on the real axis, see Figs. 5(a) and\n5(c). The eigenvalues of JCatz1;2are purely imaginary, iden-\ntifying the fixed points are centers , according to the stan-\ndard classification [39]. Closed trajectories represent PT-\nsymmetric dynamics with balanced loss and gain: the spin\nsystem gains and loses equal amounts of energy from the non-\nconservative term i\fSyon they <0 andy >0 segments of\ntrajectories, respectively.As the driving parameter j\fjis increased, z1;2move towards\neach other until they eventually collide at j\fj=jhxj, which\nmarks the point of PT symmetry breaking. In the regime\nofbrokenPT symmetry,jhxj<j\fj, the energy eigenvalues\nare imaginary, E1;2=\u0006ip\n\f2\u0000h2x, and no closed trajectories\nare possible. The eigenvalues of the Jacobian JC(z1;2) in this\nregime are real and of the same sign, defining z1;2assink\nandsource nodes [39]. All trajectories follow circle arcs con-\nnecting the fixed points with coordinates in three-dimensional\nspace\u0010\n0;\u0006p\n1\u0000(hx=\f)2;\u0000hx=\f\u0011\n, which are now out of the xz\nplane, see Figs. 5(b) and 5(d).\nWe emphasize that in the linear Hamiltonian (B1) the ef-\nfects of damping can always be fully compensated by the\nappropriate choice of the applied STT. For instance, the\nPT-symmetric Hamiltonian (B2) describes a single spin\nplaced in external magnetic field H=H0(hx;\u000b\f; 0)and STT\nj=\rH0(\u0000\u000bhx;\f;0). Note that in the general case of nonlin-\near spin Hamiltonian, dissipation cannot be completely can-\nceled by STT. However, in many magnetic systems dissipative\nforces are extremely weak, which justifies the approximation\nof zero damping.7\nAPPENDIX C. INTEGRALS OF MOTION\nThere exists a remarkable similarity between spin dynam-\nics in the regime of unbroken PT symmetry and that of a\nfully Hermitian system. Indeed, it follows from Eq. 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Anosov, V . I. Arnold, and S. Kh. Aranson. Ordinary di \u000ber-\nential equations and smooth dynamical systems, Vol. 1 (Springer\nScience & Business Media, 1997)."    },    {        "title": "0708.3323v1.Enhancement_of_the_Gilbert_damping_constant_due_to_spin_pumping_in_noncollinear_ferromagnet_nonmagnet_ferromagnet_trilayer_systems.pdf",        "content": "arXiv:0708.3323v1  [cond-mat.mes-hall]  24 Aug 2007Enhancement of the Gilbert damping constant due to spin pump ing in non-collinear\nferromagnet / non-magnet / ferromagnet trilayer systems\nTomohiro Taniguchi1,2, Hiroshi Imamura2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577,\n2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan\n(Dated: October 29, 2018)\nWe analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-\ncollinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert\ndamping constant depends both on the precession angle of the magnetization of the free layer and\non the direction of the magntization of the fixed layer. We find the condition to be satisfied to\nrealize strong enhancement of the Gilbert damping constant .\nPACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es\nThere is currently great interest in the dynamics of\nmagnetic multilayers because of their potential applica-\ntions in non-volatile magnetic random access memory\n(MRAM) and microwave devices. In the field of MRAM,\nmuch effort has been devoted to decreasing power con-\nsumption through the use of current-induced magnetiza-\ntion reversal (CIMR) [1, 2, 3, 4, 5, 6, 7]. Experimentally,\nCIMR is observed as the current perpendicular to plane-\ntype giant magnetoresistivity (CPP-GMR) of a nano pil-\nlar, in which the spin-polarized current injected from the\nfixed layer exerts a torque on the magnetization of the\nfree layer. The torque induced by the spin current is\nutilized to generate microwaves.\nThe dynamics of the magnetization Min a ferromag-\nnet under an effective magnetic field Beffis described by\nthe Landau-Lifshitz-Gilbert (LLG) equation\ndM\ndt=−γM×Beff+α0M\n|M|×dM\ndt,(1)\nwhereγandα0are the gyromagnetic ratio and the\nGilbert damping constant intrinsic to the ferromagnet,\nrespectively. The Gilbert damping constant is an im-\nportant parameter for spin electronics since the critical\ncurrent density of CIMR is proportional to the Gilbert\ndamping constant [8, 9] and fast-switching time magne-\ntization reversal is achieved for a large Gilbert damp-\ning constant [10]. Several mechanisms intrinsic to ferro-\nmagnetic materials, such as phonon drag [11] and spin-\norbit coupling [12], have been proposed to account for\nthe origin of the Gilbert damping constant. In addition\nto these intrinsic mechanisms, Mizukami et al.[13, 14]\nand Tserkovnyak et al.[15, 16] showed that the Gilbert\ndampingconstantinanon-magnet(N) /ferromagnet(F)\n/ non-magnet(N) trilayersystem is enhanced due to spin\npumping. Tserkovnyak et al.[17] also studied spin pump-\ning in a collinear F/N/F trilayer system and showed that\nenhancement of the Gilbert damping constant depends\non the precession angle of the magnetization of the free\nlayer.\nOn the other hand, several groups who studied CIMR\nin a non-collinear F/N/F trilayer system in which theFIG. 1: (Color online) The F/N/F trilayer system is schemat-\nically shown. The magnetization of the F 1layer (m1) pre-\ncesses around the z-axis with angle θand angular velocity ω.\nThe magnetization of the F 2layer (m2) is fixed with tilted\nangleρ. The precession of the magnetization in the F 1layer\npumpsspin current Ipump\nsintotheNandF 2layer, andcreates\nthe spin accumulation µNin the N layer. The spin accumu-\nlation induces the backflow spin current Iback(i)\ns(i= 1,2).\nmagnetization of the free layer is aligned to be perpen-\ndicular to that of the fixed layer have reported the reduc-\ntion of the critical current density [5, 6, 7]. Therefore, it\nis intriguing to ask how the Gilbert damping constant is\naffected by spin pumping in non-collinear F/N/F trilayer\nsystems.\nIn this paper, we analyze the enhancement of the\nGilbert damping constant due to spin pumping in non-\ncollinear F/N/F trilayer systems such as that shown in\nFig. 1. Following Refs. [15, 16, 17, 18], we calculate the\nspin current induced by the precession of the magnetiza-\ntion of the free layer and the enhancement of the Gilbert\ndamping constant. We show that the Gilbert damping\nconstant depends not only on the precession angle θof\nthe magnetization of a free layer but also on the angle ρ\nbetweenthemagnetizationsofthefixedlayerandthepre-\ncession axis. The Gilbert damping constant is strongly\nenhanced if angles θandρsatisfy the condition θ=ρor\nθ=π−ρ.\nThe system we consider is schematically shown in Fig.\n1. A non-magnetic layer is sandwiched between two fer-\nromagnetic layers, F 1and F 2. We introduce the unit2\nvectormito represent the direction of the magnetiza-\ntion of the i-th ferromagnetic layer. The equilibrium\ndirection of the magnetization m1of the left free fer-\nromagnetic layer F 1is taken to exist along the z-axis.\nWhen an oscillatingmagnetic field is applied, the magne-\ntization of the F 1layer precesses around the z-axis with\nangleθ. The precession of the vector m1is expressed\nasm1= (sinθcosωt,sinθsinωt,cosθ), whereωis the\nangular velocity of the magnetization. The direction of\nthe magnetization of the F 2layer,m2, is assumed to be\nfixed and the angle between m2and thez-axis is repre-\nsented byρ. The collinear alignment discussed in Ref.\n[17] corresponds to the case of ρ= 0,π.\nBefore studying spin pumping in non-collinear sys-\ntems, we shall give a brief review of the theory of\nspin pumping in a collinear F/N/F trilayer system [17].\nSpin pumping is the inverse process of CIMR where the\nspin current induces the precession of the magnetization.\nContrary to CIMR, spin pumping is the generation of\nthe spin current induced by the precession of the mag-\nnetization. The spin current due to the precession of the\nmagnetization in the F 1layer is given by\nIpump\ns=/planckover2pi1\n4πg↑↓m1×dm1\ndt, (2)\nwhereg↑↓is a mixing conductance [18, 19] and /planckover2pi1is the\nDirac constant. Spins are pumped from the F 1layer\ninto the N layer and the spin accumulation µNis cre-\nated in the N layer. Spins also accumulate in the F 1\nand F 2layers. In the ferromagnetic layers the trans-\nverse component of the spin accumulation is assumed to\nbe absorbed within the spin coherence length defined as\nλtra=π/|k↑\nFi−k↓\nFi|, wherek↑,↓\nFiis the spin-dependent\nFermi wave number of the i-th ferromagnet. For fer-\nromagnetic metals such as Fe, Co and Ni, the spin co-\nherence length is a few angstroms [20]. Hence, the spin\naccumulation in the i-th ferromagnetic layer is aligned to\nbe parallel to the magnetization, i.e., µFi=µFimi. The\nlongitudinal component of the spin accumulation decays\non the scale of spin diffusion length, λFi\nsd, which is of the\norder of 10 nm for typical ferromagnetic metals [21].\nThe difference in the spin accumulation of ferromag-\nnetic and non-magnetic layers, ∆ µi=µN−µFimi(i=\n1,2), induces a backflow spin current, Iback(i)\ns, flowing\ninto both the F 1and F 2layers. The backflow spin cur-\nrentIback(i)\nsis obtained using circuit theory [18] as\nIback(i)\ns=1\n4π/braceleftbigg2g↑↑g↓↓\ng↑↑+g↓↓(mi·∆µi)mi\n+g↑↓mi×(∆µi×mi)/bracerightbig\n,(3)\nwhereg↑↑andg↓↓are the spin-up and spin-down con-\nductances, respectively. The total spin current flowing\nout of the F 1layer is given by Iexch\ns=Ipump\ns−Iback(1)\ns\n[17]. The spin accumulation µFiin the F ilayer is ob-\ntained by solving the diffusion equation. We assume\nthat spin-flip scattering in the N layer is so weak thatwe can neglect the spatial variation of the spin current\nwithin the N layer, Iexch\ns=Iback(2)\ns. The torque τ1\nacting on the magnetization of the F 1layer is given by\nτ1=Iexch\ns−(m1·Iexch\ns)m1=m1×(Iexch\ns×m1). For\nthe collinear system, we have\nτ1=g↑↓\n8π/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\nm1×dm1\ndt,(4)\nwhereν= (g↑↓−g∗)/(g↑↓+g∗) is the dimensionless\nparameter introduced in Ref. [17]. The Gilbert damping\nconstant in the LLG equation is enhanced due to the\ntorqueτ1asα0→α0+α′with\nα′=gLµBg↑↓\n8πM1dF1S/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\n,(5)\nwheregLis the Land´ e g-factor,µBis the Bohr magneton,\ndF1is the thickness of the F 1layer andSis the cross-\nsection of the F 1layer.\nNext, we move on to the non-collinear F/N/F trilayer\nsystem with ρ=π/2, in which the magnetization of the\nF2layer is aligned to be perpendicular to the z-axis. Fol-\nlowing a similar procedure, the LLG equation for the\nmagnetization M1in the F 1layer is expressed as\ndM1\ndt=−γeffM1×Beff+γeff\nγ(α0+α′)M1\n|M1|×dM1\ndt,(6)\nwhereγeffandα′are the effective gyromagnetic ratio\nand the enhancement of the Gilbert damping constant,\nrespectively. The effective gyromagnetic ratio is given by\nγeff=γ/parenleftbigg\n1−gLµBg↑↓νcotθcosψsinωt\n8πMdF1Sǫ/parenrightbigg−1\n,(7)\nwhere cosψ= sinθcosωt=m1·m2and\nǫ= 1−ν2cos2ψ−ν(cot2θcos2ψ−sin2ψ+sin2ωt).(8)\nThe enhancement of the Gilbert damping constant is ex-\npressed as\nα′=gLµBg↑↓\n8πMdF1S/parenleftbigg\n1−νcot2θcos2ψ\nǫ/parenrightbigg\n.(9)\nItshouldbenotedthat, fornon-collinearsystems, both\nthegyromagneticratioandtheGilbert dampingconstant\nare modified by spin pumping, contrary to what occurs\nin collinear systems. The modification of the gyromag-\nnetic ratio and the Gilbert damping constant due to spin\npumping can be explained by considering the pumping\nspincurrentandthe backflowspincurrent[SeeFigs. 2(a)\nand 2(b)]. The direction of the magnetic moment car-\nried by the pumping spin current Ipump\nsis parallel to\nthe torque of the Gilbert damping for both collinear and\nnon-collinear systems. The Gilbert damping constant is\nenhanced by the pumping spin current Ipump\ns. On the\notherhand, the directionofthe magneticmoment carried3\nFIG. 2: (Color online) (a) Top view of Fig. 1. The dotted\ncircle in F 1represents the precession of magnetization M1\nand the arrow pointing to the center of this circle represent s\nthe torque of the Gilbert damping. The arrows in Ipump\nsand\nIback(1)\nsrepresent the magnetic moment of spin currents. (b)\nThe back flow Iback(1)\nshas components aligned with the di-\nrection of the precession and the Gilbert damping.\nby the backflow spin current Iback(1)\nsdepends on the di-\nrection of the magnetization of the F 2layer. As shown in\nEq. (3), the backflowspin current in the F 2layerIback(2)\ns\nhas a projection on m2. Since we assume that the spin\ncurrent is constant within the N layer, the backflow spin\ncurrent in the F 1layerIback(1)\nsalso has a projection on\nm2. For the collinear system, both Ipump\nsandIback(1)\ns\nare perpendicular to the precession torque because m2\nis parallel to the precession axis. However, for the non-\ncollinear system, the vector Iback(1)\nshas a projection on\nthe precession torque, as shown in Fig. 2(b). Therefore,\nthe angular momentum injected by Iback(1)\nsmodifies the\ngyromagnetic ratio as well as the Gilbert damping in the\nnon-collinear system.\nLet us estimate the effective gyromagnetic ratio using\nrealistic parameters. According to Ref. [17], the con-\nductancesg↑↓andg∗for a Py/Cu interface are given\nbyg↑↓/S= 15[nm−2] andν≃0.33, respectively. The\nLand´ eg-factor is taken to be gL= 2.1, magnetization is\n4πM= 8000[Oe] and thickness dF1= 5[nm]. Substitut-\ning these parameters into Eqs. (7) and (8), one can see\nthat|γeff/γ−1| ≃0.001. Therefore, the LLG equation\ncan be rewritten as\ndM1\ndt≃ −γM1×Beff+(α0+α′)M1\n|M1|×dM1\ndt.(10)\nThe estimated value of α′is of the order of 0.001. How-\never, we cannot neglect α′since it is of the same order\nas the intrinsic Gilbert damping constant α0[22, 23].\nExperimentally, the Gilbert damping constant is mea-\nsuredasthe width ofthe ferromagneticresonance(FMR)\nabsorptionspectrum. LetusassumethattheF 1layerhas\nno anisotropy and that an external field Bext=B0ˆzisapplied along the z-axis. We also assume that the small-\nangle precession of the magnetization around the z-axis\nis excited by the oscillating magnetic field B1applied in\nthexy-plane. The FMR absorption spectrum is obtained\nas follows [24]:\nP=1\nT/integraldisplayT\n0dtαγMΩ2B2\n1\n(γB0−Ω)2+(αγB0)2,(11)\nwhere Ω is the angular velocity of the oscillating mag-\nnetic field, T= 2π/Ω andα=α0+α′. Sinceαis very\nsmall, the absorption spectrum can be approximately ex-\npressedasP∝α0+∝an}bracketle{tα′∝an}bracketri}htandthehighestpointofthepeak\nproportional to ∝an}bracketle{t1/(α0+α′)∝an}bracketri}ht, where∝an}bracketle{tα′∝an}bracketri}htrepresents the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant. In Fig. 3(a), the time-averaged value\n∝an}bracketle{tα′∝an}bracketri}htfor a non-collinear system in which ρ=π/2 is plot-\nted by the solid line as a function of the precession an-\ngleθ. The dotted line represents the enhancement of\nthe Gilbert damping constant α′for the collinear system\ngiven by Eq. (5). The time-averaged value of the en-\nhancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}httakes\nits maximum value at θ= 0,πfor the collinear system\n(ρ= 0,π). Contrary to the collinear system, ∝an}bracketle{tα′∝an}bracketri}htof the\nnon-collinear system in which ρ=π/2 takes its maxi-\nmum value at θ=π/2.\nAs shown in Fig. 2(b), the backflow spin current\ngives a negative contribution to the enhancement of the\nGilbert damping constant. This contribution is given by\nthe projection of the vector Iback(1)\nsonto the direction\nof the torque of the Gilbert damping, which is repre-\nsented by the vector m1×˙m1. Therefore, the condition\nto realize the maximum value of the enhancement of the\nGilbert damping is satisfied if the projection of Iback(1)\ns\nontom1×˙m1takes the minimum value; i.e., θ=ρor\nθ=π−ρ.\nWe can extend the above analysis to the non-collinear\nsystemwith anarbitraryvalue of ρ. After performingthe\nappropriate algebra, one can easily show that the LLG\nequation for the magnetization of the F 1layer is given\nby Eq. (6) with\nγeff=γ/bracketleftBigg\n1−gLµBg↑↓νsinρsinωt(cotθcos˜ψ−cscθcosρ)\n8πMdS˜ǫ/bracketrightBigg−1\n(12)\nα′=gLµBg↑↓\n8πMdS/braceleftBigg\n1−ν(cotθcos˜ψ−cscθcosρ)2\n˜ǫ/bracerightBigg\n,\n(13)\nwhere cos ˜ψ= sinθsinρcosωt+ cosθcosρ=m1·m2\nand\n˜ǫ=1−ν2cos2˜ψ\n−ν{(cotθcos˜ψ−cscθcosρ)2−sin2˜ψ+sin2ρsin2ωt}.\n(14)\nSubstituting the realistic parameters into Eqs. (12) and\n(14), we can show that the effective gyromagnetic ratio4\n\u0013\n\u0003\u0013\u0011\u0013\u0013\u0015\u001a\u0003\u0013\u0011\u0013\u0013\u0016\u0015\u0003\u0013\u0011\u0013\u0013\u0016\u001a\n\u000bD\f\n\u000bE\f\u0013\u0011\u0013\u0013\u0016\u001a\n\u0013\u0011\u0013\u0013\u0016\u0015\n\u0013\u0011\u0013\u0013\u0015\u001a/c50/c0f/c12 /c50\n/c51/c1c/c41\n/c1e\n\u0013 /c50/c0f/c12 /c50\n/c52\u0013/c50/c0f/c12/c50\n/c51/c1c/c41\n/c1e\nFIG. 3: (Color online) (a) The time-averaged value of the en-\nhancement of the Gilbert damping constant α′is plotted as a\nfunction of the precession angle θ. The solid line corresponds\nto the collinear system derived from Eq. (9). The dashed\nline corresponds to the non-collinear system derived from E q.\n(5). (b) The time-averaged value of the enhancement of the\nGilbert damping constant α′of the non-collinear system is\nplotted as a function of the precession angle θand the an-\ngleρbetween the magnetizations of the fixed layer and the\nprecession axis.γeffcan be replaced by γin Eq. (6) and that the LLG\nequation reduces to Eq. (10). Figure 3(b) shows the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant ∝an}bracketle{tα′∝an}bracketri}htof Eq. (13). Again, the Gilbert\ndamping constant is strongly enhanced if angles θandρ\nsatisfy the condition that θ=ρorθ=π−ρ.\nIn summary, we have examined the effect of spin\npumping on the dynamics of the magnetization of mag-\nnetic multilayers and calculated the enhancement of the\nGilbertdampingconstantofnon-collinearF/N/Ftrilayer\nsystems due to spin pumping. The enhancement of the\nGilbert damping constant depends not only on the pre-\ncession angle θof the magnetization of a free layer but\nalso on the angle ρbetween the magnetizations of the\nfixed layerand the precession axis, as shown in Fig. 3(b).\nWe have shown that the θ- andρ-dependence of the en-\nhancement of the Gilbert damping constant can be ex-\nplained by analyzing the backflow spin current. The con-\ndition to be satisfied to realizestrongenhancement of the\nGilbert damping constant is θ=ρorθ=π−ρ.\nThe authors would like to acknowledge the valuable\ndiscussions we had with Y. Tserkovnyak, S. Yakata, Y.\nAndo, S. Maekawa, S. Takahashi and J. Ieda. This work\nwas supported by CREST and by a NEDO Grant.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[4] A. Deac, K. J. Lee, Y. Liu, O. Redon, M. Li, P. Wang,\nJ. P. Nozi´ eres, and B. Dieny, J. Magn. Magn. Mater.\n290-291 , 42 (2005).\n[5] A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys.\nLett.84, 3897 (2004).\n[6] K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86,\n022505 (2005).\n[7] T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl.\nPhys. Lett. 89, 172504 (2005).\n[8] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[9] J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J. M. George,\nG. Faini, J. B. Youssef, H. L. LeGall, and A. Fert, Phys.\nRev. B67, 174402 (2003).\n[10] R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett.\n92, 088302 (2004).\n[11] H. Suhl, IEEE Trans. Magn. 34, 1834 (1998).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[13] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn.\nMater.239, 42 (2002).[14] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[15] Y. Tserkovnyak and A. Brataas and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[16] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n[18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n[19] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[20] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[21] J. Bass and W. P. Jr., J. Phys.: Condens. Matter 19,\n183201 (2007).\n[22] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[23] F. Schreiber, J. Pflaum, Th. M¨ uhge, and J. Pelzl, Solid\nState Commun. 93, 965 (1995).\n[24] S. V. Vonsovskii, ed., FERROMAGNETIC RESO-\nNANCE (Israel Program for Scientific Translations Ltd.,\nJersalem, 1964)."    },    {        "title": "2307.07751v2.Switching_current_distributions_in_ferromagnetic_anomalous_Josephson_junctions.pdf",        "content": "SCD in anomalous JJs\nSwitching current distributions in ferromagnetic anomalous Josephson\njunctions\nC. Guarcello\n ,1, 2,a)F.S. Bergeret\n ,3, 4,b)and R. Citro\n1, 2, 5, c)\n1)Dipartimento di Fisica “E.R. Caianiello”, Universit` a di Salerno, Via Giovanni Paolo II, 132,\nI-84084 Fisciano (SA), Italy\n2)INFN, Sezione di Napoli Gruppo Collegato di Salerno, Complesso Universitario di Monte S. Angelo,\nI-80126 Napoli, Italy\n3)Centro de F´ ısica de Materiales, Centro Mixto CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5,\n20018 San Sebasti´ an, Spain\n4)Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebasti´ an,\nSpain\n5)CNR-SPIN c/o Universit´ a degli Studi di Salerno, I-84084 Fisciano (Sa), Italy\nWe investigate the switching current distributions of ferromagnetic anomalous Josephson junctions subjected\nto a linearly increasing bias current. Our study uncovers a significant correlation between the position\nof the switching current distributions and crucial system parameters, such as the strength of the spin-orbit\ncoupling and the Gilbert damping parameter. This indicates that these parameters can be directly determined\nthrough experimental measurements. By conducting a comprehensive analysis of the interplay among noise,\nmagnetization, phase dynamics, and the statistical properties of the switching current distribution, we\ndeepen our understanding of these intriguing cryogenic spintronics devices. These findings hold potential\nfor applications in the field of quantum computing architectures and information processing technologies.\nFerromagnetic anomalous Josephson junctions\n(FAJJs), formed by combining superconductivity with\nferromagnetic materials, have garnered significant\nattention due to their unique properties and potential\napplications in spintronics, quantum technologies, and\ninformation processing1–6. These junctions exhibit\nintriguing phenomena, including anomalous phase shifts\nand magnetization reversal induced by the flow of\nsupercurrent, making them promising candidates for a\nwide range of cutting-edge technological advancements.\nThe combination of superconductivity and\nferromagnetism in these junctions offers exciting\npossibilities for spintronics applications7. The\nexchange interaction between the superconducting\nand ferromagnetic regions enables the control of the\nmagnetization direction through the manipulation of\nsuperconducting phases. This unique functionality holds\ngreat promise for the development of spintronics memory\nelements, magnetic field sensors, and logic circuits with\nlow power consumption and high-speed operation.\nSince the work by Shukrinov et al.8, several\nstudies have explored the dynamics of FAJJs, for\ninstance focusing on the current-induced magnetization\nreversal phenomenon, in both short junctions9–19\nand superconducting quantum interference devices\n(SQUIDs)20–22. An interested reader may refer to\nRefs. 23 and 24 for a comprehensive review and further\ndetails.\nNoise influences the behavior of these systems,\nplaying an important role in both understanding their\na)Electronic mail: Author to whom correspondence should be\naddressed: cguarcello@unisa.it\nb)Electronic mail: fs.bergeret@csic.es\nc)Electronic mail: rocitro@unisa.itoperation and harnessing their full potential, although\nits effect has been little studied to date. Nonetheless,\nstochastic thermal fluctuations can profoundly affect\nthe behavior and the performance of these devices25,26.\nUnderstanding the interplay between noise and the\ncomplex dynamics of AFJJs is essential for optimizing\ntheir operation and ensuring the reliability of spintronics\nand quantum devices.\nIn the broader context of Josephson junctions (JJs),\nthe study of the switching current distribution (SCD)\nis of fundamental importance to characterize their\nbehaviour27. It provides insights into the dynamics of\nthe superconducting phase and the interplay between\nnoise and junction parameters, and it was throughly\ndiscussed, both theoretically and experimentally, in many\nworks about Josephson devices27–31. The SCD in\nconventional JJs can exhibit a wide range of behaviors,\nranging from narrow and well-defined distributions to\nbroad and multimodal distributions. In fact, noise\nsources introduce fluctuations in the superconducting\norder parameter, leading to variations in the switching\nprocess. Consequently, the shape of the SCD reflects\nthe intertwining of noise effects and the specific\nx yz\nImax\nFIG. 1. Sketch of the AFJJ driven by a linearly ramping bias\ncurrent, Ibias(t).arXiv:2307.07751v2  [cond-mat.supr-con]  11 Oct 2023SCD in anomalous JJs 2\ncharacteristics of the junction.\nCharacterizing the SCD and understanding the role of\nnoise in shaping its properties are crucial for designing\nand optimizing the performance of FAJJs for their\nvarious applications. In this Letter, we analyze the\nimpact of thermal noise on the behavior of FAJJs, with\na focus on the SCDs. In particular, we demonstrate how\nthey can reveal information on intrinsic characteristics\nof the junction, such as the Rashba coupling strength\nand the phenomenological Gilbert damping parameter.\nThis work adds a fundamental piece to the study of\nnoise-induced effects in magnetic Josephson systems,\nwhich plays a key role in understanding the actual\nresponse of a real device; indeed, we focus on the\nswitching dynamics under a linearly ramping bias\ncurrent, moving significantly away from the conventional\nscenarios considered so far, in which the phenomenon\nof pulse-controlled magnetization reversal8,12,14,19,25,26,32\nand the response to ac currents9,11,16had been largely\nconsidered.\nThe setup that we consider consists of a FAJJ, see\nFig. 1, with a thin ferromagnetic film with an out-of-\nplane magnetic anisotropy and a Rashba-like SOC25.\nDue to the interplay between the exchange field and\nthe SOC, the ferromagnetic junction presents a current-\nphase relation with the form33,34Iφ=Icsin(φ−φ0).\nHere, Icis the critical current of the junction, φis the\nJosephson phase difference, and φ0is the anomalous\nphase shift , which has been experimentally measured first\nin Josephson devices including topological insulators and\nAl/InAs heterostructures or nanowires35–38. The value\nofφ0depends on different system parameters, such as\nthe Rashba coupling α39,40, the transparency of S/F\ninterfaces, the spin relaxation, and the disorder degree.\nFor our purposes, the exact dependence of φ0on these\nparameters is not as relevant as the device geometry.\nIf we assume a two-dimensional SOC with magnetic\nmomenta in the plane of the F film (in particular, we are\nassuming a monodomain ferromagnetic arrangement),\nand the bias current flows in x-direction, the phase shift\nφ0is proportional to the y-component of the magnetic\nmoment according to33,34,41,42\nφ0=r my, (1)\nwhere my=My/M, with M=q\nM2x+M2y+M2z\nbeing the modulus of the magnetic moment, and the\nparameter rdepends on α(for the explicit dependence of\nron various system parameters, including the strength\nof the spin-orbit interaction, refer to Refs. 33 and 41).\nEquation (1) evidently establishes a direct coupling\nbetween the magnetic moment and the supercurrent.\nThe time evolution of the magnetic moment can be\ndescribed in terms of the Landau–Lifshitz–Gilbert (LLG)\nequation43,44\ndM\ndτ=γ\nM\u0012\nM×dM\ndτ\u0013\n−grM×Heff, (2)where grdenotes the gyromagnetic ratio. The first\nterm on the right-hand side accounts for the dissipation\nthrough the phenomenological dimensionless Gilbert\ndamping parameter γ, while the second term represents\nthe precession around Heff, which components can be\ncalculated as45\nHeff,i=−1\nV∂F\n∂Mi, with i=x, y, z. (3)\nHere, Vis the volume of the F layer and\nF=−EJφIbias+Es(φ, φ 0) +EM (4)\nis the free energy of the system. Here, EJ= Φ 0Ic/(2π)\n(with Φ 0being the flux quantum), Ibiasis the external\ncurrent in units of Ic,Es(φ, φ 0) =EJ[1−cos(φ−φ0)],\nandEM=−KV\n2\u0000Mz\nM\u00012is the magnetic energy that\ndepends on the anisotropy constant K. In the following,\nwe use the parameter ε=EJ/(KV) to indicate the ratio\nbetween the energy scales of the system. In Eq. (2)\nwe are neglecting a second-derivative term proportional\nto an angular momentum relaxation time τa46–49. On\ntimescales shorter than τa, this term is demonstrated\nto give nutation oscillations on top of the precession\nmotion. Instead, on timescales longer than τa, the usual\nLLG equation is expected to work well in describing the\nmagnetic evolution. This is exactly our case, being τa\nwell below both the inverse Josephson frequency and the\nramp time, tmax\nFrom Eq. (3), we obtain the effective magnetic field\nHeff=K\nM[εrsin (φ−rmy) ˆy+mzˆz], (5)\nwhere mx,y,z =Mx,y,z/M are the normalized\ncomponents of the magnetic moment that have to satisfy\nthe condition m2\nx+m2\ny+m2\nz= 1. The LLG equations can\nbe conveniently expressed in spherical coordinates50, so\nas to conserve the magnetic moment modulus even when\nstochastic noise contributions are taken into account.\nThen, mx,y,z can be written in terms of polar and\nazimuthal angles θandϕas\nmx(τ) = sin θ(τ) cosϕ(τ)\nmy(τ) = sin θ(τ) sinϕ(τ) (6)\nmz(τ) = cos θ(τ).\nWe can also define the θand ϕcomponents of the\nnormalized effective field as\neHeff,θ=εrsin(φ−rmy) cosθsinϕ−mzsinθ (7)\neHeff,ϕ=εrsin(φ−rmy) cosϕ. (8)\nThus, if the time is normalized to the inverse of the\nferromagnetic resonance frequency ωF=grK/M, that\nist=ωFτ, the LLG equations in spherical coordinates\nreduce to the following two coupled equations50\ndθ\ndt=1\n1 +γ2\u0010\neHeff,ϕ+γeHeff,θ\u0011\n(9)\nsinθdϕ\ndt=1\n1 +γ2\u0010\nγeHeff,ϕ−eHeff,θ\u0011\n. (10)SCD in anomalous JJs 3\n0.6 0.7 0.8 0.9 1.0010203040\niSWPDF\n0.60.70.80.91.0〈iSW〉\n0.005\nσiSW\n0.001-1.0-0.8-0.6-0.4-0.20.0\nDIS\n012345\nK0.60.70.80.91.0〈iSW〉\n0.0050.010.050.1\nσiSW\n0.001 0.005 0.005    0.01 0.05            0.1-1.0-0.8-0.6-0.4-0.20.0\nDIS\n012345\nK(a)\n(b)\n(c)\nγ\n0.01\n0.013\n0.016\n0.02\n0.025\n0.032\n0.04\n0.050.063\n0.079\n0.1\n0.13\n0.16\n0.2\n0.250.32\n0.4\n0.5\n0.63\n0.79\n0.75 0.8 0.85 0.9 0.95024681012\niSWPDF\n0.880.890.90.91〈iSW〉\n0.0340.0380.042\nσiSW\n0.01 0.05 0.1 0.5-1.2-1.1-1.0-0.9-0.8-0.7\nγS\n3.544.555.56\nK〈iSW〉\nσiSW\nKS\u000bG\f\n\u000bH\f\n\u000bI\f\n0.8 0.85 0.9 0.95024681012\niSWPDF\n0.870.880.890.90.91〈iSW〉\n0.0320.0360.040\nσiSW\n0.0 0.1 0.2 0.3 0.4 0.5-0.9-0.8-0.7\nrS\n3.644.44.8\nKr\n0.5 0r (g)\n(h)\n(i)DI\n0.001\n0.0014\n0.0019\n0.00270.0037\n0.0052\n0.0072\n0.010.014\n0.019\n0.027\n0.0370.052\n0.072\n0.1\n0.14\nFIG. 2. PDF of the switching currents iSW, see top panels, moments of the SCDs, i.e., mean values ( ⟨iSW⟩) and root mean\nsquare ( σiSW) in middle panels, whereas skewness ( S) and kurtosis ( K) in bottom panels, as a function of: (a-c) DIwith\n(r, γ) = (0 .25,0.1), (d-f) γwith ( DI, r) = (0 .05,0.1), and (i-l) r(from r= 0, red curve, to r= 0.5, purple curve, with\n∆r= 0.02) with ( DI, γ) = (0 .05,0.1). The dashed lines in panels (b-c) indicate the SCD moments obtained in the decoupled\ncase, i.e., r= 0. The legend in panel (e) [(f)] refers also to (b) and (h) [(c) and (i)].\nThe dynamics of the Josephson phase can be described\nin terms of the resistively and capacitively shunted\njunction (RCSJ) model51–54, which can be generalized\nto include the anomalous phase shift φ0=rmy55:\ndφ\ndt=ωχ+rdmy\ndt(11a)\ndχ\ndt=ω\nβc[Ibias(t) +Ith(t)−sin (φ−rmy)−χ].(11b)\nHere, time ( t) is still in units of the inverse of the\nferromagnetic resonance frequency, and ω=ωc/ωF, with\nωc= 2πIcR/Φ0being the characteristic frequency51of\nthe junction with a normal-state resistance R. The\ndamping is quantified by the McCumber parameter, βc=\n2πIcCR2/Φ0(Cis the junction capacitance), which is\nrelated to the so-called quality factor of the junction\naccording to βc=Q2. In this work, we consider a\nmoderately underdamped junction, i.e., we set Q= 5\njust in line with Refs. 13, 15, and 18. The values of\nthe other parameters are ε= 10 and ω= 1. The\nvalue of energy ratio εis expected to range from ∼100,\nfor a weak magnetic anisotropy, to ∼1, for a strong\nanisotropy8. We chose an intermediate value, in line with\nother theoretical work8, but to guess how a different value\naffects the system we observe that εcomes into play in\ni) the normalised effective field, see Eqs. (7-8) and in\nii) the thermal field intensity. Thus, we expect a larger\n(smaller) εto give faster (slower) magnetic dynamics\nand a more (less) relevant noise-induced effect, e.g., see\nRefs. 8, 10, 12, 23, and 24.\nThe noise term Ith(t) is a sort of “thermal current”\nwith the usual white-noise statistical properties that, innormalized units, can be expressed as51,56,57\n⟨Ith(t)⟩= 0 ⟨Ith(t)Ith(t′)⟩= 2DIδ(t−t′),\n(12)\nwith\nDI=kBT\nRωF\nI2c=1\nωkBT\nEJ, (13)\nbeing the dimensionless intensity of thermal current\nfluctuations. In this work we consider also magnetic field\nfluctuations, i.e., we take into account in Eq. (2) a delta-\ncorrelated stochastic “thermal field” Hthcontribution,\nwith an intensity DH= (γ εω)DI(see Ref. 25 for more\ndetails). This means that one can regulate the relative\nstrength of the two noisy mechanisms by changing the\nmagnetization energy, the Gilbert damping parameter,\nor the magnetic-resonance frequency. Thus, the system\nparameters can be optimized in such a way to make, for\ninstance, the impact of the thermal field negligible with\nrespect to the thermal current. Below, although we only\nspecify the value of DI, we point out that both thermal\ncurrent and thermal field are included.\nIn order to construct a SCD, we consider a linearly\nincreasing bias current ibias(t) =Ibias(t)/Ic=vbt, where\nvb=t−1\nmaxis the ramp speed. A measurement consists\nin slowly and linearly ramping the bias current in a time\ntmax, so that Imax≡Ib(tmax) =Ic, and to record the\ncurrent value, iSW, at which a switch occurs, namely, at\nwhich the phase particle leaves the initial potential well.\nAlternatively, one could look at the voltage drop. In\nthis way, it should be also possible to distinguish phase\ndiffusion events, in which the phase particle once escaped\nis retrapped in an adjacent minimum due to dampingSCD in anomalous JJs 4\naffecting its motion. In this case, the resulting SCD\nshould show opposite behavior with temperature, i.e., its\nwidth should increase as the temperature decreases58,59.\nIn this readout scheme, the noise influence is considered\nin the limit of the adiabatic bias regime, where the change\nof the slope of the potential induced by the bias current is\nslow enough to keep the phase particle in the metastable\nstate until the noise pushes out the particle. Since we are\ndealing with stochastic switching processes, we need to\nperform many independent experiments under the same\nconditions, in order to obtain comprehensive statistics.\nIn particular, sequences of N= 104independent\nnumerical experiments of maximum duration tmax =\n104are performed. Thus, the collection of these\naccumulated switching currents forms a SCD, whose\nmoments, i.e., mean, variance, skewness, and kurtosis,\ncan be investigated.\nWe assume that the magnetic moment initially points\ntowards the z-direction, that is M= (0,0,1) at t= 0,\nand that φ(0) = dφ(0)/dt= 0. With these initial\nconditions, we solve Eqs. (9)-(11) self-consistently, at\ndifferent system parameter values. Once the switching\nhas taken place, the Josephson phase undergoes fast\n2πrotations and a non-zero voltage drop appears. In\nthis condition, the instantaneous value of magnetic\nmoment is not so relevant, but rather the time-averaged\nvalue it takes after switching. Computing also the\nensemble average on the total amount of independent\nrealizations, we observe, in the regions of parameter\nspace under consideration, zero average after-switching\nmagnetic moments (data not shown). The readout of the\nmagnetic state of the system could be eventually achieved\nthrough a similar nondestructive scheme proposed in\nRef. 25.\nIn Fig. 2 we collect the probability distribution\nfunction (PDF) of switching currents iSW(see top panels)\nand the moments (i.e., mean value, ⟨iSW⟩, and root mean\nsquare, σiSW, in the middle panels, while skewness, S,\nand kurtosis, K, are shown in the bottom panels) of the\nSCDs, varying the main system parameters, ( DI, r, γ),\nin suitable ranges. In particular, we choose: in panels\n(a-c) DI∈[0.001−1.0] with ( r, γ) = (0 .25,0.1), in (d-f)\nγ∈[0.01−1.0] with ( DI, r) = (0 .05,0.1), and in (i-l)\nr∈[0−0.5] with ( DI, γ) = (0 .05,0.1). Skewness and\nkurtosis are obtained as\n˜S[X]=E\"\u0012X−µ\nσ\u00133#\nand ˜K[X]=E\"\u0012X−µ\nσ\u00134#\nwhere Xrepresents the random variable of the\nmeasurements, E[·] is the expectation operator, µ=\nE [X], and σ2= var ( X). These quantities can be\nestimated through the measured switching currents iSW,j\nas\nS=D\n(iSW,j− ⟨iSW⟩)3E\nσ3andK=D\n(iSW,j− ⟨iSW⟩)4E\nσ4,\nwhere ⟨iSW⟩andσare the estimates of the mean andthe standard deviation of the switching distribution,\nrespectively.\nWe first look in Fig. 2(a) at the behavior of the SCDs\nat different noise intensities DI, which is proportional\nto the temperature according to Eq. (13). We note\nthat the DIvalues are chosen in such a way that the\nmoments and the magnetic moments in Figs. 2(b-c)\nappear equally spaced on a logarithmic scale. The more\nthe noise intensity increases, the more the SCDs are\nlower, wider, and centered at smaller switching current\nvalues. The first two moments, i.e., mean value, ⟨iSW⟩,\nand root mean square, σiSW, see panel (b), show a\ntypical trend: in particular, ⟨iSW⟩tends to saturate\nat the level ∼1 at low noises and then it decreases,\nwhile σiSWincreases exponentially (i.e., linearly on a\nlog-log scale) as the noise increases. We stress that\nwe are assuming to work at temperatures above the so-\ncalled “crossover temperature”27,60, i.e., that value below\nwhich macroscopic quantum tunneling (MQT) effects\ndominate the switching dynamics: in this case, being the\nMQT probability independent of T, one would obtain\noverlapping SCDs, all centered around the crossover\ntemperature. In panel (c) we show the third and fourth\nmoments: we see that the skewness S(the kurtosis K)\ntakes the value S∼ −0.4 (K∼3) for DI= 0.001 and\ndecreases (increases) up to the value S∼ −0.8 (K∼4)\nforDI= 0.05. The obtained values are in line with\nthose already observed for other Josephson systems61.\nIn other words, increasing noise makes the SCDs more\nasymmetric and with tails more pronounced.\nIn order to understand the interplay between the\nJosephson and the magnetic systems, we enrich panels\n(b) and (c) with the SCD moments of a conventional JJ\n[i.e., setting r= 0 in Eq. (11) in order to decouple the\nJosephson and magnetic systems] marked with dashed\nlines. It is clear that all moments tend to diverge\nfrom the pure-thermal behavior for noises larger that\nDI≳0.05. The response of a FAJJ therefore cannot be\nmerely assimilated to that of a conventional JJ residing\nat a slightly higher temperature, i.e., subject to a larger\nnoise intensity. Indeed, in principle, one could assume an\neffective temperature such that the behavior of the SCDs\nin Fig. 2(a-c) could again be achieved in a conventional\nJJ. In particular, we could search for an ad-hoc additional\nnoise intensity that gives the values of ⟨iSW⟩(DI) that\nagree with those we obtained; however, by doing so,\nthe other moments would differ significantly from those\npresented in Fig. 2(b-c) (data not shown). Thus, one\ncannot merely reduce our results to what would be\nobserved considering a conventional JJ with a higher\neffective temperature.\nThe central panels of Fig. 2 report how the SCDs\ndepend on the Gilbert damping parameter, γ∈[0.01−\n1.0], imposing ( DI, r) = (0 .05,0.1). Looking at the\nPDF of the switching currents in Fig. 2(d), it is\nevident that by changing γboth the position and the\nshape of the SCDs change. Specifically, ⟨iSW⟩(σiSW)\nshow a monotonically decreasing (increasing) behaviorSCD in anomalous JJs 5\nincreasing γ, see Fig. 2(e). Instead, both SandKremain\npractically unchanged for γ≲0.2, while for higher γ\nthey tend to increase, the former, and to decrease, the\nlatter, see Fig. 2(f). The value of rsignificantly affects\nthe threshold γvalue above which the shape of the SCDs\nstarts to change appreciably. We stress that the scattered\nbehavior of the higher-moment curves can be smoothed\nout by increasing the number of numerical repetitions.\nFinally, Fig. 2(g-h) says that by increasing the SOC\nstrength, r, the SCDs tend to shift towards lower\nswitching currents. In particular, we observe that both\n⟨iSW⟩and skewness (root-mean-square and kurtosis)\ntend to decrease (increase) almost linearly by increasing\nr.\nFigure 2 says that the analysis of both the average and\nthe shape of the switching current distributions could be\nemployed to experimentally determine γorr, the other\nbeing known.\nIn conclusion, we discussed the behavior of the SCDs\nof a current-biased φ0-junction, that is a superconductor-\nferromagnet-superconductor JJ with a Rashba-like spin-\norbit coupling. We investigated the noise impact on both\nthe Josephson phase dynamics and the magnetization of\nthe system in the case of a slowly, linearly ramping bias\ncurrent. In particular, we showed how a change in the\nvarious characteristic parameters of the system impacts\nthe SCDs, also looking at the distribution moments, i.e.,\nmean values, root mean square, skewness, and kurtosis.\nIn particular, their analysis give information on the\nmain quantities of the system, i.e., the phenomenological\nGilbert damping parameter γand the SOC strength.\nHowever, the description of a concrete experiment should\nfirst proceed with a fine-tuning of the other parameters,\nsuch as Ic,ε, the saturation magnetization, ωc, and ωF\n(some values can be found in, e.g., Refs. 8–10, 17, 23, and\n24).\nThe ability to control and manipulate the\nmagnetization states through the superconducting\nphases opens up possibilities for encoding and processing\nquantum information, thus our results may contribute\nto the development of optimized spintronics.\nACKNOWLEDGMENTS\nF.S.B. acknowledges financial support from Spanish\nMCIN/AEI/ 10.13039/501100011033 through project\nPID2020-114252GB-I00 (SPIRIT) and TED2021-\n130292B-C42, and the Basque Government through\ngrant IT-1591-22. 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Lett. 46, 388 (1981).\n61A. Murphy, P. Weinberg, T. Aref, U. C. Coskun, V. Vakaryuk,\nA. Levchenko, and A. Bezryadin, Phys. Rev. Lett. 110, 247001\n(2013)"    },    {        "title": "2102.10394v2.Fast_magnetization_reversal_of_a_magnetic_nanoparticle_induced_by_cosine_chirp_microwave_field_pulse.pdf",        "content": "Fast magnetization reversal of a magnetic nanoparticle induced by cosine\nchirp microwave field pulse\nM. T. Islam,1,a)M. A. S. Akanda,1M. A. J. Pikul,1and X. S. Wang2,b)\n1)Physics Discipline, Khulna University, Khulna 9208, Bangladesh\n2)School of Physics and Electronics, Hunan University, Changsha 410082, China\nWe investigate the magnetization reversal of single-domain magnetic nanoparticle driven by the circularly polarized\ncosine chirp microwave pulse (CCMP). The numerical findings, based on the Landau-Lifshitz-Gilbert equation, reveal\nthat the CCMP is by itself capable of driving fast and energy-efficient magnetization reversal. The microwave field am-\nplitude and initial frequency required by a CCMP are much smaller than that of the linear down-chirp microwave pulse.\nThis is achieved as the frequency change of the CCMP closely matches the frequency change of the magnetization pre-\ncession which leads to an efficient stimulated microwave energy absorption (emission) by (from) the magnetic particle\nbefore (after) it crosses over the energy barrier. We further find that the enhancement of easy-plane shape anisotropy\nsignificantly reduces the required microwave amplitude and the initial frequency of CCMP. We also find that there is an\noptimal Gilbert damping for fast magnetization reversal. These findings may provide a pathway to realize the fast and\nlow-cost memory device.\nI. INTRODUCTION\nAchieving fast and energy-efficient magnetization rever-\nsal of high anisotropy materials has drawn much attention\nsince it has potential application in non-volatile data stor-\nage devices1–3and fast data processing4. For high thermal\nstability and low error rate, high anisotropy materials are\nrequired5in device application. But one of the challenging\nissues is to find out the way which can induce the fastest mag-\nnetization reversal with minimal energy consumption. Over\nthe last two decades, many magnetization reversal methods\nhas been investigated, such as by constant magnetic fields6,7,\nby the microwave field of constant frequency, either with or\nwithout a polarized electric current8–10and by spin-transfer\ntorque (STT) or spin-orbit torque (SOT)11–28. However, all\nthe means are suffering from their own limitations. For in-\nstance, in the case of external magnetic field, reversal time\nis longer and has scalability and field localization issues6.\nIn case of the constant microwave field driven magnetiza-\ntion reversal, the large field amplitude and the long rever-\nsal time are emerged as limitations29–31. In the case of the\nSTT-MRAM, the threshold current density is a large and thus,\nJoule heat which may lead the device malfunction durability\nand reliability issues32–38. Moreover, there are several stud-\nies showing magnetization reversal induced by microwaves of\ntime-dependent frequency39–45. In the study39, the magnetiza-\ntion reversal, with the assistance of external field, is obtained\nby a radio-frequency microwave field pulse. Here, a dc ex-\nternal field acts as the main reversal force. In the study40,\nto obtain magnetization reversal, the applied microwave fre-\nquency needs to be the same as the resonance frequency, and\nin the studies42,43, optimal microwave forms are constructed.\nThese microwave forms are difficult to be realized in practice.\nThe study44reports magnetization reversal induced by the mi-\ncrowave pulse, but the pulse is applied such that the magne-\ntization just crosses over the energy barrier, i.e., only positive\na)Electronic mail: torikul@phy.ku.ac.bd\nb)Electronic mail: justicewxs@hnu.edu.cnfrequency range ( +f0to 0) is employed.\nA recent study45has demonstrated that the circularly polar-\nized linear down-chirp microwave pulse (DCMP) (whose fre-\nquency linearly decreases with time from the initial frequency\n+f0to\u0000f0) can drive fast magnetization reversal of uniax-\nial nanoparticles. The working principle of the above model\nis that the DCMP triggers stimulated microwave energy ab-\nsorption (emission) by (from) the magnetization before (after)\ncrossing the energy barrier. However, the efficiency of trig-\ngered microwave energy absorption or emission depends on\nhow closely the frequency of chirp microwave pulse matches\nthe magnetization precession frequency. In DCMP-driven\ncase, the frequency linearly decreases from f0to\u0000f0with\ntime but, in fact, the decrement of magnetization precession\nfrequency is not linear46,47during magnetization reversal . So\nthe frequency of DCMP only roughly matches the magneti-\nzation precession frequency. Thus, the DCMP triggers in-\nefficient energy absorption or emission and the required mi-\ncrowave amplitude is still large.\nTherefore, to achieve more efficient magnetization reversal,\nwe need to find a microwave pulse of proper time-dependent\nfrequency that matches the intrinsic magnetization precession\nfrequency better. In this study, we demonstrate that a cosine\nchirp microwave pulse (CCMP), defined as a microwave pulse\nwhose frequency sweeps in a cosine function with time from\n+f0to\u0000f0in first half-period of the microwave pulse, is ca-\npable of driving the fast and energy efficient magnetization\nreversal. This is because the frequency change of the CCMP\nmatches the nonlinear frequency change of magnetization pre-\ncession better than the DCMP. In addition, this study empha-\nsizes how the shape anisotropy influences the required param-\neters of CCMP and how the Gilbert damping affects the mag-\nnetization reversal. We find that the increase of easy-plane\nshape anisotropy makes the magnetization reversal easier. The\nmaterials with larger damping are better for fast magnetization\nreversal. These investigations might be useful in device appli-\ncations.arXiv:2102.10394v2  [cond-mat.mes-hall]  15 Sep 20212\nmf0\n0-f(b) (a)\nτ t\nFIG. 1. (a) Schematic diagram of the system in which mrepresents\na unit vector of the magnetization. A circularly polarized cosine\n(nonlinear) chirp microwave pulse is applied onto the single domain\nnanoparticle. (b) The frequency sweeping (from +f0to\u0000f0) of a\ncosine chirp microwave pulse.\nII. ANALYTICAL MODEL AND METHOD\nWe consider a square magnetic nanoparticle of area Sand\nthickness dwhose uniaxial easy-axis anisotropy directed in\nthez-axis as shown in FIG. 1(a). The size of the nanopar-\nticle is chosen so that the magnetization is considered as a\nmacrospin represented by the unit vector mwith the mag-\nnetic moment SdM s, where Msis the saturation magnetization\nof the material. The demagnetization field can be approxi-\nmated by a easy-plane shape anisotropy. The shape anisotropy\nfield coefficient is hshape=\u0000m0(Nz\u0000Nx)Ms, and the shape\nanisotropy field is hshape=hshapemzˆz, where NzandNxare\ndemagnetization factors48,49andm0=4p\u000210\u00007N=A2is the\nvacuum magnetic permeability. The strong uniaxial mag-\nnetocrystalline anisotropy hani=hanimzˆzdominates the total\nanisotropy so that the magnetization of the nanoparticle has\ntwo stable states, i.e., mparallel to ˆzand\u0000ˆz.\nThe magnetization dynamics min the presence of circularly\npolarized CCMP is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation50\ndm\ndt=\u0000gm\u0002heff+am\u0002dm\ndt; (1)\nwhere aandgare the dimensionless Gilbert damping con-\nstant and the gyromagnetic ratio, respectively, and heffis the\ntotal effective field which includes the microwave magnetic\nfieldhmw, and the effective anisotropy field hkalong zdirec-\ntion (note that although we consider small nanoparticles that\ncan be treated as macrospins approximately, we still perform\nfull micromagnetic simulations with small meshes and full de-\nmagnetization field to be more accurate).\nThe effective anisotropy field can be expressed in terms of\nuniaxial anisotropy haniand shape anisotropy hshape ashk=\nhani+hshape= [hani\u0000m0(Nz\u0000Nx)Ms]mzˆz. Thus, the resonant\nfrequency of the nanoparticle is obtained from the well-known\nKittel formula\nf0=g\n2p[hani\u0000m0(Nz\u0000Nx)Ms]: (2).\nFor microwave field-driven magnetization reversal from the\nLLG equation, the rate of energy change is expressed as\ndE\ndt=\u0000agjm\u0002heffj2\u0000m\u0001dhmw\ndt: (3)\nThe first term is always negative since damping ais posi-\ntive. The second term can be either positive or negative for\ntime-dependent external microwave field. Therefore, the mi-\ncrowave field pulse can trigger the stimulated energy absorp-\ntion or emission, depending on the angle between the instan-\ntaneous magnetization manddhmw\ndt.\nInitially, because of easy-axis anisotropy, the magnetiza-\ntion prefers to stay in one of the two stable states, \u0006ˆz, cor-\nresponding to two energy minima. The objective of magne-\ntization reversal is to get the magnetization from one stable\nstate to the other. Along the reversal process, the magnetiza-\ntion requires to overcome an energy barrier at mz=0 which\nseparates two stable states. For fast magnetization reversal,\nthe external field is required to supply the necessary energy\nto the magnetization until crossing the energy barrier and, af-\nter crossing the energy barrier, the magnetization releases en-\nergy through damping and =or the external field is required\nto extract (by negative work done) energy from the magne-\ntization. It is mentioned that there is as intrinsic anisotropy\nfieldhanidue to the anisotropy which induces a magnetization\nnatural =resonant frequency proportional to mz. When mag-\nnetization goes from one stable state to another, the magneti-\nzation resonant frequency decreases while the magnetization\nclimbs up and becomes zero momentarily while crossing the\nenergy barrier and then increases with the opposite preces-\nsion direction while it goes down from the barrier. In princi-\nple, for fast and energy-efficient reversal, one requires a chirp\nmicrowave pulse whose frequency always matches the mag-\nnetization precession frequency to ensure the term m\u0001˙hmw\nto be maximal (minimal) before (after) crossing the energy\nbarrier. The study45, employs the DCMP (whose frequency\nlinearly decreases with time) to match the magnetization pre-\ncession frequency roughly. In fact, during the reversal from\nmz= +1 to mz=\u00001 , the decreasing of the resonant fre-\nquency ( while the spin climbs up the energy barrier) and in-\ncreasing of the resonant frequency (while it goes down from\nthe barrier) are not linear46,47. This leads us to consider a\ncosine chirp microwave pulse (CCMP) (a microwave pulse\nwhose frequency decreases non-linearly with time) in order\nto match the change of magnetization precession frequency\nclosely. Thus the CCMP might trigger more efficient stim-\nulated microwave absorptions (emissions) by (from) magne-\ntization before (after) the spin crosses the energy barrier to\ninduce fast and energy-efficient magnetization reversal.\nIn order to substantiate the above mentioned prediction,\nwe apply a circularly polarized cosine down-chirp microwave\npulse in the xyplane of the nanoparticle and solve the LLG\nequation numerically using the MUMAX3 package51. The\ncosine chirp microwave pulse (CCMP) takes the form hmw=\nhmw[cosf(t)ˆx+sinf(t)ˆy], where hmwis the amplitude of the\nmicrowave field and f(t)is the phase. Since the phase f(t)is\n2pf0cos(2pRt)t, where R(in units of GHz) is the controlling3\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49\n/s46\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s49/s48/s45/s53/s48/s53/s49/s48/s101 /s32/s40/s49/s48/s49/s53\n/s32/s74/s47/s115/s32/s109/s51\n/s41(a) (b)\n(c)\nFIG. 2. Model parameters of nanoparticle of Ms= 106A/m, Hk=\n0:75 T, g= 1:76\u00021011rad/(T\u0001s), and a=0:01. (a) Temporal evo-\nlutions of mzofV= (8\u00028\u00028)nm3driven by the CCMP (with the\nminimal hmw=0:035 T, f0=18:8 GHz and R= 0.32 GHz) (red line)\nand DCMP (with hmw=0:035 T, f0=18:8 GHz and R= 1.53 GHz\n) (blue line). For CCMP case, (b) the corresponding magnetization\nreversal trajectories and (c) temporal evolutions of mz(red lines) and\nthe energy changing rate ˙eof the magnetization against time (blue\nlines).\nparameter, the instantaneous frequency f(t)of CCMP is ob-\ntained as f(t) =1\n2pdf\ndt=f0[cos(2pRt)\u0000(2pRt)sin(2pRt)]\nwhich decreases with time from f0to final\u0000f0at a time\ndependent chirp rate h(t)(in units of ns\u00002) as shown\nin FIG. 1(b). The chirp rate takes the form h(t) =\n\u0000f0h\n(4pR)sin(2pRt)+(2pR)2tcos(2pRt)i\n.\nAccording to the applied CCMP, the second term of right\nhand side of Eq. (3), i.e., the energy changing rate can be\nexpressed as\n˙e=\u0000Hmwsinq(t)sinF(t)\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\n(4)\nwhereF(t)is the angle between mt(the in-plane component\nofmandhmw. Therefore, the microwave field pulse can trig-\nger the stimulated energy absorption (before crossing the en-\nergy barrier) with\u0000F(t)and emission (after crossing the en-\nergy barrier) with F(t).\nThe material parameters of this study are chosen from typ-\nical experiments on microwave-driven magnetization reversal\nasMs=106A=m,hani=0:75 T, g=1:76\u00021011rad=(T\u0001s),\nexchange constant A=13\u000210\u000012J=m and a=0:01. Al-\nthough, the strategy and other findings of this study would\nwork for other materials also. The cell size (2\u00022\u00022)nm3\nis used in this study. We consider the switching time win-\ndow 1 ns at which the magnetization switches/reverses to\nmz=\u00000:9.\nIII. NUMERICAL RESULTS\nWe first investigate the possibility of reversing the magne-\ntization of cubic sample (8\u00028\u00028)nm3by the cosine chirpmicrowave pulse (CCMP). Accordingly, we apply the CCMP\nwith the microwave amplitude hmw=0:045 T, initial fre-\nquency f0=21 GHz and R=1:6 ns\u00001which are same as\nestimated in the study45), to the sample and found that CCMP\ncan drive the fast magnetization reversal. Then, we search the\nminimal hmw,f0, and Rof the CCMP such that the fast re-\nversal is still valid. Interestingly, the CCMP with significantly\nsmaller parameters i.e., hmw=0:035 T, f0=18:8 GHz and\noptimal R=0:32 ns\u00001, is capable of reversing the magnetiza-\ntion efficiently shown by red line in FIG. 2(a).\nThen we intend to show how efficient the CCMP driven\nmagnetization reversal compare to DCMP driven case. For\nfair comparison, we choose the same pulse duration t(as\nshown in Fig. 1, tis the time at which the frequency changes\nfrom +f0to\u0000f0). For the CCMP, we solve cos (2pRt)\u0000\n(2pRt)sin(2pRt) =\u00001, which gives the relation t=1:307\n2pR.\nBut, in case of DCMP, we know that t=2f0\nhand for f0=\n18:8 GHz, the chirp rate becomes h=57:86 ns\u00002and hence\nfind the parameter R(=1=t)= 1.53 ns\u00001. Then we apply the\nDCMP with the hmw=0:035 T, f0=18:8 GHz (which are\nsame as CCMP-driven case), and R=1.53 ns\u00001and found that\nthe magnetization only precesses around the initial state, i.e.,\nthe DCMP is not able to reverse the magnetization as shown\nby blue line in FIG. 2(a). So, it is mentioned that the CCMP\ncan reverse the magnetization with lower energy consumption\nwhich is the desired in device application. To be more explicit,\nthe trajectories of magnetization reversal induced by CCMP\nis shown FIG. 2(b) which shows the magnetization reverses\nswiftly.For further justification of CCMP driven reversal, we\ncalculate the energy changing rate dE=dtrefers to (4) by de-\ntermining the angles F(t)andq(t)and plotted with time in\nFIG. 2(c). The stimulated energy absorption (emission) peaks\nare obtained before (after) crossing the energy barrier as ex-\npected for faster magnetization reversal. This is happened be-\ncause the frequency of the CCMP closely matches the fre-\nquency of magnetization precession frequency, i.e., before\ncrossing the energy barrier, the F(t)remains around\u000090\u000eand\nafter crossing the energy barrier F(t)around 90\u000eto maximize\nthe energy absorption and emission respectively.\nThen, this study emphasizes how shape-anisotropy field\nhshape affects the magnetization switching time, microwave\namplitude hmwand initial frequency f0of CCMP. Since de-\nmagnetization field or shape-anisotropy field hshape should\nhave significant effect on magnetization reversal process as\nit opposes the magnetocrystalline anisotropy field haniwhich\nstabilizes the magnetization along two stable states. Accord-\ningly, to induce the hshape in the sample, we choose square\ncuboid shape samples and, to increase the strength of hshape,\nthe cross-sectional area S=xyis enlarged gradually for the\nfixed thickness d(=z) =8 nm. Specifically, we focus on\nthe samples of S1=10\u000210,S2=12\u000212,S3=14\u000214,\nS4=16\u000216,S5=18\u000218,S6=20\u000220 and S7=22\u000222 nm2\nwith d=8 nm. For the samples of different S, by determining\nthe analytic demagnetization factors NzandNx48,52, the shape\nanisotropy field hshape=m0(Nz\u0000Nx)Msmzˆzare determined.\nThehshape actually opposes the anisotropy field haniand\nhence resonance frequencies f0=g\n2p[hani\u0000m0(Nz\u0000Nx)Ms]4\nTABLE I. Shape anisotropy coefficient, resonant frequency f0, simulated frequency, f0and frequency-band\nCross\u0000sectional area Shape anisotropy coefficient Resonant frequency, Simulated minimal frequency, Simulated frequency-band\nS(nm2) hshape (T) f0(GHz) f0(GHz) (GHz)\nS1 0.09606 18.3 17.8 17.8 \u000019.9\nS2 0.17718 16 14.9 14.9 \u000018.3\nS3 0.2465 14.1 13.7 13.7 \u000015.4\nS4 0.3064 12.4 12.2 12.2 \u000013.8\nS5 0.3588 11 10.7 10.7 \u000012.3\nS6 0.4049 9.6 8.8 8.8 \u000011.5\nS7 0.4459 8.5 7.7 7.4 \u00008.7\ndecreases as shown in the Table I. Then, for the samples of\ndifferent S, with the fixed hmw=0:035 T, the corresponding\nminimal f0and optimal Rof CCMP are determined through\nthe study of the magnetization reversal. The temporal evolu-\ntions of mzfor different Sare shown in FIG. 3(a) and found\nthat for S7\u001522\u000222 nm2, the magnetization smoothly reveres\nwith the shortest time. The switching time tsas a function of\nthe coefficient hshape (corresponding to S) is plotted in FIG.\n3(c). It is observed that, with the increase of hshape orS, the\ntsshows slightly increasing trend but for S7(22\u000222)nm2or\nhshape=0:4459 T , tsdrops to 0.43 ns. For further increment\nofSorhshape,tsremains constant around 0.43 ns which is\nclose to the theoretical limit (0.4 ns) refers to the study37.\nThis is because the hshape reduces the effective anisotropy\nand thus reduces the height of energy barrier (energy differ-\nence between the initial state and saddle point) which is shown\nin the FIG. 3(b). Therefore, after decreasing certain height\nof energy barrier, the magnetization reversal becomes fastest\neven with the same filed amplitude hmw=0:035 T. Due to the\nreduction of height of the energy barrier with the hshape, one\ncan expect that the hmwand f0should also decrease with the\nincrease of the anisotropy coefficient hshape and theses find-\nings are presented subsequently.\nHere we present the effect of the shape anisotropy hshape\non the microwave amplitude hmwand initial frequency f0of\nCCMP. Purposely, for each sample Sorhshape, we numerically\ndetermine (by tuning hmw,f0and optimal R) the minimally re-\nquired hmwandf0of CCMP for the the magnetization reversal\ntime window 1 ns. Interestingly, we find that the fast and effi-\ncient reversal is valid for a wide range of initial frequency f0.\nFIG. 4(a) shows the estimated frequency bands of f0by verti-\ncal dashed lines for different S( for example, f0=19:9\u001817:8\nforhshape=0:096 T) for time window 1 ns. So there is a great\nflexibility in choosing the initial frequency f0which is useful\nin device application. FIG. 3(d) shows how the minimal f0\n(red circles) and hmw(blue square) decrease with the increase\nofhshape. The decay of f0is expected as hshape reduces ef-\nfective anisotropy (refers to Eq.(2)). For more justification, in\nsame FIG. 3(d), theoretically resonant frequency (black solid\nline) and simulated minimal frequency (red circles) as func-\ntion of hshape indicate the agreement. The minimal frequency\nf0always smaller than the theoretical =resonant frequency.\nMoreover, the decreasing trend of hmwwith hshape can be\nattributed by the same reason as the height of the energy bar-\n1\n2\n3\n4\n5\n6\n72\n4\n6\n70\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s53/s49/s48/s49/s53/s50/s48/s32/s102\n/s111\n/s32/s102\n/s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41\n/s32/s72\n/s109/s119\n/s104\n/s115/s104/s97/s112/s101/s32/s40/s84 /s41/s102\n/s111/s32/s40/s71/s72/s122/s41\n/s48/s46/s48/s50/s48/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s104\n/s109/s119/s32/s40/s84/s41(a) (b)\n(c) (d)\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s52/s53/s54/s55/s56/s57/s116\n/s115/s32/s40/s110/s115/s41\n/s104\n/s115/s104/s97/s112/s101/s32/s40/s84 /s41FIG. 3. (a) Temporal evolution of mzinduced by CCMP (with hmw=\n0:035 T fixed) for different cross-sectional area, S. (b) The energy\nlandscape Ealong the line f=0. The symbols i and s represent the\ninitial state and saddle point. (c) tsas a function of hshape. (d) The\nminimal f0(red dotted) and hmw(blue square) as a function of hshape\nwhile switching time window 1 ns.\n0\n1\n32\n4\n5\n760\n1\n2\n3(a) (b)\nFIG. 4. (a) Minimal switching time tsas a function of estimated fo\nof CCMP with fixed hmwandRcorresponding to different S. (b)\nMinimal tsas a function of Gilbert damping afor different S.\nrier decreases with hshape (refers to FIG. 3(b)). For the larger\nhshape or lower height of energy barrier, the smaller microwave\nfield hmwcan induce the magnetization reversal swiftly.\nThe Gilbert damping parameter, ahas also a crucial effect\non the magnetization magnetization dynamics and hence re-\nversal process and reversal time53–55. In case of CCMP-driven5\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121 /s109\n/s120/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120(a) (b)\nFIG. 5. Magnetization reversal trajectories of biaxial shape (10\u0002\n10\u00028)nm3driven by CCMP for (a) a=0:010. (b) a=0:045.\nmagnetization reversal, smaller (larger) ais preferred while\nthe magnetization climbing the energy barrier (the magnetiza-\ntion goes down to stable states). Therefore, it is meaningful to\nfind the optimal afor samples of different Sat which the re-\nversal is fastest. For fixed hmw=0:035 T, using the optimal f0\nandRcorresponding to S0,S1,S2andS3, we study the CCMP-\ndriven magnetization reversal as a function of afor different\nS. FIG. 4(b) shows the dependence of switching time on the\nGilbert damping for different S. For each S, there is certain\nvalue or range of afor which the switching time is minimal.\nFor instance, the switching time is lowest at a=0:045 for\nthe sample of S1=10\u000210 nm2. To be more clear, one may\nlook at FIG. 5(a) and FIG. 5(b) which show the trajectories\nof magnetization reversal for a=0:01 and a=0:045 respec-\ntively and observed that for a=0:045, the reversal path is\nshorter. This is because, after crossing over the energy bar-\nrier, the larger damping dissipates the magnetization energy\npromptly and thus it leads to faster magnetization reversal.\nThis finding suggests that larger ashows faster magnetiza-\ntion reversal.\nIV. DISCUSSIONS AND CONCLUSIONS\nThis study investigates the CCMP-driven magnetization re-\nversal of a cubic sample at zero temperature limit and found\nthat the CCMP with significantly smaller hmw=0:035 T,\nf0=18:8 GHz and R=0:32 ns\u00001than that of DCMP (i.e.,\nhmw=0:045 T, f0=21 GHz and R= 1.6 ns\u00001) can drive fast\nmagnetization reversal. Since the frequency change of CCMP\nclosely matches the magnetization precession frequency and\nthus it leads fast magnetization reversal with lower energy-\ncost. Then we study the influence of demagnetization field =\nshape anisotropy field hshape, on magnetization reversal pro-\ncess and the optimal parameters of CCMP. Interestingly we\nfind that, with the increase of hshape, the parameters hmwand\nf0of CCMP decreases with increasing hshape.\nSo, we search a set of minimal parameters of CCMP for\nthe fast ( ts\u00181 ns) magnetization reversal of the sample 22 \u0002\n22\u00028 nm3, and estimated as hmw=0:03 T, f0=7:7 GHz\nandR=0:24 ns\u00001which are (significantly smaller than that\nof DCMP) useful for device application. This is happened\nbecause with increase of hshape, the effective anisotropy field\ndecreases and thus the energy barrier (which separates two\nstable states) decreases. In addition, it is observed that thematerials with the larger damping are better for fast magneti-\nzation reversal. There is a recent study56reported that ther-\nmal effect assists the magnetization reversal i.e., thermal ef-\nfect reduces the controlling parameters of chirp microwave\nfield pulse. Thus, it is expected that the parameters of CCMP\nalso might be reduced further at room temperature. To gener-\nate such a cosine down-chirp microwave pulse, several recent\ntechnologies46,47are available. Therefore, the strategy of the\ncosine chirp microwave chirp driven magnetization reversal\nand other findings may lead to realize the fast and low-cost\nmemory device.\nACKNOWLEDGMENTS\nThis work was supported by the Ministry of Education\n(BANBEIS, Grant No. SD2019972). X. S. W. acknowledges\nthe support from the Natural Science Foundation of China\n(NSFC) (Grant No. 11804045) and the Fundamental Research\nFunds for the Central Universities.\nAppendix A: Calculation of ˙e\nIn this Appendix, we show the details of the derivation of ˙e\nin Eq. 4. The rate of change of hmwis\n˙hmw=dhmw\ndt\n=d\ndt(hmw[cosf(t)ˆx+sinf(t)ˆy])\n=hmw[\u0000sinf(t)ˆx+cosf(t)ˆy]df\ndt\n=hmw[\u0000sinf(t)ˆx+cosf(t)ˆy]\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nThe magnetization is given by\nm=mxˆx+myˆy\n=sinq(t)cosfm(t)ˆx+sinq(t)sinfm(t)ˆy\nwhere q(t)is the polar angle and fm(t)is the azimuthal angle\nof the magnetization m.\nSubstituting mxand˙hmwin Eq. 3, we get,\n˙e=\u0000m\u0001˙hmw\n=hmwsinq(t)[\u0000sinf(t)cosfm(t)+cosf(t)sinfm(t)]\u0001\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\n=hmwsinq(t)sin(f(t)\u0000fm(t))\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nDefining F(t) =fm(t)\u0000f(t), we have finally\n˙e=hmwsinq(t)sinF(t)\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nwhereh\nf(t)\nt\u0000d\ndt\u0010\nf(t)\nt\u0011\nti\nrepresents w(t).6\nREFERENCES\n1S. Sun, C. B. Murray, D. Weller, L. Folks, and A. 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The interplay of these currents with\ntiny gradients of either the magnetic field or the temperatur e can induce a rotation of the magnetic\npattern for j > j c. Either a rotation by a finite angle of up to 15◦or – for larger gradients –\na continuous rotation with a finite angular velocity is induc ed. We use Landau-Lifshitz-Gilbert\nequations extended by extra damping terms in combination wi th a phenomenological treatment\nof pinning forces to develop a theory of the relevant rotatio nal torques. Experimental neutron\nscattering data on the angular distribution of skyrmion lat tices suggests that continuously rotating\ndomains are easy to obtain in the presence of remarkably smal l currents and temperature gradients.\nPACS numbers:\nI. INTRODUCTION: SPINTORQUES AND\nSKYRMION LATTICES\nManipulating magnetic structures by electric current\nis one of the main topics in the field of spintronics. By\nstrong current pulses one can, for example, switch mag-\nnetic domains in multilayer devices1,2, induce microwave\noscillations in nanomagnets3or move ferromagnetic do-\nmain walls4,5. The latter effect may be used to develop\nnew types of non-volatile memory devices6. It is there-\nfore a question of high interest to study the coupling\nmechanisms of currents to magnetic structures7,8.\nHere, the recent discovery9,10ofthe so-called skyrmion\nlattice in chiral magnets like MnSi provides a new oppor-\ntunity for studying the manipulation of magnetism by\nelectric currents both experimentally and theoretically.\nThe skyrmions in MnSi form a lattice of magnetic whirls,\nsimilar to the superfluid whirls forming the vortex lattice\nin type-II superconductors. While in ordinary ferromag-\nnets, currents couple only to the canted spin configura-\ntions at domain walls, the peculiar magnetic structure\nof the skyrmion lattice allows for an efficient bulkcou-\npling. Furthermore, the smooth magnetic structure of\nthe skyrmion lattice decouples efficiently from the un-\nderlying atomic lattice and from impurities. As a conse-\nquence, itwasobserved10that the criticalcurrentdensity\nneeded to affect the magnetic structure was more than\nfive orders of magnitude smaller than in typical spin-\ntorque experiments.\nThese low current densities open opportunities for new\ntypes of experiments to study quantitatively the physics\nof spin transfer torques. Due to the much lower cur-\nrent densities it is now possible to perform spintorque\nexperiments in bulk materials and thus avoid the surface\neffects that dominate in nanoscopic samples. Moreover\nfor smaller currents the effects of heating and Oersted\nmagnetic fields created by the current are suppressed.\nFigure 1: Schematic plot of the forces on a skyrmion lattice\nperpendicular and parallel to the current flowing in vertica l\ndirection. For a static, non-moving skyrmion lattice the re d\nhorizontal arrows correspond to the Magnus force and the\ngreen vertical arrows to dissipative forces. In the presenc e of\na temperature or field gradient, these forces change smoothl y\nacross a domain, thereby inducing rotational torques which\ndepend sensitively on the relative orientation of current a nd\ngradient (and on the direction in which the skyrmion lattice\nmoves). Small black arrows: local orientation of the magnet i-\nzation projected into the plane perpendicular to the magnet ic\nfieldB. Ineachunitcell themagnetization windsoncearound\nthe unit sphere.\nIn this paper we suggest experiments and develop a\ntheory with the goal to exploit the rotational motion in-\nsteadofjust translationalmotionto investigatethe inter-\nplay of electric currents and moving magnetic structures.\nOurtheoryisdirectlymotivatedbyrecentexperiments10,\nwhere a change of orientation of the skyrmion lattice as a\nfunction ofthe applied electric currentwas observedwith\nneutron scattering. In Ref. [10] we have shown that the2\nrotation arises from the interplay of a tiny thermal gradi-\nent parallel to the current and the Magnus forces arising\nfromthespintorquecouplingofcurrentandskyrmionlat-\ntice. For example, the rotation angle could be reversed\nby reversing either the current direction or the direction\nof the thermal gradient.\nThe basic idea underlying the theoretical analysis of\nour paper is sketched in Fig. 1. In the presence of an\nelectric current several forces act on the skyrmion lat-\ntice. First, dissipative forces try to drag the skyrmion\nlattice parallel to the (spin-) current. Second, the inter-\nplay of dissipationless spin-currents circulating around\neach skyrmion and the spin-currents induced by the elec-\ntric current lead to a Magnus force oriented perpendic-\nular to the current for a static skyrmion lattice (for the\nrealistic case of moving skyrmions the situation is more\ncomplicated). In the presence of any gradient across the\nsystem (e.g. a temperature or field gradient), indicated\nby the color gradient, these forces will vary in strength\nacross a skyrmion domain.\nAs in the experiment, we assume that the gradients\nare tiny: on the length scale set by the skyrmion distance\nthe gradients have negligible effects. However, multiply-\ning the tiny gradient with a large length, i.e., the size of\na domain of the skyrmion lattice (which can be11several\nhundred µm), one obtainsa sizablevariationofthe forces\nacross the domain. These inhomogeneous forces can give\nrisetorotationaltorques. Whetherthetorquearisesfrom\nthe Magnus forcesor the dissipative forces depends, how-\never, on the relative orientation of current and gradient\nand also on the direction in which the skyrmion lattice\ndrifts. Fig. 1 givesa simple example: if, forexample, cur-\nrent and gradient are parallel to each other (right panel)\ntheforcesperpendiculartothecurrentdirection(redhor-\nizontal arrows) give rise to rotational torques while the\nparallel forces do not contribute. The situation is re-\nversed when current and gradient are perpendicular (left\npanel).\nWe therefore suggest to use the rotation of magnetic\nstructures as a function of the relative orientation of cur-\nrent and further gradients as a tool to explore the cou-\npling of magnetism and currents. We will show that the\nresulting rotations depend very sensitively both on the\nrelative size of the various forces affecting the skyrmion\ndynamics and on how these forces depend on the induced\ngradients. While we apply our theory here to skyrmion\nlattices, our theoretical approaches can also be used for\nother complex magnetic textures and our results should\nalso have ramifications for other setups12,13. Quantita-\ntively, we will only study the role of gradients induced\nby changes in temperature or magnetic field but other\noptions are also possible. For example, macroscopic vari-\nations of the cross section of a sample will lead to gradi-\nents in the current density. Also changes in the chemical\ncomposition or strain in the sample can induce gradients.\nIt is also essential to investigate the effect of pin-\nningofthe magneticstructurebyinhomogeneitiesarising\nfrom crystalline imperfections. Inhomogeneities distortthe perfect skyrmion lattice and lead to forces prohibit-\ning (up to a very small creep) the motion of the mag-\nnetic structure as long as the current is below a criti-\ncal value, j < jc. Also for j/greaterorsimilarjc, inhomogeneities in-\nduce an effective, velocity dependent frictional force on\nthe moving skyrmion lattice connected to local, time-\ndependent distortions of the skyrmion lattice. Pinning\nhas widely been studied both experimentally and the-\noretically for charge density waves and vortex lattices\nin superconductors14–17. As the dynamics of skyrmions\ndiffers qualitatively (and quantitatively) from these two\ncases it is not clear which of these results can be trans-\nferred to skyrmion lattices. Due to the non-linear depen-\ndence of the pinning forces on the velocity, they can not\nbe described by a simple damping term. Within this pa-\nper we will not try to develop a theory ofpinning but will\ninstead use a simple phenomenologicalansatz to describe\nand discuss pinning effects.\nRotational torques can also arise in the absence of the\ntypes of gradients discussed above. In Ref. [18] we have\nstudied the role of distortions of the skyrmion lattice by\ntheunderlyingatomiclatticeextendingthemethodsused\nbyThiele19torotationaltorques(thismethodwillalsobe\nused below). Such distortions indeed induce small rota-\ntional torques in a macroscopicallyhomogeneous system,\ni.e. withoutanyexternalgradients. Similarly,alsodistor-\ntions induced by disorder can induce rotational torques\nwithout external gradients as has been discussed in the\nseminal paper by Hauger and Schmid14. But all these\neffects are very small and have notbeen observed in the\nexperimental setup of Ref. [10] as no rotation has been\nobserved in the absence of gradients. Therefore they will\nbe neglected in the following.\nIn the following we will first describe briefly the rele-\nvant Ginzburg-Landau model and the Landau-Lifshitz-\nGilbert equation used to model the dynamics of the\nskyrmions. Here we include a novel damping term α′\nrecently introduced in Refs. [20,21] (we also add the cor-\nresponding β′term). We then derive effective equations\nfor the translational and rotational mode where pinning\nphysics is taken into account by an extra phenomeno-\nlogical term. This allows to develop predictions both for\nstaticrotationsbyafiniteangleandcontinuousrotations.\nIn the light of our results we interpret experimental re-\nsults on the angular distribution of skyrmion lattices in\nthe presence of currents and gradients.\nII. SETUP\nA. Ginzburg-Landau model\nThe starting point of our analysis is the standard\nGinzburg-Landau model of a chiral magnet in the pres-\nence of a Dzyaloshinskii-Moriya interaction22,23. After a\nrescaling of the length r, the local magnetization M(r)\nand the magnetic field Bthe free energy functional re-3\nduces to9\nF=γF/integraldisplay\nd3r/bracketleftbig\n(1+t)M2+(∇M)2\n+2M·(∇×M)+M4−B·M/bracketrightbig\n,(1)\nHeret∝T−TMF\ncparametrizesthe distance to the mean-\nfield phase transition at B= 0 from a phase with helical\nmagnetic order ( t <0) to a paramagnetic phase ( t >\n0)22,23. In the presence of weak disorder t(and strictly\nspeaking also the prefactors of all other terms) fluctuates\nslightly as a function of r.\nThe skyrmion lattice (stabilized by thermal fluctua-\ntions) exists for a small temperature and field range9.\nIt is translationally invariant parallel to Band shows a\ncharacteristic winding of the magnetization in the plane\nperpendicular to B, see Fig. 1.\nB. Landau-Lifshitz-Gilbert equation\nTo describe the dynamics of the orientation ˆΩ(r,t) =\nM(r,t)/|M(r,t)|of the magnetization M(r,t) in the\npresence of spin-transfer torques due to electric cur-\nrentsweuse the standardLandau-Lifshitz-Gilbert(LLG)\nequation,7,8,24extended by a new dissipative term20,21\n(∂t+vs∇)ˆΩ=−ˆΩ×Heff+αˆΩ×/parenleftBig\n∂t+β\nαvs∇/parenrightBig\nˆΩ\n−α′/bracketleftBig\nˆΩ·/parenleftbig\n∂iˆΩ×(∂t+β′\nα′vs∇)ˆΩ/parenrightbig/bracketrightBig\n∂iˆΩ.(2)\nHerevsis an effective spin velocity parallel to the spin\ncurrent density. More precisely, for smooth magnetic\nstructures with constant amplitude of the magnetization\nit is given by the ratio of the spin current25and the size\nof the local magnetization, |M|. In a good metal (for ex-\nample, MnSi) vsis expected to be parallel to the applied\nelectric current and to depend only weakly on temper-\nature and field. The magnetization precesses in the ef-\nfective magnetic field Heff≈ −1\nMδF\nδˆΩ. Strictly speaking\nEq. (2) is only valid for a constant amplitude of the mag-\nnetization, |M|= const. Since |M|varies only weakly9\nin the skyrmion phase, we use as a further approxima-\ntionHeff≈ −1\nMδF\nδM∂M\n∂ˆΩwhereMis the average local\nmagnetization, M2=∝angb∇acketleftM2∝angb∇acket∇ight.\nThe last two terms in Eq. (2) describe dissipation.\nαis called the Gilbert damping and βparametrizes\nthe dissipative spin transfer torque. The new damping\nterm proportional to α′was introduced (for β′= 0) in\nRefs. [20,21]. It arises from the ohmic damping of elec-\ntrons coupled by Berry phases to the spin texture as can\nbe seen by rewriting Eq. (2) in the form\n−δF\nδˆΩ=MˆΩ×(∂t+vs∇)ˆΩ+αM/parenleftBig\n∂t+β\nαvs∇/parenrightBig\nˆΩ\n+MˆΩ×α′/bracketleftBig\nEe\ni+β′\nα′(vs×Be)i/bracketrightBig\n∂iˆΩ.(3)whereEe\ni=ˆΩ·(∂iˆΩ×∂tˆΩ) can be interpreted as the\nemergent electric field and Be\ni=1\n2ǫijkˆΩ·(∂jˆΩ×∂kˆΩ) as\nthe emergent magnetic field26,27. These fields describe\nthe forces on the electrons arising from Berry phases\nwhich they pick up when their spin adiabatically follows\nˆΩ(r,t). They couple to the spin rather to the charge:\nelectrons with magnetic moment parallel (antiparallel)\ntoˆΩcarry the ”emergent charge” −1/2 (+1/2), respec-\ntively. For vs= 0 the change of the free energy density\nis given by\n∂tF=δF\nδˆΩ∂tˆΩ=−αM(∂tˆΩ)2−α′M(Ee)2.(4)\nwhich shows that the last term describes the dissi-\npated power ∝(Ee)2arising from the emergent electric\nfield.α′Mis therefore approximately given by the spin-\nconductivity σs.\nWe have also added a new β′–term. The presence of\nsuch a term becomes evident if one considers the special\ncaseofaGalileaninvariantsystem. In thiscase, allforces\nhave to cancel when the magnetic structure is comoving\nwiththe conductionelectrons, ˆΩ(r,t) =ˆΩ(r−vst). This\nis only possible for α=βandα′=β′. Solids are not\nGalilean invariant and therefore β′is different from α′\nbut one can, nevertheless, expect that the two quantities\nare of similar order of magnitude.\nWhich of the damping terms will dominate? As\npointed out in Refs. [20,21], the naive argument, that the\nα′terms are suppressed compared to the αterms as they\ncontain two more derivatives, is not correct. The dis-\ntance of skyrmions is9proportional to 1 /λSO, whereλSO\nparametrizes the strength of spin-orbit coupling. While\ntheα′term has two more gradients compared to the α\nterm, the contribution arising from α′is, nevertheless,\nof the same order in powers of λSO, if we assume that\nαarises only from spin-orbit coupling, α∝λ2\nSO, while\nα′∝λ0\nSO(ohmic damping (see above) does not require\nspin-orbit effects). As furthermore αis proportional to a\nscattering rate while α′is proportional to a conductivity\nand therefore the scattering time20,21,α′andβ′might\nbe the dominating damping terms in good metals.\nIII. DYNAMICS OF SKYRMIONS\nOur goal is to describe both the drift and the rotation\nofthe skyrmion lattice in the limit of small currentdensi-\nties and small magnetic or thermal gradients. We there-\nforeassumethat vsissmallcomparedtoallcharacteristic\nvelocity scales of the skyrmion lattice (e.g. Tc−Tmulti-\nplied with the skyrmion distance). The gradients should\nbe so small that the total change across a domain of ra-\ndiusrdremains small, rd∇λ≪λwhereλisBorTc−T\nfor magnetic or thermal gradients, respectively. In this\nlimit, both the drift velocities vd/lessorsimilarvsand the angular\nvelocity ∂tφ∝vs·∇λcharacterizing rotational motion\nremain small. Below we will show, that even ∂tφrd, the4\nvelocity at the boundary of the domain remains small in\nthe considered limit.\nWe can therefore neglect macroscopic deformations of\nthe magnetic structure and consider the following ansatz\nˆΩ(r,t) =Rφ(t)·ˆΩ0/parenleftBig\nR−1\nφ(t)·(r−vdt)/parenrightBig\n(5)\nHereˆΩ0(r) describes the static skyrmion lattice, Rφis\na matrix describing a rotation by the angle φaround the\ndirection of the skyrmion lines (i.e. around the field di-\nrection when anisotropies are neglected, which will be\nassumed in the following) and vdtdescribes the loca-\ntion of the center of the skyrmion domain. This ansatz\ndescribes a magnetic domain which rotates around its\ncenter, while the center is moving with the velocity vd.\nWhen the torque forces are too weak to induce a steady-\nstate rotation, such that ∂tφ= 0, we will study rotations\nby the finite angle φas in the experiment of Ref. [9].\nA. Drift of domains\nTo obtain an equation for the drift velocity vdwe fol-\nlow Thiele19and project Eq. (3) onto the translational\nmode by multiplying Eq. (3) with ∂iˆΩand integrating\nover a unit cell (UC). We thereby obtain to order ( ∇λ)0\n(where no rotations occur) an equation for the force per\n2d magnetic unit cell (and per length)28\nG×(vs−vd)+D(˜βvs−˜αvd)+Fpin= 0 (6)\nGi=/integraldisplay\nUCd2rMBe\ni=GˆBi,G= 4πMW\nDij=/integraldisplay\nUCd2rM∂iˆΩ∂jˆΩ=DPij\nD′=/integraldisplay\nUCd2rM(Be)2\n˜α=α+α′D′/Dand˜β=β+β′D′/D\nHere the first term describes the Magnus force which is\nproportionalto the topologicalwinding number Wwhich\nis for the skyrmion lattice exactly given by W=−1.\nGis called the gyromagnetic coupling vector following\nThiele19. The second term are the dissipative forces with\nthe projector Pinto the plane perpendicular to B,P=\n(1−ˆB·ˆBT).\nBesides the forces discussed above, also pinning forces,\ndescribed by the last term in Eq. (6), have to be con-\nsidered. Formally, they are encoded in spatial fluctua-\ntions of δF/δˆΩin Eq. (3). The Thiele approach, used\nabove, which considers only a global shift (or a global\nrotation18, see below) of the magnetic structure does not\ncapture these pinning effects as for a perfectly rigid mag-\nnetic structure, random pinning forces average to zero,\nsuch that no net effect remains in the limit of a large\ndomain. To describe pinning, it is necessary14,15to take\ninto account that the magnetic structure adjusts locallyto the pinning forces, a complicated problem for which\npresently no full solution exists17,29and which is far be-\nyond the scope of the present paper. Instead, we use a\nphenomenological ansatz and write for a finite drift ve-\nlocityvd\nFpin=−4πMvpinf(vd/vpin)ˆvd (7)\nto describe a net pinning force, which is oriented op-\nposite to the direction of motion. Its strength, which\ndepends both on the number (and nature) of defects re-\nsponsible for pinnning and the elastic properties of the\nskyrmion lattice, is parametrized by the ‘pinning veloc-\nity’vpin. The function f(x) withf(x→0) = 1 and\nf(x→ ∞) =xνparametrizes the non-linear dependence\nof the pinning force on the velocity. Presently, it is not\nclear to what extent f(x) depends on microscopic details\nand also the exponent νis not known. For large driving\nvelocities, however, pinning becomes less and less impor-\ntant (ν <1)16,17,29. If the driving forces are smaller\nthan the force 4 πMvpin, needed to depin the lattice, vd\nvanishes and the pinning forces cancel exactly the driv-\ning forces. Note that we do not consider creep, i.e. a\ntinymotiondrivenbythermal(orquantum)fluctuations,\nwhich occurs even in the pinning regime17. If the dissi-\npative forces can be neglected, it is in principle possible\nto obtain f(x) from a measurement of the velocity of the\nskyrmion lattice27.\nIn the limit vs≫vpin, whereFpincan be neglected,\nwe solve Eq. (6) for vs⊥Bto obtain\nvd=˜β\n˜αvs+˜α−˜β\n˜α3(D/G)2+ ˜α/parenleftBig\nvs+ ˜αD\nGˆB×vs/parenrightBig\n(8)\nwith ˜α=α+α′D′/Dand˜β=β+β′D′/D.\nB. Rotational torques\nBy symmetry, a small uniform current cannot induce\nany rotational torques on a skyrmion lattice with per-\nfect sixfold rotation symmetry and therefore all effects\narise from gradients. To derive an equation for the ro-\ntational torques which determine the rotations around\ntheBaxis, we follow18a similar procedure as used for\nthe translations by multiplying (3) by the generator of\nrotations applied to ˆΩ\n∂φˆΩ=ˆB׈Ω−(ˆB(∆r×∇))ˆΩ (9)\nwith∆r=r−vdtandintegratingover r. Thisprocedure\nleads to several types of contributions.\nFor the first type of contribution, we observe that the\nsecond term in Eq. (9), linear in ∆ r, is much larger than\nthe first one which we can therefore neglect whenever\nthe second term contributes. The second term induces\ntorques of the form r×fwhere the force fiis obtained\nby multiplying ∇iˆΩwith the terms of Eq. (3). In the5\npresence of gradients of the parameter λwe obtain\n/integraldisplay\nˆB·[r×f(λ(r))]≈/integraldisplay/parenleftBig\nˆB·[r×∂λf]/parenrightBig\n(r·∇λ)\n≈A\n4πˆB·[∇λ×∂λ/integraldisplay\nf] (10)\nwhereAis the area of the domain. Here it is essential to\ntake the derivative with respect to λforfixedvdreflect-\ning that due to the rigidity of the skyrmion crystal vdis\nconstant across the domain. As the sum of all relevant\nforces vanishes [Eq. (6)],/summationtext\nifi(λ,vd) = 0, one obtains\nd\ndλ/summationtext\nifi= 0 while∂\n∂λ/summationtext\nifi/vextendsingle/vextendsingle\nvdis finite. In Eq. (10)\nwe have implicitely assumed a symmetrically shaped do-\nmain, where integrals odd in rvanish. In general, there\nwill also be a shape dependent torque Tshapearising even\nin the absence of a gradient. As its sign is random, it\ncan easily be distinguished from the other torques (and\nappears to be relatively small in the MnSi experiments9).\nMore difficult is the question what happens at the inter-\nface of different domains or when a domain comes close\nto the surface ofthe sample. Nominally surfaceforcesare\nsuppressed by a factor proportional to 1 /√\nAcompared\nto the bulk terms considered above but the relevant pref-\nactors are difficult to estimate. We will neglect in the\nfollowing formulas both extra surface forces and shape\ndependent torques.\nA different contribution arises from the time deriva-\ntives∂tˆΩ=∂tφ∂φˆΩ−(vd∇)ˆΩin Eq. (3). The contri-\nbution proportional to vdis of the form discussed above.\nThe term proportionalto ∂tφleads to extra torquesinde-\npendent of ∇λ. By combiningthe linearterm in ∆ rfrom\n∂φˆΩwith the second term of Eq. (9) we obtain for exam-\nple the contribution α∂tφ/integraltext\nM[(ˆB[∆r×∇])Ω]2which is\nalso linear in A. Physically this term describes the fric-\ntional torque which is linear in the angular velocity ∂tφ.\nThe frictional torque per volume is proportional to Abe-\ncause the velocity and therefore the frictional forces grow\nlinearly with the distance from the center of the rotating\ndomain.\nFinally, a contribution exists which is independent of\nthe gradients ∇λ, the angular velocity ∂tφand ofvs.\nThis contribution describesthat in the absence ofanyex-\nternal perturbation the skyrmion lattice has a preferred\norientation relative to the atomic lattice. Such terms ex-\npressthat angularmomentum canbe transferreddirectly\nfrom the skyrmion lattice to the underlying atomic lat-\nticemediatedbyspin-orbitcouplingandsmallanisotropy\nterms (not included in Eq. (1)). These terms have been\ndiscussed in detail in Ref. [18]. This torque per unit cell\nTL=−/integraldisplay\nUCd2rδF\nδˆΩ(ˆGrotˆΩ) =−∂FUC\n∂φ≈ −χsin(6φ)\n(11)\ncan be expressed by the change of free energy per unit\ncell,FUC, upon rotation by the angle φ, where φ= 0\nreflects the equilibrium position and sin6 φreflects the\nsixfold symmetry of the skyrmion lattice. As has been\ndiscussed in Ref. [10], the absolute value of χin materials-1.5-1-0.500.511.5\nλ  Vs-15-10-5051015 φ  φ \n-1.5-1-0.500.511.50\n-1.5-1-0.500.511.5\n ω γ ω γ ∆.\nFigure 2: Rotation angle φ(in units of 1◦) and angular ve-\nlocity ¯ω(times the prefactor γ) as a function of ∇λ·Vsde-\ntermined from Eq. (13).\nlike MnSi is tiny as it arises only to high order in spin-\norbitcoupling and, in contrastto allother terms, it is not\nlinear in the size of the domain. Nevertheless, we have\nto consider this term, as it is the leading contribution\narising to zeroth order in ∇λandvs.\nBalancing all torques (per unit cell) we obtain as our\ncentral result\n0 =TL+TG+Tpin+TD (12)\nTG=A\n4π∇λ·/bracketleftbigg∂(Gvs)\n∂λ−∂G\n∂λvd/bracketrightbigg\nTpin=A\n4π∇λ·[ˆB׈vd]∂Fpin\n∂λ, Fpin≡ |Fpin|\nTD=−A˜αD\n2π∂tφ\n−A\n4π∇λ·/bracketleftbigg\nˆB×/parenleftBig∂(D˜βvs)\n∂λ−∂(D˜α)\n∂λvd/parenrightBig/bracketrightbigg\nThe direction of the torques, which depends on the rel-\native orientation of velocities and currents, is for vd= 0\n(and∂tφ= 0) fully consistent with the simple picture\nshown in Fig. 1: the dissipative torques TDarise when\ngradient and current are perpendicular to each other\nwhile the reactive torque TGarising from the Magnus\nforce is activated for a parallel alignment of gradients\nand currents. For finite vd, however, this simple intu-\nitive picture cannot be used especially as some of the\ntorques tend to cancel when vdapproaches vs.\nC. Rotation angle and angular velocity\nEq. (12) can be rewritten in the compact form\nsin6φ=−γ∂tφ+∇λ·Vs (13)\nwhereγ=A˜αD\n2πχand the vector Vs=Vs[vs] can be ob-\ntainedbyfirstsolvingEq.(6)toobtain vdasafunctionof6\nvs. This function is inserted into Eq. (12) which, finally,\nis devided by −χ. The function Vs[vs] withVs[0] = 0 is\nproportional to the area Aof the domain and encodes all\ninformation how the current couples to small gradients\nand includes contributions from Magnus forces, dissipa-\ntive forces and pinning.\n1. Dependence on size of gradients\nQualitatively, three different regimes have to be dis-\ntinguished. For j < jc, when pinning forces cancel all\nreactive and dissipative forces, there is neither a motion\nnor a rotation of the skyrmion lattice, Vs= 0,φ= 0,\nwithin our approximation. Note, however, that it is\nwell known from the physics of charge density waves or\nvortices17that even below jca slow creep motion is pos-\nsible. Whether during this creep also rotations are possi-\nble is unclear, but the rather sharp onset of the rotation\nin the experiments of Ref. 10, see Fig. 8, seems to con-\ntradict a scenario of pronounced rotations during creep.\nForj > jc, the domains move and Vswill generally be\nfinite. In this case, one can control the size and direction\nof rotations by the size of ∇λas shown in Fig. 2. For\n|∇λ·Vs|<1, one obtains a solution where ∂tφ= 0 but\nthe gradients induce a rotation by a finite angle\nφ=1\n6arcsin∇λ·Vs, (14)\nwhich grows upon increasing ∇λfrom zero until it\nreachesthe maximalpossiblevalue π/12 = 15◦(rotations\nby an average angle of 10◦have already been observed10,\nsee Fig. 8). For |∇λ·Vs|>1 the domain rotates (see\nFig. 2) with the (average) angular velocity\n¯ω=/radicalbig\n(∇λ·Vs)2−1\nγ(15)\nand Eq. (13) is solved by\nφ(t) =1\n3arctan/bracketleftBigg\n1+γ¯ωtan(3¯ωt)/radicalbig\n1+γ2¯ω2/bracketrightBigg\n.(16)\ndisplayed in the inset of Fig. 3. As both γandVsare\nlinear in the area Aof the domain, ¯ ω≈(∇λ·Vs)/γbe-\ncomesindependent ofthe domainsizefor A→ ∞. Inthis\nlimit, the domain rotates continuously, φ= ¯ωt. Close to\nthe threshold, ∇λ·Vs= 1, however, the rotation be-\ncomes very slow close to an angle of 15◦(plus multiples\nof 60◦).\nA way to detect the rotation of the magnetization is\nto exploit the emergent electric field Eewhich obtains\na contribution proportional to ∝∂tφand can be mea-\nsured in a Hall experiment27. In Fig. 3 we therefore\nshow the modulus of the Fourier components, |cn|=\n|/integraltext\nei6¯ωnt∂tφdt|of∂tφas a function of ∇λ·Vs. At the\nthreshold, all Fourier components are of equal weight\nwhile for large gradients the rotation gets more uniform.0 1 2 3 4 5\nλ Vs0 010 1020 2030 3040 4050 5060 60cnc0c1c2c3c4c5\n0 0.1 0.2 0.3 0.4 0.5\nωt/2π060120180φ1.01\n1.3\n10∆.\nFigure 3: Inset: Rotation angle (in units of 1◦) as a function\nof time for three values of ∇λVs>1, see Eq. (16). For\ntorques close to the value where rotations sets in, the rotat ion\nis strongly anharmonic. This can also be seen by considering\nthe Fourier coefficients cn=|/integraltext2π/6¯ω\n0∂tφein6¯ωtdt|shown in\nthe main panel as a function of ∇λVs.\nFor fixed ¯ ωthe velocities at the boundary of the do-\nmain,vb= ¯ωrd, grow linearly with the radius of the do-\nmainrd. As we have assumed that the gradients across\nthe sample and therefore also across a single domain are\nsmall,rd∇λ≪λ, the velocities nevertheless remain\nsmall,vb≪ |Vs|λ/γ/lessorsimilarvs/˜α. While our estimate does\nnot rule out that vbcan become somewhat larger than vs\norvd, we expect that the typical situation is that the ve-\nlocityvbarising from the rotation remains smaller than\nthe overall drift velocity of the domain vd. This estimate\nalso implies that violent phenomena like the breakup of\ndomains due to the rotation will probably not occur.\n2. Domain size dependence and angular distribution\nIn a real system, there will always be a distribution\nof domain sizes A. BothVsandγare linear in Aand\ntherefore both the rotation angle (14) and the angular\nvelocity (15) will in general depend on the domain size\nand therefore on the distribution of domains.\nOnly in the limit |∇λ·Vs| ≫1, the dependence on A\ncancels in Eq. (15) and all domains rotate approximately\nwiththesameangularvelocity. For |∇λ·Vs|/lessorsimilar1onewill\nin general obtain a distribution of rotation angles which\ncan be calculated from the distribution of domain sizes\nPd(A). For the static domains only angles up to 15◦are\npossible with\nPs\nφ=/integraldisplayAc\n0dAPd(A)δ(φ−arcsin(A/Ac)\n6)\n= 6Accos(6φ)Pd(Acsin(6φ)) for 0 ≤φ≤π\n12(17)7\n0 10 20 30 40 50 60\n φ 02468PφA0/Ac=0.25\nA0/Ac=0.5\nA0/Ac=1.0\nA0/Ac=2.0\nA0/Ac=4.0\nFigure 4: Angular distribution Pφof the rotation angle of the\nskyrmion lattice for various values of A0/Ac∝ ∇λ(see text).\nHere we assumed a distribution of domain sizes of the form\nPd(A) =e−A/A0A\nA2\n0. While static domains contribute only for\n0≤φ≤15◦one obtains a smooth angular distribution when\none takes the rotating domains with A > A cinto account.\nwhereAc=A/(∇λ·Vs) is the size of a ‘critical’ domain\nwhich just starts to rotate continuously.\nThe continuously rotating domains also have a non-\ntrivial angular distribution as their rotation will be\nslowed down when the counterforces are strongest, i.e.,\nforφ= 15◦, see inset of Fig. 3. The angular distribution,\nPr\nφ, of the rotating domains is calculated from distribu-\ntion ofdomainsizes, Pd(A), andthe angulardistribution,\npr\nφ(A), of a single domain\nPr\nφ=/integraldisplay∞\nAcdAPd(A)pr\nφ(A)\npr\nφ(A) =1\nT/integraldisplayT\n0δ(φ−φ(t))dt=1\nT∂tφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nφ(t)=φ\n=3\nπ/radicalbig\nA2−A2c\nA−Acsin6φ(18)\nwhereT= 2π/(6¯ω). While both Ps\nφandPr\nφare non-\nanalytic at φ= 15◦, the total distribution, Pφ=Ps\nφ+Pr\nφ\nissmoothfor φ >0andnormalizedto1,/integraltext2π/6\n0Pφdφ= 1.\nIn Fig. 4 we show Pφassuming the domain distribution\nPd(A) =e−A/A0A\nA2\n0for various values of A0/Ac.\nIn elastic neutron scattering, the skyrmion phase is\nobservedby six Braggspots forming a regularhexagonin\naplaneperpendiculartothemagneticfield. Arotationof\nthe skyrmion domain results in a rotation of these Bragg\nspots. Therefore the angular distribution Pφof rotation\nangles is directly observable (see Sec. IIID below) by\nmeasuring the scattering intensity as a function of angle.\nBy comparing angulardistributions for different strength\nof the current or gradient, one can – at least in principle\n– obtain not only Acasa function of ∇λorjbut alsothedistribution of domain sizes. The latter can be extracted\nmost easily in the regime where most of the domains\ndo not rotate continuously by plotting Pφ/cos6φas a\nfunction of sin6 φusing Eq. (17).\n3. Dependence on strength of current\nWhile the behavior of φand ¯ωas a function of ∇λis\nrather universal and independent of microscopic details,\nits dependence on the strength of the current for fixed\n∇λis much more complex. As discussed above, Vs= 0\nforj < jc. Directly at jc, when the domain starts to\nmove with vd≈0,Vsjumps to the finite value\nVs|vs=vpin=−A\n4πχ/bracketleftbigg/parenleftbigg\n−∂Gvs\n∂λ+Gvs\nFpin∂Fpin\n∂λ/parenrightbigg\n+ˆB×/parenleftBigg\n∂D˜βvs\n∂λ−D˜βvs\nFpin∂Fpin\n∂λ/parenrightBigg/bracketrightBigg\n.(19)\nNote that the jump is independent of αandα′as well as\nof their gradients, as the skyrmions are not moving di-\nrectlyatthe depinningtransition(seeFig.6). Depending\non the direction and size of ∇λ, the jump of Vseither\nleads to a jump of the rotation angle for |∇λ·Vs|<1 or\nimmediately to a continuous rotation for |∇λ·Vs|>1.\nUpon increasing the current, ∇λ·Vscan either in-\ncrease, decrease or even change its sign depending on\n(i) the direction of ∇λand (ii) on the question which\nof the forces changes most strongly when varying λ(i.e.,\ntemperature or magnetic field).\nMotivatedbyexistingexperimentaldata(discussedbe-\nlow in Sec. IIID) we study the case of a temperature gra-\ndient,λ=t, based on the following assumptions. First,\nwe assume that all damping constants are temperature\nindependent (this assumption is relaxed later). Second,\nwe need also a theory for the temperature dependence of\nthe pinning force. Here we use the experimental obser-\nvation27that the critical current is almost temperature\nindependent at least for a certain range of temperatures.\nWithin our theory, Eqs. (6) and (7), this implies that\nall temperature dependence of Fpin(i.e., the dependence\non the parameter tin Eq. (1)) arises from the tempera-\nture dependence of the magnetization Mwhich we cal-\nculate from the Ginzburg-Landau theory (1). From the\nGinzburg-Landautheory, we obtain alsothe temperature\ndependence of the other parameters, see Fig. 7.\nIn Fig. 5 we show a typical result (for temperature-\nindependent dissipation constants) for the rotation angle\nand angular velocity of a skyrmion domain as a function\nofvsin the presence of a temperature gradient. For a\ntemperature gradientperpendicular to the current(lower\npanel of Fig. 5), the rotation angle increases after the\ninitial jump. For the gradient parallel to the current,\nhowever, we obtain that the rotation angle dropsafter\nthe initial jump (upper panel). For larger values of vs\nthe angle rises again until it reaches its maximal value of8\n-30 -20 -10 0 10 20 30vs-15-10-5051015φφ\n-2-1012\nω  γω γ\nvs\n∆t\n-10 -5 0 510vs-15-10-5051015φφ\n-2-1012\nω  γω γ\nt    vs\n∆\nFigure 5: Rotation angle φ(in units of 1◦) and angular veloc-\nity,γ¯ω, as a function of vsfor a temperature gradient parallel\n(∇t∝ba∇dblvs,∇t= (−0.1,0,0), upper panel) and perpendicular\n(∇t⊥vs,∇t= (0,−0.05,0), lower panel) to the current\n(α= 0.2,β= 0.45,α′= 0.01,β′= 0.2,A/χ= 200,t=\n−1,B= (0,0,1/√\n2),vpin= 1,f= 1). For both geometries\none observes a jump of φatvs≈vpinfrom zero to a finite\nrotation angle. After the initial jump the rotation angle in -\ncreases for the perpendicular configuration (panel b) while for\nthe parallel arrangement first a drop and then an increase up\nto the maximal angle of 15◦occurs. For larger vsa contin-\nuous rotation characterized by the angular velocity ¯ ωsets in\nfor both configurations. For the calculation we assumed that\nthe damping parameters and vpinare independent of t.\n15◦. This qualitative shape of the curve appears to be\nrather independent of the precise values of the various\nparameters ifwe assume that all damping parameters\nare temperature independent.\nIn Fig. 6 we plot the rotation angle for small cur-\nrent densities taking an extra effect into account which is\npresent in the experiments described in Ref. [10]: as the\ntemperature gradients are induced by the currents, they\ngrowquadraticallywith vs. This doesnot giverisetoany\nqualitative changes. The thin blue curve Fig. 6 thereby\nreflects the same physics as the corresponding curve in-1 0 1vs-15 -15-10 -10-5 -50 05 510 1015 15φφ (   α = 0.035   t)\nφ (   α = 0 )\n∆ ∆∆∆t      vs\n-1 0 1vs-15 -15-10 -10-5 -50 05 510 1015 15φφ (   α = 0.035   t)\nφ (   α = 0 )\nt     vs\n∆\n∆∆\n∆\nFigure 6: Rotation angle φ(in units of 1◦) as a function of vs\nfor a temperature gradient parallel ( ∇t∝ba∇dblvs, upper panel) and\nperpendicular ( ∇t⊥vs, lower panel) to the current. The pa-\nrameters are the same as in Fig. 5 with two exceptions. First,\nwe have taken into account that in the experiments of Ref. 10\nthe temperature gradient grow with the square of the applied\ncurrent, ∇t= (−0.1v2\ns,0,0) and∇t= (0,−0.05v2\ns,0)), for\ncurrent parallel an perpendicular to vs, respectively. For the\nthin blue curve we assumed (as in Fig. 5) that the damping\nconstants are independent of twhile for the thick green curve\na weak temperature dependence of the damping constant α,\n∇α= 0.035∇t, was assumed. This parameter has been cho-\nsen to reflect the experimental observation, see Fig. 8. For\neven stronger currents (not measured experimentally and no t\nshown in the figure) the size of the torque drops again and a\nfinite rotation angle is obtained for 1 .57/lessorsimilarvs/lessorsimilar2.53 in the\nparallel configuration with the temperature dependent damp -\ning constant.\nFig. 5 (note the different scale on the xaxis). The thick\ngreen curve of Fig. 6 shows that one can, however, ob-\ntain qualitative different results (an increase rather than\na reduction of the rotation angle after the initial jump\nforTgradients parallel to the current, upper panel) by\nincluding a small temperature dependence of the Gilbert\ndamping α. As we will discuss in Sec. IIID, this can9\n-1.1 -1 -0.9 -0.8t468101214 -G,  D ,  D’-G\nD\nD'\nFigure 7: Change of G,DandD′defined in Eq. (6) with\ntemperature t. The applied magnetic field is h/√\n2(0,0,1).\nDashed lines are for h= 0.9 and continuous lines for h= 1.1.\nreproduce qualitatively the experimentally observed be-\nhavior.\n4. Dependence on orientation of gradients\nFig. 5 shows that the rotational torques on the sys-\ntem depend strongly on the relative orientation of gradi-\nent and current. More importantly, one probes different\nphysical mechanism for gradients parallel or perpendicu-\nlartothecurrent. Thiseffectwasalreadydiscussedinthe\nintroduction, see Fig. 1, where, however, only the simple\ncase of a static domain without pinning was described.\nIn reality, the situation is more complex. All directional\ninformation is encoded in the function Vs(vs) which can\nbe obtained by first solving Eq. (6) to obtain vdand then\ncomparing Eqs. (12) and (13). Unfortunately, a rather\nlarge number of unknown parameters (most importantly,\nthe pinning forces and their dependence on λ) enters the\ndescription. Therefore we will discuss in the following\nonly a few limiting cases.\nA drastically simplified picture occurs in regimes when\nonly two forces dominate in Eq. (6). For example, close\nto the pinning transition, the Magnus force is of the\nsame order as the pinning force while the two dissipa-\ntive forces are typically much smaller. In this case one\ncan use Eq. (6) to show that ˆvdbecomes proportional\ntoˆB×(vs−vd). Thus, for an λ-independent vs, both\nthe reactive rotational coupling vector and the rotational\npinning vector become proportional to ∇λ·(vs−vd)\n(here we neglect a possible λ-dependence of vs). There-\nfore theratioof the component of Vsparallel ( Vs/bardbl) and\nperpendicular ( Vs⊥) tovsdepends only on the directionin which the skyrmion lattice drifts.\nVs/bardbl\nVs⊥≈(vs−vd)/bardbl\n(vs−vd)⊥=−v⊥\nd\nv/bardbl\nd(20)\nThe ratioVs/bardbl\nVs⊥can be obtained experimentally by mea-\nsuring the rotation angle or the angular velocity for ∇λ\nparallel and perpendicular to the current, from which\none can obtain directlyVs/bardbl\nVs⊥using Eqs. (14) and (15).\nFor small angles, arcsin x≈x, for example, one obtains\nVs/bardbl\nVs⊥directly from the ratio of the two rotation angles. A\ndifferent, but probably more precise way to determine\nthis ratio is to find experimentally the “magic angle”\nφmof gradient vs. current, where all rotations vanish,\n∇λ·Vs= 0. In this case one obtains\nVs/bardbl\nVs⊥=1\ntanφm(21)\nThis should allow for a quantitative determination of\nv⊥\nd/v/bardbl\nd. Asv/bardbl\ndcan be measured independently us-\ning emergent electric fields generated by the motion of\nskyrmions27, one can obtain the complete information\non the drift motion by combining both experiments. It\nis also instructive to compare skyrmions and vortices in\na superconductor. Vortices and skyrmions follow essen-\ntially the same equation of motions, Eq. (6). The rele-\nvant parameters (and therefore also the pinning physics)\nare, however, rather different. For vortices in conven-\ntional superconductors17,29the dissipation is very large\nDα≫ G. Therefore, vortices drift – up to small correc-\ntions – predominantly perpendicular to the current while\nfor magnetic skyrmions we expect that at least not too\nclose to the depinning transition, the motion is domi-\nnantly parallel to the current.\nIn the limit where the pinning forces can be neglected,\ni.e.,vs≫vpin, to linear order in ˜βand ˜αthe vector Vs\nis given by\nVs=−A\n4πχ/parenleftBig\nˆB×vs/parenrightBig/parenleftBigg\n(˜β−˜α)∂G\n∂λD\nG+∂D(˜β−˜α)\n∂λ/parenrightBigg\n(22)\n=−A\n4πχ/parenleftBig\nˆB×vs/parenrightBig1\nG∂\n∂λ/parenleftBig\nDG(˜β−˜α)/parenrightBig\n(23)\nHere we also neglected a possible λ-dependence of vs. In\nthis limit the rotation can be induced primarily by gra-\ndients perpendicular to vsreflecting that the motion of\nskyrmions is mainly parallel to the current, see Eq. (20)\nand Eq. (8). This is also consistent with the behavior\nshown in Fig. 5 where we used a two-times smaller gradi-\nentfortheperpendicularconfigurationandobtainednev-\nertheless an onset of the rotational motion for values of\nvsmuch smaller than in the parallel configuration. Note\nthat in a Galilean invariant system, ˜ α=˜β, no torques\ncan be expected.10\nD. Experimental situation\nOur study is directly motivated by recent neutron\nscattering experiments in the skyrmion lattice phase of\nMnSi10. In the presence of a sufficiently large current,\na rotation of the magnetic diffraction pattern by a finite\nangle was observed when simultaneously a temperature\ngradient was present (only temperature gradients paral-\nlel to the current have been studied). The rotation angle\ncould be reversed by reversing either the direction of the\ncurrent, the direction of the magnetic field or the direc-\ntion of the temperature gradient. This clearly showed\nthat rotational torques in the experiment were driven by\nthe interplay of gradients and currents as studied in this\npaper.\nIn Fig. 8a we reproduce Fig. 3 (A) of Ref. [10], which\nshows the average rotation angle (defined as the maxi-\nmum of the azimuthal distribution of the scattering in-\ntensity) as a function of current density. Above a critical\ncurrent, j > jc, the rotation sets in. The rotation angle\ninitially increases abruptly, followed by a slower increase\nfor larger current densities. When comparing these re-\nsults with our theory one has to take into account that\nthe temperaturegradientin theexperiment wasnotinde-\npendent of the strength of the applied electrical current\ndensity as it originated in the resistive heating in the\nsample. Therefore the temperature gradient was grow-\ning with j2(i.e. the heating rate due to the electric cur-\nrent). This was taken into account in Fig. 6 as discussed\nabove. For a full quantitative comparison of theory and\nexperiment, it would be desirable to have data, where\nthe applied current as well as both the strength and the\ndirection of the gradients are changed independently. As\nsuch data is presently not available, we restrict ourselves\nto a few more qualitative observations.\nIn our theory we expect a jump of the rotation angle\natjc, which depends on the domain size. This appears\nto be consistent with the steep increase of the rotation\nangle as observed experimentally at jc, especially when\ntaking into account the experimental results are subject\nto a distribution of domain sizes.\nInterestingly, the experimentally observed increase of\nthe rotation angle after its initial jump is apparently\nnotconsistent with the predictions from the extended\nLandau-Lifshitz-Gilbert equation shown in Eq. (2) ifwe\nassumeα,α′,β,β′are independent of temperature. As\nshown in Fig. 6, we can, however, describe the experi-\nmentally observed behavior if we assume a weak temper-\nature dependence of the Gilbert damping.\nAn important question concerns, whether the existing\nexperiments already include evidence of some larger do-\nmainsthatrotatecontinuously. Fig.8ashowsthatforthe\nlargest currents averagerotation angles of up to 10◦have\nbeen obtained. As this is rather close to the maximally\npossiblevalueof15◦forstaticdomains, thissuggeststhat\ncontinuously rotating domains are either already present\nin the system or may be reached by using slightly larger\ncurrents or temperature gradients.We have therefore investigated the angular distribu-\ntion of the scattering pattern using the same set of ex-\nperimental data analyzed in Ref. [10] (technical details\nof the experimental setup are reported in this paper). In\nFig. 8b weshowthe azimuthal intensity distribution with\nand without applied current. Already for zero current\na substantial broadening of the intensity distribution is\nobserved. The origin of this broadening are demagneti-\nzation effects which lead to small variations of the orien-\ntation of the local magnetic fields in the sample tracked\nclosely by the skyrmions. It has been shown11that this\neffect can be avoided in thin samples when illuminating\nonly the central part of this sample. For the existing\ndata this implies that a quantitative analysis of Pφis not\npossible. We observed that the measured experimental\ndistribution of angles extents up just to 15◦. Therefore,\nfrom the present data we can neither claim nor exclude\nthat continuously rotating domains already exist for this\nset of data but slightly larger current densities or gradi-\nents should be sufficient to create those.\nIV. CONCLUSIONS\nThemagneticskyrmionlattices,firstobservedinMnSi,\nhave by now been observed in a wide range of cubic,\nchiral materials including insulators30,31, doped semicon-\nductors32andgoodmetals9,33. Thisisexpectedfromthe-\nory: in any material with B20 symmetry, which would\nbe ferromagnetic in the absence of spin-orbit coupling,\nweak Dzyaloshinskii Moriya interaction induce skyrmion\nlattices in a small magnetic field. While in bulk they are\nonly stabilized in a small temperature window by ther-\nmal fluctuations close to the critical temperature, they\nare much more stable in thin films34,35.\nFrom the viewpoint of spintronics, such skyrmions\nare ideal model systems to investigate the coupling of\nelectric-, thermal- or spin currents to magnetic textures:\n(i) the coupling by Berry phases to the quantized wind-\ningnumber providesa universalmechanism tocreateeffi-\ncientlyMagnusforces,(ii)skyrmionlatticecanbemanip-\nulated by extremely small forces induced by ultrasmall\ncurrents10,27, (iii) the small currents imply that also new\ntypes of experiments (e.g., neutron scattering on bulk\nsamples) are possible.\nWe think that the investigation of the rotational dy-\nnamics of skyrmion domains provides a very useful\nmethod to learn in more detail which forces affect the dy-\nnamics of the magnetic texture. As we have shown, the\nrotational torques can be controlled by both the strength\nand the direction of field- or temperature gradients in\ncombination with electric currents. They react very sen-\nsitively not only on the relative strength of the various\nforces but also on how the forces depend on temperature\nand field.\nWhile some aspects of the theory, e.g. the dependence\non the strength of the gradients, can be worked out in\ndetail, many other questions remain open. An important11\nFigure 8: a) Average rotation angle ∆ φ(in units of 1◦) of the\nskyrmion lattice in MnSi measured by neutron scattering in\nthe presence of an electric current and a temperature gradie nt\nparallel to the current. The figure is taken from Ref. [10]\nwhere further details on the experimental setup can be found .\nb) Angular distribution Pφof the intensity normalized to 1\nfor currents of strength j= 0 (black diamonds) and j≈\n−2.07·106A/m2forT= 27.4K (red circles). The lines are\nGaussian fits servingas aguide to theeye. The distribution o f\nangles extentsup the maximally possible rotation angle of 1 5◦\nwhich suggests that some of the larger domains are rotating\nwith finite angular velocity for this parameter range.question is, for example, to identify the leading damping\nmechanisms and their dependence on temperature and\nfield. Also an understanding of the interplay of pinning\nphysics, damping and the motion of magnetic textures\nis required to control spin torque effects. Here future\nrotation experiments are expected to give valuable in-\nformation. Furthermore, it will be interesting to study\nthe pinning physics in detail and to learn to what ex-\ntent skyrmions and vortices in superconductors behave\ndifferently.\nOnewaytoobservetherotationoftheskyrmionlattice\nis to investigate the angular distribution of the neutron\nscattering pattern as discussed in Sec. IIIC. This does,\nhowever, only provide indirect evidence on the expected\ncontinuous rotation of the skyrmion lattice. Therefore\nit would be interesting to observe the continuous rota-\ntion more directly. For example, one can use that time-\ndependent Berry phases arising from moving skyrmions\ninduce “emergent” electrical fields which can be directly\nmeasured27in a Hall experiment. Here it would be in-\nteresting to observe higher harmonics in the signal which\nare expected to appear close to the threshold where con-\ntinuous rotations set in, see Fig. 3.\nIn future, it might also be interesting to use instead of\nelectrical current other methods, e.g. pure spin currents\northermalcurrents, tomanipulateskyrmionlattices(e.g.\nin insulators). We expect that also in such systems the\ninvestigationofrotationalmotiondrivenbygradientswill\ngive useful insight in the control of magnetism beyond\nthermal equilibrium.\nAcknowledgments\nWe gratefully acknowledgediscussionswith M. Halder,\nM. 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In contrast, small-scale domain structure is aff ected by the damping. The difference in\nsmall-scale structure arises from energy dissipation due t o the damping.\nPACS numbers: 89.75.Kd,89.75.Da,75.10.Hk\nI. INTRODUCTION\nCoarseningorphase-orderingdynamicsisobservedina\nwidevarietyofsystems. Whenasystemisquenchedfrom\na disordered phase to an ordered phase, many small do-\nmainsareformed, andtheygrowwithtime. Forexample,\nin the case of an Ising ferromagnet, up-spin and down-\nspin domains are formed, and the characteristic length\nscale increases with time. The Ising spins can be inter-\npreted as two different kinds of atoms in the case of a\nbinary alloy. At the late stage of domain growth in these\nsystems, characteristic length L(t) follows a power-law\ngrowth law,\nL(t)∼tn, (1)\nwherenis the growth exponent. The growth laws in\nscalarfieldshavebeenderivedbyseveralgroups: n= 1/2\nfornon-conservedscalarfields, and n= 1/3forconserved\nscalar fields [1–8].\nSimilar coarsening dynamics and domain growth have\nbeen observed alsoin Bose-Einstein condensates (BECs).\nThe characteristic length grows as L(t)∼t2/3in two-\ndimensional (2D) binary BECs and ferromagnetic BECs\nwith an easy-axis anisotropy [9–11]. The same growth\nexponent n= 2/3 is found in classical binary fluids in\nthe inertial hydrodynamic regime [1, 12]. It is remark-\nable that the same growth law is found in both quan-\ntum and classical systems. It should be also noted that\ndomain formation and coarsening in BECs occur even\nwithout energy dissipation. The dynamics in a ferro-\nmagnetic BEC can be described not only by the so-\ncalled Gross-Pitaevskii equation, which is a nonlinear\nSchr¨ odinger equation, but also approximately by a mod-\nified Landau-Lifshitz equation in which the interaction\nbetween superfluid flow and local magnetization is incor-\nporated[13–15]. Ifenergydissipationexists, theequation\nchanges to an extended Landau-Lifshitz-Gilbert (LLG)\nequation [9, 15, 16]. The normal LLG equation is usu-\nally used to describe spin dynamics in a ferromagnet.\nThe LLG equation includes a damping term which is\ncalled the Gilbert damping. When the system has an\neasy-axis anisotropy, the damping has the effect to directa spin to the easy-axis direction. The Gilbert damping\nin the LLG equation corresponds to energy dissipation\nin a BEC. In other words, domain formation without en-\nergy dissipation in a BEC implies that domains can be\nformed without the damping in a ferromagnet. However,\nthe LLG equation without the damping describes merely\nthe precession of magnetization with a ferromagnetic in-\nteraction.\nInthispaper, wefocusonwhateffectsthedampinghas\nondomainformationanddomaingrowth. UsingtheLLG\nequation (without flow terms), we investigate the mag-\nnetic domain growth in a 2D system with an easy-axis\nanisotropy. Since our system is simpler than a BEC, we\ncan also give simpler discussions on what causes domain\nformation. When the easy axis is perpendicular to the\nx-yplane, the system is an Ising-like ferromagnetic film,\nand domains in which the zcomponent of each spin has\nalmostthesamevalueareformed. Inordertoobservedo-\nmain formation both in damping and no-damping cases,\nwe limit the initial condition to almost uniform in-plane\nspins. Actually, without the damping, domain formation\ndoes not occur from an initial configuration of spins with\ntotally random directions. Without the damping, the z\ncomponent is conserved. The damping breaks the con-\nservation of the zcomponent as well as energy. Here,\nwe should note that the growth laws for conserved and\nnonconserved scalar fields cannot simply be applied to\nthe no-damping and damping cases, respectively, in our\nsystem. Although the zcomponent corresponds to the\norderparameterofascalarfield, oursystemhastheother\ntwo components. It is uncertain whether the difference\nin the number of degrees of freedom can be neglected in\ndomain formation.\nThe restofthe paperis organizedas follows. In Sec. II,\nwe describe the model and numerical procedures. Ener-\ngies and the characteristic length scale are also intro-\nduced in this section. Results of numerical simulations\nare shown in Sec. III. Domain patterns at different times\nand the time evolution of energies and the average do-\nmain size are demonstrated. Scaling behavior is con-\nfirmed in correlation functions and structure factors at\nlate times. In Sec. IV, we discuss why domain formation2\ncan occur even in the no-damping case, focusing on an\nalmost uniform initial condition. Finally, conclusions are\ngiven in Sec. V.\nII. MODEL AND METHOD\nThe model we use in numerical simulations is the LLG\nequation, which is widely used to describe the spin dy-\nnamics in ferromagnets. The dimensionless normalized\nform of the LLG equation is written as\n∂m\n∂t=−m×heff+αm×∂m\n∂t, (2)\nwheremis the unit vector of spin, αis the dimensionless\nGilbert damping parameter. We here consider the 2D\nsystemlyinginthe x-yplane,andassumethatthesystem\nhas a uniaxial anisotropy in the zdirection and that no\nlong-range interaction exists. Then, the dimensionless\neffective field is given by\nheff=∇2m+Canimzˆz, (3)\nwhereCaniis the anisotropy parameter, and ˆzis the unit\nvector in the zdirection.\nEquation (2) is mathematically equivalent to\n∂m\n∂t=−1\n1+α2m×heff+α\n1+α2m×(m×heff).\n(4)\nIn numerical simulations, we use a Crank-Nicolson\nmethod to solve Eq. (4). The initial condition is given as\nspins that are aligned in the xdirection with a little ran-\ndom noises: mx≃1 andmy≃mz≃0. Simulations are\nperformed in the 512 ×512 lattice with periodic bound-\nary conditions. Averages are taken over 20 independent\nruns.\nThe energy in this system is written as\nE=Eint+Eani\n=1\n2/integraldisplay\ndr(∇m(r))2−1\n2Cani/integraldisplay\ndrmz(r)2,(5)\nwhich gives the effective field as heff=−δE/δm. The\nfirst and second terms are the interfacial and anisotropy\nenergies, respectively. When Cani>0, thezcomponent\nbecomes dominant since a large m2\nzlowers the energy.\nWe take Cani= 0.2 in the simulations. The damping\nparameter αexpresses the rate of energy dissipation. If\nα= 0, the spatial average of mzas well as the energy E\nis conserved.\nConsidering mzas the order parameter of this system,\nwe here define the characteristic length scale Lof a do-\nmain pattern from the correlation function\nG(r) =1\nA/integraldisplay\nd2x/angb∇acketleftmz(x+r)mz(x)/angb∇acket∇ight,(6)\nwhereAis the area of the system and /angb∇acketleft···/angb∇acket∇ightdenotes an\nensemble average. The average domain size Lis defined\nby the distance where G(r), i.e., the azimuth average of\nG(r), first drops to zero, and thus, G(L) = 0.\nFIG. 1. (Color online) Snapshots of z-component mzat time\nt= 102((a) and (b)), 103((c) and (d)), and 104((e) and\n(f)). Snapshots (g) and (h) are enlarged parts of (e) and (f),\nrespectively. Profiles (i) and (j) of mzare taken along the\nbottom lines of snapshots (g) and (h), respectively. Left an d\nright columns are for the no-damping ( α= 0) and damping\n(α= 0.03) cases, respectively.\nIII. SIMULATIONS\nDomain patterns appear, regardless of the damping\nparameter α. The snapshots of the no-damping ( α= 0)\nand damping ( α= 0.03) cases are demonstrated in the\nleft and right columns of Fig. 1, respectively. Domain\npatterns at early times have no remarkable difference be-\ntweenthe twocases. Thecharacteristiclengthscalelooks\nalmostthesamealsoatlatertimes. However,asshownin\nthe enlarged snapshots at late times, difference appears\nespecially around domain walls. Domain walls, where\nmz≃0, are smooth in the damping case. However, in\nthe no-damping case, they look fuzzy. The difference ap-\npears more clearly in profiles of mz(Figs. 1(i) and 1(j)).\nWhile the profile in the damping case is smooth, that\nin the no-damping case is not smooth. Such an uneven\nprofile makes domain walls look fuzzy.\nThe difference in domain structure is closely connected\nwith energydissipation, which is shownin Fig. 2. The in-\nterfacial energy, which is the first term of Eq. (5), decays\nforα= 0.03 but increases for α= 0 in Fig. 2 (a). In con-3\n0 2000 4000 6000 8000 10000t00.020.040.060.080.1Eint α = 0\nα = 0.03(a)\n0 2000 4000 6000 8000 10000t-0.1-0.08-0.06-0.04-0.020Eaniα = 0\nα = 0.03(b)\nFIG. 2. (Color online) Time dependence of (a) the inter-\nfacial energy Eintand (b) the anisotropy energy Eani. The\ninterfacial energy increases with time in the no-damping ca se\n(α= 0) and decreases in the damping case ( α= 0.03). The\nanisotropy energy decreases with time in both cases.\ntrast, the anisotropy energy, which comes from the total\nofm2\nz, decreases with time for both α= 0 and α= 0.03.\nIn other words, the energy dissipation relating to the in-\nterfacial energy mainly causes the difference between the\ndamping and no-damping cases. In the damping case,\nthe interfacial energy decreases with time after a shot-\ntime increase as domain-wall structure becomes smooth.\nHowever, in the no-damping case, the interfacial energy\nincreases with time to conserve the total energy that is\ngiven by Eq. (5). This corresponds to the result that\nthe domain structure does not become smooth in the no-\ndamping case.\nBeforediscussinggrowthlaws, we shouldexaminescal-\ning laws. Scaled correlation functions of mzat different\ntimes are shown in Fig. 3. The functions look pretty\nsimilar in both damping and no-damping cases, which\nreflects the fact that the characteristic length scales in\nboth cases looks almost the same in snapshots. At late\ntimes, the correlation functions that are rescaled by the\naverage domain size L(t) collapse to a single function.\nHowever, the scaled correlation functions at early times\n(t= 100 and 1000) do not agree with the scaling func-\ntion especially in the short range. The disagreement at\nearly times is related with the unsaturation of mz. How\nmzsaturates is reflected in the time dependence of the0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(a)\n0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(b)\nFIG.3. (Color online) Scaledcorrelation functions atdiffe rent\ntimes in (a) no-damping ( α= 0) and (b) damping ( α= 0.03)\ncases. The correlation functions at late times collapse to a\nsingle function, however, the ones at early times do not.\nanisotropy energy which is shown in Fig. 2(b). At early\ntimes (t/lessorsimilar1000),Eanidecays rapidly. This implies that\nmzis not saturated enough in this time regime. The de-\ncreasein theanisotropyenergyslowsatlatetimes. In the\nlate-time regime, mzis sufficiently saturated except for\ndomain walls, and the decrease in the anisotropy energy\nis purely caused by domain growth. This corresponds to\nthe scaling behavior at late times.\nIn Fig. 4, the average domain size Lis plotted for\nthe damping and no-damping cases. In both cases,\nthe average domain size grows as L(t)∼t1/2at late\ntimes, although growth exponents at early times look\nliken= 1/3. Since scaling behavior is confirmed only\nat late times, the domain growth law is considered to be\nL(t)∼t1/2rather than t1/3in this system. In our pre-\nvious work, we saw domain growth as L(t)∼t1/3in a\nBEC without superfluid flow [9], which was essentially\nthe same system as the present one. However, the time\nregion shown in Ref. [9] corresponds to the early stage\n(t/lessorsimilar1830) in the present system.\nAlthough the growth exponent is supposed to be n=\n1/3 for conserved scalar fields, the average domain size\ngrows as L(t)∼t1/2, in our system, at late times even\nin the no-damping case. This implies that our system\nwithout damping cannot be categorized as a model of a4\n100 1000 10000t10100 Lα = 0\nα = 0.03\nt1/2\nt1/3\nFIG. 4. (Color online) Time dependence of the average do-\nmain size Lforα= 0 and 0 .03. In both damping and no-\ndamping cases, domain size grows as L(t)∼t1/2at late\ntimes. Before the scaling regime, early-time behavior look s\nas ifL(t)∼t1/3.\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3S(k)/L(t)2\nt =  6000\nt =  8000\nt = 10000\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3\nt =  6000\nt =  8000\nt = 10000(a) (b)\nk-3k-3\nFIG. 5. (Color online) Scaling plots of the structure factor\nscaled with L(t) at different times in (a) no-damping ( α= 0)\nand (b) damping ( α= 0.03) cases. In both cases, S(k)∼k−3\nin the high- kregime. However, they gave different tails in the\nultrahigh- kregime.\nconserved scalar field. Although we consider mzas the\norder parameter to define the characteristic length scale,\nthe LLG equation is described in terms of a vector field\nm.\nScaling behavior also appears in the structure factor\nS(k,t), which is given by the Fourier transformation of\nthe correlation function G(r). According to the Porod\nlaw, the structure factor has a power-law tail,\nS(k,t)∼1\nL(t)kd+1, (7)\nin the high- kregime [1]. Here, dis the dimension of\nthe system. Since d= 2 in our system, Eq. (7) leads\ntoS(k,t)/L(t)2∼[kL(t)]−3. In Fig. 5, S(k,t)/L(t)2is\nplotted as a function of kL(t). The data at different late\ntimes collapse to one curve, and they show S(k)∼k−3in the high- kregime (kL∼10) in both the damping and\nno-damping cases. In the ultrahigh- kregime (kL∼100),\ntails are different between the two cases, which reflects\nthe difference in domain structure. Since domain walls\nare fuzzy in the no-damping case, S(k) remains finite.\nHowever, in the damping case, S(k) decays faster in the\nultrahigh- kregime, which is related with smooth domain\nwalls.\nIV. DISCUSSION\nWe here have a naive question: Why does domain\npattern formation occur even in the no-damping case?\nWhenα= 0, Eq. (2) is just the equation of the pre-\ncession of spin, and the energy Eas well as mzis con-\nserved. We here discuss why similar domain patterns are\nformed from our initial condition in both damping and\nno-damping cases.\nUsing the stereographic projection of the unit sphere\nof spin onto a complex plane [17], we rewrite Eq. (4) as\n∂ω\n∂t=−i+α\n1+α2/bracketleftbigg\n∇2ω−2ω∗(∇ω)2\n1+ωω∗−Caniω(1−ωω∗)\n1+ωω∗/bracketrightbigg\n,\n(8)\nwhereωis a complex variable defined by\nω=mx+imy\n1+mz. (9)\nEquation (8) implies that the effect of the Gilbert damp-\ning is just a rescaling of time by a complex constant [17].\nThe fixed points of Eq. (8) are |ω|2= 1 and ω= 0.\nThelinearstabilityanalysisaboutthesefixedpointsgives\nsome clues about domain formation.\nAt the fixed point ω= 1,mx= 1 and my=mz= 0,\nwhich corresponds to the initial condition of the numer-\nical simulation. Substituting ω= 1 +δωinto Eq. (8),\nwe obtain linearized equations of δωandδω∗. Perform-\ning Fourier expansions δω=/summationtext\nkδ˜ωkeik·randδω∗=/summationtext\nkδ˜ω∗\n−keik·r, we have\nd\ndt/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n=/parenleftbigg\n˜α1(Cani−k2) ˜α1Cani\n˜α2Cani˜α2(Cani−k2)/parenrightbigg/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n,\n(10)\nwhere ˜α1=1\n2(−i+α)/(1+α2), ˜α2=1\n2(i+α)/(1+α2),\nk= (kx,ky), andk=|k|. The eigenvalues of the 2 ×2\nmatrix of Eq. (10) are\nλ(k) =α\n2(1+α2)(Cani−2k2)±/radicalbig\n4k2(Cani−k2)+α2C2\nani\n2(1+α2).\n(11)\nEven when α= 0,λ(k) has a positive real part for\nk <√Cani. Thus, the uniform pattern with mx= 1\nis unstable, and inhomogeneous patterns can appear.\nThe positive real parts of Eq. (11) for α= 0 and\nα= 0.03 have close values, as shown in Fig. 6. This cor-\nresponds to the result that domain formation in the early5\n0 0.1 0.2 0.3 0.4 0.5\nk00.020.040.060.080.1λ(k)α = 0\nα = 0.03\nFIG. 6. (Color online) Positive real parts of λ(k) that is given\nby Eq. (11), which has a positive real value for k <√Cani.\nThe difference between α= 0 and α= 0.03 is small.\nstage has no remarkable difference between the damping\n(α= 0.03) and no-damping ( α= 0) cases (See Fig. 1).\nFrom the view point of energy, the anisotropy energy\ndoes not necessarily keep decaying when α= 0. For con-\nservation of energy, it should be also possible that both\nanisotropy and interfacial energies change only a little.\nBecause of the instability of the initial state, mzgrows,\nand thus, the anisotropy energy decreases.\nThe initial condition, which is given as spins aligned in\nonedirection with somenoisesin the x-yplane, is the key\nto observe domain pattern formation in the no-damping\ncase. Actually, if spins have totally random directions,\nno large domains are formed in the no-damping case,\nalthough domains are formed in damping cases ( α >0)\nfrom such an initial state.\nWhenω= 0,mx=my= 0 and mz= 1, which is also\none of the fixed points. Substituting ω= 0 +δωinto\nEq. (8) and performing Fourier expansions, we have the\nlinearized equation of δ˜ωk,\nd\ndtδ˜ωk=i−α\n1+α2(k2+Cani)δ˜ωk. (12)\nThis implies that the fixed point is stable for α >0 andneutrally stable for α= 0. Although mz=−1 corre-\nsponds to ω→ ∞, the same stability is expected for\nmz=−1 by symmetry.\nSincetheinitialconditionisunstable, the z-component\nof spin grows. Moreover, linear instability is similar for\nα= 0 and α= 0.03. Since mz=±1 are not unstable,\nmzcan keep its value at around mz=±1. This is why\nsimilar domain patters are formed in both damping and\nno-damping cases. The main difference between the two\ncases is that mz=±1 are attracting for α >0 and neu-\ntrally stable for α= 0. Since mz=±1 are stable and at-\ntractingin the dampingcase, homogeneousdomainswith\nmz=±1 are preferable, which leads to a smooth profile\nofmzsuch as Fig. 1(j). In the damping case, mz=±1\nare neutrally stable (not attracting) fixed points, which\ndoes not necessarily make domains smooth.\nV. CONCLUSIONS\nWe have investigated the domain formation in 2D vec-\ntor fields with an easy-axis anisotropy, using the LLG\nequation. When the initial configuration is given as al-\nmost uniform spins aligned in an in-plane direction, sim-\nilar domain patterns appear in the damping ( α/negationslash= 0) and\nno-damping ( α= 0) cases. The average domain size\ngrows as L(t)∼t1/2in late times which are in a scal-\ning regime. The damping gives no remarkable effects\non domain growth and large-scale properties of domain\npattern. In contrast, small-scale structures are different\nbetween the two cases, which is shown quantitatively in\nthe structure factor. This difference is induced by the re-\nduction of the interfacial energy due to the damping. It\nshould be noted that the result and analysis especially\nin the no-damping case are valid for a limited initial\ncondition. Although domains grow in a damping case\neven from spins with totally random directions, domain\ngrowth cannot occur from such a random configuration\nin the no-damping case.\nACKNOWLEDGMENTS\nThis work was supported by MEXT KAKENHI\n(No. 26103514, “Fluctuation & Structure”).\n[1] A. Bray, Adv. Phys. 43, 357 (1994)\n[2] I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids\n19, 35 (1961)\n[3] C. Wagner, Z. Elektrochem 65, 581 (1961)\n[4] T. Ohta, D. Jasnow, and K. Kawasaki, Phys. Rev. Lett.\n49, 1223 (1982)\n[5] D. A. Huse, Phys. Rev. B 34, 7845 (1986)\n[6] A. J. Bray, Phys. Rev. Lett. 62, 2841 (1989)\n[7] A. J. Bray, Phys. Rev. B 41, 6724 (1990)\n[8] A. J. Bray and A. D. Rutenberg, Phys. Rev. E 49, R27\n(1994)[9] K. Kudo and Y. Kawaguchi, Phys. Rev. A 88, 013630\n(2013)\n[10] J. Hofmann, S. S. Natu, and S. Das Sarma, Phys. Rev.\nLett.113, 095702 (2014)\n[11] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116,\n025301 (2016)\n[12] H. Furukawa, Phys. Rev. A 31, 1103 (1985)\n[13] A. Lamacraft, Phys. Rev. A 77, 063622 (2008)\n[14] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85,\n1191 (2013)\n[15] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012)6\n[16] K. Kudo and Y. Kawaguchi, Phys. Rev. A 84, 043607\n(2011)\n[17] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53,\n2497 (1984)"    },    {        "title": "1708.03424v1.Gradient_expansion_formalism_for_generic_spin_torques.pdf",        "content": "Gradient expansion formalism for generic spin torques\nAtsuo Shitade\nRIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan\n(Dated: August 23, 2021)\nWe propose a new quantum-mechanical formalism to calculate spin torques based on the gradient\nexpansion, which naturally involves spacetime gradients of the magnetization and electromagnetic\n\felds. We have no assumption in the small-amplitude formalism or no di\u000eculty in the SU(2) gauge\ntransformation formalism. As a representative, we calculate the spin renormalization, Gilbert damp-\ning, spin-transfer torque, and \f-term in a three-dimensional ferromagnetic metal with nonmagnetic\nand magnetic impurities being taken into account within the self-consistent Born approximation.\nOur results serve as a \frst-principles formalism for spin torques.\nI. INTRODUCTION\nSpin torques have been investigated both theoreti-\ncally and experimentally in the \feld of magnetic spin-\ntronics since the celebrated discovery of the current-\ninduced magnetization reversal by the spin transfer\ntorque (STT)1{5. When an electric \feld is applied to\na ferromagnetic metal with magnetic structures such as\ndomain walls and skyrmions, the spin-polarized current\n\rows, and electron spin is transferred to the magnetiza-\ntion via the exchange interaction. Furthermore, the so-\ncalled\f-term arises from spin relaxation6{11. Electronic\ncontributions to spin torques in a ferromagnetic metal\nwithout spin-orbit interactions (SOIs) are expressed by\n~ \u001c=\u0000~s_~ n\u0000~\u000b~ n\u0002_~ n\u0000(~js\u0001~@)~ n\u0000\f~ n\u0002(~js\u0001~@)~ n;(1)\nin which~ nis the magnetization which is dynamical and\nnonuniform. sand\u000bare the spin renormalization and\nelectronic contribution to the Gilbert damping, respec-\ntively. The third and fourth terms are the STT and \f-\nterm driven by the spin-polarized current ~js. In the pres-\nence of SOIs, another spin torque called the spin-orbit\ntorque is allowed even without magnetic structures12{14.\nIn real materials, both magnetic structures and SOIs do\nexist, and hence a systematic formalism to calculate these\nspin torques is desired15.\nTo calculate spin torques quantitatively, a quantum-\nmechanical formalism is desirable. It is di\u000ecult to take\ninto account spin relaxation systematically in the semi-\nclassical Boltzmann theory8,10,13{15, and phenomenolog-\nical treatment may even lead to incorrect results on the\n\f-term8. The small-amplitude formalism, in which small\ntransverse \ructuations around a uniform state are as-\nsumed, is quantum-mechanical but cannot be applied\nto the \fnite-amplitude dynamics except for simple cases\nwithout SOIs9. The SU(2) gauge transformation formal-\nism, where a magnetic structure is transformed to a uni-\nform state, is also quantum-mechanical and correct5,11,12.\nHowever, we should be careful when we deal with mag-\nnetic impurities11. Magnetic impurities become dynami-\ncal and nonuniform by the SU(2) gauge transformation,\nwhich yields the additional SU(2) gauge \feld. If this con-\ntribution is not taken into account, the Gilbert damping\nvanishes.Here we propose a new quantum-mechanical formalism\nto calculate generic spin torques based on the gradient ex-\npansion. As a representative, we calculate four terms in\nEq. (1) in a three-dimensional (3d) ferromagnetic metal\nwith nonmagnetic and magnetic impurities. The gradient\nexpansion is a perturbation theory with respect to space-\ntime gradients16,17as well as electromagnetic \felds18{20\nin terms of the Wigner representations of the Keldysh\nGreen's functions. The former two terms in Eq. (1) are\nlinear responses of electron spin to a temporal gradient\nof the magnetization, and the latter two are the second-\norder responses to a spatial gradient and an electric \feld.\nAs mentioned in Ref. 10, it is a natural extension of the\nsemiclassical Boltzmann theory8,10,13{15. We do not have\nto pay any attention to the SU(2) gauge \feld even in the\npresence of magnetic impurities, SOIs, and sublattice de-\ngrees of freedom as in antiferromagnets.\nII. GRADIENT EXPANSION\nIn this Section, we review the gradient expansion of\nthe Keldysh Green's function with external gauge \felds\nbeing taken into account. We do not rely on any spe-\nci\fc form of the Hamiltonian, which may be disordered\nor interacting. Furthermore, gauge \felds may be abelian\nor nonabelian. The gradient expansion was already car-\nried out up to the in\fnite order in the absence of gauge\n\felds16,17and in the abelian case18,19and up to the \frst\norder in the nonabelian case20. Although we are inter-\nested in the abelian case, we give rigorous derivation up\nto the fourth order in the nonabelian case with the help\nof the nonabelian Stokes theorem21,22.\nA. Locally covariant Keldysh Green's function\nWhen we carry out the gradient expansion, it is es-\nsential to keep the local gauge covariance. First, let us\nexplain its meaning here. Under a gauge transformation\n 0(x) =V(x) (x) for a \feld  (x), gauge \feldsA\u0016(x), a\nlocally gauge-covariant quantity ~A(x), and the Keldysh\nGreen's function ^G(x1;x2) transform as\nA0\n\u0016(x) =V(x)A\u0016(x)Vy(x)\u0000i~[@\u0016V(x)]Vy(x);(2a)arXiv:1708.03424v1  [cond-mat.mes-hall]  11 Aug 20172\n~A0(x) =V(x)~A(x)Vy(x); (2b)\n^G0(x1;x2) =V(x1)^G(x1;x2)Vy(x2): (2c)\nThe Green's function ^G(x1;x2) with the hat symbol is\ngauge-covariant in the sense of Eq. (2c). However, in\nthe Wigner representation de\fned later in Eq. (6), the\ncenter-of-mass coordinate X12\u0011(x1+x2)=2 is the only\ncoordinate, and hence the Green's function should be de-\n\fned as locally gauge-covariant with respect to X12. It\ncan be achieved by introducing the Wilson line,\nW(x1;x2)\u0011Pexp\u0014\n\u00001\ni~Zx1\nx2dy\u0016A\u0016(y)\u0015\n; (3)\nwhich transforms in the same way as the Green's func-\ntion, i.e.,W0(x1;x2) =V(x1)W(x1;x2)Vy(x2).Pis\nthe path-ordered product. The locally gauge-covariant\nGreen's function ~G(x1;x2) with the tilde symbol is then\nde\fned by18{20\n~G(x1;x2)\u0011W(X12;x1)^G(x1;x2)W(x2;X12);(4)\nwhich transforms as\n~G0(x1;x2) =V(X12)~G(x1;x2)Vy(X12); (5)\ninstead of Eq. (2c). Similarly to Eq. (4), all the two-\npoint quantities with the hat symbol should be replaced\nby those with the tilde symbol.\nB. Gauge-covariant Wigner representation\nNext, we de\fne the Wigner representation of the lo-\ncally gauge-covariant Green's function18{20,\n~G(X12;p12)\u0011Z\ndDx12ep12\u0016x\u0016\n12=i~~G(x1;x2);(6)whereX12\u0011(x1+x2)=2 andx12\u0011x1\u0000x2are the center-\nof-mass and relative coordinates, respectively, and p12is\nthe relative momentum. Dis the spacetime dimension.\nDynamics of the Green's function is determined by the\nDyson equation involving convolution, which is a two-\npoint quantity de\fned by\n\\A\u0003B(x1;x2)\u0011Z\ndDx3^A(x1;x3)^B(x3;x2);(7)\nfor any two-point quantities ^Aand ^B. Therefore, we have\nto \fnd the Wigner representation of the locally gauge-\ncovariant convolution,\n~A(X12;p12)?~B(X12;p12)\u0011^A\u0003B(X12;p12):(8)\nSince the Wigner representation is just the Fourier trans-\nformation with respect to x12, convolution turns into the\nsimple product ~A(p12)~B(p12) for a translationally invari-\nant system in the absence of gauge \felds; otherwise, it\nbecomes noncommutative and is called the Moyal prod-\nuct. It is evaluated by expanding Eq. (8) with respect\nto the relative coordinates x13andx32as in Appendix A\nand is expressed by\n~A?~B=~A~B+ (i~=2)PD(~A;~B) + (i~=2)PF(~A;~B)\n+ (1=2!)(i~=2)2PD2(~A;~B) + (i~=2)2PD\u0003F(~A;~B) + (1=2!)(i~=2)2PF2(~A;~B); (9a)\nPD(~A;~B)\u0011DX\u0015~A@p\u0015~B\u0000@p\u0015~ADX\u0015~B; (9b)\nPF(~A;~B)\u0011(F\u0016\u0017@p\u0016~A@p\u0017~B+ 2@p\u0016~AF\u0016\u0017@p\u0017~B+@p\u0016~A@p\u0017~BF\u0016\u0017)=4; (9c)\nPD2(~A;~B)\u0011DX\u00151DX\u00152~A@p\u00151@p\u00152~B\u00002DX\u00151@p\u00152~A@p\u00151DX\u00152~B+@p\u00151@p\u00152~ADX\u00151DX\u00152~B; (9d)\nPD\u0003F(~A;~B)\u0011[F\u0016\u0017(DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)\n+ 2(DX\u0015@p\u0016~AF\u0016\u0017@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~AF\u0016\u0017DX\u0015@p\u0017~B)\n+ (DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)F\u0016\u0017]=4; (9e)\nPF2(~A;~B)\u0011(F\u00161\u00171F\u00162\u00172@p\u00161@p\u00162~A@p\u00171@p\u00172~B+ 4@p\u00161@p\u00162~AF\u00161\u00171F\u00162\u00172@p\u00171@p\u00172~B\n+@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00161\u00171F\u00162\u00172+ 4F\u00161\u00171@p\u00161@p\u00162~AF\u00162\u00172@p\u00171@p\u00172~B\n+ 4@p\u00161@p\u00162~AF\u00161\u00171@p\u00171@p\u00172~BF\u00162\u00172+ 2F\u00161\u00171@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00162\u00172)=42: (9f)3\nHere a covariant derivative and a \feld strength are de-\n\fned by\nDX\u0016~A(X;p)\u0011@X\u0016~A(X;p)\n+ [A\u0016(X);~A(X;p)]=i~; (10a)\nF\u0016\u0017(X)\u0011@X\u0016A\u0017(X)\u0000@X\u0017A\u0016(X)\n+ [A\u0016(X);A\u0017(X)]=i~: (10b)\nFor simplicity, the arguments X;p are omitted in Eq. (9)\nand below.\nAfter all, the Moyal product is regarded as a pertur-\nbation theory with respect to spacetime gradients as well\nas \feld strengths, but not to gauge \felds. Thus, the\ngauge covariance of the results is guaranteed. PDand\nPFdenote the \frst-order contributions with respect to\nspacetime gradients Dand \feld strengths F, respectively.\nPD\u0003Fis the mixed second-order contribution involving D\nandF. We also write down the second order with respect\ntoFin Eq. (9f), which may be useful for studying other\nnonlinear responses in the future. In order to derive PF2,\nwe need the fourth order with respect to x13andx32and\nobtain many other terms. All the terms up to the fourth\norder are written in Eq. (A8).\nC. Gradient expansion up to the second order\nHere we derive the gradient expansion of the Keldysh\nGreen's function. We focus on the abelian case and as-\nsume a static and uniform \feld strength. Similarly to\nEq. (9a), we expand the Green's function and self-energy\nas23,24\n~G=~G0+ (~=2)~GD+ (~=2)~GF+ (1=2!)(~=2)2~GD2\n+ (~=2)2~GD\u0003F+ (1=2!)(~=2)2~GF2; (11a)\n~\u0006 =~\u00060+ (~=2)~\u0006D+ (~=2)~\u0006F+ (1=2!)(~=2)2~\u0006D2\n+ (~=2)2~\u0006D\u0003F+ (1=2!)(~=2)2~\u0006F2: (11b)\nNote that ~G0is the unperturbed Green's function with\ndisorder or interactions being taken into account. ~GP\nand ~GP\u0003Q(P;Q =D;F) are the \frst and second orders\nwith respect to spacetime gradients or \feld strengths,\nrespectively. By substituting these into the left Dyson\nequation,\n(~L\u0000~\u0006)?~G= 1; (12)\nin which ~Lis the Lagrangian, we get ~G0= (~L\u0000 ~\u00060)\u00001\nand\n~G\u00001\n0~GP=~\u0006P~G0\u0000iPP(~G\u00001\n0;~G0); (13a)~G\u00001\n0~GP\u0003Q=~\u0006P\u0003Q~G0\u0000i2PP\u0003Q(~G\u00001\n0;~G0)\n+ [~\u0006Q~GP+iPP(~\u0006Q;~G0)\u0000iPP(~G\u00001\n0;~GQ)\n+ (P$Q)]: (13b)\nThe self-energies are determined self-consistently.\nTo calculate the expectation values, the lesser Green's\nfunction is necessary. In the real-time representation, the\nGreen's function and self-energy are of matrix forms23,24,\n~G=\u0014\nGR2G<\n0GA\u0015\n; (14a)\n~\u0006 =\u0014\n\u0006R2\u0006<\n0 \u0006A\u0015\n; (14b)\nin which R, A, and <indicate the retarded, advanced,\nand lesser components, respectively. For the \frst order,\nthe lesser one can be written as\nG<\nP=\u0006[(GR\nP\u0000GA\nP)f(\u0000p0) +G<(1)\nPf0(\u0000p0)];(15a)\n\u0006<\nP=\u0006[(\u0006R\nP\u0000\u0006A\nP)f(\u0000p0) + \u0006<(1)\nPf0(\u0000p0)];(15b)\nin which the upper and lower signs indicate boson and\nfermion, respectively, and f(\u0018) = (e\u0018=T\u00071)\u00001is the\ndistribution function at temperature T. By introducing\nPP(~A;~B)\u0011\u0011IJ\nP@I~A@J~B(I;J=X\u0016;p\u0017) with\u0011X\u0016p\u0017\nD =\n\u0000\u0011p\u0017X\u0016\nD =\u000e\u0016\n\u0017and\u0011p\u0016p\u0017\nF =F\u0016\u0017, we obtain an equivalent\nform of Eq. (13a)23,24,\n(GR\n0)\u00001GR\nP=\u0006R\nPGR\n0\u0000i\u0011IJ\nP@I(GR\n0)\u00001@JGR\n0; (16a)\n(GA\n0)\u00001GA\nP=\u0006A\nPGA\n0\u0000i\u0011IJ\nP@I(GA\n0)\u00001@JGA\n0; (16b)\n(GR\n0)\u00001G<(1)\nP=\u0006<(1)\nPGA\n0+i\u0011Ip0\nPf@I(GR\n0)\u00001(GR\n0\u0000GA\n0)\n\u0000[(GR\n0)\u00001\u0000(GA\n0)\u00001]@IGA\n0g: (16c)\nSimilarly, for the second order, we obtain the lesser\nGreen's function,\nG<\nP\u0003Q=\u0006[(GR\nP\u0003Q\u0000GA\nP\u0003Q)f(\u0000p0) +G<(1)\nP\u0003Qf0(\u0000p0)\n+G<(2)\nP\u0003Qf00(\u0000p0)]; (17a)\n\u0006<\nP\u0003Q=\u0006[(\u0006R\nP\u0003Q\u0000\u0006A\nP\u0003Q)f(\u0000p0) + \u0006<(1)\nP\u0003Qf0(\u0000p0)\n+ \u0006<(2)\nP\u0003Qf00(\u0000p0)]; (17b)\nand an equivalent form of Eq. (13b),\n(GR\n0)\u00001GR\nP\u0003Q=\u0006R\nP\u0003QGR\n0+ [\u0006R\nQGR\nP+i\u0011IJ\nP@I\u0006R\nQ@JGR\n0\u0000i\u0011IJ\nP@I(GR\n0)\u00001@JGR\nQ+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@K(GR\n0)\u00001@J@LGR\n0; (18a)4\n(GA\n0)\u00001GA\nP\u0003Q=\u0006A\nP\u0003QGA\n0+ [\u0006A\nQGA\nP+i\u0011IJ\nP@I\u0006A\nQ@JGA\n0\u0000i\u0011IJ\nP@I(GA\n0)\u00001@JGA\nQ+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@K(GA\n0)\u00001@J@LGA\n0; (18b)\n(GR\n0)\u00001G<(1)\nP\u0003Q=\u0006<(1)\nP\u0003QGA\n0+\u0010\n\u0006R\nQG<(1)\nP+ \u0006<(1)\nQGA\nP+i\u0011IJ\nP@I\u0006<(1)\nQ@JGA\n0\u0000i\u0011Ip0\nP[@I\u0006R\nQ(GR\n0\u0000GA\n0)\u0000(\u0006R\nQ\u0000\u0006A\nQ)@IGA\n0]\n\u0000i\u0011IJ\nP@I(GR\n0)\u00001@JG<(1)\nQ+i\u0011Ip0\nPf@I(GR\n0)\u00001(GR\nQ\u0000GA\nQ)\u0000[(GR\n0)\u00001\u0000(GA\n0)\u00001]@IGA\nQg\n\u0000\u0011Ip0\nP\u0011KL\nQf@I@K(GR\n0)\u00001@L(GR\n0\u0000GA\n0) +@L[(GR\n0)\u00001\u0000(GA\n0)\u00001]@I@KGA\n0g+ (P$Q)\u0011\n; (18c)\n(GR\n0)\u00001G<(2)\nP\u0003Q=\u0006<(2)\nP\u0003QGA\n0+ [i\u0011Ip0\nP\u0006<(1)\nQ@IGA\n0+i\u0011Ip0\nP@I(GR\n0)\u00001G<(1)\nQ+ (P$Q)]\n+\u0011Ip0\nP\u0011Kp0\nQf@I@K(GR\n0)\u00001(GR\n0\u0000GA\n0) + [(GR\n0)\u00001\u0000(GA\n0)\u00001]@I@KGA\n0g: (18d)\nNote that the left and right Dyson equations are equiv-\nalent as explicitly proved in Appendix B. Generally, the\nnth-order lesser Green's function with respect to a tem-\nporal gradient or an electric \feld involves the nth deriva-\ntive of the distribution function.\nD. Spin torques\nSpin torques are proportional to the spin expectation\nvalue. The spin expectation value is given by\nh~ \u001bi=\u0006i~ZdDp\n(2\u0019~)Dtr~ \u001bG<\n=\u0006i~ZdDp\n(2\u0019~)Dtr~ \u001b[G<\n0+ (~=2)G<\nD+ (~=2)G<\nF\n+ (1=2!)(~=2)2G<\nD2+ (~=2)2G<\nD\u0003F\n+ (1=2!)(~=2)2G<\nF2]: (19)\nAmong many terms in Eq. (19), G<\nDyields the spin renor-\nmalization and Gilbert damping. G<\nFyields the spin-\norbit torque in the presence of SOIs. In order to calculate\nthe STT and \f-term driven by an electric \feld, G<\nD\u0003Fis\nnecessary. Equations (15)-(19) are our central results for\ncalculating spin torques in generic systems.\nIII. APPLICATION TO A 3D\nFERROMAGNETIC METAL\nAs an example, we explicitly calculate spin torques in\na 3d ferromagnetic metal,\nH(X;~ p) =~ p2=2m\u0000\u0016\u0000J~ n(X)\u0001~ \u001b; (20)\nwith the chemical potential \u0016and [~ n(X)]2= 1. We take\ninto account nonmagnetic and magnetic impurities,\nVimp(~X) =NiX\njvi\u000e(~X\u0000~Xij)+NsX\njvs~ mj\u0001~ \u001b\u000e(~X\u0000~Xsj): (21)\nMagnetic impurities are assumed to be isotropic, namely,\nma\ni= 0;ma\nimb\nj=\u000eab=3 after average over the magnetiza-\ntion directions. The same system was studied within the\nBorn approximation in the literature9,11. Here we em-\nploy the self-consistent Born approximation, but it does\nnot change any results quantitatively.\nWe only have to calculate the momentum integrals of\nG<\n0,G<\nD,G<\nF, andG<\nD\u0003Fwith Eqs. (15)-(18) in order.\nFirst, the unperturbed Green's function is given by\n(GR\n0)\u00001(X;\u0018;~ p ) =\u0018\u0000H(X;~ p)\u0000\u0006R\n0(X;\u0018)\n=\u0018R(\u0018)\u0000x+JR(\u0018)~ n(X)\u0001~ \u001b; (22a)\nGR\n0(X;\u0018;~ p ) =\u0000R\n0(\u0018;~ p)[\u0018R(\u0018)\u0000x\n\u0000JR(\u0018)~ n(X)\u0001~ \u001b]; (22b)\n\u0000R\n0(\u0018;~ p) =f[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2g\u00001;(22c)\nwith\u0018R(\u0018)\u0011\u0018+\u0016\u0000\u0006R\n00(\u0018),JR(\u0018)\u0011J\u0000\n\u0006R\n03(\u0018), andx\u0011~ p2=2m. Note that the self-energy\nis expressed as \u0006R\n0(X;\u0018) = \u0006R\n00(\u0018) + \u0006R\n03(\u0018)~ n(X)\u0001\n~ \u001bdue to the spin rotation symmetry. For conve-\nnience, we de\fne gas the momentum integral of\nGmutiplied by 4 \u0019(~2=2m)3=2and introduce \ri\u0011\nniv2\ni(2m=~2)3=2=4\u0019;\rs\u0011nsv2\ns(2m=~2)3=2=4\u0019. The mo-\nmentum integral and self-energy are obtained by self-\nconsistently solving\ngR\n0(X;\u0018)\u00114\u0019\u0012~2\n2m\u00133=2Zd3p\n(2\u0019~)3GR\n0(X;\u0018;~ p )\n=IR\n11(\u0018)\u0000IR\n01(\u0018)~ n(X)\u0001~ \u001b; (23a)\n\u0006R\n0(X;\u0018) =\rigR\n0(X;\u0018) +\rs~ mi\u0001~ \u001bgR\n0(X;\u0018)~ mj\u0001~ \u001b\n=(\ri+\rs)gR\n00(\u0018)\n+(\ri\u0000\rs=3)gR\n03(\u0018)~ n(X)\u0001~ \u001b; (23b)\nin which we de\fne\nIR\nmn(\u0018)\u0011Z\u0003\n0pxdx\n\u0019[\u0018R(\u0018)\u0000x]m[JR(\u0018)]2n\u0000m\u00001\nf[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2gn:(24)5\nSecond, the \frst-order Green's functions are given by\nGR\nD(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006R\nD(X;\u0018)GR\n0(X;\u0018;~ p )\n+ 2JR(\u0018)[\u0000R\n0(\u0018;~ p)]2fJR(\u0018)\u0018R0(\u0018)\u0000[\u0018R(\u0018)\u0000x]JR0(\u0018)g~ n(X)\u0002_~ n(X)\u0001~ \u001b\n+ 2[JR(\u0018)]2[\u0000R\n0(\u0018;~ p)]2(pi=m)~ n(X)\u0002@Xi~ n(X)\u0001\u001b; (25a)\nG<(1)\nD(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006<(1)\nD(X;\u0018)GA\n0(X;\u0018;~ p )\n\u0000i[JR(\u0018) +JA(\u0018)]j\u0000R\n0(\u0018;~ p)j2[j\u0018R(\u0018)\u0000xj2\u0000jJR(\u0018)j2]_~ n(X)\u0001~ \u001b\n+ [JR(\u0018) +JA(\u0018)]j\u0000R\n0(\u0018;~ p)j2f[\u0018R(\u0018)\u0000x]JA(\u0018)\u0000[\u0018A(\u0018)\u0000x]JR(\u0018)g~ n(X)\u0002_~ n(X)\u0001~ \u001b\n\u0000i@X0[GR\n0(X;\u0018;~ p ) +GA\n0(X;\u0018;~ p )]; (25b)\nGR\nF(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006R\nF(X;\u0018)GR\n0(X;\u0018;~ p ); (25c)\nG<(1)\nF(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006<(1)\nF(X;\u0018)GA\n0(X;\u0018;~ p )\n+iFj0(pj=m)f2GR\n0(X;\u0018;~ p )GA\n0(X;\u0018;~ p )\u0000[GR\n0(X;\u0018;~ p )]2\u0000[GA\n0(X;\u0018;~ p )]2g: (25d)\nThe self-energies are expressed as \u0006R\nD(X;\u0018)\u0011\u0006R\nD2(\u0018)~ n(X)\u0002_~ n(X)\u0001~ \u001b;\u0006<(1)\nD(X;\u0018)\u0011\u0006<(1)\nD1(\u0018)_~ n(X)\u0001~ \u001b+\u0006<(1)\nD2(\u0018)~ n(X)\u0002\n_~ n(X)\u0001~ \u001band obtained by solving the following sets of linear equations,\ngR\nD2(\u0018) =IR\n01(\u0018)\u0006R\nD2(\u0018)=JR(\u0018) + 2[\u0018R0(\u0018)IR\n02(\u0018)\u0000JR0(\u0018)IR\n12(\u0018)]=JR(\u0018); (26a)\n\u0006R\nD2(\u0018) =(\ri\u0000\rs=3)gR\nD2(\u0018); (26b)\"\ng<(1)\nD1(\u0018)\ng<(1)\nD2(\u0018)#\n=\"\nJ<(1)\n1(\u0018)\u0000J<(1)\n2(\u0018)\nJ<(1)\n2(\u0018)J<(1)\n1(\u0018)#\"\n\u0006<(1)\nD1(\u0018)\n\u0006<(1)\nD2(\u0018)#\n\u00002i\"\n[JR(\u0018) +JA(\u0018)]J<(1)\n1(\u0018)=2\u0000[IR\n01(\u0018) + c:c:]=2\n[JR(\u0018) +JA(\u0018)]J<(1)\n2(\u0018)=2#\n; (26c)\n\"\n\u0006<(1)\nD1(\u0018)\n\u0006<(1)\nD2(\u0018)#\n=(\ri\u0000\rs=3)\"\ng<(1)\nD1(\u0018)\ng<(1)\nD2(\u0018)#\n: (26d)\nHere we de\fne\nJ<(1)\n1(\u0018)\u0011Z\u0003\n0pxdx\n\u0019j\u0018R(\u0018)\u0000xj2\u0000jJR(\u0018)j2\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2; (27a)\nJ<(1)\n2(\u0018)\u0011Z\u0003\n0pxdx\n\u0019i[\u0018R(\u0018)\u0000x]JA(\u0018)\u0000i[\u0018A(\u0018)\u0000x]JR(\u0018)\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2; (27b)\nJ<(1)\n3(\u0018)\u0011J<(1)\n2(\u0018)\u00002\n3[iJR(\u0018)\u0000iJA(\u0018)]Z\u0003\n0pxdx\n\u0019x\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2; (27c)\nJ<(1)\n4(\u0018)\u0011Z\u0003\n0pxdx\n\u00192\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2\u0000\njJR(\u0018)j2\n+2\n3x\u001aJR(\u0018)\n[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2+JA(\u0018)\n[\u0018A(\u0018)\u0000x]2\u0000JA2(\u0018)\u001b\nf[\u0018R(\u0018)\u0000x]JA(\u0018) + [\u0018A(\u0018)\u0000x]JR(\u0018)g\u0013\n:\n(27d)\nThese momentum integrals in Eqs. (24) and (27) are explicitly calculated in Appendix C. The other components\nvanish, i.e., \u0006R\nF(X;\u0018) =gR\nF(X;\u0018) = 0 and \u0006<(1)\nF(X;\u0018) =g<(1)\nF(X;\u0018) = 0.\nThird, the second-order Green's functions are given by\nGR\nD\u0003F(~X;\u0018;~ p ) =GR\n0(~X;\u0018;~ p )\u0006R\nD\u0003F(~X;\u0018)GR\n0(~X;\u0018;~ p )\n\u00002[\u0000R\n0(\u0018;~ p)]2\u0000\nfJR(\u0018)\u0018R0(\u0018)\u0000[\u0018R(\u0018)\u0000x]JR0(\u0018)g\u000eij\n\u0000JR0(\u0018)pipj=m\u0001\nFj0@Xi~ n(~X)\u0001~ \u001b=m; (28a)\nG<(1)\nD\u0003F(~X;\u0018;~ p ) =GR\n0(~X;\u0018;~ p )\u0006<(1)\nD\u0003F(~X;\u0018)GA\n0(~X;\u0018;~ p )\n\u00002j\u0000R\n0(\u0018;~ p)j2\u0000\nf[\u0018R(\u0018)\u0000x]JA(\u0018)\u0000[\u0018A(\u0018)\u0000x]JR(\u0018)g\u000eij\n\u0000[JR(\u0018)\u0000JA(\u0018)]pipj=m\u0001\nFj0@Xi~ n(~X)\u0001~ \u001b=m\n+ 4ij\u0000R\n0(\u0018;~ p)j2\u0000\njJR(\u0018)j2\u000eij+ [\u0000R\n0(\u0018;~ p)JR(\u0018) + \u0000A\n0(\u0018;~ p)JA(\u0018)]6\n\u0002f[\u0018R(\u0018)\u0000x]JA(\u0018) + [\u0018A(\u0018)\u0000x]JR(\u0018)gpipj=m\u0001\nFj0~ n(~X)\u0002@Xi~ n(~X)\u0001~ \u001b=m; (28b)\nwhere we drop the X0dependence in ~ n(X) because we are interested in the STT and \f-term only. The self-\nenergies are expressed as \u0006R\nD\u0003F(~X;\u0018)\u0011\u0006R\nD\u0003F1(\u0018)Fi0@Xi~ n(~X)\u0001~ \u001b=m; \u0006<(1)\nD\u0003F(~X;\u0018)\u0011\u0006<(1)\nD\u0003F1(\u0018)Fi0@Xi~ n(~X)\u0001~ \u001b=m +\n\u0006<(1)\nD\u0003F2(\u0018)Fi0~ n(~X)\u0002@Xi~ n(~X)\u0001~ \u001b=m and obtained by solving the following sets of linear equations,\ngR\nD\u0003F1(\u0018) =IR\n01(\u0018)\u0006R\nD\u0003F1(\u0018)=JR(\u0018)\n\u00002f[3JR(\u0018)\u0018R0(\u0018)\u00002\u0018R(\u0018)JR0(\u0018)]IR\n02(\u0018)\u0000JR(\u0018)JR0(\u0018)IR\n12(\u0018)g=3JR3(\u0018); (29a)\n\u0006R\nD\u0003F1(\u0018) =(\ri\u0000\rs=3)gR\nD\u0003F1(\u0018); (29b)\"\ng<(1)\nD\u0003F1(\u0018)\ng<(1)\nD\u0003F2(\u0018)#\n=\"\nJ<(1)\n1(\u0018)\u0000J<(1)\n2(\u0018)\nJ<(1)\n2(\u0018)J<(1)\n1(\u0018)#\"\n\u0006<(1)\nD\u0003F1(\u0018)\n\u0006<(1)\nD\u0003F2(\u0018)#\n+ 2i\"\nJ<(1)\n3(\u0018)\nJ<(1)\n4(\u0018)#\n; (29c)\n\"\n\u0006<(1)\nD\u0003F1(\u0018)\n\u0006<(1)\nD\u0003F2(\u0018)#\n=(\ri\u0000\rs=3)\"\ng<(1)\nD\u0003F1(\u0018)\ng<(1)\nD\u0003F2(\u0018)#\n: (29d)\nThe spin expectation value is given by\nh~ \u001bi(X) =\u0006i~Zd\u0018\n2\u0019~Zd3p\n(2\u0019~)3tr~ \u001bG<(X;\u0018;~ p )\n=h~ \u001bi0(X)\u0000~\n4\u00192J\u00122mJ\n~2\u00133=2\n[\u001c\u000b\u0000\u001cren~ n(X)\u0002]_~ n(X)\n\u00001\n4\u00192J\u00122mJ\n~2\u00131=2\nFi0[\u001c\f\u0000\u001cSTT~ n(~X)\u0002]@Xi~ n(~X); (30a)\nh~ \u001bi0(X)\u00111\n2\u00192\u00122m\n~2\u00133=2\n~ n(X)Z\nd\u0018f(\u0018)[\u0000=gR\n03(\u0018)]; (30b)\n\u001c\u000b\u0011J\u00001=2Z\nd\u0018[\u0000f0(\u0018)][ig<(1)\nD1(\u0018)=2]; (30c)\n\u001cren\u0011J\u00001=2Z\nd\u0018ff(\u0018)[\u0000=gR\nD2(\u0018)]\u0000[\u0000f0(\u0018)][ig<(1)\nD2(\u0018)=2]g; (30d)\n\u001c\f\u0011\u0000J1=2Z\nd\u0018ff(\u0018)[\u0000=gR\nD\u0003F1(\u0018)] + [\u0000f0(\u0018)][g<(1)\nD\u0003F1(\u0018)=2i]g; (30e)\n\u001cSTT\u0011J1=2Z\nd\u0018[\u0000f0(\u0018)][g<(1)\nD\u0003F2(\u0018)=2i]; (30f)\nand spin torques by\n~ \u001c(X) =~ n(X)\u0002Jh~ \u001bi(X)\n=\u0000~\n4\u00192\u00122mJ\n~2\u00133=2\n[\u001cren+\u001c\u000b~ n(X)\u0002]_~ n(X)\n\u00001\n4\u00192\u00122mJ\n~2\u00131=2\n\u0002Fi0[\u001cSTT+\u001c\f~ n(~X)\u0002]@Xi~ n(~X): (31)\n\u001crenand\u001c\u000bare the dimensionless spin renormalization\nand Gilbert damping, while \u001cSTT and\u001c\fare the STT\nand\f-term. To evaluate Eqs. (30c)-(30f), we carry out\nnumerical integrals by putting the energy unit J= 1, the\nmomentum cuto\u000b \u0003 = 103, and temperature T= 10\u00003\nand dividing the energy interval j\u0018j<5 into 217subinter-\nvals. In Fig. 1, we show their chemical-potential depen-dences for di\u000berent \riand\rs. We also show the previous\nresults obtained by the small-amplitude9and the SU(2)\ngauge transformation formalisms11at zero temperature,\n\u001cren=(\u00163=2\n+\u0000\u00163=2\n\u0000)=3J3=2; (32a)\n\u001c\u000b=\rs(\u00161=2\n++\u00161=2\n\u0000)2=3J3=2; (32b)\n\u001cSTT=1\n3J1=2X\n\u001b\u001b\u00163=2\n\u001b\n\ri\u00161=2\n\u001b+\rs(2\u00161=2\n\u0000\u001b+\u00161=2\n\u001b)=3\n!2J1=2(3\ri+ 5\rs)=9(\ri+\rs)2\u0011\u001cSTT1;(32c)\n\u001c\f=2\rs(\u00161=2\n++\u00161=2\n\u0000)\u001cSTT=3J; (32d)\nwith\u0016\u001b\u0011\u0016+\u001bJ. Note that \u001cSTT1in Eq. (32c) is \u001cSTT\nfor the\u0016!1 limit. Our results completely coincide\nwith these previous ones.7\n 0 1 2 3 4 5 6\n-1-0.5  0 0.5  1 1.5  2 2.5  3(a)\n3τren\nµγi=4e-3, γs=4e-3\nγi=4e-3, γs=8e-3\nγi=8e-3, γs=4e-3\nγi=8e-3, γs=8e-3\n 0 2 4 6 8 10 12\n-1-0.5  0 0.5  1 1.5  2 2.5  3(b)\n3τα/γs\nµ 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n-1-0.5  0 0.5  1 1.5  2 2.5  3(c)\nτSTT/τSTT∞\nµ 0 0.5 1 1.5 2 2.5 3 3.5\n-1-0.5  0 0.5  1 1.5  2 2.5  3(d)\n3τβ/2γsτSTT\nµ\nFIG. 1. (Color online) Chemical-potential dependences of (a) 3 \u001cren, (b) 3\u001c\u000b=\rs, (c)\u001cSTT=\u001cSTT1, and (d) 3 \u001c\f=2\rs\u001cSTTfor\ndi\u000berent\riand\rs. Points are obtained by the gradient expansion formalism and numerical calculation of Eqs. (30c)-(30f),\nwhile lines are obtained by the small-amplitude9and the SU(2) gauge transformation formalisms11.\nIV. DISCUSSION AND SUMMARY\nLet us clarify how the SU(2) gauge transformation\nformalism corresponds to the gradient expansion for-\nmalism. In the former, the Green's function is diag-\nonalized by a unitary matrix U\u0011~ u\u0001~ \u001bwith~ u\u0011\n[sin\u0012=2 cos\u001e;sin\u0012=2 sin\u001e;cos\u0012=2], which transforms the\nmagnetization ~ n\u0011[sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012] to~ zand\nyields the SU(2) gauge \feld A\u0016\u0011 \u0000iUy@X\u0016U=~ u\u0002\n@X\u0016~ u\u0001~ \u001b5,11,12. Thus, spacetime gradients of the magne-\ntization are described by this SU(2) gauge \feld. In the\nlatter,@X\u0016(GR\n0)\u00001/@X\u0016~ n\u0001~ \u001bin Eqs. (16) and (18) is\ntransformed to Uy@X\u0016(GR\n0)\u00001U/2~ z\u0002~A\u0016\u0001~ \u001band thus\nplays the same role as the SU(2) gauge \feld.\nWe emphasize that there is another source of the SU(2)\ngauge \feld in the SU(2) gauge transformation formal-\nism11. Magnetic impurities, which are quenched in the\noriginal frame, become dynamical and nonuniform in the\nadiabatic frame and yield the SU(2) gauge \feld. If this\ncontribution is not taken into account, the Gilbert damp-\ning vanishes, and the \f-term is not fully reproduced when\nmagnetic impurities are anisotropic. In the gradient ex-\npansion, we do not rely on the SU(2) gauge transforma-tion and hence do not encounter such di\u000eculty.\nIn summary, we demonstrated how the gradient ex-\npansion works for calculating spin torques quantum-\nmechanically. We derived the gradient expansion up to\nthe fourth order with respect to the relative coordinates\nwith abelian or nonabelian gauge \felds being taken into\naccount, which enables us to investigate nonlinear re-\nsponses with respect to spacetime gradients as well as\n\feld strengths. We applied this formalism to a 3d ferro-\nmagnetic metal with nonmagnetic and magnetic impuri-\nties and successfully reproduced the previous results on\nthe spin renormalization, Gilbert damping, STT, and \f-\nterm. The greatest advantage of our formalism is that we\ndo not assume any assumptions such as small transverse\n\ructuations or su\u000ber from the SU(2) gauge \feld arising\nfrom magnetic impurities or SOIs. Our central results\nEqs. (15)-(19) serve as \frst-principles formulas of spin\ntorques.\nACKNOWLEDGMENTS\nWe thank J. Fujimoto and G. Tatara for discussion\nand reading our manuscript. This work was supported\nby RIKEN Special Postdoctoral Researcher Program.\nAppendix A: Derivation of Eq. (9)and more\nIn this Appendix, we evaluate the Wigner representation of convolution Eq. (8),\n^A\u0003B(X12;p12) =Z\ndDx12Z\ndDx3ep12\u0016x\u0016\n12=i~W(X12;x1)^A(x1;x3)^B(x3;x2)W(x2;X12)\n=Z\ndDx12Z\ndDx3ZdDp13\n(2\u0019~)DZdDp32\n(2\u0019~)Dep12\u0016x\u0016\n12=i~e\u0000p13\u0016x\u0016\n13=i~e\u0000p32\u0016x\u0016\n32=i~\n\u0002W(X12;x1)W(x1;X13)~A(X13;p13)W(X13;x3)\n\u0002W(x3;X32)~B(X32;p32)W(X32;x2)W(x2;X12) (A1)\n=Z\ndDx12Z\ndDx3ZdDp13\n(2\u0019~)DZdDp32\n(2\u0019~)Dep12\u0016x\u0016\n12=i~e\u0000p13\u0016x\u0016\n13=i~e\u0000p32\u0016x\u0016\n32=i~\n\u0002[W(X12;x1)W(x1;X13)W(X13;X12)][W(X12;X13)~A(X13;p13)W(X13;X12)]8\n(a)\nX12\nX32X13x1\nx3 x2(b)\nX12\nX32X13x1\nx3 x2(c)v\n-1\n-11\n1\nu\nS2S3S1\nFIG. 2. Transformation from (a) the Wilson line corresponding to Eq. (A1) to (b) that to Eq. (A2). The plane spanned by\nx1;x2;x3is divided into three planes, S1,S2, andS3. (c) Three planes are parametrized by y=X12+ux32=2 +vx13=2.\n\u0002[W(X12;X13)W(X13;x3)W(x3;X32)W(X32;X12)][W(X12;X32)~B(X32;p32)W(X32;X12)]\n\u0002[W(X12;X32)W(X32;x2)W(x2;X12)]: (A2)\nHere we insert the identities W(X13;X12)W(X12;X13) =W(X32;X12)W(X12;X32) = 1, which corresponds to trans-\nformation of the Wilson line as shown in Fig. 2. We de\fne three planes S1,S2, andS3as in Fig. 2(b), whose origin\nisX12. To evaluate the \frst factor of the integrand in Eq. (A2), we use the nonabelian Stokes theorem21,22,\nW(X12;x1)W(x1;X13)W(X13;X12) =Sexp\u0014\n\u00001\n2i~Z\nS1dy\u0016dy\u0017W(X12;y)F\u0016\u0017(y)W(y;X 12)\u0015\n=Sexp\u00141\n8i~Z1\n0duZu\n0dv(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)W(X12;y)F\u0016\u0017(y)W(y;X 12)\u0015\n;(A3)\nin whichSis the surface-ordered product21,22, andy=X12+ux32=2+vx13=2. By using the Taylor expansion around\nX12, we obtain\nF\u0016\u0017(y) =e(ux\u0015\n32+vx\u0015\n13)@X\u0015\n12=2F\u0016\u0017\n=F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)@X\u0015\n12F\u0016\u0017=2\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)@X\u00151\n12@X\u00152\n12F\u0016\u0017=8 +O(x3); (A4a)\nW(y;X 12) =1\u00001\n2i~Z1\n0dw(ux\u0016\n32+vx\u0016\n13)[A\u0016+w(ux\u0017\n32+vx\u0017\n13)@X\u0017\n12A\u0016=2]\n+1\n4(i~)2Z1\n0dw1Zw1\n0dw2(ux\u00161\n32+vx\u00161\n13)(ux\u00162\n32+vx\u00162\n13)A\u00161A\u00162+O(x3)\n=1\u0000(ux\u0016\n32+vx\u0016\n13)A\u0016=2i~\u0000(ux\u0016\n32+vx\u0016\n13)(ux\u0017\n32+vx\u0017\n13)@X\u0017\n12A\u0016=8i~\n+ (ux\u00161\n32+vx\u00161\n13)(ux\u00162\n32+vx\u00162\n13)A\u00161A\u00162=8(i~)2+O(x3); (A4b)\nW(X12;y)F\u0016\u0017(y)W(y;X 12) =F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)(@X\u0015\n12F\u0016\u0017+ [A\u0015;F\u0016\u0017]=i~)=2\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)[@X\u00151\n12@X\u00152\n12F\u0016\u0017+ [@X\u00152\n12A\u00151;F\u0016\u0017]=i~\n+ 2[A\u00151;@X\u00152\n12F\u0016\u0017]=i~+ [A\u00151;[A\u00152;F\u0016\u0017]]=(i~)2]=8 +O(x3)\n=F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)DX\u0015\n12F\u0016\u0017=2\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)DX\u00151\n12DX\u00152\n12F\u0016\u0017=8 +O(x3)\n=e(ux\u0015\n32+vx\u0015\n13)DX\u0015\n12=2F\u0016\u0017: (A4c)\nHere and below we omit the argument X12for simplicity. Then, we express Eq. (A3) as\nW(X12;x1)W(x1;X13)W(X13;X12) =1 +1\n8i~Z1\n0duZu\n0dv(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)DX\u0015\n12F\u0016\u0017=29\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)DX\u00151\n12DX\u00152\n12F\u0016\u0017=8]\n+1\n64(i~)2Z1\n0du1Zu1\n0dv1Zu1\n0du2Zu2\n0dv2\n\u0002(x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172+O(x5)\n=1 + (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)F\u0016\u0017=16i~+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)(2x\u0015\n32+x\u0015\n13)DX\u0015\n12F\u0016\u0017=96i~\n+ (x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172=512(i~)2\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[6x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 2x\u00151\n13x\u00152\n13]\n\u0002DX\u00151\n12DX\u00152\n12F\u0016\u0017=1536i~+O(x5): (A5)\nThe other factors in Eq. (A2) are similarly obtained as\nW(X12;X13)~A(X13;p13)W(X13;X12) =ex\u0015\n32DX\u0015\n12=2~A(p13); (A6a)\nW(X12;X32)~B(X32;p32)W(X32;X12) =e\u0000x\u0015\n13DX\u0015\n12=2~B(p32); (A6b)\nW(X12;X13)W(X13;x3)W(x3;X32)W(X32;X12) =1 + (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)F\u0016\u0017=8i~\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)(x\u0015\n32\u0000x\u0015\n13)DX\u0015\n12F\u0016\u0017=32i~\n+ (x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172=128(i~)2\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[4x\u00151\n32x\u00152\n32\u00003(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 4x\u00151\n13x\u00152\n13]\n\u0002DX\u00151\n12DX\u00152\n12F\u0016\u0017=768i~+O(x5); (A6c)\nW(X12;X32)W(X32;x2)W(x2;X12) =1 + (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)F\u0016\u0017=16i~\n\u0000(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)(x\u0015\n32+ 2x\u0015\n13)DX\u0015\n12F\u0016\u0017=96i~\n+ (x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172=512(i~)2\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[2x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 6x\u00151\n13x\u00152\n13]\n\u0002DX\u00151\n12DX\u00152\n12F\u0016\u0017=1536i~+O(x5): (A6d)\nThus, the integrand in Eq. (A2) is given by\nO(1) = ~A(p13)~B(p32); (A7a)\nO(x) =[x\u0015\n32DX\u0015\n12~A(p13)~B(p32)\u0000x\u0015\n13~A(p13)DX\u0015\n12~B(p32)]=2; (A7b)\nO(x2) =(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[F\u0016\u0017~A(p13)~B(p32) + 2 ~A(p13)F\u0016\u0017~B(p32) +~A(p13)~B(p32)F\u0016\u0017]=16i~ (A7c)\n+ [x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)~B(p32)\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)DX\u00152\n12~B(p32)\n+x\u00151\n13x\u00152\n13~A(p13)DX\u00151\n12DX\u00152\n12~B(p32)]=8; (A7d)\nO(x3) =(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)fF\u0016\u0017[x\u0015\n32DX\u0015\n12~A(p13)~B(p32)\u0000x\u0015\n13~A(p13)DX\u0015\n12~B(p32)]\n+ 2[x\u0015\n32DX\u0015\n12~A(p13)F\u0016\u0017~B(p32)\u0000x\u0015\n13~A(p13)F\u0016\u0017DX\u0015\n12~B(p32)]\n+ [x\u0015\n32DX\u0015\n12~A(p13)~B(p32)\u0000x\u0015\n13~A(p13)DX\u0015\n12~B(p32)]F\u0016\u0017g=32i~ (A7e)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[(2x\u0015\n32+x\u0015\n13)DX\u0015\n12F\u0016\u0017~A(p13)~B(p32) + 3(x\u0015\n32\u0000x\u0015\n13)~A(p13)DX\u0015\n12F\u0016\u0017~B(p32)\n\u0000(x\u0015\n32+ 2x\u0015\n13)~A(p13)~B(p32)DX\u0015\n12F\u0016\u0017]=96i~ (A7f)\n+ [x\u00151\n32x\u00152\n32x\u00153\n32DX\u00151\n12DX\u00152\n12DX\u00153\n12~A(p13)~B(p32)\u00003x\u00151\n32x\u00152\n32x\u00153\n13DX\u00151\n12DX\u00152\n12~A(p13)DX\u00153\n12~B(p32)\n+ 3x\u00151\n32x\u00152\n13x\u00153\n13DX\u00151\n12~A(p13)DX\u00152\n12DX\u00153\n12~B(p32)\u0000x\u00151\n13x\u00152\n13x\u00153\n13~A(p13)DX\u00151\n12DX\u00152\n12DX\u00153\n12~B(p32)]=48; (A7g)\nO(x4) =(x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)\n\u0002[F\u00161\u00171F\u00162\u00172~A(p13)~B(p32) + 4 ~A(p13)F\u00161\u00171F\u00162\u00172~B(p32) +~A(p13)~B(p32)F\u00161\u00171F\u00162\u00172\n+ 4F\u00161\u00171~A(p13)F\u00162\u00172~B(p32) + 4 ~A(p13)F\u00161\u00171~B(p32)F\u00162\u00172+ 2F\u00161\u00171~A(p13)~B(p32)F\u00162\u00172]=512(i~)2(A7h)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)fF\u0016\u0017[x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)~B(p32)\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)DX\u00152\n12~B(p32)10\n+x\u00151\n13x\u00152\n13~A(p13)DX\u00151\n12DX\u00152\n12~B(p32)] + 2[x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)F\u0016\u0017~B(p32)\n\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)F\u0016\u0017DX\u00152\n12~B(p32) +x\u00151\n13x\u00152\n13~A(p13)F\u0016\u0017DX\u00151\n12DX\u00152\n12~B(p32)]\n+ [x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)~B(p32)\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)DX\u00152\n12~B(p32)\n+x\u00151\n13x\u00152\n13~A(p13)DX\u00151\n12DX\u00152\n12~B(p32)]F\u0016\u0017g=128i~ (A7i)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)f(2x\u00151\n32+x\u00151\n13)DX\u00151\n12F\u0016\u0017[x\u00152\n32DX\u00152\n12~A(p13)~B(p32)\u0000x\u00152\n13~A(p13)DX\u00152\n12~B(p32)]\n+ 3(x\u00151\n32\u0000x\u00151\n13)[x\u00152\n32DX\u00152\n12~A(p13)DX\u00151\n12F\u0016\u0017~B(p32)\u0000x\u00152\n13~A(p13)DX\u00151\n12F\u0016\u0017DX\u00152\n12~B(p32)]\n\u0000(x\u00151\n32+ 2x\u00151\n13)[x\u00152\n32DX\u00152\n12~A(p13)~B(p32)\u0000x\u00152\n13~A(p13)DX\u00152\n12~B(p32)]DX\u00151\n12F\u0016\u0017g=96i~ (A7j)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)f[6x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 2x\u00151\n13x\u00152\n13]DX\u00151\n12DX\u00152\n12F\u0016\u0017~A(p13)~B(p32)\n+ [8x\u00151\n32x\u00152\n32\u00006(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 8x\u00151\n13x\u00152\n13]~A(p13)DX\u00151\n12DX\u00152\n12F\u0016\u0017~B(p32)\n+ [2x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 6x\u00151\n13x\u00152\n13]~A(p13)~B(p32)DX\u00151\n12DX\u00152\n12F\u0016\u0017g=1536i~ (A7k)\n+ [x\u00151\n32x\u00152\n32x\u00153\n32x\u00154\n32DX\u00151\n12DX\u00152\n12DX\u00153\n12DX\u00154\n12~A(p13)~B(p32)\u00004x\u00151\n32x\u00152\n32x\u00153\n32x\u00154\n13DX\u00151\n12DX\u00152\n12DX\u00153\n12~A(p13)DX\u00154\n12~B(p32)\n+ 6x\u00151\n32x\u00152\n32x\u00153\n13x\u00154\n13DX\u00151\n12DX\u00152\n12~A(p13)DX\u00153\n12DX\u00154\n12~B(p32)\u00004x\u00151\n32x\u00152\n13x\u00153\n13x\u00154\n13DX\u00151\n12~A(p13)DX\u00152\n12DX\u00153\n12DX\u00154\n12~B(p32)\n+x\u00151\n13x\u00152\n13x\u00153\n13x\u00154\n13~A(p13)DX\u00151\n12DX\u00152\n12DX\u00153\n12DX\u00154\n12~B(p32)]=384: (A7l)\nNow we can carry out the integrals by putting p13=p12+q13;p32=p12+q32. Sinceq13andq32appear in the forms\nof~A(p12+q13) and ~B(p12+q32),x\u0016\n13andx\u0016\n32can be replaced with i~@p12\u0016acting on ~Aand ~B, respectively. Finally,\nwe obtain\n~A?~B=~A~B+ (i~=2)PD(~A;~B) + (i~=2)PF(~A;~B) + (1=2!)(i~=2)2PD2(~A;~B)\n+ (i~=2)2PD\u0003F(~A;~B) + (1=3)(i~=2)2PDF(~A;~B) + (1=3!)(i~=2)3PD3(~A;~B)\n+ (1=2!)(i~=2)2PF2(~A;~B) + (1=2!)(i~=2)3PD2\u0003F(~A;~B) + (1=3)(i~=2)3PD\u0003DF(~A;~B)\n+ (1=2!)(1=3)(i~=2)3PD2F(~A;~B) + (1=4!)(i~=2)4PD4(~A;~B); (A8a)\nPD(~A;~B)\u0011DX\u0015~A@p\u0015~B\u0000@p\u0015~ADX\u0015~B; (A8b)\nPF(~A;~B)\u0011(F\u0016\u0017@p\u0016~A@p\u0017~B+ 2@p\u0016~AF\u0016\u0017@p\u0017~B+@p\u0016~A@p\u0017~BF\u0016\u0017)=4; (A8c)\nPD2(~A;~B)\u0011DX\u00151DX\u00152~A@p\u00151@p\u00152~B\u00002DX\u00151@p\u00152~A@p\u00151DX\u00152~B+@p\u00151@p\u00152~ADX\u00151DX\u00152~B; (A8d)\nPD\u0003F(~A;~B)\u0011[F\u0016\u0017(DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)\n+ 2(DX\u0015@p\u0016~AF\u0016\u0017@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~AF\u0016\u0017DX\u0015@p\u0017~B)\n+ (DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)F\u0016\u0017]=4; (A8e)\nPDF(~A;~B)\u0011[DX\u0015F\u0016\u0017(2@p\u0016~A@p\u0015@p\u0017~B+@p\u0015@p\u0016~A@p\u0017~B)\n+ 3(@p\u0016~ADX\u0015F\u0016\u0017@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015F\u0016\u0017@p\u0017~B)\n\u0000(@p\u0016~A@p\u0015@p\u0017~B+ 2@p\u0015@p\u0016~A@p\u0017~B)DX\u0015F\u0016\u0017]=4; (A8f)\nPD3(~A;~B)\u0011DX\u00151DX\u00152DX\u00153~A@p\u00151@p\u00152@p\u00153~B\u00003DX\u00151DX\u00152@p\u00153~A@p\u00151@p\u00152DX\u00153~B\n+ 3DX\u00151@p\u00152@p\u00153~A@p\u00151DX\u00152DX\u00153~B\u0000@p\u00151@p\u00152@p\u00153~ADX\u00151DX\u00152DX\u00153~B; (A8g)\nPF2(~A;~B)\u0011(F\u00161\u00171F\u00162\u00172@p\u00161@p\u00162~A@p\u00171@p\u00172~B+ 4@p\u00161@p\u00162~AF\u00161\u00171F\u00162\u00172@p\u00171@p\u00172~B\n+@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00161\u00171F\u00162\u00172+ 4F\u00161\u00171@p\u00161@p\u00162~AF\u00162\u00172@p\u00171@p\u00172~B\n+ 4@p\u00161@p\u00162~AF\u00161\u00171@p\u00171@p\u00172~BF\u00162\u00172+ 2F\u00161\u00171@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00162\u00172)=42; (A8h)\nPD2\u0003F(~A;~B)\u0011[F\u0016\u0017(DX\u00151DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u00002DX\u00151@p\u00152@p\u0016~A@p\u00151DX\u00152@p\u0017~B\n+@p\u00151@p\u00152@p\u0016~ADX\u00151DX\u00152@p\u0017~B) + 2(DX\u00151DX\u00152@p\u0016~A@p\u00151F\u0016\u0017@p\u00152@p\u0017~B\n\u00002DX\u00151@p\u00152@p\u0016~AF\u0016\u0017@p\u00151DX\u00152@p\u0017~B+@p\u00151@p\u00152@p\u0016~AF\u0016\u0017DX\u00151DX\u00152@p\u0017~B)\n+ (DX\u00151DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u00002DX\u00151@p\u00152@p\u0016~A@p\u00151DX\u00152@p\u0017~B11\n+@p\u00151@p\u00152@p\u0016~ADX\u00151DX\u00152@p\u0017~B)F\u0016\u0017]=4; (A8i)\nPD\u0003DF(~A;~B)\u0011fDX\u00151F\u0016\u0017[2(DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u0000@p\u00152@p\u0016~ADX\u00152@p\u00151@p\u0017~B)\n+ (DX\u00152@p\u00151@p\u0016~A@p\u00152@p\u0017~B\u0000@p\u00151@p\u00152@p\u0016~ADX\u00152@p\u0017~B)]\n+ 3[(DX\u00152@p\u0016~ADX\u00151F\u0016\u0017@p\u00151@p\u00152@p\u0017~B\u0000@p\u00152@p\u0016~ADX\u00151F\u0016\u0017DX\u00152@p\u00151@p\u0017~B)\n\u0000(DX\u00152@p\u00151@p\u0016~ADX\u00151F\u0016\u0017@p\u00152@p\u0017~B\u0000@p\u00151@p\u00152@p\u0016~ADX\u00151F\u0016\u0017DX\u00152@p\u0017~B)]\n\u0000[(DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u0000@p\u00152@p\u0016~ADX\u00152@p\u00151@p\u0017~B)\n+ 2(DX\u00152@p\u00151@p\u0016~A@p\u00152@p\u0017~B\u0000@p\u00151@p\u00152@p\u0016~ADX\u00152@p\u0017~B)]DX\u00151F\u0016\u0017g=4; (A8j)\nPD2F(~A;~B)\u0011fDX\u00151DX\u00152F\u0016\u0017[6@p\u0016~A@p\u00151@p\u00152@p\u0017~B+ 3(@p\u00152@p\u0016~A@p\u00151@p\u0017~B+@p\u00151@p\u0016~A@p\u00152@p\u0017~B)\n+ 2@p\u00151@p\u00152@p\u0016~A@p\u0017~B] + [8@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u00151@p\u00152@p\u0017~B\u00006(@p\u00152@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u00151@p\u0017~B\n+@p\u00151@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u00152@p\u0017~B) + 8@p\u00151@p\u00152@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u0017~B] + [2@p\u0016~A@p\u00151@p\u00152@p\u0017~B\n+ 3(@p\u00152@p\u0016~A@p\u00151@p\u0017~B+@p\u00151@p\u0016~A@p\u00152@p\u0017~B) + 6@p\u00151@p\u00152@p\u0016~A@p\u0017~B]DX\u00151DX\u00152F\u0016\u0017g=16; (A8k)\nPD4(~A;~B)\u0011DX\u00151DX\u00152DX\u00153DX\u00154~A@p\u00151@p\u00152@p\u00153@p\u00154~B\u00004DX\u00151DX\u00152DX\u00153@p\u00154~A@p\u00151@p\u00152@p\u00154DX\u00154~B\n+ 6DX\u00151DX\u00152@p\u00153@p\u00154~A@p\u00151@p\u00152DX\u00153DX\u00154~B\u00004DX\u00151@p\u00152@p\u00153@p\u00154~A@p\u00151DX\u00152DX\u00153DX\u00154~B\n+@p\u00151@p\u00152@p\u00153@p\u00154~ADX\u00151DX\u00152DX\u00153DX\u00154~B: (A8l)\nThe arguments X;p are omitted. Equations (A8f), (A8g), and (A8i)-(A8l) are not written in Eq. (9).\nAppendix B: Equivalence of the left and right Dyson equations\nEquations (13), (16), and (18) are derived from the left Dyson equation. From the right Dyson equation,\n~G?(~L\u0000~\u0006) = 1; (B1)\nwe also obtain\n~GP~G\u00001\n0=~G0~\u0006P\u0000iPP(~G0;~G\u00001\n0); (B2a)\n~GP\u0003Q~G\u00001\n0=~G0~\u0006P\u0003Q\u0000i2PP\u0003Q(~G0;~G\u00001\n0) + [ ~GP~\u0006Q+iPP(~G0;~\u0006Q)\u0000iPP(~GQ;~G\u00001\n0) + (P$Q)]; (B2b)\nGR\nP(GR\n0)\u00001=GR\n0\u0006R\nP\u0000i\u0011IJ\nP@IGR\n0@J(GR\n0)\u00001; (B3a)\nGA\nP(GA\n0)\u00001=GA\n0\u0006A\nP\u0000i\u0011IJ\nP@IGA\n0@J(GA\n0)\u00001; (B3b)\nG<(1)\nP(GA\n0)\u00001=GR\n0\u0006<(1)\nP+i\u0011Ip0\nPf@IGR\n0[(GR\n0)\u00001\u0000(GA\n0)\u00001]\u0000(GR\n0\u0000GA\n0)@I(GA\n0)\u00001g; (B3c)\nand\nGR\nP\u0003Q(GR\n0)\u00001=GR\n0\u0006R\nP\u0003Q+ [GR\nP\u0006R\nQ+i\u0011IJ\nP@IGR\n0@J\u0006R\nQ\u0000i\u0011IJ\nP@IGR\nQ@J(GR\n0)\u00001+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@KGR\n0@J@L(GR\n0)\u00001; (B4a)\nGA\nP\u0003Q(GA\n0)\u00001=GA\n0\u0006A\nP\u0003Q+ [GA\nP\u0006A\nQ+i\u0011IJ\nP@IGA\n0@J\u0006A\nQ\u0000i\u0011IJ\nP@IGA\nQ@J(GA\n0)\u00001+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@KGA\n0@J@L(GA\n0)\u00001; (B4b)\nG<(1)\nP\u0003Q(GA\n0)\u00001=GR\n0\u0006<(1)\nP\u0003Q+\u0010\nG<(1)\nP\u0006A\nQ+GR\nP\u0006<(1)\nQ+i\u0011IJ\nP@IGR\n0@J\u0006<(1)\nQ\u0000i\u0011Ip0\nP[@IGR\n0(\u0006R\nQ\u0000\u0006A\nQ)\u0000(GR\n0\u0000GA\n0)@I\u0006A\nQ]\n\u0000i\u0011IJ\nP@IG<(1)\nQ@J(GA\n0)\u00001+i\u0011Ip0\nPf@IGR\nQ[(GR\n0)\u00001\u0000(GA\n0)\u00001]\u0000(GR\nQ\u0000GA\nQ)@I(GA\n0)\u00001g\n\u0000\u0011Ip0\nP\u0011KL\nQf@I@KGR\n0@L[(GR\n0)\u00001\u0000(GA\n0)\u00001] +@L(GR\n0\u0000GA\n0)@I@K(GA\n0)\u00001g+ (P$Q)\u0011\n; (B4c)\nG<(2)\nP\u0003Q(GA\n0)\u00001=GR\n0\u0006<(2)\nP\u0003Q+ [\u0000i\u0011Ip0\nP@IGR\n0\u0006<(1)\nQ\u0000i\u0011Ip0\nPG<(1)\nQ@I(GA\n0)\u00001+ (P$Q)]\n+\u0011Ip0\nP\u0011Kp0\nQf@I@KGR\n0[(GR\n0)\u00001\u0000(GA\n0)\u00001] + (GR\n0\u0000GA\n0)@I@K(GA\n0)\u00001g: (B4d)12\nThus, the left and right Dyson equations seem di\u000berent from each other but in fact are equivalent. Both equations\nlead to\nGR\nP=GR\n0\u0006R\nPGR\n0+i\u0011IJ\nPGR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0; (B5a)\nGA\nP=GA\n0\u0006A\nPGA\n0+i\u0011IJ\nPGA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0; (B5b)\nG<(1)\nP=GR\n0\u0006<(1)\nPGA\n0\u0000i\u0011Ip0\nPfGR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0+@I(GR\n0+GA\n0)g: (B5c)\nand\nGR\nP\u0003Q=GR\n0\u0006R\nP\u0003QGR\n0+fGR\n0\u0006R\nQGR\n0\u0006R\nPGR\n0+i\u0011IJ\nPGR\n0[\u0000@I\u0006R\nQGR\n0@J(GR\n0)\u00001\u0000@I(GR\n0)\u00001GR\n0@J\u0006R\nQ\n+ \u0006R\nQGR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001+@I(GR\n0)\u00001GR\n0\u0006R\nQGR\n0@JGR\n0+GR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0\u0006R\nQ]GR\n0\n+ (P$Q)g+\u0011IJ\nP\u0011KL\nQGR\n0f\u0000@I@K(GR\n0)\u00001GR\n0@J@L(GR\n0)\u00001\n\u0000@J(GR\n0)\u00001GR\n0@I@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001\u0000@L(GR\n0)\u00001GR\n0@I@K(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\n+@I@K(GR\n0)\u00001GR\n0[@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001+@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001]\n+ [@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001+@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001]GR\n0@I@K(GR\n0)\u00001\n\u0000@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001\u0000@I(GR\n0)\u00001GR\n0@K(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001\n\u0000@I(GR\n0)\u00001GR\n0@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\u0000@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\n\u0000@K(GR\n0)\u00001GR\n0@I(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\u0000@K(GR\n0)\u00001GR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001g\n\u0002GR\n0; (B6a)\nGA\nP\u0003Q=GA\n0\u0006A\nP\u0003QGA\n0+fGA\n0\u0006A\nQGA\n0\u0006A\nPGA\n0+i\u0011IJ\nPGA\n0[\u0000@I\u0006A\nQGA\n0@J(GA\n0)\u00001\u0000@I(GA\n0)\u00001GA\n0@J\u0006A\nQ\n+ \u0006A\nQGA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001+@I(GA\n0)\u00001GA\n0\u0006A\nQGA\n0@JGA\n0+GA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0\u0006A\nQ]GA\n0\n+ (P$Q)g+\u0011IJ\nP\u0011KL\nQGA\n0f\u0000@I@K(GA\n0)\u00001GA\n0@J@L(GA\n0)\u00001\n\u0000@J(GA\n0)\u00001GA\n0@I@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\u0000@L(GA\n0)\u00001GA\n0@I@K(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\n+@I@K(GA\n0)\u00001GA\n0[@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001+@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001]\n+ [@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001+@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001]GA\n0@I@K(GA\n0)\u00001\n\u0000@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\u0000@I(GA\n0)\u00001GA\n0@K(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\n\u0000@I(GA\n0)\u00001GA\n0@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\u0000@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001GA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\n\u0000@K(GA\n0)\u00001GA\n0@I(GA\n0)\u00001GA\n0@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\u0000@K(GA\n0)\u00001GA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001g\n\u0002GA\n0; (B6b)\nG<(1)\nP\u0003Q=GR\n0\u0006<(1)\nP\u0003QGA\n0+\u0010\nGR\n0(\u0006R\nQGR\n0\u0006<(1)\nP+ \u0006<(1)\nQGA\n0\u0006A\nP)GA\n0+i\u0011IJ\nPGR\n0[\u0000@I\u0006<(1)\nQGA\n0@J(GA\n0)\u00001\u0000@I(GR\n0)\u00001GR\n0@J\u0006<(1)\nQ\n+ \u0006<(1)\nQGA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001+@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0\u0006<(1)\nQ+@I(GR\n0)\u00001GR\n0\u0006<(1)\nQGA\n0@J(GA\n0)\u00001]GA\n0\n+i\u0011Ip0\nPGR\n0f@I(\u0006R\nQ+ \u0006A\nQ)\u0000\u0006R\nQGR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]\u0000@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0\u0006A\nQgGA\n0\n+\u0011Ip0\nP\u0011KL\nQGR\n0f\u0000@I@K[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@L(GA\n0)\u00001\u0000@K(GR\n0)\u00001GR\n0@I@L[(GR\n0)\u00001+ (GA\n0)\u00001]\n+@K(GR\n0)\u00001GR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@L(GA\n0)\u00001+@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\n+@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]gGA\n0\u0000i\u0011Ip0\nP@I(GR\nQ+GA\nQ) + (P$Q)\u0011\n; (B6c)\nG<(2)\nP\u0003Q=GR\n0\u0006<(2)\nP\u0003QGA\n0+fi\u0011Ip0\nPGR\n0[@I(GR\n0)\u00001GR\n0\u0006<(1)\nQ\u0000\u0006<(1)\nQGA\n0@I(GA\n0)\u00001]GA\n0+ (P$Q)g\n+\u0011Ip0\nP\u0011Kp0\nQ\u0000\nGR\n0f\u0000@I@K[(GR\n0)\u00001\u0000(GA\n0)\u00001]\u0000@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@K(GA\n0)\u00001\n\u0000@K[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@I(GA\n0)\u00001+@I(GR\n0)\u00001GR\n0@K[(GR\n0)\u00001+ (GA\n0)\u00001]\n+@K(GR\n0)\u00001GR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]gGA\n0+@I@K(GR\n0\u0000GA\n0)\u0001\n: (B6d)13\nAppendix C: Momentum integrals in Eqs. (24)and (27)\nIn this Appendix, we give the explicit forms of the momentum integrals de\fned above. Here we introduce\nD0(E)\u0011\u0000Z\u0003\n0pxdx\n\u00191\nE\u0000x= 2[\u00031=2\u0000E1=2tanh\u00001(\u0003=E)1=2]=\u0019; (C1a)\nD1(E) =Z\u0003\n0pxdx\n\u00191\n(E\u0000x)2= [\u00031=2=(\u0003\u0000E)\u0000E\u00001=2tanh\u00001(\u0003=E)1=2]=\u0019; (C1b)\nandER\n\u0006(\u0018)\u0011\u0018R(\u0018)\u0006JR(\u0018). Equation (24) is given by\nIR\n01(\u0018) =[D0(ER\n+(\u0018))\u0000D0(ER\n\u0000(\u0018))]=2; (C2a)\nIR\n11(\u0018) =\u0000[D0(ER\n+(\u0018)) +D0(ER\n\u0000(\u0018))]=2; (C2b)\nIR\n02(\u0018) =\u0000[D0(ER\n+(\u0018))\u0000D0(ER\n\u0000(\u0018))\u0000JR(\u0018)D1(ER\n+(\u0018))\u0000JR(\u0018)D1(ER\n\u0000(\u0018))]=4; (C2c)\nIR\n12(\u0018) =\u0000[JR(\u0018)D1(ER\n+(\u0018))\u0000JR(\u0018)D1(ER\n\u0000(\u0018))]=4; (C2d)\nIR\n22(\u0018) =[D0(ER\n+(\u0018))\u0000D0(ER\n\u0000(\u0018)) +JR(\u0018)D1(ER\n+(\u0018)) +JR(\u0018)D1(ER\n\u0000(\u0018))]=4; (C2e)\nand Eq. (27) by\nJ<(1)\n1(\u0018) =1\n2\u0014D0(ER\n+(\u0018))\nER\n+(\u0018)\u0000EA\n\u0000(\u0018)+D0(ER\n\u0000(\u0018))\nER\n\u0000(\u0018)\u0000EA\n+(\u0018)\u0015\n+ c:c:; (C3a)\nJ<(1)\n2(\u0018) =i\n2\u0014D0(ER\n+(\u0018))\nER\n+(\u0018)\u0000EA\n\u0000(\u0018)\u0000D0(ER\n\u0000(\u0018))\nER\n\u0000(\u0018)\u0000EA\n+(\u0018)\u0015\n+ c:c:; (C3b)\nJ<(1)\n3(\u0018) =J<(1)\n2(\u0018)\u00001\n3[iJR(\u0018)\u0000iJA(\u0018)]\n\u0002\u001aER\n+(\u0018)D0(ER\n+(\u0018))\nJR(\u0018)[ER\n+(\u0018)\u0000EA\n+(\u0018)][ER\n+(\u0018)\u0000EA\n\u0000(\u0018)]\u0000ER\n\u0000(\u0018)D0(ER\n\u0000(\u0018))\nJR(\u0018)[ER\n\u0000(\u0018)\u0000EA\n\u0000(\u0018)][ER\n\u0000(\u0018)\u0000EA\n+(\u0018)]+ c:c:\u001b\n;(C3c)\nJ<(1)\n4(\u0018) =1\n3\u001a[ER\n+(\u0018)]2\u0000j\u0018R(\u0018)j2+ 2jJR(\u0018)j2\nJR(\u0018)[ER\n+(\u0018)\u0000EA\n+(\u0018)][ER\n+(\u0018)\u0000EA\n\u0000(\u0018)]D0(ER\n+(\u0018))\n\u0000[ER\n\u0000(\u0018)]2\u0000j\u0018R(\u0018)j2+ 2jJR(\u0018)j2\nJR(\u0018)[ER\n\u0000(\u0018)\u0000EA\n\u0000(\u0018)][ER\n\u0000(\u0018)\u0000EA\n+(\u0018)]D0(ER\n\u0000(\u0018))\n\u0000ER\n+(\u0018)\nER\n+(\u0018)\u0000EA\n+(\u0018)D1(ER\n+(\u0018))\u0000ER\n\u0000(\u0018)\nER\n\u0000(\u0018)\u0000EA\n\u0000(\u0018)D1(ER\n\u0000(\u0018))\u001b\n+ c:c: (C3d)\n1L. 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Bortolotti1, R. Lebourgois4, J-C. Mage1, G. de Loubens3, O. \nKlein3, V. Cros1 and A. Fert1 \n1Unité Mixte de Physique CNRS/Thales and Université Paris -Sud, 1 avenue Augustin Fresnel, \nPalaiseau, France  \n2Université de Br etagne Occidentale, LMB -CNRS, Brest, France  \n3Service de Physique de l’Etat Condensé, C EA/CNRS, Gif-sur-Yvette, France  \n4Thales Research and Technology, 1 avenue Augustin Fresnel, Palaiseau, France  \n \nKey words:  YIG, FMR, Inverse Spin Hall Effect, spin waves  \n \nAbstract:  \nHigh  quality nanometer -thick (20 nm , 7 nm and 4 nm) epitaxial YIG films have been grown on GGG \nsubstrates using pulsed laser deposition. The Gilbert damping coefficient for the 20 nm thick films is \n2.3 x 10-4 which is the lowest value reported for sub -micrometric thick films. We demonstrate Inverse \nspin Hall effect (ISHE) detection of propagating spin waves using Pt. The amplitu de and the lineshape \nof the ISHE voltage correlate  well to the increase of the Gilbert damping when decreasing thickness of \nYIG. Spin Hall effect based loss -compensation experiments have been conducted but no change in the \nmagnetization dynamics could be d etected.  \n \n \n \n* Contact author :  \nAbdel madjid Anane  \nabdelmadjid.anane@thalesgroup.com  \n  2  \nAmong all magnetic materials, Yttrium Iron Garnet Y 3Fe5O12 (YIG) has been the one that had the most \nprominent role in understanding high frequency magnetization dynamics. Because of its unique \nproperties, bulk YIG crystal was the prototypal material for ferromagnetic resonance (FMR) studies in \nthe mid -twentieth ce ntury. The attractive properties of YIG include: high Curie temperature , ultra low \ndamping (the lowest among all materials at room temperature), electrical insulation, high chemical \nstability and easy synthesis in single crystalline form. Micrometer thick films of YIG were first grown \nusing liquid phase epitaxy (LPE)1, and paved the way for the emergence of a large variety of \nmicrowave devices for high -end analogue electronic applic ations throughout the 1970’s  2. \nMore recently, the interest in emerging large -scale integrated circuit technologies for beyond CMOS \napplications has fostered  new paradigms for data processing . Many of them are based on state \nvariables other than the electron charge and may eventually allow for unforeseen functionalities. \nCoding the information in a spin wave (SW) is among the most promising routes under investigation  \nand has been referred to as magnonics  3. Exciting and detecting spin waves has been ma inly achieved  \nthrough inductive coupling with radiofrequenc y (rf) anten nas but this technology remains \nincompatible with large scale integration4. Disruptive  solution s merging magnonics and spintronics \nhave been recently proposed  where spin transfer torque  (STT)  and magnetoresistive effects  would be \nused to couple to the SW s. For instance, using a STT -nano -oscillator  in a nanocontact geometry,  \ncoherent SWs emission in Ni81Fe19 (Py) thin metallic layer has been recently demonstrated and probed \nby micro -focused  Brilloui n light scattering (BLS)  5.  \nYIG is often considered as the best medium for SW propagation because of its very small  Gilbert \ndamping coefficient (2x10-5 for bulk YIG) . Being an electrical insulator, electron mediated angular \nmomentum transfer can only  occur at the  interface between YIG and a metallic layer . In that context , \nmetals with  large Spin Orbit Coupling (SOC)  like Pt  where  a pure spin current can be generated \nthrough Spin Hall Effect (SHE) 6 have been used to excite7 or amplify8,9  propagating SWs  through \nloss compensation  in YIG . Moreover, d etection of SW  can be achie ved using the  Inverse Spin Hall \nEffect (ISHE) . In ISHE , the flow of a pure spin current from the YIG into the large SOC metal \ngenerates a dc voltage . The ISHE voltage is proportional to the Spin Hall angle () and the effective \nspin mixing conductance  ( ) that is in play in the physics of spin pumping . As for the SOC \nmaterials, u p to recently, mainly Pt has been used , however it can be observed that  other 5 d heavy \nmetals such as Ta 10, W 11 or CuBi 12 are also very promising . \nAs the  amount of angular momentum transferred from (to) the YIG magnetic film per unit volume \nscales with 1/  (where t is the YIG thickness) , it is necessary to reduce the YIG thickness as much as \npossible while keeping its magnetic properties . Indeed , the threshold current density in the SOC metal \nfor the macrospin  mode excitation is expressed as13 :  (Eq. 1) where  is the spin \nHall angle, the electron charge,  the gyromagnetic ratio ,  the FMR frequency and  the YIG’s 3 saturation magnetization . Furthermore, a better understanding of the physics involved in spin \nmomentum transfer at the YIG/metal interface would be achieved by reducing the YIG film thickness \nbelow the exchange length (~ 10 nm)  14 . Up to now, sub micrometer -thick YIG films have been \nmainly grown by  LPE but the ultimate thickn ess are around  200 nm15. To further reduce the \nthickness, other growth  methods are to be considered . Pulsed laser deposition (PLD) is the most \nversatile technique for oxide films epitax y. Several  groups have worked on PLD grown YIG16,17,18,19 \nbut it is only recently that the films quality is approaching that of LPE20,21. \nIn this letter , we present PLD growth of ultrathin YIG films with various thickness (20 nm, 7 nm and 4 \nnm) on Gadolinium Gallium  Garnet  (GGG)  (111) substrates . Structural and magnetic  characterizations  \nand FMR  measurements  demonstrate the high quality of our nanometer -thick YIG films  comparable to \nstate of the art LPE films . The growth has been performed using a frequency tripled (\n = 355 nm) \nNd:YAG laser  and a stochiometric polycrystalline YIG pellet . The pulse rate was 2.5  Hz and the \nsubstrate -target distance was 4 4 mm. Prior to the YIG deposition, the GGG substrate  is annealed  at \n700°C under an oxygen pressure of 0.4 mbar. Growth temperature is then set to be 650 °C, and oxygen \npressure to 0.25 mbar. After the film deposition , samples are cooled down to room temperat ure under \n300 mbar of O 2. The YIG thickness is measured for each sample  using X -ray reflectometry which \nyield a precision better than 0.3 nm. The surface morphology and roughness have been studied by \natomic force microscopy ( AFM ). RMS roughness has been measured over 1 µm2  ranges between 0.2 \nnm and 0.3 nm  for all films  (Fig 1a) . As often with PLD growth, d roplets  are present  on the film \nsurface, here the ir lateral sizes  are below 100 nm and their density is very low (~ 0.1 µm-2). X-Ray \nDiffraction  (XRD)  spectr a using Cu K\n 1 radiation  show that the growth is along the (111) direction . \nOnly peaks characteristic of YIG and the GGG substrate are observed ( see Fig 1b ). The YIG lattice \nparameter is very close to that of the substrate and can only be resolved ev entually at large diffraction \nangle s. For the 20 nm YIG  film (Fig 1b ,1a), refinement using EVA software on the 888 reflection \nyields a cubic lattice parameter of 1.2459 nm, to be compared to 1.2376  nm for the bulk YIG22. For \nthinner films (between 4 and 15 nm) , it was not possible to distinguish the YIG peaks from those of \nthe substrate  (Fig 1d, 1e). This  sharp variation of the XRD spectra  with respect to the film thickness  \ntends to point  towards a critical thickness  for strain relaxation. It is however worth noting that the \ncubic lattice parameter of the 20 nm thick film is larger than the bulk lattice parameter  but also of that \nto the GGG substrate (1.2383 nm) . A slight o ff-stoichiometry (either oxygen vaca ncies or cation \ninterstitials)  is probably at the origin of this observation . Pole figure  measurements  have been \nperformed to gain insight s into the in -plane crystal structure, but it was not possible to resolve , at this \nstage,  the film s peaks from the substrate peaks  from w hich we infer that  the growth is epitaxial and the \nfilm single crystalline . \nFrom SQUID magnetometry with in -plane  magnetic field , we measure  a magnetization of 4\nMs = \n2100 G \n 50 G at room temperature  for both the 20 and 7 nm films . This value is independently  4 confirmed by out -of plan e FMR resonance  while the tabulated bulk value  for YIG  is 4\nMs = 1760 G. \nA similar increase of the PLD grown YIG magnetization have been reported and attributed to an  off-\nstoichiome try19. The coercitive  fields are extremely small , about 0.2 Oe  (which is the experimental \nresolution ) and the saturation field is 5 Oe. There is no evidence for in -plane magnetic anisotropy . The \noverall magnetic signature is that of an ultra -soft material . Note that for the thinnest films (4 nm) we \nmeasure a decrease of the saturation  magnetization  to roughly 1700 G . Finally, we emphasize that the \nstructural and magnetic properties of the samples are well reproducible with respect to the elaboration \nconditions.  \nFMR fields and linewidths were measured at frequencies in the range 1 -40 GHz using high sensitive \nwideband r esonance spectrometer with a nonresonant microstrip transmission line.  The FMR is \nmeasured via the derivative of microwave power absorption using a small rf exciting field. Resonance \nspectra were recorded with the applied static magnetic field  oriented in plane . During the magnetic \nfield sweeps , the amplitude of the modulation field was appreciably smaller than the FMR linewidth. \nThe amplitude of the excit ation field h rf is about 1 mOe , which corresponds to the linear response \nregime. A phase -sensitive detector with lock -in detection was used. The field derivative of the \nabsorbed power  is proportional to the field derivative of the imaginary part of the rf susceptibility: \n (where  and \n″ is the imaginary part of the susceptibility of the uniform \nmode) . Typical resonance curve s are  plotted in Fig. 2a 2b . In Fig. 2c, we  show the frequency \ndependence of the peak -to-peak linewidth  for three different  YIG thicknesses , i.e., 20, 7 and 4 nm . As \nfor the 20 nm  YIG film, we find a linear dependence of the FMR linewidth with rf frequency  while for \nthe thinnest films , we do find an almost linear increase in the low frequency range (< 12 GHz) and \nthen a saturation  of the linewidth with frequency . Such qualitative difference depending on the \nthickness is reminiscent of the qualitative difference discussed earlier in the X -ray diffraction data  (Fig \n1). The linear dependence  of the resonance linewidth  is expected with in the frame of the Landau -\nLifshitz Gilbert equation and allow for a straightforward  calculation  of the intrinsic Gilbert damping \ncoefficient ( for the 20 nm thick film). The zero frequency intercept of the fitting line , \nusually referred to a s the extrinsic lin ewidth  23,24, is found to be  \nH0  =1.4 Oe. We emphasize that our \nvalue for the  intrinsic damping on the 20 nm thick film is among the best ever reported  independ ently \nof the growth technique  and is only outperformed by the 1.3 µm film used by Y. Kajiwara  et al. 7. As \nfor the extrinsic damping, our value s are still a bit larger than those obtained for 200 nm thick films \ngrown by  LPE (\n H0= 0.4 Oe) 10. The saturation of the linewidth with increasing  excitation  frequency  \nobserved for  thinnest films ( t = 7 and 4 nm) is usually  ascribed to two -magnon s scattering  due to the \ninterfaces 25. An estimation of the intrinsic damping in such thin films is  thus not correct. \nNevertheless , and only for the sake of comparison, considering frequencies under 6 GHz , we can \nrough ly estimate the low frequency Gilbert damping  to be  1.610-3 for the 7 nm and 3.810-3 the 4nm \nYIG films . However, it is  worth mentioning that for those  two thinnest films;  samples  sliced  from the 5 same substrate  can give different linewidths ( up to a factor of 3) with for some of them up to 2 \nabsorption s lines. This observation point s to a slight lateral non-homogeneity  in the chemical \ncomposition24. The data presented in figure 2 are tho se of the best samples showing a single absorption \nline. For the 20  nm thick films , all samples have only one resonance line and the dispersion of \nlinewidths is within 5%.   \nIn order to characterize the conversion of propagating SWs in YIG into a charge current in a normal \nadjacent metal with large SOC , we perform ISHE detection of SW  with the strip geometry used by \nChumak et al.26. In our sample design (see Fig 3a) , SWs  excitation is achieved using a patterned 100 \nµm wide  Au stripline  antenna  whereas the ISHE voltage is measured on a 13 nm thick Pt strip ( 0.2 \nmm x  5 mm ) located at  100 µm away from the Au stripe  and parallel to it . The metallic Pt strip is \ndeposited using dc magnetron sputtering  and lift -off. Prior to  the Pt deposition , an in -situ O2/Ar-\nplasma is used to  remove  the photo -resist residue s and increase the ISHE voltage.  This cleaning step \nhas been shown recently to improve the ISHE signal by one order of magnitude27,28. Measurement s of \nthe ISHE signal is performed either using a lock -in (with a 5 kHz TTL modulation  of the rf power ) or \na nano -voltmeter.  In order to increase the output ISHE signa l, we chose a specific configuration with a \nmagnetic field at 45° from  to the SW propagation direction  (cf Fig. 3a). This configuration  has the \nadvantage of provid ing a good coupling of the YIG film  to the rf field under the antenna while still \nhaving a sig nificant spin polarization  () that is orthogonal  to the measured ISHE electrical field . We \nshould point out  that our choice of magnetic field direction implies that the propagating SWs are \nneither Damon -Eshbach modes  where k \n M nor backward volume modes where  k // M.  \nIn Fig 3b, we display the  ISHE voltage (without any geometrical correction) as a function of the in \nplane magnetic field  measured  on a 20 nm thick YIG under a 10 mW rf excitation at 1 GHz. The sign \nof voltage peak (occurring at magnetic fields that resonantly excite the magnetization) reverses when \nthe applied field is reversed as expected from the ISHE symmetry 29 : where  \nis the pure spin current that flows through the YIG/Pt interface , is the spin polarization vector \nparallel to the dc magnetization direction  and  is the unit vector parallel to the Pt strip.  A close -up \non the ISHE signal lineshape for the different thickness es is plotted in Fig . 3c 3d 3e. In accordance \nwith FMR study, the linewidth increases with decreasing YIG thickness. The spectral  lineshap es are \nalmost symmetrical confirming previous reports on YIG films  grown using LPE  30. The ISHE \nmaximum voltage decreases when the film thickness  is decrease d. Going from 20 nm to 4 nm this \ndecrease is as large as two order s of magn itudes  and correlates to the increase of the Gilbert damping . \nA decrease of the ISHE signal when increasing damping is expected.  Indeed, a s we are in the weak \nexcitation regime, the ISHE voltage is expected to scale with 1/\n2, see for instance supplementary  \nmate rials of Ref [7]. If we consider the Gilbert damping obt ained from  FMR on the bar e YIG samples,  \na more dramatic decrease of the I SHE signal  than the one observed is expected.  In fact here, one \nshould consider the effective damping of the Pt/YIG stack as spin -pumping will increase the YIG 6 effective damping under the Pt strip. We should ag ain point out  that our measurement geometry relies  \non propagating SWs  and therefor e they are subject to an exponential decay with distance . The length \nscale of this exponential decay is different for the three  thicknesses  owing to the difference in the \ndamping parameter ; hence a quantitative interpretation of the voltage amplitude is to be avoided here.  \nWe thus succeeded  to grow high quality ultrathin YIG film and demonstrated that a n efficient  spin \nangular momentum transfer from the YIG film toward the Pt layer . The nex t objective is to induce an \nmodification on the effective YIG damping coefficient via interfacial spin injection using SHE , as it \nhas been  reported in Py/Pt31,32 system . Using YIG/Pt,  Kajiwara et al.  have shown that even without rf \nexcitation, spin injection induced by  a dc current in Pt can generate propagating SWs ; the threshold  \ncurrent density has been estimated to be 4.4 x  108 A.m-2. Such unexpectedly low value has been \nattributed to the presence of an easy –axis surface anisotropy13. Hence, w e have performed experiments  \nwhere  a large  dc bias current is applied on the Pt electrode  while pumping spin waves  in the 20 nm \nthick YIG film  with TTL modulated rf excitation field . The VISHE linewidth  is measured at the TTL \nmodulation frequency using a lock -in. We  expected to increase or decrease  this linewidth  depending \non the dc current  polarity  but w e have not been able to see any sizable effect even for current densities \nas large as 6 x 109 Am-2, within a 0.2 Oe resolution, the measured linewidth remains absolutely \nunchanged . Theory  predicts that th e threshold dc current for the onset of magnetic excitations scales \nwith the Gilbert  damping and the thickness  of the ferromagnetic insulator  (Eq. 1), the smaller the \n product the lower  the threshold current for SW excitation  is. In Kajiwara et al. ’s experiment  \n nm; in our case  nm (considering the spin pumping contribution  to the \ndamping ). We therefore applie d a dc current that is roughly 2  orders of magnitude  larger than the \nexpected threshold current . One possible explanation  is that in our films , surface anisotropy is absent \nand therefore the pumped angular momentum is spread over many excitations modes33. Further \ninvestigations are under progress to clarify this point.  \nIn summary,  we have fabricated PLD grown YIG on GGG (111) substrates with t hickness as low as 4 \nnm. We present here a comparative study on three  different thicknesses : 4, 7 and 20 nm . The Gilbert \ndamping coefficient for the 20 nm thick films is 2.3 x 10-4 which is the lowest value reported for sub -\nmicrometric thick films. We demonstrate ISHE detection of SWs for ultra thin YIG film. The \namplitudes of the ISHE voltage correlate s well to the increase of the Gilbert damping when decreasing \nthickness. Owing to extremely low product  of the 20 nm film that is almost 10 time s smaller \nthan the one reported by Kajiwara et al.  we expected to observe  compensation of the damping by spin \ncurrent injection through the SHE  but our preliminary results on the VISHE  linewidth did not reveal any \neffect on the magnetization dynamics.  \n \nAcknowledgement:  \nThis work has been supported by ANR -12-ASTR -0023 Trinidad   7  \nReferences:  \n \n1 J. E. Mee, J. L. Archer, R.H. Meade, and T.N. Hamilton,  Applied Physics Letters  10 \n(1967).  \n2 J. P. Castera,  Journal of Applied Physics 55 (6), 2506 (1984).  \n3 A. Khitun, M. Q. Bao, and K. L. Wang,  Journal of Physics D -Applied Physics 43 \n(26), 10 (2010).  \n4 V. Vlaminck and M. Bailleul,  Physical Review B 81 (1) (2010).  \n5 Vladislav E. Demidov, Sergei Urazhdin, and Sergej O. Demokritov,  Nature  Materials \n9 (12), 984 (2010).  \n6 J. E. Hirsch,  Physical Review Letters 83 (9), 1834 (1999).  \n7 Y. Kajiwara, K. Harii, S. 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Tatara,  Applied Physics Letters 88 (18) \n(2006).  \n30 T. Tashiro, R. Takahashi, Y. Kajiwara, K. Ando, H. Nakaya ma, T. Yoshino, D. \nKikuchi, and E. Saitoh, presented at the Conference on Spintronics V, San Diego, CA, \n2012 (unpublished).  \n31 K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh,  \nPhysical Review Letters 101 (3), 4 (2008).  \n32 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. \nSchmitz, and S. O. Demokritov,  Nature Materials 11 (12), 1028 (2012).  \n33 M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. \nMancoff, M. A. Yar, a nd J. Akerman,  Nature Nanotechnology 6 (10), 635 (2011).  9 Figure Captions :  \nFigure 1 : \n(color online). (a) 1 µm x 1µm AFM surface topography of a 20nm YIG film on GGG (111) \n(RMS roughness=0.23  nm). (b) XRD  : \n-2\n scan of the same film using Cu -K\n1 radiation. \nThis spectrum shows that the growth is along the (111) direction with no evidence of parasitic \nphases. The YIG peak are best resolved at high diffraction angle, a zoom around the 888 GGG \npeak is shown in panel (c). \n(c) (d) (e) XRD diffraction s pectrum centered on GGG (888) reflection for different YIG \nthicknesses: 20, 7 and 4 nm. For the 2 thinnest films, the YIG peak is masked by the substrate \npeak . The cubic lattice parameter of the 20 nm thick YIG film (obtained from 888 peak) is \nslightly lar ger than that of the substrate ( aYIG=1.2459 nm , aGGG=1.2383  nm) \n \nFigure 2 : \n(color online) ( a) (b). FMR absorption derivative spectra of 20 and 4 nm thick YIG films at an \nexcitation frequency of 6 GHz.  \n(c) rf excitation frequency dependence of FMR absorption linewidth measured on different \nYIG film thicknesses with an in -plane oriented static field. The black continuous line is a \nlinear fit on the 20  nm thick film from w hich a Gilbert damping coefficient of 2. 3 x 10-4 can \nbe inferred  ( . The damping of the 7 nm and 4 nm films is \nsignificantly larger but must off all the frequency dependence is not linear (see text for \ndiscussion).   \n \nFigure 3 : \n(a) Schematic illustration of the experimental setup: spin waves are excited through the YIG \nwaveguide with a microstrip antenna. The detection is performed by measuring ISHE voltage \ndc signal on a ~200µm wide Pt stripe. (b) External dc magnetic field dependen ce of ISHE \nvoltage measured on Pt electrode showing the polarity inversion when the magnetic field is \nreversed. The peaks occur at the FMR conditions as verified through S 11 spectroscopy on the \nAu antenna (not shown here).  \n(c). (d). (e). ISHE voltage for d ifferent YIG film thicknesses around the resonance magnetic \nfield; microwave frequency is f=3GHz. The peaks can be fitted to a lorentzian shape and the \nextracted linewidth are respectively:  19.2 , 13.2 and 4.6 Oe. The dramatic increase of the \nISHE signal w ith thickness is discussed in the text.   10  \n \nFig 1  \n \n \n  \n11  \nFig 2  \n  \n12  \nFig 3  \n"    },    {        "title": "2305.10111v1.Material_Parameters_for_Faster_Ballistic_Switching_of_an_In_plane_Magnetized_Nanomagnet.pdf",        "content": "arXiv:2305.10111v1  [cond-mat.mes-hall]  17 May 2023Journal of the Physical Society of Japan FULL PAPERS\nMaterial Parameters for Faster Ballistic Switching of an In -plane Magnetized\nNanomagnet\nToshiki Yamaji1*and Hiroshi Imamura1 †\n1National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nHigh-speed magnetization switching of a nanomagnet is nece ssary for faster information processing. The ballistic\nswitching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switch-\ning speed of the ballistic switching can be increased by incr easing the magnitude of the pulsed magnetic field. However\nit is difficult to generate a strong and short magnetic field pulse in a sm all device. Here we explore another direction\nto achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturati on\nmagnetization, and the Gilbert damping constant. We perfor m the macrospin simulations for the ballistic switching of\nin-plane magnetized nano magnets with varying material par ameters. The results are analyzed based on the switching\ndynamics on the energy density contour. We show that the puls e width required for the ballistic switching can be re-\nduced by increasing the magnetic anisotropy constant or by d ecreasing the saturation magnetization. We also show that\nthere exists an optimal value of the Gilbert damping constan t that minimizes the pulse width required for the ballistic\nswitching.\n1. Introduction\nIn modern information technologies huge amount of data\nare represented as the direction of the magnetization in a sm all\nmagnet such as magnetic grains in magnetic tapes or hard\ndisk drives. To write information on the conventional mag-\nnetic recording media an external magnetic field is applied i n\nthe opposite direction of the magnetization to switch the di -\nrection of the magnetization. During the switching the mag-\nnetization undergoes multiple precessions around the loca l ef-\nfective field consisting of the external field, anisotropy fie ld,\nand demagnetizing field. The typical switching time or write\ntime is of the order of nanoseconds.\nTo meet the growing demand for fast information process-\ning it is important to develop a faster switching scheme. The\nballistic switching is a promising candidate for high-spee d\nswitching, and much e ffort has been devoted to developing\nthe ballistic switching both theoretically1–8)and experimen-\ntally.9–16)In ballistic switching a pulsed magnetic field is ap-\nplied perpendicular to the easy axis to induce the large-ang le\nprecession around the external magnetic field axis. The dura -\ntion of the pulse is set to a half of the precession period. Aft er\nthe pulse the magnetization relaxes to the equilibrium dire c-\ntion opposite to the initial direction. The switching speed of\nthe ballistic switching can be increased by increasing the m ag-\nnitude of the pulsed field. However, it is di fficult to generate a\nstrong and short field pulse in a small device. It is desired to\nfind a way to speed up the ballistic switching without increas -\ning magnetic field.\nThe magnetization dynamics of the ballistic switching is\ndetermined by the torques due to the external magnetic field,\nthe uniaxial anisotropy field, the demagnetizing field, and t he\nGilbert damping. The torques other than the external mag-\nnetic field torque are determined by the material parameters\nsuch as the anisotropy constant, the saturation magnetizat ion,\nand the Gilbert damping constant. There is room to speed up\n*toshiki-yamaji@aist.go.jp\n†h-imamura@aist.go.jpthe ballistic switching by designing the appropriate mater ial\nparameters.\nIn the early 2000s the several groups each independently\nreported the optical microscope measurements of the ballis -\ntic switching by picosecond pulse magnetic field.9–13)Then\nthe mechanism of a ballistic switching was analyzed in terms\nof the nonlinear dynamics concepts such as a fixed point, at-\ntractors, and saddle point.2, 3, 6)Especially the minimal field\nrequired for a ballistic switching was investigated by comp ar-\ning the so-called Stoner-Wohlfarth (SW) type.2, 3)The damp-\ning constant dependence of the minimal switching field was\nalso studied.2)The characteristics of the parameters of a pulse\nmagnetic field, i.e., magnitude, direction, and rise /fall time on\nthe mechanism of a ballistic switching had been also studied\nby the simulations and experiments.6, 7, 14, 15)\nAs described above, in 2000s and 2010s a ballistic switch-\ning technique had received much attention for the fast magne -\ntization reversal with ringing suppression by fine-tuning t he\nmagnetic pulse parameters. Due to the recent advance of an\nultra-fast measurement17)the studies of a ballistic switching\nhave attracted much attention again. Last year the in-plane\nmagnetization switching dynamics as functions of the pulse\nmagnetic field duration and amplitude was calculated and\nanalyzed by using the conventional Landau-Lifshitz-Gilbe rt\n(LLG) equation and its inertial form, the so-called iLLG\nequation.16)The solutions of both equations were compared\nin terms of the switching characteristics, speed and energy\ndensity analysis. Both equations return qualitatively sim ilar\nswitching dynamics. However the extensive material param-\neter dependences of a ballistic switching region have not\nyet been sufficiently explored. Therefore it is worth clearing\nthe extensive material parameter dependences of the ballis tic\nswitching of an in-plane magnetized nanomagnet.\nIn this paper, we study the ballistic switching of an in-\nplane magnetized nanomagnet with systematically varying\nthe material parameters by using the macrospin simulations .\nThe results show that the pulse width required for the bal-\nlistic switching can be reduced by increasing the magnetic\n1J. Phys. Soc. Jpn. FULL PAPERS\nHp\nmz\nyx(a)\n(c) my at t = 10 ns (b) \n(d) 0 200 400-1 01\nt [ps]my\ntp [ps]0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tutSW \n0 1 -1 \n0 1 2 3 4 502.55.010.0\n7.5\ntp [ps]Hp [T] \nFig. 1. (a) Schematic illustration of the in-plane magnetized nano magnet.\nThe pulse field, Hp, is applied along the x-direction. The initial direction of\nthe magnetization is in the positive y-direction. (b) Gray scale map of myat\nt=10 ns as a function of the pulse field width, tp, and Hp. The black and\nwhite regions represent the success and failure of switchin g. The parameters\nareµ0Ms=0.92 T,µ0HK=0.1 T, andα=0.023. (c) Typical example of\nthe time evolution of mywhen the magnetization switches ( Hp=5 T and tp\n=0.4 ps). The switching time, tSW, is defined as the time when mychanges\nthe sign. (d) tpdependence of tSWalong the dashed horizontal line at Hp=5\nT shown in Fig. 1(b). tlandtuare 3.15 ps and 3.93 ps, respectively. tSWat\ntl≤tp≤tuis 1.7 ps.\nanisotropy constant or by decreasing the saturation magnet i-\nzation. There exists an optimal value of the Gilbert damping\nconstant that minimizes the pulse width required for ballis -\ntic switching. The simulation results are intuitively expl ained\nby analyzing the switching trajectory on the energy density\ncontour.\n2. Model and Method\nIn this section we show the theoretical model, the numer-\nical simulation method, and the analysis using the trajecto ry\nin the limit ofα→0. The macrospin model of the in-plane\nmagnetized noanomagnet and the equations we solve to simu-\nlate the magnetization dynamics are given in Sec. 2.1. In Sec .\n2.2 we show that the switching conditions can be analyzed by\nusing the trajectory on the energy density contour in the lim it\nofα→0 if theα≪1.\n2.1 Macrospin Model Simulation\nFigure 1(a) shows the schematic illustration of the in-\nplane magnetized nanomagnet. The pulsed magnetic field,\nHp, is applied along the x-direction. The unit vector m=\n(mx,my,mz) indicates the direction of the magnetization. The\nsize of the nanomagnet is assumed to be so small that the dy-\nnamics of mcan be described by the macrospin LLG equation\ndm\ndt=−γm×/parenleftBigg\nHeff−α\nγdm\ndt/parenrightBigg\n, (1)\nwhere tis time,γis the gyromagnetic ratio, αis the Gilbert\ndamping constant. The e ffective field, Heff=Hp+HK+Hd,\ncomprises the pulse field, Hp, the anisotropy field, HK, andthe demagnetizing field, Hd. The anisotropy field and the de-\nmagnetizing field are defined as\nHK=/bracketleftbig2K/(µ0Ms)/bracketrightbigmyey, (2)\nand\nHd=µ0Msmzez, (3)\nrespectively, where Kis the uniaxial anisotropy constant, µ0\nis the magnetic permeability of vacuum, Msis the saturation\nmagnetization, and ejis the unit vector along the j-axis ( j=\nx,y,z).\nThe switching dynamics are calculated by numerically\nsolving the LLG equation. The initial ( t=0) direction is set\nasmy=1. The rectangular shaped pulse magnetic field with\nduration of tpis applied at t=0. The time evolution of magne-\ntization dynamics are calculated for 10 ns. Success or failu re\nof switching is determined by whether my<−0.5 att=10\nns.\nFigure 1(b) shows the gray scale plot of myatt=10 ns\non the tp-Hpplane. Following Ref. 16 the parameters are as-\nsumed to beµ0Ms=0.92 T, K=2.3 kJ/m3, i.e.µ0HK=\n0.1 T, andα=0.023. The black and white regions represent\nthe success and failure of switching, respectively. The wid e\nblack region at upper right of Fig. 1(b) represents the balli stic\nswitching region (BSR). A typical example of the time evolu-\ntion of mywhen the magnetization switches is shown in Fig.\n1(c). The switching time, tSW, is defined as the time when my\nchanges the sign. Figure 1(d) shows the tpdependence of tSW\nalong the horizontal line shown in Fig. 1(b), i.e. at Hp=5\nT. The BSR indicated by shade appears between tl=3.15\nps and tu=3.93 ps, where tSW=1.7 ps independent of tp.\nThe lower and upper boundary of the BSR are represented by\ntlandtu, respectively. We investigate the material parameter\ndependence of tlandtuwith keeping Hp=5 T.\n2.2 Analysis of the Switching Conditions for α≪1\nIf the Gilbert damping constant is much smaller than unity\nthe approximate value of tlandtucan be obtained without\nperforming macrospin simulations. In the limit of α→0, the\ntrajectory is represented by the energy contour because the en-\nergy is conserved during the motion of m. The energy density,\nE, of the nanomagnet is defined as18)\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ), (4)\nwhereθandφare the polar and azimuthal angles of the mag-\nnetization, respectively. The color plot of the energy dens ity\ncontour is shown in Fig. 2. The separatrix representing the\nenergy contour with E=Kis indicated by the white curve,\nwhich is expresses as\n1\n2µ0M2\nscos2θ−Ksin2θsin2φ=0. (5)\nThe green dot indicates the initial direction of matt=0. The\nblack curve represents the trajectory of munder the pulse field\nofHpin the limit ofα→0. Under the pulse field the energy\ndensity is given by\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ. (6)\n2J. Phys. Soc. Jpn. FULL PAPERS\n01 5 4 3 26E/K\ntltu\nθ\nφ\nFig. 2. (Color online) Color plot of the energy density contour give n by\nEq. (4).θandφare the polar and azimuthal angles of the magnetization, re-\nspectively. The material parameters, MsandKare same as in Fig. 1. The\nseparatrix given by Eq. (5) is indicated by the white curve. T he initial direc-\ntion of mis indicated by the green dot at ( θ,φ)=(π/2,π/2). The black curve\nrepresents the trajectory of the magnetization under the fie ld of Hp=5 T in\nthe limit ofα→0, which is given by Eq. (7). The yellow stars indicate the\nintersection points of the separatrix and the trajectory, w hich correspond to tp\n=tlandtu. If the pulse is turned o ffattl≤t≤tu, the magnetization switches\nballistically. The yellow triangle indicates the turning p oint of the trajectory\nof the magnetization near mz=1, at whichφ=0.\nSince the energy density of the initial direction, θ=φ=π/2,\nisE=0, the trajectory under the pulse field is expressed as\n1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ=0. (7)\nThe yellow stars indicate the points where the trajectory\ncrosses the separatrix surrounding the equilibrium point a t\nφ=−π/2. The upper and lower points indicates the direc-\ntion of mat the end of the pulse with tp=tuandtl, re-\nspectively. The corresponding angles ( θl,φl) and (θu,φu) can\nbe obtained by solving Eqs. (5) and (7) simultaneously. If\ntl≤tp≤tu, the magnetization relaxes to the equilibrium di-\nrection at (θ,φ)=(π/2,−π/2) after the pulse to complete the\nswitching. We can obtain the approximate expressions of tl\nandtuas follows. Assuming that the pulse field is much larger\nthan the other fields, the angular velocity of the precession ,ω,\nis approximated as γHp/(1+α2), and tlandtuare analytically\nobtained as\ntl=π−2θturn\nω−1\n2∆θ\nω, (8)\nand\ntu=π−2θturn\nω+1\n2∆θ\nω, (9)\nwhere∆θ=θu−θl, andθturnis the polar angle at the turning\npoint (φ=0) indicated by the yellow triangle.3. Results and Discussion\nIn this section we discuss the dependence of the BSR on\nthe material parameters by analyzing the numerical simula-\ntion results and Eqs. (8) and (9). The results for the variati on\nof the magnetic anisotropy constant, K, saturation magnetiza-\ntion, Ms, and the Gilbert damping constant, α, will be given\nin Secs. 3.1, 3.2, and 3.3, respectively.\n3.1 Anisotropy Constant Dependence of the BSR\nFigure 3(a) shows the anisotropy constant, K, dependence\nof the BSR. The parameters are Hp=5 T,µ0Ms=0.92 T, and\nα=0.023. The simulation results of tlandtuare indicated\nby the orange and blue dots, respectively. The analytical ap -\nproximations of tlandtuobtained by solving Eqs. (5),(7),(8),\nand (9) are represented by the orange and blue curves, respec -\ntively. The simulation and analytical results agree well wi th\neach other because the Gilbert damping constant is as small a s\n0.023. As shown in Fig. 3(a), tlis a monotonically decreasing\nfunction of Kwhile tuis a monotonically increasing function\nofK. As a result the width of the BSR, tu-tl, is a monoton-\nically increasing function of Kas shown in the inset of Fig.\n3(a).\nIn the left panel of Fig. 3(b) the separatrix and the trajecto ry\nwithα=0 for K=2.3 kJ/m3are shown by the blue and\nblack curves, respectively. The same plot for K=9.3 kJ/m3\nis shown in the right panel. As shown in these panels, the\nincrease of Kdoes not change the trajectory much. However,\nthe increase of Kchanges the separatrix significantly through\nthe second term of Eq. (5). Assuming that the angular velocit y\nof the precession is almost constant, the spread of the area\nsurrounded by the separatrix results in the spread of the tim e\ndifference between tlandtu. As a result the BSR is spread by\nthe increase of Kas shown in Fig. 3(a)\n3.2 Saturation Magnetization Dependence of the BSR\nFigure 4(a) shows the saturation magnetization dependence\nof the BSR obtained by the numerical simulation and the ana-\nlytical approximations. The horizontal axis represents th e sat-\nuration magnetization in unit of T, i.e µ0Ms. The parameters\nareHp=5 T,K=2.3 kJ/m3, andα=0.023. The symbols are\nthe same as in Fig. 3(a). The lower boundary of the BSR, tl,\nincreases as theµ0Msincreases while the upper boundary of\nthe BSR, tu, decreases with increase of µ0Ms. Therefore, the\nfaster switching is available for smaller Ms. Theµ0Msdepen-\ndence of the BSR ( tu-tl) is also shown in the inset of Fig. 4(a).\nThe BSR decreases with increase of µ0Ms. In other words, the\nwider BSR is obtained for smaller Ms.\nIn the right panel of Fig. 4(b) the separatrix and the trajec-\ntory withα=0 forµ0Ms=0.35 T are shown by the blue and\nblack curves, respectively. The same plot for µ0Ms=0.92 T is\nshown in the left panel. As shown in these panels, the increas e\nofMsdoes not change the trajectory much but decrease the\nseparatrix significantly through the first term of Eq. (5). As -\nsuming that the angular velocity of the precession is almost\nconstant, the reduction of the area surrounded by the separa -\ntrix results in the reduction of the time di fference between tl\nandtu. As a result the BSR decreases with increase of Msas\nshown in Fig. 4(a)\n3J. Phys. Soc. Jpn. FULL PAPERStl, t u [ps] \n0 10 20 30 40 2.03.04.05.0\nK [kJ/m3]ballistic switching region (a)\ntltu\ntu - t l [ps] \nK [kJ/m 3]0 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 \n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ(b) K = 2.3 kJ/m3K = 9.3 kJ/m3\ntltltutu\nFig. 3. (Color online) (a) Anisotropy constant, K, dependence of the BSR\n(orange shade). Simulation results of tlandtuare plotted by the orange and\nblue dots, respectively. The analytical results are indica ted by the solid curves\nwith the same color. The parameters are Hp=5 T,µ0Ms=0.92 T, andα=\n0.023. In the inset the simulation and analytical results of the width of the\nBSR, tu-tl, are plotted by the dots and the solid curve, respectively. ( b)\nTypical examples of the trajectory of the magnetization (bl ack curve) and the\nseparatrix (blue curve). The left and right panels show the r esults for K=2.3\nkJ/m3andK=9.3 kJ/m3, respectively. The orange and blue stars indicate\nthe direction at t=tlandtu, respectively. The green dots indicate the initial\ndirection of m.\n3.3 Gilbert Damping Constant Dependence of the BSR\nFigure 5(a) shows the simulation results of the Gilbert\ndamping constant, α, dependence of the BSR. The width of\nthe BSR is shown in the inset. The symbols are the same as\nin Fig. 3(a). The approximate values obtained by Eqs. (8) and\n(9) are not shown because the αis not limited toα≪1. The\nparameters are Hp=5 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. There exists an optimal value of αthat minimizes tl. The\noptimum value in Fig. 5 (a) is αopt=0.35.\nTo understand the mechanism for minimization of tlat a\ncertain value ofαone need to consider two di fferent effects of\nαon the magnetization dynamics. One e ffect is the decrease\nof the precession angular velocity with increase of α. The pre-\ncession angular velocity around the e ffective field of Heffis\ngiven by (γHeff)/(1+α2), which decreases with increase of α.\nThis effect causes the increase of tlandtu.\nThe other effect is the increase of the energy dissipation rate\nwith increase ofα. Let us consider the trajectory in the cases\nof small damping ( α=0.023) and large damping ( α=αopt).\nIn Fig. 5 (b) the typical examples of the trajectory for the\nsmall damping are shown by the yellow and green curves\nand dots on the energy density contour. The pulse widths are\ntp=tl(=3.15 ps) and 3.14 ps. The trajectories during the\npulse are represented by the solid curves and the trajectori es\nafter the pulse are represented by the dots. The white curve\nshows the separatrix and the black dot indicates the initial di-\ntl, t u [ps] \n2.03.04.05.0\n0.0 0.3 0.6 0.9 1.2\nμ0Ms [T]ballistic switching region \ntltu(a)\n(b) μ0Ms = 0.92 T μ0Ms = 0.35 T\n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ\ntl tltu tu1.5tu - t l [ps] \n0123\n0.0 0.3 0.6 0.9 1.2 \nμ0Ms [T] 1.5 \nFig. 4. (Color online) (a) Saturation magnetization dependence of the\nBSR. The horizontal axis represents the saturation magneti zation in unit of T,\ni.eµ0Ms. The parameters are Hp=5 T, K=2.3 kJ/m3, andα=0.023. The\nsymbols are the same as in Fig. 3 (a). (b) Typical examples of t he trajectory\nof the magnetization (black curve) and the separatrix (blue curve). The right\nand left panels show the results for µ0Ms=0.35 T and 0.92 T, respectively.\nThe symbols are the same as in Fig. 3 (b).\nrection. The yellow and green stars indicate the points wher e\nthe trajectories cross the separatrix surrounding the targ et and\ninitial states, respectively. The arrows indicate the dire ction\nof the movement of the magnetization. For the small damp-\ning, even very close to the separatrix around the target stat e at\nthe end of the pulse, the magnetization flows to the sepatrari x\naround the initial state and relax to the initial state after many\nprecessions with the slow energy dissipation.\nFigure 5 (c) shows the tpdependence of tSWat the large\ndamping (α=αopt). All parameters except αare the same\nas in Fig. 1 (d). t′\nl,tl, and tuare 0.82 ps, 1.98 ps, and 4.54\nps, respectively. t′\nlis the time when for the large damping the\nmagnetization goes across the e ffective separatrix around the\ninitial state during the pulse duration. In Fig. 5 (d) the typ ical\nexamples of the trajectory for the large damping are shown\nby the yellow ( tp=0.9 ps), green ( tp=tl=1.98 ps), and\npurple ( tp=4.55 ps) curves and dots on the energy density\ncontour. The symbols are the same as in Fig. 5 (b). In the\nregion 1 ( tp<t′\nl) of Fig. 5 (c) the magnetization is located on\nthe inside of the effective separatrix at the end of the pulse and\nreturn to the initial state. The trajectory is not shown in Fi g. 5\n(d). In the region 2 ( t′\nl≤tp<tl) of Fig. 5 (c) after the pulse is\nremoved the magnetization move toward the target state unde r\nthe effective field of Heffand goes across the separatrix. Then\nthe magnetization finishes the switching. The typical examp le\nof the trajectory is shown by the yellow curve and dots in Fig.\n5 (d). In the region 3 ( tl≤tp≤tu) after the pulse is removed\nthe magnetization ballistically switches. The typical exa mple\n4J. Phys. Soc. Jpn. FULL PAPERS\ntl, t u [ps] \n2.04.06.08.0\n0.0 0.2 0.4 0.6 0.8\nαballistic switching region \ntltu(a)\n0\ntu - t l [ps] \n0.00123\n0.2 0.4 0.6 0.8 \nα45\nαopt \nαopt\n01 5 4 3 26E/K \nα = 0.023 \ntp = t l = 3.15 ps \ntp = 3.14 ps (b)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n< <<\n< <\nˑ\nˑ\n(c)\ntp [ps] 0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tu1 2 4 3\ntl’\n01 5 4 3 26E/K \nα = 0.35 \ntp = 0.9 ps \ntp = t l = 1.98 ps \ntp = 4.55 ps (d)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n<\n<\n<ˑˑ <\n< <\nˑ\nFig. 5. (Color online) (a) Simulation results of the tlandtuas a function\nofα. The parameters are Hp=5.0 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. The symbols are the same as in Fig. 3 (a). (b) The trajectori es atα=\n0.023 with tp=3.15 ps (yellow) and 3.14 ps (green) on the energy density\ncontour. The trajectory during the pulse is represented by t he solid curve. The\ntrajectory after the pulse is represented by the dots. The wh ite curve shows\nthe separatrix.The direction of the trajectory is indicate d by the arrow. The\nstar indicates the intersection point of the trajectory and the separatrix. The\ninitial direction is indicated by the black dot. (c) tpdependence of tSWat\nα=αopt. All parameters except αare the same as in Fig. 1 (d). t′\nl,tl, and tu\nare 0.82 ps, 1.98 ps, and 4.54 ps, respectively. tSWattl≤tp≤tuis 1.98 ps.\n(d) The trajectories at α=αoptwith tp=0.9 ps (yellow), 1.98 ps (green), and\n4.55 ps (purple) on the energy density contour. The symbols a re the same as\nin Fig. 5 (b).\nof the trajectory is shown by the green curve and dots in Fig.5 (d). As explained in Sec. 2.1, in this study the tSWis defined\nas the time when mychanges the sign, in other words, the\nmagnetization reaches the turning point ( φ=0) on the energy\ndensity contour. Therefore the tlfor the large damping is equal\nto the ballistic tSW, i.e. the time when under the pulse field the\nmagnetization reaches the turning point. The ballistic tSWat\nthe region 3 is 1.98 ps.\nAs described above, for the large damping the magnetiza-\ntion can relax to the target state even at greater distances f rom\nthe separatrix by moving under Heffwith the fast energy dis-\nsipation. The effect can be regarded as the e ffective spread\nof the separatrix and results in the decrease of tland the in-\ncrease of tuwith increase ofα. Therefore, there exists the\noptimal value ofαthat minimizes tlwhile tumonotonically\nincreases with increase of αas shown in Fig. 5(a). In the re-\ngion 4 ( tp>tu) after the pulse is removed the magnetization\nmoves toward the separatrix around the initial state under Heff\nand relaxes to the initial state. We find that the BSR for the\nlarge damping can be explained by the anisotropic spread of\nthe effective separatrix with increasing α, which is fundamen-\ntally due to the breaking of the spatial inversion symmetry o f\nthe spin dynamics. The broken symmetry of the spatial inver-\nsion of the spin dynamics for the large damping can be easily\nconfirmed by comparing Fig. 5 (c) with Fig. 1 (d).\n4. Summary\nIn summary, we study the material parameter dependence\nof the ballistic switching region of the in-plane magnetize d\nnanomagnets based on the macrospin model. The results show\nthat the pulse width required for the ballistic switching ca n be\nreduced by increasing the magnetic anisotropy constant or b y\ndecreasing the saturation magnetization. The results also re-\nvealed that there exists an optimal value of the Gilbert damp -\ning constant that minimizes the pulse width required for the\nballistic switching. The simulation results are explained by\nanalyzing the trajectories on the energy contour. The resul ts\nare useful for further development of the high-speed inform a-\ntion processing using the ballistic switching of magnetiza tion.\nThis work is partially supported by JSPS KAKENHI Grant\nNumber JP20K05313.\n1) L. He and W. D. Doyle: J. Appl. Phys. 79(1996) 6489.\n2) Z. Z. Sun and X. R. Wang: Phys. Rev. B 71(2005) 174430.\n3) D. Xiao, M. Tsoi, and Q. Niu: Journal of Applied Physics 99(2006)\n013903.\n4) Y . Nozaki and K. Matsuyama: Jpn. J. Appl. Phys. 45(2006) L758.\n5) Y . Nozaki and K. Matsuyama: Journal of Applied Physics 100(2006)\n053911.\n6) Q. F. Xiao, B. C. Choi, J. Rudge, Y . K. Hong, and G. Donohoe: J ournal\nof Applied Physics 101(2007) 024306.\n7) P. P. Horley, V . R. Vieira, P. M. Gorley, V . K. Dugaev, J. Ber akdar, and\nJ. Barna´ s: Journal of Magnetism and Magnetic Materials 322(2010)\n1373.\n8) Y . B. Bazaliy: Journal of Applied Physics 110(2011) 063920.\n9) T. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and T . Rasing:\nNature 418(2002) 509.\n10) I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr,\nG. Ju, B. Lu, and D. Weller: Nature 428(2004) 831.\n11) H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat:\nPhys. Rev. Lett. 90(2003) 017204.\n12) W. K. Hiebert, L. Lagae, J. Das, J. Bekaert, R. Wirix-Spee tjens, and\nJ. De Boeck: Journal of Applied Physics 93(2003) 6906.\n13) W. K. Hiebert, L. Lagae, and J. De Boeck: Phys. Rev. B 68(2003)\n5J. Phys. Soc. Jpn. FULL PAPERS\n020402.\n14) H. W. Schumacher: Appl. Phys. Lett. 87(2005) 042504.\n15) N. Kikuchi, Y . Suyama, S. Okamoto, O. Kitakami, and T. Shi matsu:\nAppl. Phys. Lett. 104(2014) 112409.\n16) K. Neeraj, M. Pancaldi, V . Scalera, S. Perna, M. d’Aquino , C. Serpico,\nand S. Bonetti: Phys. Rev. B 105(2022) 054415.17) K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagstr ¨ om, S. S.\nP. K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Dein ert,\nI. Ilyakov, M. Chen, M. Bawatna, V . Scalera, M. d’Aquino, C. S erpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti: Nat. Ph ys.17\n(2021) 245.\n18) W. F. Brown: Phys. Rev. 130(1963) 1677.\n6"    },    {        "title": "1810.06471v1.Localized_spin_waves_in_isolated__kπ__skyrmions.pdf",        "content": "Localized spin waves in isolated k\u0019skyrmions\nLevente Rózsa,1,\u0003Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: October 16, 2018)\nThelocalizedmagnonmodesofisolated k\u0019skyrmionsonafield-polarizedbackgroundareanalyzed\nbasedontheLandau–Lifshitz–Gilbertequationwithinthetermsofanatomisticclassicalspinmodel,\nwith system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion\norderka higher number of excitation modes are found, including modes with nodes in the radial\neigenfunctions. Itisshownthatatlowfields 2\u0019and 3\u0019skyrmionsaredestroyedviaaburstinstability\nconnected to a breathing mode, while 1\u0019skyrmions undergo an elliptic instability. At high fields all\nk\u0019skyrmions collapse due to the instability of a breathing mode. The effective damping parameters\nof the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge\nin the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic\ninstability. It is shown that the breathing modes of k\u0019skyrmions may become overdamped at higher\nGilbert damping values.\nI. INTRODUCTION\nMagneticskyrmionsarelocalizedparticle-likespincon-\nfigurations [1], which have become the focus of intense\nresearch activities over the last years due to their promis-\ning applications in spintronic devices [2–5]. While their\nparticle-like properties make them suitable to be used\nas bits of information, the collective excitations of the\nspins constituting the magnetic skyrmion, known as spin\nwaves or magnons, open possible applications in the field\nof magnonics [6].\nThese spin wave modes were first investigated theo-\nretically [7–10] and experimentally [11–13] in skyrmion\nlattice phases, where the interactions between the\nskyrmions lead to the formation of magnon bands. If\na skyrmion is confined in a finite-sized nanoelement, it\nwill possess discrete excitation frequencies [14–17]. Al-\nthough such geometries have also been successfully ap-\nplied to the time-resolved imaging of the dynamical mo-\ntion of magnetic bubble domains [18, 19], in such a case\nit is not possible to distinguish between the excitations\nof the particle-like object itself and spin waves forming\nat the edges of the sample [14]. In order to rule out\nboundary effects, the excitations of isolated skyrmions\nhave to be investigated, as was performed theoretically\nin Refs. [20–23]. It was suggested recently [24] that\nthe experimentally determined excitation frequencies in\nthe Ir/Fe/Co/Pt multilayer system may be identified as\nspin wave modes of isolated skyrmions, rather than as\nmagnons stemming from an ordered skyrmion lattice.\nIn most investigations skyrmions correspond to sim-\nple domains with the magnetization in their core point-\ning opposite to the collinear background. However, it\nwas shown already in Ref. [25] that the Dzyaloshinsky–\nMoriya interaction [26, 27] responsible for their stabiliza-\ntion may also lead to the formation of structures where\nthe direction of the magnetization rotates multiple times\n\u0003rozsa.levente@physnet.uni-hamburg.debetween the center of the structure and the collinear re-\ngion. Such target states or k\u0019skyrmions, where kis the\nnumber of sign changes of the out-of-plane magnetization\nwhen moving along the radial direction, have also been\ninvestigated in constricted geometries [28–32]. The ex-\nperimental observation of localized spin structures with\nmultiple rotations has been mainly restricted to systems\nwith negligible Dzyaloshinsky–Moriya interaction so far\n[19, 33, 34], where the formation of domain structures is\nattributed to the magnetostatic dipolar interaction.\nThe collapse of isolated k\u0019skyrmions and their cre-\nationinnanodotsbyswitchingtheexternalfielddirection\nwas recently investigated in Ref. [35]. It was found that\nduring the creation process the skyrmions display signif-\nicant size oscillations resembling breathing eigenmodes.\nIn Ref. [25], the stability of k\u0019skyrmions was studied\nin a system with a ferromagnetic ground state, and it\nwas found that applying the external field opposite to\nthe background magnetization leads to a divergence of\nthe skyrmion radius at a critical field value, a so-called\nburst instability. This instability can be attributed to a\nsign change of one of the eigenvalues of the energy func-\ntional expanded around the k\u0019skyrmion configuration,\nintrinsically related to the dynamics of the system. How-\never, the spin wave frequencies of isolated k\u0019skyrmions\nremain unexplored.\nBesides the excitation frequencies themselves, the life-\ntime of spin waves is also of crucial importance in\nmagnonics applications. This is primarily influenced by\nthe Gilbert damping parameter \u000b[36], the value of which\ncan be determined experimentally based on resonance\nlineshapes measured in the collinear state [11, 19, 24].\nIt was demonstrated recently [23] that the noncollinear\nspin structure drastically influences the effective damp-\ning parameter acting on the spin waves, leading to mode-\ndependent and enhanced values compared to the Gilbert\ndamping parameter. This effect was discussed through\nthe example of the 1\u0019skyrmion in Ref. [23], but it is also\nexpected to be observable for k\u0019skyrmions with higher\norderk.\nHere the localized spin wave frequencies of isolated k\u0019arXiv:1810.06471v1  [cond-mat.mes-hall]  15 Oct 20182\nskyrmions are investigated in a classical atomistic spin\nmodel. The parameters in the Hamiltonian represent the\nPd/Fe/Ir(111) model-type system, where the properties\nof skyrmions have been studied in detail both from the\nexperimental [37, 38] and from the theoretical [35, 39–\n41] side. The paper is organized as follows. The classical\natomistic spin Hamiltonian and the method of calculat-\ning the eigenmodes is introduced in Sec. IIA, while the\nangular momentum and nodal quantum numbers charac-\nterizing the excitations are defined in Sec. IIB within the\nframework of the corresponding micromagnetic model.\nEigenfrequencies equal to or approaching zero are dis-\ncussed in Sec. IIC, and the effective damping param-\neters are introduced in Sec. IID. The eigenmodes of k\u0019\nskyrmions with k= 1;2;3are compared in Sec. IIIA, the\ninstabilities occurring at low and high field values are dis-\ncussed in connection to magnons with vanishing frequen-\ncies in Sec. IIIB, and the effective damping parameters\nof the different modes are calculated for vanishing and\nhigher values of the Gilbert damping in Secs. IIIC and\nIIID, respectively. A summary is given in Sec. IV.\nII. METHODS\nA. Atomistic model\nThe system is described by the classical atomistic\nmodel Hamiltonian\nH=\u00001\n2X\nhi;jiJSiSj\u00001\n2X\nhi;jiDij(Si\u0002Sj)\n\u0000X\niK(Sz\ni)2\u0000X\ni\u0016sBSi; (1)\nwith theSiunit vectors representing the spins in\na single-layer triangular lattice; J,Dij, andKde-\nnoting nearest-neighbor Heisenberg and Dzyaloshinsky–\nMoriya exchange interactions and on-site magnetocrys-\ntalline anisotropy, respectively; while \u0016sandBstand\nfor the spin magnetic moment and the external mag-\nnetic field. The numerical values of the parameters are\ntaken from Ref. [35], being J= 5:72meV;D=jDijj=\n1:52meV;K= 0:4meV, and\u0016s= 3\u0016B, describing the\nPd/Fe/Ir(111) system. The energy parameters were de-\ntermined based on measuring the field-dependence of 1\u0019\nskyrmion profiles in the system by spin-polarized scan-\nning tunneling microscopy in Ref. [38].\nDuring the calculations the external field Bis ori-\nented along the out-of-plane zdirection. The equilib-\nriumk\u0019skyrmion structures are determined from a rea-\nsonable initial configuration by iteratively rotating the\nspinsSitowards the direction of the effective magnetic\nfieldBeff\ni=\u00001\n\u0016s@H\n@Si. The iteration is performed un-\ntil the torque acting on the spins, Ti=\u0000Si\u0002Beff\ni,\nbecomes smaller at every lattice site than a predefinedvalue, generally chosen to be 10\u00008meV=\u0016B. The calcula-\ntions are performed on a lattice with periodic boundary\nconditions, with system sizes up to 256\u0002256for the\nlargestk\u0019skyrmions in order to avoid edge effects and\nenable the accurate modeling of isolated skyrmions.\nOnce the equilibrium configuration S(0)\niis determined,\nthe spins are rotated to a local coordinate system ~Si=\nRiSiusing the rotational matrices Ri. In the local coor-\ndinatesystemtheequilibriumspindirectionsarepointing\nalong the local zaxis, ~S(0)\ni= (0;0;1). The Hamiltonian\nin Eq. (1) is expanded up to second-order terms in the\nsmall variables ~Sx\ni;~Sy\nias (cf. Ref. [23])\nH\u0019H0+1\n2\u0010\n~S?\u0011T\nHSW~S?\n=H0+1\n2\u0002~Sx~Sy\u0003\u0014A1A2\nAy\n2A3\u0015\u0014~Sx\n~Sy\u0015\n:(2)\nThematrix products areunderstoodtorunoverlattice\nsite indices i, with the matrix components reading\nA1;ij=\u0000~Jxx\nij+\u000eij X\nk~Jzz\nik\u00002~Kxx\ni+ 2~Kzz\ni+\u0016s~Bz\ni!\n;(3)\nA2;ij=\u0000~Jxy\nij\u0000\u000eij2~Kxy\ni; (4)\nA3;ij=\u0000~Jyy\nij+\u000eij X\nk~Jzz\nik\u00002~Kyy\ni+ 2~Kzz\ni+\u0016s~Bz\ni!\n:(5)\nThe energy terms in the Hamiltonian are ro-\ntated to the local coordinate system via ~Jij=\nRi[JI\u0000Dij\u0002]RT\nj;~Ki=RiKRT\nj;and ~Bi=RiB,\nwhereIis the 3\u00023identity matrix, Dij\u0002is the ma-\ntrix describing the vector product with Dij, andKis\nthe anisotropy matrix with the only nonzero element be-\ningKzz=K.\nThe spin wave frequencies are obtained from the lin-\nearized Landau–Lifshitz–Gilbert equation [36, 42]\n@t~S?=\r0\n\u0016s(\u0000i\u001by\u0000\u000b)HSW~S?=DSW~S?;(6)\nwith\u001by=\u0014\n0\u0000iIs\niIs0\u0015\nthe Pauli matrix in Cartesian\ncomponents and acting as the identity matrix Isin the\nlattice site summations. The symbol \r0denotes the gyro-\nmagnetic ratio \r=ge\n2mdivided by a factor of 1+\u000b2, with\ngthe electron gfactor,ethe elementary charge, mthe\nelectron’s mass, and \u000bthe Gilbert damping parameter.\nEquation (6) is rewritten as an eigenvalue equation by\nassuming the time dependence ~S?(t) =e\u0000i!qt~S?\nqand\nperforming the replacement @t!\u0000i!q.\nSince thek\u0019skyrmions represent local energy minima,\nHSWin Eq. (2) is a positive semidefinite matrix. For\n\u000b= 0the!qfrequenciesof DSWarerealandtheyalways\noccurin\u0006!qpairsonthesubspacewhere HSWisstrictly\npositive, for details see, e.g., Ref. [23]. In the following,\nwe will only treat the solutions with Re !q>0, but their3\nRe!q<0pairs are also necessary for constructing real-\nvalued eigenvectors of Eq. (6). The zero eigenvalues are\ndiscussed in Sec. IIC.\nAs is known from previous calculations for 1\u0019\nskyrmions [21–23], the localized excitation modes of k\u0019\nskyrmions are found below the ferromagnetic resonance\nfrequency!FMR =\r\n\u0016s(2K+\u0016sB). During the numerical\nsolution of Eq. (6) these lowest-lying eigenmodes of the\nsparse matrix DSWare determined, as implemented in\nthemontecrystal atomistic spin simulation program\n[43].\nB. Micromagnetic model\nThe atomistic model described in the previous Sec-\ntion enables the treatment of noncollinear spin structures\nwhere the direction of the spins significantly differs be-\ntween neighboring lattice sites. This is especially impor-\ntant when discussing the collapse of k\u0019skyrmions on the\nlattice as was performed in Ref. [35]. Here we will dis-\ncuss the micromagnetic model which on the one hand is\napplicable only if the characteristic length scale of non-\ncollinear structures is significantly larger than the lattice\nconstant, but on the other hand enables a simple classi-\nfication of the spin wave modes.\nThe free energy functional of the micromagnetic model\nis defined as\nH=Z\nAX\n\u000b=x;y;z(rS\u000b)2+K(Sz)2\u0000MBSz\n+D(Sz@xSx\u0000Sx@xSz+Sz@ySy\u0000Sy@ySz)dr;\n(7)\nwhere for the Pd/Fe/Ir(111) system the following pa-\nrameter values were used: A= 2:0pJ/m is the ex-\nchange stiffness,D=\u00003:9mJ/m2is the Dzyaloshinsky–\nMoriya interaction describing right-handed rotation [39],\nK=\u00002:5MJ/m3is the easy-axis anisotropy, and M=\n1:1MA/m is the saturation magnetization.\nThe equilibrium spin structure S(0)=\n(sin \u0002 0cos \b 0;sin \u0002 0sin \b 0;cos \u0002 0)ofk\u0019skyrmions will\nbe cylindrically symmetric, given by \b0(r;') ='+\u0019\ndue to the right-handed rotational sense and\n\u00020(r;') = \u0002 0(r), which is the solution of the\nEuler–Lagrange equation\nA\u0012\n@2\nr\u00020+1\nr@r\u00020\u00001\nr2sin \u0002 0cos \u0002 0\u0013\n+jDj1\nrsin2\u00020\n+Ksin \u0002 0cos \u0002 0\u00001\n2MBsin \u0002 0= 0: (8)\nThe skyrmion order kis encapsulated in the bound-\nary conditions \u00020(0) =k\u0019;\u00020(1) = 0. Equation (8) is\nsolved numerically in a finite interval r2[0;R]signifi-\ncantly larger than the equilibrium k\u0019skyrmion size. A\nfirst approximation to the spin structure is constructed\nbased on the corresponding initial value problem usingthe shooting method [25], then iteratively optimizing the\nstructure using a finite-difference discretization.\nThe spin wave Hamiltonian may be determined anal-\nogously to Eq. (2), by using the local coordinate system\n\u0002 = \u0002 0+~Sx;\b = \b 0+1\nsin \u0002 0~Sy. ThematricesinEqs.(3)-\n(5) are replaced by the operators\nA1=\u00002A\u0012\nr2\u00001\nr2cos 2\u0002 0\u0013\n\u00002jDj1\nrsin 2\u0002 0\n\u00002Kcos 2\u0002 0+MBcos \u0002 0; (9)\nA2= 4A1\nr2cos \u0002 0@'\u00002jDj1\nrsin \u0002 0@'; (10)\nA3=\u00002A\u001a\nr2+\u0014\n(@r\u00020)2\u00001\nr2cos2\u00020\u0015\u001b\n\u00002jDj\u0012\n@r\u00020+1\nrsin \u0002 0cos \u0002 0\u0013\n\u00002Kcos2\u00020+MBcos \u0002 0: (11)\nDue to the cylindrical symmetry of the structure, the\nsolutions of Eq. (6) are sought in the form ~S?(r;';t ) =\ne\u0000i!n;mteim'~S?\nn;m(r), performing the replacements @t!\n\u0000i!n;mand@'!im. For each angular momentum\nquantum number m, an infinite number of solutions in-\ndexed bynmay be found, but only a few of these are\nlocated below !FMR =\r\nM(\u00002K+MB), hence repre-\nsenting localized spin wave modes of the k\u0019skyrmions.\nThe different nquantum numbers typically denote solu-\ntions with different numbers of nodes, analogously to the\nquantum-mechanical eigenstates of a particle in a box.\nBecause of the property HSW(m) =H\u0003\nSW(\u0000m)and\nHSWbeing self-adjoint, the eigenvalues of HSW(m)\nandHSW(\u0000m)coincide, leading to a double degeneracy\napart from the m= 0modes. The\u0006!qeigenvalue pairs\nofDSWdiscussed in Sec. IIA for the atomistic model at\n\u000b= 0in this case can be written as !n;m=\u0000!n;\u0000m.\nHowever, considering only the modes with Re !n;m>0,\none has!n;m6=!n;\u0000mindicating nonreciprocity or an\nenergy difference between clockwise ( m < 0) and coun-\nterclockwise ( m> 0) rotating modes [17, 23].\nFor finding the eigenvectors and eigenvalues of the\nmicromagnetic model, Eq. (6) is solved using a finite-\ndifference method on the r2[0;R]interval. For treat-\ning the Laplacian r2in Eqs. (9) and (11) the improved\ndiscretization scheme suggested in Ref. [44] was applied,\nwhich enables a more accurate treatment of modes with\neigenvalues converging to zero in the infinite and contin-\nuous micromagnetic limit.\nThe spin wave modes of the atomistic model discussed\nin Sec. IIA were assigned the (n;m)quantum numbers,\nwhich are strictly speaking only applicable in the mi-\ncromagnetic limit with perfect cylindrical symmetry, by\nvisualizingthe real-spacestructureofthe numericallyob-\ntained eigenvectors.4\nC. Goldstone modes and instabilities\nSince the translation of the k\u0019skyrmions on the\ncollinear background in the plane costs no energy, the\nspin wave Hamiltonian HSWpossesses two eigenvectors\nbelonging to zero eigenvalue, representing the Goldstone\nmodes of the system. Within the micromagnetic descrip-\ntion of Sec. IIB, these may be expressed analytically as\n[21–23]\n\u0010\n~Sx;~Sy\u0011\n=e\u0000i'\u0012\n\u0000@r\u00020;i1\nrsin \u0002 0\u0013\n;(12)\n\u0010\n~Sx;~Sy\u0011\n=ei'\u0012\n\u0000@r\u00020;\u0000i1\nrsin \u0002 0\u0013\n:(13)\nEquations (12) and (13) represent eigenvectors of the\ndynamical matrix DSWas well. From Eqs. (2) and (6) it\nfollows that the eigenvectors of HSWandDSWbelong-\ning to zero eigenvalue must coincide, HSW~S?=0,\nDSW~S?=0, because (\u0000i\u001by\u0000\u000b)in Eq. (6) is an in-\nvertible matrix. Because from the solutions of the equa-\ntion of motion (6) we will only keep the ones satisfying\nRe!n;m>0, the eigenvectors from Eqs. (12) and (13)\nwill be denoted as the single spin wave mode !0;\u00001= 0.\nSince the eigenvectors and eigenvalues are determined\nnumerically in a finite system by using a discretization\nprocedure, the Goldstone modes will possess a small fi-\nnite frequency. However, these will not be presented\nin Sec. IIIA together with the other frequencies since\nthey represent a numerical artifact. For the 1\u0019and\n3\u0019skyrmions the !0;1eigenmode has a positive fre-\nquency and an eigenvector clearly distinguishable from\nthat of the !0;\u00001translational mode. However, for the\n2\u0019skyrmion both the !0;\u00001and the!0;1eigenfrequen-\ncies ofDSWare very close to zero, and the correspond-\ning eigenvectors converge to Eqs. (12) and (13) as the\ndiscretization is refined and the system size is increased.\nThis can occur because DSWis not self-adjoint and its\neigenvectors are generally not orthogonal. In contrast,\nthe eigenvectors of HSWremain orthogonal, with only a\nsingle pair of them taking the form of Eqs. (12) and (13).\nIn contrast to the Goldstone modes with always zero\nenergy, the sign change of another eigenvalue of HSW\nindicates that the isolated k\u0019skyrmion is transformed\nfrom a stable local energy minimum into an unstable\nsaddle point, leading to its disappearance from the sys-\ntem. Such instabilities were determined by calculating\nthe lowest-lying eigenvalues of HSWin Eq. (2). Due\nto the connection between the HSWandDSWmatrices\nexpressed in Eq. (6), at least one of the precession fre-\nquencies!qwill also approach zero at such an instability\npoint.\nD. Effective damping parameters\nFor finite values of the Gilbert damping \u000b, the spin\nwaves in the system will decay over time as the systemrelaxes to the equilibrium state during the time evolu-\ntion described by the Landau–Lifshitz–Gilbert equation.\nThe speed of the relaxation can be characterized by the\neffective damping parameter, which for a given mode q\nis defined as\n\u000bq;eff=\f\f\f\fIm!q\nRe!q\f\f\f\f: (14)\nAs discussed in detail in Ref. [23], \u000bq;effis mode-\ndependent and can be significantly higher than the\nGilbert damping parameter \u000bdue to the elliptic polar-\nization of spin waves, which can primarily be attributed\nto the noncollinear spin structure of the k\u0019skyrmions.\nFor\u000b\u001c1,\u000bq;effmay be expressed as\n\u000bq;eff\n\u000b=X\ni\f\f\f~S(0);x\nq;i\f\f\f2\n+\f\f\f~S(0);y\nq;i\f\f\f2\nX\ni2Imh\u0010\n~S(0);x\nq;i\u0011\u0003~S(0);y\nq;ii;(15)\nwheretheeigenvectorsinEq.(15)arecalculatedat \u000b= 0\nfrom Eq. (6). Equation (15) may also be expressed by\nthe axes of the polarization ellipse of the spins in mode\nq, see Ref. [23] for details.\nForhighervaluesof \u000b, thecomplexfrequencies !qhave\nto be determined from Eq. (6), while the effective damp-\ning parameters can be calculated from Eq. (14). Also for\nfinite values of \u000bfor each frequency with Re !q>0there\nexists a pair with Re !q0<0such that!q0=\u0000!\u0003\nq[23].\nThe spin waves will be circularly polarized if A1=A3\nandAy\n2=\u0000A2in Eq. (2), in which case the dependence\nof!qon\u000bmay simply be expressed by the undamped\nfrequency!(0)\nqas\nRe!q(\u000b) =1\n1 +\u000b2!(0)\nq; (16)\njIm!q(\u000b)j=\u000b\n1 +\u000b2!(0)\nq: (17)\nThese relations are known for uniaxial ferromagnets;\nsee, e.g., Ref. [45]. In the elliptically polarized modes of\nnoncollinear structures, such as k\u0019skyrmions, a devia-\ntion from Eqs. (16)-(17) is expected.\nIII. RESULTS\nA. Eigenmodes\nThe frequencies of the localized spin wave modes of the\n1\u0019,2\u0019, and 3\u0019skyrmion, calculated from the atomistic\nmodel for\u000b= 0as described in Sec. IIA, are shown in\nFig. 1. For the 1\u0019skyrmion six localized modes can be\nobserved below the FMR frequency of the field-polarized\nbackground in Fig. 1(a), four of which are clockwise ro-\ntating modes ( m < 0), one is a gyration mode rotating\ncounterclockwise ( m= 1), while the final one is a breath-\ning mode (m= 0). The excitation frequencies show good5\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175\n(b)\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175\n(c)\nFIG. 1. Frequencies of localized spin wave modes at \u000b= 0for\n(a) the 1\u0019, (b) the 2\u0019, and (c) the 3\u0019skyrmion. Selected spin\nwave modes are visualized in contour plots of the out-of-plane\nspin component and denoted by open symbols connected by\nlines in the figure, the remaining modes are denoted by con-\nnected dots.quantitative agreement with the ones calculated from the\nmicromagnetic model for the same system in Ref. [23].\nCompared to Ref. [21], the additional appearance of the\neigenmodes with m= 1;\u00004;\u00005can be attributed to the\nfinite value of the anisotropy parameter Kin the present\ncase. Increasing the anisotropy value makes it possible\nto stabilize the skyrmions at lower field values, down to\nzerofieldatthecriticalvalueinthemicromagneticmodel\njKcj=\u00192D2\n16A, where the transition from the spin spiral\nto the ferromagnetic ground state occurs at zero exter-\nnal field [46]. Since the excitation frequencies decrease\nat lower field values as shown in Fig. 1(a), this favors\nthe appearance of further modes. Simultaneously, the\nFMR frequency increases with K, meaning that modes\nwith higher frequencies become observable for larger uni-\naxial anisotropy. For each angular momentum quantum\nnumberm, only a single mode ( n= 0) appears.\nIn the case of the 2\u0019skyrmion an increased number of\neigenmodes may be seen in Fig. 1(b). This can mainly be\nattributed to the appearance of spin waves with higher\nangular momentum quantum numbers both for clockwise\n(up tom=\u000017) and counterclockwise (up to m= 12)\nrotational directions. Furthermore, in this case modes\nwithn= 1node in the eigenfunction can be observed\nas well. The same trend continues in the case of 3\u0019\nskyrmionsinFig.1(c), thelargenumberofinternaleigen-\nmodes can be attributed to angular momentum quantum\nnumbers ranging from m=\u000022tom= 16, as well as to\nspin wave eigenvectors with up to n= 2nodes. The dif-\nferent rotational directions and numbers of nodes are il-\nlustrated in Supplemental Videos 1-4 [47] via the square-\nshaped modes ( n= 0;1,m=\u00064) of the 3\u0019skyrmion at\nB= 0:825T.\nThe increase of possible angular momentum quantum\nnumbers for higher skyrmion order kas well as for de-\ncreasing magnetic field Bmay be qualitatively explained\nby an increase in the skyrmion size. Modes with a given\nvalue ofmindicate a total of jmjmodulation periods\nalong the perimeter of the skyrmion; for larger skyrmion\nsizes this corresponds to a modulation on a longer length\nscale, which has a smaller cost in exchange energy.\nThe breathing modes of the 3\u0019skyrmion with dif-\nferent numbers of nodes are visualized in Fig. 2 at\nB= 1T. The results shown in Fig. 2 are obtained\nfrom the micromagnetic model in Sec. IIB, which is in\ngood quantitative agreement with the atomistic calcu-\nlations at the given field. All the eigenmodes display\nthree peaks of various heights, while they decay expo-\nnentially outside the 3\u0019skyrmion. As can be seen in\nFig. 2, the peaks are localized roughly around the re-\ngions where the spins are lying in-plane, indicated by\nthe domain walls (DW) between pairs of dashed lines.\nThe widths of the domain walls were determined by ap-\nproximating the 3\u0019skyrmion profile with linear func-\ntions close to the inflection points rj;\u00020;j;j= 1;2;3\nwhere the spins are lying in-plane, and calculating where\nthese linear functions intersect integer multiples of \u0019in6\n0 10 20 30 40 50-2-023\n-0.100.000.10\nFIG. 2. Comparison between the 3\u0019skyrmion profile (left\nvertical axis) and the eigenvectors of the breathing modes\n(m= 0) with different numbers of nodes n= 0;1;2(right\nvertical axis). The calculations were performed using the mi-\ncromagnetic model described in Sec. IIB at B= 1T, the\nlattice constant is a= 0:271nm. Double arrows between ver-\ntical dashed lines indicate the extensions of the domain walls\nin the structure.\n\u00020. Thus, the domain walls are located between the in-\nnerRin;j=rj+[@r\u00020(rj)]\u00001[(4\u0000j)\u0019\u0000\u00020;j]and outer\nRout;j=rj+ [@r\u00020(rj)]\u00001[(3\u0000j)\u0019\u0000\u00020;j]radii. Such\na description was used to calculate the skyrmion radius\nin, e.g., Ref. [46], and it was also applied for calculating\nthe widths of planar domain walls [48].\nThe nodes of the eigenmodes are located roughly be-\ntween these domain walls, meaning that typically excita-\ntion modes with n= 0;:::;k\u00001nodes may be observed\nink\u0019skyrmions, in agreement with the results in Fig. 1.\nA higher number of nodes would require splitting a single\npeak into multiple peaks, the energy cost of which gen-\nerally exceeds the FMR frequency, thereby making these\nmodes unobservable. The sign changes in the ~Sx\nn;meigen-\nvectors mean that the different modes can be imagined\nas the domain walls breathing in the same phase or in\nopposite phase, as can be seen in Supplemental Videos\n5-7 [47]. Note that eigenmodes with higher nquantum\nnumbers may also be observed for skyrmions confined in\nnanodots [14–16] where the peaks of the eigenmodes may\nalso be localized at the edge of the sample, in contrast\nto the present case where isolated k\u0019skyrmions are dis-\ncussed on an infinite collinear background.\nItisalsoworthnotingthatthelowest-lyingnonzerogy-\nration mode is n= 0;m= 1for the 1\u0019and3\u0019skyrmions,\nwhile it isn= 1;m= 1for the 2\u0019skyrmion, see Fig. 1.\nAs already mentioned in Sec. IIC, numerical calculations\nfor the 2\u0019skyrmion indicate both in the atomistic and\nthemicromagneticcasethatbyincreasingthesystemsize\nor refining the discretization the eigenvectors of both the\nn= 0;m=\u00001and then= 0;m= 1modes ofDSW\nin Eq. (6) converge to the same eigenvectors in Eqs. (12)\nand(13)and 0eigenvalue, whichcorrespondtothetrans-\nlational Goldstone mode in the infinite system. This dif-ference can probably be attributed to the deviation in\nthe value of the topological charge, being finite for 1\u0019\nand3\u0019skyrmions but zero for the 2\u0019skyrmion [35].\nB. Instabilities\nSkyrmions with different order kdeviate in their low-\nfield behavior. Since the considered Pd/Fe/Ir(111) sys-\ntem has a spin spiral ground state [38], decreasing the\nmagnetic field value will make the formation of domain\nwallsenergeticallypreferableinthesystem. Inthecaseof\nthe1\u0019skyrmion this means that the lowest-lying eigen-\nmode ofHSWin Eq. (2), which is an elliptic mode with\nm=\u00062, changes sign from positive to negative, occur-\nring between B= 0:650T andB= 0:625T in the present\nsystem. This is indicated in Fig. 1(a) by the fact that the\nfrequency of the n= 0;m=\u00002eigenmode of DSWin\nEq. (6) converges to zero. This leads to an elongation of\nthe skyrmion into a spin spiral segment which gradually\nfills the ferromagnetic background, a so-called strip-out\nor elliptic instability already discussed in previous publi-\ncations [21, 46]. In contrast, for the 2\u0019and3\u0019skyrmions\nthe lowest-lying eigenmode of HSWis a breathing mode\nwithm= 0, which tends to zero between B= 0:800T\nandB= 0:775Tforbothskyrmions. Thisisindicatedby\nthe lowest-lying n= 0;m= 0mode ofDSWin Fig. 1(b)\nfor the 2\u0019skyrmion, which is the second lowest after the\nn= 0;m= 1mode for the 3\u0019skyrmion in Fig. 1(c). This\nmeans that the radius of the outer two rings of 2\u0019and\n3\u0019skyrmions diverges at a finite field value, leading to a\nburst instability. Such a type of instability was already\nshown to occur in Ref. [25] in the case of a ferromagnetic\nground state at negative field values, in which case it also\naffects 1\u0019skyrmions.\nAt the burst instability, modes with n= 0and all\nangular momentum quantum numbers mappear to ap-\nproach zero because of the drastic increase in skyrmion\nradius decreasing the frequency of these modes as dis-\ncussed in Sec. IIIA. A similar effect was observed for the\n1\u0019skyrmion in Ref. [22] when the critical value of the\nDzyaloshinsky–Moriya interaction, jDcj=4\n\u0019p\nAjKj, was\napproached at zero external field from the direction of\nthe ferromagnetic ground state. In contrast, the ellip-\ntic instability only seems to affect the n= 0;m=\u00002\nmode, while other mvalues and the nonreciprocity are\napparently weakly influenced.\nIn the atomistic model, skyrmions collapse when their\ncharacteristic size becomes comparable to the lattice\nconstant. For the 1\u0019,2\u0019, and 3\u0019skyrmions the col-\nlapse of the innermost ring occurs at Bc;1\u0019\u00194:495T,\nBc;2\u0019\u00191:175T, andBc;3\u0019\u00191:155T, respectively [35].\nAs can be seen in Figs. 1(b), 1(c), and 3, this instabil-\nity is again signaled by the n= 0;m= 0eigenfrequency\ngoing to zero, but in contrast to the burst instability,\nthe other excitation frequencies keep increasing with the\nfield in this regime. Figure 3 demonstrates that close to\nthe collapse field the excitation frequency may be well7\n4.45 4.46 4.47 4.48 4.49 4.50020406080100\nFIG. 3. Frequency of the breathing mode n= 0;m = 0of\nthe 1\u0019skyrmion close to the collapse field. Calculation data\nare shown by open symbols, red line denotes the power-law\nfitf0;0=Af(Bc;1\u0019\u0000B)\ff.\napproximated by the power law f0;0=Af(Bc;1\u0019\u0000B)\ff,\nwithAf= 175:6GHz\nT\ff,Bc;1\u0019= 4:4957T, and\ff= 0:23.\nC. Effective damping parameters in the limit of\nlow\u000b\nThe effective damping parameters \u000bn;m;effwere first\ncalculated from the eigenvectors obtained at \u000b= 0fol-\nlowing Eq. (15). The results for the 1\u0019,2\u0019, and 3\u0019\nskyrmions are summarized in Fig. 4. As discussed in\nRef. [23], the \u000bn;m;effvalues are always larger than the\nGilbert damping \u000b, and they tend to decrease with in-\ncreasing angular momentum quantum number jmjand\nmagnetic field B. The spin wave possessing the high-\nest effective damping is the n= 0;m= 0breathing\nmode both for the 1\u0019and 2\u0019skyrmion, but it is the\nn= 0;m= 1gyration mode for the 3\u0019skyrmion for a\nlarge part of the external field range where the struc-\nture is stable. Excitation pairs with quantum num-\nbersn;\u0006mtend to decay with similar \u000bn;m;effvalues to\neach other, with \u000bn;jmj;eff< \u000bn;\u0000jmj;eff, where clockwise\nmodes (m < 0) have lower frequencies and higher effec-\ntive damping due to the nonreciprocity.\nThe effective damping parameters drastically increase\nand for the lowest-lying modes apparently diverge close\nto the burst instability, while no such sign of nonan-\nalytical behavior can be observed in the case of the\n1\u0019skyrmion with the elliptic instability. For the same\nn;mmode, the effective damping parameter tends to in-\ncrease with skyrmion order kaway from the critical field\nregimes; for example, for the n= 0;m= 0mode at\nB= 1:00T one finds \u000b0;0;eff;1\u0019= 2:04,\u000b0;0;eff;2\u0019= 5:87,\nand\u000b0;0;eff;3\u0019= 10:09.\nClose to the collapse field, the effective damping pa-\nrameter of the n= 0;m= 0breathing mode tends to\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.5\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100FIG.4. Effectivedamping parameters calculatedaccordingto\nEq. (15) for the eigenmodes of the (a) 1\u0019, (b) 2\u0019, and (c) 3\u0019\nskyrmions, plotted on a logarithmic scale. The corresponding\nexcitation frequencies are shown in Fig. 1.\ndiverge as shown in Figs. 4(b), 4(c), and 5 for the 2\u0019,\n3\u0019, and 1\u0019skyrmions, respectively. Similarly to the\neigenfrequency converging to zero in Fig. 3, the criti-\ncal behavior of the effective damping may be approxi-\nmated by a power-law fit \u000b0;0;eff=A\u000b(Bc;1\u0019\u0000B)\u0000\f\u000b\nas shown in Fig. 5, this time with a negative exponent8\n4.45 4.46 4.47 4.48 4.49 4.50024681012\nFIG. 5. Effective damping parameter \u000b0;0;effof the breathing\nmoden= 0;m= 0of the 1\u0019skyrmion close to the collapse\nfield. The corresponding excitation frequencies are shown in\nFig. 3. Calculation data are shown by open symbols, red line\ndenotes the power-law fit \u000b0;0;eff=A\u000b(Bc;1\u0019\u0000B)\u0000\f\u000b.\ndue to the divergence. The fitting yields the parameters\nA\u000b= 0:96T\f\u000b,Bc;1\u0019= 4:4957T, and\f\u000b= 0:23. Natu-\nrally, the critical field values agree between the two fits,\nbut interestingly one also finds \ff=\f\u000bup to two digits\nprecision. Rearranging Eq. (14) yields\n\u000b0;0;eff\n\u000bRe!0;0=1\n\u000bjIm!0;0j; (18)\nwhere the left-hand side is proportional to\n(Bc;1\u0019\u0000B)\ff\u0000\f\u000bwhich is approximately constant\ndue to the exponents canceling. This indicates that\nwhile Re!0;0diverges close to the collapse field,\njIm!0;0j=\u000bremains almost constant at low \u000bvalues.\nD. Damping for higher \u000bvalues\nDue tothe divergences oftheeffective damping param-\neters found at the burst instability and collapse fields, it\nis worthwhile to investigate the consequences of using a\nfinite\u000bvalue in Eq. (6), in contrast to relying on Eq. (15)\nwhich is determined from the eigenvectors at \u000b= 0. The\n\u000bdependence of the real and imaginary parts of the !0;0\nbreathingmodefrequencyofthe 1\u0019skyrmionisdisplayed\nin Fig. 6, at a field value of B= 1T far from the el-\nliptic and collapse instabilities. As shown in Fig. 6(a),\nunlike circularly polarized modes described by Eq. (16)\nwhere Re!qdecreases smoothly and equals half of the\nundamped value at \u000b= 1, the Re!0;0value for the ellip-\ntically polarized eigenmode displays a much faster decay\nand reaches exactly zero at around \u000b\u00190:58. According\ntoEq.(14), thisindicatesthatthecorrespondingeffective\ndamping parameter \u000b0;0;effdiverges at this point.\nSince the real part of the frequency disappears, the\n!q0=\u0000!\u0003\nqrelation connecting Re !q>0and Re!q0<0\n0.0 0.2 0.4 0.6 0.8 1.00102030405060\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6FIG. 6. (a) Frequency f0;0=Re!0;0=2\u0019and (b) inverse\nlifetimejIm!0;0jof then= 0;m = 0breathing mode of\nthe 1\u0019skyrmion at B= 1T as a function of the Gilbert\ndamping parameter \u000b. The solutions of Eq. (6) for the ellip-\ntically polarized eigenmode of the 1\u0019skyrmion are compared\nto Eqs. (16)-(17) which are only valid for circularly polarized\nmodes.\nsolutions of Eq. (6) discussed in Sec. IID no longer holds,\nand two different purely imaginary eigenfrequencies are\nfound in this regime as shown in Fig. 6(b). This is analo-\ngous to overdamping in a classical linear harmonic oscil-\nlator, meaningthatthepurelyprecessionalfirst-orderdif-\nferential equation describing circularly polarized modes\nis transformed into two coupled first-order differential\nequations [23] with an effective mass term for the breath-\ning mode of k\u0019skyrmions. This implies that when per-\nforming spin dynamics simulations based on the Landau–\nLifshitz–Gilbert equation, the value of the Gilbert damp-\ning parameter has to be chosen carefully if the fastest\nrelaxation to the equilibrium spin structure is required.\nThe high effective damping of the breathing mode in the\n\u000b\u001c1limit (cf. Fig. 4(a)) ensures that the inverse\nlifetime of the elliptically polarized excitations remains\nlarger for a wide range of \u000bvalues in Fig. 6(b) than what\nwould be expected for circularly polarized modes based9\n0.85 0.90 0.95 1.00 1.05 1.10 1.1505101520\n0.85 0.90 0.95 1.00 1.05 1.10 1.150.000.020.040.060.080.10\nFIG. 7. (a) Frequency f0;0=Re!0;0=2\u0019and (b) inverse\nlifetimejIm!0;0jof then= 0;m= 0breathing mode of the\n2\u0019skyrmion at \u000b= 0:1as a function of the external magnetic\nfieldB. The solutions of Eq. (6) for the elliptically polarized\neigenmode of the 2\u0019skyrmion are compared to Eqs. (16)-(17)\nwhich are only valid for circularly polarized modes.\non Eq. (17). Note that contrary to Sec. IIIB, Re !0;0be-\ncomingzeroinFig.6(a)doesnotindicateaninstabilityof\nthe system, since stability is determined by the eigenval-\nues of the matrix HSWin Eq. (2) which are independent\nof\u000b.\nSince the disappearance of Re !0;0and the bifurcation\nof Im!0;0occurs as the excitation frequency becomes\nsmaller, it is expected that such an effect may also be ob-\nservedatafixed \u000bvalueastheexternalfieldisdecreased.\nThis is illustrated for the n= 0;m= 0breathing mode\nof the 2\u0019skyrmion in Fig. 7 at \u000b= 0:1. For this interme-\ndiate value of the damping, the breathing mode becomes\noverdamped around B= 0:875T, which is significantly\nhigher than the burst instability between B= 0:775T\nandB= 0:800T (cf. Fig. 1(b) and the circularly polar-\nized approximation in Fig. 7(a)). This means that the\nlowest-lying breathing mode of the 2\u0019skyrmion cannot\nbe excited below this external field value. In Fig. 7(b) it\ncan be observed that contrary to the circularly polarizedapproximation Eq. (17) following the field dependence of\nthe frequency, for the actual elliptically polarized eigen-\nmodejIm!0;0jisalmostconstantforallfieldvaluesabove\nthebifurcationpoint. Althoughasimilarobservationwas\nmade at the end of Sec. IIIC as the system approached\nthe collapse field at \u000b= 0, it is to be emphasized again\nthat no instability occurs where Re !0;0disappears in\nFig. 7(a).\nIV. CONCLUSION\nIn summary, the localized spin wave modes of k\u0019\nskyrmions were investigated in an atomistic spin model,\nwith parameters based on the Pd/Fe/Ir(111) system. It\nwas found that the number of observable modes increases\nwith skyrmion order k, firstly because of excitations with\nhigher angular momentum quantum numbers mforming\nalong the larger perimeter of the skyrmion, secondly be-\ncause of nodes appearing between the multiple domain\nwalls. It was found that the 2\u0019and3\u0019skyrmions un-\ndergo a burst instability at low fields, in contrast to the\nelliptic instability of the 1\u0019skyrmion. At high field val-\nues the innermost ring of the structure collapses in all\ncases, connected to an instability of a breathing mode.\nThe effective damping parameters of the excitation\nmodes were determined, and it was found that for the\nsamen;mmode they tend to increase with skyrmion\norderk. The effective damping parameter of the n=\n0;m= 0breathing mode diverges at the burst and\ncollapse instabilities, but no such effect was observed\nin case of the elliptic instability. For higher values of\nthe Gilbert damping parameter \u000ba deviation from the\nbehavior of circularly polarized modes has been found,\nwith the breathing modes becoming overdamped. It was\ndemonstrated that such an overdamping may be observ-\nable in 2\u0019and3\u0019skyrmions for intermediate values of\nthe damping significantly above the burst instability field\nwhere the structures themselves disappear from the sys-\ntem.\nThe results presented here may motivate further ex-\nperimental and theoretical studies on k\u0019skyrmions, of-\nfering a wider selection of localized excitations compared\nto the 1\u0019skyrmion, thereby opening further possibilities\nin magnonics applications.\nACKNOWLEDGMENTS\nThe authors would like to thank A. Siemens for fruit-\nful discussions. Financial support for this work from the\nAlexander von Humboldt Foundation, from the Deutsche\nForschungsgemeinschaft via SFB 668, from the European\nUnion via the Horizon 2020 research and innovation pro-\ngram under Grant Agreement No. 665095 (MAGicSky),\nand from the National Research, Development and Inno-\nvation Office of Hungary under Project No. K115575 is\ngratefully acknowledged.10\n[1] A. N. Bogdanov and D. A. Yablonski ˘i, Sov. Phys. JETP\n68, 101 (1989).\n[2] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[3] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B.\nJungfleisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L.\nWang, Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis,\nNat. Phys. 13, 162 (2017).\n[4] P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von\nBergmann, and R. Wiesendanger, Nat. Nanotechnol. 12,\n123 (2017).\n[5] F. Büttner, I. Lemesh, M. Schneider, B. Pfau, C. M.\nGünther, P. Hessing, J. Geilhufe, L. Caretta, D. Engel,\nB. Krüger, J. 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Schäfer, Magnetic Domains (Springer,\nBerlin, 1998)."    },    {        "title": "1201.3553v1.Magnetic_vortex_echoes__application_to_the_study_of_arrays_of_magnetic_nanostructures.pdf",        "content": "arXiv:1201.3553v1  [cond-mat.mes-hall]  17 Jan 2012Magnetic vortex echoes: application to the study of arrays o f magnetic nanostructures\nF. Garcia1, J.P. Sinnecker2, E.R.P. Novais2, and A.P. Guimar˜ aes2∗\n1Laborat´ orio Nacional de Luz S´ ıncrotron, 13083-970, Camp inas, SP, Brazil and\n2Centro Brasileiro de Pesquisas F´ ısicas, 22290-180, Rio de Janeiro, RJ, Brazil\n(Dated: December 13, 2018)\nWe propose theuse ofthe gyrotropic motion of vortexcores in nanomagnets toproduce amagnetic\necho, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g.,\nnanodisks, is magnetized with an in-plane ( xy) field, and after a time τa field pulse inverts the\ncore magnetization; the echo is a peak in Mxyatt= 2τ. Its relaxation times depend on the\ninhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant α.\nIts feasibility is demonstrated using micromagnetic simul ation. To illustrate an application of the\nechoes, we have determined the inhomogeneity and measured t he magnetic interaction in an array\nof nanodisks separated by a distance d, finding a d−ndependence, with n≈4.\nPACS numbers: 75.70.Kw,75.78.Cd,62.23.Eg,76.60.Lz\nThe interest in magnetic vortices and their properties\nand applications has grown steadily in the last years[1–\n4]. Vortices have been observed, for example, in disks\nand ellipses having sub-micron dimensions[5]. More re-\ncently, the question of the intensity of the coupling be-\ntweenneighbordiskswith magneticvortexstructureshas\nattracted an increasing interest[6–9].\nAt thevortexcorethe magnetizationpointsperpendic-\nularlyto the plane: this characterizesits polarity, p= +1\nfor the + zdirection and p=−1 for−z. The direc-\ntion of the moments in the vortex defines the circulation:\nc=−1 for clockwise (CW) direction, and +1 for CCW.\nIf removed from the equilibrium position at the center\nof the nanodisk by, for example, an in-plane field, and\nthen left to relax, a vortex core will perform a gyrotropic\nmotion, with angular frequency ω, given for thin disks[3]\nbyωG≈(20/9)γMsβ(β=h/Ris the aspect ratio)[10].\nWe propose in this paper that, manipulating the dy-\nnamic properties of the vortex in an analogous way as it\nis done in Nuclear Magnetic Resonance (NMR), a new\nphenomenon results, the magnetic vortex echo (MVE),\nsimilar to the spin echo observed in NMR[11]. This new\necho may provide information on fundamental properties\nof arrays of nanodisks, e.g., their inhomogeneity and in-\nteractions. Despite the fact that applications of vortices\nnecessarilyinvolvearrays,mostoftherecentpublications\ndeal with the analysis of individual nanodisks or arrays\nwith a few elements. Therefore, the possibility of char-\nacterizing large arrays is of much interest. In this paper\nwe have examined the motion of vortex cores in an ar-\nray of nanometric disks under the influence of a pulsed\nmagnetic field, using micromagnetic simulation.\nLet us consider an array of nanodisks where the vor-\ntex cores precess with a distribution of angular frequen-\ncies centered on ω0, of width ∆ ω, arising from any type\nof inhomogeneity (see note [10]). We assume that the\nfrequencies vary continuously, and have a Gaussian dis-\ntribution P(ω) with mean square deviation ∆ ω. To sim-\nplify we can assume that the polarization of every vortex\nFIG. 1. (Color online) Diagram showing the formation of\nmagnetic vortex echoes; the disks are described from a refer -\nence frame that turns with the average translation frequenc y\nω0: a) disks with in-plane magnetizations Mialong the same\ndirection (defined by the white arrows); b) after a time τ\nthe disks on the left, center and right have turned with fre-\nquencies, respectively, lower, higher and equal to ω0; c) the\npolarities of the vortex cores are reversed, and the ωiof the\nvortex cores (and of the Mi) change sign, and d) after a sec-\nond interval τthe cores (and Mi) are again aligned, creating\nthe echo.\nis the same: pi= +1. This is not necessary for our argu-\nment, but, if required, the system can be prepared; see\nref. [12] and the references therein.\nSince the direction of rotation of the magnetic vortex\ncores after removal of the in-plane field is defined exclu-\nsively by p, all the cores will turn in the same direction;\nas the vortex core turns, the in-plane magnetization of\nthe nanodot also turns.2\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s45/s49/s48/s49/s45/s49/s48/s49/s45/s49/s48/s49/s45/s49/s48/s49\n/s32/s32\n/s116/s32/s40/s110/s115/s41/s32/s61/s32/s49/s48/s32/s110/s109 /s44/s32 /s61/s32/s50/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s46/s48/s48/s53/s77/s32/s40/s49/s48/s45/s49/s52\n/s65/s47/s109/s41\n/s32/s61/s32/s50/s48/s32/s110/s109/s44/s32 /s61/s32/s50/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s32/s61/s32/s50/s48/s32/s110/s109 /s44/s32 /s61/s32/s49/s48/s32/s110/s115 /s44/s32 /s61/s32/s52/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s46/s48/s48/s49\n/s100/s99/s98/s97/s32/s61/s32/s49/s48/s32/s110/s109 /s44/s32 /s61/s32/s51/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48\n/s32\nFIG. 2. (Color online) Micromagnetic simulation of magneti c\nvortex echoes, for 100 nanodisks, with d= infinity, and a)\nσ= 10nm, τ= 30ns, α= 0; b) σ= 20nm, τ= 10ns\nandτ= 40ns (two pulses), and α= 0.001; c)σ= 20nm,\nτ= 20ns, α= 0; d)σ= 10nm, τ= 20ns, α= 0.005. The\ninversion pulses ( Bz=−300mT) are also shown (in red).\n/s48/s46/s48/s48/s48 /s48/s46/s48/s48/s49 /s48/s46/s48/s48/s50 /s48/s46/s48/s48/s51 /s48/s46/s48/s48/s52 /s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s32/s32/s49/s47/s84\n/s50/s32/s40/s110/s115/s45/s49\n/s41\n/s68/s97/s109/s112/s105/s110/s103/s32/s99/s111/s110/s115/s116/s97/s110/s116/s32\nFIG. 3. (Color online) Variation of the inverse of the re-\nlaxation times T2(diamonds) obtained by fitting the curves\nof echo intensity versus time interval τtoM0exp(−τ/T2),\nas a function of α, for simulations made with D= 250nm,\nσ= 10nm, separation infinite; the continuous line is a linear\nleast squares fit.\nLet us assume that all the vortex cores have been dis-\nplaced from their equilibrium positions alongthe positive\nxaxis[13], by a field B. The total in-plane magnetiza-\ntion (that is perpendicular to the displacement of the\ncore) will point along the yaxis, therefore forming an\nangleθ0= 0 att= 0. Using the approach employed in\nthe description of magnetic resonance (e.g., see [14, 15]),one can derive the total in-plane magnetization:\nMy(t) =My(0)/integraldisplay∞\n−∞e/bracketleftBig\n−1\n2(ω−ω0)2\n∆ω2/bracketrightBig\n∆ω√\n2πcos(ωt)dω(1)\nan integral that is[16] the Fourier transform of the func-\ntionP(ω); usingT∗\n2= 1/∆ω:\nMy(t) =My(0)exp/parenleftbigg\n−1\n2t2\nT∗2\n2/parenrightbigg\ncos(ω0t) (2)\nThis result shows that the total magnetization tends to\nzero, as the different contributions to My(t) get gradu-\nally out of phase. This decay is analogous to the free\ninduction decay (FID) in NMR; its characteristic time is\nT∗\n2= 1/∆ω.\nAfter a time τ, the angle rotated by each vortex core\nwill beωτ; if att=τwe invert the polarities of the vor-\ntices in the array, using an appropriate pulse, the motion\nof the cores will change direction (i.e., ω→ −ω), and one\nobtains:\nMy(t−τ) =My(0)/integraldisplay∞\n−∞e−1\n2(ω−ω0)2\n∆ω2\n∆ω√\n2πcos[ω(τ−t)]dω(3)\nThe magnetization at a time t > τis then:\nMy(t) =My(0)e[−1\n2(t−2τ)2\nT∗2\n2]e(−t−τ\nT2)cos(ω0t) (4)\nThis means that the magnetization component My(t) in-\ncreases for τ < t < 2τ, reaching a maximum at a time\nt= 2τ: this maximum is the magnetic vortex echo , anal-\nogous to the spin echo observed in magnetic resonance,\nwhich has important applications in NMR, including in\nMagnetic Resonance Imaging (MRI)[14, 15] (Fig. 1). In\nthe case of the NMR spin echo, the maximum arises from\nthe refocusing of the in-plane components of the nuclear\nmagnetization.\nIn Eq. 4 we have included a relaxation term contain-\ning the time constant T2- also occurring in NMR -, to\naccount for a possible decay of the echo amplitude with\ntime; its justification will be given below.\nIn the arrayof nanodisks, there will be in principle two\ncontributionsto the defocusing ofthe magnetization, i.e.,\ntwo mechanisms for the loss of in-plane magnetization\nmemory: 1) the spread in values of βandH(see note\n[10]), producing an angular frequency broadening term\n∆ω, and 2) irreversible processes that are characterized\nby a relaxation time T2: thus 1/T∗\n2= ∆ω+1/T2.\nThe second contribution is the homogeneous term\nwhose inverse, T2, is the magnetic vortex transverse re-\nlaxationtime, analogousto thespin transverserelaxation\ntime (or spin-spin relaxation time) T2in magnetic reso-\nnance. The irreversible processes include a) the interac-\ntion between the disks, which amounts to random mag-\nnetic fields that will increase or decrease ωof a given3\ndisk, producing a frequency spread of width ∆ ω′= 1/T′\n2,\nand b) the loss in magnetization (of rate 1 /Tα) arising\nfrom the energy dissipation related to the Gilbert damp-\ning constant α. Identifying Tαto the NMR longitudinal\nrelaxation time T1, one has [14]: 1 /T2= 1/T′\n2+1/2Tα.\nTherefore the relaxation rate 1 /T∗\n2is given by:\n1\nT∗\n2= ∆ω+1\nT2= ∆ω+1\nT′\n2+1\n2Tα(5)\n1/T∗\n2is therefore the total relaxation rate of the in-plane\nmagnetization, composed of a) ∆ ω, the inhomogeneity\nterm, and b) 1 /T2, the sum ofall the other contributions,\ncontaining1 /T′\n2, duetotheinteractionbetweenthedisks,\nand 1/Tα, the rate of energy decay. The vortex cores\nwill reach the equilibrium position at r= 0 after a time\nt∼Tα, therefore there will be no echo for 2 τ≫Tα.\nThe vortex echo maximum at t= 2τ, from Eq. 4, is\nMy(2τ)∝exp(−τ/T2); one should therefore note that\nthe maximum magnetization recovered at a time 2 τde-\ncreases exponentially with T2, i.e., this maximum is only\naffected by the homogeneous part of the total decay rate\ngiven by Eq. 5. In other words, the vortex echo cancels\nthe loss in Mydue to the inhomogeneity ∆ ω, but it does\nnot cancel the decrease in Mydue to the interaction be-\ntween the nanodisks (the homogeneous relaxation term\n1/T′\n2), or due to the energy dissipation (term 1 /2Tα).\nNote also that if one attempted to estimate the inho-\nmogeneityofan arrayofnanodotsusing anothermethod,\nfor example, measuring the linewidth of a FMR spec-\ntrum, one would have the contribution of this inhomo-\ngeneity together with the other terms that appear in Eq.\n5, arising from interaction between the dots and from\nthe damping. On the other hand, measuring the vor-\ntex echo it would be possible to separate the intrinsic\ninhomogeneity from these contributions, since T2can be\nmeasured separately, independently of the term ∆ ω.T2\ncan be measured by determining the decay of the echo\namplitude for different values of the interval τ.\nThe Fourier transform of either the vortex free induc-\ntiondecayorthetimedependenceoftheecho My(t) gives\nthe distribution of gyrotropic frequencies P(ω).\nFor the experimental study of vortex echoes, the se-\nquence of preparation (at t=0) and inversion fields (at\nt=τ) should of course be repeated periodically, with a\nperiodT≫Tα. As in pulse NMR, this will produce\nechoes on every cycle, improving the S/N ratio of the\nmeasured signals. Also note that the time T∗\n2can be ob-\ntained either from the initial decay (FID, Eq. 2) or from\nthe echo (Eq. 4), but T2can only be obtained from the\nMVE.\nIn order to demonstrate the MVE, we have performed\nmicromagnetic simulations of an assembly of 100 mag-\nnetic nanodisks employing the OOMMF code[17]. The\nsimulated system was a square array of 10 ×10 disks,\nthickness 20nm, with distance dfrom center to center.\nIn order to simulate the inhomogeneity of the system,we have introduced a Gaussian distribution of diame-\nters, centered on 250nm and mean square deviation σ;\nσ= 10nm corresponds to ∆ ω≈1.6×108s−1. The\ndisks were placed at random on the square lattice. The\ninitial state of the disks ( p= +1 and c=−1) was pre-\npared by applying an in-plane field of 25mT; the po-\nlarity was inverted with a Gaussian pulse of amplitude\nBz=−300mT, with width 100ps. The results for the\ncased=∞were simulations made on the disks one at a\ntime, adding the individual magnetic moments µi(t).\nWe have successfully demonstrated the occurrence of\nthe magnetic vortex phenomenon, and have shown its\npotential as a characterization technique. The simula-\ntions have confirmed the occurrence of the echoes at the\nexpected times ( t= 2τ). For different values of σ, the\nT∗\n2time, and consequently the duration of the FID and\nthe width of the echo are modified (Fig. 2a, 2c); increas-\ningαresults in a faster decay of the echo intensity as a\nfunction of time (Fig. 2a, 2d). We have also obtained\nmultiple echoes, by exciting the system with two pulses\n(Fig. 2b)[18].\nFig. 3 shows the dependence of T2onαforσ= 10nm;\nessentially the same result is obtained for σ= 20nm,\nsinceT2does not depend on ∆ ω(Eq. 5). Taking a linear\napproximation, 1 /T2=Aα, and since for d=infinity\nthere is no interaction between the disks, 1 /T2= 1/2Tα,\nand therefore:\n1\nTα= 2Aα (6)\nFrom the least squares fit (Fig. 3), A= 1.6×1010s−1.\nThis relation can be used to determine experimentally α,\nmeasuring T2with vortex echoes, for an array of well-\nseparated disks.\nRecently some workers have analyzed the important\nproblem of the interaction between disks exhibiting mag-\nnetic vortices, obtaining that it varies with a d−ndepen-\ndence: Vogel and co-workers [6], using FMR, obtained\nfor a 4×300 array a dependence of the form d−6, the\nsame found by Sugimoto et al. [8] using a pair of disks\nexcited with rf current. Jung et al. [7] studying a pair of\nnanodisks with time-resolved X-ray spectroscopy, found\nn= 3.91±0.07 and Sukhostavets et al. [9], also for\na pair of disks, in this case studied by micromagnetic\nsimulation, obtained n= 3.2 and 3.7 for the xandy\ninteraction terms, respectively.\nAs a first approximation one can derive the depen-\ndence of the contribution to 1 /T∗\n2related to the distance\nbetween the disks as:\nT∗\n2=B+Cd−n(7)\nFrom our simulations, and using Eq. 7 we found, from\nthe best fit, that this interaction varies as d−n, withn=\n3.9±0.1, in a good agreement with [7] and reasonable\nagreement with Sukhostavets et al.[9].4\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48/s54/s46/s53\n/s32/s32/s84\n/s50/s42/s32/s40/s110/s115/s41\n/s100/s45/s52\n/s32/s40/s49/s48/s45/s49/s49\n/s110/s109/s45/s52\n/s41/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s32/s32/s77 /s97 /s103 /s110 /s101 /s116/s105/s99 /s32/s109 /s111 /s109 /s101 /s110 /s116/s32/s120 /s32/s49 /s48 /s49/s52/s32\n/s40/s65 /s32/s109 /s50\n/s41\n/s84/s105/s109/s101/s32/s40/s110/s115/s41\nFIG. 4. (Color online) Variation of the relaxation times\nT∗\n2versusd−4for an array of 10 ×10 magnetic nanodisks\nwith a distribution of diameters centered on D= 250nm\n(σ= 10nm), damping constant α= 0.001 and separation\ndbetween their centers; the continuous line is a linear least\nsquares fit. The inset shows a vortex echo simulation for the\narray, with d= 500nm, τ= 30ns, α= 0.001.\nDetermining 1 /T′\n2has allowed us to obtain the inten-\nsity of the interaction between the disks as a function\nof separation dbetween them. Substituting Eq. 6 and\nEq. 7 in Eq. 5, we can obtain the expression for the\ninteraction as a function of d:\n1\nT′\n2=1\nB+Cd−n−Aα−∆ω≈dn\n|C|−Aα−∆ω; (8)\n(approximation valid for dsmall). In Fig. 4 we show the\nresults of the simulations with σ= 10nm and α= 0.001.\nAssuming n= 4 and making a linear squares fit, we\nobtained B= 6.15×10−9s,C=−4.03×10−35s m4.\nA new phenomenon, the magnetic vortex echo, anal-\nogous to the NMR spin echo, is proposed and demon-\nstrated here through micromagnetic simulation. Appli-\ncationsofthe magneticvortexechoincludesthe measure-\nment of the inhomogeneity, such as, distribution of di-\nmensions, aspect ratios, defects, and perpendicular mag-\nnetic fields and so on, in a planar array of nanodisks or\nellipses; it may be used to study arrays of nanowires or\nnanopillars containing thin layers of magnetic material.\nThese properties cannot be obtained directly, for exam-\nple, from the linewidth of FMR absorption.\nThe MVE is a tool that can be used to evaluate the\ninteractionbetween the elements ofalargearrayofnano-\nmagnets with vortex ground states. It can also be used\nto determine the Gilbert damping constant αin thesesystems.\nThe authors would like to thank G.M.B. Fior for the\ncollaboration; we are also indebted to the Brazilianagen-\ncies CNPq, CAPES, FAPERJ, FAPESP.\n∗Author to whom correspondence should be\naddressed: apguima@cbpf.br\n[1] A. P. Guimar˜ aes, Principles of Nanomagnetism\n(Springer, Berlin, 2009)\n[2] C. L. Chien, F. Q. Zhu, and J.-G. Zhu, Physics Today\n60, 40 (2007)\n[3] K. Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev.\nLett.100, 027203 (2008)\n[4] F. Garcia, H. Westfahl, J. Schoenmaker, E. J. Carvalho,\nA. D. Santos, M. Pojar, A. C. Seabra, R. Belkhou,\nA. Bendounan, E. R. P. Novais, and A. P. Guimar˜ aes,\nAppl. Phys. Lett. 97, 022501 (2010)\n[5] E. R. P. Novais, P. Landeros, A. G. S. Barbosa, M. D.\nMartins, F. Garcia, and A. P. Guimar˜ aes, J. Appl. Phys.\n110, 053917 (2011)\n[6] A. Vogel, A. Drews, T. Kamionka, M. Bolte, and\nG. Meier, Phys. Rev. Lett. 105, 037201 (2010)\n[7] H. Jung, K.-S. Lee, D.-E. Jeong, Y.-S. Choi, Y.-S.\nYu, D.-S. Han, A. Vogel, L. Bocklage, G. Meier, M.-\nY. Im, P. Fischer, and S.-K. Kim, Sci. Rep. 59, 1\n(2011/08/10/online)\n[8] S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Bar-\nman, and Y. Otani, Phys. Rev. Lett. 106, 197203 (2011)\n[9] O.V.Sukhostavets,J. M.Gonzalez, andK.Y.Guslienko,\nAppl. Phys. Express 4, 065003 (2011)\n[10] The sources of inhomogeneity are the spread in radii, in\nthickness or the presence of defects. An external perpen-\ndicular field Haddsacontributionto ω[19],ω=ωG+ωH,\nwithωH=ω0p(H/Hs), where pis the polarity and Hs\nthe field that saturates the nanodisk magnetization. A\ndistribution ∆ His another source of the spread ∆ ω\n[11] E. L. Hahn, Phys. Rev. 80, 580 (1950)\n[12] R. Antos, M. Urbanek, and Y. Otani, J. Phys.: Conf.\nSeries200, 042002 (2010)\n[13] If the disks have different circulations ( c=±1) the cores\nwill be displaced in opposite directions, butthe effect will\nbe the same, since the Miwill all point along the same\ndirection.\n[14] C. P. Slichter, Principles of Magnetic Resonance, 3. ed.\n(Springer, Berlin, 1990)\n[15] A. P. Guimar˜ aes, Magnetism and Magnetic Resonance in\nSolids(John Wiley & Sons, New York, 1998)\n[16] T. Butz, Fourier Transformation for Pedestrians\n(Springer, Berlin, 2006)\n[17] Available from http://math.nist.gov/oommf/\n[18] These echoes, however, are not equivalent to the stimu-\nlated echoes observed in NMR with two 90opulses[11]\n[19] G. de Loubens, A. Riegler, B. Pigeau, F. Lochner,\nF. Boust, K. Y. Guslienko, H. Hurdequint, L. W.\nMolenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberke-\nvich, N.Vukadinovic,andO.Klein,Phys.Rev.Lett. 102,\n177602 (2009)"    },    {        "title": "0809.4644v2.Damping_and_magnetic_anisotropy_of_ferromagnetic_GaMnAs_thin_films.pdf",        "content": "Anisotropic Magnetization Relaxa tion in Ferromagnetic GaMnAs \nThin Films \n \nKh.Khazen, H.J.von Bardeleben, M.Cubukcu, J.L.Cantin \nInstitut des Nanosciences de Paris,  \nUniversité Paris 6, UMR 7588 au CNRS \n140, rue de Lourmel, 75015 Paris, France \n \nV.Novak, K.Olejnik, M.Cukr \nInstitut of Physics, Academy of Sciences,  \nCukrovarnicka 10, 16253 Praha, Czech Republic \n \nL.Thevenard, A. Lemaître \nLaboratoire de Photonique et  des Nanostructures, CNRS \nRoute de Nozay, 91460 Marcoussis, France \n \n \nAbstract: \n \n The magnetic properties of annealed, epitaxial Ga 0.93Mn 0.07As layers under tensile and \ncompressive stress have been investigat ed by X-band (9GHz) and Q-band (35GHz) \nferromagnetic resonance (FMR) spectroscopy. From  the analysis of the linewidths of the \nuniform mode spectra the FMR Gilbert damping factor α has been determined. At T=4K we \nobtain a minimum damping factor of α = 0.003 for the compressively stressed layer. Its value \nis not isotropic. It has a minimum value for th e easy axes orientations of the magnetic field \nand increases with the measuring temperature. It s average value is for both type of films of \nthe order of 0.01 in spite of strong differences  in the inhomogeneous linewidth which vary \nbetween 20 Oe and 600 Oe for the layers grown on GaAs and GaInAs substrates respectively .   \n \n \nPACS numbers: 75.50.Pp, 76.50.+g, 71.55.Eq  \nIntroduction: \n \nThe magnetic properties of ferromagnetic Ga 1-xMn xAs thin films with Mn \nconcentrations between x=0.03 and 0.08 have been studied in great detail in the recent years \nboth theoretically and experimentally. For recent reviews see references [1, 2].  A \nparticularity of GaMnAs ferro magnetic thin films as comp ared to conventional metal \nferromagnetic thin films is the predominance of  the magnetocrystalline anisotropy fields over \nthe demagnetization fields. The strong anisotropy fields are not directly related to the crystal \nstructure of GaMnAs but are induced by the la ttice mismatch between the GaMnAs layers and \nthe substrate material on which they are grow n. When grown on (100) GaAs substrates the \ndifference in the lattice constants induces biaxial strains  of ≈ 0.2% which give rise to \nanisotropy fields of several 103 Oe. The   low value of the de magnetization fields (~300Oe) is \nthe direct consequence of the small spin conc entration in diluted magnetic semiconductors \n(DMS) which for a 5% Mn doping leads to a saturation magnetization of only  40 emu/cm3. \nAs the strain is related to the lattice mismat ch it can be engineer ed by choosing different \nsubstrate materials. The two systems which have been investigated most often are (100) GaAs \nsubstrates and (100)GaInAs pa rtially relaxed buffer layers. These two cases correspond to \ncompressive and tensile strained Ga MnAs layers respectively [3].  \nThe static micro-magnetic pr operties of GaMnAs layers can be determined by \nmagnetization, transport, magneto-optical and ferromagnetic resonance techniques. For the \ninvestigation of the ma gnetocrystalline anisotropies the ferromagnetic resonance spectroscopy \n(FMR) technique has been shown to be partic ularly well adapted [2, 4]. The dynamics and \nrelaxation processes of the magnetization of such layers have hardly been investigated up to \nnow [5-7]. The previous FMR studies on this  subject concerned either unusually low doped \nGaMnAs layers [5, 7] or employed a single microwave frequency [6] which leads to an \noverestimation of the damping factor. The knowle dge and control of the relaxation processes \nis in particular important for device applications as they de termine for example the critical \ncurrents necessary for current induced magne tization switching. It is thus important to \ndetermine the damping factor for state of the art samples with high Curie temperatures of T C ≈ \n150K, such as those used in this work.  Anothe r motivation of this work is the search for a \npotential anisotropy of the ma gnetization relaxation in a dilu ted ferromagnetic semiconductor in which the m agnetocrystalline anisotro pies are strong and dom inant over the \ndemagnetization contribution.  \nThe intr insic sm all angle m agnetization re laxation is generally described by one \nparam eter, the Gilbert d amping coefficient α, which is defined by the Landau Lifshitz Gilbert \n(LLG) equation of m otion for the m agnetization: \n \n⎥⎦⎤\n⎢⎣⎡× +⎥⎦⎤\n⎢⎣⎡× −= ⋅dtsdMeffH MdtMdrr rrr\nγα\nγ1   eq.1 \nwith M the m agnetization, H eff the effective m agnetic field, α the dam ping fa ctor, γ the \ngyrom agnetic ratio and s the uni t vector parallel to M.   \nThe dam ping factor α is generally assum ed to be a scal ar quantity [8, 9] . It is defined \nfor sm all angle precess ion relaxatio n   which  is the case of FMR experim ents. This param eter \ncan be experim entally determ ined by FMR spectr oscopy either from  the angular variation of \nthe linewidth or from  the variati on of the uniform  mode linewidth ∆Hpp with th e microwave \nfrequency. In this second case the linewidth is given  by: \n \nω\nγω ⋅\n⋅⋅ + ∆= ∆+ ∆= ∆\nMGH H H Hin inpp\n2 hom hom hom32)(  eq.2 \nWith ∆Hpp the first derivative p eak-to -peak  linewid th of  the uniform mode of \nLorentzian lineshape, ω the angular m icrowave frequency an d G the Gil bert dam ping factor \nfrom which the m agnetization independent damping factor α can be deduced as α=G/γM. In  \neq. 2 it is assum ed that the m agnetiz ation and th e applied magnetic f ield are collin ear which is \nfulfilled for high symm etry direction s in GaMnAs  such as [001], [110] and [100]. Ot herwise a \n1/cos ( θ-θH) term  has to be added to eq.2 [8]. \n∆Hinhom is the inhom ogeneous, frequency indepe ndent linewidth; it can be further \ndecom posed in three con tributions, re lated to the crysta lline imperf ection of  the f ilm [10]: \n \nint\ninthom HHH H HHr\nH\nHr\nH\nHr\nin ∆⋅ + ∆⋅ + ∆⋅∂= ∆δδφδφδθθδ  eq.3 \n \nThese three term s were introduced to take in to account a slight m osaic structure of the \nmetallic thin f ilms def ined by the polar angles  (θ, φ) and their distributions ( ∆θ, ∆φ)  - \nexpressed in the first two term s in eq.3- and a distribution of  the internal anisotropy fields H int – the last term of eq.3. In the case of  homo epitaxial III-V films obtained by MBE growth like \nGaMnAs on GaAs, films of high crystalline qu ality are obtained [LPN] and only the third \ncomponent ( ∆Hint) is expected to play an important role.  \n Practically, the variation of the FMR linewid th with the microwave frequency can be   \nmeasured with resonant cavity systems at different frequencies between 9GHz and 35GHz; \nthe minimum requirement -used also in this work - is the use of two frequencies. We disposed \nin this work of 9GHz and 35GHz spectrometers. The linewidth is decomposed in a frequency independent inhomogeneous part and a linear fr equency dependent homogeneously broadened \npart. For most materials the inhomogeneous fr action of the linewidth is strongly sample \ndependent and depends further on the interface quali ty and the presence of cap layers. It can \nbe smaller but also much larger than the intrinsic linewidth. In Ga\n0.95Mn 0.05As single films \ntotal X-band linewidths be tween 100Oe and 1000Oe have been encountered. These \nobservations indicate already the impor tance of inhomogeneous broadening. The \nhomogeneous linewidth will depend on the intrinsic sample properties. This approach supposes that the inhomogeneous  linewidth is frequency independent and the homogenous \nlinewidth linear dependent on the frequency, two assumptions generally valid for high \nsymmetry orientations of the a pplied field for which the magne tization is parallel to the \nmagnetic field.  \nIt should be underlined that  in diluted magnetic semiconductor (DMS) materials like \nGaMnAs the damping parameter is not only determined by the sample composition x Mn [5]. It \nis expected to depend as we ll on (i) the magnetic compensati on which will vary with the \ngrowth conditions, (ii) the (hol e) carrier concentration respon sible for the ferromagnetic Mn-\nMn interaction which is influenced by the presence of native donor defects like arsenic \nantisite defects or Mn interstitial ions [11] and (iii) the valence bandstructure, sensitive to the \nstrain in the film. Due to the high out-of –plane and in-plane anisotropy of the magnetic \nparameters [12] which further vary with the applied field and the temperature a rather \ncomplex situation with an anisotropic and te mperature dependent da mping factor can be \nexpected in GaMnAs.  \nWhereas the FMR Gilbert damping factor has been determined for many metallic \nferromagnetic thin films [8] only three experimental FMR studies have been published for GaMnAs thin films up to now [5-7]. In ref.[ 5,7] low doped GaMnAs laye rs with a critical \ntemperature of 80K which do not correspond to the high quality, standa rd layers available \ntoday were studied. In the ot her work [6] higher doped layers  were investigated but the \nexperiments were limited to a single microw ave frequency (9GHz) and thus no frequency dependence could be studied.  In this work we present the results of FMR studies at 9GHz and \n35 GHz on two high quality GaMnAs layers with  optimum critical temperatures: one is a \ncompressively strained layer grown on a GaAs buffer layer and the othe r a tensile strained \nlayer grown on a (Ga,In)As buffer layer. Due to  the opposite sign of the strains the easy axis \nof magnetization is in-plane [ 100] in the first case and out-o f-plane [001] in the second. The \nGaMnAs layers have been annealed ex-situ after their growth in order to  reduce the electrical \nand magnetic compensation, to homogenize the laye rs and to increase the Curie temperature \nto ≈ 130K. Such annealings have become a st andard procedure for improving the magnetic \nproperties of low temperature molecular b eam epitaxy (LTMBE) grown GaMnAs films. \nIndeed, the low growth temperature required to incorporate the high Mn concentration \nwithout the formation of precipitates gives rise to native defect the conc entration of which can \nbe strongly reduced by the annealing.  \n \nExperimental details \nA first sample consisting of a Ga 0.93Mn 0.07As layer of 50nm thickness has been grown \nat 250° C by low temperature molecular beam epitaxy on a semi-insulating (100) oriented \nGaAs substrate. A thin GaAs buffer layer has been grown before the deposition of the \nmagnetic layer. The second sample, a 50 nm thick Ga 0.93Mn 0.07As layer have been grown \nunder very similar conditions on a partially relaxed (100) Ga 0.902In0.098As buffer layer; for \nmore details see ref. [13]. After the growth the structure was thermally annealed at 250° C for \n1h under air or nitrogen gas fl ow. The Curie temperatures were 157K and 130K respectively. \nBased on conductivity measuremen ts the hole concentratio n is estimated in the 1020cm-3 \nrange. \nThe FMR measurements were performed with Bruker X-band and Q-band \nspectrometers under standard conditions: mW microwave power and 100 KHz field \nmodulation. The samples were measured at te mperatures between 4K and 170K. The angular \nvariation of the FMR spectra was measured in  the two rotation planes (110) and (001). The \npeak-to peak linewidth of the first derivati ve spectra were obtained from a lineshape \nsimulation. The value of the st atic magnetization M(T) had been determined by a commercial \nsuperconducting quantum interference device (SQUID) magnetometer. A typical hysteresis \ncurve is shown in the inset of fig.8. \n \nExperimental results: The saturation magnetizations of the two laye rs and the magneto crystalline anisotropy \nconstants which had been previously de termined by SQUID and FMR measurements \nrespectively are given in table I. The anisotropy constants had been determined in the whole \ntemperature range but for clarity only its values  at T=55K and T=80K are given in table I. We \nsee that the dominant anisotropy constant K 2⊥ are of different sign with -55000 erg/cm3 to \n+91070 erg /cm3 and that the other three constants ha ve equally opposite signs in the two \ntypes of layers. The easy axes of magnetization are the in-pla ne [100] and the out-of-plane \n[001] direction respectively. Howe ver the absolute values of th e total effective perpendicular \nanisotropy constant Ku=K 2⊥ +K 4⊥ are less different for the two samples: -46517erg/cm3 and \n+57020erg/cm3 respectively. More detailed inform ation on the measurements of these \nmicromagnetic parameters will be published elsewhere.  \nFor the GaMnAs/GaAs layers the peak-to-peak linewidth of the first derivative \nuniform mode spectra has been strongly re duced by the thermal annealing; in the non \nannealed sample the X-band linewidth was highl y anisotropic with va lues between 150Oe and \n500Oe  at T=4K. After annealing it is reduced to  an quasi isotropic average value of 70Oe at \nX-band. Quite differently, for the GaMnAs/GaInA s system the annealing process decreases \nthe linewidth of the GaMnAs layers only marginally. Although full angular dependencies \nhave been measured by FMR we will analyze only the linewidth of the four high symmetry field orientations H//[001], H //[100], H//[1-10], H//[ 110] corresponding to the hard and easy \naxes of magnetization. As will be shown below, in spite of rather similar high critical temperatures (157K/130K) the linewidth are drastically di fferent for the two cases.  \n \n1. GaMnAs on GaAs \n In fig. 1a and 1b we show typical low te mperature FMR spectra at X-band and Q-band \nfrequencies for the hard [001] /intermediate  [100] axis orientation of the applied magnetic \nfield. The spectra are characterized by excelle nt signal to noise ra tios and well defined \nlineshapes.  We see that at both frequencies the lineshape is close to a Lorentzian. In addition \nto the main mode one low intensity spin wave  resonance is observed at both frequencies at \nlower fields (not shown). \n The linewidth at X-band (fig.2) is of  the order of 50Oe to 75Oe with a weak \norientation and temperature dependence. Above T>130K, close to the criti cal temperature, the \nlinewidth increase strongly. At Q-band we observe a systematic increase by a factor of two of \nthe total linewidth (fig.3) with an increase d temperature and orient ation dependence. As generally observed in GaMnAs, the easy axis orie ntation gives rise to th e lowest linewidth. At \nQ-band the lineshape is perfectly Lorentzian (f ig.1b). These linewidth are among the smallest \never reported for GaMnAs thin films, which re flects the high crystal line and magnetic quality \nof the film. \n To determine the damping factor α we have plotted the frequency dependence of the \nlinewidth for the different orientations and at various temperatures. An example is given in \nfig. 4 for T=80K; this allows us to determin e the inhomogeneous linewidth obtained from a \nlinear extrapolation to zero frequency and the damping factor from the slope. The \ninhomogeneous linewidth at T=80K is of the order of 30 Oe, i.e. 50% of the total linewidth at \nX-band. This shows that the approximation ∆Hinhom<< ∆Hhomo which had been previously \nused [5] to deduce the damping factor from a single (X-band) frequenc y measurement is not \nfulfilled here.  \n The temperature dependence of the inhomogeneous linewidth is shown in fig.5. \nSimilar trends as for the total linewidth in the non annealed films are observed: the linewidth \nis high at the lowest temperatures, decreases with increasing temperat ures up to 120K   and \nincreases again close to T C.   \n From the slope of the linewidth variati on with microwave frequency we obtain the \ndamping factor α (fig.6). Its high temperature value is of the order of 0.010 but we observe a \nsystematic, linear variation with the temperatur e and a factor two difference between the easy \naxis orientation [100] and the hard axis orientation [001].  \n \n \n2. GaMnAs on GaInAs \n Similar measurements have been performed on the annealed tensile strained layer. In \ntensile strained GaMnAs films the easy axis  of magnetization ([001]) coincides with the \nstrong uniaxial second order anisotropy directio n.  For that reason no FMR resonance can be \nobserved at temperatures below T=80K for the easy axis orientation H// [001] at X-band. For \nthe other three orientations the resonances  can be observed at X-band in the whole \ntemperature range 4K to T C. Due to the strong temperature dependence of the anisotropy \nconstants and the parallel decr ease of the internal anisotropy fields the easy axis resonance \nbecomes observable at X-band for temperatures above 80K. In the films on GaInAs much \nhigher linewidth are encountered th an in films on GaAs, the values are up to ten times higher \nindicating a strong inhomogeneity in this film. A second low fiel d resonance is systematically \nobserved at X-band and Q-band; it is equally attributed to a spin wave resonance.   Figures 7a and 7b show typical FMR spectra  at X- and Q-band re spectively. At both \nfrequencies the lineshape can no longer be simu lated by a Lorentzian but has changed into a \nGaussian lineshape. \n Contrary to the first cas e of GaMnAs/GaAs the X-band linewidth varies monotonously \nin the whole temperature region (fig.8). We observe a linewidth of ~600Oe at T=4K, which \ndecreases only slowly with temperature; the linewidth becomes minimal in the 100 K to 140K \nrange. The Curie temperature “s een” by the FMR spectroscopy is s lightly higher as compared \nto the one measured by SQUID due to th e presence of the applied magnetic field. \n At low temperature the Q-ba nd linewidth vary strongly w ith the orientation of the \napplied field with values be tween 500Oe and 700Oe. The lowest  value is observed for the \neasy axis orientation. They decrease as at X-band only slowly with increasing temperature \nand increase once again when approaching the Curie temperature. At Q-band the easy axis \nFMR spectrum, which is also accompanied by a str ong spin wave spectrum at lower fields, is \nobservable in the whole temperature range. \n For this sample we observe especially at Q-band a systematic difference between the cubic axes [100], [001] linewidth and the one for the in-plane [110] and [1-10] field \norientations (fig.8). The most surprising observation is that for temperatures below T<100K \nthe linewidth for H//[100] and H//[110] are co mparable at X-band and Q-band and thus an \nanalysis in the simple model discussed above is not possible. We attribute this to much higher \ncrystallographic/magnetic inhomogeneities, which mask the homogenous linewidth. The \norigin of the strong inhomogeneity is still unclear.  The only orientation for which in the whole temperature range a systematic increas e in the linewidth between X-and Q-band is \nobserved is the H//[1-10] orienta tion. We have thus analyzed th is variation (fig.10) according \nto eq.1.  \n In spite of important differences in the lin ewidth the slope varies only weakly which \nindicates that the inhomogeneous linewidth is  very temperature dependent and decreases \nmonotonously with increasing temperature from 570Oe to 350Oe.   \nIn the high temperature range (T ≥100K) the easy axis orientation could also be \nanalyzed (fig.11). The inhomogeneous  linewidth are lower than for the hard axis orientation at \nthe same temperatures and are in the 300Oe range (fig.12). The homogenous linewidth at \n9Ghz is in the 50Oe range which is close to the values determined in the first case of \nGaMnAs/GaAs.  \n From the slope (fig.13) we obtain the da mping factor which for the hard axis \norientation is α=0.010 in the whole temperature range. Th is value is comparable to the one measured for the GaMnAs/GaAs film for H//[110]. The damping factor for the easy axis \norientation is lower but   increases close to T C as in the previous case.  \n \nDiscussion: \n  An estimation of the FMR intrinsic damping factor in a ferromagnetic GaMnAs thin \nfilm has been made within a model of localiz ed Mn spins coupled by p-d kinetic exchange \nwith the itinerant-spin of holes treated by the 6-band Kohn-Luttinger Hamiltonian [5]. Note, that these authors take for the effective kinetic exchange field the value in the mean-\nfield approximation, i.e. H\neff=JN<S>, so that their calculation are made within the random \nphase approximation (RPA). RPA calculations of α have been made by Heinrich et al. [14] \nand have recently been used by Tserkovnyak et al .[15] for numerical app lications to the case \nof Ga 0.95Mn 0.05As.  Both models however, are phenomenological and include an \nadjustable parameter: the quasiparticle lifetime Γ for the holes in [5] and the spin-flip \nrelaxation T 2 in [15]. These models do not take into account neither multi-magnon \nscattering nor any damping beyond the RPA. It has been argued elsewhere [16], that in diluted \nmagnetic semiconductors such affects are only impor tant at high temperatur e (i.e. at T>Tc). In \nparticular, the increase of α in the vicinity of Tc may be attributed to such effects that are \nbeyond the scope of the models of references  [5] and [14]. At low temperatures T<<Tc \nhowever, where the corrections to the RPA are expected to be negligib le, the models of [5, \n14,15] provide us with a numerical value of α in agreement with our experiments if we \nintroduce reasonable values of   these parameters. For GaMnAs films with metallic \nconductivity, Mn concentrations of x= 0.05 and hole concentr ations of 0.5 nm-3 (5x1020cm-3) \nSinova et al [5] predict an isotropi c  low temperature damping factor α between 0.02 and 0.03 \ndepending on the quasiparticle life time broa dening. Tserkovnyak et al [15] found a similar \nvalue of α ≈ 0.01 for the isotropic damping factor fo r a typical GaMnAs film with 5% Mn \ndoping and full hole polarization. \n Both predicted values are of the same orde r of magnitude as the experimental values \ndetermined in this study. Our results for GaMn As/GaAs show further that the damping factor \nis not isotropic as generally a ssumed but is anisotropic with a lowest value for the in-plane  \neasy axis orientations of the applied magnetic field H//[100], H//[1-10] and an increase of up \nto a factor of two for the hardest axis orientat ion H// [001]. Intrinsic anisotropic damping is \nrelated to the fact that the free energy density depends on the orientation of the magnetization which in the case of GaMnAs is related to th e anisotropy of the p-hole Fermi surface. We have shown (table I) that the anisotropy of the magnetocristalline constants and the related \nfields are important in these strained layers and it is thus not surprisi ng to find also anisotropy \nof the damping factor. For further discussions on this subject see reference [9]. The system \nmight also contain extrinsic an isotropies related to the pres ence of lattice defects. Their \ninfluence can be deduced from the value and anisotropy of the inhomogeneous linewidth. In \nthe case of the compressively strained layers (G aMnAs/GaAs) we see that their value is small \nand rather isotropic quite differen tly from the tensile strained film. It is in the first case that \nour measurements show a factor of two anisot ropy of the Gilbert damping factors. A further \nindication for the intrinsic charac ter of the anisotropy is the fact that the damping factor for \nthe perpendicular orientation has the highest value. In this case any contribution from two \nmagnon scattering will be minimized. Anyway, such contributions are generally only \nimportant at low frequency measurements in the 2-6GHz range but even there they were \nfound to be negligible [7]. \n Additional material related parameters are expected to further influence the damping \nfactor. As the spin flip relaxation times will depend on the sample properties and in particular \nthe presence of scattering centers we will not expect to find a unique damping factor even for GaMnAs/GaAs samples with the same Mn composition x. More likely, different damping \nfactors are expected to be found in  real films and their values might be used to assess the film \nquality. In this sense the GaMnAs/GaAs film studi ed here is of course “better” than the one \non GaInAs in line with the strong difference in the sample inhomogeneities.  \n The inhomogeneous linewidth originates from spatial inhomogeneities in the local \nmagnetic anisotropy fields and inhomogeneities in the local exchange interactions. Given the particular growth conditions of these films, low temper ature molecular beam epitaxy, \ninhomogeneities can not be expected to be neglig ible in these materials.  If we had analyzed \nour X-band results of the GaMnAs/GaAs film in the spirit of ref. [5], i.e. assuming a \nnegligible inhomoge neous linewidth - ( ∆H\ninhom=0) -, we would have obtained artificially \nincreased damping factors. A further contribution might be expected from the intrinsic \ndisorder in these films: as GaMnAs is a diluted magnetic semiconductor with random \ndistribution of the Mn ions, this  disorder will even for crysta llographically perfect crystals \ngive some importance to this term.  \n In the previous studies of  the FMR damping factor in GaMnAs/GaAs single films \nhigher values have been reported. Matsuda et  al [6] found damping factors between 0.02 and \n0.06 in the T=10K to T=20K temperature range. They  observed the same tend encies as in this \nwork concerning the anisotropy and temperature dependence of α: the lowest damping factor is seen for the easy axis orientation and its va lues increases with increasing temperatures. The \nfilms of their study were however significantly different: (i) the Mn doping concentration was \nlower, x=0.03 and thus the hole concentratio n was equally lower and (ii) the critical \ntemperature of the annealed film was only T=80K. The anisotropy in the inhomogeneous \nlinewidth at T=20K was equally much higher, varying between 30Oe for the easy axis to \n250Oe for the hardest axis [110]. In a second study Sinova et al [5] have measured an \nannealed GaMnAs/GaAs sample with a similar composition (x= 0.08) and critical temperature \n(TC=130K) as the one studied here. They deduced a damping factor of the order of α=0.025 \nwith only a slight temperature dependence betw een 4K and 80K and an increase close to T C. \nHowever, these measurements were done at one  (X-band) microwave frequency only and the \nnumerical value of α was obtained by assuming a neglig ible inhomogeneous linewidth. As \nexplained above, the value  of α can be expected to be overestimated in this case.In the \nphotovoltage measurements of ref.7 the dampi ng factor of a low (x=0.02) doped GaMnAs \nlayer has been determined with microwave fr equencies from 2 to 19.6GHz but for one field \norientation H//[001] (h ard axis) and one temperature (T =9K) only. Interestingly, their \nmeasurements show a linear behavior even in  the low frequency range down to 4GHz which \ndemonstrates the negligible contribution from two magnon scattering in this case. \n The intrinsic damping factor α plays also an important role in the critical currents \nrequired to switch the magnetization in FM/NM/ FM trilayers [5]. However, in trilayer \nstructures interface and spin pumping effects wi ll add to the intrinsic damping factor of the \nferromagnetic material and give rise to an incr eased effective damping fa ctor. Sinova et al [5] \nhave estimated the critical current for real istic GaMnAs layers: based on a value of α= 0.02, \nthey estimated the critica l current density to J C=105A/cm2. It should be noted that the damping \nfactor involved in the domain wall motion [ 17, 18] is by definition different form the FMR \ndamping factor. Both are however linearly related with αFMR<αDW [17, 18]. First observations \nof current induced magne tization switching in Ga 0.956Mn 0.044As/GaAs/Ga 0.967Mn 0.033As \ntrilayers confirm these theoretical predictions [19]. These authors observed a critical current \ndensity of ≈  105A/cm2 which would have been predicted from Slonczewski’s formula [20] for \na  domain wall damping factor of αDW=0.002. \n \nConclusion:  \n  We have determined the intrinsic FM R Gilbert damping factor for annealed \nGa0.93Mn 0.07As thin films with high critical temperatures. To evaluate the influence of the \nstrain the two prototype cases of  compressive and tensile strained  layers were studied.   In \nboth cases we find an average damping factor of the order of 0.01. We thus see that the sign \nof the strain does not seem to  influence the damping factor strongly.  The homogeneity of the \nfilms as judged from the inhomogeneous linewid th is much higher in the case of GaAs \nsubstrates than for GaInAs substrates. This must  be attributed to the high dislocation density \nin the GaInAs layer [13]. In the case of the GaMnAs/GaAs layers, where the small linewidth \nallows a finer analysis of the data, we observe  an anisotropy of the da mping factor, which has \nthe lowest value for the easy axis orientation. This value of α[1−10]=0.003 is an order of \nmagnitude lower than the previous reported va lues. The few experiment al results available \nseem to indicate that the damping factor decr eases with increasing Mn concentration. This \ncorresponds well to the th eoretical predictions by Sinova et al [5] for the case of small quasi-\nparticle lifetime broadening. It wi ll be interesting to test this behavior further in more highly \ndoped layers (x ≈0.15), which become now available.  \n \nAcknowledgment: We thank Alain Mauger of the IMPMC laboratory of the University Paris \n6 and Bret Heinrich from the Simon Frazer University in Vancouver for many helpful \ndiscussions. \n \nReferences : \n \n[1] T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A.H. MacDonald, Rev.Mod.Phys. 78, \n809 (2006). \n[2] X. Liu, and J. K. Furdyna , J. Phys.: Condens. Matter 18, R245 (2006). \n[3] X. Liu, Y. Sasaki and J. K. Furdyna, Phys.Rev.B 67 , 205204 (2003). \n[4] K.Khazen, H.J.von Bardeleben, J.L.Cantin, L.  Thevenard, L. Largeau, O. Mauguin, and A. \nLemaître,  Phys.Rev. B77, 165204 (2008) . \n[5] J.Sinova, T.Jungwirth, X.Liu, Y.Sa saki, J.K.Furdyna, W.A.Atkinson, and \nA.H.MacDonald, Phys.Rev. B69, 085209 (2004). \n [6] Y.H.Matsuda, A.Oiwa, K.Ta naka, and H.Munekata, Physica B 376-377,668 (2006) \n[7] A.Wirthmann, X.Hui, N.Mecking, Y.S.Gu i, T.Chakraborty, C.M.Hu, M.Reinwald, \nC.Schüller, and W.Wegschneider, Appl.Phys.Lett.92, 232106 (2008) \n[8] M. Farle, Rep.Prog.Phys. 61, 755 (1998). [9] B.Heinrich, « Spin relaxation in magnetic me tallic layers and multilayers », in Ultrathin \nMagnetic Structures III, ed. by J.A.C.Bland, and B.Heinrich, Springe r Verlag (Berlin 2005), \np.143 \n[10] C.Chappert,K.LeDang,P.Beauvilain,H .Hurdequint, and D.Renard, Phys.Rev.B34 , 3192 \n(1986) \n[11] F. Glas, G. Patriarche, L. Largeau, A. Lemaître, Phys. Rev. Lett., 93, 086107 (2004) \n[12] L.Thevenard, L.Largeau,, O.Mauguin, A.Lemaitre, K.Khazen and H.J.von Bardeleben, \nPhys. Rev.B75 ,195218 (2006) \n[13] L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A. Lemaître, N. Vernier and J. Ferré, Phys. Rev. B 73, 195331 (2006)  \n[14] B.Heinrich, D.Fraitova, V.Kambersky , Phys. Stat. Solidi 23, 501 (1967). \n[15] T.Y.Tserkovnyak, G.A.Fiete, B.I.Halperin, Appl.Phys.Lett. 84, 5234 (2004) \n[16] A.Mauger, D.L. Mills, Phys.Rev. B29, 3815 (1984) \n[17] S.C.Chen and HL.Huang, I EEE Transactions on Magnetics 33, 3978 (1997) \n[18] H.L.Huang, V.L.Sobolev, and S.C.Chen,  J.Appl.Phys. 81, 4066(1997) \n[19] D.Chiba, Y.Sato, T.Kita, F. Matsukura, and H.Ohno, Phys.Rev.Lett. 93, 216602(2004) \n[20] J.C.Slonczewski, J.Magn.Magn.Mater. 159, L1(1996) \n \n \nFigure Captions: \n \nFigure 1a: X-band FMR spectrum of the GaMnAs /GaAs film taken at T=20K and for H// \n[001]; the peak-to-peak linewidth is 60Oe. The experimental  spectrum is shown by circles and \nthe Lorentzian lineshape simulation by a line. \n \n \nFigure 1b: Q-band FMR spectrum of the GaMnAs /GaInAs film taken at T=80K and for H// \n[100]; the peak to peak  linewidth is 120Oe. The experimental spectrum is shown by circles \nand the Lorentzian lineshape simulation by a line \n \nFigure 2 (color on line): X-band peak-to-peak li ne widths for the GaMnAs/GaAs film for the \nfour main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] \nred circles, H//[100] blue upper triangles.   \nFigure 3 color on line): Q-band peak-to-peak line widths for the GaMnAs/GaAs film for the \nfour main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] \nred circles, H//[100] blue upper triangles. \n \nFigure 4 (color on line): Peak-t o-peak linewidth at 9GHz and 35GHz for the GaMnAs/GaAs \nfilm; T=80K and H//[001] black squares, H//[1-10] olive lowe r triangles, H//[110] red circles, \nH//[100] blue upper triangles \n \nFigure 5 (color on line): GaMnAs/GaAs inhomogeneous linewidth as a function of temperature for four orientations of the applie d field: H//[001] black squares, H//[1-10] olive \nlower triangles, H//[110]  red circles, H//[100] blue upper triangles \n \nFigure 6 (color on line): damping factor α as a function of temperature and magnetic field \norientation; H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] blue upper \ntriangles, H//[100] red circles; the maximum erro r in the determination of the linewidth is \nestimated to 10G which co rresponds to an error in α of  0.001 \n \nFigure 7a: X-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard \naxis); the low field spin wave resonance (SW) is of high intensity in this case; circles: \nexperimental points, line: simulation with Gaussian lineshape \n \nFigure 7b: Q-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard \naxis); the low field spin wave resonance (SW) is of high intensity in this case; circles: experimental points, line: simulation with Gaussian lineshape \n \n \nFigure 8 (color on line): GaMnAs/GaInAs:  X-band FMR linewidth as a function of \ntemperature for the four orientations of the applied magnetic field: H//[001] black squares, \nH//[1-10] olive lower triangles, H//[110] red circ les, H//[100] blue upper triangles; the easy \naxis FMR spectrum is not observable below T=100K ; a typical hysteresis  curve as measured \nby SQUID is shown in the inset. \n Figure 9 (color on line): GaMnAs/GaInAs : Q-band FMR linewidth as a function of \ntemperature for the four orientations of the applied magnetic field : H//[001] black squares, \nH//[1-10] olive lower triangl es, H//[110] red circles, H//[100] blue upper triangles \n \nFigure 10 (color on line): GaMnAs/GaInAs : FM R linewidth as a function of microwave \nfrequency for H//[1-10] at different temper atures; T=10K (black squares), T=25K (red \ncircles), T=55K(black spades), T=80K( blue stars), T=115K (red triangles) \n \nFigure 11 (color on line): GaMnAs/GaInAs: FMR linewidth as a function of microwave \nfrequency for H//[001] (easy ax is)  at different temperatur es; T=100K (red circles), T=120K \n(blue spades),T=130K(black squares) \n \nFigure 12: GaMnAs/GaInAs inhomogeneous linewi dth as a function of temperature for two \norientations of the applied field: H//[001]  squares, H//[1-10]  lower triangles,  \n \nFigure 13 : GaMnAs/GaInAs damping factor  α as a function of temperature and magnetic \nfield orientation; H//[001]  squa res, H//[1-10]  lower triangles.  \n \nTables \n \n \nGa 0.93Mn 0.07As/GaAs G a0.93Mn 0.07As/Ga 0.902In0.098As \nMs(T=4K)= 47e mu/cm3 Ms(T=4K)= 38 e mu/cm3 \nTC =157K  (SQUID) TC=130K  (SQUID) \nAnisotropy constants (T=80K) Anisotropy constants (T=55K) \nK2⊥ = - 55000 e rg/cm3 K 2⊥ = +91070 e rg/cm3 \nK2// =2617 er g/cm3 K 2// = -2464 erg/cm3 \nK4⊥ = 8483  erg/cm3 K 4⊥ = -34050  erg/cm3 \nK2// =2590 er g/cm3 K 2// = -1873er g/cm3 \nEasy axis of magnetization Easy axis of magnet ization \n[100] 4K<T<T C [001] 4K<T< T C \n \n \nTable I \n \nTable caption \n \nTable I: M icrom agnetic  param eters of the two sam ples studied in this work: s aturation \nmagnetization M s at T=4K, critical tem perature T C, magneto crystalline anisotropy constants \nof second and fourth order K 2⊥ , K 2//, K 4⊥, K 4// at T=55K and T=80K respectively and the \norientation of the easy axis for m agnetization. \n \n \n \n \n \n \n \n            \n \n7500 7600 7700 7800 7900 8000SW\n  FMR Signal (arb.u.)\nMagnetic Field H ( Oe)\n \n \nFigure 1a \n \n10000 1020 0 10400 1060 0 10800 1100 0\n  FMR Signal (arb.u.)\nMagnetic Field H (Oe)\n \nFigure 1b \n 0 20 40 60 80 100 120140 160050100150200250300Linew idth ∆H (Oe)\nTempera ture (K) 001\n 110\n 100 1-10\n \nFigure 2 \n \n0 20 40 60 80 100 120 140160050100150200250300Linew idth ∆H (Oe)\nTemperature ( K) 001\n 110\n 100 1-10\n \nFigure 3 \n \n 0 5 10 15 20 25 30 35 4004080120160200Linewi dth ( G)\nFreq uency  (GHz)\n  \n \n \nFigure 4 \n \n20 40 60 80 100 120 140 160020406080100120∆Hinhom (Oe)\nTemperature (K) 001\n 110\n 100\n 1-10\n \n \nFigure 5 \n \n 0 20 40 60 80 100 120140 1600.0040.0060.0080.0100.0120.0140.0160.0180.020Damping factor α\nTemperature (K ) 001\n 110\n 100\n 1-10\n \nFigure 6 \n \n4000 5000 6000 7000 8000SW\n  FMR S ignal (arb. u.)\nMagnetic Field (Oe )\n \nFigure 7a \n 13000 14000 15000 16000 17000SW\n  FMR Signal (arb. u.)\nMagnet ic Field  (Oe)\n \n \nFigure 7b \n \n0 20 406 08 0 100120140300400500600700800\n  Linewidth ∆H (Oe)\nTemperature (K) [110]\n [100]\n [1-10]\n [001]-500 0 500-40040 M (emu/cm2)\n Magnetic F ield (O e)\n \nFigure 8 \n 0 204 0 608 0 100 120 140300400500600700800\n  Linewidth ∆H (Oe)\nTemperat ure (K ) [110]\n [100]\n [1-10]\n [001]\n \n \nFigure 9 \n \n0 5 10 15 20 25 30 35 40350400450500550600650700750\n115K80K55K25K10K\n  Linewidth ∆H (Oe)\nFreque ncy (GHz)\n \n \nFigure 10 \n 0 5 10 15 20 25 30 35 40200250300350400450500\n100K120K130K\n  Linewidth ∆H (Oe)\nFreque ncy (GHz)\n \n \nFigure 11 \n \n0 204 0 608 0 100120140200300400500600\n  ∆Hinhom (Oe)\nTemperat ure (K ) [001]\n [1-10]\n \n \n \nFigure 12 \n 0 204 06 0 80 100 120 1400.0040.0060.0080.0100.0120.0140.0160.0180.020Dam ping factor α\n  \nTemperature (K)\n \n \nFigure 13 \n \n \n \n "    },    {        "title": "2401.00714v1.Calculation_of_Gilbert_damping_and_magnetic_moment_of_inertia_using_torque_torque_correlation_model_within_ab_initio_Wannier_framework.pdf",        "content": "Calculation of Gilbert damping and magnetic moment of inertia\nusing torque-torque correlation model within ab initio Wannier\nframework\nRobin Bajaj1, Seung-Cheol Lee2, H. R. Krishnamurthy1,\nSatadeep Bhattacharjee2,∗and Manish Jain1†\n1Centre for Condensed Matter Theory,\nDepartment of Physics,\nIndian Institute of Science,\nBangalore 560012, India\n2Indo-Korea Science and Technology Center,\nBangalore 560065, India\n(Dated: January 2, 2024)\n1arXiv:2401.00714v1  [cond-mat.mtrl-sci]  1 Jan 2024Abstract\nMagnetization dynamics in magnetic materials are well described by the modified semiclassical\nLandau-Lifshitz-Gilbert (LLG) equation, which includes the magnetic damping ˆαand the magnetic\nmoment of inertia ˆItensors as key parameters. Both parameters are material-specific and physi-\ncally represent the time scales of damping of precession and nutation in magnetization dynamics.\nˆαandˆIcan be calculated quantum mechanically within the framework of the torque-torque corre-\nlation model. The quantities required for the calculation are torque matrix elements, the real and\nimaginary parts of the Green’s function and its derivatives. Here, we calculate these parameters\nfor the elemental magnets such as Fe, Co and Ni in an ab initio framework using density functional\ntheory and Wannier functions. We also propose a method to calculate the torque matrix elements\nwithin the Wannier framework. We demonstrate the effectiveness of the method by comparing it\nwith the experiments and the previous ab initio and empirical studies and show its potential to\nimprove our understanding of spin dynamics and to facilitate the design of spintronic devices.\nI. INTRODUCTION\nIn recent years, the study of spin dynamics[1–5] in magnetic materials has garnered\nsignificant attention due to its potential applications in spintronic devices and magnetic\nstorage technologies[6–9]. Understanding the behaviour of magnetic moments and their\ninteractions with external perturbations is crucial for the development of efficient and reliable\nspin-based devices. Among the various parameters characterizing this dynamics, Gilbert\ndamping[10] and magnetic moment of inertia play pivotal roles. The fundamental semi-\nclassical equation describing the magnetisation dynamics using these two crucial parameters\nis the Landau-Lifshitz-Gilbert (LLG) equation[11, 12], given by:\n∂M\n∂t=M× \n−γH+ˆα\nM∂M\n∂t+ˆI\nM∂2M\n∂t2!\n(1)\nwhere Mis the magnetisation, His the effective magnetic field including both external\nand internal fields, ˆαandˆIare the Gilbert damping and moment of inertia tensors with\nthe tensor components defined as αµνand Iµν, respectively, and γis the gyromagnetic\n∗s.bhattacharjee@ikst.res.in\n†mjain@iisc.ac.in\n2ratio. Gilbert damping, ˆαis a fundamental parameter that describes the dissipation of\nenergy during the precession of magnetic moments in response to the external magnetic\nfield. Accurate determination of Gilbert damping is essential for predicting the dynamic\nbehaviour of magnetic materials and optimizing their performance in spintronic devices. On\nthe other hand, the magnetic moment of inertia, ˆIrepresents the resistance of a magnetic\nmoment to changes in its orientation. It governs the response time of magnetic moments to\nexternal stimuli and influences their ability to store and transfer information. The moment\nof inertia[13] is the magnetic equivalent of the inertia in classical mechanics[14, 15] and acts\nas the magnetic inertial mass in the LLG equation.\nExperimental investigations of Gilbert damping[16–23] and moment of inertia involve\nvarious techniques, such as ferromagnetic resonance (FMR) spectroscopy[24, 25], spin-torque\nferromagnetic resonance (ST-FMR), and time-resolved magneto-optical Kerr effect (TR-\nMOKE)[26, 27]. Interpreting the results obtained from these techniques in terms of the\nLLG equation provide insights into the dynamical behaviour of magnetic materials and can\nbe used to extract the damping and moment of inertia parameters. In order to explain\nthe experimental observations in terms of more macroscopic theoretical description, various\nstudies[28–34] based on linear response theory and Kambersky theory have been carried out.\nLinear response theory-based studies of Gilbert damping and moment of inertia involve\nperturbing the system and calculating the response of the magnetization to the pertur-\nbation. By analyzing the response, one can extract the damping parameter. Ab initio\ncalculations based on linear response theory[33] can provide valuable insights into the mi-\ncroscopic mechanisms responsible for the damping process. While formal expression for the\nmoment of inertia in terms of Green’s functions have been derived within the Linear response\nframework[11], to the best of our knowledge, there hasn’t been any first principle electronic\nstructure-based calculation for the moment of inertia within this formalism.\nKambersky’s theory[35–37] describes the damping phenomena using a breathing Fermi\nsurface [38] and torque-torque correlation model [39], wherein the spin-orbit coupling acts\nas the perturbation and describes the change in the non-equilibrium population of electronic\nstates with the change in the magnetic moment direction. Gilmore et al.[32, 34] have reported\nthe damping for ferromagnets like Fe, Ni and Co using Kambersky’s theory in the projector\naugmented wave method[40].\nThe damping and magnetic inertia have been derived within the torque-torque correlation\n3model by expanding the effective dissipation field in the first and second-time derivatives\nof magnetisation[29–31]. In this work, the damping and inertia were calculated using the\ncombination of first-principles fully relativistic multiple scattering Korringa–Kohn–Rostoker\n(KKR) method and the tight-binding model for parameterisation[41]. However, there hasn’t\nbeen any full ab intio implementation using density functional theory (DFT) and Wannier\nfunctions to study these magnetic parameters.\nThe expressions for the damping and inertia involves integration over crystal momentum\nkin the first Brillouin zone. Accurate evaluation of the integrals involved required a dense k-\npoint mesh of the order of 106−108points for obtaining converged values. Calculating these\nquantities using full ab initio DFT is hence time-consuming. To overcome this problem,\nhere we propose an alternative. To begin with, the first principles calculations are done on\na coarse kmesh instead of dense kmesh. We then utilize the maximally localised Wannier\nfunctions (MLWFs)[42] for obtaining the interpolated integrands required for the denser k\nmeshes. In this method, the gauge freedom of Bloch wavefunctions is utilised to transform\nthem into a basis of smooth, highly localised Wannier wavefunctions. The required real\nspace quantities like the Hamiltonian and the torque-matrix elements are calculated in the\nWannier basis using Fourier transforms. The integrands of integrals can then be interpolated\non the fine kmesh by an inverse Fourier transform of the maximally localised quantities,\nthereby enabling the accurate calculations of the damping and inertia.\nThe rest of this paper is organized as follows: In Sec. II, we introduce the expressions for\nthe damping and the inertia. We describe the formalism to calculate the two key quantities:\nGreen’s function and torque matrix elements, using the Wannier interpolation. In Sec. III,\nwe describe the computational details and workflow. In Sec. IV, we discuss the results for\nferromagnets like Fe, Co and Ni, and discuss the agreement with the experimental values\nand the previous studies. In Sec. V, we conclude with a short summary and the future\nprospects for the methods we have developed.\nII. THEORETICAL FORMALISM\nFirst, we describe the expressions for Gilbert damping and moment of inertia within the\ntorque-torque correlation model. Then, we provide a brief description of the MLWFs and\nthe corresponding Wannier formalism for the calculation of torque matrix elements and the\n4Green’s function.\nA. Gilbert damping and Moment of inertia within torque-torque correlation\nmodel\nIf we consider the case when there is no external magnetic field, the electronic structure\nof the system can be described by the Hamiltonian,\nH=H0+Hexc+Hso=Hsp+Hso (2)\nThe paramagnetic band structure is described by H0andHexcdescribes the effective lo-\ncal electron-electron interaction, treated within a spin-polarised (sp) local Kohn-Sham\nexchange-correlation (exc) functional approach, which gives rise to the ferromagnetism. Hso\nis the spin-orbit Hamiltonian. As we are dealing with ferromagnetic materials only, we can\nclub the first two terms as Hsp=H0+Hexc. During magnetization dynamics, (when the\nmagnetization precesses), only the spin-orbit energy of a Bloch state |ψnk⟩is affected, where\nnis the band index of the state. The magnetization precesses around an effective field\nHeff=Hint+Hdamp+HI, where Hintis the internal field due to the magnetic anisotropy\nand exchange energies, Hdampis the damping field, and HIis the inertial field, respectively.\nFrom Eqn. (1), we can see that the damping field Hdamp =α\nMγ∂M\n∂t, while HI=I\nMγ∂2M\n∂t2.\nEquating these damping and inertial fields to the effective field corresponding to the change\nin band energies as magnetization processes, we obtain the mathematical description of the\nGilbert damping and inertia. It was proposed by Kambersky [39] that the change of the\nband energies∂εnk\n∂θµ(θ=θˆndefines the vector for the rotation) can be related to torque\noperator or matrix depending on how the Hamiltonian is being viewed Γµ= [σµ,Hso], where\nσµare the Pauli matrices. Eventually, within the so-called torque-torque correlation model,\nthe Gilbert damping tensor can be expressed as follows:\nαµν=g\nmsπZ Z\u0012\n−d f(ϵ)\ndϵ\u0013\nTr[Γµ(IG)(Γν)†(IG)]d3k\n(2π)3dϵ (3)\nHere trace Tris over the band indices, and msis the magnetic moment. Recently, Thonig\net al [30] have extended such an approach to the case of moment of inertia also, where they\n5deduced the moment of inertia tensor components to be,\nIνµ=gℏ\nmsπZ Z\nf(ϵ)Tr[Γν(IG)(Γµ)†∂2\n∂ϵ2(RG)\n+Γν∂2\n∂ϵ2(RG)(Γµ)†(IG)]d3k\n(2π)3dϵ (4)\nHere f(ϵ) is the Fermi function, ( RG) and ( IG) are the real and imaginary parts of Green’s\nfunction G=(ϵ+ιη− H)−1with ηas a broadening parameter, mis the magnetization in\nunits of the Bohr magneton, Γµ= [σµ,Hso] is the µthcomponent of torque matrix element\noperator or matrix, µ=x, y, z .αis a dimensionless parameter, and I has units of time,\nusually of the order of femtoseconds.\nTo obtain the Gilbert damping and moment of inertia tensors from the above two k-\nintegrals calculated as sums of discrete k-meshes, we need a large number of k-points, such\nas around 106, and more than 107, for the converged values of αand I respectively. The\nreason for the large k-point sampling in the first Brillouin zone (BZ) is because Green’s\nfunction becomes more sharply peaked at its poles at the smaller broadening, ηwhich need\nto be used. For I, the number of k-points needed is more than what is needed for αbecause\n∂2RG/∂ϵ2has cubic powers in RGand/or IGas:\n∂2RG\n∂ϵ2= 2\u0002\n(RG)3−RG(IG)2−IGRGIG−(IG)2RG\u0003\n(5)\nWe note that to carry out energy integration in αit is sufficient to consider a limited\nnumber of energy points within a narrow range ( ∼kBT) around the Fermi level. This\nis mainly due to the exponential decay of the derivative of the Fermi function away from\nthe Fermi level. However, the integral for I involves the Fermi function itself and not its\nderivative. Consequently, while the Gilbert damping is associated with the Fermi surface, the\nmoment of inertia is associated with the entire Fermi sea. Therefore, in order to adequately\ncapture both aspects, it is necessary to include energy points between the bottom of the\nvalence band and the Fermi level.\n6B. Wannier Interpolation\n1.Maximally Localised Wannier Functions (MLWFs)\nThe real-space Wannier functions are written as the Fourier transform of Bloch wave-\nfunctions,\n|wnR⟩=v0\n(2π)3Z\nBZdke−ιk.R|ψnk⟩ (6)\nwhere |ψnk⟩are the Bloch wavefunctions obtained by the diagonalisation of the Hamiltonian\nat each kpoint using plane-wave DFT calculations. v0is the volume of the unit cell in the\nreal space.\nIn general, the Wannier functions obtained by Eqn. (6) are not localised. Usually, the\nFourier transforms of smooth functions result in localised functions. But there exists a phase\narbitrariness of eιϕnkin the Bloch functions because of independent diagonalisation at each\nk, which messes up the localisation of the Wannier functions in real space.\nTo mitigate this problem, we use the Marzari-Vanderbilt (MV) localisation procedure[42–\n44] to construct the MLWFs, which are given by,\n|wnR⟩=1\nNX\nqNqX\nm=1e−ιq.RUq\nmn|ψmq⟩ (7)\nwhere Uq\nmnis a (Nq×N) dimensional matrix chosen by disentanglement and Wannierisation\nprocedure. Nare the number of target Wannier functions, and Nqare the original Bloch\nstates at each qon the coarse mesh, from which Nsmooth Bloch states on the fine k-mesh\nare extracted requiring Nq>Nfor all q,Nis the number of uniformly distributed qpoints\nin the BZ. The interpolated wavefunctions on a dense k-mesh, therefore, are given via inverse\nFourier transform as:\n|ψnk⟩=X\nReιk.R|wnR⟩ (8)\nThroughout the manuscript, we use qandkfor coarse and fine meshes in the BZ, respec-\ntively.\n7FIG. 1: The figure shows the schematic of the localisation of the Wannier functions on a\nRgrid. The matrix elements of the quantities like Hamiltonian on the Rgrid are\nexponentially decaying. Therefore, most elements on the Rgrid are zero (shown in blue).\nWe can hence do the summation till a cutoff Rcut(shown in red) to interpolate the\nquantities on a fine kgrid.\n2.Torque Matrix elements\nAs described in the expressions of αµνand Iνµin Eqns. (3) and (4), the µthcomponent of\nthe torque matrix is given by the commutator of µthcomponent of Pauli matrices and spin-\norbit coupling matrix i.e.Γµ= [σµ,Hso]. Physically, we define the spin-orbit coupling (SOC)\nand spin-orbit torque (SOT) as the dot and cross products of orbital angular momentum and\nspin angular momentum operator, respectively, such that Hso=ξℓ.where ξis the coupling\namplitude. Using this definition of Hso, one can show easily that −ι[σ,Hso] = 2ξℓ×which\nrepresents the torque.\nThere have been several studies on how to calculate the spin-orbit coupling using ab\ninitio numerical approach. Shubhayan et al. [45] describe the method to obtain SOC matrix\nelements in the Wannier basis calculated without SO interaction, using an approximation of\nweak SOC in the organic semiconductors considered in their work. Their method involves\nDFT in the atomic orbital basis, wherein the SOC in the Bloch basis can be related to\nthe SOC in the atomic basis. Then, by the basis transformation, they get the SOC in\nthe Wannier basis calculated in the absence of SO interaction. Farzad et al. [46] calculate\nthe SOC by extracting the coupling amplitude from the Hamiltonian in the Wannier basis,\ntreating the Wannier functions as atomic-like orbitals.\n8We present a different approach wherein we can do the DFT calculation in any basis (plane\nwave or atomic orbital). Unlike the previous approaches, we perform two DFT calculations\nand two Wannierisations: one is with spin-orbit interaction and finite magnetisation (SO)\nand the other is spin-polarised without spin-orbit coupling (SP). The spin-orbit Hamiltonian,\nHsocan then be obtained by subtracting the spin-polarized Hamiltonian, Hspfrom the full\nHamiltonian, HasHso=H−H sp. This, however, can only be done if both the Hamiltonians,\nHandHsp, are written in the same basis. We choose to use the corresponding Wannier\nfunctions as a basis. It should however be noted that, when one Wannierises the SO and\nSP wavefunctions, one will get two different Wannier bases. As a result, we can not directly\nsubtract the HandHspin these close but different Wannier bases. In order to do the\nsubtraction, we find the transformation between two Wannier bases , i.e. express one set of\nWannier functions in terms of the other. Subsequently, we can express the matrix elements\nofHandHspin the same basis and hence calculate Hso. In the equations below, the Wannier\nfunctions, the Bloch wavefunctions and the operators defined in the corresponding bases in\nSP and SO calculations are represented with and without the tilde ( ∼) symbol, respectively.\nTheNSO Wannier functions are given by:\n|wnR⟩=1\nNX\nqNqX\nm=1e−ιq.RUq\nmn|ψmq⟩ (9)\nwhere Uq\nmnis a (Nq× N) dimensional matrix. The wavefunctions and Wannier functions\nfrom the SO calculation for a particular qandRare a mixture of up and down spin states\nand are represented as spinors:\n|ψnq⟩=\n|ψ↑\nnq⟩\n|ψ↓\nnq⟩\n |wnR⟩=\n|w↑\nnR⟩\n|w↓\nnR⟩\n (10)\nThe ˜NsSP Wannier functions are given by:\n|˜ws\nnR⟩=1\nNX\nq˜Ns\nqX\nm=1e−ιq.R˜Uqs\nmn|˜ψs\nmq⟩ (11)\nwhere s=↑,↓.˜Uqs\nmnis a ( ˜Ns\nqטNs) dimensional matrix. Since the spinor Hamiltonian\ndoesn’t have off-diagonal terms corresponding to opposite spins in the absence of SOC, the\nwavefunctions will be |˜ψs\nnq⟩=|˜ψnq⟩⊗|s⟩. The combined expression for ˜Uqfor˜N↑+˜N↓=˜N\n9FIG. 2: This figure shows the implementation flow chart of the theoretical formalism\ndescribed in Sec. II\nSP Wannier functions is:\n˜Uq=\n˜Uq↑0\n0˜Uq↓\n (12)\nwhere ˜Uqis˜NqטNdimensional matrix with ˜Nq=˜N↑\nq+˜N↓\nq. Dropping the sindex for\nSP kets results in the following expression for ˜NSP Wannier functions:\n|˜wnR⟩=1\nNX\nq˜NqX\nm=1e−ιq.R˜Uq\nmn|˜ψmq⟩ (13)\n10We now define the matrix of the transformation between SO and SP Wannier bases as:\nTRR′\nmn=⟨˜wmR|wnR′⟩\n=1\nN2X\nqq′˜Nq,Nq′X\np,l=1eι(q.R−q′.R′)˜Uq†\nmp⟨˜ψpq|ψlq′⟩Uq′\nln\n=1\nN2X\nqq′eι(q.R−q′.R′)[˜Uq†Vqq′Uq′]mn (14)\nHereVqq′\npl=⟨˜ψpq|ψlq′⟩. Eqn. (14) is the most general expression to get the transformation\nmatrix. We can reduce this quantity to a much simpler one using the orthogonality of\nwavefunctions of different q. Eqn. (14) hence becomes,\nTRR′\nmn=1\nN2X\nqeιq.(R−R′)[˜Uq†(NVq)Uq]mn\n=1\nNX\nqeιq.(R−R′)[˜Uq†VqUq]mn (15)\nwhere Vq\npl=⟨˜ψpq|ψlq⟩. Using this transformation, we write SP Hamiltonian in SO Wannier\nbases as:\n(Hsp)RR′\nmn=⟨wmR|Hsp|wnR′⟩\n=X\nplR′′R′′′⟨wmR|˜wpR′′⟩\n⟨˜wpR′′|Hsp|˜wlR′′′⟩⟨˜wlR′′′|wnR′⟩\n=X\nplR′′R′′′(T†)RR′′\nmp(˜Hsp)R′′R′′′\npl TR′′′R′\nln (16)\nSince Wannier functions are maximally localised and generally atomic-like, the major con-\ntribution to the overlap TRR′\nmnis for R=R′. Therefore, we can write TRR\nmn=T0\nmn. The\nreason is that it depends on relative R−R′, we can just consider overlaps at R=0. Eqn.\n16 becomes,\n(Hsp)RR′\nmn=X\npl(T†)0\nmp(˜Hsp)RR′\nplT0\nln (17)\nTherefore, we write the Hsoin Wannier basis as,\n(Hso)RR′\nmn=HRR′\nmn−(Hsp)RR′\nmn (18)\n11The torque matrix elements in SO Wannier bases are given by,\n(Γµ)RR′\nmn= (σµHso)RR′\nmn−(Hsoσµ)RR′\nmn (19)\nConsider ( σµHso)RR′\nmnand insert the completeness relation of the Wannier functions, and\nalso neglecting SO matrix elements between the Wannier functions at different sites because\nof their being atomic-like.\n(σµHso)RR′\nmn=P\npR′′(σµ)RR′′\nmp(Hso)R′′R′\npn\n= (σµ)RR′\nmp(Hso)0\npn (20)\n(σµ)RR′\nmpis calculated by the Fourier transform of the spin operator written in the Bloch\nbasis, just like the Hamiltonian.\n(σµ)RR′\nmp=1\nNX\nqe−ιq.(R′−R)\u0002\nUq†(σµ)qUq\u0003\nmp(21)\nWe interpolate the SOT matrix elements on a fine k-mesh as follows:\n(Γµ)k\nmn=X\nR′−Reιk.(R′−R)(σµ)RR′\nmn (22)\nThis yields the torque matrix elements in the Wannier basis. In the subsequent expres-\nsions, WandHsubscripts represent the Wannier and Hamiltonian basis, respectively. In\norder to rotate to the Hamiltonian gauge, which diagonalises the Hamiltonian interpolated\non the fine kmesh using its matrix elements in the Wannier basis.\n(HW)k\nmn=X\nR′−Reιk.(R′−R)HRR′\nmn (23)\n(HH)k\nmn=\u0002\n(Uk)†(HW)kUk\u0003\nmn(24)\nHere Uk(not to be confused with Uq) are the matrices with columns as the eigenvectors of\n(HW)k, and ( HH)k\nmn=ϵmkδmn. We use these matrices to rotate the SOT matrix elements\nin Eqn. (22) to the Hamiltonian basis as:\n(Γµ\nH)k\nmn=\u0002\n(Uk)†(Γµ\nW)kUk\u0003\nmn(25)\n123.Green’s functions\nThe Green’s function at an arbitrary kandϵon a fine k-mesh in the Hamiltonian basis\nis given by:\nGk\nH(ϵ+ιη) = (ϵ+ιη−(HH)k)−1(26)\nwhere ηis a broadening factor and is caused by electron-phonon coupling and is generally\nof the order 5 −10 meV. Gk\nH(ϵ+ιη) is aN × N dimensional matrix.\nTherefore, we can calculate RG,IGand∂2RG/∂ϵ2as defined in Eqn. (5) and hence, α\nand I.\nIII. COMPUTATIONAL DETAILS\nPlane-wave pseudopotential calculations were carried out for the bulk ferromagnetic tran-\nsition metals bcc Fe, hcp Co and fcc Ni using Quantum Espresso package[47, 48]. The\nconventional unit cell lattice constants ( a) used for bcc Fe and fcc Ni were 5.424 and 6.670\nbohrs, respectively and for hcp Co, a=4.738 bohr and c/a=1.623 were used. The non-\ncollinear spin-orbit and spin-polarised calculations were performed using fully relativistic\nnorm-conserving pseudopotentials. The kinetic energy cutoff was set to 80 Ry. Exchange-\ncorrelation effects were treated within the PBE-GGA approximation. The self-consistent\ncalculations were carried out on 16 ×16×16 Monkhorst-Pack Grid using Fermi smearing of\n0.02 Ry. Non-self-consistent calculations were carried out using the calculated charge den-\nsities on Γ-centered 10 ×10×10 coarse k-point grid. For bcc Fe and fcc Ni, 64 bands were\ncalculated and hcp Co 96 bands were calculated (because there are two atoms per unit cell\nfor Co). We define a set of 18 trial orbitals sp3d2,dxy,dxz, and dyzfor Fe, 18 orbitals\nper atom s,panddfor Co and Ni, to generate 18 disentangled spinor maximally-localized\nWannier functions per atom using Wannier90 package [43].\nFrom the Wannier90 calculations, we get the checkpoint file .chk, which contains all the\ninformation about disentanglement and gauge matrices. We use .spnand.eigfiles generated\nbypw2wannier90 to get the spin operator and the Hamiltonian in the Wannier basis. We\nevaluate the SOT matrix elements in the Wannier Basis.\nWe get αby simply summing up on a fine- kgrid with appropriate weights for the k-\nintegration, and we use the trapezoidal rule in the range [-8 δ,8δ] for energy integration\n1310−610−410−2100\n10−610−410−210010−610−410−210010\n10\n10\n10\n10\n101\n0\n1−2\n−2\n−310−410−310−210−1100101\n10−410−310−210−1100101\nFe Ni Co(a) (b) (c)\nFIG. 3: (a), (b) and (c) shows the αvsηfor Fe, Ni and Co, respectively. Damping\nconstants calculated using the Wannier implementation are shown in blue. Damping\ncalculated using the tight binding method based on Lorentzian broadening and Green’s\nfunction by Thonig et al[29] are shown in brown and green, respectively. Comparison with\ndamping constants calculated by Gilmore et al[32] using local spin density approximation\n(LSDA) are shown in red. The dotted lines are guides to the eye.\naround the Fermi level where δis the width of the derivative of Fermi function ∼kBT. We\nconsider 34 energy points in this energy range. We perform the calculation for T= 300K.\nFor the calculation of I, we use a very fine grid of 400 ×400×400k-points. For η >0.1,\nwe use 320 energy points between VBM and Fermi energy. For 0 .01< η < 0.1, we use\n3200-6400 energy points for the energy integration.\nTABLE I\nMaterial η(meV) -I (fs) α(×10−3) τ(ps)\nFe 6 0.210 3.14 0.42\n8 0.114 2.77 0.26\n10 0.069 2.51 0.17\nNi 10 2.655 34.2 0.48\nCo 10 0.061 1.9 0.21\n14FIG. 4: Schematic explaining the dependence of intraband and interband contribution in\nαwith η.\nIV. RESULTS AND DISCUSSION\nA. Damping constant\nIn this section, we report the damping constants calculated for the bulk iron, cobalt and\nnickel. The magnetic moments are oriented in the z-direction. For reference direction z, the\ndamping tensor is diagonal resulting in the effective damping constant α=αxx+αyy.\nIn Fig. 3, we report the damping constants calculated by the Wannier implementation as\na function of broadening, ηknown to be caused by electron-phonon scattering and scattering\nwith impurities. We consider the values of ηranging from 10−6to 2 eV to understand the\nrole of intraband and interband transitions as reported in the previous studies[29, 32]. We\nnote that the experimental range is for the broadening is expected to be much smaller with\nη∼5−10 meV. The results are found to be in very good agreement with the ones calculated\nusing local spin density approximation (LSDA)[32] and tight binding paramterisation[29].\nThe expression for Gilbert damping[3] is written in terms of the imaginary part of Green’s\nfunctions. Using the spectral representation of Green’s function, Ank(ω), we can rewrite Eqn.\n15(3) as:\nαµν=gπ\nmsX\nnmZ\nTµ\nnm(k)T∗ν\nnm(k)Snmdk (27)\nwhere Snm=R\nη(ϵ)Ank(ϵ)Amk(ϵ)dϵis the spectral overlap. Although we are working in the\nbasis where the Hamiltonian is diagonal, the non-zero off-diagonal elements in the torque\nmatrix lead to both intraband ( m=n) and interband ( m̸=n) contributions. For the\nsake of simple physical understanding, we consider the contribution of the spectral overlaps\nat the Fermi level for both intraband and interband transitions in Fig. 4. But in the\nnumerical calculation temperature broadening has also been considered. For the smaller\nη, the contribution of intraband transitions decreases almost linearly with the increase in\nηbecause the overlaps become less peaked. Above a certain η, the interband transitions\nbecome dominant and the contribution due to the overlap of two spectral functions at\ndifferent band indices m and n becomes more pronounced at the Fermi level. Above the\nminimum, the interband contribution increases till η∼1 eV. Because of the finite Wannier\norbitals basis, we have the accurate description of energy bands only within the approximate\nrange of ( ϵF−10, ϵF+ 5) eV for the ferromagnets in consideration. The decreasing trend\nafter η∼1 eV is, therefore, an artifact.\n10−2\n0.000.050.100.150.20\n10−1100-0.001 0.000 0.001 0.002\n10−1100\nFIG. 5: Plot showing moment of inertia, −I versus broadening, η. The moment of inertia\nin the range 0 .03−3.0 eV is shown as an inset. The values using the Wannier\nimplementation and the tight binding method[30] are shown in blue and green, respectively.\n160.020.040.060.08\nCo(a)\n10−210−11000.0010−1100−0.0075−0.0050−0.0050−0.0025 0.000\n10−210−1100−\n0.00.51.0 1.01.5\nNi(b)\n0.000.010.02\n100101\nFIG. 6: (a) and (b) show negative of the moment of inertia, −I versus broadening, ηfor\nCo and Ni, respectively. The values using the Wannier implementation and the tight\nbinding method are shown in magenta and cyan, respectively. The moment of inertia is\nshown as an inset in the range 0 .03−2.0 eV and 0 .03−3.0 eV for Co and Ni, respectively.\nB. Moment of inertia\nIn Fig. 5, we report the values for the moment of inertia calculated for bulk Fe, Co\nand Ni. Analogous to the damping, the inertia tensor is diagonal, resulting in the effective\nmoment of inertia I = Ixx+ Iyy.\nThe behaviour for I vs ηis similar to that of the damping, with smaller and larger ηtrends\narising because of intraband and interband contributions, respectively. The overlap term\nin the moment of inertia is between the ∂2RG/∂ϵ2andIGunlike just IGin the damping.\nIn Ref. [30], the moment of inertia is defined in terms of torque matrix elements and the\noverlap matrix as:\nIµν=−gℏ\nmsX\nnmZ\nTµ\nnm(k)T∗ν\nnm(k)Vnmdk (28)\nwhere Vnmis an overlap function, given byR\nf(ϵ)(Ank(ϵ)Bmk(ϵ) +Bnk(ϵ)Amk(ϵ))dϵand\nBmk(ϵ) is given by 2( ϵ−ϵmk)((ϵ−ϵmk)2−3η2)/((ϵ−ϵmk)2+η2)3. There are other notable\nfeatures different from the damping. In the limit η→0, the overlap Vmnreduces to 2 /(ϵmk−\nϵnk)3. For intraband transitions ( m=n), this leads to I → −∞ . In the limit η→ ∞ ,\nVmn≈1/η5which leads to I →0. The behaviour at these two limits is evident from Fig. 5.\nThe large τ(small η) behaviour is consistent with the expression I = −α.τ/2πderived by\n17104\n−10−310−2−10−1100\n10−2−10−110−1100101102\nFIG. 7: The damping, magnetic moment of inertia and relaxation rate are shown as a\nfunction of broadening, ηin blue, green and red, respectively. The grey-shaded region\nshows the observed experimental relaxation rate, τ, ranging from 0 .12 to 0 .47 ps. The\ncorresponding range of ηis shown in purple and is 6 −12 meV. This agrees with the\nexperimental broadening in the range of 5 −10 meV, arising from electron-phonon\ncoupling. The numbers are tabulated in Table I\nF¨ahnle et al. [49]. Here τis the Bloch relaxation lifetime. The behaviour of τas a function\nofηusing the above expression in the low ηlimit is shown in Fig. 7. Apart from these\nlimits, the sign change has been observed in a certain range of ηfor Fe and Co. This change\nin sign can be explained by the Eqn. (5). In the regime of intraband contribution, at a\ncertain η, the negative and positive terms integrated over ϵandkbecome the same, leading\nto zero inertia. Above that η, the contribution due to the negative terms decreases until the\ninterband contribution plays a major role leading to maxima in I (minima in −I). Interband\ncontribution leads to the sign change from + to - and eventually zero at larger η.\nThe expression I = −α.τ/2πderived from the Kambersky model is valid for η <10 meV,\nwhich indicates that damping and moment of inertia have opposite signs. By analyzing the\nrate of change of magnetic energy, Ref. [11] shows that Gilbert damping and the moment\nof inertia have opposite signs when magnetization dynamics are sufficiently slow (compared\ntoτ).\nExperimentally, the stiffening of FMR frequency is caused by negative inertia. The\nsoftening caused by positive inertia is not observed experimentally. This is because the\nexperimentally realized broadening, ηcaused by electron-phonon scattering and scattering\nwith impurities, is of the order of 5 −10 meV. The values of Bloch relaxation lifetime, τ\n18measured at the room temperature with the FMR in the high-frequency regime for Ni 79Fe21\nand Co films of different thickness, range from 0 .12−0.47 ps. The theoretically calculated\nvalues for Fe,Ni and Co using the Wannier implementation for the ηranging from 5 −10\nmeV are reported in Table. I and lies roughly in the above-mentioned experimental range\nfor the ferromagnetic films.\nV. CONCLUSIONS\nIn summary, this paper presents a numerical method to obtain the Gilbert damping\nand moment of inertia based on the torque-torque correlation model within an ab initio\nWannier framework. We have also described a technique to calculate the spin-orbit coupling\nmatrix elements via the transformation between the spin-orbit and spin-polarised basis. The\ndamping and inertia calculated using this method for the transition metals like Fe, Co and\nNi are in good agreement with the previous studies based on tight binding method[29, 30]\nand local spin density approximation[32]. We have calculated the Bloch relaxation time\nfor the approximate physical range of broadening caused by electron-phonon coupling and\nlattice defects. The Bloch relaxation time is in good agreement with experimentally reported\nvalues using FMR[27]. The calculated damping and moment of inertia can be used to study\nthe magnetisation dynamics in the sub-ps regime. In future studies, we plan to use the\nWannier implementation to study the contribution of spin pumping terms, arising from\nthe spin currents at the interface of ferromagnetic-normal metal bilayer systems due to\nthe spin-orbit coupling and inversion symmetry breaking to the damping. 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Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: December 4, 2018)\nA temperature gradient applied to a ferromagnetic metal induces not only independent \rows of\nelectrons and magnons but also drag currents because of their mutual interaction. In this paper, we\npresent a microscopic study of the electron \row induced by the drag due to magnons. The analysis\nis based on the s-dmodel, which describes conduction electrons and magnons coupled via the s-d\nexchange interaction. Magnetic impurities are introduced in the electron subsystem as a source\nof spin relaxation. The obtained magnon-drag electron current is proportional to the entropy of\nmagnons and to \u000b\u0000\f(more precisely, to 1 \u0000\f=\u000b), where\u000bis the Gilbert damping constant and\n\fis the dissipative spin-transfer torque parameter. This result almost coincides with the previous\nphenomenological result based on the magnonic spin-motive forces, and consists of spin-transfer\nand momentum-transfer contributions, but with a slight disagreement in the former. The result\nis interpreted in terms of the nonequilibrium spin chemical potential generated by nonequilibrium\nmagnons.\nI. INTRODUCTION\nTransport phenomena in ferromagnetic metals exhibit\nsurprisingly rich physics as unveiled by intensive stud-\nies in spintronics. This is largely because they involve\ntransport of not only charge and heat but also spin\nangular momentum. In the presence of magnetization\ntextures, applying an electric current induces magneti-\nzation dynamics because of spin-transfer torques that\nthe spin current of electrons exerts on the magnetiza-\ntion [1, 2]. In turn, a time-dependent magnetization\ninduces spin and charge currents of electrons via spin-\nmotive forces that are reciprocal to the spin-transfer\ntorques [3]. Even when the (equilibrium) magnetization\nis uniform, its thermal/quantum \ructuations, i.e., spin\nwaves or magnons, can interact with electrons. More-\nover, transport through an inhomogeneous region induces\nnonequilibrium spin accumulation, both in electrons and\nmagnons, which then induce di\u000busion spin currents. The\nconcept of \\spin chemical potential\" [4] and \\magnon\nchemical potential\" [5] have been introduced to describe\nsuch e\u000bects.\nOne of the important e\u000bects in the interplay of elec-\ntrons and magnons in transport phenomena are drag ef-\nfects. When subjected to a temperature gradient, elec-\ntrons and magnons start to \row, \frst independently, and\nthen by dragging with each other. Thermoelectric mea-\nsurements indicate the presence of magnon-drag contri-\nbutions in Fe [6], NiCu [7], NiFe [8], and in Fe, Co and\nNi [9]. Theoretical studies include both phenomenolog-\nical [7, 10, 11] and microscopic [12] ones. In particular,\nphenomenological studies based on the spin-motive force\npicture [10, 11] indicate the importance of the dissipative\n\fparameter, which stems from spin relaxation of elec-trons. Microscopic treatment of spin-relaxation e\u000bects\nrequires the consideration of so-called vertex corrections,\nbeyond the simple self-energy (damping or scattering-\ntime) e\u000bects, as noted in the study of current-induced\nspin torque [13, 14], but such studies are not available\nyet for the drag e\u000bects. In a related work, which stud-\nies spin torques due to magnons, a careful treatment of\nthe spin-relaxation e\u000bects revealed an additional contri-\nbution not obtained in a phenomenological analysis [15].\nTherefore, one may expect an analogous situation also in\nmagnon-drag transport phenomena.\nIn this paper, we present a microscopic analysis of\nmagnon-drag electric current (or electron \row) induced\nby a temperature gradient. Using as a microscopic model\nthes-dmodel that describes conduction electrons inter-\nacting with magnons, we calculate the electric current\ncaused by magnons that are driven by the temperature\ngradient. The temperature gradient is treated by its me-\nchanical equivalent, a \fctitious gravitational \feld, intro-\nduced by Luttinger [16]. The obtained result consists\nof two terms, which may be interpreted as due to the\nspin-transfer e\u000bect and the momentum-transfer e\u000bect,\nas in the phenomenological theory [11]. However, as to\nthe former (spin-transfer e\u000bect), there is a quantitative\ndi\u000berence, and our result is proportional to \u000b\u0000\f(or\n1\u0000\f=\u000b), where\u000bis the Gilbert damping constant. It\nvanishes, and changes sign, at \u000b=\f, which agrees with\nthe intuitive notion that the case \u000b=\fis very spe-\ncial. Although this is mostly of conceptual importance,\nit may acquire a practical one if one can determine the\nvalue of\f=\u000b from magnon-drag experiments. We inter-\npret the results in terms of the spin chemical potential\ninduced by magnons. In the course of our study, we give\nan argument that justi\fes the Luttinger's argument byarXiv:1812.00720v1  [cond-mat.mes-hall]  3 Dec 20182\nan explicit calculation.\nThe organization of the paper is as follows. In Sec. II,\nwe describe the microscopic model and some calcula-\ntional tools such as Green's functions. In Sec. III, we out-\nline the microscopic calculation of magnon-drag electron\n\row. The result is discussed in terms of spin-motive force\nand spin chemical potential. In Sec. IV, we revisit the\nphenomenological theory based on the spin-motive force,\nand compare the result with our microscopic result. In\nSec. V, we give an alternative analysis which \\derives\"\nthe spin chemical potential. Details of the microscopic\ncalculations are presented in Appendices A and B. In Ap-\npendix C, we reanalyze the phenomenological theory in\nanother way using the stochastic Landau-Lifshitz-Gilbert\nequation.\nII. MODEL\nA. Hamiltonian\nWe consider a system consisting of conduction elec-\ntrons and magnons in a ferromagnetic metal with uniform\nequilibrium magnetization. The Hamiltonian is given by\nH=H0\nel+Hmag+Hsd; (1)\nH0\nel=Z\ndr\u00141\n2m(@icy)(@ic) +cy(Vimp\u0000\u0016)c\u0015\n;(2)\nHmag=X\nq!qay\nqaq; (3)\nHsd=\u0000JsdZ\ndrcy(S\u0001\u001b)c; (4)\nwherec=t(c\";c#) andcy= (cy\n\";cy\n#) are annihilation and\ncreation operators of the electrons, aqanday\nqare those\nof magnons, mand\u0016are the mass and the chemical\npotential of the electrons, !q=Jq2+ \u0001 is the magnon\ndispersion with exchange sti\u000bness Jand energy gap \u0001,\nS=Sn(jnj= 1) is the localized spin with magnitude\nS,\u001b= (\u001bx;\u001by;\u001bz) are Pauli matrices, and Jsdis the\ns-dexchange coupling constant. We consider low enough\ntemperature and assume Sis constant. Hereafter we use\nM\u0011JsdSandninstead ofS. ForVimp, we consider\nboth nonmagnetic and magnetic impurities,\nVimp(r) =uiX\ni\u000e(r\u0000Ri) +usX\njSj\u0001\u001b\u000e(r\u0000R0\nj);\n(5)\nwhereSjis the impurity spin located at position R0\nj. We\naverage over the impurity positions, RiandR0\nj, as usual,\nand the impurity spin directions,\nS\u000b\niS\f\nj=\u000eij\u000e\u000b\f\u0002\u001a\nS2\n?(\u000b=\f=x;y)\nS2z(\u000b=\f=z): (6)The s-dexchange interaction describes the exchange-\nsplitting in the electron spectrum, and the electron-\nmagnon scattering,\nHsd=\u0000MZ\ndrcy\u001bzc+Hel\u0000mag; (7)\nHel\u0000mag=MZ\ndr\u00141\ns0aya^\u001bz\u0000r\n2\ns0\u0000\na^\u001b\u0000+ay^\u001b+\u0001\u0015\n;\n(8)\nwheres0=S=r3\n0is the spin density of the magnetization,\nr0the lattice constant, ^\u001b=cy\u001bc, and ^\u001b\u0006= (^\u001bx\u0006i^\u001by)=2.\nThe total Hamiltonian is given by\nH=Hel+Hmag+Hel\u0000mag; (9)\nHel=Z\ndr\u00141\n2m(@icy)(@ic) +cy(Vimp\u0000\u0016)c\u0000Mcy\u001bzc\u0015\n:\n(10)\nB. Green's function\nThe Green's functions of electrons Gk\u001b(i\"n) and\nmagnonsDq(i\u0017l) are given by\nGk\u001b(i\"n) =1\ni\"n+\u0016\u0000k2=2m+\u001bM\u0000\u0006\u001b(i\"n);(11)\nDq(i\u0017l) =1\ni\u0017l\u0000!q\u0000\u0005q(i\u0017l); (12)\nwith Matsubara frequencies, \"n= (2n+ 1)\u0019Tand\u0017l=\n2\u0019lT, and self-energies, \u0006 \u001b(i\"n) and \u0005 q(i\u0017l), for the elec-\ntrons and magnons, respectively.\nFIG. 1. (a) Self-energy of electrons, \u0006. (b) Self-energy of\nmagnons, \u0005. (c) Spin vertex \u0003\u000brenormalized by impurity-\nladder corrections. (d) Four-point vertex \u0000 \u001b\u001b0, which we\ncall the di\u000busion-type vertex correction, or simply, the dif-\nfusion propagator. The solid (wavy) lines represent electron\n(magnon) propagators, and the dashed line with a cross rep-\nresents impurity scattering.3\nWe assume the electron self-energy is dominated by im-\npurity scattering and treat it in the Born approximation\n[Fig. 1 (a)]. Thus, \u0006R\n\u001b(\") = \u0006\u001b(\"+i0) =\u0000i\r\u001b, with\n\r\u001b=\u0019(\u00001\u0017\u001b+ \u00002\u0017\u0016\u001b)\u00111\n2\u001c\u001b; (13)\nand\n\u00001=niu2\ni+nsu2\nsS2z; \u00002= 2nsu2\nsS2\n?: (14)\nHere,ni(ns) is the concentration of nonmagnetic (mag-\nnetic) impurities, and \u0017\u001bis the density of states of spin- \u001b\nelectrons.\nThe magnon self-energy comes from the electron-\nmagnon scattering [Fig. 1 (b)]. Expanding with respect\nto the wave vector qand the frequency \u0017of magnons, we\nwrite\n\u0005q(\u0017+i0) =\u0000\u0014\u000eS\nS+i\u000b\nz\u0015\n\u0017\u0000\u000eJq2+O(\u00172;q4):(15)\nHere,\u000eS,\u000eJandz\u0011S=(S+\u000eS) are the renormaliza-\ntion constants for spin, the exchange sti\u000bness, and wave\nfunction, respectively, of the localized spins. Also, \u000bis\nthe Gilbert damping constant calculated as [13]\n\u000b=\u0019nsu2\nsh\n2S2z\u0017\"\u0017#+S2\n?(\u00172\n\"+\u00172\n#)i\nz=s0: (16)\nHere and hereafter, we assume the s-dexchange coupling\nMis much larger than the spin-relaxation rate [15].\nAs seen fromHel\u0000mag[Eq. (8)], the natural expan-\nsion parameter in the electron-magnon problem is s\u00001\n0\n(orS\u00001). In this paper, we focus on the leading contri-\nbutions, which are O(s\u00001\n0). (As seen below, we need two\nelectron-magnon scattering vertices in the magnon-drag\nprocess, giving\u0018(s\u00001=2\n0)2=s\u00001\n0.) Since\u000eSand\u000eJare\nO(s\u00001\n0), we setz= 1 and\u000eJ= 0 in the magnon Green's\nfunction.\nIII. MICROSCOPIC CALCULATION\nA. Thermal linear-response theory\nTo treat the temperature gradient in the linear re-\nsponse theory, we introduce Luttinger's (\fctitious) gravi-\ntational potential  , which couples to the energy density\nh(r) of the system [16]. The coupling is described by the\nHamiltonian,\nH0=Z\ndrh(r) (r;t): (17)\nWe consider the case,  (r;t) = Q;!ei(Q\u0001r\u0000!t), whereQ\nand!are the wave vector and the frequency of  , and\nwrite the linear response of a physical quantity Ato as\nhAi \n!=\u0000hA;h(\u0000Q)i!+i0 Q;!; (18)\nFIG. 2. Feynman diagrams for Kij(i!\u0015) [Eq. (26)], which\ndescribe the magnon-drag processes. The solid (wavy) lines\nrepresent the electron (magnon) Green's functions. (a) Pro-\ncesses for Q=0. The gray triangles are de\fned in Fig. 1 (c).\n(b) Additional processes that contribute when Q6=0. The\ngray square represents the di\u000busion propagator \u0000 \u001b\u001b0de\fned\nin Fig. 1 (d). The diagrams in (b) vanish for Q=0, but\ncontribute for \fnite Qand lead to Eq. (57).\nwhereh(\u0000Q) is the Fourier component of h(r). The\nresponse function hA;Bi!+i0is obtained from\nhA;Bii!\u0015\u0011ZT\u00001\n0d\u001cei!\u0015\u001chT\u001cA(\u001c)Bieq; (19)\nby the analytic continuation, i!\u0015!!+i0, whereA\nandBare arbitrary operators. Here, Tis the tempera-\nture andh\u0001\u0001\u0001ieqrepresents the average in thermal equi-\nlibrium. Hereafter we use h\u0001\u0001\u0001i instead ofh\u0001\u0001\u0001ieqfor\nsimplicity. Using the continuity equation,\n@th(r) +@ijQ\ni= 0; (20)\nwhich de\fnes the heat-current density jQ\ni, we rewrite\nEq. (18) as a linear response to ( \u0000@i ) [17],\nhAi \n!=Ki(!+i0)\u0000Ki(0)\ni!\u0012\n\u0000@j \u0000@jT\nT\u0013\nQ;(21)\nKi(i!\u0015) =hA;jQ\ni(\u0000Q)ii!\u0015: (22)\nHere, we introduced the temperature gradient @iT\nthrough the combination, \u0000@i \u0000@iT=T. This is justi\fed\nfor operators Aof which the average vanishes naturally\nin the equilibrium state, where @iT=T +@i = 0 holds\n[16, 17]. Therefore, the response to ( \u0000@iT=T) is obtained\nas the response to ( \u0000@i ) [16].\nB. Magnon-drag process\nSpecializing to the present model, Eq. (9), we \fnd from\nEq. (20) that the heat-current density jQ\niconsists of two4\nparts,jQ\ni=jQ\nel;i+jQ\nmag, one for the electrons ( jQ\nel;i) and\none for magnons,\njQ\nmag;i=\u0000J\u0002\n_ay(@ia) + (@iay)_a\u0003\n; (23)\nwhere _a=@ta.\nIn this paper, we are interested in the magnon-drag\nprocess, which corresponds to taking the magnon heat-\ncurrent density jQ\nmag;iforjQ\niin Eq. (22). As for Ain\nEq. (22), we focus on the electron (number) current den-\nsity,\njel;i=~\n2mi\u0002\ncy(@ic)\u0000(@icy)c\u0003\n: (24)Therefore, we consider\nhjel;iidrag=Kij(!+i0)\u0000Kij(0)\ni!\u0012\n\u0000@j \u0000@jT\nT\u0013\n;\n(25)\nKij(i!\u0015) =hjel;i(Q) ;jQ\nmag;j(\u0000Q)ii!\u0015; (26)\ni.e., the correlation function between the electron (num-\nber) current and the magnon heat current. Here jel;i(Q)\nandjQ\nmag;j(\u0000Q) represent their respective Fourier com-\nponents. The combination \u0000@j \u0000@jT=T in Eq. (25) in-\ndicates that the current vanishes in the equilibrium state,\nin which\u0000@j \u0000@jT=T = 0 (Einstein-Luttinger relation)\nholds. We will verify this form by an explicit calculation\nin Secs. V-A and V-B.\nThe relevant magnon-drag processes are shown dia-\ngrammatically in Fig. 2 (a). These are the leading con-\ntribution with respect to 1 =s0, and expressed as\nKij(i!\u0015) =2M2\ns0TX\nl;quj\u001a\u0012\ni\u0017l+i!\u0015\n2\u0013\nDq(i\u0017l+i!\u0015)Dq(i\u0017l)\u00001\n2[Dq(i\u0017l+i!\u0015) +Dq(i\u0017l)]\u001b\nEi; (27)\nwhereui= 2Jqiis the magnon velocity, !\u0015is the Matsubara frequency of the external perturbation  , and we have\nsetQ=0for simplicity. The terms linear in Dqare \\corrections\" arising from the \u000e-function in the relation [17],\nhT\u001ca(\u001c) _ayi=\u0000hT\u001c_a(\u001c)ayi=d\nd\u001cD(\u001c) +\u000e(\u001c): (28)\nThese terms, combined with the \frst term ( \u0018DqDq) in the curly brackets, lead to f\u0001\u0001\u0001g =f!q+1\n2[\u0005q(i\u0017l+i!\u0015) +\n\u0005q(i\u0017l)]gDqDq. This amounts to making a replacement, i\u0017l+i!\u0015=2!!q, in the \frst term if the self-energies are\nneglected.\nThe last factorEiin Eq. (27) is the electron part coming from the electron triangles in Fig. 2 (a),\nEi=TX\nn;kvi\u0002\nGk#(i\"n+i!\u0015) \u0003\u0000\n#\"Gk\u0000q;\"(i\"n\u0000i\u0017l) \u0003+\n\"#Gk#(i\"n)\n+Gk\"(i\"n+i!\u0015) \u0003+\n\"#Gk+q;#(i\"n+i\u0017l+i!\u0015) \u0003\u0000\n#\"Gk\"(i\"n)\u0003\n; (29)\nwherevi=ki=mis the electron velocity and \u0003\u0006\n\u001b\u001b0's are\nthe renormalized spin ( \u001b\u0006) vertices; see Appendix A for\nthe de\fnition. After the analytic continuations, i\u0017l!\u0017\nandi!\u0015!!+i0, an expansion is made with respect to!and/or\u0017. From Eq. (25), we are primarily interested\nin the!-linear terms. The factor !comes either from the\nmagnon part or from the electron part. Hence we write\nKij(!+i0)\u0000Kij(0)\n'2M2\ns0i!\n2\u0019(Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013\n\u0017X\nqujDR\nq(\u0017)DA\nq(\u0017)E(2)\ni\u00002Z\nd\u0017n(\u0017)\u0017X\nqujIm\"\nDR\nq(\u0017)DR\nq(\u0017)E(1)\ni\n\u0000i!#\n\u00001\n2Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013X\nquj\u0002\nDR\nq(\u0017) +DA\nq(\u0017)\u0003\nE(2)\ni+ 2Z\nd\u0017n(\u0017)X\nqujIm\"\nDR\nq(\u0017)E(1)\ni\n\u0000i!#)\n;(30)\nwheren(\u0017) = (e\u0017=kBT\u00001)\u00001is the Bose-Einstein dis- tribution function. The terms in the second line are the5\ncorrections mentioned above. E(1)\niis obtained fromEiby\nthe analytic continuation, i(\u0017l+!\u0015)!\u0017+!+i0 and\ni\u0017l!\u0017+i0, andE(2)\nibyi(\u0017l+!\u0015)!\u0017+!+i0 and\ni\u0017l!\u0017\u0000i0. In the term with E(2)\ni,!is picked up from\nthe magnon part, whereas in the term with E(1)\ni,!is\nobtained from the electron part.\nAt this point, it is worth noting that not only DRDA\nbut alsoDRDRappears in Eq. (30) for the pair of\nmagnon propagators. This is not surprising in diagram-\nmatic calculations as being done here, but seems incom-\npatible with the spin-motive force picture, in which there\nshould be a causal relationship between the magnetiza-\ntion dynamics and the resulting current (see Sec. IV).\nWe retain low-order terms with respect to \u0017, which is\njusti\fed because the magnon energy \u0017is typically small\ncompared to the electron Fermi energy. Deferring the de-\ntails to Appendix B, the electron part has been calculated\nas\nE(2)\ni=1\n(2M)2\u001b\"\u0000\u001b#\ne2f2\fel\u0017\u0000i!(1 +i\fel)gqi;(31)\nE(1)\ni'1\n(2M)2\u001b\"\u0000\u001b#\ne2(\u0000i!qi); (32)\nwhere\u001b\u001b=e2(v2\nF\u001b=3)\u0017\u001b\u001c\u001bis the conductivity of elec-trons with spin \u001b,\n\fel=\u0019nsu2\ns\nMh\n(S2\n?+S2z)\u0017++P\u00001(S2\n?\u0000S2z)\u0017\u0000i\n;(33)\nis the so-called \fparameter that parametrizes the dissi-\npative corrections to the spin-transfer torque [13, 14] and\nto the Berry-phase spin-motive force [18, 19]. We de\fne\n\u0017\u0006=\u0017\"\u0006\u0017#andP= (\u001b\"\u0000\u001b#)=(\u001b\"+\u001b#). For the present\npurpose, we can discard the !-linear term inE(2)\ni. It will\nbe used in Sec. IV when we discuss the spin-motive force.\nThe magnon part is calculated by using\n1\n2\u0019Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013\n\u00172X\nquiqjDR\nqDA\nq=1\n2\u000bTSmag\u000eij;\n(34)\n1\n\u0019Z\nd\u0017n(\u0017)\u0017X\nquiqjImh\u0000\nDR\nq\u00012i\n=Emag\u000eij;(35)\n1\n\u0019Z\nd\u0017n(\u0017)X\nquiqjIm\u0002\nDR\nq\u0003\n= \n mag\u000eij;(36)\nwhereEmag=P\nq!qn(!q) is the energy density, \n mag=\nkBTP\nqln(1\u0000e\u0000~!q=kBT) is the thermodynamic poten-\ntial density, andSmag=\u0000@\nmag=@T is the entropy den-\nsity of magnons. Thus the magnon-drag electron (num-\nber) current is obtained as\nhjel;iidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0014\nEmag\u0000\fel\n\u000bTSmag\u0000\nmag\u0015\n(\u0000@i )\n=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag(\u0000@i ); (37)\nwhere we used \n mag=Emag\u0000TSmag. Note that \n mag,\nwhich arises as \\corrections\" here, turned the energy\nEmaginto the entropy TSmag, and the result depends on\nmagnons only through their entropy. This is the main\nresult of this paper.\nC. Result\nA physical result is obtained by replacing @i by\n@iT=T,\nhjelidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag\u0012\n\u0000rT\nT\u0013\n(38)\n=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nr\nmag: (39)\nIn the second line, we noted Smag=\u0000@\nmag=@T and as-\nsumed that \n magis position (r) dependent only throughthe local temperature, T=T(r).\nThe obtained magnon-drag current, Eq. (39), is pro-\nportional to \u001b\"\u0000\u001b#[20]. This indicates that the magnons\nexert on the electrons a spin-dependent force,\nF\u001b=\u0000\u001b\n2s0\u0012\n1\u0000\fel\n\u000b\u0013\nr\nmag; (40)\nwhere\u001b= 1 or\u00001 depending on the electron spin projec-\ntion,\u001b=\"or#. Some discussion will be given in Sec. IV\nin relation to the spin-motive force.\nEquation (40) has the form of total gradient, suggest-\ning that it is of di\u000busive nature and is induced by a spin-\ndependent, nonequilibrium chemical potential,\n\u000e\u0016\u001b=\u001b\n2s0\u0012\n1\u0000\fel\n\u000b\u0013\n\u000e\nmag; (41)\nwhere\u000e\nmagis the deviation of \n magfrom its thermal-\nequilibrium value. In Sec. V-B, we will give a further\nanalysis that supports this picture of the spin chemical\npotential.6\nIV. PHENOMENOLOGY BASED ON\nSPIN-MOTIVE FORCE\nIn this section, we revisit the phenomenology based\non the spin-motive force along the lines of Refs. [10, 11],\nand compare the result with the microscopic result. The\nphysical pictures that emerge from the microscopic study\nare also discussed.\nWhen the magnetization vector nvaries in space and\ntime, it exerts a spin-dependent force, \u0006Fi, on electrons,\nwhere\nFi=~\n2[n\u0001(_n\u0002@in)\u0000\f_n\u0001@in]: (42)\nThis is called the spin-motive force. The \frst term is\nthe \\Berry phase term\" and the second term with a\ndimensionless coe\u000ecient \fis the dissipative correction,\nwhich we call the \f-term [18, 19, 21]. ( \fis equal to\n\fel[Eq. (33)], but we continue to use these notations;\n\felfor the microscopically-calculated one, and \ffor the\nphenomenologically-introduced one.) These e\u000bects are\nreciprocal to the current-induced spin torques; the former\nis reciprocal to the spin-transfer torque, and the latter to\nits dissipative correction.\nSpin waves, or magnons, can also be the origin of the\nspin-motive force. Although they are \ructuations, they\nwill induce a net electron current\nhjel;iismf=\u001b\"\u0000\u001b#\ne2hFii; (43)\nif the average survives, hFii6= 0. This will contribute\nto the magnon-drag electron current. Here we assume\na uniformly magnetized state at equilibrium, njeq=\n^z, and consider small \ructuations \u000enaround it, such\nthatn= ^z+\u000en. With magnon operators, fa;ayg=\n(s0=2)1=2(\u000enx\u0006i\u000eny), we rewrite Fias\nFi=i\n2s0\u0002\n\u0000_ay@ia+ (@iay)_a\u0003\n\u0000\f\n2s0\u0002\n_ay@ia+ (@iay)_a\u0003\n:\n(44)\nAs noted previously [10, 11], the second term is essen-\ntially the magnon heat current jQ\nmag;i[Eq. (23)]. Here\nwe note that the \frst term is expressed by the magnon\nenergy,hmag=i(ay_a\u0000_aya)=2, and the magnon current\njmag;i=\u0000iJ[ay@ia\u0000(@iay)a]. Thus,\nFi=1\n2s0\u001a\n@ihmag+1\n2J@\n@tjmag;i+\f\nJjQ\nmag;i\u001b\n:(45)\nLet us evaluate each term in Eq. (45) for a steady\nstate with a temperature gradient. Since the \frst term\nhas a spatial derivative @i, we evaluate it in the local\nequilibrium state as hhi=Emag(T), which depends on r\nthrough the local temperature T=T(r). This leads to\n@ihhmagi= (@Emag=@T)(@iT). The second term vanishes\nin the steady state because of the overall time deriva-\ntive. The third term is evaluated as hjQ\nmag;ii=\u0000\u0014@iTwith the magnon heat conductivity \u0014. This is calculated\nusing, e.g., the Kubo-Luttinger formula as [22]\n\u0014=1\nTZd\u0017\n2\u0019\u0012\n\u0000@n\n@\u0017\u0013\n\u00172X\nqu2\nxDR\nqDA\nq=J\n\u000bSmag;(46)\nwhere we used Eq. (34). This expression for \u0014in terms\nof magnon entropy also follows from an intuitive argu-\nment. Following Drude, one may express the magnon\nheat-current density at position xas [23]\njQ\nx(x) =1\n2X\nqux!q[n(x\u0000ux\u001c)\u0000n(x+ux\u001c)];(47)\nwheren(x) is the Bose distribution function de\fned with\na local temperature T(x),\u001c= (2\u000b!q)\u00001is the lifetime\nof magnons, and the temperature gradient is assumed in\nthexdirection. The \frst term represents the energy \row\nfrom the left region, and the second term from the right,\nwhich are due to magnons that experienced their last\ncollision at x\u0006ux\u001c; the factor 1/2 is there because half of\nmagnons at x\u0006ux\u001c(namely, those with qx>0 orqx<0)\npropagate to x. Expanding as n(x\u0000ux\u001c)\u0000n(x+ux\u001c)'\n\u00002ux\u001c(@n=@T )(@T=@x ) and using Eq. (B16), one has\n\u0014=1\n2\u000b@\n@TX\nqu2\nxn(!q) =J\n\u000bSmag; (48)\nin agreement with Eq. (46).\nTaken together, we obtain\nhFii=1\n2s0\u001a\n\u0000@Emag\n@T+\f\n\u000bSmag\u001b\n(\u0000@iT): (49)\nThe same result has been obtained by other methods; see\nAppendix C. Therefore, we may conclude that any (phe-\nnomenological) theories starting from the spin-motive\nforce lead to Eq. (49). The \frst term is somewhat di\u000ber-\nent from the one obtained in Ref. [11], and gives a slight\nrevision to it (see Appendix C-3).\nWe now compare Eq. (49) with the microscopic result,\nEq. (38). One readily sees a disagreement in the \frst\nterm, namely, the entropy Smagappears in the micro-\nscopic result instead of @Emag=@T in the phenomenolog-\nical result.\nTo identify the the origin of the di\u000berence, let us look\nat the Feynman diagram. To calculate the spin-motive\nforce, one calculates the electric current induced by mag-\nnetization dynamics [18]. This can be done by consider-\ning small \ructuations \u000enaround the uniform magnetiza-\ntion, and look at the second-order (nonlinear) response\nto\u000en[24]. This is expressed diagrammatically in Fig. 3\n(a), and the response function is given by E(2)in Eq. (31).\nTherefore, the induced current is calculated as\nhjel;ii=2M2\ns0E(2)\niaq;\u0017+!a\u0003\n\u0000q;\u0000\u0017; (50)\nwherefa;a\u0003gis a classical (c-number) counterpart of\nfa;aygde\fned just above Eq. (44), and the subscripts7\nFIG. 3. Feynman diagrams for the electric current in-\nduced by magnetization dynamics. Arrows in the electron\nlines (solid lines) are suppressed for simplicity. (a) Nonlin-\near response to the (classical) magnetization dynamics. The\nwavy lines represent the perturbations due to (classical) mag-\nnetization. Because of causality (retarded response), the in-\ncoming Matsubara frequencies should satisfy the conditions,\n\u0017l+!\u0015>0 and\u0000\u0017l>0 [25]. (b) Part of the diagram of\nthe present magnon-drag process (Fig. 2). The wavy lines\nrepresent (quantum) magnon propagators. Note that the\n\row of the Matsubara frequency in the lower magnon line\nis reversed compared to (a). The same causality relation as\n(a) leads to the analytic continuation, D(i\u0017l+i!\u0015)D(i\u0017l)!\nDR(\u0017+!)DA(\u0017), for the pair of magnon propagators, and\nthis is associated with E(2)\nigiven by Eq. (31).\nindicate their wave vector and frequency. This leads to\nEq. (44), hence to Eq. (42). Therefore, the spin-motive\nforce is described by the \u0017- and!-linear terms inE(2)\ni.\nThe appearance of E(2)\ni(originally from the magnon-drag\ncalculation) in the nonlinear response here is due to the\nmatching of the causality relationship; see Fig. 3 (b) and\nthe caption thereof.\nOn the other hand, in the present magnon-drag pro-\ncess, the \frst term comes from the !-linear term inE(1)\ni,\nnot from the !-linear term inE(2)\ni; the latter is irrelevant\nfor the magnon-drag DC electron current. Since E(1)\niis\naccompanied by DRDR(notDRDA), the physical in-\nterpretation of this term (in the magnon-drag current)\ndoes not necessarily rely on the causal relationship to\nthe magnetization dynamics. In fact, the spin-transfer\nprocess may be understood to occur in the quasi- or\nlocal-equilibrium situation, as will be discussed in the\nparagraph containing Eq. (64).\nV. SPIN CHEMICAL POTENTIAL\nIn this section, we give an alternative argument that\nintroduces a spin chemical potential. This is intended to\ncomplement the heuristic discussion in Sec. III-C.\nOur strategy here is as follows. From the viewpoint\nof microscopic theory, statistical quantities such as the\nchemical potential and temperature, which characterize\nthe distribution function, cannot be easily handled. In-\nstead, we can disturb the system by \\mechanical\" per-\nturbations (which are described by the Hamiltonian and\nthus controllable theoretically) and then observe the re-\nsult. By examining how the distribution function is de-formed, we may read o\u000b the change of statistical param-\neters such as chemical potential and temperature. For\nexample, an inhomogeneous potential (or electric \feld)\ninduces a density modulation. This e\u000bect is described\nby an inhomogeneous change of chemical potential, and\nappears in the current as a di\u000busion current [21].\nIn the following, we examine the possibility that the\nmagnon-drag e\u000bects are described in a similar manner.\nWe \frst illustrate the procedure using a simple model\n(Sec. V-A), and then consider the present problem of\nmagnon-drag process (Sec. V-B). In both cases, we take\nthe \feld as a mechanical perturbation.\nA. Electron-only process: E\u000bective temperature\nWe begin by reviewing the relation between the grav-\nitational \feld @i and temperature gradient, @iT. For\nsimplicity, we consider a (spin-unpolarized) free elec-\ntron system subject to nonmagnetic impurities, forget-\nting about magnons and even the magnetization (ex-\nchange splitting). We calculate the electron density \u000enel\nand current density hjel;ii \n!induced by the disturbance\n having \fnite Q(and!). In this case, it is essential\nto consider the di\u000busion-type vertex corrections [Fig. 1\n(d)], hence the diagrams shown in Fig. 4. The results are\ngiven by\n\u000enel=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u0011(\"); (51)\nhjel;ii \n!=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u001b(\")(\u0000@i )Q\n\u0000Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"D(\") [@i\u0011(\")]Q; (52)\nwhere\n\u0011(\") =\u0000\u0017(\")D(\")Q2\nD(\")Q2\u0000i! Q; (53)\ndescribes \\di\u000busive corrections\" which arise since Qis\n\fnite. We de\fned e2\u001b(\") =e2(v2\nF=3)\u0017(\")\u001c(\") andD(\") =\n(v2\nF=3)\u001c(\"), which are the Boltzmann conductivity and\nthe di\u000busion constant, respectively, evaluated at energy\n\"(measured from the chemical potential \u0016).\nIf we consider a local modi\fcation of temperature, T!\nT+\u000eT(r), the electron density changes by\n\u000enel=Z\nd\"\u0017(\")[f(\";T+\u000eT)\u0000f(\";T) ]\n'\u000eT\nTZ\nd\"\u0017(\")\"\u0012\n\u0000@f\n@\"\u0013\n: (54)\nIn the `slow' limit !!0, Eq. (51) may be compared with\nEq. (54), and we may identify the e\u000bective temperature\nchange\u000eTQby\n\u0000lim\n!!0D(\")Q2\nD(\")Q2\u0000i! Q=\u000eTQ\nT: (55)8\nFIG. 4. Feynman diagrams for the electron density \u000enel\n[Eq. (51)] and the current density jel;i[Eq. (52)] induced by\nrT.helandjQ\nel;iare the Hamiltonian density and heat cur-\nrent density, respectively, of the conduction electrons. The\nshaded rectangle represents the di\u000busion-type ladder vertex\ncorrection due to impurities [Fig. 1(d)], which describes dif-\nfusive motion of the electrons.\nThis is nothing but the Einstein-Luttinger relation,  Q+\n\u000eTQ=T= 0, that holds in the equilibrium state (under\na static potential,  Q). Using this \u000eTQ, we may rewrite\nEq. (52) as\nhjel;ii \n!=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u001b(\")(\u0000@i )Q\n\u0000Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"D(\")\u0017(\")\u0012\n\u0000@iT\nT\u0013\nQ\n=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u001b(\")\u0012\n\u0000@i \u0000@iT\nT\u0013\nQ:(56)\nThis shows the \\equivalence\" of the mechanical force @i \nand the statistical force @iT=T, and forms a basis of Lut-\ntinger's thermal linear-response theory.\nB. Magnon-drag process: Spin chemical potential\nLet us apply a similar procedure to the magnon-drag\nprocess. For this purpose, we calculate the magnon-drag\nelectron current in response to a spatially-modulated po-\ntential, / Qei(Q\u0001r\u0000!t), with \fnite wave vector Q. As\nin the preceding subsection, we consider the di\u000busion-\ntype vertex corrections and the diagrams in Fig. 2 (b).\nThe result is obtained as\nhjel;iidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag(\u0000@i )\n\u0000@i\u0010\nD\"\u000en\"\nel+D#\u000en#\nel\u0011\n; (57)\nwhere\n\u000en\u001b\nel=\u001b\n2s0\u001b\u001b\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmagQ2\nD\u001bQ2\u0000i! Q;(58)\nis the change of the electron density (of spin \u001b) caused by\nthe perturbation  Q, andD\u001b= (v2\nF\u001b=3)\u001c\u001bis the di\u000busionconstant. From the form of Eq. (58), it is natural to\nregard the density change \u000en\u001b\nelas caused by the change of\nthe electrons' chemical potential, instead of temperature\nas in Eq. (54). Namely, Eq. (58) in the `slow' limit, !!\n0, may be compared with\n\u000en\u001b\nel=Z\nd\"\u0017\u001b(\")[f(\";T\u0016+\u000e\u0016\u001b)\u0000f(\";T;\u0016) ]\n'Z\nd\"\u0017\u001b(\")\u0012\n\u0000@f\n@\"\u0013\n\u000e\u0016\u001b\n'\u0017\u001b\u000e\u0016\u001b; (59)\nwhere\u000e\u0016\u001bis the change in (spin-dependent) chemical\npotential. From the comparison, we may identify [26]\n\u000e\u0016\u001b=\u001b\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\nTSmag Q; (60)\n=\u001b\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\nTSmag\u0012\n\u0000\u000eTQ\nT\u0013\n: (61)\nIn the second line, we used the Einstein-Luttinger rela-\ntion, Q=\u0000\u000eTQ=T. Note that Eq. (61) is consistent\nwith Eq. (41). Using Eq. (61) for \u000e\u0016\u001bin Eq. (59), we\nrewrite Eq. (57) as\nhjel;iidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag(\u0000@i )\n+1\ne2\u001b\"(\u0000@i\u000e\u0016\") +1\ne2\u001b#(\u0000@i\u000e\u0016#) (62)\n=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag\n\u0002\u0012\n\u0000@i \u0000@iT\nT\u0013\n: (63)\nThis reproduces the form of Eq. (25).\nThe nonequilibrium chemical potential \u000e\u0016\u001bis spin de-\npendent,\u000e\u0016\"=\u0000\u000e\u0016#(because of the overall factor\n\u001b=\u00061). Thus the electrons feel the e\u000bects of the\nnonequilibrium magnons as a \\spin chemical potential\",\nor spin accumulation, \u0016s=\u0016\"\u0000\u0016#. This is quite natural\nsince the local change \u000eTin temperature modulates the\nmagnon density, and the balance of the \\reaction\"\nm +e\"\u001de#; (64)\nshifts in the left or the right direction. Here, m, e\"and\ne#represent a magnon, an electron with spin up, and\nan electron with spin down, respectively. If we focus on\nthe electrons ( e\"ande#), this is precisely controlled by\nthe chemical-potential di\u000berence, \u0016\"\u0000\u0016#. This process\ncorresponds to the \frst term (the spin-transfer term).\nThe absence of the causality relationship, as discussed at\nthe end of Sec. IV, may be due to the local equilibrium\nnature of this process.\nThe second term (proportional to \fel=\u000b) acts in the\nopposite way; it increases the density of up-spin (down-\nspin) electrons in the hotter (colder) region. Let us inter-\npret this e\u000bect in terms of momentum transfer process.9\nFor this, we consider the e\u000bects of magnon \row. The\nmagnons \row from the hotter to the colder region, and\nwill scatter electrons into the colder region. If a magnon\nis absorbed by an electron, the scattered electron has\ndown spin and \rows downstream. This means that the\ndown-spin electrons \row to colder regions and this e\u000bect\nwill increase the density of down-spin electrons in the\ncolder region. There is also a reverse process: if a down-\nspin electron emits a magnon and \rips its spin, and if the\nmagnon \rows downstream, the up electron will \row up-\nstream. This process will increase the density of up-spin\nelectrons in the hotter region.\nVI. SUMMARY\nIn this paper, we studied magnon-drag electron \row in-\nduced by a temperature gradient. The analysis is based\non a microscopic model that contains spin relaxation,\nand on the linear response theory due to Luttinger that\nexploits a gravitational potential  . The obtained re-\nsult is physically interpreted in terms of the spin-transfer\nprocess and the momentum-transfer process from the\nmagnons to the electrons. It is found that the e\u000bect\nof nonequilibrium magnons yields a nonzero spin chem-\nical potential of the electrons. In the process, we gave\na microscopic procedure that leads to the Luttinger's\nform of the response, namely, a combination of the form,\n\u0000@i \u0000@iT=T. We supplemented the analysis with a phe-\nnomenological one that is based on the spin-motive force,\nand found that the agreement with the microscopic result\nis good for the dissipative \f-term, but di\u000bers slightly for\nthe Berry-phase (spin-transfer) term.\nACKNOWLEDGEMENT\nWe are grateful to Y. Imai for fruitful discussions.\nValuable comments by G. E. W. Bauer and J. P. Here-\nmans are also appreciated. This work is supported by\nJSPS KAKENHI Grant Numbers 25400339, 15H05702\nand 17H02929. TY is supported by a Program for Lead-\ning Graduate Schools \\Integrative Graduate Education\nand Research in Green Natural Sciences\". RD is a mem-\nber of the D-ITP consortium, a program of the Nether-\nlands Organisation for Scienti\fc Research (NWO) that\nis funded by the Dutch Ministry of Education, Culture\nand Science (OCW). This work is in part funded by\nthe Stichting voor Fundamenteel Onderzoek der Materie\n(FOM) and the European Research Council (ERC).\nAppendix A: Vertex corrections\nIn this Appendix, we calculate the vertex corrections\nto the electron spin \u001b\u0006due to impurity potentials in\nthe ladder approximation. The renormalized vertex \u0003\u0006\n\u001b\u0016\u001bsatis\fes\n(\u0003\u0006\n\u001b\u0016\u001b)ab= (\u001b\u0006)\u001b\u0016\u001b+ \u00000Yab\n\u001b\u0016\u001b(\u0003\u0006\n\u001b\u0016\u001b)ab; (A1)\nwhere (\u001b+)\"#= (\u001b\u0000)#\"= 1 (other elements vanish),\n\u00000=niu2\ni\u0000nsu2\nsS2z; (A2)\nandYab\n\u001b\u0016\u001b=P\nkGa\nk\u001bGb\nk\u0016\u001bwith \u0016\u001b=\u0000\u001b. We write the\nGreen's function as Ga\nk\u001b= (i\"a+\u001bM\u0000~2k2=2m\u0000\u0006a\n\u001b)\u00001,\nwherea,bspecify retarded (R) or advanced (A), namely,\na= R for\"a>0, anda= A for\"a<0. Writing the\nself-energy as\n\u0006a\n\u001b= \u00001ga\n\u001b+ \u00002gb\n\u0016\u001b; (A3)\nwith \u0000 1=niu2\ni+nsu2\nsS2z, \u00002= 2nsu2\nsS2\n?[Eq. (14)], and\nga\n\u001b=P\nkGa\nk\u001b, we evaluate Yab\n\u001b\u0016\u001bas\nYab\n\u001b\u0016\u001b=gab\n\u001b\u0016\u001b\ni\"ba\u00002\u001bM+ \u0006ab\n\u001b\u0016\u001b; (A4)\nwhere\"ba=\"b\u0000\"a,gab\n\u001b\u001b0=ga\n\u001b\u0000gb\n\u001b0and \u0006ab\n\u001b\u001b0= \u0006a\n\u001b\u0000\u0006b\n\u001b0.\nThen, from Eq. (A1), we obtain\n(\u0003\u0006\n\u001b\u0016\u001b)ab=(\u001b\u0006)\u001b\u0016\u001b\n1\u0000\u00000Yab\n\u001b\u0016\u001b=i\"ba\u00002\u001bM+ \u0006ab\n\u001b\u0016\u001b\ni\"ba\u00002\u001bM+ \u0001ab\n\u001b\u0016\u001b(\u001b\u0006)\u001b\u0016\u001b;\n(A5)\nwhere \u0001ab\n\u001b\u001b0= \u0001a\n\u001b\u0000\u0001b\n\u001b0with\n\u0001a\n\u001b= (\u0000 1\u0000\u00000)ga\n\u001b+ \u00002ga\n\u0016\u001b\n= 2nsu2\ns(S2zga\n\u001b+S2\n?ga\n\u0016\u001b): (A6)\nExplicitly, \u0003+and \u0003\u0000are given by\n(\u0003+\n\"#)ab=i\"ba\u00002M+ \u0006ab\n\"#\ni\"ba\u00002M+ \u0001ab\n\"#; (A7)\n(\u0003\u0000\n#\")ab=i\"ba+ 2M+ \u0006ab\n#\"\ni\"ba+ 2M+ \u0001ab\n#\": (A8)\n(Other elements vanish, \u0003+\n#\"= \u0003\u0000\n\"#= 0, etc.) Therefore,\nGa\nk\"(\u0003+\n\"#)abGb\nk#=Ga\nk\"\u0000Gb\nk#\ni\"ba\u00002M+ \u0001ab\n\"#; (A9)\nGa\nk#(\u0003\u0000\n#\")abGb\nk\"=Ga\nk#\u0000Gb\nk\"\ni\"ba+ 2M+ \u0001ab\n#\": (A10)\nFor example,\nGR\nk\"(\u0003+\n\"#)RAGA\nk#=GR\nk\"\u0000GA\nk#\n\u00002M+ \u0001RA\n\"#\n'\u00001\n2M \n1 +\u0001RA\n\"#\n2M!\n\u0000\nGR\nk\"\u0000GA\nk#\u0001\n;\n(A11)\nGR\nk#(\u0003\u0000\n#\")RAGA\nk\"=GR\nk#\u0000GA\nk\"\n2M+ \u0001RA\n#\"\n'1\n2M \n1\u0000\u0001RA\n#\"\n2M!\n\u0000\nGR\nk#\u0000GA\nk\"\u0001\n:\n(A12)10\nIn the second lines, we assumed that \u0001ab's, which are on\nthe order of spin relaxation rate, are much smaller than\nthe exchange splitting M.\nAppendix B: Details of microscopic calculation\nIn this Appendix, we present the details of the calcu-\nlation of the magnon-drag electron current. It is divided\ninto the electron part and the magnon part.\n1. Electron part\nAs described in the text, the electron part, given\nby Eq. (29), contributes in two di\u000berent ways, E(2)\ni\u0011Ei(q;\u0017\u0000i\u0011;!+ 2i\u0011) andE(1)\ni\u0011 Ei(q;\u0017+i\u0011;!+i\u0011),\nwhere\u0011is a positive in\fnitesimal. For the magnon-drag\ncontribution, the former is calculated by setting != 0\nand retaining the \u0017-linear terms, and the latter by set-\nting\u0017= 0 and retaining the !-linear terms. They are\ngiven, respectively, by\nE(2)\ni=\u0017\n2\u0019q`(E0\ni`+E00\ni`); (B1)\nwith\nE0\ni`=\u0000iX\nkviv`h\nGR\n#n\n(\u0003\u0000\n#\")RR\u0000\nGR\n\"\u00012(\u0003+\n\"#)RA\u0000(\u0003\u0000\n#\")RA\u0000\nGA\n\"\u00012(\u0003+\n\"#)AAo\nGA\n#\n+GR\n\"n\n(\u0003+\n\"#)RR\u0000\nGR\n#\u00012(\u0003\u0000\n#\")RA\u0000(\u0003+\n#\")RA\u0000\nGA\n#\u00012(\u0003\u0000\n#\")AAo\nGA\n\"i\n; (B2)\nE00\ni`=\u0000X\nkviv`Imh\nGR\n#(\u0003\u0000\n#\")RR\u0000\nGR\n\"\u00012(\u0003+\n\"#)RRGR\n#+GR\n\"(\u0003+\n\"#)RR\u0000\nGR\n#\u00012(\u0003\u0000\n#\")RRGR\n\"i\n; (B3)\nand\nE(1)\ni'1\n2\u0019q`X\nkviv`h\n\u0000GR\n#(\u0003\u0000\n#\")RA\u0000\nGA\n\"\u00012(\u0003+\n\"#)AAGA\n#+GR\n\"(\u0003+\n\"#)RR\u0000\nGR\n#\u00012(\u0003\u0000\n#\")RAGA\n\"i\n; (B4)\nwhereGR(A)\n\u001b =GR(A)\nk\u001b(0).\nTo calculateE0\ni`, we use Eqs. (A9)-(A10) and the approximations as in Eqs. (A11)-(A12) valid for weak spin\nrelaxation (compared to M). With short notations, \u0001ab= \u0001ab\n\"#and ~\u0001ab= \u0001ab\n#\", we write\nE0\ni`'i\n(2M)2X\nkviv`\" \n1 +\u0001RA\u0000~\u0001RR\n2M!\n(GR\n#\u0000GR\n\")(GR\n\"\u0000GA\n#)\u0000 \n1 +\u0001AA\u0000~\u0001RA\n2M!\n(GR\n#\u0000GA\n\")(GA\n\"\u0000GA\n#)\n\u0000 \n1 +\u0001RA\u0000~\u0001AA\n2M!\n(GR\n\"\u0000GA\n#)(GA\n#\u0000GA\n\") + \n1 +\u0001RR\u0000~\u0001RA\n2M!\n(GR\n\"\u0000GR\n#)(GR\n#\u0000GA\n\")#\n'i\n(2M)2X\nkviv`\u0014\u00011\n2MGR\n\"GA\n\"\u0000\u00012\n2MGR\n#GA\n#\u00002iIm\b\n(GR\n\"\u0000GR\n#)2\t\u0015\n; (B5)\nwhere \u0001 1\u0011\u0001RA\u0000\u0001RR+~\u0001RA\u0000~\u0001AAand \u0001 2\u0011\u0001RA\u0000\n\u0001AA+~\u0001RA\u0000~\u0001RR, and we retained the leading terms\nwith respect to the electron damping. On the other hand,\nE00\ni`is calculated as\nE00\ni`'\u00002\n(2M)2X\nkviv`Im\u0002\n(GR\n\"\u0000GR\n#)2\u0003\n: (B6)Therefore, we have\nE0\ni`+E00\ni`=i\n(2M)2X\nkviv`\u0014\u00011\n2MGR\n\"GA\n\"\u0000\u00012\n2MGR\n#GA\n#\u0015\n=i\u000ei`\n(2M)22\u0019\ne2\u0014\u00011\n2M\u001b\"\u0000\u00012\n2M\u001b#\u0015\n: (B7)\nHere we noted\nX\nkviv`GR\n\u001bGA\n\u001b=\u000ei`v2\nF\u001b\n3\u0019\u0017\u001b\n\r\u001b=\u000ei`2\u0019\ne2\u001b\u001b; (B8)11\nwith\u001b\u001b=e2(v2\nF\u001b=3)\u001c\u001bbeing the conductivity of spin-\n\u001belectrons. From Eq. (A6), we have \u0001 1= 2\u0001RA\n##and\n\u00012= 2\u0001RA\n\"\"with \u0001RA\n\u001b\u001b=\u00004\u0019insu2\ns(S2z\u0017\u001b+S2\n?\u0017\u0016\u001b), and\nthus\n\u00011\n2M\u001b\"\u0000\u00012\n2M\u001b#=\u00002i\fel(\u001b\"\u0000\u001b#); (B9)\nwhere\felis given by Eq. (33). Using these relations in\nEq. (B1), we obtain\nE(2)\ni=\fel\u0017\n2M2qi\u001b\"\u0000\u001b#\ne2: (B10)\nThe!-linear terms inE(2)\ni[as given in Eq. (31)], which\ncontributes to the spin-motive forces, can be obtained in\na similar way.\nSimilarly, we obtain\nE(1)\ni'1\n2\u00191\n(2M)2q`X\nkviv`\u0002\nGR\n\"GA\n\"\u0000GR\n#GA\n#\u0003\n'1\n(2M)2qi\u001b\"\u0000\u001b#\ne2: (B11)\n2. Magnon part\nFor the magnon part, we encounter the following inte-\ngrals,\nI1=1\n2\u0019Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013\n\u00172X\nquiqjDR\nq(\u0017)DA\nq(\u0017);(B12)\nI2=1\n\u0019Z\nd\u0017n(\u0017)\u0017X\nquiqjImh\u0000\nDR\nq(\u0017)\u00012i\n; (B13)\nI3=1\n\u0019Z\nd\u0017n(\u0017)X\nquiqjIm\u0002\nDR\nq\u0003\n; (B14)\nTo calculate I1, we useDR\nq(\u0017)DA\nq(\u0017)'(\u0019=\u000b\u0017 )\u000e(\u0017\u0000!q).\nThen,\nI1'1\n2\u000bX\nq!quiqj\u0012\n\u0000@n\n@\u0017\u0013\n\u0017=!q\n=1\n2\u000bT@\n@TX\nqn(!q)uiqj: (B15)\nBy noting ( @=@qi)kBTln\u0000\n1\u0000e\u0000~!q=kBT\u0001\n=n(!q)ui, we\nsee\nX\nqn(!q)uiqj=X\nqqj@\n@qikBTln\u0010\n1\u0000e\u0000~!q=kBT\u0011\n=\u0000\u000eij\nmag; (B16)\nwhere\n\nmag=kBTX\nqln\u0010\n1\u0000e\u0000~!q=kBT\u0011\n; (B17)is the thermodynamic potential of magnons. Therefore,\nI1=\u00001\n2\u000b\u000eijT@\n@T\nmag=1\n2\u000b\u000eijTSmag; (B18)\nwhereSmag=\u0000@\nmag=@T is the entropy (density) of\nmagnons.\nForI2, we useui(DR)2=@DR=@qiand ImDR\nq(\u0017)'\n\u0000\u0019\u000e(\u0017\u0000!q), and calculate as\nI2=\u0000\u000eij\n\u0019Z\nd\u0017n(\u0017)\u0017X\nqIm\u0002\nDR\nq\u0003\n'\u000eijX\nq!qn(!q)\n=\u000eijEmag: (B19)\nSimilarly,I3is calculated as\nI3'\u0000X\nquiqjn(!q) =\u000eij\nmag: (B20)\nAppendix C: Semi-classical analysis based on\nspin-motive force\nIn this Appendix, we calculate\nhFii=1\ns0\b\nImh_ay@iai\u0000\fReh_ay@iai\t\n; (C1)\nsemi-classically using the stochastic Landau-Lifshitz-\nGilbert (LLG) equation. This method has been used in\nthe calculation of magnonic spin torques [27, 28].\n1. Formulation\nThe stochastic LLG equation is given by\n_n=\u0000Jn\u0002@2\nin+n\u0002h\u0000\u000bn\u0002_n; (C2)\nwherenis the magnetization unit vector, and his\nthe Langevin noise \feld that satis\fes the \ructuation-\ndissipation theorem,\nhhi(r;t)hj(r0;t0)i= 2\u000bs0T\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);(C3)\nwhereTis the temperature. We consider the case that\nthe temperature is nonuniform and assume Tin Eq. (C3)\nis position-dependent, T=T(r), and calculatehFiithat\nis proportional to @iT.\nIn the complex notation, a= (s0=2)1=2(\u000enx+i\u000eny)\nandh=hx+ihy, Eq. (C2) becomes\ni_a= (\u0000J@2\ni+ \u0001)a+\u000b_a\u00001p2s0h(r;t); (C4)\nwhere \u0001 is the magnon energy gap, and hsatis\fes\nhh(r;t)h\u0003(r0;t0)i= 4\u000bs0T(r)\u000e(r\u0000r0)\u000e(t\u0000t0):(C5)12\nUsing the retarded Green's function DRthat satis\fes\n\u0002\ni@t+J@2\ni\u0000\u0001\u0000\u000b@t\u0003\nDR=\u000e(r\u0000r0)\u000e(t\u0000t0);(C6)\nEq. (C4) is solved as\na(r;t) =\u00001p2s0Z\ndt0Z\ndr0DR(r\u0000r0;t\u0000t0)h(r0;t0):\n(C7)\nIn the Fourier representation, DR\nq(\u0017) = (\u0017\u0000!q+i\u000b\u0017)\u00001,\nwhere!q=Jq2+ \u0001, it reads\naq(\u0017) =\u00001p2s0DR\nq(\u0017)h(q;\u0017); (C8)anda\u0003is given by the complex conjugate of Eq. (C7).\nFor a quantum system (in the present case, magnons),\nwe consider the Fourier transform of Eq. (C3) with re-\nspect to time, wherein the temperature is replaced as\nT!\u0017\n2coth\u0017\n2T=\u0017[n(\u0017) +1\n2] for the Fourier component\nof frequency \u0017. Its gradient is thus replaced as\n@iT!\u0017\u0012@n\n@T\u0013\n@iT: (C9)\n2. Calculation of hFii\nTo obtainhFii, it is su\u000ecient to calculate\n_ay@ia\u000b\n.\nWith Eq. (C8), this proceeds as follows,\n\n_ay(r;t)@ia(r;t)\u000b\n=\u001c\n@t\u0012\n\u00001p2s0Z\ndt1Z\ndr1DR(r\u0000r1;t\u0000t1)h(r1;t1)\u0013\u0003\n\u0002@i\u0012\n\u00001p2s0Z\ndt2Z\ndr2DR(r\u0000r2;t\u0000t2)h(r2;t2)\u0013\u001d\n=1\n2s0ZZ\ndt1dt2ZZ\ndr1dr2\u0000\n@tDR(r\u0000r1;t\u0000t1)\u0001\u0003\u0000\n@iDR(r\u0000r2;t\u0000t2)\u0001\nhh\u0003(r1;t1)h(r2;t2)i\n=4\u000bs0\n2s0Z\ndt1Z\ndr1\u0000\n@tDR(r\u0000r1;t\u0000t1)\u0001\u0003\u0000\n@iDR(r\u0000r1;t\u0000t1)\u0001\nT(r1)\n= 2\u000bZ\ndt1Z\ndr1X\nq;q0;q1ZZd\u0017d\u00170\n(2\u0019)2i\u00170iqiDA\nq0(\u00170)DR\nq(\u0017)Tq1ei(q0\u0000q+q1)\u0001r1e\u0000i(\u0017\u0000\u00170)(t\u0000t1)ei(q\u0000q0)\u0001r\n= 2\u000bX\nq;q1Zd\u0017\n2\u0019i\u0017\u0001i\u0010\nqi+q1;i\n2\u0011\nDR\nq+q1=2(\u0017)DA\nq\u0000q1=2(\u0017)Tq1eiq1\u0001r; (C10)\nwhereDA\nq(\u0017)\u0011\u0000\nDR\nq(\u0017)\u0001\u0003= (\u0017\u0000!q\u0000i\u000b\u0017)\u00001. We are interested in the term linear in q1, which, combined with Tq1,\ngives the temperature gradient. Thus,\n\n_ay(r;t)@ia(r;t)\u000b\n'\u000bX\nq;q1Zd\u0017\n2\u0019i\u0017DRDAiq1;iTq1eiq1\u0001r+ 2\u000bX\nq;q1Zd\u0017\n2\u0019i\u0017iqiuj2iImh\u0000\nDR\u00012DAiq1;j\n2Tq1eiq1\u0001r;(C11)\nwhereDR=DR\nq(\u0017) andDA=DA\nq(\u0017). With the replacement (C9), we obtain\n\n_ay(r;t)@ia(r;t)\u000b\n'\u000b(@iT)@\n@T(\niX\nqZd\u0017\n2\u0019\u00172n(\u0017)DRDA\u00002X\nqZd\u0017\n2\u0019qiuj\u00172n(\u0017) Imh\u0000\nDR\u00012DAi)\n: (C12)\nUsing the relations,\nX\nqZd\u0017\n2\u0019\u00172n(\u0017)DRDA'X\nqZd\u0017\n2\u0019\u00172n(\u0017)\u0001\u0019\n\u000b\u0017\u000e(\u0017\u0000!q)\n=1\n2\u000bEmag; (C13)\nand\nX\nqZd\u0017\n2\u0019\u00172n(\u0017)qiujImh\u0000\nDR\u00012DAi=X\nqZd\u0017\n2\u0019\u00172n(\u0017)qiujIm\u0014\nDR1\n2i\u000b\u0017\u0000\nDA\u0000DR\u0001\u0015\n'X\nqZd\u0017\n2\u0019\u00172n(\u0017)qiujIm\u00141\n2i\u000b\u0017DRDA\u0015\n'\u00001\n4\u000b2X\nqqiujn(!q)\n=1\n4\u000b2\nmag\u000eij; (C14)13\nwhere \n magis given by Eq. (B17), we obtain\n\n_ay@ia\u000b\n=1\n2\u0012@iT\nT\u0013\nT@\n@T\u0014\niEmag\u00001\n\u000b\nmag\u0015\n=1\n2\u0012@iT\nT\u0013\u0014\niT@\n@TEmag+1\n\u000bTSmag\u0015\n:(C15)\nFrom Eq. (C1), this leads to\nhFii=1\n2s0\u001a\n\u0000@Emag\n@T+\f\n\u000bSmag\u001b\n(\u0000@iT): (C16)\n3. Comparison with the previous study\nTo compare the phenomenological result (C16) ob-\ntained here with the one obtained previously [11], letus consider the case, T\u001d\u0001, where every quantity\nshows power-law dependence on temperature T. In\nthis case,T(@Emag=@T)'(1 +d=2)EmagandTSmag'\n(1 + 2=d)Emag, and Eq. (C16) becomes\nhFii=\u00001\n2s0\u0012\n1 +d\n2\u0013\u0012\n1\u00002\nd\f\n\u000b\u0013\nEmag\u0012\n\u0000@iT\nT\u0013\n:\n(C17)\nCompared with the result of Ref. [11], the coe\u000ecient of\n\f=\u000b is di\u000berent by a factor of 2.\n[1] D. C. Ralph and M. D. Stiles, J. Mag. Mag. Mat. 320,\n1190 (2008).\n[2] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[3] V. Korenman, J. L. Murray, and R. E. Prange, Phys.\nRev. B 16, 4032 (1977); G. E. Volovik, J. Phys. C 20,\nL83 (1987); A. Stern, Phys. Rev. Lett. 68, 1022 (1992); S.\nE. Barnes and S. 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Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[14] R. A. Duine, A. S. N\u0013 u~ nez, J. Sinova and A. H. MacDon-\nald, Phys. Rev. B 75, 214420 (2007).\n[15] T. Yamaguchi, and H. Kohno, J. Phys. Soc. Jpn. 86,\n063706 (2017).\n[16] J. M. Luttinger, Phys. Rev. 135, A1505 (1964).\n[17] H. Kohno, Y. Hiraoka, M. Hatami, and G.E.W. Bauer,\nPhys. Rev. B 94, 104417 (2016).\n[18] R. A. Duine, Phys. Rev. B 77, 014409 (2008).[19] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[20] It is possible to argue this di\u000berently if we note\n(\u001b\"\u0000\u001b#)\fel= (\u001b\"\u0000\u001b#)\fs+ (\u001b\"+\u001b#)\fc;\nwhere\n\fs=\u0019nsu2\ns\nM(S2\n?+S2z)\u0017+;\n\fc=\u0019nsu2\ns\nM(S2\n?\u0000S2z)\u0017\u0000:\nThus Eq. (39) consists of a term proportional to \u001b\"\u0000\u001b#\nand a term proportional to \u001b\"+\u001b#. This suggests the\nspin-dependent force that magnons exert on electrons,\nF\u001b=\u001bFs+Fc, contains a spin-independent part Fcas\nwell, where\nFs=\u00001\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\nr\nmag;\nFc=\u00001\n2s0\u0012\n\u0000\fc\n\u000b\u0013\nr\nmag:\nSimilarly, the spin-dependent nonequilibrium chemical\npotential,\u000e\u0016\u001b=\u001b\u000e\u0016 s+\u000e\u0016c, contains a spin-independent\npart\u000e\u0016c, where\n\u000e\u0016s=1\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\n\u000e\nmag;\n\u000e\u0016c=1\n2s0\u0012\n\u0000\fc\n\u000b\u0013\n\u000e\nmag:\nThe presence of the charge current (the term proportional\nto\fc) seems interesting from the viewpoint of the mo-\nmentum transfer e\u000bect. However, the sign of \fchere is\nnot de\fnite and it seems di\u000ecult to give a clear physical\nmeaning. It vanishes when the magnetic impurities are\nisotropic,S2\n?=S2z, anyway. Hence we will not pursue\nthis aspect in this paper.\n[21] For example, see: J. Shibata and H. Kohno, Phys. Rev.\nB84, 184408 (2011).14\n[22] Y. Imai and H. Kohno, J. Phys. Soc. Jpn. 87, 073709\n(2018).\n[23] N. W. Aschcroft and N. D. Mermin: Solid State Physics\n(Saunders College, Philadelphia, 1976).\n[24] H. Kohno et al. , inProceedings of ISQM-Tokyo '08 , pp.\n111-117, Eds. S. Ishioka and K. Fujikawa (World Scien-\nti\fc, 2009) (arXiv:0912.1676).[25] H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710\n(2007).\n[26] If a comparison is made with Eq. (54), we see that \u000eT=\n0, namely, the electron temperature is not changed by\nthe coupling to nonequilibrium magnons.\n[27] A. A. Kovalev, Phys. Rev. B 89, 241101(R) (2014).\n[28] S. K. Kim and Y. Tserkovnyak, Phys. Rev. B 92,\n020410(R) (2015)."    },    {        "title": "1703.10903v1.Spin_Seebeck_effect_in_Y_type_hexagonal_ferrite_thin_films.pdf",        "content": "arXiv:1703.10903v1  [cond-mat.mtrl-sci]  31 Mar 2017Spin Seebeck effect in Y-type hexagonal ferrite thin films\nJ. Hirschner,1K. Kn´ ıˇ zek,1,∗M. Maryˇ sko,1J. Hejtm´ anek,1R. Uhreck´ y,2\nM. Soroka,2J. Burˇ s´ ık,2A. Anad´ on Barcelona,3and M. H. Aguirre3\n1Institute of Physics ASCR, Cukrovarnick´ a 10, 162 00 Prague 6, Czech Republic.\n2Institute of Inorganic Chemistry ASCR, 250 68 ˇReˇ z near Prague, Czech Republic.\n3Instituto de Nanociencia de Arag´ on, Universidad de Zaragoz a, E-50018 Zaragoza, Spain\nSpin Seebeck effect (SSE) has been investigated in thin films o f two Y-hexagonal ferrites\nBa2Zn2Fe12O22(Zn2Y) and Ba 2Co2Fe12O22(Co2Y) deposited by a spin-coating method on\nSrTiO 3(111) substrate. The selected hexagonal ferrites are both f errimagnetic with similar magnetic\nmoments at room temperature and both exhibit easy magnetiza tion plane normal to c-axis. Despite\nthat, SSE signal was only observed for Zn2Y, whereas no signi ficant SSE signal was detected for\nCo2Y. We tentatively explain this different behavior by a pre sence of two different magnetic ions in\nCo2Y, whose random distribution over octahedral sites inte rferes the long range ordering and en-\nhances the Gilbert damping constant. The temperature depen dence of SSE for Zn2Y was measured\nand analyzed with regard to the heat flux and temperature grad ient relevant to the SSE signal.\nKeywords:\nI. INTRODUCTION\nSpintronics is a multidisciplinary field which involves\nthe study of active manipulation of spin degrees of free-\ndom in solid-state systems [1]. Thermoelectricity con-\ncerns the ability of a given material to produce voltage\nwhen temperature gradient is present, thus converting\nthermal energy to electric energy [2]. The emerging re-\nsearch field of spin caloritronics, which may be regarded\nas interconnection of spintronics and thermoelectricity,\ncombines spin-dependent charge transport with energy\nor heat transport. One of the core elements of spin\ncaloritronics is the spin-Seebeck effect discovered in 2008\nby Uchida et al.[3]. The spin-Seebeck effect (SSE) is\na combination of two phenomena - the generation of a\nspin current by a temperature gradient applied across a\nmagnetic material, and a conversion of the spin current\nto electrical current by means of the inverse spin Hall\neffect (ISHE) [4] in the attached metallic thin layer. A\nnecessarycondition for the observationof SSE is that the\ndirections of the spin current, magnetic moments of the\nmagnetic material, and electrical current in the metal-\nlic layer, are mutually perpendicular. Since the resulting\nelectric field is related to temperature gradient, it is pos-\nsible in the regime of linear response to define a spin\nSeebeck coefficient SSSE=EISHE/∇T.\nAs regards the magnetic material as a source of the\nspincurrent,itismoreconvenienttouseinsulatorsrather\nthan conductors, in order to avoid parasitic signals such\nas a planar or anomalous Nernst effect [5]. There are\nthree main types of magnetic insulators possessing crit-\nical temperature TCabove the room temperature: gar-\nnets, spinels, and hexagonal ferrites. So far, most of the\nSSE experiments employed iron-based garnet because of\ntheir very low Gilbert damping constant, i.e.slow decay\n∗corresponding author: knizek@fzu.czof spin waves, since this decay limits the thickness of the\nmagnetic layer that actively generates the spin flow.\nIn this work we have focused on Y-hexaferrites as\nmagnetic material, namely Ba 2Zn2Fe12O22(Zn2Y) and\nBa2Co2Fe12O22(Co2Y). Their mass magnetizations at\nroomtemperatureare42.0emu/gforZn2Yand34emu/g\nfor Co2Y [6], which are higher than 27.6 emu/g of yt-\ntrium ferrite garnet Y 3Fe5O12[7]. Since a positive cor-\nrelation between SSE and the saturation magnetization\nhas also been proposed [8], Y-hexaferrites appear to be\na suitable material for the spin current generation in the\nspin-Seebeck effect.\nThe crystal structure of Y-hexaferrites belongs to the\ntrigonalspace group R3mand is composed ofalternating\nstacksofS (spinel Me 2Fe4O8, Me =Zn or Coin our case)\nand T (Ba 2Fe8O14) blocks along the hexagonal c-axis.\nThe magnetic configuration of Y-hexaferrites is usually\nferrimagnetic, with spin up orientation in octahedral 3 a,\n3band 18hsites and spin down in tetrahedral 6 cT, 6cS\nand octahedral 6 csites, see Fig. 1.\nMagnetocrystallineanisotropyis observedin all hexag-\nonal ferrites, which means that their induced magnetisa-\ntion has a preferred orientation within the crystal struc-\nture, either with an easy axis of magnetisation in the\nc-direction or with an easy plane of magnetisation per-\npendicular to c-direction, the latter being the case of the\nselected Y-hexaferrites. Due to their direction of easy\ngrow lying in ab-plane, hexaferrites inherently tend to\ngrow with their c-axis perpendicular to the film plane\nwhen deposited as thin films. Since the magnetization\nvector in SSE element should lie in parallel to the film\nsurface, the hexaferrites with an easy plane of magneti-\nsation are more suitable for the SSE experiment.\nThe principal difference between Zn2Y and Co2Y\ncomes from a different site preferences and magnetic\nproperties of Zn2+and Co2+. Zn2+ion is non-magnetic\n(d10) and occupies preferentially the tetrahedral sites.\nSince both Fe3+in tetrahedral sites have spin down ori-\nentation, thesubstitutionofZn2+tothesesitesmaximize2\n    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n T \nS /g3 /g3 /g127/g374/g1006/g122/g3 /g18/g381/g1006/g122/g3\n/g1007/g258/g3/g3/g894/g381/g272/g410/g895/g3 → /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0001/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286\u0002/g127/g374/g3 /g38/g286/g3\n/g1010/g272/g3/g3/g894/g381/g272/g410/g895/g3 ← /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1005/g1012/g346/g3/g894/g381/g272/g410/g895 /g3→→→ /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0003/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g1007/g271/g3/g3/g894/g381/g272/g410/g895/g3 → /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0000/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g1005/g1012/g346/g3/g894/g381/g272/g410/g895 /g3→→→ /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272/g3/g3/g894/g381/g272/g410/g895/g3 ← /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0004/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g3 /g3 /g3 /g3\n/g3\nFIG. 1: One formula unit of Ba 2Zn2Fe12O22or\nBa2Co2Fe12O22structure with alternating structural\nblocksSandT. Shown are Fe, Co and Zn polyhedra and Ba\ncations (magenta bullets). The description includes Wycko ff\npositions, types of polyhedra (tetrahedral or octahedral) ,\narrows indicating spin direction of the collinear ferrimag netic\nstructure, and the preferential occupation of sites.\ntheoverallmagneticmomentandthesaturationmagneti-\nsation at low temperature reaches 18.4 µB(theoretical\nlimit considering 5 µBperFe3+would be 20 µB). How-\never, because of the relatively low critical temperature\nTC∼130◦C, the magnetization at room temperature is\nonly about 10.7 µB(42 emu/g). Co2+ion is in the low\nspin state (LS, t6\n2ge1\ng) and occupies preferentially the oc-\ntahedral sites. The resulting magnetic moment depends\non the actual distribution of Co between octahedral sites\noccupied by Fe3+with spin up or spin down orienta-\ntion, nevertheless generally will be much lower than in\nZn2Y and the typical saturation magnetization is around\n10µB. On the other hand, since the critical temperature\nTC∼340◦C of Co2Y is higher, the magnetic moment\natTroomaround 8.6 µBis not so different from that of\nZn2Y, see e.g.the review paper [6].\nSpin-SeebeckeffectinY-hexaferritewasstudiedforthe\ncompound of stoichiometry Ba 2−xSrxZn2Fe12O22(x=\n1.5) [9]. In this study it was observed, that the magni-\ntude of SEE is proportional to bulk magnetization even\nthrough the successive magnetic transitions among vari-\nous helimagnetic and ferrimagnetic phases. M-type hex-\naferrite BaFe 12O19was studied in [10]. Since M-type\nhexaferrite have strong anisotropy with an easy axis of\nmagnetisation in the c-direction, a proper substrate and\ndeposition procedure must be selected in order to grow\nthe thin films with the c-axis oriented parallel to the sur-\nface. The advantage of M-type is its high coercive field,\nwhich makes the resulting SSE element self-biased, thus\nproducing SEE signal even without presence of magnetic\nfield. Spin-Seebeck effect was also studied in Fe 3O4with\nspinel structure, which may be in some context consid-\nered as the simplest structural type of hexagonal ferrites.\nLarge coercive fields and high saturation magnetisationISHEEr\nMr/g115\nsJrΔ/g100/g460/g410/g460/g87/g410\n/g122/g882/g346/g286/g454/g258/g296/g286/g396/g396/g349/g410/g286\n/g94/g396/g100/g349/g75/g1007/g400/g437/g271/g400/g410/g396/g258/g410/g286\n/g4/g367/g69 /g400/g286/g393/g258/g396/g258/g410/g381/g396\n/g346/g286/g258/g410/g286/g396\nFIG. 2: Schema of the longitudinal experimental configura-\ntions. Directions of temperature gradient ( ∇T), magnetiza-\ntion (M), spin current ( Js) and electrical field resulted from\ninverse spin Hall effect ( EISHE) are shown. The meaning of\nparameters Vx,tz,dxand△Tzused in eq. 2 is also indicated.\nmakes Fe 3O4promising magnetic material for the inves-\ntigation of self-biased SSE elements [11–16]\nCurrent researchdescribes the SSE using typical quan-\ntity of spin Seebeck coefficient with unit of µV/K, which\nis in conventional thermoelectric materials used for eval-\nuating the effectiveness of the process. However, in most\nof the experimental setups the temperature sensors mea-\nsuring the temperature difference △Tare attached to\nthe measure cell itself. This implies that △Tdescribes\nnot only the thermal characteristics of the studied ma-\nterial, but the whole measurement cell instead, making\nthe quantity in unit of µV/Kphysically irrelevant to the\nspin Seebeck effect itself. This issue was studied in de-\ntails in Ref. [17]. The authors pointed out, that when\nusing the setup dependent △Tas independent variable\nthe determined SSE can be hardly comparable between\nlaboratories. In order to solve this problem, the authors\ndesigned a measurement system with precise measure-\nment of the heat flux through the sample and proposed\nusing heat flux or thermal gradient at the sample as the\nindependent variable.\nIn this work we have followed this approach and man-\nifested, that the total temperature difference △Tis not\nsuitable independent variable even for measuring within\none setup if the temperature dependent experiment is\nperformed, since the temperature evolution of thermal\nconductivity of the whole setup may be different from\nthat of the sample material itself.\nII. EXPERIMENTAL\nThin films of Ba 2Zn2Fe12O22(Zn2Y) and\nBa2Co2Fe12O22(Co2Y) were prepared by spin-coating\ntechnique on (111)-oriented, epitaxially polished3\nSrTiO 3(STO) single crystals with metalorganic\nprecursor solutions. Commercial 2-ethylhexanoates\nMe(CH 3(CH2)3CH(C2H5)COO) n(n= 2 for Me = Ba,\nCo, Zn; n = 3 for Me = Fe, ABCR, Germany) were used\nas precursors. Calculated amounts of metal precursors\nwere dissolved in iso-butanol, mixed and heated for\nseveral hours at 80◦C to accomplish homogenization.\nSubsequently a suitable amount of 2,2-diethanolamine\n(DEA) used as a modifier was added. The modifier to\nalkali earth metal molar ratio was n(DEA)/n(alkali earth\nmetal) = 2. Prior to the deposition the stock solutions\nwere usually diluted with iso-butanol to obtain films of\ndesired thickness. All reactions and handling were done\nunder dry nitrogen atmosphere to prevent reaction with\nair humidity and preliminary formation of alkaline earth\ncarbonates in solutions. Single crystals of STO were\nwashed in acetone combined with sonication and then\nannealed at 1200◦C in air for 24 hours to heal up the sur-\nface damage caused during polish treatment. Prior the\ndeposition they were treated with plasma (Zepto Plasma\ncleaner, Diener Electronic, Germany). After the drying\nat 110◦C for several minutes and pyrolysis of gel films at\n300◦C for 5 minutes, crystallization annealing was done\nat 1000◦C for 5 minutes in conventional tube furnace\nunder open air atmosphere. The deposition-annealing\ncycle was repeated ten times to obtain the desired film\nwith approximately 300 −350 nm of thickness. Final\nannealing was done in tube furnace under open air\natmosphere at 1050◦C for 5 min (Zn2Y) or 1000◦C for\n60 min (Co2Y).\nSpin Seebeck effect was measured using home-made\napparatus. A longitudinal configuration was used, in\nwhich the directions of the spin current, magnetic mo-\nments and electrical current are mutually perpendicular\n[18], see Fig. 2. AlN plate with high thermal conduc-\ntivity was used to separate the heater and the sample\nin order to uniformly spread the heat flux over the sam-\nple area. The thermal barriers between individual parts\nof the call were treated by appropriate greases (Apiezon\ntype N, Dow Corning Varnish, Ted Pella silver paste).\nThe width of the measured sample was 2 mm, the\nlengthwas7mmandcontactdistancewasapprox. 5mm.\nThickness of the Zn2Y-hexaferrite layers was between\n300−350 nm, the thickness for Co2Y-hexaferrite layers\nranged between 150 −300 nm. Pt layer was deposited\nusing K550X Quorum Technologies sputter coater. The\nthickness of the layer was determined by internal FTM\ndetector (Tool factor 4.7), the final Pt deposition thick-\nness was ∼8 nm. The resistanceof the Pt-layermeasured\nby a 2-point technique was within the range 350 −650 Ω\nat room temperature and linearly decreased by 10 −15%\ndown to 5 K, whereas the resistance of the Y-hexaferrite\nthin layer itself was more than GΩ. Therefore, the con-\ntributionfromthe anomalousNernst effect(ANE) canbe\nconsidered as negligible due to the resistivity difference\nbetween Y-hexaferrite and Pt layers.\nThe magnetic hysteresis loops were measured within\nthe range of magnetic field from −25 to 25 kOe at room25 30 35 45 50 55 60 650100020003000400050006000\n-0.5 0.0 0.5-0.5 0.0 0.5\n00120015\n0027\n0030Co2Y\nZn2Y\n2theta (Cu, K α) counts per sec.\nomega (°) counts per sec.(0012)\n±0.45°counts per sec.\nomega (°) \n(0012)\n±0.58°\nFIG. 3: X-ray diffraction of the Ba 2Zn2Fe12O22(black line,\nZn2Y) and Ba 2Co2Fe12O22(blue line, Co2Y) thin film. The\ninsets show rocking-curve measurements. The diffraction\npeak (111) of the SrTiO 3substrate is skipped.\nFIG. 4: AFM images of surface topography of (a)\nBa2Zn2Fe12O22(calculated roughness r.m.s. = 27 nm) and\n(b) Ba 2Co2Fe12O22(r.m.s = 30 nm).\ntemperature using a SQUID magnetometer (MPMSXL,\nQuantum Design)\nThe phase purity and degree of preferred orientation\nof the thin films was checked by X-ray diffraction over\nthe angular range 10 −100◦2θusing the X-ray pow-\nder diffractometer Bruker D8 Advance (CuK α1,2radia-\ntion, secondary graphite monochromator). Atomic force\nmicroscopy AFM (Explorer, Thermomicroscopes, USA)\nwas used to evaluate surface microstructure of the thin\nfilms.\nIII. RESULTS AND DISCUSSION\nThe X-ray diffraction confirmed single phase purity of\nthe thin film and c-axis preferred orientation, quantified\nby the full-width at the half-maximum (FWHM) of the\nrocking curve as 0.45◦for Zn2Y and 0.58◦for Co2Y, see\nFig. 3. The c-lattice parameters 43.567(7) ˚A for Zn2Y4\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s50/s45/s49/s48/s49/s50/s86\n/s83/s83/s69/s32/s40 /s86/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s32 /s84/s32/s61/s32/s49/s75\n/s32 /s84/s32/s61/s32/s50/s75\n/s32 /s84/s32/s61/s32/s51/s75\n/s32 /s84/s32/s61/s32/s52/s75\n/s32 /s84/s32/s61/s32/s53/s75/s84/s32/s61/s32/s51/s48/s48/s75\n/s116/s32/s61/s32/s51/s48/s48/s32/s110/s109\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114\n/s32/s80/s97/s114/s97/s108/s108/s101/s108\nFIG. 5: Spin Seebeck signal (upper panel), and in plane and\nout of plane magnetization (lower panel), in dependence on\nmagnetic field at 300 K for Ba 2Zn2Fe12O22.\nand 43.500(9) ˚A for Co2Y, calculated using cos θ/tanθ\nextrapolation to correct a possible off-centre position of\nthe film during XRD measurement, are in good agree-\nment with literature values [19].\nFig. 4 shows AFM images of surface topography of\nZn2Y and Co2Y. Platelets with hexagonal shape can be\nidentified in both images with similar shape and size.\nCalculated roughness (r.m.s.) values are around 27 −\n30 nm.\nThe magnetic properties ofthe Y-hexaferritesthin lay-\ners werecharacterizedby magnetization curvesmeasured\nat room temperature. The magnetic moment of Zn2Y at\nTroomdetermined from the saturatedvalue of magnetiza-\ntionin parallelorientationis11 µB, seethelowerpanelof\nFig. 5, which is comparable with the expected value [6].\nThe measurement confirms that Zn2Y is a soft magnet\nwith negligible hysteresis. The saturation in the orien-\ntation parallel with the thin layer is attained already at\nlow field, whereas the saturation in the out of plane ori-\nentation, i.e.along the c-direction, is achieved at higher\nfield above 1 ∼T, in agreement with the abeasy plane\norientation.\nThe Spin Seebeck signal of Zn2Y at room tempera-\nture is displayed in the upper panel of Fig. 5 for various/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s86\n/s83/s83/s69/s32/s40 /s86/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s116/s32/s61/s32/s51/s48/s48/s32/s110/s109\n/s32/s116/s32/s61/s32/s49/s53/s48/s32/s110/s109/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s84/s32/s61/s32/s53/s32/s75\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114\n/s32/s80/s97/s114/s97/s108/s108/s101/s108\nFIG. 6: Spin Seebeck signal (upper panel) for 2 selected\nthin layers, and in plane and out of plane magnetization\n(lower panel), in dependence on magnetic field at 300 K for\nBa2Co2Fe12O22.\ntemperature gradients applied across the thin layer. The\nmeasuredvoltageispositiveinpositiveexternalmagnetic\nfield, in agreement with the positive spin Hall angle of Pt\n[20], and changes sign when switching the polarity of the\nmagneticfield. Thedependence onthe magneticfield has\nthe same shape with negligible hysteresis as the magne-\ntization in parallel orientation. The data clearly show\nlinear dependence on temperature gradient.\nThe magnetic moment of Co2Y at Troomdetermined\nfrom the saturated value of magnetization in parallel ori-\nentation is 10 µB/f.u., see the lower panel of Fig. 6.\nThis value is slightly higher than the expected moment\n[6], presumably due to relatively higher structural pref-\nerence of Co for spin down sites in the case of our thin\nfilms. The difference between the saturation in parallel\nandperpendicularorientationisbiggerinagreementwith\nhigher magnetocrystalline anisotropy of Co2Y compared\nto Zn2Y.\nHowever, despite the similar magnetic properties, the\nSSE signal for Co2Y was not observed, see the upper\npanel of Fig. 6. To explain this different behavior of\nZn2Y and Co2Y, we have considered the difference in\nthe cation distributions of the transition metal cations\nover the structure, see for details the Introduction sec-5\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s83/s83/s69/s32/s40 /s86/s47/s75/s41/s32/s84/s32/s61/s32/s53/s75 /s32/s84/s32/s61/s32/s49/s48/s48/s75/s83/s83/s69/s32/s40 /s86/s47/s75/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s84/s32/s61/s32/s50/s48/s48/s75\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s84/s32/s61/s32/s51/s48/s48/s75\nFIG. 7: Spin Seebeck signal (SSE) dependence on magnetic\nfield, divided by the total temperature difference △T, for\nBa2Zn2Fe12O22at selected temperatures.\ntion. Since Zn2+ion is a non-magnetic, the structure of\nBa2Zn2Fe12O22only contains one type of magnetic ion.\nZn cation preferentially substitutes Fe3+in two tetrahe-\ndral sites with the same direction of spin polarization,\ntherefore the total spin polarization of the unit cell does\nnot significantly fluctuate across the material. In dis-\ntinction, Ba 2Co2Fe12O22contains two types of magnetic\nions,i.e.Co2+in low spin state in addition to Fe3+\nin high spin state. Co substitute Fe3+in all octahedral\nsites, where Fe may have both directions of spin polar-\nization, without strong preference for a particular sites.\nWe tentatively propose, that this random distribution of\nCo2+over octahedral sites with various spin polariza-\ntions interferes the long range magnetic ordering across\nthe material, enhancesthe Gilbert damping constant and\npossibly results in suppressing of the SSE signal.\nSSE loops of Ba 2Zn2Fe12O22was measured at several\ntemperatures down to 5 K, see the measurements at se-\nlected temperatures 5, 100, 200 and 300 K in Fig. 7. The\noutput power of the heater was the same for all temper-\natures. The character of the loops is not changed with\nloweringtemperature, the magnitude of the signalis only\nvarying.\nIn order to investigate the temperature dependence of\nthe SSE signal of Zn2Y in more details, we have per-\nformed measurementdown to low temperature with 10 K\nstep. The output power of the heater was also kept con-\nstantduringthismeasurement. ThevalueoftheSSEwas\ndetermined by switching the magnetic field to ±0.4 T at\neach temperature and calculating the difference\nSSE=V+0.4T−V−0.4T\n2(1)\nThe resulting temperature dependence is displayed in\nthe Fig. 9 in three ways. In the upper part of the figure,\nFig. 9a, the SSE signal is divided by the total temper-\nature difference △Tdetermined over the whole measur-0 50 100 150 200 250 3000.00.51.01.52.0\n0 100 200 300050100150200250 Thermal conductivity (W/K/m)\nTemperature (K) AlN\n SrTiO3 W/K/m\nT (K)\nFIG. 8: Thermal conductivity of bulk Ba 2Zn2Fe12O22. Inset:\nthermal conductivity of AlN and SrTiO 3.\ning cell. The temperature evolution of △Tshown in the\ninset revealed, that △Tincreased several times during\ncooling. Since the output power of the heater was kept\napproximately constant, this increase should be related\nto a decrease of the thermal conductivity of the materi-\nals between the temperatures probes. To verify this as-\nsumption, we have measured thermal conductivity of the\nrelevant materials, i.e.the bulk sample Ba 2Zn2Fe12O22\nsynthesizedfromtheprecursorsusedforthethinlayerde-\nposition and compacted by isostatic pressing, AlN plate\nused to separate the heater and the sample, and the\nSrTiO 3substrate, see Fig. 8. However, thermal conduc-\ntivities of these materials weighted by their thickness in\nthe measuring cell cannot explain the evolution of △T.\nIt is obvious, that in order to explain the observed tem-\nperature dependence of △T, the thermal resistance of\nthe thermal barriers between the attached parts of the\ncell must be taken into account. We have calculated,\nthat the thermal resistance of the barriers at room tem-\nperature represents more than 50% of the total thermal\nresistance of the cell, and its percentage increases with\ntemperature.\nSince the value of the total temperature difference △T\ncannot be used as independent variable in different mea-\nsurement setups among various laboratories, another less\nsetup dependent parameter should be used instead, in\norder to normalize the measured SSE signal. We used\nthe heat flux through the sample, as it was proposed in\nRef [17]. SSE signal divided by the heat flux through\nthe sample is displayed in Fig. 7b, the corresponding\nheat flux corrected for the heat losses due to radiation,\nis shown in the inset.\nIn order to extract quantity comparable over differ-\nent measurement setups including the geometry of the\nsample, an expression for spin Seebeck effect related to\nsample dimensions and temperature difference over the\nsample itself was defined [17, 21]\nSSSE=Vxtz\ndx△Tz(2)\nwhereVxis the voltage measured, tzis the thickness6\n0.00.20.40.60.8\n0 100 200 30002040\n020406080100\n0 50 100 150 200 250 3000.00.10.20.30.40.50 100 200 300110120130\n0 100 200 3000.00.51.01.5SSE (µV/K)\na)ΔT (K)\nT (K)\nb) SSE (µV/ W)\nc) SSSE (µV/K)\nTemperature (K) Heat flux\n(mW)\nT (K)\n ΔTz(10-3K)\nT (K)\nFIG. 9: Spin Seebeck signal (SSE) dependence on tempera-\nture for Ba 2Zn2Fe12O22, (a) divided by overall temperature\ngradient, inset: temperature difference. (b) divided by hea t\nflux, inset: heat fluux. (c) calculated according to eq. 2, ins et:\ntemperature difference △Tz.\nof the magnetic material, dxis the electric contact dis-\ntance, and △Tzis the temperature difference at the mag-\nnetic material along the thickness tz, see Fig 2. With the\nknowledge of the heat flux and the thermal conductivity\nof the sample material Ba 2Zn2Fe12O22we were able to\ncalculate SSSEaccording to eq. 2, see the temperature\ndependence in Fig. 9c, the evolution of △Tzis displayed\nin the inset.\nThe correct normalization of SSE signal is important\nnot only for comparing among various measurement se-\ntups, but also for the correct determination of the tem-\nperature dependence, as it is evident by comparison of\nvarious temperature evolutions of SSE shown in Fig. 7.\nThe SSE related to the total temperature difference △T\n(Fig. 7a) shows incorrect temperature dependence influ-\nenced by the temperature dependence of the total ther-\nmal conductivity of the measuring setup. We propose\nthat the correct temperature dependence is determined\nby relating SSE to heat flux (Fig. 7b) or to temperature\ndifference at the sample △Tz(Fig. 7c). In this case, SSE\nis almost linearly increasing with lowering temperature.\nThe almost 5 ×increase of SSE at low temperaturecompared to room temperature can be partially ex-\nplained by the increased magnetization (almost 2 ×), but\nthe decrease of Gilbert damping factor αshould be of\ngreater influence in this regard. It was determined in\nthe study of the temperature dependence of SSE signal\nin Y3Fe5O12garnet (YIG) [22], that the effective prop-\nagation length of thermally excited magnons ξis pro-\nportional to T−1, and since at the same time α∼ξ−1\n[23, 24], it means that Gilbert damping factor α, which is\nexpected to suppress the SSE signal, is linearly decreas-\ning with temperature.\nIn distinction to temperature dependence of SSE in\nYIG [22], where a maximum in SSE was observed and\nexplained by the interplay of the increase of magnon ef-\nfective propagationlength and decrease ofthe total num-\nber of thermally excited magnons, we observed no max-\nimum down to low temperature. We ascribe it to the\nlower dispersion of acoustic branches in magnon spectra\nof Y-hexaferrite, which makes the influence of increasing\ntotal number of thermally excited magnons less impor-\ntant.\nFor the confrontation of the normalized room val-\nues between Y-hexaferrite and garnet, we have deter-\nmined values 21 µV/W and SSSE= 0.11µV/K for\nBa2Zn2Fe12O22, which are lower in comparison with\n46.6µV/W and 0.28 µV/K for Y 3Fe5O12[17], despite\nthe higher magnetic moment of Zn2Y. We presume, that\nit is due to the lower Gilbert damping constant αand\nhigher dispersion of acoustic branches in magnon spectra\nof Y3Fe5O12.\nIV. CONCLUSIONS\nSpin Seebeck effect (SSE) has been investigatedin thin\nfilms of two Y-hexagonal ferrites Ba 2Zn2Fe12O22(Zn2Y)\nand Ba 2Co2Fe12O22(Co2Y) deposited by spin-coating\nmethod on SrTiO 3(111) substrate. The SSE signal was\nobservedfor Zn2Y, whereasno significant SSE signal was\ndetected for Co2Y. This can be explained by a pres-\nence of two different magnetic ions in Co2Y, whose ran-\ndom distribution over octahedral sites interferes the long\nrange ordering and enhances the Gilbert damping con-\nstant. The magnitude of spin-Seebeck signal of Zn2Y\nnormalized to the temperature difference at the inves-\ntigated layer and sample dimensions ( SSSE) is compa-\nrable to the results measured on yttrium iron garnet\nY3Fe5O12.SSSEofZn2Yexhibitsmonotonicallyincreas-\ning behaviour with decreasing temperature, as a result\nof the simultaneous increase of the magnetization and\nmagnon effective propagation length.\nAcknowledgement . This work was supported by\nProject No. 14-18392S of the Czech Science Foundation\nand SGS16/245/OHK4/3T/14 of CTU Prague.7\n[1] Y. Xu, D. D. Awschalom, and J. Nitta, Handbook of\nSpintronics (Springer Netherlands, 2015).\n[2] D. M. Rowe, Thermoelectrics Handbook: Macro to Nano\n(Taylor & Francis, 2005).\n[3] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. 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Pasquale,\nT. Kikkawa, K. Uchida, and E. Saitoh, Evaluation of\nthermalgradientsinlongitudinalspinSeebeckeffectmea-\nsurements, J. Appl. Phys. 117, 17C510 (2015).\n[18] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Mu-\nrakami, and E. Saitoh, Longitudinal spin Seebeck effect:\nfrom fundamentals to applications, J. Phys.-Condens.\nMat.26, 343202 (2014).\n[19] H. S. Shin and S.-J. Kwon, X-ray powder diffraction pat-\nterns of two Y-type hexagonal ferrites, Powder Diffr. 8,\n98 (1993).\n[20] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87,\n1213 (2015).\n[21] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom,\nJ. P. Heremans, and R. C. Myers, Observation of the\nspin-Seebeck effect in a ferromagnetic semiconductor,\nNat. Mater. 9, 898 (2010).\n[22] E. J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson,\nD. A. MacLaren, G. Jakob, and M. Klaui, Influence of\nThickness and Interface on the Low-Temperature En-\nhancement of the Spin Seebeck Effect in YIG Films,\nPhys. Rev. X 6, 031012 (2016).\n[23] U. Ritzmann, D. Hinzke, and U. Nowak, Propagation of\nthermally induced magnonic spin currents, Phys. Rev. B\n89, 024409 (2014).\n[24] U. Ritzmann, D. Hinzke, A. Kehlberger, E. J. Guo,\nM. Klaui, and U. Nowak, Magnetic field control of the\nspin Seebeck effect, Phys. Rev. B 92, 174411 (2015)."    },    {        "title": "1001.2845v1.Resonance_Damping_in_Ferromagnets_and_Ferroelectrics.pdf",        "content": "arXiv:1001.2845v1  [cond-mat.other]  16 Jan 2010Resonance Damping in Ferromagnets and Ferroelectrics\nA. Widom\nPhysics Department, Northeastern University, Boston, MA U SA\nS. Sivasubramanian\nNSF Nanoscale Science & Engineering Center for High-rate Na nomanufacturing,\nNortheastern University, Boston MA USA\nC. Vittoria and S. Yoon\nDepartment of Electrical and Computer Engineering, Northe astern University, Boston, MA USA\nY.N. Srivastava\nPhysics Department and & INFN, University of Perugia, Perug ia IT\nThe phenomenological equations of motion for the relaxatio n of ordered phases of magnetized\nand polarized crystal phases can be developed in close analo gy with one another. For the case of\nmagnetized systems, thedrivingmagnetic fieldintensityto ward relaxation was developedbyGilbert.\nFor the case of polarized systems, the driving electric field intensity toward relaxation was developed\nby Khalatnikov. The transport times for relaxation into the rmal equilibrium can be attributed to\nviscous sound wave damping via magnetostriction for the mag netic case and electrostriction for the\npolarization case.\nPACS numbers: 76.50.+g, 75.30.Sg\nI. INTRODUCTION\nIt has long been of interest to understand the close\nanalogiesbetween orderedelectric polarized systems, e.g.\nferroelectricity , and ordered magnetic systems, e.g. fer-\nromagnetism . Atthe microscopiclevel, the sourceofsuch\nordering must depend on the nature of the electronic en-\nergy spectra. The relaxation mechanism into thermal\nequilibrium state must be described by local electric field\nfluctuationsforthe electricpolarizationcaseandbymag-\nnetic intensity fluctuations for the magnetization case;\nSpecifically, the field fluctuations for each case\nGpol\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Ej(r′,−iλ)∆Ei(r,t)/an}bracketri}htdλ,\nGmag\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Hj(r′,−iλ)∆Hi(r,t)/an}bracketri}htdλ,\nwherein β=¯h\nkBT,(1)\ndetermine the relaxation time tensor for both cases via\nthe fluctuation-dissipation formula1–4\nτij=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVGij(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(2)\nWe have unified the theories of relaxation in ordered po-\nlarized systems and ordered magnetized systems via the\nKubo transport time tensor in Eqs.(1) and (2).\nThe transport describing the relaxation of or-\ndered magnetization is the Landau-Lifshitz-Gilbert\nequation5–7. This equation has been of considerable\nrecent interest8–10in describing ordered magnetic reso-\nnancephenomena11–14. The equationdescribingthe elec-tric relaxation of an ordered polarization is the Landau-\nKhalatinikov-Tani equation15–17. This equation can be\nsimplymodeled18–21witheffectiveelectricalcircuits22–25.\nInformation memory applications26–29of such polarized\nsystem are of considerable recent interest30–32.\nThe unification of the magnetic Gilbert-Landau-\nLifshitz equations and the electric Landau-Khalatnikov-\nTani equations via the relaxation time tensor depends\non the notion of a nonequilibrium driving field . For the\nmagnetic case, the driving magnetic intensity Hddeter-\nmines the relaxation of the magnetization via the torque\nequation\n˙M=γM×Hd, (3)\nwhereinγisthegyromagneticratio. Fortheelectriccase,\nthe driving electric field Eddetermines the relaxation of\nthe polarization via the equation of motion for an ion of\nchargeze\nm¨r=zeEd. (4)\nThe unification of both forms of relaxation lies in the\nclose analogy between the magnetic driving intensity Hd\nand the electric driving field Ed.\nIn Sec.II the thermodynamics of ordered magnetized\nand polarized systems is reviewed. The notions of mag-\nnetostrictionand electrostrictionaregiven aprecise ther-\nmodynamic definition. In Sec.III, the phenomenology of\nthe relaxation equations are presented. The magnetic\ndriving intensity Hdand the electric driving field Edare\ndefined in terms of the relaxation time tensor Eq.(2). In\nSec.IV, we introduce the crystal viscosity tensor. From a\nKubo formula viewpoint, the stress fluctuation correlax-2\nation\nFijkl(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆σkl(r′,−iλ)∆σij(r,t)/an}bracketri}htdλ,(5)\ndetermines the crystal viscosity\nηijkl=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVFijkl(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(6)\nFor models of magnetic relaxation wherein acoustic heat-\ning dominates via magnetostriction33and for models of\nelectric relaxation wherein acoustic heating dominates\nvia electrostriction, the relaxation time tensor in Eq.(2)\ncan be related to the viscosity tensor Eq.(6). An inde-\npendent microscopic derivation of viscosity induced re-\nlaxationisgiveninAppendixA. IntheconcludingSec.V,\nthe sound wave absorption physics of the viscous damp-\ning mechanism will be noted.\nII. THERMODYNAMICS\nOur purpose is to review the thermodynamic proper-\nties of both magnetically ordered crystals and polariza-\ntion ordered crystals. The former is characterized by\na remnant magnetization Mfor vanishing applied mag-\nnetic intensity H→0 while the latter is characterized by\na remnant polarization Pfor vanishing applied electric\nfieldE→0.\nA. Magnetically Ordered Crystals\nLetwbe the enthalpy per unit volume. The funda-\nmental thermodynamic law determining the equations of\nstate for magnetically ordered crystals is given by\ndw=Tds+H·dM−e:dσ, (7)\nwhereinsistheentropyperunitvolume, Tisthetemper-\nature,eis the crystal strain and σis the crystal stress.\nThe magnetic adiabatic susceptibility is defined by\nχ=/parenleftbigg∂M\n∂H/parenrightbigg\ns,σ. (8)\nIf\nN=M\nM⇒N·N= 1 (9)\ndenotes a unit vector in the direction of the magnetiza-\ntion, then the tensor Λ ijkldescribing adiabatic magne-\ntostriction coefficients may be defined as34\n2ΛijklNl=M/parenleftbigg∂eij\n∂Mk/parenrightbigg\ns,σ=−M/parenleftbigg∂Hk\n∂σij/parenrightbigg\ns,M.(10)When the system is out of thermal equilibrium, the driv-\ning magnetic intensity is\nHd=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−τ·/parenleftbigg∂M\n∂t/parenrightbigg\n,(11)\nwherein τare the relaxation time tensor transport co-\nefficients which determine the relaxation of the ordered\nmagnetic system into a state of thermal equilibrium.\nB. Ordered Polarized Crystals\nThe fundamental thermodynamic law determining the\nequations of state for ordered polarized crystals is given\nby\ndw=Tds+E·dP−e:dσ, (12)\nwherein wis the enthalpy per unit volume, sis the en-\ntropy per unit volume, Tis the temperature, eis the\ncrystal strain and σis the crystal stress. The electric\nadiabatic susceptibility is defined by\nχ=/parenleftbigg∂P\n∂E/parenrightbigg\ns,σ. (13)\nThe tensor βijkdescribing adiabatic electrostriction co-\nefficients may be defined as34\nβijk=/parenleftbigg∂eij\n∂Pk/parenrightbigg\ns,σ=−/parenleftbigg∂Ek\n∂σij/parenrightbigg\ns,P.(14)\nThe piezoelectric tensor is closely related to the elec-\ntrostriction tensor via\nγijk=/parenleftbigg∂eij\n∂Ek/parenrightbigg\ns,σ=/parenleftbigg∂Pk\n∂σij/parenrightbigg\ns,E=βijmχmk.(15)\nWhen the system is out of thermal equilibrium, the driv-\ning electric field is\nEd=E−/parenleftbigg∂w\n∂P/parenrightbigg\ns,σ−τ·/parenleftbigg∂P\n∂t/parenrightbigg\n,(16)\nwherein τis the relaxation time tensor transport coef-\nficients which determine the relaxation of the ordered\npolarized system into a state of thermal equilibrium.\nIII. RESONANCE DYNAMICS\nHere we shall show how the magnetic intensity Hd\ndrives the magnetic resonance equations of motion in\nmagnetically ordered systems. Similarly, we shall show\nhowtheelectricfield Eddrivesthe polarizationresonance\nequations of motion for polarized ordered systems.3\nA. Gilbert-Landau-Lifshitz Equations\nThe driving magnetic intensity determines the torque\non the magnetic moments according to\n∂M\n∂t=γM×Hd. (17)\nEmploying Eqs.(11) and (17), one finds the equations for\nmagnetic resonance in the Gilbert form\n∂M\n∂t=γM×/bracketleftBigg\nH−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−/parenleftbiggα\nγM/parenrightbigg\n·∂M\n∂t/bracketrightBigg\n,(18)\nwherein the Gilbert dimensionless damping tensor αis\ndefined as\nα= (γM)τ. (19)\nOnemaydirectlysolvetheGilbert equationsforthe driv-\ning magnetic intensity according to\nHd+α·/parenleftbig\nN×Hd/parenrightbig\n=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ.(20)\nEqs.(17) and (20) expressthe magneticresonancemotion\nin the Landau-Lifshitz form.\nB. Landau-Khalatnikov-Tani Equations\nThe driving electric field gives rise to a polarization\nresponse according to\n∂2P\n∂t2=/parenleftBigg\nω2\np\n4π/parenrightBigg\nEd, (21)\nwhereinωpis the plasma frequency. A simple derivation\nof Eq.(21) may be formulated as follows. In a large vol-\numeV, the polarization due to charges {zje}is given\nby\nP=/parenleftbigg/summationtext\njzjerj\nV/parenrightbigg\n. (22)\nIf the drivingelectric field acceleratesthe chargesaccord-\ning to\nmj¨rj=zjeEd, (23)\nthen Eq.(21) holds true with the plasma frequency\nω2\np= 4πe2lim\nV→∞/bracketleftBigg/summationtext\nj(z2\nj/mj)\nV/bracketrightBigg\n= 4πe2/summationdisplay\nanaz2\na\nma,(24)\nwhereinnais the density of charged particles of type a.\nThe polarization resonance equation of motion follows\nfrom Eqs.(16) and (21) as17\n/parenleftbigg4π\nω2p/parenrightbigg∂2P\n∂t2+τ·∂P\n∂t+∂w(P,s,σ)\n∂P=E.(25)The electric field Einduces the polarization Pat reso-\nnant frequencies which are eigenvalues of the tensor Ω\nfor which\nΩ2=ω2\npχ−1\n4π≡ω2\np(ǫ−1)−1. (26)\nThedecayratesforthepolarizationoscillationsareeigen-\nvalues of the tensor Γfor which\nΓ=ω2\npτ\n4π. (27)\nIfthedecayratesarelargeonthescaleofthetheresonant\nfrequencies, then the equation of motion is over damped\nso that\nmin\njΓj≫max\niΩiimplies\nτ·∂P\n∂t+∂w(P,s,σ)\n∂P=E. (28)\nEq.(28) represents the Landau-Khalatnikov equation for\npolarized systems.\nIV. HEATING RATE PER UNIT VOLUME\nLet us here consider the heating rate implicit in relax-\nation processes. Independently of the details of the mi-\ncroscopic mechanism for generating such heat, the rates\nof energy dissipation are entirely determined byτ. Ex-\nplicitly, the heating rates per unit volume for magnetiza-\ntion and polarization are given, respectively, by\n˙qM=∂M\n∂t·τ·∂M\n∂t, (29)\nand\n˙qP=∂P\n∂t·τ·∂P\n∂t. (30)\nFinally, the notion of crystal viscosity ηijklis introduced\ninto elasticity theory35via the heating rate per unit vol-\nume from rates of change in the strain ∂e/∂t; It is\n˙qe=∂eij\n∂tηijkl∂ekl\n∂t. (31)\nCrystal viscosity is employed to describe, among other\nthings, sound wave attenuation. Our purpose is to de-\nscribe how heating rates in Eqs.(29) and (30) can be re-\nlated to the heating rate in Eq.((31)). This allows us to\nexpressthetransportcoefficients τintermsofthecrystal\nviscosity.\nA. Relaxation via Magnetostriction\nFrom the magnetostriction Eq.(10), it follows that\nmagnetic relaxation gives rise to a strain\n∂eij\n∂t=2\nMΛijklNk∂Ml\n∂t, (32)4\nand thereby to the heating rate,\n˙q=4\nM2∂Mi\n∂t(ΛmnqiNq)ηmnrs(ΛrskjNk)∂Mj\n∂t,(33)\nin virtue of Eq.(31). Employing Eqs.(29) and (33), we\nfind that the magnetic relaxation transport coefficient in\nthe magnetostriction model\nτij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk).(34)\nThe Gilbert damping tensor follows from Eqs.(19) and\n(34) as\nαij=4γ\nM(ΛmnqiNq)ηmnrs(ΛrskjNk).(35)\nThe central relaxation tensor Eq.(35) describes the mag-\nnetic relaxation in terms of the magnetostriction coeffi-\ncients and the crystal viscosity.\nB. Relaxation via Electrostriction\nFromtheelectrostrictionEq.(14), it followsthatatime\nvarying polarization gives rise to a time varying strain\n∂eij\n∂t=βijk∂Pk\n∂t, (36)\nand thereby to the heating rate,\n˙q=∂Pi\n∂tβkliηklmnβmnj∂Pj\n∂t, (37)\nin virtue of Eq.(31). Employing Eqs.(30) and (37), we\nfind that the electric relaxation transport coefficient in\nthe electroostriction model\nτij=βkliηklmnβmnj. (38)\nThecentralrelaxationtensorEq.(38) describesthe polar-\nization relaxation time tensor coefficients in terms of the\nelectrostriction coefficients and the crystal viscosity. The\nimplications ofthe electrostrictionmodel forthe Landau-\nKhalatnikov equation is to the authors knowledge a new\nresult.\nV. CONCLUSIONS\nFor ordered polarized and magnetized systems, we\nhave developed phenomenological equations of motion inclose analogywith one another. For the magnetized case,\nthe relaxation is driven by the magnetic intensity Hd\nyielding the Gilbert equation of motion7. For the polar-\nized case, the relaxation is driven by the electric field Ed\nyielding the Tani equation of motion17. In both cases,\nthe relaxation time tensor τis determined by the crystal\nviscosity as derived in the Appendix A; i.e. in Eqs.(A3)\nand (A6). The viscosity can be measured independently\nfrom the magnetic or electrical relaxation by employing\nsound absorption techniques36.\nAppendix A: Kubo formulae\nFrom the thermodynamic Eq.(10), the fluctuations in\nthe magnetic intensity are given by magnetostriction, i.e.\n∆Hk(r,t) =−/parenleftbigg2ΛijklNl\nM/parenrightbigg\n∆σij(r,t).(A1)\nEqs.(A1), (1) and (5) imply\nGmag\nij(r,r′,t) =\n4\nM2(ΛmnqiNq)Fmnrs(r,r′,t)(ΛrskjNk).(A2)\nEmploying Eqs.(A2), (2) and (6), one finds the central\nresult for the magnetic relaxation time tensor; It is\nτmag\nij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk) =αij\nγM.(A3)\nFrom the thermodynamic Eq.(14), the fluctuations in\nthe electric intensity are given by electrostriction, i.e.\n∆Ek(r,t) =−βijk∆σij(r,t). (A4)\nEqs.(A4), (1) and (5) imply\nGpol\nij(r,r′,t) =βkliFklmn(r,r′,t)βmnj.(A5)\nEmploying Eqs.(A5), (2) and (6), one finds the central\nresult for the electric relaxation time tensor; It is\nτpol\nij=βkliηklmnβmnj. (A6)\n1S. Machlup and L. Onsager, Phys. Rev. 91, 1505 (1953).\n2S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953).\n3H.B. Callen, Fluctuation, Relaxation, and Resonance inMagnetic Systems , Editor D. ter Haar, p 176, Oliver &\nBoyd Ltd., London (1962).\n4R. Kubo, M. Toda and N. Hashitsume, Statistical Physics5\nII, Nonequilibrium Statistical Mechanics Springer, Berlin\n(1998).\n5L. Landau and L. Lifshitz, Phys. Zeit. Sowjetunion 8,153\n(1935).\n6T. L. Gilbert, Armor Research Foundation Rep. No. 11\nChicago, IL. (1955).\n7T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004).\n8M. Fahnle, D. Steiauf, and J. Seib, J. Phys. D: Appl. Phys.\n41, 164014 (2008).\n9E. Rossi, O.G. Heinonen, andA.H. MacDonald, Phys. Rev.\nB 72, 174412 (2005).\n10V. Kambersky, Phys. Rev. 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B: Microelectronics and Nanometer Structures 27,\n498 (2009).\n31N. Inoue and Y. Hayashi, IEEE Trans. on Elect. Dev. 48,\n2266 (2001).\n32T. Mikolajick, C. Dehm, W. Hartner, I. Kasko, M. Kast-\nner, N. Nagel, M. Moert and C. Mazure, Microelectronics\nReliability 41, 947 (2001).\n33C. Vittoria, S. D. Yoon and A. Widom, Phys. Rev. B 81,\n014412 (2010).\n34L.D. Landau and E.M. Lifshitz, “ Electrodynamics of Con-\ntinuos Media ”, Pergamon Press, Oxford (1960).\n35L.D. Landau and E.M. Lifshitz, “ Theory of Elasticity ”,\nSecs.10, 23 and 34, Pergamon Press, Oxford (1986).\n36A.B. Bhatia, Ultrasonic Absorption , Oxford University\nPress, Oxford (1967)."    },    {        "title": "0810.4633v1.The_domain_wall_spin_torque_meter.pdf",        "content": "The domain wall spin torque-meter\nI.M. Miron, P.-J. Zermatten, G. Gaudin, S. Au\u000bret, B. Rodmacq, and A. Schuhl\nSPINTEC, CEA/CNRS/UJF/GINP,\nINAC, 38054 Grenoble Cedex 9, France\n(Dated: September 15, 2021)\nAbstract\nWe report the direct measurement of the non-adiabatic component of the spin-torque in domain\nwalls. Our method is independent of both the pinning of the domain wall in the wire as well as\nof the Gilbert damping parameter. We demonstrate that the ratio between the non-adiabatic and\nthe adiabatic components can be as high as 1, and explain this high value by the importance of\nthe spin-\rip rate to the non-adiabatic torque. Besides their fundamental signi\fcance these results\nopen the way for applications by demonstrating a signi\fcant increase of the spin torque e\u000eciency.\nPACS numbers: 72.25.Rb,75.60.Ch,75.70.Ak,85.75.-d\n1arXiv:0810.4633v1  [cond-mat.other]  25 Oct 2008The possibility of manipulating a magnetic domain wall via spin torque e\u000bects when\npassing an electrical current through it opens the way for conceptually new devices such as\ndomain wall shift register memories[1]. Early spin-torque theories[2, 3, 4] were based on a\nso called adiabatic approximation which assumed that the incoming electron's spin follows\nexactly the magnetization as it changes direction within the domain wall. Nevertheless, the\nobserved critical currents needed to trigger the domain wall motion were lower than the\nvalue predicted within this framework[5]. As \frst predicted by Zhang[6], the existence of a\nnon-adiabatic term in the extended Landau-Lifshitz-Gilbert equation leads to the vanishing\nof the intrinsic critical current. The action of this non-adiabatic torque on a DW is expected\nto be identical to that of an easy axis magnetic \feld. Micromagnetic simulations have been\nused to predict the velocity dependence on current for a DW submitted to the action of the\ntwo components of the spin-torque[7]. The quantitative measurement of this non-adiabatic\ntorque can be achieved either by demonstrating the equivalence of \feld and current in\na static regime, or by observing the complex dynamic behavior[7]. The main di\u000eculty\nof these measurements comes from the pinning of the DW by material imperfections. It\nmasks the existence of the intrinsic critical current, and in addition, above the depinning\ncurrent, obscures the DW velocity dependence on current. Moreover, most of the DW\nvelocity measurements were done using materials with in-plane magnetization[5, 8, 9, 10,\n11], where the velocity can also depend on the micromagnetic structure of the wall[12]\n(transverse wall or vortex wall). Despite the simpler micromagnetic structure of the DWs,\nvery few results were reported[13] for Perpendicular Magnetic Anisotropy (PMA) materials.\nIn this case the intrinsic pinning is much stronger, probably due to a local variation of\nthe perpendicular anisotropy. Up to now, none of the measurements were able to clearly\nevidence the equivalence between \feld and current, nor to reproduce the predicted dynamic\nbehavior; hence the value of the non-adiabatic torque is still under debate.\nIn this letter we use a novel approach for the measurement of the non-adiabatic component\nof spin-torque. Instead of measuring the DW velocity, we perform a quasistatic measurement\nof its displacement under current and magnetic \feld. In principle this method is similar to\nany quasi-static force measurement: a small displacement is created, \frst with the unknown\nforce and then with a known reference force. In our case the unknown force is caused by\nthe electric current passing through the DW while the reference force is due to an applied\nmagnetic \feld. By comparing the two displacements one directly compares the applied\n2FIG. 1: Schematic representation of the experimental setup. The inset shows an SEM picture of a\nsample.\nforces. Due to the high sensitivity of our method (able to detect DW motion down to\n\u001810\u00002nm[14]) we can study the displacement of the DW inside its pinning center. Since the\nmeasurement relies on the comparison to a reference force, the method is independent of\nthe strength of the pinning. Moreover, as the \feld and current are applied quasi-statically,\nthe damping parameter does not play any role.\nAccording to recent theories[6, 15, 16] that derived the value of the spin torque, \f(the\nratio between the non-adibatic and adiabatic torques) is given by the ratio between the\nrate of the spin-\rip of the conduction electrons and that of the s-d exchange interaction.\nGenerally, two conditions must be ful\flled to obtain a high spin-\rip rate. First it is necessary\nto have a strong crystalline \feld inside the material. The electric \felds will yield a magnetic\n\feld in the rest frame of the moving electrons. Second, a breaking of the inversion symmetry\nis needed. Otherwise the total torque of the magnetic \feld on the electron spin averages\nout, and the spin-\rip may only occur during momentum scattering[17].\nIn order to highlight these e\u000bects we have patterned samples from a Pt 3nm/\nCo0.6nm/(AlO x)2nmlayer[18]. In this case the symmetry is broken by the presence of the\n3AlO xon one side of the Co layer, and of the heavy Pt atoms on the other[19, 20]. We\nwill emphasize the importance of the spin-\rip interaction to spin torque by comparing\nresults from these samples with those for samples fabricated from a symmetric Pt 3nm/\nCo0.6nm/Pt 3nmlayer[21], where a much smaller spin-\rip rate is expected. As the only di\u000ber-\nence between the two structures is the upper layer, we expect similar growth properties for\nthe Co layer. Both samples exhibit PMA and a strong Anomalous Hall E\u000bect (AHE)[22].\nThe \flms are patterned into the shape depicted in Figure 1. This shape is well suited for a\nquasi static measurement as a constriction is created by the presence of the four wires used\nfor the AHE measurement (\fgure 1 inset). This way a DW can be pinned in a position\nwhere changes in the out of plane component of the magnetization (i.e. DW motion) can\nbe detected by electrical measurements. A current is passed through the central wire. This\ncurrent will serve to push the domain wall as well as to probe the eventual displacement.\nIn the case where the DW does not move under the action of the current, the transverse\nresistance remains unchanged and the voltage measured across the side wires (AHE) will be\nlinear with the current. If the DW moves due to the electric current, the exciting force will\ncreate resistance variations, causing a nonlinear relationship between the measured voltage\nand the applied current. A simple way to detect such nonlinearities is to apply a perfectly\nharmonic low frequency (10 Hz) ac current, and look at the \frst harmonic in the Fast Fourier\nTransform (FFT) of the measured voltage. Its value is a measure of the amplitude of the\nDW displacement at the frequency of the applied current. To quantitatively compare the\naction of a magnetic \feld to that of an electric current, the magnetic \feld is applied at\nthe same frequency and in phase (or opposition of phase) with the electric current. By\napplying current and \feld simultaneously, we ensure that their corresponding torques act on\nthe same DW con\fguration. In addition to the displacement provoked by the current, the\n\feld induced displacement will add to the value of the \frst harmonic, which can be either\nincreased if the \feld and current push the wall in the same direction, or decreased if they\nact in opposite directions.\nFigure 2 shows the dependence of the resistance variation at the frequency of the current\n(\u0001R f) on the current amplitude for di\u000berent values of the \feld amplitude. First, at low\ncurrent and \feld amplitudes the displacement is almost linear ( \u0018107A/cm2), but for higher\nvalues, the \u0001R fvaries more rapidly. A simple estimation based on the value of the resistance\nvariation compared to the total Hall resistance of a cross (1 \n) yields \u00181 nm for the\n4FIG. 2: (a) Dependence of the resistance variation on the current amplitude for several \feld\namplitudes (Pt/Co/AlO xsample). The inset shows a possible nonlinear and asymmetric\npotential well. The energy landscape can be modeled by an e\u000bective out of plane magnetic\n\feld that has negative values on one side of the equilibrium position and positive values\non the other. (b) A zoom on the small amplitude regime. The inset shows the perfect\nsuperposition obtained by shifting the curves horizontaly with 1.25 \u0001105Acm\u00002Oe\u00001.5maximum amplitude of the DW motion in the \frst regime and \u00187 nm for the second regime.\nThis behavior can be explained by the anatomy of the local pinning. The local potential\nwell trapping the DW can be considered as a superposition of the geometric pinning [23]\nand intrinsic pinning caused by defects randomly distributed inside the material[13, 21].\nBecause the potential well for the small scale displacements (below 10nm) is dominated by\nthe random intrinsic pinning rather than geometric pinning (the increase of the length of\nthe DW is small \u00181%) in the general case it should be asymmetric. We have veri\fed the\nsupposed asymmetry of the e\u000bective potential well by applying alongside the ac current and\n\feld, a dc bias \feld that changes the local potential well (inset of \fgure 3). By varying\nthis \feld we observed a reduction of the current amplitude needed to access this strongly\nnon-linear regime (\fgure 3). When the magnetic bias \feld was reversed this second regime\nwas no longer attained with the available current densities (not shown).\nThe observed dependence of \u0001R fon current and \feld (\fgures 2 and 3) is in perfect\nagreement with the characteristic features of the non-adiabatic component of the spin-torque.\nFirst, we do not observe any critical current down to the lowest current value (106A/cm2-\n\fgure 4 in [14]). Futhermore, by extrapolating the amplitude of the DW displacement (\fgure\n2), when the current is reduced, the displacement goes to zero as the current goes to zero,\nin agreement with the absence of the critical current.\nHowever, the most important feature of the \u0001R fbehavior is that the curves obtained\nfor any \feld amplitude can be obtained from the curve corresponding to zero \feld just by\nshifting it horizontally (in current): towards the lower current values when the \feld and\ncurrent act in the same direction on the DW and towards higher values when their actions\nare opposed. This means that any displacement of the DW can also be achieved with a\ndi\u000berent current if a magnetic \feld is added. The di\u000berence in current is compensated by\nthe magnetic \feld. The value of this horizontal shift gives the \feld to current correspondence.\nThe inset of \fgure 1b shows that all the curves corresponding to di\u000berent \feld amplitudes\nhave the same shape; by shifting them horizontally (using the \feld-current correspondence),\nthey all collapse on the zero \feld amplitude curve. This shows that independently of the\ndirection or strength of the applied current and \feld, as predicted by the theories, their e\u000bect\non the DW is fundamentally similar. Moreover, further evidence that this correspondence is\nintrinsic and not in\ruenced by pinning is that its value remains the same within the di\u000berent\namplitude regimes as well as when the local potential well is tuned by a constant bias \feld.\n6FIG. 3: The nonlinear regime (Pt/Co/AlO xsample). When an external bias \feld is added,\nthe e\u000bective pinning \feld changes (inset) and the nonlinear regime is reached for di\u000berent\ncurrent and \feld amplitudes. However, this does not cause any change in the \feld to current\ncorrespondence: the horizontal distance between the curves remains the same.\n7Since the motion of the DW is quasi-static the magnetization can be considered to be\nat equilibrium during motion. In this case the sum of all torques must be zero. In order\nfor the DW to remain at rest, the torque from the applied current must be compensated\nby the torque generated by the magnetic \feld. The upturn observed on the -60 Oe curve\n(\fgure 2b) determines the position of the zero amplitude point. Note that the position of\nthis point is in perfect agreement with the \feld to current correspondence obtained from the\nhorizontal shifting of the curves. By taking into account the micromagnetic structure of the\nDW (very thin 5nm Bloch wall) the two torques are integrated over the width of the wall,\nand by comparing their values (the \feld torque is easily calculated; [14]) the non-adiabatic\nterm of the spin-torque is determined. In the case of Pt/Co/AlO xstacks the current-\feld\ncorrespondence is approximately 1.25 105A/cm2to 1Oe, corresponding to a value of \f= 1.\nSimilar measurements (\fgure 1 in [14]) were also performed in the saturated state (with-\nout the DW). They con\frm that there is no contribution to the signal from the ordinary\nHall e\u000bect, but indicate a small contribution from thermoelectric e\u000bects - the Nernst-\nEttingshausen E\u000bect(NEE)[24]. The contribution from DW motion to \u0001R fis much higher\nthan the NEE for the Pt/Co/AlO xstack. In the case of Pt/Co/Pt layers we \fnd that the\namplitude of the current induced DW motion is much smaller and entirely masked by the\nNEE. When a DW is moving inside the perfectly harmonic region at the bottom of the po-\ntential well, its displacement depends linearly on the applied force. In such a scenario, the\ncurrent induced DW motion and the NEE are indistinguishable. They both lead to a linear\ndependence of the \u0001R fresponse on current. The only possibility to separate these e\u000bects,\nfor the Pt/Co/Pt layer, is to attain the high amplitude nonlinear regime of DW motion.\nThis is done by keeping the current amplitude constant and varying the \feld amplitude.\nWhen the current and \feld push the wall in the same direction, the nonlinear regime should\nbe reached for smaller \feld amplitudes, than if their actions were opposed.\nIn the presence of current induced displacements, the nonlinearities observed in the\n\u0001R fversus \feld amplitude curve should be asymmetric. Moreover the asymmetry should\ndepend on the current value. Such an asymmetry is observed (inset of \fgure 4) in the case\nof Pt/Co/AlO xsamples. In contrast to this behavior, a fully symmetric dependence that\ndoes not depend on the current amplitude is measured for the Pt/Co/Pt samples (\fgure 4).\nWe conclude that in this case the spin torque induces DW displacements smaller than the\nresolution limit of this method. This limit value leads to (supplementary notes) \f\u00140.02.\n8FIG. 4: The nonlinear response of a DW to magnetic \feld. (a) \u0001R fvs. the amplitude of the \feld\nfor three di\u000berent current densities in the case of Pt/Co/Pt layers (inset Pt/Co/AlO x).(b)\nDerivative of \u0001R fvs. the \feld amplitude for a Pt/Co/Pt sample (inset Pt/Co/AlO x).\nTheoretical estimations[6] based on a spin-\rip frequency of 1012Hz yield a value \f=0.01.\nTo clarify the di\u000berence of the spin-torque e\u000eciency in the two samples, the symmetry\nbreaking due to the presence of the AlO xsurface must be taken into account. As a metallic\n\flm gets thinner, the conduction electron's behavior resembles more and more to that of a\ntwo-dimensional electron gas. When such a gas is trapped in an asymmetric potential well,\n9the spin-orbit coupling is much stronger than in the case of a symmetric potential due to\nthe Rashba interaction[25]. This e\u000bect was \frst evidenced in nonmagnetic materials where\nthis interaction leads to a band splitting (0.15 eV for the surface states of Au (111)[26]).\nIn the case of ferromagnetic metals this e\u000bect was already proposed to contribute as an\ne\u000bective magnetic \feld[27] for certain DW micromagnetic structures, but should not have\nany e\u000bect for Bloch walls in PMA materials. The simple 1D representation used in this\ncase[27] to model the DW accounts for the coherent rotation of the spins of the incoming\nelectrons around the e\u000bective \feld, but excludes any de-coherence between electrons having\ndi\u000berent k-vector directions on the Fermi sphere (di\u000berent directions of the Rashba e\u000bective\n\feld) as well as possible spatial inhomogeneities of this \feld (surface roughness). Since the\nspin-torque is caused by the cumulative action of all conduction electrons [6], the relevant\nparameter is not the spin-\rip rate of a single electron but the relaxation rate of the out\nof equilibrium spin-density [6]. In a more realistic 2D case, in the presence of the above\nmentioned strong decoherence e\u000bects, the relaxation rate of the out of equilibrium spin-\ndensity approaches the rate of spin precession around the Rashba e\u000bective \feld. The above\nvalue of the measured spin-orbit splitting (0.15 eV) will yield in this case an e\u000bective spin-\n\rip rate of 30 \u00011012Hz, which is in excellent agreement with the order of magnitude of the\nmeasured non-adiabatic parameter, supporting this scenario.\nIn summary, a technique that allows the direct measurement of the torque from an\nelectric current on a DW was developed. We have pointed out the importance of spin-\n\rip interactions to spin torque by comparing its e\u000eciency between two di\u000berent systems.\nWe show that the Pt/Co/AlO xsample with the required symmetry properties to increase\nthe spin-\rip frequency (breaking of the inversion symmetry) shows an enhanced spin torque\ne\u000bect. A value of the order of 1 was measured for the \fparameter approaching the maximum\nvalue predicted by existing theories. This value can be explained by order of magnitude\nconsiderations on the Rashba e\u000bect observed on surface states of metals. Obtaining a high\ne\u000eciency spin torque in a low coercivity material would make possible the development\nof nanoscale devices whose magnetization could be switched at low current densities. The\norder of magnitude of the current densities would be similar to the one observed for magnetic\n10semiconductors[28], but, as the resistance is smaller, the supplied power will be lower.\n[1] S. Parkin, US Patent 309,6,834,005 (2004).\n[2] L. Berger, J. Appl. Phys. 55, 1954 (1984).\n[3] A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier, J. Appl. Phys. 95, 70497051 (2004).\n[4] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[5] M. Klui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman,\nF. Nolting, and U. Rudiger, Phys. Rev. Lett. 94, 106601 (2005).\n[6] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[7] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett 69, 990996 (2005).\n[8] G. S. D. Beach, C. Knutson, C. 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Ohno, Nature 428, 539 (2004).\n12"    },    {        "title": "1701.09110v1.Lack_of_correlation_between_the_spin_mixing_conductance_and_the_ISHE_generated_voltages_in_CoFeB_Pt_Ta_bilayers.pdf",        "content": "arXiv:1701.09110v1  [cond-mat.mes-hall]  31 Jan 2017Lack of correlation between the spin mixing conductance and the ISHE-generated\nvoltages in CoFeB/Pt,Ta bilayers\nA. Conca,1,∗B. Heinz,1M. R. Schweizer,1S. Keller,1E. Th. Papaioannou,1and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n(Dated: June 21, 2021)\nWe investigate spin pumping phenomena in polycrystalline C oFeB/Pt and CoFeB/Ta bilayers\nand the correlation between the effective spin mixing conduc tanceg↑↓\neffand the obtained voltages\ngenerated by the spin-to-charge current conversion via the inverse spin Hall effect in the Pt and Ta\nlayers. For this purpose we measure the in-plane angular dep endence of the generated voltages on\nthe external static magnetic field and we apply a model to sepa rate the spin pumping signal from the\none generated by the spin rectification effect in the magnetic layer. Our results reveal a dominating\nrole of anomalous Hall effect for the spin rectification effect with CoFeB and a lack of correlation\nbetween g↑↓\neffand inverse spin Hall voltages pointing to a strong role of th e magnetic proximity\neffect in Pt in understanding the observed increased damping . This is additionally reflected on the\npresence of a linear dependency of the Gilbert damping param eter on the Pt thickness.\nINTRODUCTION\nIn spin pumping experiments,[1, 2] the magnetization\nof a ferromagnetic layer (FM) in contact with a non-\nmagnetic one (NM) is excited by a microwave field. A\nspin current is generated and injected into the NM layer\nand its magnitude is maximized when the ferromagnetic\nresonance (FMR) condition is fulfilled. The spin cur-\nrent can be detected by using the inverse spin Hall effect\n(ISHE) for conversion into a charge current in appropri-\nate materials. The injected spin current Jsin the NM\nlayer has the form[1]\nJs=/planckover2pi1\n4πg↑↓ˆm×dˆm\ndt(1)\nwhere ˆmis the magnetization unit vector and g↑↓is the\nrealpartofthe spinmixing conductancewhich iscontrol-\nling the intensity of the generated spin current. Its value\nis sensitive to the interface properties. The generation of\nthe spin current opens an additional loss channel for the\nmagnetic system and consequently causes an increase in\nthe measured Gilbert damping parameter α:\n∆αsp=γ/planckover2pi1\n4πMsdFMg↑↓(2)\nThis expression is only valid for thick enough NM lay-\ners where no reflection of the spin current takes place\nat the interfaces. In principle, it allows the estimation\nofg↑↓by measuring the increase in damping compared\nto the intrinsic value. However, other phenomena, like\nthe magnetic proximity effect (MPE) in the case of Pt\nor interface effects depending on the exact material com-\nbination or capping layer material, can have the same\ninfluence, [7, 8] which challenges the measurement of the\ncontribution from the spin pumping. In this sense, it\nis preferable to use an effective value g↑↓\neff. Still, if thespin pumping is the main contribution to the increase\ninα, a correlation between g↑↓\neffand the measured ISHE\nvoltages is expected. A suitable approach in order to un-\nderstand the weight of MPE on the value of g↑↓\neffis the\nuse of FM/NM with varying NM metals, with presence\nand absence of the MPE effect. The measurement of ∆ α\nandg↑↓\nefftogether with the ISHE voltages generated by\nthe spin current in the NM layer can bring clarity to the\nissue.\nHowever, the generation of an additional dc voltage\nby the spin rectification effect,[3–6] which adds to the\nvoltagegeneratedbythe ISHE spin-to-chargeconversion,\ndeters the analysis of the obtained data. The spin recti-\nfication originates from the precession of the magnetiza-\ntion in conducting layers with magnetoresistive proper-\nties, mainly Anisotropic Magnetoresistance (AMR) and\nAnomalous Hall Effect (AHE). Information about the\nphysics behind the measured voltage can only be ob-\ntained after separation of the different contributions. For\nthis purpose, we made use of the different angulardepen-\ndenciesofthecontributionsunderin-planerotationofthe\nexternal magnetic field.\nEXPERIMENTAL DETAILS\nHere, we report on results on polycrystalline\nCo40Fe40B20/Pt,Ta bilayers grown by rf-sputtering on Si\nsubstrates passivated with SiO 2. CoFeB is a material\nchoice for the FM layer due to its low damping proper-\nties and easy deposition.[9, 10] A microstrip-based VNA-\nFMR setup was used to study the damping properties. A\nmore detailed description of the FMR measurement and\nanalysis procedure is shown in previous work.[7, 10] A\nquadrupole-based lock-in setup described elsewhere[11]\nwas used in order to measure the ISHE generated volt-\nage. The dependence of the voltage generated during the\nspin pumping experiment on the in-plane static external2\nfield orientation is recorded for a later separation of the\npure ISHE signal from the spin rectification effect.\nGILBERT DAMPING PARAMETER AND SPIN\nMIXING CONDUCTANCE\nFigure1showsthedependenceoftheeffectivedamping\nparameter αeff(sum ofall contributions)onthe thickness\ndof the NM metal for a CoFeB layer with a fixed thick-\nness of 11 nm. The case d= 0 nm represents the case of\nreference layers with Al capping. From previous studies\nit is known that the use of an Al capping layer induces\na large increase of damping in Fe epitaxial layers.[7] For\npolycrystallineNiFe andCoFeBlayersthis is notthe case\nand it allows the measurement of the intrinsic value α0.\n[8]\nThe observed behavior differs strongly for Pt and\nTa. In the Pt case a large increase in damping is ob-\nserved with a sharp change around d= 1 nm and a\nfast saturation for larger thicknesses. This is quali-\ntatively very similar to our previous report on Fe/Pt\nbilayers.[7] From the measured ∆ αwe extract the value\ng↑↓\neff= 6.1±0.5·1019m−2. This value is larger than the\nonereportedpreviouslyinourgroup[8]forthinnerCoFeB\nlayers with larger intrinsic damping 4 .0±1.0·1019m−2\nandalsolargerthanthevaluereportedbyKim et al.[12],\n5.1·1019m−2. The impact of the Ta layer on damping\nis very reduced and, consequently, a low value for g↑↓\neffof\n0.9±0.3·1019m−2is obtained. This value is now smaller\nthan the one reported by Kim et al.1.5·1019m−2) in-\ndicating that the difference between CoFeB/Pt and Ta\nis larger in our case. A reference has also to be made\nto the work of Liu et al.on CoFeB films thinner than\nin this work. [13] There, no value for the spin mixing\nconductance is provided, but the authors claim a vanish-\nFIG. 1. (Color online) Dependence of the effective Gilbert\ndamping parameter αeffon the thickness of the NM metal.\nA large increase in damping is observed for the Pt case while\na very small but not vanishing increase is observed for Ta.\nFrom the change ∆ αthe effective spin mixing conductance\ng↑↓\neffis estimated using Eq. 2.ing impact on αfor the Ta case. On the contrary the\nincrease due to Pt is almost three times larger than ours,\npointing to a huge difference between both systems. In\nany case, the trend is similar, only the relative difference\nbetween Ta and Pt changes.\nA closer look to the data allows to distinguish a region\nin the Pt damping evolution prior to the sharp increase\nwhere a linear behavior is recognized ( d <1 nm). A lin-\near thickness dependence of αin spin-sink ferromagnetic\nfilms and in polarized Pt has been reported. [14, 15] The\nincreasein damping due to spin currentabsorptionin the\nPt with ferromagnetic order can then be described by:\n∆α= ∆αMPE·dPt/dPt\nc (3)\nwhere ∆αMPEis the total increase in damping due only\nto the magnetic proximity effect in Pt, dPtis the thick-\nness of the Pt layer and dPt\ncis a cutoff thickness which\nis in the order of magnitude of the coherence length in\nferromagnetic layers.[15, 16]\nThe inset in Fig. 1 shows a fit of Eq. 3 from where\ndPt\nc= 0.8nm isobtained assumingavalue ∆ αMPE= 1.2.\nThe value is in qualitative agreement with the reported\nthickness where MPE is present in Pt, ( dPt\nMPE≤1 nm\n[17, 18]) and is lower than the one reported for Py/Pt\nsystems.[14]\n/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48\n/s40/s98/s41\n/s80/s116/s32/s68/s97/s116/s97\n/s32/s70/s105/s116\n/s32/s83/s121/s109/s109/s101/s116/s114/s105/s99\n/s32/s65/s110/s116/s105/s115/s121/s109/s109/s101/s116/s114/s105/s99/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41\n/s84/s97\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s50/s48/s48/s50/s48\n/s32/s32/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41\n/s181\n/s48/s72/s32/s40/s109/s84/s41/s40/s97/s41\nFIG. 2. (Color online) Voltage spectra measured for (a)\nCoFeB/Ta and (b) CoFeB/Pt at 13 GHz. The solid line is a\nfit to Eq. 4. The symmetric voltage Vsymand antisymmetric\nvoltageVantisymcontributions are separated and plotted inde-\npendently (dashed lines). The voltage signal is dominated b y\nVantisymin the Pt case and by Vsymin the Ta case.3\nThe increase of damping due to spin pumping is de-\nscribed by an exponential dependence and explains the\nsharp increase at dPt= 1 nm. However, the fast increase\ndoes not allow for a deep analysis and it is pointing to a\nspin diffusion length in Pt not larger than 1 nm.\nIn any case, this point has to been treated with care.\nThe contribution of MPE to damping can be easily un-\nderestimated and consequently also the value for dPt\nc. In\nany case, the value can be interpreted as a lower limit\nfor ∆αMPE. If this is substracted, under the assumption\nthat the rest of increase is due to spin pumping, the spin\nmixing conductance due only to the this effect would be\ng↑↓\neff= 4.9±0.5·1019m−2.\nELECTRICAL DETECTION OF SPIN PUMPING\nFigure 2(a),(b) shows two voltage measurements\nrecorded at 13 GHz for a NM thickness of 3 nm and\na nominal microwave power of 33 dBm. The measured\nvoltage is the sum of the contribution of the ISHE effect\nand of spin rectification effect originating from the dif-\nferent magnetoresistive phenomena in the ferromagnetic\nlayer. While the spin rectification effect generates both a\nsymmetric and an antisymmetric contribution, [3–5] the\npure ISHE signal is only symmetric. For this reason a\nseparation of both is carried out by fitting the voltage\nspectra (solid line) to\nVmeas=Vsym(∆H)2\n(H−HFMR)2+(∆H)2+\n+Vantisym−2∆H(H−HFMR)\n(H−HFMR)2+(∆H)2(4)\nwhere ∆ HandHFMRare the linewidth and the reso-\nnance field, respectively. The dotted lines in Fig. 2 show\nthe two contributions. When comparing the data for Pt\nand Ta some differences are observed. First of all, the\nabsolute voltage values are smaller for the Pt cases and,\nmoreimportant, therelativeweightofbothcontributions\nis different. While the first point is related to the differ-\nent conductivity of Ta and Pt, the second one is related\nto the intrinsic effect causing the voltage. We calculate\nthe ratio S/A = Vsym/Vantisymfor all the measurements\nand the results are shown in Fig. 3(a) as a function of the\nNM thickness. While the antisymmetric contribution is\ndominating in the Pt samples with a S/A ratio smaller\nthan 1 for the samples with Pt, the opposite is true for\nthe Ta case. Since the ISHE signal is contributing only\ntoVsymit might be concluded that spin pumping is tak-\ning place stronger in the Ta system. However, since also\nthe spin rectificationeffect has a symmetric contribution,\nthis conclusion cannot be supported. Furthermore, since\nthe spin Hall angle θSHEhas opposite sign in these two\nmaterials, also the ISHE signal should have it. In appar-ent contradiction to this, we observe that both symmet-\nric contributions have the same sign in (a) and (b). This\npoints to the fact that for Pt, Vsymis dominated by the\nspin rectification effect, which does not change sign and\novercompensates a smaller ISHE signal. All these con-\nsiderations have the consequence that it is not possible\nto extract complete information of the origin of the mea-\nsured voltage by analyzing single spectra. For the same\nreason, the large increase in S/A for Ta for d= 5 nm or\nthe change in sign for Pt with the same thickness cannot\nbe correctly explained until the pure ISHE signal is not\nseparated from the spin rectification effect. As already\npointed out in recent papers[3–5, 11, 19], an analysis of\nthe angular dependence (in-plane or out-of-plane) of the\nmeasured voltages can be used to separate the different\ncontributions.\nIn any case, before proceeding it has to be proven that\nallthe measurementswereperformed in the linearregime\nwith small cone angles for the magnetization precession.\nThe measurements performed out of this regime would\nhave a large impact on the linewidth and a Gilbert-like\ndampingwouldnotbeguaranteed. Figure3(b) showsthe\ndependence of the voltage amplitude on the microwave\nnominal power proving indeed that the measurements\nwere carried on in the linear regime.\nFIG. 3. (Color online) (a) Dependence of the ratio S/A\n=Vsym/Vantisymon the thickness of the NM layer. (b)\nDependence of the total voltage on the applied microwave\npower proving the measurements were carried out in the lin-\near regime.4\nSEPARATION OF THE ISHE SIGNAL FROM\nTHE SPIN RECTIFICATION VOLTAGE\nWe performed in-plane angular dependent measure-\nments of the voltage and Eq. 4 was used to extract\nVsym,antisymfor each value of the azimuthal angle φ\nspanned between the direction of the magnetic field and\nthe microstrip antenna used to excite the magnetization.\nWe used a model based on the work of Harder et al.[3] to\nfit the dependence. This model considers two sources for\nthe spin rectification, which are the Anisotropic Mag-\nnetoresistance (AMR) and the Anomalous Hall Effect\n(AHE):\nVsym=Vspcos3(φ)+\n+VAHEcos(Φ)cos( φ)+Vsym\nAMR−⊥cos(2φ)cos(φ)\n+Vsym\nAMR−/bardblsin(2φ)cos(φ)\nVantisym=VAHEsin(Φ)cos( φ) +Vantisym\nAMR−⊥cos(2φ)cos(φ)\n+Vantisym\nAMR−/bardblsin(2φ)cos(φ)\n(5)\nHere,VspandVAHEare the contributions from spin\npumping (pure ISHE) and from AHE, respectively. Φ\nis the phase between the rf electric and magnetic fields\nin the medium. The contribution from the AMR is di-\nvided in one generating a transverse ⊥(with respect to\nthe antenna) or longitudinal /bardblvoltage. In an ideal case\nwith perfect geometry and point-like electrical contacts\nVsym,antisym\nAMR−/bardblshould be close to zero.\nFigure 4 shows the angular dependence of Vsym(top)\nandVantisym(bottom) for the samples with NM thick-\nness of 3 nm. The lines are a fit to the model which\nis able to describe the dependence properly. From the\ndata it can be clearly concluded that while the values\nofVantisymare comparable, with the difference resulting\nfrom the different resistivity of Pt and Ta, the values\nofVsymare much larger for Ta. The values obtained\nfrom the fits for the different contributions are plotted\nin Fig. 5 as a function of the thickness of the NM layer.\nThe value of Φ is ruling the lineshape of the electrically\nmeasured FMR peak[20] which is always a combination\nof a dispersive ( D, antisymmetric) and a Lorentzian ( L,\nsymmetric) contribution in the form D+iL. In order\nto compare the relative magnitudes of the different con-\ntributions independently of Φ we compute the quantities\nVAMR−/bardbl,⊥=/radicalbigg/parenleftBig\nVantisym\nAMR−/bardbl,⊥/parenrightBig2\n+/parenleftBig\nVsym\nAMR−/bardbl,⊥/parenrightBig2\nwhich it\nis equivalent to√\nD2+L2and we show them together\nwithVAHEandVsp. This step is important to allow for\ncomparison of the different contributions independent of\nthe value of Φ.\nSeveral conclusions can be extracted from Fig. 5. First\nof all, the spin rectification effect in CoFeB systems is al-\nmost fully dominated by the AHE. AMR plays a veryFIG. 4. (Color online) Angular dependence of Vsym(top)\nandVantisym(bottom) for CoFeB/Pt,Ta samples with NM\nthickness of 3 nm. The lines are a fit to the model described\nin Eq. 5.\nminor role. This is a difference with respect to NiFe\nor Fe. [4, 11, 20] This is correlated with the very large\nAHE reported in CoFeB films. [21, 22] Second, the volt-\nages generated by the spin pumping via the ISHE are\nlarger in the case of Ta and of opposite sign as expected\nfrom the different sign of θSHEin both materials. This\nsolves the apparent contradiction observed by the posi-\ntive symmetric contributions in both materials as shown\nin Fig. 2(a) and (b) and confirms the interpretation than\ninthecaseofPtthesymmetriccontributionisdominated\nby the spin rectification effect with opposite sign to the\nISHE signal. Again, this shows that the interpretation\nusing single spectra may lead to confusion and that angle\ndependent measurements are required.\nThe evolution of the spin rectification voltages with\nNM thickness shows a saturation behavior in both cases\nfor small thicknesses and a decrease with the NM layer\nthickness compatible with a dominant role of the re-\nsistance of the CoFeB layer. This is expected from\nthe resistivity values for amorphous CoFeB layers, 300-\n600µm·cm,[23] which are much larger than for β-Ta\n(6-10µm·cm) or sputtered Pt (100-200 µm·cm).[24, 25]\nHowever, the dependence does not completely agree with\nthe expected behavior[19] 1 /dNMpointing out to addi-\ntional effects like a variation of the conductivity of Pt for\nthe thinner layers.\nConcerning the correlation of the absolute values of\nthe ISHE-generated voltages and the spin Hall angles in\nboth materials, unfortunately the scatter in θSHEvalues\nin the literature is very large.[26] Howeverthis is reduced\nif we consider works were θSHEwas measured simultane-\nously for Pt and Ta in similar samples. In YIG/Pt,Ta5\nsystems[27, 28] it was determined that |θPt\nSHE|>|θTa\nSHE|\nwith a relative difference of around 30% which it is at\nodds with our results. On the contrary, in CoFeB/Pt,Ta\nbilayers|θTa\nSHE|= 0.15>|θPt\nSHE|= 0.07 is reported.[13]\nHowever the difference is not large enough to cover com-\npletely the difference in our samples. In order to ex-\nplain this point together with the absolute low value in\nCoFeB/Pt we have to take into account the possibility of\na certain loss of spin current at the interface FM/Pt or at\nthe very first nanometer, the latter due to the presence\nof a static magnetic polarization due to the proximity ef-\nfect. With this the spin current effectively being injected\nin Pt would be lower than in the Ta case.\nThe data does not allow for a quantitative estimation\nof the spin diffusion length λsd, but in any case the evo-\nlution is only compatible with a value for Pt not thicker\nthan 1 nm, similarto reportedvalues forsputtered Pt[25]\nand a a value of a few nm for Ta, also compatible with\nliterature.[28]\nAn important point is the lack of correlation of g↑↓\neff\nand the expected generated spin current using Eq. 1 with\nthe absolute measured ISHE voltage that results from\nthe spin-to-charge current conversion, obtained after the\nseparationfromthe overimposedspin rectificationsignal.\nThis is true even if we substract the MPE contribution\nassumed for Eq. 3. The same non-mutually excluding\nexplanations are possible here: ∆ αin Pt in mainly due\nto the MPE, or the spin current pumped into Pt van-\nishes at or close to the interface. The first alternative\nwould render Eq. 2 unuseful since most of the increase\nin damping is not due to spin pumping as long as the\nMPE is present. The second would reduce the validity\nof Eq. 1 to estimate the current injected in Pt and con-\nverted into a charge current by the ISHE. In any case,\nCoFeB/Ta shows very interesting properties, with strong\nspin pumping accompanied by only a minor impact on\nα.\nLet us discuss the limitations of the model defined in\nEq. 5 and the suitability to describe the measurements.\nFirst of all, the model assumes a perfect isotropic mate-\nrial. The anisotropy in CoFeB is known to be small but\nnot zero and a weak uniaxial anisotropy is present. The\neffect onthe angulardependenceisnegligible. Themodel\nassumes also a perfect geometry and point-like electrical\ncontacts to measure the voltages. Our contacts are ex-\ntended (∼200µm) and a small misalignment is possible\n(angle between the antenna and the imaginary line con-\nnecting the electrical contacts may not be exactly 90◦).\nThis is the most probable reason for the non-vanishing\nsmall value for Vsym,antisym\nAMR−/bardbl. Nevertheless, the angular\ndependence of the measured voltage is well described by\nthe model and no large deviations are observed.\nFIG. 5. (Color online) NM thickness dependence of the dif-\nferent contributions to the measured voltages extracted fr om\nthe angular dependence of VsymandVantisymfor Ta (top) and\nPt (bottom).\nCONCLUSIONS\nIn summary, we made use of in-plane angular de-\npendent measurements to separate ISHE-generated from\nspin rectification voltages and we compare the absolute\nvalues and thickness dependence for Pt and Ta. Differ-\nently to other materials, the spin rectification signal in\nCoFeB is almost fully dominated by AHE. No correlation\nbetween the observed spin mixing conductance via FMR\nmeasurement and the spin pumping signal is obtained\npointing to a dominant role of the magnetic proximity\neffect in the increase in damping with Pt.\nACKNOWLEDGEMENTS\nFinancial support by M-era.Net through the\nHEUMEM project and by the Carl Zeiss Stiftung\nis gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and\nB. I. Halperin, Rev. Mod. Phys. 77, No. 4, 1375 (2005).\n[2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[3] M. Harder, Y. Gui, and C.-M. Hu, arXiv:1605.00710v1\n[cond-mat.mtrl-sci] (2016).6\n[4] W. T. Soh, B. Peng, and C. K. Ong, J. Phys. D:\nAppl. Phys. 47, 285001 (2014).\n[5] Y. Gui, L. Bai, and C. Hu, Sci. China-Phys. Mech. As-\ntron.56, 124 (2013).\n[6] W. Zhang, , B. Peng, F. Han, Q. Wang, W. T. Soh,\nC. K. Ong, and W. Zhang, Appl. Phys. Lett. 108, 102405\n(2016).\n[7] A. Conca, S. Keller, L. Mihalceanu, T. 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BenYoussef, Phys.Rev.B. 87, 174417 (2013)."    },    {        "title": "1709.04911v2.Intrinsic_Damping_Phenomena_from_Quantum_to_Classical_Magnets_An_ab_initio_Study_of_Gilbert_Damping_in_Pt_Co_Bilayer.pdf",        "content": "Intrinsic Damping Phenomena from Quantum to Classical Magnets:\nAn ab-initio Study of Gilbert Damping in Pt/Co Bilayer\nFarzad Mahfouzi,1,\u0003Jinwoong Kim,1, 2and Nicholas Kioussis1,y\n1Department of Physics and Astronomy, California State University, Northridge, CA, USA\n2Department of Physics and Astronomy, Rutgers University, NJ, USA\nA fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented\nin the framework of the Keldysh Green function approach using ab initio electronic structure cal-\nculations. In contrast to previous calculations of classical Gilbert damping ( \u000bGD), we demonstrate\nthat\u000bGDin the quantum case does not diverge in the ballistic regime due to the \fnite size of\nthe total spin, S. In the limit of S!1 we show that the formalism recovers the torque correla-\ntion expression for \u000bGDwhich we decompose into spin-pumping and spin-orbital torque correlation\ncontributions. The formalism is generalized to take into account a self consistently determined de-\nphasing mechanism which preserves the conservation laws and allows the investigation of the e\u000bect\nof disorder. The dependence of \u000bGDon Pt thickness and disorder strength is calculated and the\nspin di\u000busion length of Pt and spin mixing conductance of the bilayer are determined and compared\nwith experiments.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nMagnetic materials provide an intellectually rich arena\nfor fundamental scienti\fc discovery and for the invention\nof faster, smaller and more energy-e\u000ecient technologies.\nThe intimate relationship of charge transport and mag-\nnetic structure in metallic systems on one hand, and the\nrich physics occurring at the interface between di\u000berent\nmaterials in layered structures on the other hand, are the\nhallmark of the \rourishing research \feld of spintronics.1{5\nRecently, intense focus has been placed on the signi\f-\ncant role played by spin-orbit coupling (SOC) and the ef-\nfect of interfacial inversion symmetry breaking on the dy-\nnamics of the magnetization in ferromagnet (FM)-normal\nmetal (NM) bilayer systems. Of prime importance to this\n\feld is the (precessional) magnetization damping phe-\nnomena, usually treated phenomenologically by means\nof a parameter referred to as Gilbert damping constant,\n\u000bGD, in the LandauLifshitzGilbert (LLG) equation of\nmotiond~ m=dt =\r~ m\u0002~B+\u000bGD~ m\u0002d~ m=dt , which de-\nscribes the rate of the angular momentum loss of the\nFM.6Here,~ mis the unit vector along the magnetization\ndirection and ~Bis an e\u000bective magnetic \feld.\nIn FM/NM bilayer devices the e\u000bect of the NM on the\nGilbert damping of the FM is typically considered as an\nadditive e\u000bect, where the total Gilbert damping can be\nseparated into an intrinsic bulk contribution and an inter-\nfacial component due to the presence of the NM.7,8While\nthe interfacial Gilbert damping is usually attributed to\nthe loss of angular momentum due to pumped spin cur-\nrent into the NM,9,10in metallic bulk FMs the intrin-\nsic Gilbert damping constant is described by the cou-\npling between the conduction electrons and the (time-\ndependent) magnetization degree of freedom.11\nThe conventional approach to determine the Gilbert\ndamping constant involves calculating the imaginary part\nof the time-dependent susceptibility of the FM in thepresence of conduction electrons in the linear response\nregime.12{14In this case, the time-dependent magneti-\nzation term in the electronic Hamiltonian leads to the\nexcitation of electrons close to the Fermi surface trans-\nferring angular momentum to the conduction electrons.\nThe excited electrons in turn relax to the ground state\nby interacting with their environment, namely through\nphonons, photons and/or collective spin/charge excita-\ntions. These interactions are typically parameterized\nphenomenologically by the broadening of the energy lev-\nels,\u0011=~=2\u001c, where\u001cis the relaxation time of the elec-\ntrons close to the Fermi surface. The phenomenological\ntreatment of the electronic relaxation is valid when the\nenergy broadening is small which corresponds to clean\nsystems, i.e., \u0011D(EF)\u00141, whereD(EF) is the den-\nsity of states per atom at the Fermi energy. In the case\nof large\u0011[\u0011D(EF)&1)] however, this approach vio-\nlates the conservation laws and a more accurate descrip-\ntion of the relaxation mechanism that preserves the en-\nergy, charge and angular momentum conservation laws\nare required.15The importance of including the vertex\ncorrections has already been pointed out in the literature\nwhen the Gilbert damping is dominated by the interband\ncontribution,16{18i.e.,when there is a signi\fcant number\nof states available within the energy window of \u0011around\nthe Fermi energy.\nIn this paper we investigate the magnetic damping phe-\nnomena through a di\u000berent Lens in which the FM is as-\nsumed to be small and quantum mechanical. We show\nthat in the limit of large magnetic moments we recover\ndi\u000berent conventional expressions for the Gilbert damp-\ning of a classical FM. We calculate the Gilbert damping\nfor a Pt/Co bilayer system versus the energy broaden-\ning,\u0011and show that in the limit of clean systems and\nsmall magnetic moments the FM damping is governed\nby a coherent dynamics. We show that in the limit of\nlarge broadening \u0011 > 1meV which is typically the case\nat room temperature, the relaxation time approximationarXiv:1709.04911v2  [cond-mat.mes-hall]  14 Nov 20172\nfails. Hence, we employ a self consistent approach pre-\nserving the conservation laws. We calculate the Gilbert\ndamping versus the Pt and Co thicknesses and by \ftting\nthe results to spin di\u000busion model we calculate the spin\ndi\u000busion length and spin mixing conductance of Pt.\nII. THEORETICAL FORMALISM OF\nMAGNETIZATION DAMPING\nFor a metallic FM the magnetization degree of freedom\nis inherently coupled to the electronic degrees of freedom\nof the conduction electrons. It is usually convenient to\ntreat each degree of freedom separately with the corre-\nsponding time-dependent Hamiltonians that do not con-\nserve the energy. However, since the total energy of the\nsystem is conserved, it is possible to consider the total\nHamiltonian of the combined system and solve the corre-\nsponding stationary equations of motion. For an isolated\nmetallic FM the wave function of the coupled electron-\nmagnetic moment con\fguration system is of the form,\njm\u000b~ki=jS;mi\nj\u000b~ki, where the parameter Sdenotes\nthe total spin of the nano-FM ( S! 1 in the classi-\ncal limit),m=\u0000S:::; +S, are the eigenvalues of the\ntotal Szof the nano-FM,\nrefers to the Kronecker prod-\nuct, and\u000bdenotes the atomic orbitals and spin of the\nelectron Bloch states. The single-quasi-particle retarded\nGreen function and the corresponding density matrix can\nbe obtained from,19\n\u0012\nE\u0000i\u0011\u0000^H~k\u0000HM\u00001\n2S^\u0001~k^~ \u001b\u0001~S\u0013\n^Gr\n~k(E) =^1;(1)\nand\n^\u001a~k=ZdE\n\u0019^Gr\n~k(E)\u0011f(E\u0000HM)^Ga\n~k(E): (2)\nHere,HM=\r~B\u0001~S, is the Hamiltonian of the nano-\nFM in the presence of an external magnetic \feld ~Bwith\neigenstates,jS;mi,\ris the gyromagnetic ratio, f(E) is\nthe Fermi-Dirac distribution function, ^~ \u001bis the vector of\nthe Pauli matrices, ^H~kis the non-spin-polarized Hamilto-\nnian matrix in the presence of spin orbit coupling (SOC),\nand^\u0001~kis the~k-dependent exchange splitting matrix, dis-\ncussed in detail in Sec. III. We employ the notation that\nbold symbols operate on jS;mibasis set and symbols\nwith hat operate on the j\u000b~kis. Here, for simplicity we\nignore explicitly writing the identity matrices ^1 and 1as\nwell as the Kronecker product symbol in the expressions.\nA schematic description of the FM-Bloch electron en-\ntangled system and the damping process of the nano-FM\nis shown in Fig. 1. The presence of the magnetic Hamil-\ntonian in the Fermi distribution function in Eq. (2) act-\ning as a chemical potential leads to transition between\nmagnetic statesjS;mialong the direction in which the\nmagnetic energy is minimized19. The transition rate of\nthe FM from the excited states, jS;mi, to states with\nFIG. 1: (Color online) Schematic representation of the com-\nbined FM-Bloch electron system. The horizontal planes de-\nnote the eigenstates, jS;miof the total Szof the nano-FM\nwith eigenvalues m=\u0000S;\u0000S+ 1;:::; +S. For more details\nsee Fig. 2 in Ref.19\n.\nlower energy ( i.e.the damping rate) can be calculated\nfrom19,\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (3)\nwhere,\nT\u0006\nm=1\n2SNX\n~kTrel[^\u0001~k^\u001b\u0007S\u0006\nm^\u001a~k;m;m\u00061]: (4)\nHere,Nis the number of ~k-points in the \frst Brillouin\nzone,Trel, is the trace over the Bloch electron degrees\nof freedom,S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061), and ^\u001b\u0007\u0011\n^\u001bx\u0007i^\u001by.\nThe precessional Gilbert damping constant can be de-\ntermined from conservation of the total angular mo-\nmentum by equating the change of angular momen-\ntum per unit cell for the Bloch electrons, Tm, and\nthe magnetic moment obtained from LLG equation,\n\u000bGDMtotsin2(\u0012)=2, which leads to,\n\u000bGD(m) =\u00002\nMtot!sin2(\u0012m)Tm\n\u0011\u0000S2\nMtot!(S(S+ 1)\u0000m2)Tm: (5)\nHere, cos(\u0012m) =mp\nS(S+1), is the cone angle of precession\nandMtotis the total magnetic moment per unit cell in\nunits of1\n2g\u0016Bwithgand\u0016Bbeing the Land\u0013 e factor and\nmagneton Bohr respectively. The Larmor frequency, !,\ncan be obtained from the e\u000bective magnetic \feld along\nthe precession axis, ~!=\rBz.\nThe exact treatment of the magnetic degree of freedom\nwithin the single domain dynamical regime o\u000bers a more\naccurate description of the damping phenomena that can\nbe used even when the classical equation of motion LLG\nis not applicable. However, since in most cases of in-\nterest the FM behaves as a classical magnetic moment,\nwhere the adiabatic approximation can be employed to\ndescribe the magnetization dynamics, in the following\ntwo sections we consider the S!1 limit and close to\nadiabatic regime for the FM dynamics.3\nA. Classical Regime: Relaxation Time\nApproximation\nThe dissipative component of the nonequilibrium elec-\ntronic density matrix, to lowest order in @=@t, can be\ndetermined by expanding the Fermi-Dirac distribution\nin Eq. (2) to lowest order in [ HM]mm0=\u000emm0m~!.\nPerforming a Fourier transformation with respect to the\ndiscrete Larmor frequency modes, m!\u0011i@=@t , we \fnd\nthat, ^\u001adis\nneq(t) =1\n\u0019~\u0011^Gri@^Ga=@t, where ^Gr=\u0002\nEF\u0000i\u0011\u0000\n^H(t)\u0003\u00001and ^Ga= (^Gr)yare the retarded and advanced\nGreen functions calculated at the Fermi energy, EF, and\na \fxed time t.\nThe energy absorption rate of the electrons can\nbe determined from the expectation value of the\ntime derivative of the electronic Hamiltonian, E0\ne=\n<(Tr(^\u001adis\nneq(t)@^H=@t )), where<() refers to the real part.\nCalculating the time-derivative of the Green function and\nusing the identity, \u0011^Gr^Ga=\u0011^Ga^Gr==(^Gr), where,=()\nrefers to the anti-Hermitian part of the matrix, the torque\ncorrelation (TC) expression for the energy excitation rate\nof the electrons is of the form,\nE0\ne=~\n\u0019NX\nkTrh\n=(^Gr)@^H\n@t=(^Gr)@^H\n@ti\n: (6)\nIn the case of semi-in\fnite NM leads attached to the FM,\nusing,=(^Gr) =^Gr^\u0000^Ga=^Ga^\u0000^Gr, Eq.(6) can be written\nas\nE0\ne=~\n\u0019NX\nkTrh\n^\u0000@^Gr\n@t^\u0000@^Ga\n@ti\n(7)\nwhere, ^\u0000 =\u0011^1 + ( ^\u0006r\u0000^\u0006a)=2i, with ^\u0006r=abeing the\nretarded=advanced self energy due to the NM lead at-\ntached to the FM which describes the escape rate of\nelectrons from/to the reservoir. It is useful to separate\nthe dissipation phenomena into local andnonlocal compo-\nnents as follows. Applying the unitary operator, ^U(t) =\nei!^\u001bzt=2ei\u0012^\u001bx=2e\u0000i!^\u001bzt=2= cos(\u0012\n2)^1 +isin(\u0012\n2)(^\u001b+ei!t+\n^\u001b\u0000e\u0000i!t), to \fx the magnetization orientation along z\nwe \fnd,\n@(^U^Gr\n0^Uy)\n@t\u0019!\n2sin(\u0012)\u0010\n^G0ei!t+^G0ye\u0000i!t\u0011\n;(8)\nwhere we have ignored higher order terms in \u0012and,\n^G0= [^Gr\n0;^\u001b+]\u0000^Gr\n0[^H0;^\u001b+]^Gr\n0: (9)\nHere, [;] refers to the commutation relation, ^H0is the\ntime independent terms of the Hamiltonian, and ^Gr=a\n0\nrefers to the Green function corresponding to magnetiza-\ntion alongz-axis. Using Eq. (7) for the average energyabsorption rate we obtain,\nE0\ne=~!2\n2\u0019Nsin2(\u0012)X\nkTr\u0010\n^\u0000^G0^\u0000^G0y\u0011\n=\u0000~!2\n2\u0019Nsin2(\u0012)X\nk<\u0010\nTr\u0010\n^\u0000[^Gr\n0;^\u001b+]^\u0000[^Ga\n0;^\u001b\u0000]\n+=(^Gr)[^H0;^\u001b+]=(^Gr)[^H0;^\u001b\u0000]\n\u00002 [=(^Gr\n0);^\u001b+]^\u0000^Ga\n0[^H0;^\u001b\u0000]\u0011\u0011\n: (10)\nIn the absence of the SOC, the \frst term in Eq.\n(10) is the only non-vanishing term which corresponds\nto the pumped spin current into the reservoir [i.e.\nISz=~Tr(^\u001bz^\u0000^\u001adis\nneq)=2] dissipated in the NM (no back\n\row). This spin pumping component is conventionally\nformulated in terms of the spin mixing conductance20,\nISz=~g\"#sin2(\u0012)=4\u0019, which acts as a nonlocal dissi-\npation mechanism. The second term, referred to as the\nspin-orbital torque correlation11,21(SOTC) expression for\ndamping, is commonly used to calculate the intrinsic con-\ntribution to the Gilbert damping constant for bulk metal-\nlic FMs. The third term arises when both SOC and the\nreservoir are present. It is important to note that the\nformalism presented above is valid only in the limit of\nsmall\u0011(ballistic regime). On the other hand, in the case\nof large\u0011, typical in experiments at room temperature,\nthe results may not be reliable due to the fact that in\nthe absence of metallic leads a \fnite \u0011acts as a \fctitious\nreservoir that yields a nonzero dissipation of spin cur-\nrent even in the absence of SOC. A simple approach to\nrectify the problem is to ignore the e\u000bect of \fnite \u0011in\nthe spin pumping term in calculating the Gilbert damp-\ning constant. A more accurate approach is to employ\na dephasing mechanism that preserves the conservation\nlaws, which we refer it to as conserving torque correlation\napproach discussed in the following subsection.\nB. Classical Regime: Conserving Dephasing\nMechanism\nRather than using the broadening parameter, \u0011, as a\nphenomenological parameter, we determine the self en-\nergy of the Bloch electrons interacting with a dephas-\ning bath associated with phonons, disorder, etc. using a\nself-consistent Green function approach22. Assuming a\nmomentum-relaxing self energy given by,\n^\u0006r=a\nint(E;t) =1\nNX\nk^\u0015k^Gr=a\nk(E;t)^\u0015y\nk; (11)\nwhere ^\u0015kis the interaction coupling matrix, the dressed\nGreen function, ^Gr=a\nk(E;t) , and corresponding self en-\nergy, ^\u0006r=a\nint(E;t), are calculated self-consistently. This\nwill in turn yield a renormalized broadening matrix,\n^\u0000int==(^\u0006r\nint), which is the vertex correction modi\f-\ncation of the in\fnitesimal initial broadening \u00110.4\nThe nonequilibrium density matrix is calculated from\n^\u001adis\nneq(k;t) =~\n\u0019^Gr\nk^\u0000int^Ga\nk\u0010@^Hk(t)\n@t+^Saa\nt\u0011\n^Ga\nk;(12)\nwhere the time derivative vertex correction term is\n^Saa\nt=1\nNX\nk^\u0015k^Ga\nk\u0010@^Hk(t)\n@t+^Saa\nt\u0011\n^Ga\nk^\u0015y\nk: (13)\nThe energy excitation rate for the Bloch electrons then\nreads,\nE0\ne=~\n\u0019NX\nk<h\nTr\u0010\u0010@^Hk(t)\n@t+^Sar\nt\u0011\n^\u001adis\nneq(k;t)\u0011i\n;(14)\nwhere\n^Sar\nt=1\nNX\nk^\u0015k^Ga\nk\u0010@^Hk(t)\n@t+^Sar\nt\u0011\n^Gr\nk^\u0015y\nk: (15)\nThe vertex correction Eqs. (13) and (15) can be solved\neither exactly by transforming them into a system of lin-\near equations or by solving them self consistently. Due to\nthe large number of orbitals and atoms per unit cell for\nthe Co/Pt bilayer the latter approach is computationally\nmore e\u000ecient. In the following numerical calculations we\nassume ^\u0015k=\u0015int^1 to be a constant independent of kand\nof orbitals, which can be viewed as the root mean square\nvalue of a random on-site potential, \u0015int=p\nhV2\nrandi,\nwhereh:::idenotes an ensemble averaging, where the self\nenergy in Eq. (11) corresponds to the self consistent Born\napproximation.\nC. Gilbert Damping Calculation\nHaving determined the energy absorption rate of the\nBloch electrons due to the precessing FM, from conserva-\ntion of energy one can deduce the energy dissipation rate\nof the FM from, E0\nM=\u0000E0\ne. Using the LLG equation\nof motion the energy dissipation rate per unit cell of the\nprecessing FM can be obtained from\nE0\nM=1\n2Mtot~!@mz\n@t=\u00001\n2\u000bGDMtot~!2sin2(\u0012);(16)\nwhere~ mis the unit vector along the magnetization of\nthe FM. The Gilbert damping parameter can then be\nobtained from\n\u000bGD=2E0\ne\nMtot~!2sin2(\u0012): (17)\nIII. COMPUTATIONAL SCHEME\nThe spin-polarized density functional theory calcula-\ntions for the hcp Co(0001)/fcc Pt(111) bilayer were car-\nried out using the Vienna ab initio simulation package(VASP)23,24. The pseudopotential and wave functions\nare treated within the projector-augmented wave (PAW)\nmethod25,26. Structural relaxations were carried using\nthe generalized gradient approximation as parameter-\nized by Perdew et al.27when the largest atomic force\nis smaller than 0.01 eV/ \u0017A. The plane wave cuto\u000b energy\nis 500 eV and a 14 \u000214\u00021kpoint mesh is used in the\n2D BZ sampling. The Pt( m)/Co(n) bilayer is modeled\nemploying the slab supercell approach along the [111]\nconsisting of mfcc Pt monolayers (MLs) (ABC stacking)\n(m=1, 2,:::, 6),nhcp Co MLs (AB stacking) Co ( n= 6),\nand a 25 \u0017A thick vacuum region separating the periodic\nslabs. The in-plane lattice constant of the hexagonal unit\ncell was set to the experimental value of 2.505 \u0017A for bulk\nCo.\nThe Gilbert damping constant was calculated us-\ning the tight-binding parameters obtained from VASP-\nWannier90 calculations28with a 250\u0002250k-mesh for\nthe bilayer and 250 \u0002250\u0002250k-mesh for bulk Co. The\nelectron Hamiltonian, ^H~k, and exchange splitting, ^\u0001~k,\nmatrices in Eq. (1) in the Wannier basis have the form\n^H~k=^HSOC+1\n2X\n~ n1\nD~ n(^H\"\"\n~ n+^H##\n~ n)e2i\u0019~ n\u0001~k(18)\n^\u0001~k=1\n2X\n~ n1\nD~ n(^H\"\"\n~ n\u0000^H##\n~ n)e2i\u0019~ n\u0001~k; (19)\nwhere ^HSOC is the SOC Hamiltonian matrix, ^H\"\"\n~ nand\n^H##\n~ nare the spin-majority and spin-minority matrices,\n~ n= (n1;n2;n3) are integers denoting the lattice vectors,\nD~ nis the degeneracy of the Wigner-Seitz grid point, and\nki2[0;1].\nThe ^H\"\"\n~ nand ^H##\n~ nare determined from spin-polarized\nVASP-Wannier90 calculations without SOC. On the\nother hand, ^HSOC, is determined from VASP-Wannier90\nnon-spin-polarized calculations with SOC as the follow-\ning.\nUsing the identity, Tr[^Li^Lj] =\u000eijl(l+1)(2l+1)\n3, where\n^Liis the angular momentum operator of orbital land\ni;j=x;y;z , the SOC strength of the Ith atom can be\ncalculated from\n\u0018I\nl=3Tr[^HP\nll;II^Li^\u001bi]\nl(l+ 1)(2l+ 1): (20)\nHere, the superscript Pdenotes the paramagnetic Hamil-\ntonian,IIare the on-site Hamiltonian matrix elements\nfor atomIat~ n= (0;0;0), andllis the block Hamiltonian\nmatrix corresponding to orbital l. The result is indepen-\ndent of the direction of the angular momentum operator.\nWe \fnd that \u0018Pt\nd=0.5 eV and \u0018Co\nd=70 meV for the d-\norbitals of Pt and Co, respectively, that are somewhat\nsmaller than the values considered in the literature7(i.e.\n\u0018Pt\nd=0.65 eV,\u0018Co\nd=85 meV). The SOC Hamiltonian can\nin turn be written as,\nhI;lmsj^HSOCjI0;l0m0s0i=1\n2\u000ell0\u000eII0\u0018I\nlX\nihlmj^Lijlm0i^\u001bi\nss0:\n(21)5\n0 45 90 135 1800.0280.02850.0290.02950.030.03050.0310.03150.032\nCone Angle (deg)Gilbert Damping\nStudent Version of MATLAB\nFIG. 2: (Color online). Gilbert damping versus precessional\ncone angle calculated from Eq. (5) for Pt(1 ML)/Co(6 ML)\nbilayer system for S=60 and\u0011=1 meV, respectively.\nIV. RESULTS AND DISCUSSION\nThe Gilbert damping constant (calculated from\nEq. (5))versus the precessional cone angle, \u0012m=\ncos\u00001\u0010\nmp\nS(S+1)\u0011\n, for the Pt(1 ML)/Co(6 ML) bilayer\nis shown in Fig. 2 with \u0011=1 meV and S=60. We \fnd\nthat the Gilbert damping is relatively independent of the\ncone angle. The small angular dependence of the Gilbert\ndamping is material dependent and it could increase or\ndecrease upon increasing the cone angle, depending on\nthe material.\nIn order to see the transition from quantum mechanical\nto classical dynamical regimes, in Fig. 3 we present the\nGilbert damping constant of the Pt(1 ML)/Co(6 ML)\nbilayer versus the broadening parameter, \u0011, for di\u000ber-\nent values of the total spin Sof the FM. For the case\nof \fniteSwe used Eq. (5) while for S=1we used\nthe TC expression Eq. (6). We \fnd that for \fnite S\nthe Gilbert damping value exhibits a peak in the small \u0011\nregime (clean system) where the peak value increases lin-\nearly withSand shifts to smaller broadening value with\nincreasingS. The underlying origin of the \u000bGD(\u0011) behav-\nior withSin coherent regime can be understood in terms\nof the coherent transport of quasi-particles along the aux-\niliary direction min Fig. 1, where the auxiliary current\n\row (damping rate) depends linearly on the chemical po-\ntential di\u000berence between the \frst ( m= +S) and last\nlayers (m=\u0000S) which is simply 2 S. This suggests that\nin the limit of in\fnite Sand ballistic regime \u0011!0 the\nintrinsic Gilbert damping diverges. It was shown that\nthe problem of in\fnite Gilbert damping in the ballistic\nregime can be removed by taking into account the collec-\ntive excitations.29,30\nAs we discussed in Sec. II A, the relaxation time ap-\nproximation is valid only in the small \u0011limit and it vi-\nolates the conservation law when \u0011is large. In order\n10−410−210010−310−210−1100\nBroadening, η, (eV)Gilbert Damping\n  \nS=10S=30S=60S=∞\nStudent Version of MATLABFIG. 3: (Color online) Gilbert damping versus broadening\nparameter for the Pt(1 ML)/Co(6 ML) bilayer for di\u000berent\nvalues of the total spin Sof the FM nanocluster. Eqs. (5)\nand (6) were used to calculate the Gilbert damping for \fnite\nand in\fnite S, respectively.\nto quantify the validity of the relaxation time approxi-\nmation, in Fig. 4 we display the Gilbert damping ver-\nsus broadening, \u0011;or interaction parameter, \u0015int, for\nthe Pt(1 ML)/Co(6 ML) bilayer with S=1, using\n(i)torque correlation(TC) expression (Eq. (6)); (ii)the\nspin-orbital torque correlation (SOTC) expression (sec-\nond term in Eq. (10);) and (iii)the conserving TC expres-\nsion (Eq. (14)). The upper horizontal-axis refers to the\ninteraction strength \u0015intof the conserving TC method\nand the lower one refers to the broadening parameter,\n\u0011. The calculations show that for \u0011 >20 meV the TC\nresults deviate substantially from those of the conserv-\ning TC method. Ignoring the spin pumping contribu-\ntion to the Gilbert damping in Eq. (10) and considering\nonly the SOTC component increases the range of the va-\nlidity of the relaxation time approximation. Therefore,\nthe overestimation of the Gilbert damping using the TC\nmethod can be attributed to the disappearance of elec-\ntrons (pumped spin current) in the presence of the \fnite\nnon-Hermitian term, i\u0011^1, in the Hamiltonian.\nWe have used the conserving TC approach to calculate\nthe e\u000bect of \u0015inton the Gilbert damping as a function of\nthe Pt layer thickness for the Pt( m)/Co(6 ML) bilayer.\nAs an example, we display in Fig. 5 the results of Gilbert\ndamping versus Pt thickness for \u0015int= 1eVwhich yields\na Gilbert damping value of 0.005 for bulk Co ( m= 0 ML)\nand is in the range of 0.00531,32to 0.01133{35reported\nexperimentally. Note that this large \u0015intvalue describes\nthe Gilbert damping in the resistivity-like regime which\nmight not be appropriate to experiment, where the bulk\nGilbert damping decreases with temperature, suggesting\nthat it is in the conductivity regime.36\nFor a given \u0015intwe \ftted the ab initio calculated\nGilbert damping versus Pt thickness to the spin di\u000bu-6\n10−410−210010−310−210−1100\nBroadening, η, (eV)Gilbert Damping10−1100Interaction Strength, λint, (eV)\nConserving TC MethodSOTC MethodTC Method\nStudent Version of MATLAB\nFIG. 4: (Color online). Gilbert damping of Pt(1 ML)/Co(6\nML) bilayer versus the broadening parameter \u0011(lower ab-\nscissa) and interaction strength, \u0015int, (upper abscissa), using\nthe torque correlation (TC), spin-orbital torque correlation\n(SOTC), and conserving TC expressions given by Eqs. (6),\n(10) and (14), respectively.\nsion model,37{39\n\u000bPt=Co =\u000bCo+ge\u000b\n\"#VCo\n2\u0019MCodCo(1\u0000e\u00002dPt=Lsf\nPt):(22)\nHere,ge\u000b\n\"#is the e\u000bective spin mixing conductance, dCo\n(dPt) is the thickness of Co (Pt), VCo= 10:5\u0017A3\n(MCo= 1:6\u0016B) is the volume (magnetic moment) per\natom in bulk Co, and Lsf\nPtis the spin di\u000busion length\nof Pt. The inset of Fig. 5 shows the variation of the ef-\nfective spin mixing conductance and spin di\u000busion length\nwith the interaction strength \u0015int. In the di\u000busive regime\n\u0015int>0:2eV,Lsf\nPtranges between 1 to 6 nm in agree-\nment with experiment \fndings which are between 0.5 and\n10 nm33,40. Moreover, the e\u000bective spin mixing conduc-\ntance is relatively independent of \u0015intoscillating around\n20 nm\u00002, which is approximately half of the experimen-\ntal value of\u001935 - 40 nm\u00002.33,41On the other hand,\nin the ballistic regime ( \u0015int<0.2 eV), although the er-\nrorbar in \ftting to the di\u000busion model is relatively large,\nthe value of Lsf\nPt\u00190.5 nm is in agreement with Ref.7and\nexperimental observation40.\nV. CONCLUDING REMARKS\nWe have developed an ab initio -based electronic struc-\nture framework to study the magnetization dynamics ofa nano-FM where its magnetization is treated quantum\nmechanically. The formalism was applied to investigate\nthe intrinsic Gilbert damping of a Co/Pt bilayer as a\n0 1 2 300.0050.010.0150.02\nPt Thickness, dPt (nm)Gilbert Damping\n  \n10−210−110002468\nInteraction Strength, λint (eV)Spin Diffusion Length (nm)100101102103\ng↑↓eff (nm−2)\nStudent Version of MATLAB\nFIG. 5: (Color online). Ab initio values (circles) of Gilbert\ndamping versus Pt thickness for Pt( mML)/Co(6 ML) bilayer\nwheremranges between 0 and 6 and \u0015int= 1eV. The dashed\ncurve is the \ft of the Gilbert damping values to Eq. (22).\nInset: spin di\u000busion length (left ordinate) and e\u000bective spin\nmixing conductance, ge\u000b\n\"#, (right ordinate) versus interaction\nstrength. The errorbar for ge\u000b\n\"#is equal to the root mean\nsquare deviation of the damping data from the \ftted curve.\nfunction of energy broadening. We showed that in the\nlimit of small Sand ballistic regime the FM damping is\ngoverned by coherent dynamics, where the Gilbert damp-\ning is proportional to S. In order to study the e\u000bect of\ndisorder on the Gilbert damping we used a relaxation\nscheme within the self-consistent Born approximation.\nTheab initio calculated Gilbert damping as a function of\nPt thickness were \ftted to the spin di\u000busion model for a\nwide range of disorder strength. In the limit of large dis-\norder strength the calculated spin di\u000busion length and\ne\u000bective spin mixing conductance are in relative agree-\nment with experimental observations.\nAcknowledgments\nThe work is supported by NSF ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\nGrant No. 1160504 and by NSF-Partnership in Research\nand Education in Materials (PREM) Grant No. DMR-\n1205734.7\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.Kioussis@csun.edu\n1Ioan Mihai Miron, Gilles Gaudin, Stphane Au\u000bret,\nBernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan Vo-\ngel and Pietro Gambardella, Current-driven spin torque in-\nduced by the Rashba e\u000bect in a ferromagnetic metal layer,\nNat. Mater. 9, 230234 (2010).\n2Ioan Mihai Miron, Kevin Garello, Gilles Gaudin, Pierre-\nJean Zermatten, Marius V. 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Hillebrands1\n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universität Kaiserslautern, D-67663 Kaisersl autern,\nGermany\n2)Graduate School Materials Science in Mainz, Gottlieb-Daim ler-Strasse 47,\nD-67663 Kaiserslautern, Germany\n3)Innovent e.V., Prüssingstraße 27B, 07745 Jena, Germany\n(Dated: 7 May 2018)\nWe present an experimental study of spin-wave excitation an d propagation in\nmicrostructured waveguides patterned from a 100 nm thick yt trium iron garnet\n(YIG)/platinum (Pt) bilayer. The life time of the spin waves is found to be more than\nan order of magnitude higher than in comparably sized metall ic structures despite\nthe fact that the Pt capping enhances the Gilbert damping. Ut ilizing microfocus\nBrillouin light scattering spectroscopy, we reveal the spi n-wave mode structure for\ndifferent excitation frequencies. An exponential spin-wav e amplitude decay length of\n31µm is observed which is a significant step towards low damping , insulator based\nmicro-magnonics.\n1The concept of magnon spintronics, i.e., the transport and m anipulation of pure spin\ncurrents in the form of spin-wave quanta, called magnons, ha s attracted growing interest\nin the recent years1–12. One of the key advantages of magnon spin currents is their la rge\ndecay length which can be several orders of magnitude higher than the spin diffusion length\nin conventional spintronic devices based on spin-polarize d electron currents. Considering\npossible applications, the miniaturization of magnonic ci rcuits is of paramount importance.\nUp to now, downscaling has been achieved using metallic ferr omagnets like NiFe or Heusler\ncompounds2–7. But even the best metallic ferromagnets exhibit a damping w hich is two or-\nders of magnitude larger than for Yttrium Iron Garnet (YIG), a ferrimagnetic insulator1,13,14.\nHowever, to the best of our knowledge, as high quality YIG film s could only be grown with\nthicknesses in the range of microns, no microstructured YIG devices have been fabricated so\nfar. A big step forward has been taken with the recent introdu ction of methods to produce\nhigh quality, low damping YIG films with thicknesses down to s everal nanometers9,15–17.\nIn this Letter, we show that microscaled waveguides (see Fig . 1) can be fabricated from\nliquid phase epitaxy (LPE) grown YIG films of 100nm thickness whose high quality has\nbeen confirmed by ferromagnetic resonance spectroscopy (FM R). Studying the excitation\nand propagation of spin-waves in these waveguides by microf ocus Brillouin light scattering,\nwe demonstrate that the damping of the unstructured film can b e preserved during the\nstructuring process.\nAnother key feature of magnon spintronics is its close relat ionship to a multitude of phys-\nical phenomena like spin-pumping, spin-transfer torque, s pin Seebeck effect, and (inverse)\nspin Hall effect, which allow for the amplification, generati on and transformation between\ncharge currents and magnonic currents7–12,15–24. Hetero-structures of YIG covered with a\nthin layer of platinum (Pt) have proven to show these effects w hich opens a way to a new class\nof insulator based spintronics. Therefore, we directly stu dy bilayers of YIG/Pt, providing a\nbasis for further studies utilizing the described effects.\nThe used YIG film is prepared by liquid phase epitaxy from a PbO -B2O3-FeO3flux\nmelt using a standard isothermal dipping technique with a gr owth rate of 20nm/min. The\nincorporation of Pb and Pt ions into the garnet lattice allow s for a low relative lattice\nmismatch of 3·10−4.\nWe determine the magnetic properties of the film using FMR and compare the results\nto measurements performed after the deposition of a 9nmPt film onto YIG using plasma\n2FIG. 1. Sample schematic: In a 5µm wide waveguide patterned from a bilayer of YIG/Pt\n(100nm /9nm), spin waves are excited using the dynamic Oersted fields of a microwave current\nflowing in a copper antenna. An external bias field Hextis applied along the short axis of the\nwaveguide. The spin-wave intensity is detected using micro focus Brillouin light scattering spec-\ntroscopy.\ncleaning and RF sputtering. From the resonance curve HFMR(fFMR), a saturation magne-\ntization of Ms= 144±2kA/mhas been determined for the pure YIG film. We find that\nthe deposition of Pt slightly reduces the resonance field µ0HFMR(for example by 1mT for\nfFMR= 7.0GHz ) compared to the pure YIG film. This shift agrees with the rece nt findings\nof Ref. 15, where a proximity induced ferromagnetic orderin g of Pt combined with a static\nexchange coupling to YIG has been proposed as possible expla nation.\nFigure 2 shows the ferromagnetic resonance linewidth (FWHM )µ0∆Hwith and with-\nout Pt and the corresponding fits to evaluate the effective Gil bert damping parameter α\naccording to22\nµ0∆H=µ0∆H0+2αfFMR\nγ(1)\nwith the gyromagnetic ratio γ= 28GHz /T. The Gilbert damping αincreases by almost a\nfactor of 5 due to the deposition of Pt: from (2.8±0.3)×10−4to(13.0±1.0)×10−4. The\ninhomogeneous linewidth µ0∆H0is unchanged within the accuracy of the fit ( 0.16±0.02mT\nand0.14±0.04mT , respectively). Please note that the increase of the dampin g cannot be\nexplained exclusively by spin pumping from YIG into Pt. Othe r interface effects, like the\nalready mentioned induced ferromagnetic ordering of Pt in c ombination with a dynamic\nexchange coupling may play a role15,23. Using the spin mixing conductance for YIG/Pt\n(g↑↓≈1.2×1018m−2from22,24), we find that the expected increase in Gilbert damping due\nto spin pumping11,20–22isαsp= 1.25×10−4,i.e., it is by a factor of 8 smaller than the\n3/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s89/s73/s71/s61/s50/s46/s56/s32/s120/s49/s48/s45/s52/s32/s89/s73/s71/s47/s80/s116/s32/s98/s105/s108/s97/s121/s101/s114\n/s32/s112/s117/s114/s101/s32/s89/s73/s71/s181\n/s48/s32 /s72 /s32/s40/s109/s84/s41\n/s32/s102\n/s70/s77/s82/s40/s71/s72/s122/s41/s89/s73/s71/s47/s80/s116/s61/s49/s51/s46/s48/s32/s120/s49/s48/s45/s52\nFIG. 2. Linewidth µ0∆Has a function of the ferromagnetic resonance frequency fFMRfor the\npure YIG film (blue circles) and the YIG/Pt bilayer (red squar es). The deviations from the linear\nincrease of µ0∆HwithfFMR(fit according to Eqn. 1) are mainly due to parasitic modes cau sing a\nsmall systematical error in the measurement of the linewidt h.\nmeasured increase. This clearly demonstrates the importan ce of additional effects15,23. As\nshown recently in Ref.15, this additional damping can be strongly reduced by the intr oduction\nof a thin copper (Cu) layer in between YIG and Pt, which does no t significant influence the\nspin-pumping efficiency.\nThe micro structuring of the YIG/Pt waveguide is achieved us ing a negative protective\nresist mask pattered by electron beam lithography and physi cal argon ion beam etching. As\nlast production step, a microwave antenna (width 3.5µm,510nm thickness) made of copper\nis deposited on top of the waveguide (see Fig. 1).\nTo experimentally detect the spin waves in the microstructu red waveguide, we employ\nmicrofocus Brillouin light scattering spectroscopy (BLS)3–8. This method allows us to study\nthe spin-wave intensity as a function of magnetic field and sp in-wave frequency. In addition,\nit provides a spatial resolution of 250nm , which is not available in experiments using spin\npumping and inverse spin Hall effect8–10as these methods integrate over the detection area\n(and also over the complete spin-wave spectrum12).\nTo achieve an efficient spin-wave excitation, we apply a stati c magnetic field of 70mT\nperpendicular to the long axis of the waveguide. The dynamic Oersted field of a microwave\ncurrent passing through the antenna exerts a torque on the st atic magnetization. This config-\nuration results in an efficient excitation of Damon-Eshbach l ike spin waves which propagate\n4/s51/s46/s52 /s51/s46/s53 /s51/s46/s54 /s51/s46/s55 /s51/s46/s56/s48/s46/s49/s49/s32/s99/s101/s110/s116/s101/s114/s32/s114/s101/s103/s105/s111/s110\n/s32/s101/s100/s103/s101/s32/s114/s101/s103/s105/s111/s110/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s66/s76/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s102\n/s77/s87/s32/s40/s71/s72/s122/s41\n/s49 /s49 /s32/s181 /s109\n/s97/s110/s116/s101/s110/s110/s97/s89/s73/s71/s47/s80/s116 \n/s72 \n/s101/s120/s116 \nFIG. 3. Normalized BLS intensity (log scale) as a function of t he applied microwave frequency fMW\n(external field µ0Hext= 70mT ). The blue line (circular dots) shows the spectrum measured at\nthe edges of the waveguide (see inset). The red line (rectang ular dots) is an average of the spectra\nrecorded in the center of the waveguide.\nperpendicular to the static magnetization. A microwave pow er of0dBm (pulsed, duration\n3µs, repetition 5µs) in the quasi-linear regime, where nonlinearities are not s ignificantly\ninfluencing the spin-wave propagation, has been chosen. To o btain a first characterization\nof the excitation spectrum, BLS spectra as a function of the a pplied microwave frequency\n(fMW) have been taken at different positions across the width of th e waveguide at a distance\nof11µm from the antenna. Figure 3 shows the spectrum of the edge re gions (blue circles)\nand of the center of the waveguide (red squares, see sketch in the inset). The main excitation\nin the center of the waveguide takes place at frequencies bet weenfMW= 3.49−3.66GHz\nand we will refer to these spin-wave modes as the waveguide modes . At the borders of the\nwaveguide, edge modes have their resonance around fMW= 3.44GHz . The reason for the\nappearance of these edge modes is the pronounced reduction o f the effective magnetic field\nHeffat the edges by the demagnetization field and the accompanyin g inhomogeneity of the\nz-component of the static magnetization. This situation has been analyzed experimentally\nand theoretically in detail for metallic systems25,26.\nTo get a better understanding of the nature of the involved sp in-wave modes, mode profiles\nat different excitation frequency measured at a distance of 6µm from the antenna are shown\nin Fig. 4. The evolution of the modes can be seen clearly: for f requencies below fMW=\n3.45GHz , the spin-wave intensity is completely confined to the edges of the waveguide. In\n5/s48 /s49 /s50 /s51 /s52 /s53/s51/s46/s52/s48/s51/s46/s52/s53/s51/s46/s53/s48/s51/s46/s53/s53/s51/s46/s54/s48/s51/s46/s54/s53\n/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s122 /s32/s97/s108/s111/s110/s103/s32/s119/s105/s100/s116/s104/s32/s111/s102/s32/s119/s97/s118/s101/s103/s117/s105/s100/s101/s32/s40/s181/s109/s41/s102\n/s77/s87/s40/s71/s72/s122/s41\n/s48/s48/s46/s51/s48/s46/s53/s48/s46/s56/s49/s32\n/s66/s76/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\nFIG. 4. BLS intensity (linear scale) as a function of the posit ion along the width of the waveguide for\ndifferent excitation frequencies fMW(µHext= 70mT ). Frequencies below 3.45GHz show strongly\nlocalized edge modes which start to extend into the center of the waveguide for frequencies between\n3.45−3.50GHz . For higher fMW, waveguide modes appear which have their local intensity ma xima\nin the center of the waveguide. The dashed lines indicate the calculated minimal frequencies of the\nwaveguide modes shown in Fig. 5.\nthe range fMW= 3.45−3.50GHz , the maximum of the intensity is also located near the\nedges, but two additional local maxima closer to the center o f the waveguide appear. For\nfrequencies in the range of 3.50−3.57GHz , three spin-wave intensity maxima symmetrically\ncentered around the center of the waveguide are observed. Th is mode is commonly labeled\nas the third waveguide mode n= 3(ndenotes the number of maxima across the width of\nthe waveguide). For higher fMW, only one intensity maximum is found in the center of the\nwaveguide (first waveguide mode, n= 1).\nFor the waveguide modes, we can compare the experimental res ults to theoretical consid-\nerations. The theory for spin waves in thin films27with the appropriate effective field from\nmicromagnetic simulations and a wave-vector quantization over the waveguide’s short axis\nprovides an accurate description of the spin-wave mode disp ersions4–6. Figure 5 shows the\ndispersion relations and the excitation efficiencies of the w aveguide modes n= 1,3,5. Only\nodd waveguide modes can be efficiently excited4,5(even modes have no net dynamic mag-\nnetic moment averaged over the width of the waveguide) using direct antenna excitation.\nThe minimal frequencies of these three modes are indicated a s dashed lines for comparison\nin Fig. 4. Comparing Fig. 4 and Fig. 5, we find a reasonable agre ement between theory and\nexperiment for the first and the third waveguide mode. The n= 5and higher waveguide\n6/s51/s46/s52/s51/s46/s53/s51/s46/s54/s51/s46/s55/s51/s46/s56/s51/s46/s57\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s49/s49/s102/s32/s40/s71/s72/s122/s41/s32/s49\n/s32/s51\n/s32/s53/s119/s97/s118/s101/s103/s117/s105/s100/s101/s32/s109/s111/s100/s101/s69/s120/s99/s46/s32/s101/s102/s102/s46/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s41\n/s32/s87/s97/s118/s101/s32/s118/s101/s99/s116/s111/s114 /s32/s107\n/s120/s32/s40/s114/s97/s100/s47/s181/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s32\nFIG. 5. (color online) Dispersion relations and amplitude e xcitation efficiencies for the first three\nodd waveguide modes of a transversally magnetized YIG waveg uide and an antenna width of 3.5µm\n(external field µHext= 70mT , width of waveguide 5µm, further parameters see28).\nmodes are not visible in the experiment. Due to the fact that t he excitation efficiency and\nthe group velocity of the spin-wave modes decreases with inc reasingn(see Ref. 4 and 5 for\ndetails), this can be attributed to a small amplitude of thes e modes.\nTo visualize the influence of the spatial decay of the spin wav es on the mode composition,\nFig. 6 (a) shows 2D spin-wave intensity maps for two exemplar y excitation frequencies. For\nfMW= 3.45GHz , edge modes can be detected for distances larger than 20µm. Different\nhigher order waveguide modes are also excited, but they can o nly be detected within 5µm\nfrom the antenna. From this findings, we can conclude that the edge modes are dominating\nthe propagation in this frequency range because of their hig h group velocities (proportional\nto the decay length) compared to the available waveguide mod es (n≥5).\nThe situation is completely different for fMW= 3.60GHz . Here, the preferably excited\nn= 1waveguide mode is interfering with the weaker n= 3waveguide mode causing a\nperiodic beating effect3–5of the measured spin-wave intensity. In this frequency rang e, no\nsignificant contribution of modes confined to the edges is vis ible.\nAn important parameter for magnonic circuits and applicati ons is the exponential de-\ncay length δampof the spin-wave amplitude. To determine δexper\nampforfMW= 3.60GHz , we\nintegrate the spin-wave intensity over the width of the wave guide (Fig. 6 (b)) and obtain\nδexper\namp= 31µm which is substantially larger than the reported decay len gths in metallic mi-\ncrostructures made of Permalloy or Heusler compounds3,4. This value can be compared\n7FIG. 6. (a) BLS intensity maps (linear scale) for two different excitation frequencies ( µ0Hext=\n70mT ). (b) Integrated BLS intensity (logarithmic scale) over the width of the waveguide for fMW=\n3.60GHz including a fit to determine the exponential amplitude decay length (δamp= 31µm).\nto the expected theoretical value δtheo\namp=vgτwherevgis the group velocity and τis\nthe life time of the spin wave. The Gilbert damping of the unpa tterned YIG/Pt bilayer\nα= 1.3·10−3measured by FMR corresponds to a life time τ≈28ns for our experi-\nmental parameters. The group velocity vgcan be deduced from the dispersion relations\nin Fig. 5 or from dynamic micromagnetic simulations yieldin gvg≈1.0−1.1µm/ns, thus\nδtheo\namp= 28−31µm. The agreement with our experimental findings δexper\namp= 31µm is excel-\nlent, especially if one considers that the plain film values o fαandMs, which might have\nbeen changed during the patterning process, have been used f or the calculation. This indi-\ncates that possible changes of the material properties due t o the patterning have only an\nnegligible influence on the decay length of the waveguide mod es and that the damping of the\nspin waves due to the Pt capping is well described by the measu red increase of the Gilbert\ndamping.\nTo conclude, we presented the fabrication of micro-magnoni c waveguides based on high\nquality YIG thin films. Spin-wave excitation and propagatio n of different modes in a mi-\ncrostructured YIG/Pt waveguide was demonstrated. As expec ted, the enhancement of the\nGilbert damping due to the Pt deposition leads to a reduced li fe time of the spin waves com-\npared to the pure YIG case. However, the life time of the spin w aves in the YIG/Pt bilayer is\nstill more than an order of magnitude larger than in the usual ly used microstructured metallic\n8systems. This leads to a high decay length reaching δexper\namp= 31µm for the waveguide modes.\nOne can estimate that the achievable decay length for a simil ar microstructured YIG/Pt\nwaveguide is δamp= 100 µm if a Cu interlayer is introduced to suppress the damping eff ects\nwhich are not related to spin pumping15(α=αYIG+αsp). Going further, from YIG thin\nfilms having the same damping than high quality, micron thick LPE films ( α≈4×10−5,\nµ0∆H≈0.03mT , Ref. 13 and 14), the macroscopic decay length of δAmp= 1mm for\nmicro-magnonic waveguides of pure YIG might be achieved.\nOur studies show that downscaling of YIG preserving its high quality is possible. Thus,\nthe multitude of physical phenomena reported for macroscop ic YIG can be transferred to\nmicrostructures which is the initial step to insulator base d, microscaled spintronic circuits.\nREFERENCES\n1A.A. Serga, A.V. Chumak, and B. Hillebrands, J. Phys. D: Appl . Phys. 43, 264002 (2010).\n2B. Lenk, H. Ulrichs, F. Garbs, and M. 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Porter, Interagency Report NISTIR 6376, National Institute of\nStandards and Technology, Gaithersburg, MD (Sept 1999).\n11"    },    {        "title": "1907.01045v1.Magnon_decay_theory_of_Gilbert_damping_in_metallic_antiferromagnets.pdf",        "content": "Magnon decay theory of Gilbert damping in metallic antiferromagnets\nHaakon T. Simensen, Akashdeep Kamra, Roberto E. Troncoso, and Arne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: July 3, 2019)\nGilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a\ntheoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals through\ns-dexchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets\nowing to their bipartite nature, in addition to providing analytic expressions for the damping parameters. Magnon-\ninduced intraband electron scattering is found to predominantly cause magnetization damping, whereas the Néel\nfield is found to be damped via disorder. Depending on the conduction electron band structure, we predict that\nmagnon-induced interband electron scattering around band crossings may be exploited to engineer a strong Néel\nfield damping.\nIntroduction.— The dynamical properties of a harmonic\nmode are captured by its frequency and lifetime [ 1,2]. While\nthe eigenfrequency is typically determined by the linearized\nequations of motion, or equivalently by a non-interacting de-\nscription of the corresponding quantum excitation, the lifetime\nembodies rich physics stemming from its interaction with one\nor more dissipative baths [ 1,3]. Dissipation plays a central\nrole in the system response time. In the context of magnetic\nsystems employed as memories, the switching times decrease\nwith increasing damping thereby requiring a stronger dissi-\npation for fast operation [ 4–6]. The dissipative properties of\nthe system also result in rich phenomena such as quantum\nphase transitions [ 7–10]. Furthermore, the formation of hybrid\nexcitations, such as magnon-polarons [ 11–18] and magnon-\npolaritons [ 19–24], requires the dissipation to be weak with\nrespect to the coupling strengths between the two participating\nexcitations [ 25]. Therefore, in several physical phenomena that\nhave emerged into focus in the recent years [ 12,16,26–30],\ndamping not only determines the system response but also the\nvery nature of the eigenmodes themselves. Understanding,\nexploiting and controlling the damping in magnets is thus a\nfoundational pillar of the field.\nThe success of Landau-Lifshitz-Gilbert (LLG) phenomenol-\nogy [ 31,32] in describing ferromagnetic dynamics has inspired\nvigorous e \u000borts towards obtaining the Gilbert damping param-\neter using a wide range of microscopic theories. The quantum\nparticles corresponding to magnetization dynamics - magnons\n- provide one such avenue for microscopic theories and form\nthe central theme in the field of magnonics [ 33,34]. While\na vast amount of fruitful research has provided a good under-\nstanding of ferromagnets (FMs) [ 35–54], analogous studies on\nantiferromagnets (AFMs) are relatively scarce and have just\nstarted appearing [ 55,56] due to the recently invigorated field\nof antiferromagnetic spintronics [ 57–62]. Among the ongoing\ndiscoveries of niches borne by AFMs, from electrically and\nrapidly switchable memories [ 63], topological spintronics [ 60],\nlong range magnonic transport [ 64] to quantum fluctuations\n[65], an unexpected surprise has been encountered in the first\nprinciples evaluation of damping in metallic AFMs. Liu and\ncoworkers [ 56] and another more recent first-principles study\n[66] both found the magnetization dissipation parameter to bemuch larger than the corresponding Néel damping constant,\nin stark contrast with previous assumptions, exhibiting richer\nfeatures than in FMs. An understanding of this qualitative\ndi\u000berence as well as the general AFM dissipation is crucial\nfor the rapidly growing applications and fundamental novel\nphenomena based on AFMs.\nHere, we accomplish an intuitive and general understanding\nof the Gilbert damping in metallic AFMs based on the magnon\npicture of AFM dynamics. Employing the s-d, two-sublattice\nmodel for a metallic AFM, in which the dandselectrons\nconstitute the magnetic and conduction subsystems, we derive\nanalytic expressions for the Gilbert damping parameters as\na function of the conduction electron density of states at the\nFermi energy and s-dexchange strength. The presence of spin-\ndegenerate conduction bands in AFMs is found to be the key\nin their qualitatively di \u000berent damping properties as compared\nto FMs. This allows for absorption of AFM magnons via s-\ndexchange-mediated intraband conduction electron spin-flip\nprocesses leading to strong damping of the magnetization as\ncompared to the Néel field [ 67]. We also show that interband\nspin-flip processes, which are forbidden in our simple AFM\nmodel but possible in AFMs with band crossings in the conduc-\ntion electron dispersion, result in a strong Néel field damping.\nThus, the general qualitative features of damping in metallic\nAFMs demonstrated herein allow us to understand the Gilbert\ndamping given the conduction electron band structure. These\ninsights provide guidance for engineering AFMs with desired\ndamping properties, which depend on the exact role of the\nAFM in a device.\nModel.— We consider two-sublattice metallic AFMs within\nthes-dmodel [ 35,36,44]. The delectrons localized at lat-\ntices sites constitute the magnetic subsystem responsible for\nantiferromagnetism, while the itinerant selectrons form the\nconduction subsystem that accounts for the metallic traits. The\ntwo subsystems interact via s-dexchange [Eq. (3)]. For ease of\ndepiction and enabling an understanding of qualitative trends,\nwe here consider a one-dimensional AFM (Fig. 1). The re-\nsults within this simple model are generalized to AFMs with\nany dimensionality in a straightforward manner. Furthermore,\nwe primarily focus on the uniform magnetization dynamics\nmodes.arXiv:1907.01045v1  [cond-mat.mes-hall]  1 Jul 20192\nFIG. 1: Schematic depiction of our model for a metallic AFM.\nThe red and blue arrows represent the localized delectrons\nwith spin up and down, respectively, thereby constituting the\nNéel ordered magnetic subsystem. The green cloud illustrates\nthe delocalized, itinerant selectrons that forms the conduction\nsubsystem.\nAt each lattice site i, there is a localized delectron with spin\nSi. The ensuing magnetic subsystem is antiferromagnetically\nordered (Fig. 1), and the quantized excitations are magnons\n[68,69]. Disregarding applied fields for simplicity and as-\nsuming an easy-axis anisotropy along the z-axis, the magnetic\nHamiltonian, Hm=˜JP\nhi;jiSi\u0001Sj\u0000KP\ni(Sz\ni)2, wherehi;ji\ndenotes summation over nearest neighbor lattice sites, is quan-\ntized and mapped to the sublattice-magnon basis [69]\nHm=X\nqh\nAq\u0010\nay\nqaq+by\nqbq\u0011\n+By\nqay\nqby\nq+Bqaqbqi\n; (1)\nwhere we substitute ~=1,Aq=(2˜J+2K)SandBq=\n˜JS e\u0000iq\u0001aP\nh\u000eieiq\u0001\u000e, where S=jSij,ais the displacement\nbetween the two atoms in the basis, and h\u000eidenotes sum-\nming over nearest neighbor displacement vectors. aqandbq\nare bosonic annihilation operators for plane wave magnons\non the A and B sublattices, respectively. We diagonalize\nthe Hamiltonian [Eq. 1] through a Bogoliubov transforma-\ntion [ 69] toHm=P\nq!q\u0010\n\u000by\nq\u000bq+\fy\nq\fq\u0011\n;with eigenenergies\n!q=q\nA2q\u0000jBqj2. In the absence of an applied field, the\nmagnon modes are degenerate.\nTheselectron conduction subsystem is described by a tight-\nbinding Hamiltonian that includes the “static” contribution\nfrom the s-dexchange interaction [Eq. (3)] discussed below:\nHe=\u0000tX\nhi;jiX\n\u001bcy\ni\u001bcj\u001b\u0000JX\ni(\u00001)i\u0010\ncy\ni\"ci\"\u0000cy\ni#ci#\u0011\n:(2)\nHere ci\u001bis the annihilation operator for an selectron at site\niwith spin\u001b.t(>0)is the hopping parameter, and J(>0)\naccounts for s-dexchange interaction [Eq. (3)]. The (\u00001)i\nfactor in the exchange term reflects the two-sublattice nature of\nthe AFM. The conduction subsystem unit cell consists of two\nbasis atoms, similar to the magnetic subsystem. As a result,\nthere are four distinct electron bands: two due to there being\ntwo basis atoms per unit cell, and twice this due to the two\npossible spin polarizations per electron. Disregarding applied\nfields, these bands constitute two spin-degenerate bands. We\nlabel these bands 1 and 2, where the latter is higher in energy.The itinerant electron Hamiltonian [Eq. (2)] is diagonalized\ninto an eigenbasis (c1k\u001b;c2k\u001b)with eigenenergies \u000f1k=\u0000\u000fk\nand\u000f2k= +\u000fk, where\u000fk=p\nJ2S2+t2j\rkj2, where\rk=P\nh\u000eie\u0000ik\u0001\u000e. The itinerant electron dispersion is depicted in Fig.\n2.\nThe magnetic and conduction subsystems interact through\ns-dexchange interaction, parametrized by J:\nHI=\u0000JX\niSi\u0001si; (3)\nwhere si=P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is the spin of the itinerant elec-\ntrons at site i, where \u001bis the vector of Pauli matrices. The term\nwhich is zeroth order in the magnon operators, and thus ac-\ncounts for the static magnetic texture, is already included in He\n[Eq. (2)]. To first order in magnon operators, the interaction\nHamiltonian can be compactly written as\nHe\u0000m=X\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n+h.c.;(4)\nwhere\u0015and\u001aare summed over the electron band indices. As\ndetailed in the Supplemental material, WA;\u0015\u001a\nkk0qandWB;\u0015\u001a\nkk0q, both\nlinear in J, are coe \u000ecients determining the amplitudes for\nscattering between the itinerant electrons and the aqandbq\nmagnons, respectively. Specifically, when considering plane\nwave states, WA=B;\u0015\u001a\nkk0qbecomes a delta function, thereby enforc-\ning the conservation of crystal momentum in a translationally\ninvariant lattice. Inclusion of disorder or other many-body\ne\u000bects results in deviation of the eigenstates from ideal plane\nwaves causing a wave vector spread around its mean value [ 2].\nThe delta function, associated with an exact crystal momentum\nconservation, is thus transformed to a peaked function with\nfinite width (\u0001k). The\u0015\u001acombinations 11and22describe\nintraband electron scattering, while 12and21describe in-\nterband scattering. Intraband scattering is illustrated in Fig.\n2. Interband scattering is prohibited within our model due to\nenergy conservation, since the uniform q=0magnon energy\nis much smaller than the band gap.\nThe scattering described by He\u0000m[Eq. (4)] transfers spin\nangular momentum between the magnetic and conduction sub-\nsystems. The itinerant electrons are assumed to maintain a\nthermal distribution thereby acting as a perfect spin sink. This\nis consistent with a strong conduction electron spin relaxation\nobserved in metallic AFMs [ 70,71]. As a result, the magnetic\nsubsystem spin is e \u000bectively damped through the s-dexchange\ninteraction.\nGilbert damping.— In the Landau-Lifshitz-Gilbert (LLG)\nphenomenology for two-sublattice AFMs, dissipation is ac-\ncounted via a 2\u00022 Gilbert damping matrix [ 72]. Our goal here\nis to determine the elements of this matrix in terms of the\nparameters and physical observables within our microscopic\nmodel. To this end, we evaluate the spin current “pumped”\nby the magnetic subsystem into the sconduction electrons,\nwhich dissipate it immediately within our model. The angu-\nlar momentum thus lost by the magnetic subsystem appears\nas Gilbert damping in its dynamical equations [ 72,73]. The3\n-\n/2 -\n /4 0\n /2\n /4\ne\ne\nkF,1a/epsilon1=µ1\nm\ne\ne\nm\n/epsilon1=µ2\nkF,2a\nFIG. 2: The selectron dispersion in metallic AFM model,\nwhere the red and blue dispersions depict electron bands 1 and\n2, respectively. Illustrations of intraband electron-magnon\nscattering at two di \u000berent Fermi levels, \u00161and\u00162, are added.\nThe depicted momentum transfer is exaggerated for clarity.\nsecond essential ingredient in identifying the Gilbert damping\nmatrix from our microscopic theory is the idea of coherent\nstates [ 74,75]. The classical LLG description of the magne-\ntization is necessarily equivalent to our quantum formalism,\nwhen the magnetic eigenmode is in a coherent state [ 74–76].\nDriving the magnetization dynamics via a microwave field,\nsuch as in the case of ferromagnetic resonance experiments,\nachieves such a coherent magnetization dynamics [73, 77].\nThe spin current pumped by a two-sublattice magnetic sys-\ntem into an electronic bath may be expressed as [78]\nIz=Gmm(m\u0002˙m)z+Gnn(n\u0002˙n)z\n+Gmn\u0002(m\u0002˙n)z+(n\u0002˙m)z\u0003;(5)\nwhere mand nare the magnetization and Néel field nor-\nmalized by the sublattice magnetization, respectively. Here,\nGi j=\u000bi j\u0002(M=j\rj), where\u000bi jare the Gilbert damping co-\ne\u000ecients,\ris the gyromagnetic ratio of the delectrons\nandMis the sublattice magnetization. Considering the uni-\nform magnetization mode, Izis the spin current operator\nIz=i[He\u0000m;Sz][79], where Sz=P\niSz\ni. We get\nIz=iX\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n\u0000h.c.:(6)\nThe expectation value of this operator assuming the uniform\nmagnetization mode to be in a coherent state corresponds to\nthe spin pumping current [Eq. (5)].\nIn order to evaluate the spin pumping current from Eq. (6),\nwe follow the method employed to calculate interfacial spin\npumping current into normal metals in Refs. [ 73,77,78], and\nthe procedure is described in detail therein. Briefly, this method\nentails assuming the magnetic and conduction subsystems to\nbe independent and in equilibrium at t=\u00001, when the mu-\ntual interaction [Eq. (4)] is turned on. The subsequent timeevolution of the coupled system allows evaluating its physical\nobservables in steady state. The resulting coherent spin-current\ncorresponds to the classical spin current Izthat can be related\nto the motion of the magnetization and the Néel field [Eq. (5)].\nAs a last step, we identify expressions for (m\u0002˙m)z,(m\u0002˙n)z\nand(n\u0002˙n)zin terms of coherent magnon states, which enables\nus to identify the Gilbert damping coe \u000ecients\u000bmm,\u000bnnand\n\u000bmn.\nResults.— Relegating the detailed evaluation to Supplemen-\ntal Material, we now present the analytic expression obtained\nfor the various coe \u000ecients [Eq. (5)]. A key assumption that\nallows these simple expressions is that the electronic density of\nstates in the conduction subsystem does not vary significantly\nover the magnon energy scale. Furthermore, we account for a\nweak disorder phenomenologically via a finite scattering length\nlassociated with the conduction electrons. This results in an\ne\u000bective broadening of the electron wavevectors determined by\nthe inverse electron scattering length, (\u0001k)=2\u0019=l. As a result,\nthe crystal momentum conservation in the system is enforced\nonly within the wavevector broadening. By weak disorder we\nmean that the electron scattering length is much larger than\nthe lattice parameter a. Ifkandk0are the wave vectors of the\nincoming and outgoing electrons, respectively, we then have\n(k\u0000k0)a=(\u0001k)a\u001c1. This justifies an expansion in the wave\nvector broadening (\u0001k)a. The Gilbert damping coe \u000ecients\nstemming from intraband electron scattering are found to be\n\u000bmm=\u000b0(\u0018J)\u0000\u000b0(\u0018J)\n40BBBBBBBB@1+\u00182\nJ\u0010\n\u00182\nJ+8\u00004 cos2(kFa)\u0011\n\u0010\n\u00182\nJ+4 cos2(kFa)\u001121CCCCCCCCA[(\u0001k)a]2;\n\u000bnn=\u000b0(\u0018J)\n40BBBBBB@1+sin2(kFa)\ncos2(kFa)\u00182\nJ\u0010\n\u00182\nJ+4 cos2(kFa)\u00111CCCCCCA[(\u0001k)a]2:\n(7)\nwhere\u0018J=JS=t,kFis the Fermi momentum and ais the lattice\nparameter, and where\n\u000b0(\u0018J)=\u0019v2J2\n8g2(\u0016)j˜Vj24 cos2(kFa)\n\u00182\nJ+4 cos2(kFa): (8)\nHere, vis the unit cell volume, g(\u000f)is the density of states\nper unit volume, \u0016is the Fermi level, and !0is the energy of\ntheq=0magnon mode. ˜Vis a dimensionless and generally\ncomplex function introduced to account for the momentum\nbroadening dependency of the scattering amplitudes. It satisfies\n˜V(0)=1and0\u0014j˜V(\u0001k)j\u00141within our model. These analytic\nexpressions for the Gilbert damping parameters constitute one\nof the main results of this letter.\nDiscussion.– We straightaway note that \u000bnn=\u000bmm\u0018\n[(\u0001k)a]2\u001c1.\u000bnnis strictly dependent upon (\u0001k)a, and is non-\nzero only if there is disorder and a finite electron momentum\nbroadening. \u000bmmis large even when considering a perfectly\nordered crystal. This latter result is in good accordance with\nrecent first-principles calculations in metallic AFMs [ 56,66].\nWe moreover observe that both \u000bmmand\u000bnnare quadratic\ninJandg(\u0016). This result is shared by Gilbert damping ow-\ning to spin-pumping in insulating ferrimagnet |normal metal4\ne\n e\nm\nkFa/epsilon1=µ\nFIG. 3: A schematic depiction of magnon-induced interband\nscattering in a band crossing at the Fermi level.\n(NM) and AFM |NM bilayers with interfacial exchange cou-\npling [ 78]. Metallic AFMs bear a close resemblance to these\nbilayer structures. There are however two main di \u000berences:\nThes-dexchange coupling exists in the bulk of metallic AFMs,\nwhereas it is localized at the interface in the bilayer structures.\nAdditionally, the itinerant electron wave functions are qual-\nitatively di \u000berent in metallic AFMs and NMs, owing to the\nmagnetic unit cell of the AFM. Indeed, these di \u000berences turn\nout to leave prominent signatures in the Gilbert damping in\nmetallic AFMs.\nThe uniform mode magnon energy is much smaller than the\nelectron band gap within our simple model. Interband scat-\ntering is thus prohibited by energy conservation. However,\nin real AFM metals, the electron band structure is more com-\nplex. There may for instance exist band crossings [ 80–82].\nIn such materials, magnon-induced interband electron scatter-\ning should also contribute to Gilbert damping, depending on\nthe position of the Fermi surface. Motivated by this, we now\nconsider Gilbert damping stemming from interband scattering,\nwhile disregarding the energy conservation for the moment,\nlabeling the coe \u000ecients\u000bI\nmmand\u000bI\nnn. We then find the same\nexpressions as in Eq. (7) with the roles of \u000bI\nmm;nninterchanged\nwith respect to \u000bmm;nn. This implies that \u000bI\nnnis large and inde-\npendent of electron momentum broadening, whereas \u000bI\nmmis\nproportional to the electron momentum broadening squared.\nAlthough arriving at this result required disregarding the en-\nergy conservation constraint, the qualitative e \u000bect in itself is\nnot an artifact of this assumption. In materials with a band\ncrossing, as depicted in Fig. 3, \u000bI\nnn=\u000bI\nmm> \u000b nn=\u000bmmis a gen-\neral result. This generic principle derived within our simple\nmodel provides valuable guidance for designing materials with\nan engineered Gilbert damping matrix.\nWe now provide a rough intuitive picture for the damping\ndependencies obtained above followed by a more mathemati-\ncal discussion. Consider a conventional di \u000braction experiment\nwhere an incident probing wave is able to resolve the two\nslits only when the wavelength is comparable to the physical\nseparation between the two slits. In the case at hand, the wave-\nfunctions of electrons and magnon participating in a scatteringprocess combine in a way that the wavenumber by which the\nconservation of crystal momentum is violated becomes the\nprobing wavenumber within a di \u000braction picture. Therefore,\nthe processes conserving crystal momentum have vanishing\nprobing wavenumber and are not able to resolve the opposite\nspins localized at adjacent lattice sites. Therefore, only the aver-\nage magnetization is damped leaving the Néel field una \u000bected.\nWith disorder, the probing wavenumber becomes non-zero and\nthus also couples to the Néel field. The interband scattering,\non the other hand, is reminiscent of Umklapp scattering in a\nsingle-sublattice model and the probing wavenumber matches\nwith the inverse lattice spacing. Therefore, the coupling with\nthe Néel field is strong.\nThe Gilbert damping in metallic AFMs here considered is\ncaused by spin pumping from the magnetic subsystem into\nthesband. This spin pumping induces electron transitions\nbetween spin\"/#states among the selectrons. The Gilbert\ndamping coe \u000ecients depend thus on transition amplitudes pro-\nportional to products of itinerant electron wave functions such\nas y\n\u0015k\"(x) \u001ak0#(x). The damping e \u000bect on sublattice A depends\non this transition amplitude evaluated on the A sublattice, and\nequivalently for the B sublattice. Assuming without loss of gen-\nerality that site i=0belongs to sublattice A, we find in the one-\ndimensional model that the damping on sublattice A is a func-\ntion ofP\njcos2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj), whereas the damping\non sublattice B is a function ofP\njsin2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj).\nEquivalently, by straightforwardly using that m=(mA+mB)=2\nandn=(mA\u0000mB)=2, this analysis predicts that \u000bmmis a\nfunction ofP\nj y\n\u0015k\"(xj) \u001ak0#(x), whereas\u000bnnis a function of\nP\njcos\u0010\u0019xj\na\u0011\n y\n\u0015k\"(xj) \u001ak0#(x). Assuming plane wave solutions\nof the electron wave functions, and if we consider intraband\nscattering only, we more concretely find that \u000bmmis a function\nof(1\u0000i(\u0001k)a), where iis the imaginary unit, whereas \u000bnnis a\nfunction of ( \u0001k)a. This coincides well with Eq. (7).\nAbove, we presented a discussion of interband scattering in\nthe minimal model where the band gap artificially was set to\nzero. In this limit, the upper electron band is a continuation\nof the lower band with a \u0006\u0019=amomentum shift. We may then\nwrite 2k\u001b= 1;k+\u0019=a;\u001b. Under the assumption of a disappear-\ning band gap, momentum-conserving interband scattering at\nmomentum kis therefore equivalent to intraband scattering be-\ntween kandk\u0006\u0019=a. This is the exact phase shift which results\nin a small\u000bmmand a large \u000bnnconsistent with the discussion\nabove. In real metallic AFMs with complex band structures,\nthe exact wave function relations unveiled above do not apply.\nHowever, interband transition amplitudes will undoubtedly\ncarry a position dependent phase. This position dependence\nresults in a dephasing of transition amplitudes at neighboring\nlattice sites, which gives rise to a non-negligible \u000bnn. The pre-\ncise damping coe \u000ecients in real metallic AFMs depend on the\ndetailed electron wave functions. We may however generally\nconclude that \u000bI\nnn=\u000bI\nmm>\u000b nn=\u000bmm.\nConclusion.— We have provided a microscopic derivation\nof Gilbert damping resulting from magnon decay through s-d\nexchange interaction in metallic antiferromagnets. Analytic5\nexpressions for Gilbert damping coe \u000ecients resulting from in-\ntraband electron scattering are presented, while Gilbert damp-\ning resulting from interband electron scattering is discussed on\na conceptual level. We find that intraband electron scattering\ngives rise to a large magnetization damping and a negligible\nNéel field damping. The intraband Néel field damping is pro-\nportional to the inverse electron scattering length squared, and\ndisappears exactly if there is no crystal disorder. 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Bodnar, L. Šmejkal, I. Turek, T. Jungwirth, O. Gomonay,\nJ. Sinova, A. A. Sapozhnik, H.-J. Elmers, M. Kläui, and M. Jour-\ndan, “Writing and reading antiferromagnetic Mn2Au by Néel\nspin-orbit torques and large anisotropic magnetoresistance,” Na-\nture Communications 9, 348 (2018).\n[82] D. F. Shao, G. Gurung, S. H. Zhang, and E. Y . Tsymbal, “Dirac\nNodal Line Metal for Topological Antiferromagnetic Spintron-\nics,” Physical Review Letters 122, 77203 (2019)."    },    {        "title": "2107.11699v1.Electron_Phonon_Scattering_governs_both_Ultrafast_and_Precessional_Magnetization_Dynamics_in_Co_Fe_Alloys.pdf",        "content": "1 \n Electron -Phonon Scattering governs both Ultrafast and Precessional Magnetization \nDynamics in Co -Fe Alloys  \nRamya Mohan1, Victor  H. Ortiz2, Luat Vuong2, Sinisa Coh2, Richard B. Wilson1,2* \n1Materials Science & Engineering Program, University of California, Riverside  \n2Department of Mechanical Engineering, University of California, Riverside  \n* Corresponding Author : rwilson@ucr.edu  \n \n \nAbstract  \nRecent investigations have advanced the understanding of how structure -property relationships \nin ferromagnetic metal alloys affect the magnetization dynamics on nanosecond time -scales . A \nsimilar understanding for magnetization dynamics on femto - to pico -second time -scales does not \nyet exist. To address this, we perform time -resolved magneto optic Kerr effect (TRMOKE) \nmeasurements of magnetization dynamics in Co -Fe alloys on femto - to nano-second  regimes . \nWe show that Co -Fe compositions that exhibit low Gilbert damping parameters also feature \nprolonged ultrafast demagnetization upon photoexcitation. We analyze our experimental TR -\nMOKE data with the three-temperature -model (3TM) and the Landau -Lifshitz -Gilbert equation . \nThese analyses  reveal a strong compositional dependence of the dynamics across all time -scales \non the strength of electron -phonon interactions. Our findings are beneficial to the spintronics  and \nmagnonics community , and wil l aid in the quest  for energy -efficient magnetic storage  \napplications.  \nIntroduction  \nLaser excitation of  a magnetic metal causes energy to cascade from photoexcited electrons into \nspin and vibrational degrees of freedom1–3. In ferromagnetic 3d transiti on metals such as Fe, Co, \nand Ni, the rapid increase in thermal energy stored by spin degrees of freedom causes  \nfemtosecond   quenching of the magnetization2,3, followed by a partial recover over the next few \npicoseconds . Subsequently, on nanosecond time-scales , a temperature induced change in \nequilibrium properties causes  oscillatory precessions of the magnetic moment .  \nBoth ultrafast  and precessional magnetization dynamics involve energy exchange between  \nmagnetic and vibrational degrees of freedom . The energy exchange is mediated by quasi -particle \ninteractions . The strength of quasi -particle  interactions in a ferromagnet depends on e lectronic \nband structure4,5. In 3d ferromagnetic alloys, the electronic energy bands  near the Fermi -level \nvary strong ly with composition6. Several  recent investigations  of nanosecond precessional \ndynamics  in ferromagnetic alloy s have explored  the relationship between electronic band \nstructure, quasi -particle interactions , and magnetic damping6–8. Schoen et al.  report a n intrinsic 2 \n damping parameter less than 10-3 for Co 0.25Fe0.756, which is unusually low for  a metal . They \nconclude  that the low damping  in Co0.25Fe0.75 is a result of  a minimization in the density of states \nat the Fermi -level, which decreases the rate of electron -phonon scattering . \nResearchers have not yet reached a  unified understanding of how quasi -particle interactions \ngovern  the magnetization dynamics in the  femtosecond regime2,9–15. Some studies have \nhypothesized that spin-flips caused by electron -phonon interactions are key drivers of \nfemtose cond  magnetization dynamics9,11. Other experimental and theoretical studies have \nexplored the importance of electron -magnon interactions12–15. Encouraged by the recent \nadvances  in the materials science of nanosecond precessional dynamics6–8, we study the \ncompositional dependence of ultrafast magnetization dynamics in Co -Fe alloys. Our  study’s  goal \nis to understand the relationship between electronic band structure , quasi -particle interactions, \nand femto -magnetism properties of ferromagnetic metal alloys.   \nWe perform time -resolved magneto optic Kerr effect  (TR-MOKE)  measurements to characterize  \nthe magnetization dynamics of thin CoxFe1-x alloy  films (capped and seeded with Ta/Cu layers on \na sapphire substrate)  on femto - to nanosecond time-scale s. See Methods for details on sample \ngeometry.  We observe that the ultrafast magnetization dynamics are a strong function of Co -\nconcentration , see Figure . 1a. The ultrafast dynamics of Co xFe1-x differ most significantly from \nthose of Co and Fe at a composition of  x = 0.25.  We also analyze the time -resolved macroscopic \nprecessional dynamics and report the effective damping parameter of our samples , see Figure  2a. \nAfter linewidth analyses, f or CoxFe1-x, we observe  that the  Gilbert  damping parameter varies \nfrom  3.6 ×10−3 to 5.6 ×10−3 for compositions between x = 0 and 1, with a minimum value of \n1.5 ×10−3 at x = 0.25 , in good agreement with previously reported results , see Figure 3b .  \nTo determine the strength and composition dependence of electron -magnon and electron -phonon  \nquasi -particle interactions , we analyze our ultrafast magnetization dynamics  data with a three-\ntemperature -model (3TM)2,16. Our results reveal a strong composition al dependence of  the \nelectron -phonon energy transfer coefficient, 𝑔𝑒𝑝,  suggesting that  the variation in the ultrafast \ndynamics  in Co xFe1-x alloys  occurs primarily due to electron -phonon scattering. We draw this \nconclusion because t he value of 𝑔𝑒𝑝 depends on the rate of phonon emission by hot electrons 17. \nElectron -phonon scattering is also predicted  to govern the dampin g of nanosecond precessional \ndynamics  6,18,19. Therefore, o ur results  demonstrate  that the same  microscopic electron -phonon \ninteractions  responsible for Gilbert damping also play a dominant role in femto -magnetism \nproperties of ferromagnetic alloys.  \nResults  \nUltrafast  Magnetization  Dynamics  \nWe plot the normalized u ltrafast magnetization dynamics  response , ∆M(t), for Co, Fe, and \nCo0.25Fe0.75 as a function of time delay in Figure . 1a. Data for the rest of the Co -Fe compositions \nare plotted  in Supplementary Figure  1. All our measurements were performed with an incident 3 \n laser fluence less than ~15 J/m2. This is a sufficiently  small fluence  for the dynamics in our \nexperiments to follow a  linear regime.  In other words, decreasing the incident f luence by a factor \nof two decreases the optical signal by a factor of two, but does not change the time-dependence  \nof the signal .    \nWe use a polar TR -MOKE configuration t o measure  the ultrafast magnetization dynamics  at \nfemtosecond time delays.  A schemati c of our experimental setup is shown in Supplementary  \nFigure  2a. We apply a n external 2.2 Tesla (T) field perpendicular to the plane of the sample  \nusing  an electromagnet (GMW 3480). This external field is strong enough to effectively \novercome the in-plane shape anisotropy of the Co -Fe alloys and saturate the moment in the out -\nof-plane direction. Since the equilibrium orientation of the moment is in the out -of-plane \ndirection , both, before and after laser irradiation, this geometry allows us to  quan tify the  \nfemtosecond  demagnetization  response  of the Co -Fe alloys , without the presence of macroscopic  \nprecessional dynamics , see schematic in Fig ure 1b.  \nUpon excitation with the pump pulse, the magnetic moment decreases on a sub -picosecond time-\nscale due to the flow of energy from electrons to magnons2,3,16,20,21. Then, on picosecond time-\nscale s, the magnetization partially recovers as energy is transferr ed to the lattice  and temperature \ngradients across the film thickness relax.  After a few picoseconds, the  magnetic film reaches  a \nnew equilibrium at an elevated temperature.  Ultrafast dynamics with sub -picosecond  \ndemagnetization followed by picosecond  re-magnetization  are commonly categorized as “type I” \ndynamics , and are characteristic of 3d ferromagnetic metals such as Fe, Co, and Ni9.  \nTo elucidate  how the de - and re -magnetization dynamics change with composition, we define \ntwo data descriptors : τD and R. We define  the demagnetization time , τD, as the delay time where \nd∆M(t)/dt reaches its maximum value.  We define R as the ratio of the maximum of 𝛥𝑀(𝑡) to \n𝛥𝑀(𝑡≈10ps). We plot τD  and R as a function of composition in Figure  3a. τD varies weakly \nwith composition and has a minimum  value  of 40 fs at x = 0.25. In contrast , we observe that R \nvaries strongly with composition and is a maximum  of 4 at x = 0.25.  \nNanosecond Precessional Dynamics   \nWe show measurements of the  macroscopic  precessional dynamics of Fe, Co, and Co 0.25Fe0.75 in \nFigure  2a. Data for the other Co -Fe compositions are plotted in Supplementary  Figure  3. We use \na polar TR -MOKE experimental setup, with an obliquely angled external magnetic field, to \nmeasure the macroscopic precessional dynamics of our samples. A schematic of our \nexperimental setup is shown in Supplementary  Figure 2b. Tilting the electromagnet to an angle \nof 11° , with respect to the plane of the sample, allows us to apply a canted  external  magnetic \nfield so that the magnetic moment has an out -of-plane component. The equilibrium orientation of \nthe moment depends on the balancing between the applied  external  field and the thin -film shape \nanisotropy field.  The shape anisotropy field in the z -direction is proportional to the  out-of-plane \ncomponent of the  magnetic moment. Upon heating, the  total magnetic moment decreases . This \ndecrease results in an ultrafast change to the  out-of-plane  anisotropy field  and equilibrium 4 \n orientation . As a result, t he magnetic moment will precess to a new equilibrium orientation , see \nschematic in Figure  2b. Our polar TR -MOKE setup detects  changes in the out -of-plane moment , \nso we can sensitively measure the frequency and amplitude of the precessional dynamics.  \nWe collect between  6 and 12 TR-MOKE scans  of precessional dynamics for each sample .  Each \nof these scans is co llected  with a different applied external magnetic field , ranging from 0. 2 T to \n2.2 T. The TR -MOKE signals include precessional dynamics  in addition with a background \nrelated to temperature -induced demagnetization. To analyze the precessional dynamics,  we \nsubtract  the background with  a biexponential  decay  function . We fit the resulting dataset with a \ndamped harmonic function,  V(t)=Asin(ωt+∅)exp (−t/τ). Our fits yield unique  values of A \n(amplitude),  ∅ (the initial phase of the oscillation), T (period), and τ (the exponential decay time \nof the precession). Using these values,  we determine  the effective dimensionless damping \nparameter , αeff = ω.τ-1. \nThe resonance frequency  is a function of applied external magnetic field  and magnetic moment,  \n𝜔=γ √Heff(Heff+μ0Ms). Here, ɣ is the gyromagnetic ratio,  μ0 is the vacuum permeability, \nHeff is the out -of-plane component of the external magnetic field  as measured by a Hall probe , \nand Ms is the saturation magnetization of the sample . We derive the magnetic moment of the \nsample by treating Ms as a fit parameter . We also perform VSM measurements of the moment  of \nsome of the samples and find that the magnetic moment obtained is in good agreement with the \nvalue that we derive by fitting our precessional dynamics data . See Supplementary  Figure 4 for \nmore details . \nThe effective damping parameter α eff that we deduce from our precessional dynamics \nmeasurements includes effects from damping and  inhomogeneous broadening. The effect of \ninhomogeneous  broadening is independent of the applied field at high frequencies22. To obtain \nthe Gilbert  damping parameter  intrinsic to the sample  geometry  (not intrinsic to the material) , we \nplot the effective linewidth, αeff∙f,  as a function of frequency, and linearly fit  to the equation , \nαeff∙f=α∙f+∆H, where ∆H  is the inhomogeneous broadening component and α is the Gilbert \ndamping parameter . Further details can be found in Supplementary Figure 5. \nIn contra st to prior investigations that performed FMR measurements in the frequency range \nfrom 16 -18 GHz8 and 40 GHz6, our TR -MOKE experimental setup allows us to study dynamics \nat frequencies as large as 90 GHz. At such high frequency, we can be confident that our \nmeasured Gilbert damping parameter is dominate d by the intrinsic linewidth over \ninhomogeneous  broadening effects.  \nThe Gilbert damping parameter we observe of α = 1.5 ×10−3 for Co 0.25Fe0.75 is amongst the \nlowest ever reported for a ferromagnetic metal. Schoen et al. report α=2.1 ×10−3 for \nCo0.25Fe0.75.  After accounting for radiative and spin -pumping contributions, they estimate an \nintrinsic damping parameter for Co0.25Fe0.75 to be αint=5 ×10−4 .  Lee et al. 8 performed FMR \nmeasurements of Co0.25Fe0.75 epitaxial films and report α=1.4 ×10−3. Wei et al. report α=5 \n 1.5 ×10−3 for Fe 0.75Al0.25 films 7. We note that our measured damping parameter likely \nincludes  significant contributions from spin -pumping into the adjoining Ta /Cu layers, but we  did \nnot experimentally examine the effect s of spin -pumping in our samples.  \nAnalysis  and Discussion  \nThe c omparison of 𝑅 and 𝛼 in Figure  3a and Figure  3b reveals  that the two quantities depend on \ncomposition in a similar manner.  R is at a maximum and 𝛼 is at a minimum at x = 0.25 .  Fe and \nCoxFe1-x alloys with x ≥ 0.5 have small R and high 𝛼.  Alternatively, C oxFe1-x  alloys with  0.1< x \n< 0.5 have both high 𝑅 and low 𝛼. To confirm this correlation , we performed a hierarchical \ncluster analysis of the raw data  at both femtosecond and nanosecond time-scale s. The clustering \nalgorithm divides  the Co -Fe alloys into groups based on similarit ies in the dynamics data . The \nclustering results as a function of composition are nearly  identical  when based on the femto -\n/pico -second time -scale data vs. the nanosecond time -scale data.  We include further details on \nthe clustering analysis in  Supplementary  Note 1 and Supplementary Figure 6 .  \nWe now explain the correlation between ultrafast and precessional dynamics by considering how \nelectronic scattering processes depend on composition. Similar to prior studies of damping in \nCo-Fe alloys6,7,23, our results for  𝛼 vs. x are in good agreement with the “breathing Fermi \nsurface ” model for damping24. In this model , spin -orbit coupling causes the Fermi -level to shift \nwith the precessi ons of the magnetic moment25. A shift in the equilibrium Fermi -level leads to a \nnonequilibrium electron population . As the Fermi -level repopulates, i ntra-band electron -phonon \nscattering  transfers energy to the lattice .  The “breathing Fermi surface” model predicts  that the  \ndamping parameter  is directly  proportional to 𝐷(𝜀𝑓), because  more electronic states near 𝜀𝑓 leads \nto higher  rates of electron -phonon scattering . We observe  that the  𝛼 value  for Co0.25Fe0.75 is \n~2.5x lower th an 𝛼 for Fe. Density functional theory predicts a ~2x difference in 𝐷(𝜀𝑓) for \nCo0.25Fe0.75 vs. Fe, see Supplementary  Note 2  or Ref.6. Therefore, like prior studies of Co -Fe \nalloys6,7,23, we conclude that intra -band electron -phono n scattering governs precessional \ndamping.  \nTo better understand how composition affects  electron -magnon and electron -phonon energy \ntransfer  mechanisms , we analyze our 𝛥𝑀(𝑡) data with a phenomenological three temperature \nmodel  (3TM) , see Figure  4. The 3TM describes how heat flows  between electrons, phonons, and \nmagnons after laser excitation of the Co-Fe sample . (See Methods for  additional  details. ) The \n3TM predicts  that τD depends on two groupings of model parameters: 𝜏𝑒𝑚≈𝐶𝑚/𝑔𝑒𝑚 and 𝜏𝑒𝑝≈\n𝐶𝑒/𝑔𝑒𝑝.  Here 𝐶𝑚 and 𝐶𝑒 are the magnon and electron heat-capacity per unit volume, and 𝑔𝑒𝑚 \nand 𝑔𝑒𝑝 are the energy transfer coefficients from electrons to magnons an d phonons, \nrespectively.  We estimate v alues for 𝐶𝑒 vs. composition using the Sommerfeld model together \nwith the electronic density of states vs. composition reported in Ref.6. The 3TM also predicts that \nthe parameter R is determined by the following grouping of parameters: 𝑅= 𝐶𝑝𝑔𝑒𝑚/𝐶𝑚𝑔𝑒𝑝 16, \nwhere 𝐶𝑝 is the phonon heat -capacity per unit volume .  We assume  that the  value of  𝐶𝑝 is 3.75 6 \n MJ m-3 K-1 for Co, Fe and Co -Fe alloys.  With these estimates for 𝐶𝑒 and 𝐶𝑝, and other relevant \nmodel parameters, summarized in Supplementary  Table 1, we can deduce unique values for \n𝐶𝑚/𝑔𝑒𝑚 and 𝐶𝑝/𝑔𝑒𝑝  as a function of composition from our TR-MOKE  data, see Figure  4b. \nBased on our 3TM analysis, we conclude that the strong composition dependence of R is due to \nthe composition dependence of 𝑔𝑒𝑝. Boltzmann rate -equation modelling of the nonequilibrium \nelectron dynamics after photoexcitation predicts that the electron -phonon energy -trans fer \ncoefficient  is 𝑔𝑒𝑝=[𝜋ℏ𝑘𝐵𝐷(𝜀𝐹)]𝜆⟨𝜔2⟩ 5. Here,  𝜆⟨𝜔2⟩ is the second frequency moment of the \nEliashberg function  and is a measure of the strength of electron -phonon interactions . Most of the \ncomposition al dependence we observe in 𝑔𝑒𝑝 is explained by the composition al dependence of \n𝐷(𝜀𝑓). To show this, we include a prediction for 𝑔𝑒𝑝 in Figure  4b.  Our prediction uses the \n𝐷(𝜀𝑓) vs. x reported  in6  and treats 𝜆⟨𝜔2⟩ as a composition independent fit parameter . We find \n𝜆⟨𝜔2⟩=260 meV2 provides an excellent fit to our data .  The best-fit value for 𝜆⟨𝜔2⟩ is in good \nagreement with  𝜆⟨𝜔2⟩≈𝜆𝑅Θ𝐷22⁄=280 meV2.  Here, 𝜆𝑅 is derived from electrical resistivity \ndata for Fe  26, and Θ𝐷=470𝐾 is the Debye temperature of Fe.   \nBefore beginning our experimental study, we hypothesized that the energy transfer coefficient \nbetween electrons and magnons, \nemg , would be correlated with the phase -space for electron -\nmagnon scattering . We expected the phase -space for electron -magnon scattering to be a  strong \nfunction of band -structure near the Fermi -level 12–15.  We also expected the phase -space to be \nminimized at a composition of x = 0.25, because of the minimum in the density of states at the \nfermi -level.   To explore how the phase -space for electron -magnon scattering depends on \ncomposition, we performed density functional theory calculations for the electronic band \nstructure with x = 0 and x = 0.25, see Supplementa ry Note 2.  Our DFT calculations suggest  that \nthe phase -space for electron -magnon scattering is an order of magnitude higher for x = 0 vs. \n0.25.  However, we do not see evidence  that this large theoretical difference in electron -magnon \nscattering phase -space  affects ultrafast dynamics . The time -scale for magnons to heat up after \nphotoexcitation, \n/em m emCg , decreases monotonically with increasing x, and does display  \nstructure near x ~ 0.25.  \nSeveral  theoretical models predict a strong correlation between τ D and αint. For example,  \nKoopmans et al.  predicts τ D will be inversely proportional to α by assuming  that the  dissipative \nprocesses responsible for damping also drive ultrafast demagnetization  27.  Alternatively, Fähnle \net al.  predict s that  τD should be proportional to αint 28. In our experiments on Co -Fe thin films, w e \nobserve only a weak correlation between τD and αint. While α int varies  with composition  by a \nfactor of three , τD for 8 of the 9 compositions we study fall within 20% of 75 fs.  The τD value we \nobtained for Fe (=  76 fs) agrees well with experimental results reported in 9,12,29. \n \n 7 \n Conclusions  \nWe have measured the magnetization dynamics of Co xFe1-x thin-films , and we observe that  both \nultrafast and precessional dynamics of Co 0.25Fe0.75 differ significantly from Co and Fe . When the \nmoment of Co0.25Fe0.75 is driven away from its equilibrium  orientation , the time -scale for  the \nmoment  to return to equilibrium is 3 -4x as long as for Fe or Co.  Similarly, when spins of \nCo0.25Fe0.75 are driven into a nonequilibrium state by ultrafast laser heating, the time -scale for \nthermalization with the lattice is 2 -3x as long as for Fe or Co. Through  3TM analyses, we \ndemonstrate that this occurs  primarily  due to the effect of  the electronic band -structure on \nelectron -phonon interactions , consistent with the “breathing Fermi surface” theory . Our findings \nare of fundamental importance to the field of ul trafast magnetism, which seeks to control \nmagnetic order on femto - to picosecond time-scale s. Such control requires  a thorough  \nunderstanding of how and why energy is exchanged between electronic, spin, and vibrational \ndegrees of freedom. Prior studies have  shown  that 𝑔𝑒𝑝 is correlated with a wide range of physical \nproperties, e.g the superconducting transition temperature30, electrical resistivity 26, \nphotoelectron emission31, and the laser fluence required for ablation32. To our knowledge, o ur \nstudy provides the first demonstration that 𝑔𝑒𝑝 in ferromagnetic metals is also correlated to the \nGilbert damping parameter 𝛼.  \nOur findings  also have implications for the ongoing search for magnetic materials with ultrafast \nmagnetic switching functionality. Atomistic spin dynamics simulations predict that the energy \nrequired for ultrafast electrical or optical switching of rare -earth ferromagnetic  alloys, e.g. \nGdFeCo, is governed by the electron -phonon energy transfer coefficient33. To date, most studies \naimed at exploring the materials science of ultrafast switching have used alloy composition as a \nway to control magnetic  properties  34–37. Our work  suggests  an alternative strategy for reducing \nthe energy requirements for ultrafast magnetic switching. The alloy composition should be \nchosen  to minimize the electronic density of states at the Fermi -level. Such metals will have \nlower electron -phonon energy trans fer coefficients, and therefore more  energy efficient ultrafast \nswitching  33.  \nFinally , our findings  offer a new route for discovering ferro magnetic materials with ultra -low \ndamping  as a result of low 𝑔𝑒𝑝. Current methods for identifying low damping materials involve \nlabor -intensive ferromagnetic resonance measurements of one alloy composition at a time.  \nAlternatively, high-throughput  localized measurements of  ultrafast demagnetization  dynamics  of \nsamples produced using combinatorial techniques38 would allow promising alloy compounds \nwith weak electron -phonon interactions to be rapidly identified  39–41.  \n \n \n \n 8 \n Materials and Methods  \nSample Preparation  \nWe sputter deposit the Co -Fe samples onto sapphire substrates with a direct current (DC) \nmagnetron sputtering system (Orion, AJA International). The base pressure prior to deposition is \nless than 3.5  × 10-7 torr. We sputter with an Ar gon pressure of ~3.5 × 10-3 torr. The geometry of \nthe samples is sapphire/Ta(2nm)/Cu(3nm)/Co xFe1-x(15nm)/Cu(3nm)/Ta(1nm). The Co xFe1-x layer \nis deposited  by co -sputtering two 4N purity Co and Fe targets at different powers. We chose this \nfilm geometry to mimic the samples in Ref.6 which demonstrated  low damping at x = 0.25.  \nTo ensure an accurate thickness of each layer in our samples, we calibrate the deposition rates of \neach metal by sputtering individual Co, Fe, Ta, and Cu films onto SiO 2/Si substrates and/or BK -7 \nglass substrates. We use picosecond acoustics42 and time-domain thermo -reflectance  (TDTR) \nmeasurements43,44 to determine the thicknesses of these individual films. We validate the \ncomposition of the Co -Fe alloy layer by perf orming Energy Dispersive X -Ray Spectroscopy \n(EDS) analyses with a scanning electron microscope ( FEI Nova Nano SEM 450) at an operating \nvoltage of 15 kV and working distance of 14 mm. We analyze the EDS data using Aztec Synergy  \nsoftware ( Oxford Instruments ). \nTime -Resolved MOKE Experimental Setup  \nWe use a pump/probe laser system to perform TR -MOKE measurements of the magne tization \ndynamics. The pulsed laser is a Ti:sapphire oscillator with an 80 MHz repetition rate. The laser \nbeam is split into a pump and probe beam, that are modulated to frequencies of 10.7 MHz and \n200 Hz , respectively. A time -delayed  pump beam irradiates the sample surface and heats the \nmetal film. The ultrafast heating causes a change in the magnetic moment. We measure the time -\nevolution of the magnetic moment by monitoring the polarization of the probe beam reflected of f \nthe sample surface. The reflected probe beam’s polarization state is affected by the out -of-plane \nmagnetic moment of the sample due to the polar Kerr effect. Additional details about the MOKE \nexperiment set -up are in Ref.45. \nThe t ime-resolution of our experiment is controlled by the convolution of the intensity vs.  time \nof the pump and probe pulses. The wavelength of our pump and probe beams is tunable. \nEmploying a red  (900 nm ) pump and  blue (450 nm ) probe yields  higher time-resolution  \ncapabilities , allowing us to accurately measure the ultrafast magnetization at fe mtosecond time \ndelays . We measure the full-width -at-half-maximum ( FWHM ) of the convolution of the pump \nand probe pulses by performing an inverse Faraday effect  (IFE)  measurement on Pt . We obtain a \nFWHM  value  of 390 fs for  the convoluted pulses , and a pulse duration of 2 10 fs for the 900 nm \npump/450 nm probe beam setup . For further details on our IFE measurements and pulse duration \ncalculations, please refer to  Supplementary Figure 8.  9 \n To investigate  the precessional dynami cs on longer time -scales, we use a pump and probe \nwavelength of 783 nm. The pulse duration  for this setup  is 610 fs due to pulse broadening from a \ntwo-tint setup we use to prevent pump light from reaching the balanced detector45,46.  \nThree Temperature Modeling  \nTo determine  the electron, phonon, and magnon energy transfer coefficients,  we use t he \nphenomenological  three -temperature model  (3TM),  given by the following set of equations :  \n𝐶𝑒𝑑𝑇𝑒\n𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+ 𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑒𝑑2𝑇𝑒\n𝑑𝑧2+𝑆(𝑧,𝑡)    (1) \n𝐶𝑝𝑑𝑇𝑝\n𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+Ʌ𝑝𝑑2𝑇𝑝\n𝑑𝑧2     (2) \n𝐶𝑚𝑑𝑇𝑚\n𝑑𝑡=𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑚𝑑2𝑇𝑚\n𝑑𝑧2   (3) \n𝑆(𝑧,𝑡)= 𝑆0𝑃(𝑡)𝐴(𝑧)   (4) \nEquations  1 – 3  describe the temperature evolution of electrons (e), phonons (p) and magnons \n(m), as a function of time delay  (t). C, T, and Ʌ are the heat capacity per unit volume, \ntemperature, and thermal conductivity, respectively.  We use the density of states (DOS) at the \nFermi level  as a function of Co -concentration6 to calculate the electronic heat capacity (C e) using \nthe Sommerfeld model . We assume that the phonon -magnon energy transfer is negligible \ncompared to electron -magnon coupling, and  thus,  neglect  𝑔𝑝𝑚.  \nWe calculate the laser energy absorption by electrons  (S), as a function of depth  (z) and time \ndelay  (t), as described in Equation 4. The terms P(t) and A(z) denote the time -dependent laser \npulse intensity and the optical absorption profile as a function of stack thickness. We calculat e \nA(z) us ing the refractive indices of each metal constituent of the stack47–49. The material \nparameters that are used to numerically solve equations 1 – 4 are listed in Supplementary Table \n1. \n \n \n \n \n \n \n \n 10 \n Figures:  \n                   \nFigure 1. Ultrafast magnetization dynamics of Co, Fe, and Co 0.25Fe0.75 thin films (a) Polar TR -\nMOKE data showing ultrafast demagnetization behavior at short delay times. (b) Schematic \nillustration of the three phases of an ultrafast magnetization dynamics experiment. Stage I: A large \nexternal magnetic field  oriented normal to the plane of t he sample leads to an equilibrium moment , \n𝑀⃗⃗  in the out -of-plane direction. Stage II: Upon heating with a pump beam, ultrafast \ndemagnetization ( 𝑀′⃗⃗⃗⃗ ) occurs within ~100s of fs. Energy from hot electrons is transferred to the \nmagnons, increasing the amplitude of precession. Stage III: Over the next few picoseconds, energy \nis transferred from magnons and electrons to the lattice. Additionally, spatial t emperature gradients \nrelax. As a result, magnons cool, i.e. the average precessional amplitude of individual spins \ndecreases. As a result, the magnetization partially recovers to 𝑀′′⃗⃗⃗⃗⃗⃗ . The time -scale for the partial \nrecovery in stage III depends strongly o n the composition.    \n11 \n   \nFigure 2 . Precessional dynamics in Co, Fe, and Co 0.25Fe0.75 thin films (a)  Polar TR -MOKE data \non sub -nanosecond time-scale s. (b) Illustration of the three stages for precessional dynamics after \nlaser excitation . Stage I: Prior to laser excitation, the presence of a canted external magnetic field, \n𝐻𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ , oriented at an angle θ. This results in the orientation of the out -of-plane moments, 𝑀⃗⃗ 𝑧. Stage \nII: Laser -induced photoexcitation leads to the disorder of the magnetic moment, causing a decay \nin the net magnetization , denoted by 𝑀′⃗⃗⃗⃗ . The net torque imba lance causes macroscopic precessions \nof the magnons, towards equilibrium, 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , over several ~100s of picoseconds . Stage III : \nEventually, after ~1 ns, the magnetic moment re -equilibrates to 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . The lifetime of the magnetic \nprecessions depends o n the effective damping parameter, α eff.  The time -scale for the precessional \ndynamics to cease ( in stage III) depends strongly on composition, and is a maximum for x = 0.25.  \n12 \n  \nFigure 3.  Compositional dependence of descriptors for the ultrafast dynamics  data . (a) R \ndescribes the maximum change in the magnetic moment, i.e. how far from equilibrium spin -\ndegrees of freedom are driven after ultrafast excitation. τD describes the lag between zero delay \ntime and demagnetization, as a function of Co -concentration.  (b) α denotes the Gilbert damping \nparameter, as a function of Co concentration. Data obtained from our TR -MOKE experiments \ndescribed in this study (plotted in orange), agree reasonably with data from Ref. [6] (plotted in \ngreen). Co 0.25Fe0.75 features the largest deviation in R and α, when compared to its constituent \nelements Co and Fe.  \n \n \n \n \n \n \n \n \n \n \n \n13 \n  \nFigure 4. Analyses of Ultrafast Demagnetization Results using the Three Temperature Model \n(3TM) in Co -Fe alloys . (a) Polar TR -MOKE dataset of the Co 0.25Fe0.75 composition (black circles) \nwith best -fit results of the 3TM. The 3TM describes the temperature excursions of the electrons \n(blue curve), magnons (red curve) and phonons (green curve) after laser excitation. (b) We treat \n𝑔𝑒𝑝 and 𝑔𝑒𝑚 as fit parameters when solving the 3TM. Using literature values of C p and C m (further \ndetails available in Supplementary Table 1), we calculate and plot the electron -phonon (τ ep) and \nelectron -magnon (τ em) relaxation times, as a function of Co -concentration. The red -line is a best -\nfit value for the electron -phonon relaxation time as a function of composition, with the assumption \nof a composition -independent value for the electron -phonon coupling parameter λ . \n \n \n \n \n \n \n \n \n \n \n \n14 \n References:  \n1. Kirilyuk, A., Kimel, A. V & Rasing, T. 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Optical constants of transition metals. Phys. Rev. B  9, \n5056 –5070 (1974).  \n48. P. B. Johnson and R. W. Chri sty. Optical Constant of the Nobel Metals. Phys. Rev. B  6, 17 \n 4370 –4379 (1972).  \n49. Ordal, M. A., Bell, R. J., Alexander, R. W., Newquist, L. A. & Querry, M. R. Optical \nproperties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths. Appl. Opt.  27, 1203 \n(1988).  \n \n \n \nAcknowledgements  \nThe work by R. M., V. H. O, and R. B. W. was primarily supported by the U.S. Army Research \nLaboratory and the U.S. Army Research Office under contract/grant number W911NF -18-1-\n0364 and W911NF -20-1-0274. R. M. and R. B. W. also acknowledge support by NSF (C BET – \n1847632). The work by L. V. and S. C. was supported by the U.S. Army Research Laboratory \nand U.S. Army Research Office under contract/grant number W911NF -20-1-0274. Energy \nDispersive X -Ray Spectroscopy (EDS) analyses  were performed at the Central Fac ility for \nAdvanced Microscopy and Microanalysis (CFAMM) at UC Riverside.  \nAuthor Contributions  \nR. M. and R. B. W. designed the experiments. R. M. prepared all the samples and characterized \nthem , and performed TR-MOKE experiments . V. H. O performed VSM measurements. L. V. \nperformed hierarchical clustering analyses. S. C. performed DFT calculations. R. M. and R . B. \nW. analyzed the data and wrote the manuscript, with discussions and contributions from L. V. \nand S. C . \nAdditional Information:  Supplementary information is provided with this manuscript.  \nCompeting Interests:  The authors declare no competing interest.  \nData Availability:  The data that supports the findings of this paper are available from the \ncorresponding author upon reasonable request.  \nCorrespondence:  Correspondence and request for additional information must be addressed to \nrwilson@ucr.edu   "    },    {        "title": "1703.07154v1.Annealing_stability_of_magnetic_tunnel_junctions_based_on_dual_MgO_free_layers_and__Co_Ni__based_thin_synthetic_antiferromagnet_fixed_system.pdf",        "content": "Annealing stability of magnetic tunnel junctions based on dual MgO free\nlayers and [Co/Ni] based thin synthetic antiferromagnet fixed system\nT. Devolder,1,a)S. Couet,2J. Swerts,2E. Liu,2, 3T. Lin,2S. Mertens,2A. Furnemont,2and G. Kar2\n1)Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Universit ´e Paris-Saclay, C2N-Orsay,\n91405 Orsay cedex, France\n2)imec, Kapeldreef 75, 3001 Heverlee, Belgium\n3)Department of Electrical Engineering (ESAT), KU Leuven, 3001 Leuven, Belgium\n(Dated: 22 March 2017)\nWe study the annealing stability of bottom-pinned perpendicularly magnetized magnetic tunnel junctions based on dual\nMgO free layers and thin fixed systems comprising a hard [Co/Ni] multilayer antiferromagnetically coupled to thin a\nCo reference layer and a FeCoB polarizing layer. Using conventional magnetometry and advanced broadband ferro-\nmagnetic resonance, we identify the properties of each sub-unit of the magnetic tunnel junction and demonstrate that\nthis material option can ensure a satisfactory resilience to the 400\u000eC thermal annealing needed in solid-state magnetic\nmemory applications. The dual MgO free layer possesses an anneal-robust 0.4 T effective anisotropy and suffers only a\nminor increase of its Gilbert damping from 0.007 to 0.010 for the toughest annealing conditions. Within the fixed sys-\ntem, the ferro-coupler and texture-breaking TaFeCoB layer keeps an interlayer exchange above 0.8 mJ/m2, while the Ru\nantiferrocoupler layer within the synthetic antiferromagnet maintains a coupling above -0.5 mJ/m2. These two strong\ncouplings maintain the overall functionality of the tunnel junction upon the toughest annealing despite the gradual\ndegradation of the thin Co layer anisotropy that may reduce the operation margin in spin torque memory applications.\nBased on these findings, we propose further optimization routes for the next generation magnetic tunnel junctions.\nI. INTRODUCTION\nThe spin-transfer-torque (STT) switching of a nanomag-\nnet that shows perpendicular magnetic anisotropy is the ba-\nsic phenomenon harnessed in advanced magnetic random ac-\ncess memories. This technology is based on magnetic tun-\nnel junctions (MTJ) that rely on complex stacks that require\ncareful optimization to ensure manufacturability and oper-\nability. Thanks to their low Gilbert damping and their high\nperforming magneto-transport properties1the so-called sin-\ngle MgO free layer of composition Ta/FeCoB/MgO has be-\ncome de facto the standard storage layer. Unfortunately in\nsingle MgO free layers only one interface contributes to the\ntotal magneto-crystalline anisotropy. Besides, some diffusion\nof Ta into FeCoB happens at the high annealing temperature\nrequired for a CMOS back-end-of-line process, and this inter-\nmixing is detrimental2to the Gilbert damping. Dual MgO (i.e.\nMgO/FeCoB/spacer/FeCoB/MgO) free layers are thus gradu-\nally introduced in the context of spin-torque operated mag-\nnetic memory applications as they provides superior proper-\nties with the promise of better resilience to annealing3,4.\nOn the other side of the tunneling oxide, magnetically\nhard reference layers are needed. A synthetic antiferromag-\nnet (SAF) system is systematically used to avoid generat-\ning stray fields that would perturb the free layer operation.\nHistorically, two multilayers [Co/X] \u0002Nand [Co/X] \u0002Mwith\nX2[Ni, Pt, Pd ]separated by a Ru-based antiferrocoupler\nhave been used5to form a conventional thick SAF. Sadly, con-\nventional SAF do not scale well. Indeed the stray field com-\npensation requires the farthest multilayer to be much thicker\nthan the one nearest to MgO, with a ratio M=N that has to be\na)Electronic mail: thibaut.devolder@u-psud.frincreased substantially at low junction diameter6with the con-\nsequent growth and etching challenges. Thin SAFs, i.e. with\na single [Co/X] multilayer ( N= 1) were proposed7,8instead\nto maintain manufacturability at low junctions diameters.\nIn this paper, we optimize and evaluate the annealing re-\nsistance of bottom pinned MTJs comprising a dual MgO free\nlayer and a [Co/Ni]-based thin SAF in which the near MgO\nmagnet of the SAF has been simplified to a solitary Co layer.\nUsing broadband ferromagnetic resonance, we study the prop-\nerties of each sub-unit of the MTJ and demonstrate that this\nmaterial option can ensure a good resilience to 400\u000eC ther-\nmal annealing of the transport properties of the MTJ and the\ndamping of the free layer, despite the gradual evolution of the\nthin Co layer whose anisotropy and magnetization degrades\nduring the annealing.\nII. SAMPLE AND METHODS\nWe study bottom-pinned MTJs (Fig. 1) of the following\ncomposition: NiCr (seed) / [Ni(3)/Co(6)] \u00026/ Ru(4) / Co(6,\nRL) / TaFeCoB(8)/ FeCoB / MgO / Free layer / MgO / cap,\nwhere the Free layer is composed of Fe 60Co20B20(12) / Ta /\nFe60Co20B20(8), with the numbers being the thickness in ˚A.\nOur layer naming convention is described in Fig. 1. The Ta\nspacer within the free layer was grown with a sacrificial Mg\nlayer to avoid alloying of Ta and FeCoB during growth9. The\nTaFeCoB spacer used between the spin polarizing layer and\nthe reference layer was deposited by co-sputtering from two\nFe60Co20B20and Ta targets, with the Ta sputtering rate set\nto achieve a content above 50%. Note that the Ru thickness\nwas set at the first (and in principle the strongest) maximum of\nantiferromagnetic coupling, in contrast to the common prac-\ntice with conventional SAFs. This choice might be perceived\nas surprising as a too low Ru thickness endangers the over-arXiv:1703.07154v1  [cond-mat.mtrl-sci]  21 Mar 20172\nFeCoB&&(12Å)&&&&& & & & & & & &MgO[Ni&(3Å)/Co&(6Å)]x6Ru&(4Å)Ta&(4Å)TaFeCoB&(8Å)&&&&&&&&&&&FeCoB&&(8Å)Co&(6Å)NiCrFeCoB&&(8Å)Thin&reference&layerDual&MgO&free&layer\nHard&layersSpin&polarizerFixed&systemFerroMcouplerAnOferroMcouplerFerroMcouplerTunnel&barrier\nSeed&layerMgOCap\nFIG. 1. (Color online). Sketch of the Magnetic Tunnel Junction with\na dual MgO free layer and a [Co/Ni] multilayer based thin synthetic\nantiferromagnet fixed system. Note our layer labeling conventions\nwithin the fixed system.\nFIG. 2. (Color online). Properties of the dual MgO free layer. (a)\nto (d): Imaginary part of the free layer permeability in a field of 1\nTesla along the easy axis for the four annealing conditions. The lines\nare macrospin fits with the linewidth \u0001f=(2f)as fitting parameter.\n(e) Major hysteresis loop of the full tunnel junction after the 300\u000eC\nannealing step. (f) Ferromagnetic resonance frequency versus field\ncurves for the four annealing conditions. The four curves overlap\nwithin the experimental accuracy. (g) Half linewidth versus FMR\nfrequency for the four annealing conditions. The lines have slopes\nof 0.007 (black empty circles, 300\u000eC) and 0.01 (orange filled circles,\n400\u000eC).\nall resistance to annealing; however we will see that the use\nof this Ru thickness nominally optimizing the coupling was\nfound necessary to compensate for the fact that the Ru cou-\npling does not fully develop for ultrathin Co(6 ˚A) reference\nlayers like ours (Fig. 1). Note also that the Ru antiferrocoupler\nis sandwiched between two Co layers as this is recommended\nto maximize the interlayer exchange coupling. All samples\nhave been annealed at 300\u000eC for 30 minutes in a field of 1\nTesla, followed by a rapid thermal annealing of 10 minutes at\nTa=350, 375 or 400\u000eC.\nWe studied our samples by Vector Network Ferromag-\nnetic resonance (VNA-FMR,10), vibrating sample magnetom-\netry (VSM) and current-in-plane tunneling (CIPT). In VSM,\nthe loop of the fixed system was deduced by subtracting the\nsquare and low coercivity minor loop of the free layer (not\nTanneal\u000bFL1\n\r0\u0001fjH=0\u00160(Hk\u0000MS)TMR RA\n(\u000eC)(\u00060.001) (\u00061.5 mT) (\u00065 mT) (%) (\n:\u0016m2)\n300 0.007 8 410 111 9.0\n350 0.008 11 420 132 9.5\n375 0.009 11 405 146 9.8\n400 0.010 10 410 144 10.8\nTABLE I. Free layer and transport properties versus annealing for\nour MgO/Fe 60Co20B20/Ta/Fe 60Co20B20free layer.FIG. 3. (Color online). Loops of the fixed layers system and eigenex-\ncitations thereof after annealing at 300\u000eC [(a)-(b)], 350\u000eC [(c)-(d)]\nand 400\u000eC [(e)-(f)]. The blue loop is the experimental data, while the\nblack and gray loops are the results of zero temperature 3-macrospin\ncalculations. The cross symbol is panel (a) is there to evidence a\nslope in the loop at the approach to saturation. The corresponding\norientations of the magnetizations of the three layers of the fixed\nsystem are sketched in Fig. 4. The triplets on top of the eigenmode\nfrequency curves illustrate the modeled precession amplitudes in the\nFeCoB layer (PL, first component of the triplet), in the thin Co layer\n(RL, second component) and in the Co/Ni multilayer (HL, third com-\nponent). In the eigenmode fits, the bold line are for an increasing field\nscan like used in the experiments, while the dotted lines are for the\nreverse sweep. The inset in panel (e) is the major loop after 400\u000eC\nannealing. The field is varied between \u00061Tesla.\nshown) from the major loop of the full MTJ [Fig. 2(e)]. As the\nfree layer switching field slightly fluctuates from loop to loop,\nsome fuzziness appears inevitably at the free layer coercivity\nin the so-calculated hysteresis loop of the fixed layer system\n(blue curves, Fig. 3). We then use VNA-FMR to measure the\nMTJ eigenexcitations in out-of-plane fields up to 2.5 T and\n70 GHz. The frequency resolution of VNA-FMR is a conve-\nnient tool to identify selectively the properties of each subsys-\ntem. The canonical method was described in detail in ref. 11.\nFor instance the free layer FMR frequency versus field curve\nchanges slope at the free layer coercivity, while the fixed sys-\ntem eigenmodes undergo frequency jumps or slope inversion\nat the characteristic fields of the fixed system hysteresis loop.\nThis allows to assign each mode to one subsystem in an indis-\nputable manner. The modeling of the free layer FMR (Fig. 2)\nand the fixed layer system eigenmodes (Fig. 3, 4) will provide\nthe anisotropies and the interlayer coupling of the essential\nlayers of the MTJ at all stages of annealing (Tables I-II).\nIII. RESULTS\nLet us look in detail at the consequences of annealing. It\nimproves the tunneling magneto-resistance (TMR) and the in-\nsulator character (Resistance-Area product, RA) of the tun-\nneling oxide (Table I).\nA. Properties of the dual MgO free layers\nThe free layer minor loops remain square with no signifi-\ncant change of coercivity (not shown). The FMR frequency\nvs field curves [Fig. 2(f)] always exhibit the expected V-\nlike shape; its apex is expected at remanence at a frequency\n\r0\n2\u0019(Hk\u0000Ms)where\r0is the gyromagnetic factor. This fre-\nquency at remanence yields the effective anisotropy. As clear\nfrom Fig. 2(f), the effective anisotropy of our dual MgO free\nlayers is very stable against annealing (Table 2). This con-\ntrasts with single MgO free layers whose anisotropy usually\ndegrades above 375\u000eC of annealing9,11. Besides, the FMR\nlinewidth \u0001fof our dual MgO free layer does not evolve\ndramatically upon annealing [Fig. 2(a)-(d)]. Both the Gilbert3\ndamping and the samples’ inhomogenities contribute to the\nFMR linewidth. The separation of these two contributions\ncan be done classically by looking at the dependence of the\nfrequency linewidth versus FMR frequency. Despite our mod-\nest signal-to-noise ratio, we can ascertain that the linewidth is\nquasi-linear with the FMR frequency [see Fig. 2(g)]). Writing\n\u0001f\u0000\u0001fjH=0= 2\u000bfFMRallows to extract the Gilbert damp-\ning\u000bFLand a measurement of the inhomogeneity of the ef-\nfective anisotropy field (Table 2). Some increase of the damp-\ning parameter occurs upon annealing, however its remains be-\nlow 0.01, i.e. below the values observed in single MgO free\nlayers12,13. In addition to its contribution to the anisotropy, it\nseems that the MgO capping also acts as an efficient diffusion\nbarrier that helps to maintain a low damping within the free\nlayer. However a slight increase of the free layer inhomogene-\nity1\n\r0\u0001fjH=0is still unfortunately observed upon annealing\n(Table I).\nB. Properties of the fixed system\nIn comparison to the free layer, the fixed system seems to\nevolve more substantially upon annealing as attested by the\ngradual inclination of its hysteresis loop (Fig. 3). At 300\u000eC\na first unexpected fact is that the loop of the fixed system\ncomprises 3 apparent switching events. One may naively con-\nclude that the PL, RL and HL switch successively one by one\nsuch that the coupling through TaFeCoB would be vanishingly\nsmall; however a closer look at the amplitudes of the switch-\ning steps indicate that it cannot be the case. Another possible\nscenario would be that the switching at intermediate field [320\nmT, Fig. 3(a)] corresponds to the synchronous ”rigid” flip of\nthe 3 magnetization layers fPL, RL, HLgfrom af\"\"#g to a\nf##\"g configuration to align the magnetization of the largest\nmoment layer with the applied field, followed by another flip\nto the saturatedf\"\"\"g configuration. However, this scenario\nis too simple and we will se that the collinear alignment of\nthe magnetizations is not always the lowest energy configura-\ntion especially near the critical fields of the loops. Even in the\nsimplest case [Fig. 3(a)], the approach to saturation has a sub-\nstantial slope (see the x symbol in [Fig. 3(a)]), which indicates\nthat the state at this field does not consist in perfect collinear\nalignment of the magnetizations. A less intuitive magnetiza-\ntion process has thus to be identified.\nModeling as conducted below and summarized in Fig. 4\nwill indicate that the canted parts of the loops correspond\ntilted magnetization orientations within the fixed system.\nFrom the loops, a second finding is the occurrence of some\nloss of the HL moment between 300 and 350\u000eC annealing\n(see Table II). This loss of HL moment was found14to origi-\nnate from the diffusion of Cr from the seed layer into the HL.\nFinally there are also two more subtle points to be noticed.\nOnly two modes are detected out of the three expected from\nthe three magnetic parts of the fixed system. Besides, the fre-\nquency versus field slope of the highest frequency detected\nmode in the medium field region is opposite to that which was\nobserved in conventional SAF in which a weak ferro-coupler\nwas used (see for instance ref. 11). These two facts indicateFIG. 4. (Color online). Hysteresis loops of the fixed layers system\nafter 300\u000eC (a) and 400\u000eC (b) annealing as calculated from the mate-\nrial properties of Table II. The colored arrows sketch the orientations\nof the magnetization of the three layers of the fixed system. The cor-\nresponding field positions are indicated by the cross symbols when\ndisambiguation is needed. The loops are calculated for an increasing\nfield. In positive fields, the most stable configuration (narrow arrows)\ncan be reached only if a finite energy barrier is thermally overcome\nby the system. In the experiments related of (a), the HL passes from\nthe metastable#\"\"to the stable configuration \"##at a field position\nmarked by the vertical green line labeled ” exp. Hc ”. When further\nincreasing the field, the PL and RL partially switch and then their\nmagnetizatiosn progressively unwind (grey cross) at 500 mT. In the\nexperiments, the metastable state present above 320 mT (right of the\ngreen line) is not visited by the system.\nthat the present coupling though TaFeCoB is probably much\nstronger than that observed in the earlier Ta spacers11.\nTo get the quantitative properties of each part of the fixed\nlayers system, we have fitted the observed mode frequencies\nand hysteresis loops to that of 3 coupled macrospins (PL, RL\nand HL) of unknown anisotropies and couplings. The ground\nstate and its eigenexcitations were calculated by staying in the\nenergy minimum during a field sweep and then by linearizing\nthe dynamics about the energy minima to infer the eigenexci-\ntations. The layer thicknesses were considered constant dur-\ning annealing such that any interdiffusion process will be ac-\ncounted for by the layers’ magnetizations. The approach used\nto vary the unknown material parameters and make the simu-\nlated eigenmode frequencies fit with the experimental ones is\nsimilar to that used in ref. 11. The deduced material properties\nfor the layers within the fixed system are gathered in Table II,\nand the resulting layer-resolved modeled loops are reported in\nFig. 4.\nIt is interesting to note that for the annealing step of 300\u000eC\nthe interlayer exchange energy and the anisotropy energies\nare both very strong on both sides of the Co RL. As a result,\nwhen the interactions can not be all satisfied at the approach\nto saturation, the system prefers to partition its energy in both\nthe exchange degree of freedom and the anisotropy degree of\nfreedom by transiently forming a non-collinear configuration\n(see the most stable states in Fig. 4(a) between 460 and 600\nmT). This happens in a manner very similar to within a Bloch-\ntype domain wall, where the magnetization is fixed in the two\ndomains and prefers to rotate gradually in space instead of\nabruptly from one domain to the next.\nIV. DISCUSSION\nSeveral of our findings confirm previous conclusions: just\nlike in MTJs with a conventional SAF, the FeCoB polariz-\ning layer undergoes a gradual and moderate loss of anisotropy\nupon annealing11. Also anticipated, the anisotropy of the\n[Co/Ni] hard layer is strong and stays practically unaffected\nby annealing14. However, we can notice two substantial dif-\nferences compared to the standards set by conventional (i.e.\nthick) SAF in which a Ta texture breaking layer is used to4\nLayer polarizing layer ferro-coupler reference layer antiferro- hard layers\nRole (PL) & texture-breaker (RL) coupler (HL)\nComposition MgO/Fe 60Co20B20 TaFeCoB Co(6) Ru [Co(6)/Ni(3)] \u00026/ NiCr\nThickness (t,˚A) 8 8 6 4 54\nMS(A/m) JTaFeCoB (\u00060.03 mJ/m2)MS(A/m) JRu(\u00060.03 mJ/m2)MS(A/m)\n300\u000eC 1:1\u0002106\u00151:0 1 :1\u0002106-0.85 0:55\u0002106\n350\u000eC 1:1\u0002106\u00151:47 0 :6\u0002106-0.71 0:49\u0002106\n375\u000eC 1:1\u00021061.17 0:6\u0002106-0.61 0:49\u0002106\n400\u000eC 1:1\u00021060.8 0:6\u0002106-0.5 0:49\u0002106\nHk(A/m) Hk(A/m) Hk(A/m)\n\u00160(Hk\u0000MS)(\u000625 mT) \u00160(Hk\u0000MS)(\u000650 mT) \u00160(Hk\u0000MS)(\u000650 mT)\n300\u000eC 1:45\u00021061:35\u00021061:1\u0002106\n440 #interdependent \u0000\u0000\u0000\u0000\u0000\u0000! # 314 690\n350\u000eC 1:3\u00021060:75\u00021061:05\u0002106\n250 #interdependent \u0000\u0000\u0000\u0000\u0000\u0000! # 188 700\n375\u000eC 1:2\u00021060:65\u00021061:05\u0002106\n125 63 700\n400\u000eC 1:0\u00021060:2\u00021061:05\u0002106\n-125 -500 700\nTABLE II. Properties of the fixed layers versus annealing temperature. # indicates that the parameters cannot be independently determined.\nseparate the spin polarizing layer from the reference layer.\nFirst, the exchange couplings through TaFeCoB is par-\nticularly strong and it persists upon annealing. This has\ntwo consequences. The first is that the optical excitation of\nthefPL + PLgensemble is predicted to be pushed above\nour measurement upper limit of 70 GHz, hence above our\ndetection capability. As a result, we can only give a lower\nbound of the coupling through the TaFeCoB spacer at the\ntwo lowest annealing temperatures. The fPL +RLgensemble\nbehaves like a rigid block and the anisotropies of the PL and\nRL are somewhat interchangeable (only their weighed sum\ncan be deduced).\nThe second and more visible consequence of the very strong\ninterlayer exchange through both TaFeCoB and Ru is that\nfor some applied fields, the layers’ magnetizations cannot\nsatisfy both their anisotropy and their exchange couplings.\nThe Co RL layer often ends in a tilted orientation state, like\nin a spring magnet. The unanticipated tail in the approach to\nsaturation in the loops at positive applied field in Fig. 3(a-c)\nis the signature of the gradual unwinding of such vertical wall\nwithin the fixed system.\nSecond, the 6 ˚A thick cobalt reference layer seems to be the\nmain weakness of our MTJ: upon annealing, it quickly looses\nits anisotropy, which manifests as a loop rounding [Fig. 3(e)].\nFortunately the strong interlayer couplings maintain its\nperfect perpendicular orientation at remanence, such that the\nMTJ is functional even at the toughest annealing conditions.\nAlthough strong, the interlayer exchange coupling through\nRu is below the expected value for a Ru thickness matching\nwith the first antiferromagnetic maximum15. This indicates\nthat a thickness of 6 ˚A of Co is not sufficient to fully develop\nthe coupling, as we could confirm with thicker Co layers (not\nshown) that are however less performing when examining the\nstray field compensation of the fixed system.V. CONCLUSION\nIn summary, we have studied the evolution of perpendicu-\nlarly magnetized MTJs upon annealing up to 400\u000eC. Our sys-\ntem comprises a dual MgO free layer that is bottom-pinned\nby a thin SAF fixed system based on a single Co/Ni multi-\nlayer seeded by a NiCr buffer. The loops indicate that the\nlayers have their full remanence after all annealing steps de-\nspite the evolution towards an ever increasing propensity to\ncanting inside the fixed system, visible at finite applied field.\nThe FMR eigenmodes were modeled to reveal the evolutions\nof the layers properties. The dual MgO free layer has a ro-\nbust anisotropy and suffers only a minor increase of its Gilbert\ndamping upon annealing. The [Co/Ni] multilayer undergoes a\ndecrease of its moment but keeps an anisotropy field as strong\nas 700 mT during annealing. Conversely, the FeCoB polar-\nizing layer, and to a more dramatic extent the thin Co refer-\nence layer loose their anisotropy upon annealing. The cou-\npling through the texture breaker TaFeCoB layer and the Ru\nantiferro-coupler are fortunately large enough to ensure full\nperpendicular remanence of all layers of the MTJ.\nThis diagnosis argues however for the search of new mate-\nrial solutions that would avoid the partial loss of anisotropy\nof the reference layer that might be detrimental to device re-\nliability. Options include for instance the insertion of Pt be-\ntween Co and Ru; this was proven efficient16to strengthen the\nRu-mediated coupling and the Co anisotropy, yet studied only\nfor thicker Ru spacers so far. Alternative texture-breaking\nlayers17that would be more immune to interdiffusion upon\nannealing may also deserve to be studied as a replacement op-\ntion to TaFeCoB.\n1S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo,\nS. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nature Materials 9,\n721 (2010).\n2T. Devolder, I. Barisic, S. Eimer, K. Garcia, J.-P. Adam, B. Ockert, and\nD. Ravelosona, Journal of Applied Physics 113, 203912 (2013).5\n3H. Sato, E. C. I. Enobio, M. Yamanouchi, S. Ikeda, S. Fukami, S. Kanai,\nF. Matsukura, and H. Ohno, Applied Physics Letters 105, 062403 (2014).\n4J.-H. Kim, J.-B. Lee, G.-G. An, S.-M. Yang, W.-S. Chung, H.-S. Park, and\nJ.-P. Hong, Scientific Reports 5, 16903 (2015).\n5D. C. Worledge, G. Hu, D. W. Abraham, J. Z. Sun, P. L. Trouilloud,\nJ. Nowak, S. Brown, M. C. Gaidis, E. J. OSullivan, and R. P. Robertazzi,\nApplied Physics Letters 98, 022501 (2011).\n6M. Gottwald, J. J. Kan, K. Lee, X. Zhu, C. Park, and S. H. Kang, Applied\nPhysics Letters 106, 032413 (2015).\n7H. Gan, R. Malmhall, Z. Wang, B. K. Yen, J. Zhang, X. Wang, Y . Zhou,\nX. Hao, D. Jung, K. Satoh, and Y . Huai, Applied Physics Letters 105,\n192403 (2014).\n8Y . Tomczak, J. Swerts, S. Mertens, T. Lin, S. Couet, E. Liu, K. Sankaran,\nG. Pourtois, W. Kim, L. Souriau, S. V . Elshocht, G. Kar, and A. Furnemont,\nApplied Physics Letters 108, 042402 (2016).\n9J. Swerts, S. Mertens, T. Lin, S. Couet, Y . Tomczak, K. Sankaran, G. Pour-\ntois, W. Kim, J. Meersschaut, L. Souriau, D. Radisic, S. V . Elshocht, G. Kar,\nand A. Furnemont, Applied Physics Letters 106, 262407 (2015).10C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas,\nJournal of Applied Physics 101, 074505 (2007).\n11T. Devolder, S. Couet, J. Swerts, and A. Furnemont, Applied Physics Let-\nters108, 172409 (2016).\n12T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V . Kim,\nB. Ockert, and D. Ravelosona, Applied Physics Letters 102, 022407\n(2013).\n13T. Devolder, J.-V . Kim, F. Garcia-Sanchez, J. Swerts, W. Kim, S. Couet,\nG. Kar, and A. Furnemont, Physical Review B 93, 024420 (2016).\n14E. Liu, J. Swerts, S. Couet, S. Mertens, Y . Tomczak, T. Lin, V . Spamp-\ninato, A. Franquet, S. V . Elshocht, G. Kar, A. Furnemont, and J. D. Boeck,\nApplied Physics Letters 108, 132405 (2016).\n15J. Zhao, Y . J. Wang, Y . Z. Liu, X. F. Han, and Z. Zhang, Journal of Applied\nPhysics 104, 023911 (2008).\n16S. Bandiera, R. C. Sousa, S. Auffret, B. Rodmacq, and B. Dieny, Applied\nPhysics Letters 101, 072410 (2012).\n17T. Devolder, E. Liu, J. Swerts, S. Couet, T. Lin, S. Mertens, A. Furnemont,\nG. Kar, and J. D. Boeck, Applied Physics Letters 109, 142408 (2016)."    },    {        "title": "1812.09596v1.Spin_dynamics_of__3d__and__4d__impurities_embedded_in_prototypical_topological_insulators.pdf",        "content": "Spin dynamics of 3dand 4dimpurities embedded in prototypical\ntopological insulators\nJuba Bouaziz,\u0003Manuel dos Santos Dias, Filipe Souza Mendes Guimar~ aes, and Samir Lounis\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich and JARA, 52425 J ulich, Germany\n(Dated: September 16, 2021)\nAbstract\nTopological insulators are insulating bulk materials hosting conducting surface states. Their\nmagnetic doping breaks time-reversal symmetry and generates numerous interesting e\u000bects such\nas dissipationless transport. Nonetheless, their dynamical properties are still poorly understood.\nHere, we perform a systematic investigation of transverse spin excitations of 3 dand 4dsingle\nimpurities embedded in two prototypical topological insulators (Bi 2Te3and Bi 2Se3). The impurity-\ninduced states within the bulk gap of the topological insulators are found to have a drastic impact\non the spin excitation spectra, resulting in very high lifetimes reaching up to microseconds . An\nintuitive picture of the spin dynamics is obtained by mapping onto a generalized Landau-Lifshitz-\nGilbert phenomenological model. The \frst quantity extracted from this mapping procedure is the\nmagnetic anisotropy energy, which is then compared to the one provided by the magnetic force\ntheorem. This uncovers some di\u000eculties encountered with the latter, which can provide erroneous\nresults for impurities with a high density of states at the Fermi energy. Moreover, the Gilbert\ndamping and nutation tensors are obtained. The nutation e\u000bects can lead to a non-negligible shift\nin the spin excitation resonance in the high-frequency regime. Finally, we study the impact of\nthe surface state on the spin dynamics, which may be severely altered due to the repositioning of\nthe impurity-induced state in comparison to the bulk case. Our systematic investigation of this\nseries of magnetic impurities sheds light on their spin dynamics within topological insulators, with\nimplications for available and future experimental studies as, for instance, on the viability of using\nsuch impurities for solid-state qubits.\n1arXiv:1812.09596v1  [cond-mat.mes-hall]  22 Dec 2018I. INTRODUCTION\nThe ever-increasing need for higher storage density oriented research towards the minia-\nturization of magnetic memories, constricted by the super-paramagnetic limit1. The real-\nization of smaller magnetic bits requires materials with a high magnetic anisotropy energy\n(MAE), originating from the relativistic spin-orbit interaction. The extreme limit for high-\ndensity magnetic storage consists of a single atomic bit2, for which quantum e\u000bects can be\npredominant. Therefore, a deep fundamental understanding underlying the stability mech-\nanisms is crucial for future technological applications. Moreover, the manipulation of these\nmagnetic units relies on external time-dependent \felds, with their dynamical properties\nbeing of prime relevance as well.\nThe standard tool for probing the dynamical magnetic properties ( i.e.spin excitations)\nof single atoms is the inelastic scanning tunneling spectroscopy (ISTS). It was employed to\ninvestigate magnetic adatoms on non-magnetic surfaces3{12. The spin excitations signature\nin the di\u000berential conductance (dI\ndV, withIbeing the tunneling current and Vthe applied\nvoltage) consists of step-like features at the excitation frequencies. They are determined\nby the applied external magnetic \feld and the MAE, which can also be accessed via other\nexperimental methods such as X-ray magnetic circular dichroism (XMCD)13,14. The nature\nof both the substrate and the adsorbate play a major role in the determination of the\nresonance frequency and lifetime of the excitation.\nSeveral theoretical investigations of spin excitations of magnetic atoms deposited on non-\nmagnetic surfaces have been performed. In the limit of weak coupling ( i.e.low hybridization)\nbetween the adsorbate and the substrate, the ISTS spectra can be interpreted employing\na Heisenberg model with localized atomic moments possessing an integer (or half integer)\nspin. Such a scenario occurs when the substrate is of insulating or semi-conducting na-\nture6,15,16. When the coupling to the substrate is strong, the hybridization e\u000bects must\nbe taken into account and a more accurate description of the electronic structure is re-\nquired. This was achieved using real-space \frst-principles calculations in the framework of\nthe Korringa-Kohn-Rostoker Green function (KKR-GF) method, which was extended to the\ndynamical regime17{20relying on time-dependent density functional theory (TD-DFT) in its\nlinear response formulation21.\nTopological insulators are intermediate between metallic and insulating substrates, con-\n2sisting of bulk insulators hosting conducting topologically protected surface states22{24. The\nmagnetic doping of topological insulators breaks time-reversal symmetry and generates ex-\notic phenomena such as the quantum anomalous Hall e\u000bect25,26. In this case, one also expects\na rather low but \fnite hybridization (with the surface state) in the region of the bulk gap,\nleading to unconventional dynamical behaviour. For instance, the magnetization dynam-\nics of a ferromagnet coupled to the surface state of a three-dimensional (3D) topological\ninsulator has already been investigated, and an anomalous behaviour in the ferromagnetic\nresonance was predicted27. Other studies with a similar focus were done in Refs. 28{31.\nFurthermore, arrays of magnetic adatoms interacting with a topological surface state were\nconsidered in Ref. 32, with the surface magnons following a linear dispersion, very unusual\nfor a ferromagnetic ground state. Moreover, the electron spin resonance of single Gd ions\nembedded in Bi 2Se3was examined in Ref. 33. The temperature dependence of the g-factor\nwas investigated and the coexistence of a metallic and an insulating phase (dual character)\nwas reported.\nIn this paper, we systematically investigate the spin dynamics of 3 dand 4dsingle im-\npurities embedded in prototypical 3D topological insulators, namely Bi 2Te3and Bi 2Se3.\nThin \flm (with a topological surface state) and inversion symmetric bulk (insulating) ge-\nometries are considered. For an accurate description of the dynamical electronic properties\nof these impurities, we employ linear response TD-DFT as implemented in the KKR-GF\nmethod17,18,20. We compute the dynamical transverse magnetic susceptibility, which rep-\nresents the magnetic response of the system to frequency-dependent transverse magnetic\n\felds. It incorporates the density of spin excitations and can be connected to ISTS mea-\nsurements34. The spin excitation spectra we obtain reveals astonishing results, with lifetimes\nspanning six orders of magnitude: from picoseconds tomicroseconds for Fe and Mn impu-\nrities embedded in Bi 2Se3, respectively. These contrasting values of the lifetimes correlate\nwith the presence (or absence) of in-gap states in the impurity local density of states (LDOS)\nnear the Fermi energy35. Next we gain further insight on the magnetization dynamics by\nmapping the transverse dynamical magnetic susceptibility to the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation36. A generalized formulation of the LLG equation including\ntensorial Gilbert damping Gand nutationIis employed37. The static limit of the response\nfunction via the LLG formulation was used to extract the MAE. The latter is then compared\nto the values obtained with conventional ground state methods relying on the magnetic force\n3theorem: band energy di\u000berences38{40and torque method41. A connection between the MAE\nobtained within the linear response theory and the torque method using small deviations\nis established. Moreover, for elements with high resonance frequencies, the signature of the\nnutation is observed as a resonance shift, proving that inertial e\u000bects are relevant at such\nhigh precession rates37,42,43. Finally, we compare the LLG parameters obtained when the 3 d\nand 4dimpurities are embedded in the bulk and at the surface of Bi 2Te3. Our results show\nthat the modi\fcation of the in-gap state due to the presence of the surface state may play\na major role in the dynamics depending on the nature of the impurity.\nThis paper is structured as follows. Sec. II is dedicated to the description of the linear\nresponse TD-DFT approach employed to compute the spin excitation spectra. It also in-\ncludes the mapping of the transverse dynamical magnetic susceptibility into the generalized\nphenomenological LLG model and the di\u000berent methods used to compute the MAE. Sec. III\nis devoted to the analysis of the electronic structure and the ground state properties of 3 d\nand 4dtransition metal impurities embedded in Bi 2Te3and Bi 2Se3. In Sec. IV, we present\nthe MAE for the considered magnetic impurities and explain the discrepancies between the\ndi\u000berent methods. Sec. V contains a detailed discussion of the spin excitation spectra of\n3dand 4dimpurities embedded at the surface of both Bi 2Te3and Bi 2Se3. The \ftted LLG\nparameters are given as well, which are interpreted in terms of the impurity LDOS. Finally,\nin Sec. VI, the dynamical properties of the 3 dimpurities in the bulk and at the surface\nare compared. The contribution of the topological surface state for each impurity is then\nanalyzed.\nII. THEORETICAL DESCRIPTION\nThe description of the spin excitations of the investigated systems relies on linear re-\nsponse TD-DFT17,20,21,44. The central quantity in our approach is the dynamical magnetic\nsusceptibility, which displays poles at the excitation energies of the system. The calcula-\ntions are performed in two steps: First we determine the ground state of the system using\nconventional DFT calculations; then, we compute the dynamical response of the system\nto an external perturbing time-dependent magnetic \feld. To gain further physical insights\ninto the results, we also describe how to map the results of TD-DFT calculations onto an\nextended phenomenological LLG model. Lastly, we compare the MAE obtained from the\n4dynamical calculations with the ones computed from DFT calculations in di\u000berent ways.\nA. Density functional theory\nThe ground state DFT simulations are done using the KKR-GF method45,46in the atomic\nsphere approximation (ASA) including the full charge density, and the exchange-correlation\npotential is taken in the local spin density approximation (LSDA)47. The spin-orbit inter-\naction is included in a self-consistent fashion within the scalar relativistic approximation.\nSince we investigate impurities embedded in periodic crystals, we perform two types of cal-\nculations. The ground state of the clean host is determined \frst. Then, the impurities are\nself-consistently embedded in its crystalline structure. The host crystals investigated in this\nwork consist of Bi 2Te3and Bi 2Se3. The bulk unit cell contains \fve atoms (one quintuple\nlayer) in a rhombohedral structure (space group R \u00163m)48. The corresponding self-consistent\ncalculations employ a 30 \u000230\u000230k-mesh. The surface is simulated using a slab containing\nsix quintuple layers and 60 \u000260k-points, as in our previous work35.\nB. Time-dependent density functional theory\nThe dynamical magnetic susceptibility encodes the spin excitation spectra. It describes\nthe linear change in the spin magnetization density \u000e~M(~ r;!) upon the application of a\nfrequency-dependent external magnetic \feld \u000e~B(~ r;!) as\n\u000eM\u000b(~ r;!) =X\n\rZ\nd~ r0\u001f\u000b\r(~ r;~ r0;!)\u000eB\r(~ r0;!); (1)\nwhere\u000b;\r2fx;y;zg. For a speci\fc direction of ~M(~ r), the susceptibility tensor can be\ndivided into longitudinal and transversal blocks. In presence of the spin-orbit interaction\nor magnetic non-collinearity, the two blocks are coupled. However, for the systems that\nwe analyze in this paper, the coupling is negligible and we focus only on the transversal\nmagnetic response of systems (the xyblock when the magnetic moment is along the z-\ndirection). Within TD-DFT, the magnetic susceptibility \u001f\u000b\f(~ r;~ r0;!) is determined starting\nfrom the non-interacting magnetic susceptibility of the Kohn-Sham system, \u001fKS\n\u000b\f(~ r;~ r0;!),\n5using a Dyson-like equation17,20,21:\n\u001f\u000b\f(~ r;~ r0;!) =\u001fKS\n\u000b\f(~ r;~ r0;!) +\nX\n\r\u0016=x;yZ\nd~ r1d~ r2\u001fKS\n\u000b\r(~ r;~ r1;!)Kxc\n\r\u0016(~ r1;~ r2;!)\u001f\u0016\f(~ r2;~ r0;!);(2)\nwhere\u000b;\f;\r;\u00162fx;ygandKxc\n\r\u0016(~ r;~ r0;!) is the transverse part of the exchange-correlation\nkernel, with Kxc\n\r\u0016(~ r;~ r0;!) =\u000e\r\u0016Kxc\n?(~ r;~ r0;!). In the framework of the adiabatic LDA21,49,\nKxc\n?(~ r;~ r0;!) =\u000e(~ r\u0000~ r0) 2Bxc(~ r)=M(~ r) is frequency-independent and local in space. The\ndynamical Kohn-Sham susceptibility is evaluated from the single particle Green function\nG(~ r;~ r0;\") (de\fned in Eq. (B1)) as:\n\u001fKS\n\u000b\f(~ r;~ r0;!) =\u00001\n\u0019Z\"F\n\u00001d\"Trf\u001b\u000bG(~ r;~ r0;\"+!+ i0)\u001b\fImG(~ r0;~ r;\")\n+\u001b\u000bImG(~ r;~ r0;\")\u001b\fG(~ r0;~ r;\"\u0000!\u0000i0)g:(3)\nSince the frequency range of interest is relatively low20,44, the frequency dependence of the\nKohn-Sham susceptibility is incorporated via a Taylor expansion as\n\u001fKS\n\u000b\f(~ r;~ r0;!)\u0019\u001fKS\n\u000b\f(~ r;~ r0;0) +!d\u001fKS\n\u000b\f(~ r;~ r0;!)\nd!\f\f\f\f\f\n!=0+!2\n2d2\u001fKS\n\u000b\f(~ r;~ r0;!)\nd!2\f\f\f\f\f\n!=0:(4)\n\u001fKS\n\u000b\f(~ r;~ r0;0) being the static Kohn-Sham susceptibility. Moreover, for a system with uni-\naxial symmetry, the transversal excitations can be summarized in the spin-\rip magnetic\nsusceptibility20\n\u001f+\u0000(~ r;~ r0;!) =1\n4[\u001fxx(~ r;~ r0;!) + i\u001fxy(~ r;~ r0;!)\u0000i\u001fyx(~ r;~ r0;!) +\u001fyy(~ r;~ r0;!)]:(5)\nFurther details on the computation of the Kohn-Sham susceptibility and exchange-correlation\nkernel can be found in Refs. 17, 20, and 44. Finally, we can obtain an intuitive picture of\nthe spin excitations via the spatial average of \u001f+\u0000(~ r;~ r0;!) over a suitably-de\fned volume\nenclosing the magnetic impurity,\n\u001f+\u0000(!) =Z\nVd~ rZ\nVd~ r0\u001f+\u0000(~ r;~ r0;!); (6)\nwhich corresponds to its net response to a uniform external magnetic \feld20.\nC. Generalized Landau-Lifshitz-Gilbert equation\nIn order to develop a more intuitive picture of the magnetization dynamics, we make a\nconnection with a phenomenological model for the magnetization dynamics. We consider a\n6generalized formulation of the Landau-Lifshitz-Gilbert (LLG) equation36including a tenso-\nrial Gilbert damping G, as well as a nutation tensor Iaccounting for inertial e\u000bects37,50{52.\nThe latter can be important at relatively high frequencies37,42,43. The equation of motion of\nthe magnetic moment ~M(t) =R\nVd~ r~M(~ r;t) then reads\nd~M\ndt=\u0000\r~M\u0002 \n~Be\u000b+G\u0001d~M\ndt+I\u0001d2~M\ndt2!\n: (7)\nHere\ris the gyromagnetic ratio ( \r= 2 in atomic units) and ~Be\u000bis the e\u000bective magnetic\n\feld acting on the magnetic moment. ~Be\u000bcan be split into two contributions: ~Be\u000b=\n~Bext+~Ba, with~Bextbeing the external magnetic \feld, and ~Bais an intrinsic anisotropy\n\feld which arises due to the spin-orbit interaction20. The relation between ~Baand the\nmagnetocrystalline anisotropy energy (MAE) Kis detailed in Appendix A.\nTo establish a connection between the LLG equation and the transverse magnetic sus-\nceptibility computed using Eq. (2), we \frst consider that the local equilibrium direction is\nalong thez-axis and apply a small time-dependent transverse magnetic \feld:\n~Bext(t) =\u000eBx(t)~ ex+\u000eBy(t)~ ey; with\u000eBx(t);\u000eBy(t)\u001cj~Baj: (8)\nThen, we linearize Eq. (7) with respect to transverse components of ~Bext(t) and~M(t), which\nbecomes, in the frequency domain,\nX\n\f=x;y\u0012Ba\nz\nM\u000e\u000b\f+i!\n\rM\u000f\u000b\f+ i!G\u000b\f+!2I\u000b\f\u0013\n\u000eM\f(!) =\u000eB\u000b(!); (9)\nwith\u000f\u000b\fbeing the 2-dimensional Levi-Civita symbol ( \u000fxy= +1) and \u000eM\f(!) the\fcompo-\nnent of the frequency dependent magnetization ~M(!). The preceding equation combined\nwith Eq. (1) provides a direct connection between \u001f\u000b\f(!) obtained within TD-DFT and the\nphenomenological LLG parameters:\n8\n><\n>:(\u001fxx(!))\u00001=\u00002KSusc\nM2\u0000i!\n\rMGs\nk\u0000!2\n\rMIs\nk;\n(\u001fxy(!))\u00001=i!\n\rM(1 +Ga\nk) +!2\n\rMIa\nk;(10)\nwhereKSuscis the MAE, and the subscript indicates that this quantity is extracted from\nthe static magnetic susceptibility obtained from the TD-DFT calculations. Gs\nk(Is\nk) andGa\nk\n(Ia\nk) are the symmetric and anti-symmetric components of the Gilbert damping (nutation)\ntensor, respectively. A more detailed description of the Gilbert damping and nutation tensors\n7for the uniaxial symmetry that applies to the systems under consideration is provided in\nAppendix A. The previous equation shows in a clear fashion that the static limit of \u001fxx(!)\nis inversely proportional to the anisotropy. In the limit of small nutation, the MAE is\nconnected to the resonance frequency !LLG\nresvia (see Appendix A)\n!LLG\nres=\u0000\rq\n1 +\u0000\nGs\nk\u00012+ 2Ga\nk+\u0000\nGa\nk\u000122KSusc\nMs: (11)\nThis is the resonance frequency for precessional motion about the z-axis. Note that !LLG\nres\nis renormalized by Gs\nkandGa\nk, accounting for the damping of the precession and the renor-\nmalization of \r, respectively (see Eq. (A7)).\nD. Magnetocrystalline anisotropy\nIn absence of external magnetic \felds, the gap opening in the spin excitation spectrum is\nuniquely due to the MAE ( i.e.anisotropy \feld) breaking the SU(2) rotational symmetry20.\nThe expression of !LLG\nresin the LLG model provided in Eq. (11) shows that the resonance\nfrequency is proportional to K, which can also be computed from ground state DFT calcula-\ntions. Here, we discuss two di\u000berent ground state methods to compute this quantity relying\non the magnetic force theorem38{40,53and establish a connection with the MAE obtained\nusing linear response theory, Ksusc.\nFor uniaxial systems, the energy depends on the direction of the magnetic moment in\na simple way:E(\u0012)\u0018 K cos2\u0012, where\u0012is the angle that the magnetic moment makes\nwith thez-axis, i.e.~M=j~Mj= ^n(\u0012;') = (cos'sin\u0012;sin'sin\u0012;cos\u0012). To lowest order in\nthe phenomenological expansion, the axial symmetry renders the energy independent of the\nazimuthal angle '. It follows that the magnitude of the MAE, K, can be obtained from\ntotal energy di\u000berences for two di\u000berent orientations of the magnetization (out-of-plane and\nin-plane). However, as Kis at most a few meV's, this approach requires very accurate total\nenergies, which is computationally demanding.\nAlternatively, one can use the magnetic force theorem , which states that, if the changes\nin the charge and magnetization densities accompanying the rotation of the spin moment\nare small, the total energy di\u000berence can be replaced by the band energy di\u000berence38{40:\nKBand=EBand(0\u000e)\u0000E Band(90\u000e); (12)\n8whereEBand(\u0012) is the band energy (sum of Kohn-Sham energy eigenvalues) of the system\nwhen the spin moment makes an angle \u0012with thez-axis:\nEBand(\u0012) =Z\"F\n\u00001d\"(\"\u0000\"F)\u001a(\";\u0012): (13)\nIt contains the e\u000bect of the orientation of the magnetic moment through how the density of\nstates\u001a(\";\u0012) is modi\fed upon its rotation. This quantity is evaluated with a single non-self-\nconsistent calculation, by orienting the exchange-correlation magnetic \feld in the desired\ndirection,~Bxc(~ r) =Bxc(~ r) ^n(\u0012;') (rigid spin approximation54).\nThe MAE can also be evaluated from the magnetic torque, which corresponds to the \frst\nderivative ofEBand(\u0012) with respect to the magnetic moment direction. Using the Hellman-\nFeynman theorem, the torque reads41,55,56:\nT\u0012=@EBand\n@\u0012;\n=Z\nd~ rB xc(~ r)@^n(\u0012;')\n@\u0012\u0001~M(~ r;\u0012):(14)\nAs for the band energy calculations, the torque is also obtained from a single non-self-\nconsistent calculation, under the same approximations. It is non-vanishing if the output spin\nmagnetization density ~M(~ r;\u0012) is not collinear with the input magnetic moment direction.\nConsidering the expected form of the MAE for uniaxial symmetry, we should \fnd\nT\u0012=\u0000K Torque sin(2\u0012): (15)\nIn practice, the torque can be evaluated at di\u000berent angles \u0012. In this work, two deviation\nangles have been considered: a large deviation angle with \u0012= 45\u000e, as done in Ref. 41, and a\nsmall one near self-consistency, \u0012= 5\u000e. For such small deviations, one can connect KTorque\nto the value of the MAE obtained from the magnetic susceptibility, KSusc. It is shown in\nAppendix B that when considering a small rotation angle \u0012and a constant magnitude of\nthe exchange-correlation spin-splitting (frozen potential approximation),\nKSusc=KTorque\n1\u00004\u001fKS\n+\u0000(0)KSusc\nM2z;\n\u0018KTorque\n1 +Ba\nBxc:(16)\nThe previous expression shows that Ksusccorresponds to the KTorque (evaluated for a small\ndeviation angle) renormalized by a prefactor (1 +Ba\nBxc)\u00001. In fact, this result is similar to the\n9\u000010\u000050510LDOS (States/eV)CrMnFeCo\n\u00006\u00004\u000020246LDOS (States/eV)NbMoTc\n\u00005\u00004\u00003\u00002\u00001012\"\u0000\"F(eV)\u00006\u00004\u000020246LDOS (States/eV)NbMoTcRuPd\n\u00005\u00004\u00003\u00002\u00001012\"\u0000\"F(eV)\u000010\u000050510LDOS (States/eV)CrMnFeCo(a)(b)\n(c)(d)Bi2Te3surface\nBi2Se3surfaceBi2Te3surface\nBi2Se3surface3d4dFIG. 1. Spin-resolved LDOS for 3 dimpurities (Cr, Mn, Fe and Co) and 4 dimpurities (Nb, Mo,\nTc, Ru, Pd) embedded in a Bi 2Te3(Bi2Se3) surface. (a) 3 din Bi 2Te3, (b) 4din Bi 2Te3, (c) 3din\nBi2Se3and (d) 4din Bi 2Se3. The full lines represent the majority-spin states, with dashed lines\nfor the minority-spin ones. The energies are given with respect to the Fermi energy \"Fand the\nenergy window associated with the bulk band gap is highlighted with light blue color.\nrenormalization observed for magnetic interactions computed from the magnetic suscepti-\nbility57,58. For the systems of interest (3 dand 4dtransition metals impurities), Bais in the\nmeV range while Bxcis in the order of eV. Therefore, one expects small corrections due to\nthis renormalization, and the two quantities should be in good agreement.\n10III. ELECTRONIC STRUCTURE OF 3dAND 4dIMPURITIES IN Bi 2Te3AND\nBi2Se3\nIn this section, we brie\ry recap the discussion of the electronic structure and ground\nstate properties of 3 dimpurities embedded in the Bi 2Te3(Bi2Se3) surface already addressed\nin Ref. 35. Furthermore, we also consider 4 dimpurities which have a stronger hybridization\nwith the host electrons compared to the 3 dones. This information will be employed for\nthe analysis of their dynamical properties, such as the Gilbert damping. The LDOS of 3 d\nand 4dmagnetic impurities embedded into Bi 2Te3and Bi 2Se3(111) surfaces are shown in\nFig. 1. The bulk band gap (\u0001 gap) is depicted in light blue | with \u0001 gap\u00190:25 eV for Bi 2Te3\nand \u0001 gap\u00190:35 eV for Bi 2Se335. We consider that the impurity spin moment is oriented\nperpendicularly to the surface ( i.e.along the [111] direction). The full lines represent the\nmajority spin channel ( \"), while the dashed lines account for the the minority spin channel\n(#). All the 3 dand 4dimpurities donate electrons to the host atoms (see Table I). It can\nalso be seen in Fig. 1 that the spin splitting of the 4 dimpurities is weaker compared to the\n3dones, resulting in smaller spin moments, as listed in Table I. This is attributed to the\nStoner parameter being larger for 3 dthan for 4delements59.\nAll 3delements except Cr display a completely \flled majority-spin d-resonance. Mn and\nCr have a nearly-empty minority-spin d-resonance, resulting in a large spin moment and\nCr Mn Fe Co Nb Mo Tc Ru Pd\nQBi2Te35.154 6.160 7.282 8.448 3.488 4.717 5.892 7.147 9.421\nBi2Se34.841 5.863 6.963 8.136 3.077 4.316 5.474 6.734 9.041\nMsBi2Te33.843 4.412 3.395 2.108 1.097 2.678 2.493 0.000 0.000\nBi2Se33.671 4.421 3.482 2.231 0.906 2.574 2.534 0.564 0.578\nMlBi2Te30.065 0.050 0.260 0.883 -0.143 -0.004 0.202 0.000 0.000\nBi2Se30.008 0.024 0.144 0.942 -0.048 -0.093 0.079 0.378 0.135\nTABLE I. Ground state properties of 3 dand 4dimpurities embedded in the Bi 2Te3and Bi 2Se3\nsurfaces including: the valence charge on the impurity Q, spin moment Msand orbital moment\nMl. The spin and orbital moments are given in units of \u0016B.\n11a small orbital moment ( Ml). Fe and Co have a partially-\flled minority-spin d-resonance,\nleading to higher values for Ml, as shown in Table I. The LDOS also reveals impurity-induced\nin-gap states near the Fermi energy, which arise from the hybridization with the bulk sp\nstates of Bi 2Te3(Bi2Se3)35. When replacing the Bi 2Te3host by Bi 2Se3, the valence charge\nand the spin moment are mildly a\u000bected, in contrast to the orbital moments which are\nconsiderably altered35.\nFor 4dimpurities, both minority- and majority-spin d-resonances are partially occupied\ndue to a weak spin-splitting. The LDOS is broader and \ratter in comparison with the\n3dones, indicating a stronger hybridization with the host material, as the 4 d-orbitals are\nspatially more extended than the 3 dones, and so overlap more with the orbitals of the\nhost. In the Bi 2Te3host, Nb, Mo and Tc are found to be magnetic, while Ru, Rh and\nPd impurities were found to be nonmagnetic. The analysis of the paramagnetic LDOS (not\nshown here) reveals that, when moving in the periodic table from Tc towards Pd ( i.e.adding\nelectrons), the 4 dpeak is shifted to lower energies. This leads to a drastic decrease of the\nLDOS at\"Fand makes the Stoner criterion unful\flled. Nb has a less than half-\flled d-shell,\ninducing an orbital moment anti-parallel to its spin moment, as shown in Table I. For Mo\nand Tc, a half \flled d-shell results in the highest values for Msbetween the 4 delements.\nThese observations are in qualitative agreement with Hund's rules60. In-gap states are also\nobserved near \"F, as for the 3 dimpurities. Interestingly, in the Bi 2Se3host, Ru and Pd\nacquire a magnetic moment, while Rh remains nonmagnetic. Higher values of the LDOS at\n\"Fcompared to the Bi 2Te3host now satisfy the Stoner criterion for these elements. Pd is a\nrather peculiar case, since the increase of the LDOS at \"Fis related to the presence of an\nin-gap state in the minority-spin LDOS, as shown in Fig. 1d.\nThe electronic structure, especially in the vicinity of the Fermi energy, governs the be-\nhaviour of the MAE and spin excitations of the system. In particular, the presence of\nd-resonances near \"Fmay result in inaccuracies in the computation of the MAE. Together\nwith in-gap states, it can also induce high values of the Gilbert damping, as discussed in the\nnext sections.\n12IV. MAGNETOCRYSTALLINE ANISOTROPY OF 3dAND 4dIMPURITIES IN\nBi2Te3AND Bi 2Se3\nWe now investigate the MAE employing the di\u000berent methods discussed in Sec. II D. In\nour convention, a positive (negative) MAE stands for an in-plane (out-of-plane) easy-axis.\nIn Fig. 2a, we show the evolution of the MAE for 3 dimpurities embedded in Bi 2Te3and\nBi2Se3, respectively. For every impurity, all the methods predict the same easy-axis. In the\nBi2Te3host, Cr and Fe present an in-plane magnetic anisotropy, while Mn and Co favor an\nout-of-plane orientation. The trend is mostly accounted for by Bruno's formula61, where the\nMAE is given by the anisotropy of the orbital moment ( Ml):K/\u00102(Mx\nl\u0000Mz\nl), with\u0010\nbeing the spin-orbit interaction strength. Mn displays a small MAE, as it has a small orbital\nmoment, while the large anisotropy energies obtained for Fe and Co stem both from their\nlarge orbital moments and their substantial dependence on the spin orientation. However,\nthe results obtained for the MAE of Cr do not agree with the predictions of Bruno's formula,\nsince the MAE reaches \u00181 meV, despite a rather small anisotropy in the orbital moment of\nthe adatom (see Table. II). For the Bi 2Se3host, the anisotropy follows very similar trends in\ncomparison with the Bi 2Te3case. Nonetheless, the easy axis of Cr switches from in-plane to\nout-of-plane, while the MAE of Fe and Co present a noticeable increase, as shown in Fig. 2a.\nThese changes in the MAE are attributed to the modi\fcation of the ground state properties,\nparticularly the orbital moments (as listed in Table II), according to Bruno's formula.\nIn Fig. 2b, we show the MAE of 4 dimpurities embedded in Bi 2Te3and Bi 2Se3computed\nwith the di\u000berent approaches outlined in Section II D. For the Bi 2Te3case, all the impurities\n(Nb, Mo and Tc) display an in-plane easy-axis. Nb displays a large MAE, while Mo and\nTc have a rather small one (with the exception of KTorque (45\u000e) andKBand). For Mo, the\nsmall MAE correlates with its small orbital moment. In the Bi 2Se3host, Nb, Mo, and Tc\nare characterized by an in-plane easy-axis as well. Note that, due to a strong hybridization\nwith the host (broad LDOS in Fig. 1b and d), the MAE of Tc is drastically a\u000bected by the\nsurrounding environment. Ru and Pd acquire a magnetic moment in Bi 2Se3displaying an\nout-of-plane easy-axis. Particularly, Ru displays a very large MAE in comparison with the\nrest of the 4 delements.\nWe now focus on the reasons why di\u000berent methods may provide contrasting values for\nthe MAE (see Fig. 2). The origin of these divergences can be traced back to the features of\n13the electronic structure at the impurity site. Fig. 2a shows that the obtained MAE energies\nof Fe and Co can be separated in two groups, according to the method used to compute them:\nOne for large angle methods, including the band energy di\u000berences ( KBand [Eq. (12)]) and\nthe torque method at 45\u000e(KTorque (45\u000e) [Eq. (14)]); and the other for small perturbations,\nencompassing the torque method at 5\u000e(KTorque (5\u000e)[Eq. (14)]) and linear response theory\n(KSusc[Eq. (10)]). The results from the two methods in each group are in good agreement\nwith each other, but the results from one group do not agree with those from the other.\nThis can be understood via Table II, which lists the change in the ground state properties\nof the impurity upon 90\u000erotation of the spin moment ( z!xaxis), in a frozen potential\ncalculation. There is a large variation in the valence charge and in the spin moment of Fe\nand Co in comparison to Cr and Mn, owing to the change in the position of the 3 dpeak\nin the minority spin channel in the vicinity of \"F(see Fig. 1a and 1c). This violates the\nassumptions justifying the magnetic force theorem (in the frozen potential approximation),\nas previously observed in Ref. 62 for Co adatoms deposited on a Cu(111) surface. The\ndisagreement between the di\u000berent methods for Tc and Ru observed in Fig. 2b is attributed\nto a high occupation at \"Fas well (see Fig. 1b and 1d). An exception occurs for Nb, where\ngood agreement between the di\u000berent methods is observed. In this case, the high LDOS at\nCr Mn Fe Co Nb Mo Tc Ru Pd\n\u0001QzxBi2Te3-0.016 0.001 -0.224 -0.484 0.018 0.002 -0.287 0.000 0.000\nBi2Se3-0.001 0.000 -0.320 -0.583 -0.004 0.001 -0.319 -0.347 0.000\n\u0001Mzx\nsBi2Te3-0.016 -0.001 0.224 0.483 0.0147 -0.000 0.288 0.000 0.000\nBi2Se3-0.001 -0.000 0.320 0.582 -0.009 0.001 0.286 0.320 -0.003\n\u0001Mzx\nlBi2Te30.019 0.003 -0.323 0.484 -0.081 -0.002 -0.188 0.000 0.000\nBi2Se30.003 0.002 -0.493 0.487 -0.261 0.003 -0.284 0.285 0.008\nTABLE II. Change in the valence charge of the impurity \u0001 Qzx, spin moment \u0001 Mzx\nsand orbital\nmoment \u0001Mzx\nlfor 3dand 4dimpurities embedded in a Bi 2Te3and a Bi 2Se3surface, using the\nfrozen potential approximation. For Fe and Co, \u0001 Qzxand \u0001Mzx\nsare relatively large, invalidating\nthe use of the magnetic force theorem to compute the MAE.\n14 j.bouaziz@fz-juelich.de                                  1RuNbMoTcNbMoTcRuPd\u000010\u000050510MAE (meV)Bi2Te3Bi2Se3\nKBandKSuscKTorque(45\u0000)KTorque(5\u0000)CrMnFeCoCrMnFeCo\u000010\u000050510MAE (meV)Bi2Te3Bi2Se3\nKBandKSuscKTorque(45\u0000)KTorque(5\u0000)\n(a)(b)FIG. 2. Comparison of the MAE for (a) 3 dimpurities and (b) 4 dimpurities, embedded in a\nBi2Te3and a Bi 2Se3surface. The black curve is obtained using the band energy di\u000berences ( KBand\n[Eq. (12)]) (with a 90\u000erotation of the spin moment). The red curve shows the MAE computed\nfrom the static part of the magnetic susceptibility ( KSusc[Eq. (10)]). The green and blue curves\nare obtained using the torque method at 45\u000eand 5\u000e(KTorque (\u0012)[Eq. (14)]), respectively. Most of\nthe impurities display an in-plane magnetic anisotropy ( K>0).\n\"Fis due to the majority spin states, which are weakly a\u000bected by the spin rotation.\nThe previous analysis indicates that, if a high density of electronic states is present at\n\"F(Fe, Co, Tc and Ru), a large rotation angle may lead to large changes in the charge\ndensity and invalidate the use of the magnetic force theorem in combination with the frozen\npotential approximation. Therefore, a small deviation angle, for which the system remains\nnear self-consistency, should be considered. This can be achieved through the torque method\nor the magnetic susceptibility. The MAE obtained in these cases ( KTorque (5\u000e) andKSusc)\nshould be comparable with the one extracted for inelastic scanning tunneling spectroscopy\nmeasurements, since in such experiments the deviation of the magnetic moment from the\neasy-axis are rather small.\nV. SPIN EXCITATIONS OF 3dAND 4dIMPURITIES IN Bi 2Te3AND Bi 2Se3\nIn Sec. III, we addressed the ground state properties of 3 dand 4dimpurities embedded in\nBi2Te3and Bi 2Se3. Here, we investigate their spin dynamics, relate it to the MAE obtained\nin Sec. IV, and study the possibility of exciting and manipulating these impurities with\n150510152025!(meV)0.00.20.40.60.81.0\u00001⇡\u0000+\u0000(!) (states/meV)NbMoTc\n020406080100!(meV)0.000.010.020.030.040.05\u00001⇡\u0000+\u0000(!) (states/meV)TcRuPd05101520!(meV)0.00.51.01.52.02.53.0\u00001⇡\u0000+\u0000(!) (states/meV)CrMnFeCo\n05101520!(meV)0246810\u00001⇡\u0000+\u0000(!) (states/meV)CrMnFeCo(a)(b)\n(c)(d)0.100.150.20!(meV)0123\n012!(meV)012345\n0.00.20.4!(meV)0246810024!(meV)0.00.20.40.60.81.03d4dFIG. 3. Density of states of transverse spin excitations for magnetic impurities. The panels\nshow the results for (a) 3 dand (b) 4dimpurities embedded in Bi 2Te3, and (c) 3 dand (d) 4d\nimpurities embedded in Bi 2Se3. They present an almost-Lorentzian, with resonances located at\nthe excitation energies of the system. The dashed lines mark the resonance frequency without\ndynamical corrections, !0\nres=\u00002\rKSusc\nMs. For Mn, Co, Ru and Pd, \u001f\u0000+(!) is plotted instead, to\naccount for their easy-plane MAE.\ntime-dependent external magnetic \felds. We focus on the transverse spin excitations en-\ncoded in the dynamical magnetic susceptibility, which have been observed experimentally for\nmagnetic impurities on nonmagnetic surfaces by means of ISTS measurements3,8,11,63{65. In\nthese experiments, the spin excitations yield a step in the di\u000berential tunneling conductance\nat well-de\fned energies.\nWe show in Fig. 3 the imaginary part of \u001f+\u0000(!) (i.e.the density of states of the magnetic\nexcitations) as function of the frequency of the external \feld for both 3 dand 4dimpurities\n16embedded in Bi 2Te3and Bi 2Se3. Only the response of the magnetic impurities is considered,\nsince the induced moments in the surrounding (host) atoms are rather small. Nonetheless,\ntheir contribution is accounted for when computing the transverse exchange-correlation ker-\nnelKxc\n?at the impurity site via the spin-splitting sum rule17,20. The LLG parameters ob-\ntained by \ftting the data to Eq. (10) are given in Table III. As depicted in Fig. 3, Im \u001f+\u0000(!)\nhas a Lorentzian-like shape, and the resonance frequency ( !res) is \fnite even in absence of\nan external magnetic \feld. This resonance arises from the MAE, which breaks the SU(2)\nrotational symmetry ( i.e.no Goldstone mode), as explained previously in Sec. II D. The\nhighest resonance frequencies are obtained for Nb and Ru due to their strong anisotropy\ncombined with a small magnetic moment complying with Eq. (11), while the smallest value\nof!resis obtained for Mn impurities in Bi 2Se3. The dashed lines in Fig. 3 represent the\nresonance position obtained neglecting dynamical corrections in Eq. (11), leading to the es-\ntimate!0\nres=\u00002\rKSusc\nMs(with\r= 2 andG= 0)20. There is a qualitative agreement between\n!0\nresand the resonance position extracted from the spin excitation spectra, !res, including\ndamping and nutation. Nonetheless, their values are quantitatively di\u000berent, illustrating\nthat dynamical corrections can be of crucial importance for an accurate determination of\nthe resonance frequency.\nAnother quantity which is strongly dependent on the nature of the impurity and the\nhost is the full width at half maximum (FWHM) \u0000. This quantity is proportional to the\nsymmetric part of the Gilbert damping tensor ( Gs\nk) and provides information about the\nlifetime of the excitations66as\u001c=2\n\u0000. This lifetime ranges from picoseconds (comparable\nto lifetimes obtained at metallic surfaces20,66) to very high values reaching microseconds for\nMn in Bi 2Se3as shown in Fig. 4. Furthermore, the values of Gs\nk, shown in Table III, can be\ninterpreted in terms of the LDOS at \"F, sinceGs\nk/n#(\"F)n\"(\"F) (wheren#(\") andn\"(\")\nrepresents the LDOS of the minority and majority spin channels, respectively)44. The highest\nvalues ofGs\nkare obtained for Ru, which coincide the lowest excitation lifetime as displayed\nin Fig. 4. The anti-symmetric part of the Gilbert damping tensor Ga\nkis also displayed in\nTable III. It accounts for the renormalization of the gyromagnetic ratio, \re\u000b=\r\n1+Ga\nk(see\nAppendix A). This renormalization is attributed to the presence of a \fnite LDOS at \"Fas\nwell44.Ga\nkis negative for Cr, Nb and Ru indicating an enhancement of the gyromagnetic\nratio ( i.e.\re\u000b>2), while\re\u000b<2 for the remaining impurities. Note that the spin excitation\nspectra of Nb and Mo impurities in Bi 2Se3is not shown in Fig. 3, since for these elements\n17the Taylor expansion shown in Eq. (4) fails due to contributions from higher order terms in\nfrequency becoming too large.\nCr Mn Fe Co Nb Mo Tc Ru Pd\nMsBi2Te33.844 4.412 3.395 2.109 1.097 2.678 2.493 | |\nBi2Se33.671 4.421 3.482 2.231 0.906 2.574 2.534 0.564 0.578\nGs\nkBi2Te30.019 0.000 0.143 0.164 0.053 0.000 0.172 | |\nBi2Se30.037 0.000 0.112 0.012 0.003 0.000 0.512 0.852 0.094\nGa\nkBi2Te3-0.245 0.109 0.286 0.274 -0.087 0.096 0.099 | |\nBi2Se3-0.153 0.101 0.125 0.196 -0.021 0.134 0.081 -0.396 1.824\n!cBi2Te377.68 3439 135.7 277.4 21.91 224.5 31.64 | |\nBi2Se3283.2 1340 100.4 73.37 2.784 403.5 4.481 10.11 437.0\n\u0011cBi2Te37.154 298.3 65.66 38.39 30.36 752.2 234.4 | |\nBi2Se330.97 17820 76.31 40.19 8.703 171.5 84.93 341.8 502.5\nKSuscBi2Te30.959 -0.201 4.302 -6.725 4.091 0.417 0.353 | |\nBi2Se30.090 0.005 6.019 -5.894 5.453 0.102 3.845 -8.178 -0.431\n!LLG\nresBi2Te31.322 0.164 3.917 9.926 16.31 0.568 0.509 | |\nBi2Se30.115 0.004 6.113 8.833 24.08 0.158 5.073 55.49 1.055\n!LLG\nres\n!cBi2Te30.017 0.000 0.029 0.036 0.744 0.003 0.016 | |\nBi2Se30.000 0.000 0.063 0.125 8.836 0.000 1.132 5.487 0.002\nTABLE III. LLG parameters for 3 dand 4dimpurities embedded in the surface of Bi 2Te3(Bi2Se3),\nobtained by \ftting the TDDFT dynamical susceptibility data to Eq. (10). Msis the spin moment\nof the impurity.Gs\nkis the symmetric part and Ga\nkis the antisymmetric part of the damping tensor,\nboth unitless.KSuscis the MAE obtained from the magnetic susceptibility, in meV. !LLG\nresis the\nresonance frequency without including nutation, in meV, as de\fned in Eq. (11). A large ratio\nbetween!LLG\nresand!c=Ga\nk\nIs\nkindicates that the nutation makes a substantial contribution to !res,\nwhile\u0011c=Gs\nk\nIa\nkprovides information on the contribution of the nutation to the damping of the spin\nexcitation. Ru and Pd in Bi 2Te3were found to be nonmagnetic, so the corresponding entries are\nmarked with a dash.\n18Cr Mn Fe Co Nb Mo Tc Ru Pd10\u00001210\u0000910\u00006⌧(s)3d in Bi 2Te3\n4d in Bi 2Te3\n3d in Bi 2Se3\n4d in Bi 2Se3FIG. 4. Excitation lifetime of 3 dand 4dmagnetic impurities embedded in Bi 2Te3and Bi 2Se3. Note\nthat the lifetime axis is on a logarithmic scale. The highest excitation lifetime is obtained for Mn\nin Bi 2Se3and reaches microseconds, while the lowest one is obtained for Ru. Elements without\ndata were found to be nonmagnetic in the respective hosts.\nThe importance of the nutation can be estimated from the real part of the denominator\nof Eq. (A5). Both damping and nutation terms, Ga\nk!andIs\nk!2, contribute to the resonance.\nWhen it occurs at frequencies higher than !c=Ga\nk\nIs\nk,!rescan be substantially a\u000bected by the\nnutation. The ratio between !LLG\nresobtained using Eq. (11) (without including nutation) and\n!c(shown in Table III) is employed to evaluate the importance of this contribution. The\nsymmetric parts of the Gilbert damping and nutation tensors can be also related via43,67\nIs\nk/Gs\nk,i.e.the damping and nutation coe\u000ecients are proportional. The ratio !cis fairly\nsmall for the majority of the elements, indicating that nutation has no signi\fcant impact\non the resonant spin precession. However, for some elements such as Nb and Tc (in Bi 2Se3)\nthe nutation leads to a shift of \u00181:3 and 0:4 meV in the resonance frequency, respectively.\nFinally, the most striking element is once again Ru, with a shift of the resonance frequency\nfrom!LLG\nres= 55:49 to!res= 25:52 due to the nutation.\nVI. SURFACE AND BULK SPIN DYNAMICS\nWe now compare di\u000berent cases of 3 dand 4dmagnetic impurities embedded in a surface\nand in a bulk inversion symmetric Bi 2Te3(i.e.insulating phase with no topological surface\nstate). This enables us to disentangle the surface and bulk contributions to the spin dy-\n19namics. The analysis of the ground state properties of the 3 dimpurities embedded in bulk\nBi2Te3is given in Ref. 35. The impurity-induced electronic in-gap states are also present\nin 4dimpurities embedded in bulk Bi 2Te3. The LLG parameters obtained in the bulk (de-\nnoted with a subscript \\b\") and at the surface (denoted with a subscript \\s\") are displayed\nin Table IV. With the exception of Mn, the MAE obtained from the susceptibility di\u000bers\nconsiderably between the bulk and surface cases | Cr even has its easy-axis switched. The\noverall change in the MAE is a decrease from the surface to the bulk cases. The immediate\nenvironment of the embedded impurities is the same in bulk and at surface. However, for\nthe bulk case, the missing contribution of the surface state leads to modi\fcations in the\nelectronic structure, altering the virtual bound and the in-gap states35. This results in a\nreduction of the MAE. The spectral weight at the Fermi level is also a\u000bected leading to a\nmodi\fcation of the damping parameter44. For Cr, Fe and Tc, Gs\nkdecreases, while for Co,\nNb and Mo, it increases. Ga\nkfollows similar trends as in the surface case. Co and Nb are\nthe exception since Ga\nkswitches sign, resulting in a change of \re\u000b. The nutation is negligible\nfor most of elements, except for Nb and Co | for the latter, it leads to a noticeable shift\nof the resonance frequency from !LLG\nres= 4:24 meV to!res= 4:68 meV. In summary, Co and\nNb impurities are very sensitive to the the presence of the surface state, where the impurity\nstates display rather di\u000berent behaviours in the bulk and at the surface leading to a di\u000berent\nspin excitational nature. In contrast, Mn impurities have a similar behavior in the bulk and\nat the surface, showing that the topological surface state plays a negligible role for their spin\ndynamics.\nVII. CONCLUSIONS\nIn this paper, we employed a \frst-principles approach for the investigation of the spin\nexcitation spectra of 3 dand 4dimpurities embedded in two prototypical topological insu-\nlators, namely Bi 2Te3and Bi 2Se3. The simulations were carried out using linear response\nTD-DFT in the framework of the KKR-GF method, suitable for computing the properties\nof spin excitations at the nanoscale. A mapping onto a generalized LLG model allowed to\nextract from \frst-principles the MAE and transversal components of the Gilbert damping\nand nutation tensor. The obtained values of the MAE were then compared systematically\nto the ones obtained using the torque method and band energy di\u000berences, that rely on the\n20MsGs\nkGa\nk!c\u0011cKSusc!LLG\nres!LLG\nres\n!c\nCrs 3.844 0.018 -0.245 77.68 7.154 0.959 1.322 0.017\nCrb 3.823 0.004 -0.215 332.6 47.48 -0.824 1.090 0.003\nMns 4.412 0.000 0.109 3439 298.4 -0.201 0.164 0.000\nMnb 4.335 0.000 0.118 860.7 590.4 -0.216 0.178 0.000\nFes 3.395 0.143 0.286 135.7 65.66 4.302 3.917 0.029\nFeb 3.294 0.045 0.234 58.98 20.87 3.055 3.004 0.053\nCos 2.109 0.164 0.274 277.4 38.39 -6.725 9.926 0.037\nCob 1.977 0.307 -0.011 1.015 56.09 -2.168 4.237 4.174\nNbs 1.097 0.053 -0.087 21.91 30.36 4.091 16.31 0.769\nNbb 0.740 0.314 0.049 10.59 488.5 1.028 5.074 0.479\nMos 2.678 0.000 0.096 224.5 752.2 0.417 0.568 0.003\nMob 2.527 0.012 0.151 323.9 1083 0.454 0.624 0.002\nTcs 2.493 0.172 0.099 31.64 234.4 0.353 0.509 0.016\nTcb 2.057 0.059 0.072 12.67 29.32 0.755 1.368 0.111\nTABLE IV. LLG parameters for 3 dand 4dimpurities embedded in the surface (subscript s) and\nin the bulk (subscript b) of Bi 2Te3, obtained by \ftting the TDDFT dynamical susceptibility data\nto Eq. (10). Msis the spin moment of the impurity. Gs\nkis the symmetric part and Ga\nkis the\nantisymmetric part of the damping tensor, both unitless. KSuscis the MAE obtained from the\nmagnetic susceptibility, in meV. !LLG\nresis the resonance frequency without including nutation, in\nmeV, as de\fned in Eq. (11). A large ratio between !LLG\nresand!c=Ga\nk\nIs\nkindicates that the nutation\nmakes a substantial contribution to !res, while\u0011c=Gs\nk\nIa\nkprovides information on the contribution\nof the nutation to the damping of the spin excitation. The MAE and the Gilbert damping are\nconsiderably a\u000bected when going from surface to bulk. The largest changes occur in the case of\nthe Co impurity.\nmagnetic force theorem and the frozen potential approximation.\nAll the considered 3 dimpurities acquire a \fnite magnetic moment in both hosts, while\nthe strong hybridization of the 4 dimpurities with the host states makes them more sensitive\n21to the surrounding environment. For instance, Ru and Pd were found to be nonmagnetic\nin Bi 2Te3but became magnetic in Bi 2Se3. Furthermore, and independently from nature of\nthe orbitals (3 dor 4d), large rotation angles result in signi\fcant changes in the electronic\nproperties when a high electronic density of states is found at the Fermi energy, invalidating\nthe assumptions made to invoke the magnetic force theorem. The MAE must be then\ncomputed employing perturbative methods such as linear response theory or the torque\nmethod with small deviation angles. The MAE obtained using linear response theory is found\nto coincide with the one computed from the torque method di\u000bering only by a negligible\nrenormalization factor.\nThe spin excitation spectra of the impurities displays diverse trends. When the impurity\nvirtual bound states or in-gap states are located away from the Fermi energy, the Gilbert\ndamping is rather low and the lifetime of the excitation reaches high values compared to\nthe ones observed in metallic hosts20,66. The most striking example is a Mn impurity in\nBi2Se3, where the lifetime reaches microseconds . A contrasting situation is observed for Ru,\nwhich displays a \rat excitation resonance in conjunction with a low lifetime. Moreover, we\nfound that nutation e\u000bects can be important and lead to important shifts of the resonance\nfrequency for some elements such as Nb, Tc and Ru. Moreover, we examined the contribution\nof the surface state to the spin dynamics by comparing the LLG parameters of the impurities\nembedded in the surface with those of impurities embedded in the bulk. For Co and Nb\nimpurities, it was found that the topological surface state has a drastic impact on the\ndynamics via the spectral shift of the impurity-induced electronic in-gap states, while it\nplays a minor role for Mn impurities.\nWe provided a systematic investigation of the spin dynamics of 3 dand 4dimpurities\nembedded in topologically insulating hosts. The results obtained for excitation lifetimes of\nsome speci\fc impurities (Mn) provide insights on the dual (metal and insulator) nature of\nthese materials. In addition to that, the MAE computed employing perturbative methods\nsuch as the linear response can be compared to the one extracted from ISTS measure-\nments. Finally, several aspects remain to be uncovered from \frst principles: the zero-point\nspin \ructuations60of these impurities, which can be accessed via the dynamical magnetic\nsusceptibility, as well the spin dynamics of magnetic nanoclusters or full magnetic layers\ndeposited on topological insulators.\nAcknowledgements We thank Dr. Julen Iba~ nez-Azpiroz for fruitful discussions. This\n22work was supported by the European Research Council (ERC) under the European Union's\nHorizon 2020 research and innovation programme (ERC-consolidator grant 681405 DYNA-\nSORE). We gratefully acknowledge the computing time granted by the JARA-HPC Ver-\ngabegremium and VSR commission on the supercomputer JURECA at Forschungszentrum\nJ ulich.\nAppendix A: Phenomenological parameters from the generalized Landau-Lifshitz-\nGilbert equation\nIn this Appendix, we provide the explicit forms of the phenomenological quantities\n(anisotropy \feld, damping and nutation tensors) discussed in section II C. First, we es-\ntablish a connection between the anisotropy \feld ~Baand the magnetocrystalline anisotropy\nusing the phenomenological form of the band energy EBand. For ease of connection with\nthe LLG, we present the derivation using a vector formalism. For systems with uniaxial\nsymmetry, the expansion of the band energy in terms of the magnetization up to second\norder reads41\nEBand=E0(j~Mj) +K\nM2(~M\u0001~ en)2+::: : (A1)\nE0(j~Mj) contains the isotropic energy contributions and ~ enrepresents the direction of the\neasy-axis. The anisotropy \feld is then given by the \frst order derivative of EBandwith respect\nto~M(the longitudinal component does not a\u000bect the dynamics within the LLG):\n~Ba=\u0000@EBand\n@~M;\n=\u00002K\nM2(~M\u0001~ en)~ en:(A2)\nSecond, the Gilbert damping ( G) and nutation (I) tensors shown in section II C are rank-\n2 tensors, which can be split into a symmetric part (labeled with the superscript s) and\nan anti-symmetric part (labeled with the superscript a). Moreover, due to the uniaxial\nsymmetry, the Gilbert damping tensor has the following structure:\nG=\u00001\n\rM0\nBBB@Gs\nk\u0000Ga\nkGa\n?\nGa\nkGs\nk\u0000Ga\n?\n\u0000Ga\n?Ga\n?Gs\n?1\nCCCA: (A3)\nThe symbolkdenotes the spin dynamics parameters describing the transverse components\nof the precessional motion when the spin moment is along the [111] direction in its ground\n23state. As the system has uniaxial symmetry, the spin dynamics can be anisotropic, and we\nintroduce the symbol ?to account for this possibility. The nutation tensor has the same\nstructure:\nI=\u00001\n\rM0\nBBB@Is\nk\u0000Ia\nkIa\n?\nIa\nkIs\nk\u0000Ia\n?\n\u0000Ia\n?Ia\n?Is\n?1\nCCCA: (A4)\nThe previous decomposition of Gilbert damping and nutation tensors is identical to the\none performed on magnetic exchange interactions68,69. The trace of the the damping tensor\ncoincides with the conventional Gilbert damping constant for a cubic system36, while the\no\u000b-diagonal components account for the renormalization of \r, which controls the precession\nrate. Considering the previous forms for the Gilbert damping and nutation combined with\nEqs. (9) and (5), the spin-\rip dynamical magnetic susceptibility obtained from the LLG\nequation reads then:\n\u001fLLG\n+\u0000(!) =1\n2M\r\n\u00002K\r\nM\u0000(1 +Ga\nk+iGs\nk)!+ (\u0000Is\nk+iIa\nk)!2: (A5)\nThe resonance frequency is de\fned as@Im\u001fLLG\n+\u0000(!)\n@!\f\f\n!LLGres= 0. In absence of nutation, it can be\ncomputed analytically and is given by:\n!LLG\nres=\u0000\rq\n1 +\u0000\nGs\nk\u00012+ 2Ga\nk+\u0000\nGa\nk\u000122Ksusc\nMs: (A6)\nThe latter can be written in terms of the e\u000bective gyromagnetic ratio as:\n!LLG\nres=\u0000\re\u000br\n1 +\u0010Gs\nk\n1+Ga\nk\u001122Ksusc\nMs;with\re\u000b=\r\n1 +Ga\nk: (A7)\nAppendix B: Torque method and linear response theory\nIn this appendix, we consider small deviations of the spin moment from the equilibrium\ndirection and connect the MAE obtained within the torque method and linear response. This\nwill be done employing the retarded single-particle Green function (GF), which is de\fned\nas the resolvent of the single-particle Hamiltonian H(~ r),\n\u0000\n\"+ i0\u0000H(~ r)\u0001\nG(~ r;~ r0;\"+ i0) =\u000e(~ r\u0000~ r0): (B1)\n24To keep the notation as light as possible, we do not introduce the partition of space into\ncells around each atom, as is customary in the KKR-GF approach. The expressions can\neasily be generalized to take that aspect into account. We shall require the following two\nbasic properties (note that the GF is a spin matrix):\n@\n@\"G(~ r;~ r;\"+ i0) =\u0000Z\nd~ r0G(~ r;~ r0;\"+ i0)G(~ r0;~ r;\"+ i0); (B2)\n@\n@XG(~ r;~ r;\"+ i0) =Z\nd~ r0G(~ r;~ r0;\"+ i0)@H(~ r0)\n@XG(~ r0;~ r;\"+ i0); (B3)\nwhereXis some parameter or quantity upon which the Hamiltonian depends. Both relations\nfollow trivially from the de\fning equation of the GF (Eq. (B1)). The electronic density of\nstates is given by\n\u001a(\") =\u00001\n\u0019Im Tr\u001bZ\nd~ rG(~ r;~ r;\"+ i0); (B4)\nfrom which the connection between the GF and the band energy of the main text Ebandis\nestablished. The spin magnetization density is given by\n~M(~ r) =\u00001\n\u0019Im Tr\u001bZ\"F\n\u00001d\"~\u001bG(~ r;~ r;\"+ i0); (B5)\nand we make the assumption that the Hamiltonian depends on the direction of the spin\nmagnetization density in a coarse-grained way\nH(~ r) =H0(~ r) +Bxc(~ r) ^n(\u0012;')\u0001~\u001b: (B6)\n^n(\u0012;') being the direction of the exchange-correlation magnetic \feld. Assuming that the\neasy axis is along the z-direction, a small rotation angle \u0012in thexz-plane of ^nresults in\na torqueT\u0012given in Eq. (14). Using the de\fnition of the band energy and the density of\nstates (Eqs. (13) and (B4)), T\u0012can be expressed in terms of the GF as\nT\u0012=\u00001\n\u0019Im Tr\u001bZ\nd\"Z\nd~ r(\"\u0000\"F)@G(~ r;~ r;\"+ i0)\n@\u0012; (B7)\nRelying on Eq. (B3), the \frst order derivative of the GF with respect to \u0012can expressed in\nterm of the derivative of H(~ r) which reads:\n@H(~ r)\n@\u0012=Bxc(~ r)@^n(\u0012)\n@\u0012\u0001~\u001b:\n=Bxc(~ r) [cos\u0012\u001bx\u0000sin\u0012\u001bz](B8)\n25The combination of the previous equation with Eq. (B3) and Eq. (B7) leads to the following\nexpression for the torque:\nT\u0012=\u00001\n\u0019Im Tr\u001bZ\"F\n\u00001d\"Z\nd~ rBxc(~ r) [cos\u0012G(~ r;~ r;\" )\u001bx\u0000sin\u0012G(~ r;~ r;\" )\u001bz]: (B9)\nThe previous expression was obtained after performing a partial energy integration. Fur-\nthermore, considering a small rotation angle, then G(~ r;~ r;\" ),i.e.the Green function for the\nrotated~Bxcis related to the unperturbed Green function G0(~ r;~ r;\" ) (with~Bxc(~ r)kz-axis)\nvia a Dyson equation:\nG(~ r;~ r;\" )\u0019G0(~ r;~ r;\" ) +Z\nd~ r0G0(~ r;~ r0;\") \u0001~Bxc(~ r0)\u0001~\u001bG 0(~ r0;~ r;\"): (B10)\n\u0001~Bxc(~ r) being the change in the exchange-correlation spin-splitting given by:\n\u0001~Bxc(~ r) =Bxc(~ r) (sin\u0012;0;cos\u0012\u00001);\n\u0019Bxc(~ r)\u0012\n\u0012;0;\u0000\u00122\n2\u0013\n:(B11)\nThen, the expression of G(~ r;~ r;\" ) from Eq. (B10) is plugged back into Eq. (B9) and cos \u0012\nand sin\u0012are expanded for small \u0012as well (retaining linear terms), resulting in the following\nfrom for the torque:\nT\u0012=\u00001\n\u0019Im Tr\u001bZ\"F\n\u00001d\"Z\nd~ rBxc(~ r)Z\nd~ r0[\u001bxG0(~ r;~ r0;\")Bxc(~ r0)\u001bxG0(~ r0;~ r;\")]\u0012\n+1\n\u0019Im Tr\u001bZ\"F\n\u00001d\"Z\nd~ rBxc(~ r)\u001bzG0(~ r;~ r;\" )\u0012 :\n=Z\nd~ rBxc(~ r)\u0014Z\nd~ r0\u001fKS\nxx(~ r;~ r0;0)Bxc(~ r0)\u0000M(~ r)\u0015\n\u0012 :(B12)\n\u001fKS\nxx(~ r;~ r0;0) is the static Kohn-Sham magnetic susceptibility and M(~ r) is the magnetization\ndensity. Using the de\fnition of the spin-\rip Kohn-Sham magnetic susceptibility given in\nEq. (5) in the static limit ( i.e.\u001fKS\nxy(~ r;~ r0;0) =\u001fKS\nyx(~ r;~ r0;0) = 0) and xandy-directions are\nequivalent due to uniaxial symmetry), the torque reads:\nT\u0012=Z\nd~ rBxc(~ r)\u0002\n2\u001fKS\n+\u0000(~ r;~ r0;0)Bxc(~ r0)\u0000M(~ r)\u0003\n\u0012 : (B13)\nThe spin-splitting and the transversal exchange-correlation kernel Kxc\n?(~ r) are related via17,20:\nBxc(~ r) =Kxc\n?(~ r)M(~ r)\n2: (B14)\n26To obtain a simple result, we coarse-grain the exact equations by integrating out the spatial\ndependence and work with e\u000bective scalar quantities. This allows us to write the transversal\nexchange-correlation kernel as:\nKxc\n?=\u0000\n\u001fKS\n+\u0000(0)\u0001\u00001\u0000\u001f\u00001\n+\u0000(0): (B15)\nPlugging the two previous expressions into the coarse-grained form of Eq. (B13), T\u0012can be\nwritten in terms of the static spin-\rip magnetic susceptibilities (Kohn-Sham and enhanced)\nas:\nT\u0012=\u0000M2\n2\u0002\n\u001f\u00001\n+\u0000(0)\u0000\u001fKS\n+\u0000(0)\u001f\u00002\n+\u0000(0)\u0003\n\u0012 : (B16)\nOn one hand, considering that \u001f+\u0000(0) (static limit) obtained from TD-DFT relates to KSusc\nvia\u001f+\u0000(0) =M2\n4KSusc, Eq. (B16) can be recast into:\nT\u0012=\u0000\u0012\n2KSusc\u00008\u001fKS\n+\u0000(0)K2\nSusc\nM2\u0013\n\u0012 : (B17)\nOn the other hand, the torque T\u0012is also given by the \frst order derivative of the phenomeno-\nlogical form of the band energy as:\nT\u0012=@EBand\n@\u0012;\n=\u0000K Torque sin 2\u0012 :(B18)\nAfter expanding for a small angle, T\u0012reads:\nT\u0012=\u00002KTorque\u0012 : (B19)\nThe connection between KTorque andKSuscshown in Eq. (16) of the main text can be estab-\nlished when comparing Eq. (B17) and Eq. (B19).\n\u0003j.bouaziz@fz-juelich.de\n1Y. Shiroishi, K. Fukuda, I. Tagawa, H. 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The dilute permalloy free layer device switches much faster: the characteristic\nswitching time for a permalloy free (Ni 0.83Fe0.17) layer device is 1.18 ns, while that for a dilute\npermalloy ([Ni 0.83Fe0.17]0.6Cu0.4) free layer device is 0.475 ns. A ballistic macrospin model can\ncapture the data trends with a reduced spin-torque asymmetry parameter, reduced spin polarization\nand increased Gilbert damping for the dilute permalloy free layer relative to the permalloy devices.\nOur study demonstrates that reducing the magnetization of the free layer increases the switching\nspeed while greatly reducing the switching energy and shows a promising route toward even lower\npower magnetic memory devices compatible with superconducting electronics.\nThere is a growing interest in spin-transfer devices that\nwork in a cryogenic environment, such as for use in super-\nconducting logic and circuits [1]. While past low temper-\nature memory e\u000borts combined, for example, Josephson\nand complementary metal-oxide semiconductor devices\nin hybrid circuits or explored circuits that stored mag-\nnetic \rux quanta in superconducting loops [2, 3], these\napproaches did not simultaneously o\u000ber high speed, low\npower, and scalability. Spin-transfer torque (STT) driven\nmagnetic memory elements are known to be non-volatile,\nfast, and energy e\u000ecient [4, 5], but so far, they are almost\nexclusively being developed and tested for commercial\napplications [6], which require operation at and above\nroom temperature. Cryogenic operation with supercon-\nducting circuits change device and material requirements.\nFor example, the magnetic anisotropy energy barrier that\nstabilizes the magnetic states and permits long-term data\nretention can be greatly reduced. Large magnetoresis-\ntance also may not be essential given the sensitivity of\nsuperconducting circuits and the reduced thermal noise\nat low temperature. This makes it promising to study\nall metallic spin-valve structures, both due to their low\nimpedance and potential for fast switching [7, 8].\nSpin-transfer induced magnetization switching is fun-\ndamentally based on the transfer of angular momentum\nbetween itinerant electrons and background magnetiza-\ntion. Switching thus requires that the number of elec-\ntrons that \row through a circuit to be of order of the\nnumber of elemental magnetic moments (or spins) in the\nfree layer [9]. This requirement sets the order of magni-\ntude of the product of the current and the switching time\n(which is proportional to the total number of charges\ntransmitted) in what is known as the ballistic limit, the\n\u0003laura.rehm@nyu.edu\nyvolker.sluka@julumni.fz-juelich.de\nzandy.kent@nyu.edushort-pulse-time limit (typically pulse durations less than\nseveral nanoseconds) in which thermal energy has a min-\nimal e\u000bect on the switching dynamics [10]. Reducing\nthe magnetization density is thus expected to reduce the\nswitching current. It is also expected to increase the\nswitching speed and thus reduce the switching energy,\nwhich is a product of the power supplied and the time\nthe device is energized.\nIn this article we test this hypothesis by compar-\ning the switching characteristics of spin-valve nanopil-\nlars with in-plane magnetized dilute permalloy and undi-\nluted permalloy free layers, but otherwise the same lay-\ners, nanopillar shape and size. In both cases the layer\nstacks are deposited on a Niobium (Nb) bottom electrode\nto show that integration with superconducting materials\nis practical. We characterize the pulsed current switch-\ning thresholds in the ballistic regime for both composition\nfree layers and \fnd a signi\fcant decrease in the character-\nistic time scale from 1.18 ns for permalloy to 0.475 ns for\nthe dilute permalloy free layer. A macrospin model was\nused to \ft the switching time data with a reduced spin-\ntorque asymmetry parameter, reduced spin polarization\nand increased Gilbert damping for the dilute permalloy\nfree layer.\nWe investigated two sets of spin-valve nanopillar de-\nvices. One with an undiluted permalloy (Ni 0.83Fe0.17,\ndenoted as Py) free layer and another with a diluted\npermalloy ([Ni 0.83Fe0.17]0.6Cu0.4, denoted as PyCu) free\nlayer. The layer stacks consist of a Nb(50)/Al(8)\nbottom electrode layer, a CoFe(3) reference layer\n(RL) which is part of a synthetic antiferromagnet\n(SAF) CoFe(3)/Ru(0.8)/CoFe(3), and a 3 nm thick\nPy or PyCu free layer (FL): Nb(50)/Al(8)/IrMn\n(10)/SAF/Co(0.2)/Cu(3.5)/Co(0.2)/FL, as shown in\nFig. 1. The numbers in brackets are the layer thicknesses\nin nm. The Nb bottom electrode enables the integration\nwith superconducting circuitry, while the Al interlayer is\ncrucial for the properties of the magnetic stack: it wets\nthe surface of the Nb layer and creates a smoother sur-arXiv:1905.13262v1  [physics.app-ph]  30 May 20192\nLoops\n1\nDCBias TeePulse\nPyorPyCu\nSAF\nAl\nNb\nFIG. 1. Schematic of a spin-valve nanopillar device with an\nundiluted permalloy (Py) or diluted permalloy (PyCu) free\nlayer. The write pulses Iware applied through the capacitive\nport of a bias tee while the inductive port is used to read out\nthe state of the device.\nface, reducing the e\u000bect of N\u0013 eel \\orange peel\" coupling\nbetween layers [11] and e\u000bects of roughness on the mag-\nnetic switching characteristics.\nFollowing the deposition, the wafers were annealed at\n230\u000eC and 1 T to set the magnetization orientation of\nthe SAF. The annealed wafers were pattered into ellip-\ntically shaped nanopillars of various sizes using e-beam\nlithography and ion-milling. Here, we present results on\ndevices with a 50 nm \u0002110 nm cross-section. The de-\nvices are characterized by measuring their \feld and cur-\nrent pulse resistance hysteresis loops at 3.2 K. The state\nof the device is recorded using a lock-in technique. Small\nAC currents of 20 and 40 \u0016A are applied for the PyCu\nand Py free layer device, respectively. Figures 2a) and\n2c) show the minor loops of the Py and PyCu free layer\ndevice, respectively. The Py sample exhibits a resistance\nchange between the antiparallel (AP) and parallel (P)\nmagnetic con\fguration of 190 m\n, while the PyCu free\nlayer device exhibits a \u0001 Rof around 120 m\n. Both de-\nvices show a well-centered hysteresis with a small o\u000bset\n\feld of 6 mT. Both samples also show a bistable region\naround zero applied current and current-induced switch-\ning with 10 ns duration current pulse with pulse ampli-\ntudes of 403 \u0016A (for AP!P switching) and -523 \u0016A (for\nP!AP switching) of the PyCu free layer (Fig. 2b)) and\n480\u0016A (AP!P) and -868 \u0016A (P!AP ) for the Py free\nlayer sample (Fig. 2d)). A di\u000berence in the P !AP and\nAP!P switching current magnitude is often observed in\nspin-valves and associated with spin-torque asymmetries,\nas discussed in Refs. [12{14].\nIn order to explore high speed spin-torque switching,\nshort current pulses with durations of less than 5 ns were\nused. Pulses are applied using a pulse generator (Pi-\ncosecond Pulse Labs 10,070A) as well as an arbitrary\nwaveform generator (AWG, Keysight M8190A). The \frst\ngenerator provides the short pulses to explore the ballis-\ntic regime, while the second generator is used to apply\nlonger (20 ns) duration pulses to reset the magnetization\ndirection of the free layer. To increase the pulse ampli-\ntude resolution (below the 1 dB resolution of the pulse\nLoops_bigger\n0\nc)a)\nd)b)Py Py\nPyCu PyCuFIG. 2. Field- and current-induced magnetization switching\nof Py and PyCu free layer device at 3.2 K. Panels a) and c)\nshow \feld-induced switching of devices with Py and PyCu free\nlayer, respectively. The hysteresis loop shown in panel c) does\nnot fully close due to drift in the measurement setup. The\nexternal \feld is applied along the easy axis of the elliptically\nshaped nanopillar. Panels b) and d) display current-induced\nswitching for 10 ns long pulses of the same set of devices. No\ndata was taken along the dashed lines.\ngenerator's internal step attenuator) a voltage controlled\nvariable attenuator (RFMD RFSA2113SB) is employed.\nThe state of the device is determined by applying a small\nAC current and using a lock-in ampli\fer to determine the\ndevice resistance. The lock-in ampli\fer is operated at a 4\nkHz baseband. We use a bias-tee (Picosecond Pulse Labs\n5575A) to combine low-frequency measurement and high-\nfrequency switching pulses (see Fig. 1). Two 0 dB attenu-\nators at the 4 K and 50 K stage are utilized to thermalize\nthe center conductor of a ground signal ground (GSG)\nprobe. A small external \feld (6 mT) applied along the\nlong axis of the ellipse is used to conduct these pulse\nstudies at the midpoint of the free layer hysteresis loop.\nThe measurement procedure thus consists of applying\ntwo square pulses (reset IRSTand write Iwpulses) with\nopposite pulse amplitudes and reads after each pulse. We\nstart by applying a reset pulse to bring the device to a\nknown state, either P or AP. We then veri\fed the desired\nstate by measuring the resistance of the device. The sub-\nsequent write pulse is applied by the pulse generator and\nthe end state is again determined by measuring the de-\nvice resistance. The whole procedure is repeated about\n64 times for each write pulse amplitude and duration to\ndetermine the switching probability. We vary amplitude\nand duration of the write pulse to create the phase dia-\ngrams shown in Fig. 3. All the pulse measurements were\nperformed at 3.2 K.\nFigure 3 shows the switching phase diagrams for3\nExtented Fit_small\n2\n0.0 0.2 0.4 0.6 0.8 1.0ProbabilityAmplitude (mA) Amplitude (mA)\na)Py: AP→P1.2\n1.0\n0.8\n0.6\n0.4\nPy: P→AP\nb)1.6\n1.4\n1.2\n1.0\n0.8\nc)PyCu : AP→P0.9\n0.8\n0.7\n0.6\n0.40.5\nPyCu : P→AP\nd)0.9\n0.8\n0.7\n0.6\n0.40.5\nPulse Duration ( ns)1 2 3 4 5\nPulse Duration ( ns)1 2 3 4 5\nFIG. 3. Nanosecond pulsed current switching results at 3.2 K.\nSwitching phase diagrams of a device with Py free layer, a)\nAP!P and b) P!AP, and a PyCu free layer, c) AP !P and\nd) P!AP. The color in the plot represents the switching prob-\nability, where red corresponds to 0% and black is 100%. The\nblue points represent the 50% switching probability and the\nsolid cyan line shows the \ft to the macrospin model described\nin the main text.\nAP!P (left panels) and P !AP transitions (right panels)\nfor Py (Figs. 3(a) and (b)) and PyCu free layer device\n(Fig. 3(c) and (d)). The results from these samples di\u000ber\nsigni\fcantly. For longer pulse durations, \u00185 ns, switching\nof the PyCu free layer device occurs for lower pulse am-\nplitudes, especially for the P !AP transition (Figs. 3(d)).\nThe PyCu free layer device also switches with high prob-\nability for shorter duration pulses than the device with\nthe Py free layer, as seen by form of the switching bound-\naries (blue points in Fig. 3) for pulse durations less than\n1 ns. For the P!AP direction comparatively longer pulse\ndurations are required for switching, as discussed further\nbelow.\nIn order to understand the data trends in Fig. 3 we\nconsider a macrospin model, a simple model that pro-\nvides analytic expressions for the switching times in the\nballistic limit and how they vary with material and de-\nvice parameters [9, 15]. Since the devices are metallic\nspin-valves (in contrast to magnetic tunnel junctions),\nthe spin-transfer torque angular dependence is expected\nto be asymmetrical, to be di\u000berent for angular devia-\ntions from the P and AP states, and characterized by a\nparameter \u0003 [12], with a ratio of threshold currents given\nbyIcP!AP/IcAP!P= \u00032. Incorporating this asym-\nmetry into a model for switching of biaxial anisotropy\nmacrospins, and following the approach of Ref. [15], we\nderive an approximate formula relating the switching\nspeed 1/\u001c(\u001cbeing the switching time) to the overdrive\ncurrentI\u0000Ic. Due to the spin-torque asymmetry, P !AP\nand AP!P switching di\u000ber. While the relation for theformer case remains the same as in Ref. [15],\n\u001c\u00001=\r~P\n4e\u00160MsV1\nln\u0010\n\u0019\n2\u00120\u0011\u0000\nI\u0000IP!AP\nc\u0001\n; (1)\nfor the other switching direction we have\n\u0000\n\u00032\u001c\u0001\u00001=\r~P\n4e\u00160MsV1\nln\u0010\n\u0019\n2\u00120\u0011\u0000\nI\u0000IAP!P\nc\u0001\n;(2)\nwhere all currents are taken as positive. In these expres-\nsionsPis the spin polarization of the current, Msthe free\nlayer saturation magnetization, Vthe free layer volume,\n\rthe gyromagnetic ratio, \u00160the vacuum permeability,\n~the reduced Planck's constant and ethe magnitude of\nthe electron charge (i.e., e >0).\u00120is the initial angu-\nlar deviation of the free layer's magnetization from the\neasy axis, the deviation the moment the current pulse is\napplied, discussed further below.\nThe threshold currents for switching are (c.f. [15]):\nIP!AP\nc =4e\n~P\u00160MsV\u000b(Hk+Ms=2) (3)\nIAP!P\nc =4e\n~P\u00160MsV\u000b(Hk+Ms=2)=\u00032; (4)\nwhere\u000bis the damping and Hkis the easy axis\nanisotropy \feld. Important for our analysis, Eqs. 1 and 2\nare each are of the form\nI\u0000Ic=Ic\u001c0\n\u001c; (5)\nwhere\u001c0= ln(\u0019=(2\u00120))=(\r\u000b(Hk+Ms=2)) is independent\nof the switching direction. We therefore \ft the experi-\nmental data in Fig. 3 with Eq. 5 under the constraint that\n\u001c0is the same for both P !AP and AP!P switching di-\nrections. The \fts are displayed as cyan lines in Fig. 3\nand the corresponding \ft parameters are listed in Table\n1.\nFrom this analysis we draw the following conclusions.\nFirst, taking the ratios of \ft parameters IP!AP\nc to\nIAP!P\nc we \fnd that the spin-transfer torque asymme-\ntry is signi\fcantly reduced by diluting the free layer with\nCu: for the Py case \u0003 = 1.44, while in the PyCu free\nlayer device \u0003 = 1.16. Next, we consider the e\u000bect of the\ndilution on the P!AP switching currents and determine\nwhat this implies for the device's material parameters.\nTo this end we note that the uniaxial in-plane anisotropy\nin our samples can be assumed to be entirely due to the\ndevice shape, which is designed to be the same for each\ndevice (up to fabrication-induced sample to sample vari-\nations, of course). Comparing the P !AP switching cur-\nrents between the two devices, we obtain the relation\nIP!AP;u\nc\nIP!AP;d\nc=\u001f2Pd\u000bu\nPu\u000bd; (6)\nwhere\u001fdenotes the ratio of the saturation magnetiza-\ntionsMu\ns=Md\ns. The labels uanddstand for undiluted4\nTABLE I. Fit parameters from pulsed switching measurements in the ballistic regime and corresponding spin-torque asymmetry\nparameter \u0003 of PyCu and PyCu free layer devices. Saturation magnetization Msfor Py and PyCu layers at 3.2 K was determined\nby VSM measurements.\nSample Ic(\u0016A) \u0003 \u001c0(ns) \u00160Ms,3.2 K (mT)\nAP!P P!AP\nPyCu FL 395 \u00062 532\u00062 1.16 0.475 \u00060.007 240\nPy FL 432 \u00062 902\u00063 1.44 1.18 \u00060.01 860\nand diluted, respectively. Vibrating sample magnetome-\ntry (VSM) measurements give \u001f= 3:6 (see Table 1) and\nthus from Eq. 6 we \fndPu\u000bd\nPd\u000bu= 7:6.\nFurther analysis gives estimates of the ratio of the spin\npolarizations in the di\u000berent free layer devices and also\nan estimate of the ratio of the damping parameters. This\ncan be achieved by observing that\nIP!AP;u\nc\nIP!AP;d\nc\u001cu\n0\n\u001cd\n0\u0019Pdln\u0010\n\u0019\n2\u0012u\n0\u0011\nPuln\u0010\n\u0019\n2\u0012d\n0\u0011\u001f; (7)\nwhere the\u0019, refers to the assumption that the gyro-\nmagnetic ratios do not vary between the devices. The\nleft-hand side of Eq. 7 is obtained from the \fts to the\nexperimental data and is approximately equal to 4.21.\nThe two initial angles \u0012u(d)\n0are expectation values that\ndepend on the device shape, the respective saturation\nmagnetizations, and the temperature. With the satura-\ntion magnetizations in Table 1, the right-hand side of\nEq. 7 can be used to estimate the ratio of the spin po-\nlarizations Pd=Pu. To make this estimate, we assume a\nBoltzmann distribution of the initial magnetization state\nof the free layer to obtain\nh\u0012u(d)\n0i\u0019vuut\u0019DkT\n2\u00160\u0010\nMu(d)\ns\u00112\nV; (8)\nwherekis the Boltzmann constant and D=Mu(d)\ns\nHu(d)\nk\u001919.6\nonly depends on the device shape and is assumed to be\nsu\u000eciently similar for the two samples. The same applies\nto the device volume V. Inserting Eq. 8 into Eq. 7, we ob-\ntainPd=Pu= 0:85 forT= 3:2 K. The value depends only\nweakly on the assumed temperature, ranging from 0.85\nat 3.2 K to about 0.82 at 10 K. As a consistency check, we\ncalculateh\u0012u(d)\n0i= 0.015 (0.054) which are small enough\nfor Eq. 8 to be a good approximation. The above range of\nvalues is consistent with the reduced magnetoresistance\nobserved in the dilute free layer devices (c.f. Fig. 2).\nFinally, we can revisit Eq. 6 to estimate\u000bd\n\u000bu\u00196:5, in-\ndicating about a six-fold increase of the damping due to\nthe dilution. This is a large increase, but not entirely\nunexpected. Mathias et al. [16] found a factor of three\nincrease in the damping with 40% Cu dilution of Py at\nroom temperature. Also, Rantschler et al. [17] found\nthat the damping of Py at room temperature increasesby 0.2\u000210-3per atomic percent of Cu. The analysis and\nparticularly the very large apparent increase in damping\nmay also be associated with the lower magnetization and\nexchange sti\u000bness in the PyCu opening other dissipation\nchannels in spin-torque switching, such as the excitation\nof spin-waves, or the formation micromagnetic structure\nin the switching process.\nIn summary, we have studied nanosecond switching\nphase diagrams for spin-valve nanopillars with in-plane\nmagnetized PyCu and Py free layers at low temperature.\nThe PyCu free layer sample exhibits reduced switching\ncurrents for the parallel to antiparallel con\fguration and\nsigni\fcant speed-up of the characteristic switching time\ncompared to the Py free layer device. This results in\ngreatly reduced switching energies ( E=RI2\u001c) for the\nPyCu free layer device. While the switching energy for\nthe antiparallel to parallel con\fguration is reduced from\n53 to 18 fJ, the switching energy for the opposite switch-\ning direction shows over a seven-fold decrease for the di-\nluted sample from 230 fJ to 32 fJ.\nThe clear reduction in the energy consumption of the\nPyCu free layer device as well as its speed-up in the\nswitching characteristics makes it especially interesting\nas a low-energy data storage solution for superconduct-\ning computing. Further, our modeling suggest a means\nto further signi\fcant reductions in the switching energy\nand increases in device performance metrics. Foremost,\nlarger reductions in switching energy require low mag-\nnetization density materials with larger spin polarization\nand lower damping (for example, Heusler alloys [18{20]),\nwhich would have the added bene\ft of increasing the de-\nvice magnetoresistance while reducing the switching cur-\nrent.\nACKNOWLEDGMENTS\nWe thank Jamileh Beik Mohammadi for comments on\nthe manuscript. We thank Canon ANELVA for providing\nthe layer stacks and Spin Memory for patterning the de-\nvices. The research is based on work supported by the Of-\n\fce of the Director of National Intelligence (ODNI), Intel-\nligence Advanced Research Projects Activity (IARPA),\nvia contract W911NF-14-C0089. The views and conclu-\nsions contained herein are those of the authors and should\nnot be interpreted as necessarily representing the o\u000ecial\npolicies or endorsements, either expressed or implied, of5\nthe ODNI, IARPA, or the U.S. Government. The U.S.\nGovernment is authorized to reproduce and distribute\nreprints for Governmental purposes notwithstanding any\ncopyright annotation thereon. This document does notcontain technology or technical data controlled under ei-\nther the U.S. International Tra\u000ec in Arms Regulations\nor the U.S. Export Administration Regulations.\n[1] S. Holmes, A. L. Ripple, and M. A. Manheimer, IEEE\nTransactions on Applied Superconductivity 23, 1701610\n(2013).\n[2] T. Van Duzer, L. Zheng, S. R. Whiteley, H. Kim, J. Kim,\nX. Meng, and T. Ortlepp, IEEE Transactions on Applied\nSuperconductivity 23, 1700504 (2013).\n[3] S. Nagasawa, Y. Hashimoto, H. Numata, and S. Tahara,\nIEEE Transactions on Applied Superconductivity 5, 2447\n(1995).\n[4] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[5] L. Berger, Physical Review B 54, 9353 (1996).\n[6] A. D. Kent and D. C. Worledge, Nature Nanotechnology\n10, 187 (2015).\n[7] D. Bedau, H. Liu, J.-J. Bouzaglou, A. D. 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Aeschli-\nmann, M. M. Murnane, and H. C. Kapteyn, Proceedings\nof the National Academy of Sciences 109, 4792 (2012).\n[17] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J.\nShapiro, W. F. Egelho\u000b Jr, B. B. Maranville, D. Pu-\nlugurtha, A. P. Chen, and L. M. Connors, Journal of\nApplied Physics 101, 033911 (2007).\n[18] T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami,\nT. Miyazaki, H. Naganuma, and Y. Ando, Applied\nPhysics Letters 94, 122504 (2009).\n[19] M. J. Carey, S. Maat, S. Chandrashekariaih, J. A. Ka-\ntine, W. Chen, B. York, and J. R. Childress, Journal of\nApplied Physics 109, 093912 (2011).\n[20] S. Andrieu, A. Neggache, T. Hauet, T. Devolder, A. Hal-\nlal, M. Chshiev, A. M. Bataille, P. Le F\u0012 evre, and\nF. Bertran, Phys. Rev. B 93, 094417 (2016)."    },    {        "title": "1211.0492v1.Dynamic_Spin_Injection_into_Chemical_Vapor_Deposited_Graphene.pdf",        "content": "Page 1 of 13 \n Dynamic Spin Injection into Chemical Vapor Deposited  \nGraphene  \nA. K. Patra1,a), S. Singh2,a), B. Barin2, Y. Lee3, J.-H. Ahn3, E. del Barco2,b), E. R. \nMucciolo2 and B. Özyilmaz1,4,5,6 ,b) \n1Department of Physics, National University of Singapore, 2 Science Dri ve 3, Singapore 117542  \n2Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n3 School of Advanced Materials Science & Engineering, SKKU Advanced Institute of \nNanotechnology  (SAINT) , Sungkyunkwan University, Suwon, Republic  Kore a 440746  \n4NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n5Graphene Research Center, National University of Singapore, Singapore 117542  \n6NUS Graduate School for Integrative Sciences and Engineering (NGS), National Univer sity of \nSingapore, Singapore 117456  \n \nWe demonstrate dynamic spin injection into chemical vapor deposition (CVD)  grown \ngraphen e by s pin pumping from permalloy  (Py) layer s. Ferromagnetic resonance \nmeasurements at room temperature reveal a strong enhancement of the  Gilbert damping \nat the Py/graphene interface , exceeding that observed in even Py/platinum  interfaces.  \nSimilar results are als o shown on Co/graphene layers . This enhancement  in the Gilbert \ndamping is understood as the consequence of spin pumping at t he interface d riven by \nmagnetization dynamics .  Our observation s suggest a  strong enhancement of spin -orbit \ncoupling in CVD graphene , in agreement with earlier spin valve measurements.    \na)A. K. Patra and S. Singh contributed equally to this work.  \nb) Autho rs to whom correspondence should be addressed. Electronic mail s: delbarco@physics.ucf.edu  and \nphyob@nus.edu.sg  \n \nPACS numbers: 72.25. -b, 76.50.+g  Page 2 of 13 \n In spintronic s, where the electron’s  spin degree of freedom , rather than its charge , is \nemployed to process in formation , the efficient generation of the large spin current s stands as  a \nkey requirement for future spintronic device s and applications. Several approaches to generate \npure spin currents  have been proposed and  are being widely investigated, namely, non-local spin \ninjection  [1], spin Hall effect  [2-4], and spin pumping  [5,6]. Among these , spin pumping offers  \nthe advantage  of producing  spin current s over large ( mesoscopic ) areas [7-13] at \nferromagnet ic/non-magnetic  (FM/NM)  interface s. In addition, dynamical  spin pumping is \ninsensitive to a potential impedance mismatch at the FM/NM interface  [14], a problem  \nubiquitous  in the non -local spin injection approach.  Dynamical s pin pumping consists of  \ngenerating pure spin current (i.e., with no net charge current) away from  a ferromagnet  into a \nnon-magnetic material,  induced by the  coherent precession of the magnetization  upon application \nof microwave stimuli of frequency matching the  ferromagnetic resonance (FMR)  of the system  \n[15]. Since pure spin currents carry awa y spin angular momentum, in an FMR experiment the \ntransfer of angular momentum from the FM into the NM layer results in an  enhance ment of the \nGilbert damping in the ferromagnet [5-15]. Most  studies of dynamical spin pumping on FM/NM  \ninterfaces have made us e of Pt and Pd NM layers, since the large spin -orbit coupling in these \nsystems enables the conversion of the injected spin current into an electric voltage  across the N M \nlayer, a phenomenon known as i nverse spin Hall effect  (ISHE) . Recently, spin pumping h as been \nexperimentally demonstrated in FM/semiconductor interfaces (e.g. , GaAs  [13] and p -type Si  \n[15]). However, there is no experimental report on spin pumping in FM/graphene interface s, \nthough graphene  [16] (a two -dimensional layer  of carbon atoms ), pos sesses unique electronic \nproperties (e.g. high mobility  and gate-tunable charge carrier , among others ), and stands as an \nexcellent material for spin transport due to its large spin coherence length [17]. Page 3 of 13 \n In this Letter we report experimental FM R studies of Py and Co  films and  polycrystalline \ngraphene grown by chemical vapor deposition on Cu foils [ 18,19] (henceforth, Co/Gr and Py/Gr , \nrespectively ) performed in  a broad -band microwave coplanar waveguide (CPW)  spectrometer . \nThe observation of a remarkable  broadening of the FMR absorption peaks in the Py/ Gr (88%)  \nand Co/Gr (133%) films demonstrate a strong  increase  of the Gilbert damping in the FM layer  \ndue to spin pumping at the FM/Gr interface and the consequent loss of angular momentum  \nthrough spin inject ion into the CVD graphene  layer . To account for such a remarkable absorption  \nof angular momentum , the spin orbit coupling in CVD graphene must be orders of magnitude \nlarge r than what is predicted for pristine, exfoliated graphene . \nTo prepare the FM/Gr sample s, single layer CVD grown graphene [18, 19] was first \ntransferred onto a Si substrate with 300 nm thick SiO 2 layer . The sample was then anneal ed in a \nH2/Ar environment at 300 °C for 3 hour to remove all organic residue s. For the Py layer we chose \nNi80Fe20, a material extensively  used for magnetic thin film  studies because  of its low \nmagneto crystalline anisotropy  and its insensitivity to strain.  The FM layer  (Py/14nm, Co/15nm)  \nwas deposited on top of the graphene layer lying over the SiO 2/Si substrate by electron-beam \nevaporat ion at a base pressure of 310-7 Torr. For the purpose of FMR comparison experiments, \na control FM film of the same thickness was deposited simultaneously on the same SiO 2 wafer in \nan area where graphene was no t present . The schematic of th e FM/Gr samples  is shown in \nFig. 1-a, together with the Raman spectr um of the  CVD  graphene  before the deposition of Py  \n(Fig. 1-c). The high intens ity of the  2D peak , when  compared to the G peak , and the  weakness of \nthe D peak , suggests that graphene is sin gle layer and of high quality ( i.e. low degree of \ninhomogeneity/defects ).   Page 4 of 13 \n \n \nFig. 1 : (Color online) (a) Schematic of the FM/Gr film sample. (b) Schematic of the FMR measurement \nsetup, with the sample placed up -side-down on top of the micro -CPW . (c) Raman spectrum of CVD \ngraphene.  \nFMR measurements were carried out at room temperature with a high -frequency \nbroadband (1 -50 GHz) micro -coplanar -waveguide ( -CPW) [20] using the flip -chip method [ 21-\n23], by which the sample is placed up -side-down covering the cen tral part of the CPW  (as shown \nin Fig. 1-b), where the transmission line is constricted to increase the density of the microwave \nfield and enhance sensitivity . The CPW was covered with a 100nm -thick insulating layer of \nPMMA resist, hardened by electron bea m exposure, to avoid any influence of the CPW, made \nout of gold, on the sample dynamics. A 1.5  Tesla rotatable  electromagnet was employed to vary \nthe applied field direction from the in -plane ( = 0o) to normal -to-the film plane (  = 90o) \ndirections . Fig. 2-a shows t he angular dependence of the FMR field measured at 10  GHz for both \nPy and Py/Gr films. The rotation plane is chosen to keep the dc magnetic field , H, perpendicular \nto the microwave field felt by the sample at all times, as shown in Fig. 1-b. The resonance fiel d \nincreases as the magnetic field is directed away from the film plane (i.e. increasing ), as \nexpected for a thin film ferromagnet with in-plane shape magneto -anisotropy.  The angular \ndependence of the  FMR field ( HR) can be fitted using the resonance frequency condition given \nby the Smit and Beljers formula [ 23,24], Page 5 of 13 \n \n21HH,                                                                       (1)  \nwhere \nf2  is the angular frequency, \n/Bg  the gyromagnetic ratio, and H1 and H2 are \ngiven by  \nH1=Hcos(-)-4Meffsin2\nH2=Hcos(-)4Meffcos22K2\nMssin2\n ,                                       (2) \nwhere  is the magnetization  angle , \n 2\n2 1 cos 4 2 4 4S S S eff MK MK M M   is the \neffective demagnetization field,  Ms is the saturation magnetization, and K1 and K2 are the first \nand second order anisotropy energies, respectively . The best fit s to the data in Fig. 2-a are given \nby the parameters shown  in the third  column of Table 1 , together with the corresponding \nparameters extracted from equivalent measurements on the Co and Co/Gr films (not shown) . \n \n \nFig. 2 : (color online) (a) Angular dependence of the FMR fi elds measured on both Py and Py/Gr samples \nat f = 10 GHz with the dc magnetic field, H, applied in a plane perpendicular to the microwave field \ngenerated by  the CPW at the sample position. (b) In-plane frequency dependence  of the FMR fields for \nboth Py and  Py/Gr samples. The intercepts with the x-axis give the effective demagnetizing fields of the \nsamples.  Page 6 of 13 \n It is useful to study the resonant behavior by applying the magnetic field at  = 0o \n(parallel configuration) and   = 90 (perpendicular configuration),  since the frequency behavior \nof the FMR fields are given respectively by,  \n       \n\n\n\n\n \n\n\n\n,//,2\n//\n4) 4 (\neff Reff R R\nM HM HH\n ,             (3) \nwhere \nBg  is the gyromagnetic ratio , \n1 //,4 4A S eff H M M   , \n2 1 ,4 4A A S eff H H M M \n, with \nS A MK H1 12  and \nS A MK H2 24  the first and second \norder anisotropy fields, respectively , which relate to su rface, interface and/or magnetoelastic \nanisotropy. Note that K 1 > 0 (>> K2) provides out -of-plane anisotropy, competing with the in -\nplane shape anisotropy . Consequently, a graphical representation of the in - and out -of-plane \nfrequency response of the FMR f ields , conforming to Eq . (3), results in a linear behavior from \nwhich the slope and intercept with the magnetic field axis give  and the effective \ndemagnetiz ation field s, respectively.  The results obtained for the Py and Py/Gr samples are \nshown in Fig. 2-b and 2-c, and the extracted parameters are listed  in the third  column of Table  1, \ntogether with those extracted from the Co and Co/Gr . Note that the anisotropy fields depend on \nthe selection of the saturation magnetization, w ith theoretical values MS,Py = 9.27 kG (attending \nto a 20/80 -Ni/Fe ratio  and assuming identical densities) , and MS,Co = 17.59 kG. For the Co and \nCo/Gr films, the effective saturation magnetization ( Meff = 17.7 kG) is similar to the one \nexpected from theo ry, hence there is negligible ou t-of-plane anisotropy (K 1 ~ 0), in agreement \nwith previous studies [ 25]. The situation is different in the case of the Py and Py/Gr, where the \nsmall Py anisotropy field HA1 = 1.98 kG grows significantly in the Py/Gr ( HA1 = 3.60 kG), \nsuggesting an increase of the Py surface anisotropy due to the presence of the graphene layer ( i.e. Page 7 of 13 \n interface effect). Nevertheless, the magnetization remains in the plane of the film  for all samples . \nTheory  Sample  HR vs.  , f Damping  Changes  \nPy: Ni80Fe20 \ngeff = 2.10  \nFeFe\nS NiNi\nSFe\nSNi\nS\neffg M g MM Mg\n2.0 8.02.0 8.0\n\n \n \nMs = 9.27 kG \nFe\nSNi\nS M M Ms 2.0 8.0\n \n \nwith \n21.2 094.6  NiNi\nS g kG M\n \n0.2 016.22  FeFe\nS g kG M\n \n Py g = 2.110 \n = 0.0 113 \nG = 0.311 GHz \nK1 increases  \n(interface)  \n \nDamping increases \nby ~88% Meff = 7.30 kG \nH1 = 1.98 kG \nK1 = 0.73106 erg/cc  \nPy/Gr  g = 2.107 \n = 0.0 213 \nG = 0.585 GHz Meff,// = 5.70 kG \nHA1 = 3.60 kG \nK1 = 1.32106 erg/cc  \nCo \ng = 2.145 \nMs = 17.59  kG Co g = 2. 149  = 0.0 210 \nG = 1.11 GHz (no K1)  \nDamping increases \nby ~133% Meff = 17.7 kG \nCo/Gr  g = 2. 149  = 0.0 489 \nG = 2.59 GHz Meff = 17.5 kG \n \nTABLE  I: Parameters extracted from the analysis of the data reported in this work.  \n \nWe now focus on the FMR linewidth and its frequency dependence when the magnetic \nfield is applied parallel to the film (  = 0o), from which information about the Gilbert damping \n(i.e., spin relaxation dynamics)  can be directly extracted. The inset to  Fig. 3 shows a field \nderivate of the CPW S 21 transmission parameter obtained when exciting the FMR at 10  GHz in \nboth Py and Py/Gr samples , with HR = 1.28 kG and 1.55 kG, respectively . The peak -to-peak \ndistance repre sents the linewidth , H, of the FMR, whose behavior as a function of frequency is \nshown for both samples in the main panel of Fig. 3. A remarkable in crease of the FMR linewidth \nby 88% is observed in the Py/Gr sample , and even higher (133%) in the Co/ Gr fil ms. The change \nin the linewidth  must be attributed to a substantial enhancement of the Gilbert damping in the Page 8 of 13 \n FM film due to the influence of the graphene directly underneath.  The frequency dependence of \nthe FMR linewidth can be written as a contribution f rom two parts : \nf H H\n\n34\n0\n,                                                               ( 4) \nwhere  is the parameter of the Gilbert damping \nSM G . The first term, \n0H , accounts for \nsample -dependent in homogeneous broadening of the linewidth and is independent of frequenc y, \nwhile the second term represents the dynamical  broadening of the FMR  and scales linearly with \nfrequency .  \n \n \nFig. 3: (color online) Frequency dependence of the FRM linewidth for Py , Py/Gr , Co and Co/Gr  films \nobtained with the magnetic field applied at  = 0 (in-plane configuration). The inset shows the field \nderivatives of CPW S 21 transmission parameter (at 10  GHz)  of the Py and Py/Gr samples , from which the \nlinewidth, H, is calculated as the peak -to-peak distance.  \n Page 9 of 13 \n As observed in Fig. 3, the measured l inewidth for  both FM and FM/Gr samples increase s \nlinearly with frequency, with negligible  inhomogen eous broadening,  indicat ing that damping in \nthe FM film can be properly  explained by the phenomenological Landau-Lifshitz -Gilbert \ndamp ing model . A similar br oadening of the FMR linewidth is observed in both samples when \nthe field is applied perpendicular to the plane, excluding frequency -dependence inhomogeneous \nbroadening (e.g. two -magnon scattering produced by changes in morphology of the FM surface \n[26]), as its possible source. By fitting the data in Fig. 3 to Eq. (4) (using \n0H  = 0), the damping \nparameter s  and G are determined and given in the fourth  column of Table  1 for all studied \nsamples. The Gilbert damping increases  substantially in the FM/Gr films  as a result of the \nincreased linewidth, when compared to the values  obtained in the FM sample s (which are \ncomparable  with values given in the literature  for similar Py  and Co  films  [9,22]). This is our key \nfinding.  Remarkably , the change in the damping para meter  in the Py/Gr sample \n(\nPy GrPy /  = 0.01 ) is even  more pronounced  than those observed in  Py/Pt systems, in \nwhich the thick (when compared to graphene) heavy transition metal  Pt layer provides the  large \nspin-orbit coupling necessary to absorb (i.e. , relax ) the spin accumulation  pumped away from the \nferromagnet.  The efficiency of spin injection is usually cataloged by means of the interfacial \nspin-mixing conductance, which is proportional to the additional damping parameter, , as \nfollows:   \n FMSdMg4\n,                                                               (5) \ngiving \ng = 5.261019 m-2 for our Py/Gr sample  with the thickness of the Py film dFM = 14 nm. \nThe Py/Gr  value is substantially larger than those found in other Py/NM systems with a metallic Page 10 of 13 \n NM layer, e.g. , \ng = 2.191019 m-2 in Py(Ni 81Fe19:10nm)/Pt(10nm)  [9] or \ng = 2.11019 m-2 in \nPy(Ni 80Fe20:15nm)/Pt(15nm)  [11]. Note that in the cited experiments, the spin -diffusion length \nof the non -magnetic layer (~10  nm for Pt) is smaller than the layer thickness. This is sign ificant \nsince it explains how the Pt layer is capable of dissipating the spin accumulation generated by the \ndynamical spin pumping , and account for the loss of angular momentum in the Py . In the case of \ngraphe ne, the enhancement of the damping parameter  is more complicated to understand. In a \nstandard FM/NM metallic system , the spin current injected in to the NM layer decays mainly \nperpendicularly to the interface [ 27], causing the enhancement of the damping parameter to \ndepend on the ratio between the layer  thickness and the spin -diffusion length in the NM . \nHowever, graphene  has effectively zero thickness  and, at least theoretically , a very weak intrinsic \nspin-orbit coupling. Therefore, the spin current must  decay in a FM/Gr film parallel and not \nperpendicul ar to the interface . Furthermore, some sizable spin -orbit coupling must exist in CVD \ngraphene  films . The lat ter may also explain the generally observed very short spin relaxation \ntimes in lateral CVD graphene spin valves  [28]. Recently, small levels of hyd rogen  [29] and \ncopper adatoms [30] have been predicted to lead to a strong e nhancement of the spin-orbit \ncoupling, bringing it into  meV range. Cu adat oms are  certainly likely to be present  in the CVD  \nsamples utilized in our experiments, pointing at a possi ble explanation for the large spin \npumping effect observed in our FM/Gr films .  \n  \n \n Page 11 of 13 \n  \nReferences:  \n[1] F. J. Jadema, A. T. Filip, and B. J. van Wees: Nature  (London) 410, 345  (2001)  \n[2] 2.T. Kimura, Y. Ot ani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett.  98, 156601 \n(2007).  \n[3] S. O. Valenzuela and M. Tinkham, Nature (London ) 442, 176 (2006 ). \n[4] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. 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"    },    {        "title": "1506.00723v1.Current_Driven_Motion_of_Magnetic_Domain_Wall_with_Many_Bloch_Lines.pdf",        "content": "Journal of the Physical Society of Japan LETTERS\nCurrent-Driven Motion of Magnetic Domain Wall with Many Bloch\nLines\nJunichi Iwasaki1\u0003and Naoto Nagaosa1;2y\n1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan\n2RIKEN Center for Emergent Matter Science (CEMS),Wako, Saitama 351-0198, Japan\nThe current-driven motion of a domain wall (DW) in a ferromagnet with many Bloch lines (BLs) via the\nspin transfer torque is studied theoretically. It is found that the motion of BLs changes the current-velocity\n(j-v) characteristic dramatically. Especially, the critical current density to overcome the pinning force is\nreduced by the factor of the Gilbert damping coe\u000ecient \u000beven compared with that of a skyrmion. This\nis in sharp contrast to the case of magnetic \feld driven motion, where the existence of BLs reduces the\nmobility of the DW.\nDomain walls (DWs) and bubbles1,2)are the spin tex-\ntures in ferromagnets which have been studied inten-\nsively over decades from the viewpoints of both funda-\nmental physics and applications. The memory functions\nof these objects are one of the main focus during 70's, but\ntheir manipulation in terms of the magnetic \feld faced\nthe di\u000eculty associated with the pinning which hinders\ntheir motion. The new aspect introduced recently is the\ncurrent-driven motion of the spin textures.3,4)The \row\nof the conduction electron spins, which follow the direc-\ntion of the background localized spin moments, moves\nthe spin texture due to the conservation of the angu-\nlar momentum. This e\u000bect, so called the spin transfer\ntorque, is shown to be e\u000bective to manipulate the DWs\nand bubbles compared with the magnetic \feld. Magnetic\nskyrmion5,6)is especially an interesting object, which is\na swirling spin texture acting as an emergent particle\nprotected by the topological invariant, i.e., the skyrmion\nnumberNsk, de\fned by\nNsk=1\n4\u0019Z\nd2rn(r)\u0001\u0012@n(r)\n@x\u0002@n(r)\n@y\u0013\n(1)\nwith n(r) being the unit vector representing the direc-\ntion of the spin as a function of the two-dimensional spa-\ntial coordinates r. This is the integral of the solid angle\nsubtended by n, and counts how many times the unit\nsphere is wrapped. The solid angle and skyrmion number\nNskalso play essential role when one derives the equation\nof motion for the center of mass of the spin texture, i.e.,\nthe gyro-motion is induced by Nskin the Thiele equation,\nwhere the rigid body motion is assumed.7,8)\nBeyond the Thiele equation,7)one can derive the equa-\ntion of motion of a DW in terms of two variables, i.e.,\nthe wall-normal displacement q(t;\u0010;\u0011 ) and the wall-\nmagnetization orientation angle  (t;\u0010;\u0011 ) (see Fig. 1)\n\u0003iwasaki@appi.t.u-tokyo.ac.jp\nynagaosa@ap.t.u-tokyo.ac.jp\nψqFig. 1. Schematic magnetization distribution of DW with many\nBloch lines.\nwhere\u0010and\u0011are general coordinates specifying the\npoint on the DW:9)\n\u000e\u001b\n\u000e = 2M\r\u00001h\n_q\u0000\u000b\u0001_ \u0000vs\n?\u0000\f\u0001vs\nk(@k )i\n;(2)\n\u000e\u001b\n\u000eq=\u00002M\r\u00001h\n_ +\u000b\u0001\u00001_q+vs\nk(@k )\u0000\f\u0001\u00001vs\n?i\n;\n(3)\nHere, _ means the time-derivative. kand?indicate\nthe components parallel and perpendicular to the DW\nrespectively. Mis the magnetization, \ris the gyro-\nmagnetic ratio, and \u001b, \u0001 are the energy per area and\nthickness of the DW. vsis the velocity of the conduction\nelectrons, which produces the spin transfer torque. \u000bis\nthe Gilbert damping constant, and \frepresents the non-\nadiabatic e\u000bect. These equations indicate that qand \nare canonical conjugate to each other. This is understood\nby the fact that the generator of the spin rotation nor-\nmal to the DW, which is proportional to sin  in Fig. 1,\ndrives the shift of q. (Note that  is measured from the\n\fxed direction in the laboratory coordinates.)\nIn order to reduce the magnetostatic energy, the spins\nin the DW tend to align parallel to the DW, i.e., Bloch\nwall. When the DW is straight, this structure is coplanar\nand has no solid angle. From the viewpoint of eqs. (2)\n1arXiv:1506.00723v1  [cond-mat.mes-hall]  2 Jun 2015J. Phys. Soc. Jpn. LETTERS\nand (3), the angle  is \fxed around the minimum, and\nslightly canted when the motion of qoccurs, i.e., _ = 0.\nHowever, it often happens that the Bloch lines (BLs)\nare introduced into the DW as shown schematically in\nFig. 1. The angle  rotates along the DW and the N\u0013 eel\nwall is locally introduced. It is noted here that the solid\nangle becomes \fnite in the presence of the BLs. Also with\nmany BLs in the DW, the translation of BLs activates\nthe motion of the angle  , i.e., _ 6= 0, which leads to the\ndramatic change in the dynamics.\nIn the following, we focus on the straight DW which\nextends along x-direction and is uniform in z-direction.\nThus, the general coordinates here are ( \u0010;\u0011) = (x;z).\nq(t;x;z ) is independent of the coordinates q(t;x;z ) =\nq(t), and the functional derivative \u000e\u001b=\u000eq in eq. (3) be-\ncomes the partial derivative @\u001b=@q . In the absence of\nBLs, we set  (t;x;z ) = (t), and\u000e\u001b=\u000e in eq. (2) also\nbecomes@\u001b=@ . Then the equation of motion in the ab-\nsence of BL is\n@\u001b\n@ = 2M\r\u00001h\n_q\u0000\u000b\u0001_ \u0000vs\n?i\n; (4)\n@\u001b\n@q=\u00002M\r\u00001h\n_ +\u000b\u0001\u00001_q\u0000\f\u0001\u00001vs\n?i\n; (5)\nWith many BLs, the sliding motion of Bloch lines along\nDW, which activates _ , does not change the wall energy,\ni.e.,\u000e\u001b=\u000e in eq. (2) vanishes.2)Here, for simplicity, we\nconsider the periodic BL array with the uniform twist\n (t;x;z ) = (x\u0000p(t))=~\u0001 where ~\u0001 is the distance between\nBLs, which leads to\n0 = 2M\r\u00001h\n_q+\u000b\u0001~\u0001\u00001_p\u0000vs\n?\u0000\f\u0001~\u0001\u00001vs\nki\n;(6)\n@\u001b\n@q=\u00002M\r\u00001h\n\u0000~\u0001\u00001_p+\u000b\u0001\u00001_q+~\u0001\u00001vs\nk\u0000\f\u0001\u00001vs\n?i\n;\n(7)\nFirst, let us discuss the magnetic \feld driven motion\nwithout current. The e\u000bect of the external magnetic \feld\nHextis described by the force @\u001b=@q =\u00002MHextin\neqs. (5) and (7). vs\nkandvs\n?are set to be zero. In the\nabsence of BL, as mentioned above, the phase  is static\n_ = 0 with the slight tilt of the spin from the easy-plane,\nand one obtains from eq. (5)\n_q=\u0001\rHext\n\u000b: (8)\nThis is a natural result, i.e., the mobility is inversely\nproportional to the Gilbert damping \u000b. is determined\nby eq. (4) with this value of the velocity _ q.\nIn the presence of many BLs, eqs. (6) and (7) give the\nvelocities of DW and BL sliding driven by the magnetic\n\feld as\n_q=\u000b\n1 +\u000b2\u0001\rHext; (9)_p=\u00001\n1 +\u000b2~\u0001\rHext: (10)\nComparing eqs. (8) and (9), the mobility of the DW is re-\nduced by the factor of \u000b2since\u000bis usually much smaller\nthan unity. We also note that the velocity of the BL slid-\ning _pis larger than that of the wall _ qby the factor of\n\u000b. Physically, this means that the e\u000bect of the external\nmagnetic \feld Hextmostly contributes to the rapid mo-\ntion of the BLs along the DW rather than the motion of\nthe DW itself. These results have been already reported\nin refs.2,9,10)\nNow let us turn to the motion induced by the current\nvs. In the absence of BL, again we put _ = 0 in eqs. (4)\nand (5). Assuming that there is no pinning force or ex-\nternal magnetic \feld, i.e., @\u001b=@q = 0, one obtains from\neq. (5)\n_q=\f\n\u000bvs\n?; (11)\nand eq. (4) determines the equilibrium value of  . When\nthe pinning force @\u001b=@q =Fpinis \fnite, there appears a\nthreshold current density ( vs\n?)cwhich is determined by\nputting _q= 0 in eq. (5) as\n(vs\n?)c=\r\u0001\n2M\fFpin; (12)\nwhich is inversely proportional to \f.11)Since eq. (11) is\nindependent of vs\nk, the threshold current density\u0010\nvs\nk\u0011\nc\nis\u0010\nvs\nk\u0011\nc=1.\nIn the presence of the many BLs, on the other hand,\neqs. (6) and (7) give\n@\u001b\n@q=\u00002M\r\u00001\u00141 +\u000b2\n\u000b\u0001\u00001_q\n\u00001 +\u000b\f\n\u000b\u0001\u00001vs\n?\u0000\f\u0000\u000b\n\u000b~\u0001\u00001vs\nk\u0015\n;\n(13)\nwhich is the main result of this paper. From eq. (13), the\ncurrent-velocity characteristic in the absence of both the\npinning and the external \feld ( @\u001b=@q =0) is\n_q=1 +\u000b\f\n1 +\u000b2vs\n?\u0000\f\u0000\u000b\n1 +\u000b2\u0001~\u0001\u00001vs\nk\n'vs\n?+ (\f\u0000\u000b)\u0001~\u0001\u00001vs\nk; (14)\nwhere the fact \u000b;\f\u001c1 is used in the last step. If we\nneglect the term coming from vs\nk, the current-velocity\nrelation becomes almost independent of \u000band\fin\nsharp contrast to eq. (11). This is similar to the univer-\nsal current-velocity relation in the case of skyrmion,12)\nwhere the solid angle is \fnite and also the transverse\nmotion to the current occurs. Note that vs\nkslightly con-\ntributes to the motion when \u000b6=\f, while it does\nnot in the absence of BL. Even more dramatic is the\ncritical current density in the presence of the pinning\n2J. Phys. Soc. Jpn. LETTERS\n30\n20\n101525\n520\n1015\n5\n3.0\n2.0\n1.01.52.5\n0.50.6\n0.4\n0.20.30.5\n0.10.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6\n4000 2000 10000 8000 6000 4000 2000 10000 8000 6000qq\nqq\nt tt tw/o BL\nw/ BLs0.429\n0.707Pinning\nq(a)\n(c) (d)(b)\nFig. 2. The wall displacement qas a fucntion of tfor the DWs\nwithout BL and with BLs. (a) vs\n?= 22 :0. The inset shows the\npinning force Fpin. (b) vs\n?= 21 :0. (c) vs\n?= 0:0043. (d) vs\n?=\n0:0042.\n(@\u001b=@q =Fpin). When we apply only the current per-\npendicular to the DW, i.e., vs\nk= 0, putting _ q= 0 in\neq. (13) determines the threshold current density as\n(vs\n?)c=\r\u0001\n2M\u000b\n1 +\u000b\fFpin; (15)\nwhich is much reduced compared with eq. (12) by the\nfactor of\u000b\f\n1+\u000b\f\u001c1. Note that ( vs\n?)cin eq. (15) is even\nsmaller than the case of skyrmion12)by the factor of\n\u000b. Similarly, the critical current density of the motion\ndriven byvs\nkis given by\n\u0010\nvs\nk\u0011\nc=\r~\u0001\n2M\u000b\nj\f\u0000\u000bjFpin; (16)\nwhich can also be smaller than eq. (12).\nNext we look at the numerical solutions of q(t) driven\nby the current vs\n?perpendicular to the wall under the\npinning force. We assume the following pinning force:\n(\r\u0001=2M)Fpin(q) =v\u0003(q=\u0001) exp\u0002\n\u0000(q=\u0001)2\u0003\n(see the in-\nset of Fig. 2(a)). We employ the unit of \u0001 = v\u0003=\n1 and the parameters ( \u000b;\f) are \fxed at ( \u000b;\f) =\n(0:01;0:02). Here, we compare two DWs without BL\nand with BLs. The maximum value of the pinning force\n(\r\u0001=2M)Fpin\nmax= 0:429 determines the threshold current\ndensity (vs\n?)cas (vs\n?)c= 21:4 and (vs\n?)c= 0:00429 in the\nabsence of BL and in the presence of many BLs, respec-\ntively. In Fig. 2(a), both DWs overcome the pinning at\nthe current density vs\n?= 22:0, although the velocity of\nthe DW without BL is suppressed in the pinning poten-\ntial. At the current density vs\n?= 21:0 below the threshold\nvalue in the absence of BL, the DW without BL is pinned,\nwhile that with BLs still moves easily (Fig. 2(b)). The\nvelocity suppression in the presence of BLs is observed\nat much smaller current density vs\n?= 0:0043 (Fig. 2(c)),\nand \fnally it stops at vs\n?= 0:0042 (Fig. 2(d)).\nAll the discussion above relies on the assumption thatthe wall is straight and  rotates uniformly. When the\nbending of the DW and non-uniform distribution of BLs\nare taken into account, the average velocity and the\nthreshold current density take the values between two\ncases without BL and with many BLs. The situation\nchanges when the DW forms closed loop, i.e., the do-\nmain forms a bubble. The bubble with many BLs and\nlargejNskjis called hard bubble because the repulsive\ninteraction between the BLs makes it hard to collapse\nthe bubble.2)At the beginning of the motion, the BLs\nmove along the DW, which results in the tiny critical cur-\nrent. In the steady state, however, the BLs accumulate\nin one side of the bubble.13,14)Then, the con\fguration\nof the BLs is static and the Thiele equation is justi\fed\nas long as the force is slowly varying within the size of\nthe bubble. The critical current density ( vs)cis given by\n(vs)c/Fpin=Nsk(Nsk(\u001d1): the skyrmion number of\nthe hard bubble), and is reduced by the factor of Nsk\ncompared with the skyrmion with Nsk=\u00061.\nIn conclusion, we have studied the current-induced\ndynamics of the DW with many BLs. The \fnite _ in\nthe steady motion activated by BLs sliding drastically\nchanges the dynamics, which has already been reported\nin the \feld-driven case. In contrast to the \feld-driven\ncase, where the mobility is suppressed by introducing\nBLs, that in the current-driven motion is not necessarily\nsuppressed. Instead, the current-velocity relation shows\nuniversal behavior independent of the damping strength\n\u000band non-adiabaticity \f. Furthermore, the threshold\ncurrent density in the presence of impurities is tiny even\ncompared with that of skyrmion motion by the factor of\n\u000b. These \fndings will stimulate the development of the\nracetrack memory based on the DW with many BLs.\nAcknowledgments We thank W. Koshibae for useful discus-\nsion. This work is supported by Grant-in-Aids for Scienti\fc Re-\nsearch (S) (No. 24224009) from the Ministry of Education, Cul-\nture, Sports, Science and Technology of Japan. J. I. was supported\nby Grant-in-Aids for JSPS Fellows (No. 2610547).\n1) A. Hubert and R. Sch afer, Magnetic Domains: The Analysis\nof Magnetic Microstructures (Springer-Verlag, Berlin, 1998).\n2) A. P. Malozemo\u000b and J.C. Slonczewski, Magnetic Domain\nWalls in Bubble Materials (Academic Press, New York, 1979).\n3) J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1{L7 (1996).\n4) L. Berger, Phys. Rev. B 54, 9353{9358 (1996).\n5) S. M uhlbauer et al., Science 323, 915 (2009).\n6) X. Z. Yu et al., Nature 465, 901 (2010).\n7) A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n8) K. Everschor et al., Phys. Rev. B 86, 054432 (2012).\n9) J. C. Slonczewski, J. Appl. Phys. 45, 2705 (1974).\n10) A. P. Malozemo\u000b and J. C. Slonczewski, Phys. Rev. Lett. 29,\n952 (1972).\n11) G. Tatara et al., J. Phys. Soc. Japan 75, 64708 (2006).\n12) J. Iwasaki, M. Mochizuki and N. Nagaosa, Nat. Commun. 4,\n1463 (2013).\n13) G. P. Vella-Coleiro, A. Rosencwaig and W. J. Tabor, Phys.\nRev. Lett. 29, 949 (1972)\n14) A. A. Thiele, F. B. Hagedorn and G. P. Vella-Coleiro, Phys.\n3J. Phys. Soc. Jpn. LETTERS\nRev. B 8, 241 (1973).\n4"    },    {        "title": "1104.1625v1.Magnetization_Dissipation_in_Ferromagnets_from_Scattering_Theory.pdf",        "content": "arXiv:1104.1625v1  [cond-mat.mes-hall]  8 Apr 2011Magnetization Dissipation in Ferromagnets from Scatterin g Theory\nArne Brataas∗\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nYaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\nGerrit E. W. Bauer\nInstitute for Materials Research, Tohoku University, Send ai 980-8577, Japan and\nKavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamicsofferromagnets are often formu lated intermsof theLandau-Lifshitz-\nGilbert (LLG) equation. The reactive part of this equation d escribes the response of the magnetiza-\ntion in terms of effective fields, whereas the dissipative par t is parameterized by the Gilbert damping\ntensor. We formulate a scattering theory for the magnetizat ion dynamics and map this description\non the linearized LLG equation by attaching electric contac ts to the ferromagnet. The reactive part\ncan then be expressed in terms of the static scattering matri x. The dissipative contribution to the\nlow-frequency magnetization dynamics can be described as a n adiabatic energy pumping process\nto the electronic subsystem by the time-dependent magnetiz ation. The Gilbert damping tensor\ndepends on the time derivative of the scattering matrix as a f unction of the magnetization direction.\nBy the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated\nin terms of the quasistatic scattering matrix. The theory is formulated for general magnetization\ntextures and worked out for monodomain precessions and doma in wall motions. We prove that the\nGilbert damping from scattering theory is identical to the r esult obtained by the Kubo formalism.\nPACS numbers: 75.40.Gb,76.60.Es,72.25.Mk\nI. INTRODUCTION\nFerromagnets develop a spontaneous magnetization\nbelow the Curie temperature. The long-wavelengthmod-\nulations of the magnetization direction consist of spin\nwaves, the low-lying elementary excitations (Goldstone\nmodes) of the ordered state. When the thermal energy is\nmuch smaller than the microscopic exchange energy, the\nmagnetization dynamics can be phenomenologically ex-\npressed in a generalized Landau-Lifshitz-Gilbert (LLG)\nform:\n˙ m(r,t) =−γm(r,t)×[Heff(r,t)+h(r,t)]+\nm(r,t)×/integraldisplay\ndr′[˜α[m](r,r′)˙ m(r′,t)],(1)\nwhere the magnetization texture is described by m(r,t),\nthe unit vector along the magnetization direction at po-\nsitionrand timet,˙ m(r,t) =∂m(r,t)/∂t,γ=gµB//planckover2pi1is\nthe gyromagnetic ratio in terms of the g-factor (≈2 for\nfree electrons) and the Bohr magneton µB. The Gilbert\ndamping ˜αis a nonlocal symmetric 3 ×3 tensor that is\na functional of m. The Gilbert damping tensor is com-\nmonly approximated to be diagonal and isotropic (i), lo-\ncal (l), and independent of the magnetization m, with\ndiagonal elements\nαil(r,r′) =αδ(r−r′). (2)\nThe linearized version of the LLG equation for small-\namplitude excitations has been derived microscopically.1It has been used very successfully to describe the mea-\nsured response of ferromagnetic bulk materials and thin\nfilms in terms of a small number of adjustable, material-\nspecific parameters. The experiment of choice is fer-\nromagnetic resonance (FMR), which probes the small-\namplitude coherent precession of the magnet.2The\nGilbertdampingmodelinthelocalandtime-independent\napproximationhasimportantramifications, suchasalin-\near dependence of the FMR line width on resonance fre-\nquency, that have been frequently found to be correct.\nThe damping constant is technologically important since\nit governs the switching rate of ferromagnets driven by\nexternal magnetic fields or electric currents.3In spatially\ndependent magnetization textures, the nonlocal charac-\nter of the damping can be significant as well.4–6Moti-\nvated by the belief that the Gilbert damping constant is\nanimportantmaterialproperty, weset outheretounder-\nstand its physical origins from first principles. We focus\non the well studied and technologically important itiner-\nant ferromagnets, although the formalism can be used in\nprinciple for any magnetic system.\nThe reactive dynamics within the LLG Eq. (1) is de-\nscribed by the thermodynamic potential Ω[ M] as a func-\ntional of the magnetization. The effective magnetic field\nHeff[M](r)≡ −δΩ/δM(r) is the functional derivative\nwith respect to the local magnetization M(r) =Msm(r),\nincluding the external magnetic field Hext, the magnetic\ndipolar field Hd, the texture-dependent exchange energy,\nand crystal field anisotropies. Msis the saturation mag-\nnetization density. Thermal fluctuations can be included\nby a stochastic magnetic field h(r,t) with zero time av-2\nleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath (reservoirs) via metallic normal metal\nleads (N).\nerage,/an}b∇acketle{th/an}b∇acket∇i}ht= 0, and white-noise correlation:7\n/an}b∇acketle{thi(r,t)hj(r′,t′)/an}b∇acket∇i}ht=2kBT\nγMs˜αij[m](r,r′)δ(t−t′),(3)\nwhereMsis the magnetization, iandjare the Cartesian\nindices, and Tis the temperature. This relation is a con-\nsequence ofthe fluctuation-dissipation theorem (FDT) in\nthe classical (Maxwell-Boltzmann) limit.\nThe scattering ( S-) matrix is defined in the space of\nthe transport channels that connect a scattering region\n(the sample) to real or fictitious thermodynamic (left\nand right) reservoirs by electric contacts with leads that\nare modeled as ideal wave guides. Scattering matri-\nces are known to describe transport properties, such as\nthe giant magnetoresistance, spin pumping, and current-\ninducedmagnetizationdynamicsinlayerednormal-metal\n(N)|ferromagnet (F).8–10When the ferromagnet is part\nof an open system as in Fig. 1, also Ω can be expressed\nin terms of the scattering matrix, which has been used\nto express the non-local exchange coupling between fer-\nromagnetic layers through conducting spacers.11We will\nshow here that the scattering matrix description of the\neffective magnetic fields is valid even when the system is\nclosed, provided the dominant contribution comes from\nthe electronic band structure, scattering potential disor-\nder, and spin-orbit interaction.\nScattering theory can also be used to compute the\nGilbert damping tensor ˜ αfor magnetization dynamics.15\nThe energy loss rate of the scattering region can be ex-\npressedin termsofthe time-dependent S-matrix. To this\nend, the theory of adiabatic quantum pumping has to be\ngeneralizedtodescribedissipationinametallicferromag-\nnet. The Gilbert damping tensor is found by evaluating\nthe energy pumping out of the ferromagnet and relat-\ning it to the energy loss that is dictated by the LLG\nequation. In this way, it is proven that the Gilbert phe-\nnomenology is valid beyond the linear response regime\nof small magnetization amplitudes. The key approxima-\ntion that is necessary to derive Eq. (1) including ˜ αis the\n(adiabatic) assumption that the ferromagnetic resonance\nfrequencyωFMRthat characterizesthe magnetizationdy-\nnamics is small compared to internal energy scale set by\nthe exchange splitting ∆ and spin-flip relaxation rates\nτs. The LLG phenomenology works well for ferromag-\nnets for which ωFMR≪∆//planckover2pi1, which is certainly the case\nfor transition metal ferromagnets such as Fe and Co.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from the transfer of energy\nfromthemagneticorderparametertotheitinerantquasi-particle continuum. This requires either magnetic disor-\nder or spin-orbit interactions in combination with impu-\nrity/phonon scattering.2Since the heat capacitance of\nthe ferromagnet is dominated by the lattice, the energy\ntransferred to the quasiparticles will be dissipated to the\nlattice as heat. Here we focus on the limit in which elas-\ntic scattering dominates, such that the details of the heat\ntransfer to the lattice does not affect our results. Our ap-\nproachformallybreaks down in sufficiently clean samples\nat high temperatures in which inelastic electron-phonon\nscattering dominates. Nevertheless, quantitative insight\ncan be gained by our method even in that limit by mod-\nelling phonons by frozen deformations.12\nIn the present formulation, the heat generated by the\nmagnetization dynamics can escape only via the contacts\nto the electronic reservoirs. By computing this heat cur-\nrent through the contacts we access the total dissipa-\ntion rate. Part of the heat and spin current that es-\ncapes the sample is due to spin pumping that causes\nenergy and momentum loss even for otherwise dissipa-\ntion less magnetization dynamics. This process is now\nwellunderstood.10For sufficiently largesamples, the spin\npumping contribution is overwhelmed by the dissipation\nin the bulk of the ferromagnet. Both contributions can\nbe separated by studying the heat generation as a func-\ntion of the length of a wire. In principle, a voltage can be\nadded to study dissipation in the presence of electric cur-\nrents as in 13,14, but we concentrate here on a common\nand constant chemical potential in both reservoirs.\nAlthough it is not a necessity, results can be simpli-\nfied by expanding the S-matrix to lowest order in the\namplitude of the magnetization dynamics. In this limit\nscattering theory and the Kubo linear response formal-\nism for the dissipation can be directly compared. We\nwill demonstrate explicitly that both approaches lead to\nidentical results, which increases our confidence in our\nmethod. The coupling to the reservoirs of large samples\nis identified to play the same role as the infinitesimals in\nthe Kubo approach that guarantee causality.\nOur formalism was introduced first in Ref. 15 lim-\nited to the macrospin model and zero temperature. An\nextension to the friction associatedwith domain wall mo-\ntion was given in Ref. 13. Here we show how to handle\ngeneral magnetization textures and finite temperatures.\nFurthermore, we offer an alternative route to derive the\nGilbert damping in terms of the scattering matrix from\nthe thermal fluctuations of the effective field. We also\nexplain in more detail the relation of the present theory\nto spin and charge pumping by magnetization textures.\nOur paper is organized in the following way. In Sec-\ntion II, we introduce our microscopic model for the fer-\nromagnet. In Section III, dissipation in the Landau-\nLifshitz-Gilbert equation is exposed. The scattering the-\nory of magnetization dynamics is developed in Sec. IV.\nWe discuss the Kubo formalism for the time-dependent\nmagnetizationsin Sec. V, before concluding our article in\nSec. VI. The Appendices provide technical derivations of\nspin, charge, and energy pumping in terms of the scat-3\ntering matrix of the system.\nII. MODEL\nOur approach rests on density-functional theory\n(DFT), which is widely and successfully used to describe\nthe electronic structure and magnetism in many fer-\nromagnets, including transition-metal ferromagnets and\nferromagnetic semiconductors.16In the Kohn-Sham im-\nplementation of DFT, noninteracting hypothetical par-\nticles experience an effective exchange-correlationpoten-\ntial that leads to the same ground-statedensity as the in-\nteractingmany-electronsystem.17Asimpleyetsuccessful\nscheme is the local-densityapproximationto the effective\npotential. DFT theory can also handle time-dependent\nphenomena. We adopt here the adiabatic local-density\napproximation (ALDA), i.e. an exchange-correlationpo-\ntential that is time-dependent, but local in time and\nspace.18,19As the name expresses, the ALDA is valid\nwhen the parametric time-dependence of the problem is\nadiabatic with respect to the electron time constants.\nHere we consider a magnetization direction that varies\nslowly in both space and time. The ALDA should be\nsuited to treat magnetization dynamics, since the typical\ntime scale ( tFMR∼1/(10 GHz) ∼10−10s) is long com-\nparedtothethat associatedwith theFermi andexchange\nenergies, 1 −10 eV leading to /planckover2pi1/∆∼10−13s in transition\nmetal ferromagnets.\nIn the ALDA, the system is described by the time-\ndependent effective Schr¨ odinger equation\nˆHALDAΨ(r,t) =i/planckover2pi1∂\n∂tΨ(r,t), (4)\nwhere Ψ( r,t) is the quasiparticle wave function at posi-\ntionrand timet. We consider a generic mean-field elec-\ntronic Hamiltonian that depends on the magnetization\ndirection ˆHALDA[m] and includes the periodic Hartree,\nexchange and correlation potentials and relativistic cor-\nrectionssuchasthe spin-orbitinteraction. Impurityscat-\ntering including magnetic disorder is also represented by\nˆHALDA.The magnetization mis allowed to vary in time\nand space. The total Hamiltonian depends additionally\non the Zeeman energy of the magnetization in external\nHextand dipolar Hdmagnetic fields:\nˆH=ˆHALDA[m]−Ms/integraldisplay\ndrm·(Hext+Hd).(5)\nFor this general Hamiltonian (5), our task is to de-\nduce an expression for the Gilbert damping tensor ˜ α. To\nthis end, from the form of the Landau-Lifshitz-Gilbert\nequation (3), it is clear that we should seek an expansionin terms of the slow variations of the magnetizations in\ntime. Such an expansion is valid provided the adiabatic\nmagnetization precession frequency is much less than the\nexchange splitting ∆ or the spin-orbit energy which gov-\nerns spin relaxation of electrons. We discuss first dissi-\npation in the LLG equation and subsequently compare\nit with the expressions from scattering theory of electron\ntransport. This leads to a recipe to describe dissipation\nby first principles. Finally, we discuss the connection to\nthe Kubo linear response formalism and prove that the\ntwo formulations are identical in linear response.\nIII. DISSIPATION AND\nLANDAU-LIFSHITZ-GILBERT EQUATION\nThe energy dissipation can be obtained from the solu-\ntion of the LLG Eq. (1) as\n˙E=−Ms/integraldisplay\ndr[˙ m(r,t)·Heff(r,t)] (6)\n=−Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′˙ m(r)·˜α[m](r,r′)·˙ m(r′).(7)\nThescatteringtheoryofmagnetizationdissipationcanbe\nformulated for arbitrary spatiotemporal magnetization\ntextures. Much insight can be gained for certain special\ncases. In small particles or high magnetic fields the col-\nlective magnetization motion is approximately constant\nin space and the “macrospin” model is valid in which\nall spatial dependences are disregarded. We will also\nconsider special magnetization textures with a dynamics\ncharacterized by a number of dynamic (soft) collective\ncoordinates ξa(t) counted by a:20,21\nm(r,t) =mst(r;{ξa(t)}), (8)\nwheremstis the profile at t→ −∞.This representation\nhas proven to be very effective in handling magnetiza-\ntion dynamics of domain walls in ferromagnetic wires.\nThe description is approximate, but (for few variables)\nit becomes exact in special limits, such as a transverse\ndomain wall in wires below the Walker breakdown (see\nbelow); it becomes arbitrarily accurate by increasing the\nnumber of collective variables. The energy dissipation to\nlowest (quadratic) order in the rate of change ˙ξaof the\ncollective coordinates is\n˙E=−/summationdisplay\nab˜Γab˙ξa˙ξb, (9)\nThe (symmetric) dissipation tensor ˜Γabreads4\n˜Γab=Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′∂mst(r)\n∂ξaα[m](r,r′)·∂mst(r′)\n∂ξb. (10)\nThe equation of motion of the collective coordinates un-\nder a force\nF=−∂Ω\n∂ξ(11)\nare20,21\n˜η˙ξ+[F+f(t)]−˜Γ˙ξ= 0, (12)\nintroducing the antisymmetric and time-independent gy-\nrotropic tensor:\n˜ηab=Ms\nγ/integraldisplay\ndrmst(r)·/bracketleftbigg∂mst(r)\n∂ξa×∂mst(r)\n∂ξb/bracketrightbigg\n.(13)\nWe show below that Fand˜Γ can be expressed in terms\nof the scattering matrix. For our subsequent discussions\nit is necessary to include a fluctuating force f(t) (with\n/an}b∇acketle{tf(t)/an}b∇acket∇i}ht= 0),which has not been considered in Refs. 20,21.\nFrom Eq. (3) if follows the time correlation of fis white\nand obeys the fluctuation-dissipation theorem:\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBT˜Γabδ(t−t′). (14)\nIn the following we illustrate the collective coordinate\ndescription of magnetization textures for the macrospin\nmodel and the Walker model for a transverse domain\nwall. The treatment is easily extended to other rigid\ntextures such as magnetic vortices.\nA. Macrospin excitations\nWhen high magnetic fields are applied or when the\nsystem dimensions are small the exchange stiffness dom-\ninates. In both limits the magnetization direction and\nits low energy excitations lie on the unit sphere and its\nmagnetization dynamics is described by the polar angles\nθ(t) andϕ(t):\nm= (sinθcosϕ,sinθsinϕ,cosθ).(15)\nThe diagonal components of the gyrotropic tensor vanish\nby (anti)symmetry ˜ ηθθ= 0, ˜ηϕϕ= 0.Its off-diagonal\ncomponents are\nηθϕ=MsV\nγsinθ=−ηϕθ. (16)\nVis the particle volume and MsVthe total magnetic\nmoment. We now have two coupled equations of motion\nMsV\nγ˙ϕsinθ−∂Ω\n∂θ−/parenleftBig\n˜Γθθ˙θ+˜Γθϕ˙ϕ/parenrightBig\n= 0,(17)\n−MsV\nγ˙θsinθ−∂Ω\n∂ϕ−/parenleftBig\n˜Γϕθ˙θ+˜Γϕϕ˙ϕ/parenrightBig\n= 0.The thermodynamic potential Ω determines the ballistic\ntrajectories of the magnetization. The Gilbert damping\ntensor˜Γabwill be computed below, but when isotropic\nand local,\n˜Γ =˜1δ(r−r′)Msα/γ, (18)\nwhere˜1 is a unit matrix in the Cartesian basis and α\nis the dimensionless Gilbert constant, Γ θθ=MsVα/γ,\nΓθϕ= 0 = Γ ϕθ, and Γ ϕϕ= sin2θMsVα/γ.\nB. Domain Wall Motion\nWe focus on a one-dimensional model, in which the\nmagnetization gradient, magnetic easy axis, and external\nmagnetic field point along the wire ( z) axis. The mag-\nnetic energy of such a wire with transverse cross section\nScan be written as22\nΩ =MsS/integraldisplay\ndzφ(z), (19)\nin terms of the one-dimensional energy density\nφ=A\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−Hamz+K1\n2/parenleftbig\n1−m2\nz/parenrightbig\n+K2\n2m2\nx,(20)\nwhereHais the applied field and Ais the exchange stiff-\nness. Here the easy-axis anisotropy is parametrized by\nan anisotropy constant K1. In the case of a thin film\nwire, there is also a smaller anisotropy energy associated\nwith the magnetization transverse to the wire governed\nbyK2. In a cylindrical wire from a material without\ncrystal anisotropy (such as permalloy) K2= 0.\nWhen the shape of such a domain wall is pre-\nserved in the dynamics, three collective coordinates\ncharacterize the magnetization texture: the domain\nwall position ξ1(t) =rw(t), the polar angle ξ2(t) =\nϕw(t), and the domain wall width λw(t). We con-\nsider a head-to-head transverse domain wall (a tail-\nto-tail wall can be treated analogously). m(z) =\n(sinθwcosϕw,sinθwsinϕw,cosθw), where\ncosθw= tanhrw−z\nλw(21)\nand\ncscθw= coshrw−z\nλw(22)\nminimizes the energy (20) under the constraint that the\nmagnetization to the far left and right points towardsthe5\ndomain wall. The off-diagonal elements are then ˜ ηrl=\n0 = ˜ηlrand ˜ηrϕ=−2Ms/γ=−˜ηϕr.The energy (20)\nreduces to\nΩ =MsS/bracketleftbig\nA/λw−2Har+K1λw+K2λwcos2ϕw/bracketrightbig\n.\n(23)\nDisregarding fluctuations, the equation of motion Eq.\n(12) can be expanded as:\n2˙rw+αϕϕ˙ϕ+αϕr˙rw+αϕλ˙λw=γK2λwsin2ϕw,\n(24)\n−2 ˙ϕ+αrr˙rw+αrϕ˙ϕ+αrλ˙λw= 2γHa, (25)\nA/λ2\nw+αλr˙rw+αλϕ˙ϕ+αλλ˙λw=K1+K2cos2ϕw,\n(26)\nwhereαab=γΓab/MsS.\nWhen the Gilbert dampingtensorisisotropicandlocal\nin the basis of the Cartesian coordinates, ˜Γ =˜1δ(r−\nr′)Msα/γ\nαrr=2α\nλw;αϕϕ= 2αλw;αλλ=π2α\n6λw.(27)\nwhereas all off-diagonal elements vanish.\nMost experiments are carried out on thin film ferro-\nmagnetic wires for which K2is finite. Dissipation is es-\npecially simple below the Walker threshold, the regime\nin which the wall moves with a constant drift velocity,\n˙ϕw= 0 and23\n˙rw=−2γHa/αrr. (28)\nThe Gilbert damping coefficient αrrcan be obtained di-\nrectly from the scattering matrix by the parametric de-\npendence of the scattering matrix on the center coordi-\nnate position rw. When the Gilbert damping tensor is\nisotropic and local, we find ˙ rw=λwγHa/α. The domain\nwall width λw=/radicalbig\nA/(K1+K2cos2ϕw) and the out-\nof-plane angle ϕw=1\n2arcsin2γHa/αK2. At the Walker-\nbreakdownfield ( Ha)WB=αK2/(2γ) the sliding domain\nwall becomes unstable.\nIn a cylindrical wire without anisotropy, K2= 0,ϕwis\ntime-dependent and satisfies\n˙ϕw=−(2+αϕr)\nαϕϕ˙rw (29)\nwhile\n˙rw=2γHa\n2/parenleftBig\n2+αϕr\nαϕϕ/parenrightBig\n+αrr. (30)\nFor isotropic and local Gilbert damping coefficients,22\n˙rw\nλw=αγHa\n1+α2. (31)\nInthe nextsection, weformulatehowthe Gilbert scatter-\ning tensor can be computed from time-dependent scat-\ntering theory.IV. SCATTERING THEORY OF MESOSCOPIC\nMAGNETIZATION DYNAMICS\nScattering theory of transport phenomena24has\nproven its worth in the context of magnetoelectronics.\nIt has been used advantageously to evaluate the non-\nlocal exchange interactions multilayers or spin valves,11\nthe giantmagnetoresistance,25spin-transfertorque,9and\nspin pumping.10We first review the scattering theory\nof equilibrium magnetic properties and anisotropy fields\nand then will turn to non-equilibrium transport.\nA. Conservative forces\nConsidering only the electronic degrees of freedom in\nour model, the thermodynamic (grand) potential is de-\nfined as\nΩ =−kBTlnTre−(ˆHALDA−µˆN), (32)\nwhileµis the chemical potential, and ˆNis the number\noperator. The conservative force\nF=−∂Ω\n∂ξ. (33)\ncan be computed for an open systems by defining a scat-\nteringregionthat isconnectedby idealleadstoreservoirs\nat common equilibrium. For a two-terminal device, the\nflow of charge, spin, and energy between the reservoirs\ncan then be described in terms of the S-matrix:\nS=/parenleftbigg\nr t′\nt r′/parenrightbigg\n, (34)\nwhereris the matrix of probability amplitudes of states\nimpinging from and reflected into the left reservoir, while\ntdenotes the probability amplitudes of states incoming\nfrom the left and transmitted to the right. Similarly,\nr′andt′describes the probability amplitudes for states\nthat originate from the right reservoir. r,r′,t, andt′are\nmatricesin the space spanned by eigenstates in the leads.\nWe areinterested in the free magnetic energymodulation\nby the magnetic configuration that allows evaluation of\nthe forces Eq. (33). The free energy change reads\n∆Ω =−kBT/integraldisplay\ndǫ∆n(ǫ)ln/bracketleftBig\n1+e(ǫ−µ)/kBT/bracketrightBig\n,(35)\nwhere ∆n(ǫ)dǫis the change in the number of states at\nenergyǫand interval dǫ, which can be expressed in terms\nof the scattering matrix45\n∆n(ǫ) =−1\n2πi∂\n∂ǫTrlnS(ǫ). (36)\nCarrying out the derivative, we arrive at the force\nF=−1\n2πi/integraldisplay\ndǫf(ǫ)Tr/parenleftbigg\nS†∂S\n∂ξ/parenrightbigg\n,(37)6\nwheref(ǫ) is the Fermi-Dirac distribution function with\nchemical potential µ. This established result will be re-\nproducedandgeneralizedtothedescriptionofdissipation\nand fluctuations below.\nB. Gilbert damping as energy pumping\nHere we interpretGilbert damping asan energypump-\ning process by equating the results for energy dissipa-\ntion from the microscopic adiabatic pumping formalism\nwith the LLG phenomenology in terms of collective co-\nordinates, Eq. (9). The adiabatic energy loss rate of a\nscattering region in terms of scattering matrix at zero\ntemperature has been derived in Refs. 26,27. In the ap-\npendices, we generalize this result to finite temperatures:\n˙E=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ,t)\n∂t∂S†(ǫ,t)\n∂t/bracketrightbigg\n.(38)\nSince we employ the adiabatic approximation, S(ǫ,t) is\nthe energy-dependent scattering matrix for an instanta-\nneous (“frozen”)scattering potential at time t. In a mag-\nnetic system, the time dependence arises from its magne-\ntization dynamics, S(ǫ,t) =S[m(t)](ǫ). In terms of the\ncollective coordinates ξ(t),S(ǫ,t) =S(ǫ,{ξ(t)})\n∂S[m(t)]\n∂t≈/summationdisplay\na∂S\n∂ξa˙ξa, (39)\nwhere the approximate sign has been discussed in the\nprevious section. We can now identify the dissipation\ntensor (10) in terms of the scattering matrix\nΓab=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂ξa∂S†(ǫ)\n∂ξb/bracketrightbigg\n.(40)In the macrospin model the Gilbert damping tensor can\nthen be expressed as\n˜αij=γ/planckover2pi1\n4πMs/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂mi∂S†(ǫ)\n∂mj/bracketrightbigg\n,(41)\nwheremiis a Cartesian component of the magnetization\ndirection..\nC. Gilbert damping and fluctuation-dissipation\ntheorem\nAt finite temperatures the forces acting on the mag-\nnetization contain thermal fluctuations that are related\nto the Gilbert dissipation by the fluctuation-dissipation\ntheorem, Eq. (14). The dissipation tensor is therefore ac-\ncessible via the stochastic forces in thermal equilibrium.\nThe time dependence of the force operators\nˆF(t) =−∂ˆHALDA(m)\n∂ξ(42)\nis caused by the thermal fluctuations of the magneti-\nzation. It is convenient to rearrange the Hamiltonian\nˆHALDAinto an unperturbed part that does not de-\npend on the magnetization and a scattering potential\nˆHALDA(m) =ˆH0+ˆV(m). In the basis of scattering\nwave functions of the leads, the force operator reads\nˆF=−/integraldisplay\ndǫ/integraldisplay\ndǫ′/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫ′β/an}b∇acket∇i}htˆa†\nα(ǫ)ˆaβ(ǫ′)ei(ǫ−ǫ′)t//planckover2pi1, (43)\nwhere ˆaβannihilates an electron incident on the scatter-\ning region, βlabels the lead (left or right) and quantum\nnumbers of the wave guide mode, and |ǫ′β/an}b∇acket∇i}htis an associ-\nated scatteringeigenstateat energy ǫ′. We takeagainthe\nleft and rightreservoirsto be in thermal equilibrium with\nthe same chemical potentials, such that the expectation\nvalues\n/angbracketleftbig\nˆa†\nα(ǫ)ˆaβ(ǫ′)/angbracketrightbig\n=δαβδ(ǫ−ǫ′)f(ǫ).(44)\nTherelationbetweenthematrixelementofthescattering\npotential and the S-matrix\n/bracketleftbigg\nS†(ǫ)∂S(ǫ)\n∂ξ/bracketrightbigg\nαβ=−2πi/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫβ/an}b∇acket∇i}ht(45)follows from the relation derived in Eq. (61) below as\nwell as unitarity of the S-matrix,S†S= 1. Taking these\nrelationsintoaccount,the expectationvalueof ˆFisfound\nto be Eq. (37). We now consider the fluctuations in the\nforceˆf(t) =ˆF(t)− /an}b∇acketle{tˆF(t)/an}b∇acket∇i}ht, which involves expectation\nvalues\n/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)ˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n−/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)/angbracketrightbig/angbracketleftbig\nˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n=δα1β2δ(ǫ1−ǫ′\n2)δβ1α2δ(ǫ′\n1−ǫ2)f(ǫ1)[1−f(ǫ2)],\n(46)\nwhere we invoked Wick’s theorem. Putting everything7\ntogether, we finally find\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBTδ(t−t′)Γab, (47)\nwhere Γ abhas been defined in Eq. (40). Comparing with\nEq. (14), we conclude that the dissipation tensor Γ ab\ngoverningthe fluctuationsisidentical tothe oneobtained\nfrom the energy pumping, Eq. (40), thereby confirming\nthe fluctuation-dissipation theorem.\nV. KUBO FORMULA\nThe quality factor of the magnetization dynamics of\nmost ferromagnets is high ( α/lessorsimilar0.01). Damping can\ntherefore often be treated as a small perturbation. In\nthe presentSectionwedemonstratethat the dampingob-\ntained from linear response (Kubo) theory agrees28with\nthat ofthe scattering theory ofmagnetization dissipation\nin this limit. At sufficiently low temperatures or strong\nelastic disorder scattering the coupling to phonons may\nbe disregarded and is not discussed here.\nThe energy dissipation can be written as\n˙E=/angbracketleftBigg\ndˆH\ndt/angbracketrightBigg\n, (48)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htdenotes the expectation value for the non-\nequilibrium state. We are interested in the adiabatic\nresponse of the system to a time-dependent perturba-\ntion. In the adiabatic (slow) regime, we can at any time\nexpand the Hamiltonian around a static configuration at\nthe reference time t= 0,\nˆH=ˆHst+/summationdisplay\naδξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r).(49)\nThe static part, ˆHst, is the Hamiltonian for a magneti-\nzation for a fixed and arbitrary initial texture mst, as,\nwithout loss of generality, described by the collective\ncoordinates ξa. Since we assume that the variation of\nthe magnetization in time is small, a linear expansion in\nterms of the small deviations of the collective coordinate\nδξi(t) is valid for sufficiently short time intervals. We can\nthen employ the Kubo formalism and express the energy\ndissipation as\n˙E=/summationdisplay\naδ˙ξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r),(50)\nwhere the expectation value of the out-of-equilibrium\nconservative force\n/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r)≡∂aˆH (51)consists of an equilibrium contribution and a term linear\nin the perturbed magnetization direction:\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n(t) =/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/summationdisplay\nb/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t′).\n(52)\nHere, we introduced the retarded susceptibility\nχab(t−t′) =−i\n/planckover2pi1θ(t−t′)/angbracketleftBig/bracketleftBig\n∂aˆH(t),∂bˆH(t′)/bracketrightBig/angbracketrightBig\nst,(53)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htstis the expectation value for the wave functions\nof the static configuration. Focussing on slow modula-\ntions we can further simplify the expression by expand-\ning\nδξa(t′)≈δξa(t)+(t′−t)δ˙ξa(t), (54)\nso that\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n=/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t)+\n/integraldisplay∞\n−∞dt′χab(t−t′)(t′−t)δ˙ξb(t). (55)\nThe first two terms in this expression, /an}b∇acketle{t∂aˆH/an}b∇acket∇i}htst+/integraltext∞\n−∞dt′χab(t−t′)δξb(t),correspond to the energy vari-\nation with respect to a change in the static magnetiza-\ntion. These terms do not contribute to the dissipation\nsince the magnetic excitations are transverse, ˙ m·m= 0.\nOnly the last term in Eq. (55) gives rise to dissipation.\nHence, the energy loss reduces to29\n˙E=i/summationdisplay\nijδ˙ξaδ˙ξb∂χS\nab\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0, (56)\nwhereχS\nab(ω) =/integraltext∞\n−∞dt[χab(t)+χba(t)]eiωt/2. The\nsymmetrized susceptibility can be expanded as\nχS\nab=/summationdisplay\nnm(fn−fm)\n2/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}ht+(a↔b)\n/planckover2pi1ω+iη−(ǫn−ǫm),\n(57)\nwhere|n/an}b∇acket∇i}htis an eigenstate of the Hamiltonian ˆHstwith\neigenvalueǫn,fn≡f(ǫn),f(ǫ) is the Fermi-Dirac distri-\nbution function at energy ǫ, andηis a positive infinites-\nimal constant. Therefore,8\ni/parenleftbigg∂χS\nab\n∂ω/parenrightbigg\nω=0=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm), (58)\nand the dissipation tensor\nΓab=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm). (59)\nWe nowdemonstratethatthe dissipationtensorobtained\nfrom the Kubo linear response formula, Eq. (59), is\nidentical to the expression from scattering theory, Eq.\n(40), following the Fisher and Lee proof of the equiv-\nalence of linear response and scattering theory for the\nconductance.36\nThe static Hamiltonian ˆHst(ξ) =ˆH0+ˆV(ξ) can be\ndecomposed into a free-electron part ˆH0=−/planckover2pi12∇2/2m\nand a scattering potential ˆV(ξ). The eigenstates of ˆH0\nare denoted |ϕs,q(ǫ)/an}b∇acket∇i}ht,with eigenenergies ǫ, wheres=±\ndenotes the longitudinal propagation direction along the\nsystem (say, to the left or to the right), and qa trans-\nverse quantum number determined by the lateral con-\nfinement. The potential ˆV(ξ) scatters the particles be-tween the propagating states forward or backward. The\noutgoing (+) and incoming ( −) scattering eigenstates\nof the static Hamiltonian ˆHstare written as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n,\nwhichform anothercomplete basiswith orthogonalityre-\nlations/angbracketleftBig\nψ(±)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingleψ(±)\ns′,q′(ǫ′)/angbracketrightBig\n=δs,s′δq,q′δ(ǫ−ǫ′).33These\nwave functions can be expressed as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n= [1 +\nˆG(±)\nstˆV]|ϕs,q/an}b∇acket∇i}ht, where the retarded (+) and advanced ( −)\nGreen’s functions read ˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1. By\nexpanding Γ abin the basis of outgoing wave functions,\n|ψ(+)\ns,q/an}b∇acket∇i}ht, the energy dissipation (59) becomes\nΓab=π/summationdisplay\nsq,s′q′/integraldisplay\ndǫ/parenleftbigg\n−∂fs,q\n∂ǫ/parenrightbigg/angbracketleftBig\nψ(+)\ns,q/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂bˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns,q/angbracketrightBig\n, (60)\nwhere wave functions should be evaluated at the energy ǫ.\nLet us now compare this result, Eq. (60), to the direct scattering matrix expression for the energy dissipation,\nEq. (40). The S-matrix operator can be written in terms of the T-matrix as ˆS(ǫ;ξ) = 1−2πiˆT(ǫ;ξ), where the\nT-matrix is defined recursively by ˆT=ˆV[1+ˆG(+)\nstˆT]. We then find\n∂ˆT\n∂ξa=/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n.\nThe change in the scattering matrix appearing in Eq. (40) is then\n∂Ss′q′,sq\n∂ξa=−2πi/an}b∇acketle{tϕs,q|/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n|ϕs′,q′/an}b∇acket∇i}ht=−2πi/angbracketleftBig\nψ(−)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig\n. (61)\nSince\n/angbracketleftBig\nψ(−)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingle=/summationdisplay\ns′q′Ssq,s′q′/angbracketleftBig\nψ(+)\ns′q′(ǫ)/vextendsingle/vextendsingle/vextendsingle(62)\nandSS†= 1, we can write the linear response result,\nEq. (60), as energy pumping (40). This completes our\nproof of the equivalence between adiabatic energy pump-\ningintermsofthe S-matrixandtheKubolinearresponse\ntheory.VI. CONCLUSIONS\nWe have shown that most aspects of magnetization\ndynamics in ferromagnets can be understood in terms of\nthe boundary conditions to normal metal contacts, i.e.\na scattering matrix. By using the established numerical\nmethods to compute electron transport based on scatter-\ning theory, this opens the way to compute dissipation in\nferromagnets from first-principles. In particular, our for-9\nmalism should work well for systems with strong elastic\nscattering due to a high density of large impurity poten-\ntials or in disordered alloys, including Ni 1−xFex(x= 0.2\nrepresents the technologically important “permalloy”).\nThe dimensionless Gilbert damping tensors (41) for\nmacrospin excitations, which can be measured directly\nin terms of the broadening of the ferromagnetic reso-\nnance, havebeen evaluated for Ni 1−xFexalloysby ab ini-\ntiomethods.42Permalloy is substitutionally disordered\nand damping is dominated by the spin-orbit interaction\nin combination with disorder scattering. Without ad-\njustable parameters good agreement has been obtained\nwith the available low temperature experimental data,\nwhich is a strong indication of the practical value of our\napproach.\nIn clean samples and at high temperatures, the\nelectron-phonon scattering importantly affects damping.\nPhonons are not explicitly included here, but the scat-\ntering theory of Gilbert damping can still be used for\na frozen configuration of thermally displaced atoms, ne-\nglecting the inelastic aspect of scattering.12\nWhile the energy pumping by scattering theory has\nbeen applied to described magnetization damping,15it\ncan be used to compute other dissipation phenomena.\nThis has recently been demonstrated for the case of\ncurrent-induced mechanical forces and damping,43with\na formalism analogous to that for current-induced mag-\nnetization torques.13,14\nAcknowledgments\nWe would like to thank Kjetil Hals, Paul J. Kelly, Yi\nLiu, Hans Joakim Skadsem, Anton Starikov, and Zhe\nYuan for stimulating discussions. This work was sup-\nported by the EC Contract ICT-257159 “MACALO,”\ntheNSFunderGrantNo.DMR-0840965,DARPA,FOM,\nDFG, and by the Project of Knowledge Innovation Pro-\ngram(PKIP) of Chinese Academy of Sciences, Grant No.\nKJCX2.YW.W10\nAppendix A: Adiabatic Pumping\nAdiabatic pumping is the current response to a time-\ndependent scattering potential to first order in the time-\nvariation or “pumping” frequency when all reservoirsare\nat the same electro-chemical potential.38A compact for-\nmulation of the pumping charge current in terms of the\ninstantaneous scattering matrix was derived in Ref. 39.\nIn the same spirit, the energy current pumped out of the\nscattering region has been formulated (at zero tempera-\nture) in Ref. 27. Some time ago, we extended the charge\npumping concept to include the spin degree of free-\ndomandascertainedits importancein magnetoelectronic\ncircuits.10More recently, we demonstrated that the en-\nergyemitted byaferromagnetwith time-dependentmag-\nnetizations into adjacent conductors is not only causedby interface spin pumping, but also reflects the energy\nloss by spin-flip processes inside the ferromagnet15and\ntherefore Gilbert damping. Here we derive the energy\npumping expressions at finite temperatures, thereby gen-\neralizing the zero temperature results derived in Ref. 27\nand used in Ref. 15. Our results differ from an earlier ex-\ntension to finite temperature derived in Ref. 40 and we\npoint out the origin of the discrepancies. The magneti-\nzation dynamics must satisfy the fluctuation-dissipation\ntheorem, which is indeed the case in our formulation.\nWe proceed by deriving the charge, spin, and energy\ncurrentsintermsofthetimedependenceofthescattering\nmatrix of a two-terminal device. The transport direction\nisxand the transverse coordinates are ̺= (y,z). An\narbitrary single-particle Hamiltonian can be decomposed\nas\nH(r) =−/planckover2pi12\n2m∂2\n∂x2+H⊥(x,̺), (A1)\nwhere the transverse part is\nH⊥(x,̺) =−/planckover2pi12\n2m∂2\n∂̺2+V(x,̺).(A2)\nV(̺) is an elastic scattering potential in 2 ×2 Pauli\nspin space that includes the lattice, impurity, and\nself-consistent exchange-correlation potentials, including\nspin-orbit interaction and magnetic disorder. The scat-\nteringregionisattachedtoperfect non-magneticelectron\nwave guides (left α=Land rightα=R) with constant\npotential and without spin-orbit interaction. In lead α,\nthe transverse part of the 2 ×2 spinor wave function\nϕ(n)\nα(x,̺) and its corresponding transverse energy ǫ(n)\nα\nobey the Schr¨ odinger equation\nH⊥(̺)ϕ(n)\nα(̺) =ǫ(n)\nαϕ(n)\nα(̺), (A3)\nwherenis the spin and orbit quantum number. These\ntransverse wave guide modes form the basis for the ex-\npansion of the time-dependent scattering states in lead\nα=L,R:\nˆΨα=/integraldisplay∞\n0dk√\n2π/summationdisplay\nnσϕ(n)\nα(̺)eiσkxe−iǫ(nk)\nαt//planckover2pi1ˆc(nkσ)\nα,(A4)\nwhere ˆc(nkσ)\nαannihilates an electron in mode nincident\n(σ= +) or outgoing ( σ=−) in leadα. The field opera-\ntors satisfy the anticommutation relation\n/braceleftBig\nˆc(nkσ)\nα,ˆc†(n′k′σ′)\nβ/bracerightBig\n=δαβδnn′δσσ′δ(k−k′).\nThe total energy is ǫ(nk)\nα=/planckover2pi12k2/2m+ǫ(n)\nα. In the leads\nthe particle, spins, and energy currents in the transport10\ndirection are\nˆI(p)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†∂ˆΨ\n∂x−∂ˆΨ†\n∂xˆΨ/parenrightBigg\n,(A5a)\nˆI(s)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†σ∂ˆΨ\n∂x−∂ˆΨ†\n∂xσˆΨ/parenrightBigg\n,(A5b)\nˆI(e)=/planckover2pi1\n4mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†H∂ˆΨ\n∂x−∂ˆΨ†\n∂xHˆΨ/parenrightBigg\n+H.c.,\n(A5c)\nwhere we suppressed the time tand lead index α,σ=\n(σx,σy,σz) is a vector of Pauli matrices, and Tr sdenotes\nthe trace in spin space. Note that the spin current Is\nflows in the x-direction with polarization vector Is/Is.\nTo avoid dependence on an arbitrary global potential\nshift, it is convenient to work with heat ˆI(q)rather than\nenergy currents ˆI(ǫ):\nˆI(q)(t) =ˆI(ǫ)(t)−µˆI(p)(t), (A6)\nwhereµis the chemical potential. Inserting the waveg-uide representation (A4) into (A5), the particle current\nreads41\nˆI(p)\nα=/planckover2pi1\n4πm/integraldisplay∞\n0dkdk′/summationdisplay\nnσσ′(σk+σ′k′)×\nei(σk−σ′k′)xe−i/bracketleftBig\nǫ(nk)\nα−ǫ(nk′)\nα/bracketrightBig\nt//planckover2pi1ˆc†(nk′σ′)\nαˆc(nkσ)\nα.(A7)\nWeareinterestedinthelow-frequencylimitoftheFourier\ntransforms I(x)\nα(ω) =/integraltext∞\n−∞dteiωtI(x)\nα(t). Following Ref.\n41 we assume long wavelengths such that only the inter-\nvals withk≈k′andσ=σ′contribute. In the adiabatic\nlimitω→0 this approach is correct to leading order in\n/planckover2pi1ω/ǫF,whereǫFis the Fermi energy. By introducing the\n(current-normalized) operator\nˆc(nσ)\nα(ǫ(nk)\nα) =1/radicalBig\ndǫ(nkσ)\nα\ndkˆc(nkσ)\nα, (A8)\nwhich obey the anticommutation relations\n/braceleftBig\nˆc(nσ)\nα(ǫα),ˆc†(n′σ′)\nβ(ǫβ)/bracerightBig\n=δαβδnn′δσσ′δ(ǫα−ǫβ). (A9)\nThe charge current can be written as\nˆI(c)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\nǫ(n)\nαdǫdǫ′/summationdisplay\nnσσe−i(ǫ−ǫ′)t//planckover2pi1ˆc†(nσ)\nα(ǫ′)ˆc(nσ)\nα(ǫ). (A10)\nWeoperateinthe linearresponseregimeinwhichapplied\nvoltages and temperature differences as well as the exter-\nnally induced dynamics disturb the system only weakly.\nTransport is then governed by states close to the Fermi\nenergy. We may therefore extend the limits of the en-\nergy integration in Eq. (A10) from ( ǫ(n)\nα,∞) to (−∞\nto∞). We relabel the annihilation operators so that\nˆa(nk)\nα= ˆc(nk)\nα+denotes particles incident on the scattering\nregion from lead αandˆb(nk)\nα= ˆc(nk)\nα−denotes particles\nleavingthe scatteringregionbylead α. Using the Fourier\ntransforms\nˆc(nσ)\nα(ǫ) =/integraldisplay∞\n−∞dtˆc(nσ)\nα(t)eiǫt//planckover2pi1, (A11)\nˆc(nσ)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫˆc(nσ)\nα(ǫ)e−iǫt//planckover2pi1,(A12)\nwe obtain in the low-frequency limit41\nˆI(p)\nα(t) = 2π/planckover2pi1/bracketleftBig\nˆa†\nα(t)ˆaα(t)−ˆb†\nα(t)ˆbα(t)/bracketrightBig\n,(A13)\nwhereˆbαis a column vector of the creation operators forall wave-guidemodes {ˆb(n)\nα}. Analogouscalculations lead\nto the spin current\nˆI(s)\nα= 2π/planckover2pi1/parenleftBig\nˆa†\nασˆaα−ˆb†\nασˆbα/parenrightBig\n(A14)\nand the energy current\nˆI(e)\nα=iπ/planckover2pi12/parenleftBigg\nˆa†\nα∂ˆaα\n∂t−ˆb†\nα∂ˆbα\n∂t/parenrightBigg\n+H.c..(A15)\nNext, we express the outgoing operators ˆb(t) in terms\nof the incoming operators ˆ a(t) via the time-dependent\nscattering matrix (in the space spanned by all waveguide\nmodes, including spin and orbit quantum number):\nˆbα(t) =/summationdisplay\nβ/integraldisplay\ndt′Sαβ(t,t′)ˆaβ(t′).(A16)\nWhen the scattering region is stationary, Sαβ(t,t′) only\ndepends on the relative time difference t−t′, and its\nFourier transform with respect to the relative time is\nenergy independent, i.e.transport is elastic and can11\nbe computed for each energy separately. For time-\ndependent problems, Sαβ(t,t′) also depends on the total\ntimet+t′and there is an inelastic contribution to trans-\nport as well. An electron can originate from a lead with\nenergyǫ, pick up energy in the scattering region and end\nup in the same or the other lead with different energy ǫ′.\nThe reservoirs are in equilibrium with controlled lo-\ncal chemical potentials and temperatures. We insert the\nS-matrix (A16) into the expressions for the currents,Eqs. (A13), (A14), (A15), and use the expectation value\nat thermal equilibrium\n/angbracketleftBig\nˆa†(n)\nα(t2)ˆa(m)\nβ(t1)/angbracketrightBig\neq=δnmδαβfα(t1−t2)/2πℏ,(A17)\nwherefβ(t1−t2) = (2π/planckover2pi1)−1/integraltext\ndǫ−iǫ(t1−t2)//planckover2pi1fα(ǫ) and\nfα(ǫ) is the Fermi-Dirac distribution of electrons with\nenergyǫin theα-th reservoir. We then find\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)Sαβ(t,t1)fβ(t1−t2), (A18)\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)σˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)σSαβ(t,t1)fβ(t1−t2), (A19)\n2π/planckover2pi1/angbracketleftBig\n/planckover2pi1∂tˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2/bracketleftbig\n/planckover2pi1∂tS∗\nαβ(t,t2)/bracketrightbig\nSαβ(t,t1)fβ(t1−t2). (A20)\nNext, we use the Wigner representation (B1):\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫS/parenleftbiggt+t′\n2,ǫ/parenrightbigg\ne−iǫ(t−t′)//planckover2pi1, (A21)\nand by Taylor expanding the Wigner represented S-matrix S((t+t′)/2,ǫ) aroundS(t,ǫ), S((t+t′)/2,ǫ) =/summationtext∞\nn=0∂n\ntS(t,ǫ)(t′−t)n/(2nn!), we find\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2S(t,ǫ) (A22)\nand\n/planckover2pi1∂tS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2/parenleftbigg1\n2/planckover2pi1∂t−iǫ/parenrightbigg\nS(t,ǫ). (A23)\nThe factor 1 /2 scaling the term /planckover2pi1∂tS(t,ǫ) arises from commuting ǫwithei/planckover2pi1∂ǫ∂t/2. The currents can now be evaluated\nas\nI(c)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−fα(ǫ)/bracketrightBig\n(A24a)\nI(s)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig\nσ/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)/bracketrightBig\n(A24b)\nI(ǫ)\nα(t) =−1\n4π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1(−i/planckover2pi1∂t/2+ǫ)S†\nβα(ǫ,t)/parenrightBig/parenleftBig\ne+i∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n−1\n4π/planckover2pi1/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1(i/planckover2pi1∂t/2+ǫ)Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n,(A24c)\nwhere the adjoint of the S-matrix has elements S†(n′,n)\nβα=S∗(n,n′)\nαβ.\nWe are interested in the average (DC) currents, where simplified ex pressions can be found by partial integration\nover energy and time intervals. We will consider the total DC curren tsout ofthe scattering region, I(out)=−/summationtext\nαIα,\nwhen the electrochemical potentials in the reservoirs are equal, fα(ǫ) =f(ǫ) for allα. The averaged pumped spin and12\nenergy currents out of the system in a time interval τcan be written compactly as\nI(c)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−f(ǫ)/bracerightbigg\n, (A25a)\nI(s)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg\nσ/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†/bracerightbigg\n, (A25b)\nI(ǫ)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n+1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg/parenleftbigg\n−i/planckover2pi1∂S†\n∂t/parenrightbigg/bracerightbigg\n, (A25c)\nwhere Tr is the trace over all waveguide modes (spin\nand orbital quantum numbers). As shown in Ap-\npendix C the charge pumped into the reservoirs vanishes\nfor a scattering matrix with a periodic time dependence\nwhen,integrated over one cycle:\nI(p)\nout= 0. (A26)\nThis reflects particle conservation; the number of elec-\ntrons cannot build up in the scattering region for peri-\nodic variations ofthe system. We can showthat a similar\ncontribution to the energy current, i.e.the first line in\nEq. (A25c), vanishes, leading to to the simple expression\nI(e)\nout=−i\n2π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg∂S†\n∂t/bracerightbigg\n.\n(A27)\nExpanded to lowest order in the pumping frequency the\npumped spin current (A25b) becomes\nI(s)\nout=1\n2π/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nSS†f−i/planckover2pi1\n2∂S\n∂tS†∂ǫf/parenrightbigg\nσ/bracerightbigg\n(A28)\nThis formula is not the most convenient form to com-\npute the current to specified order. SS†also contains\ncontributions that are linear and quadratic in the pre-\ncession frequency since S(t,ǫ) is theS-matrix for a time-\ndependent problem. Instead, wewouldliketoexpressthe\ncurrent in terms of the frozenscattering matrix Sfr(t,ǫ).\nThe latter is computed for an instantaneous, static elec-\ntronic potential. In our case this is determined by a mag-\nnetization configuration that depends parametrically on\ntime:Sfr(t,ǫ) =S[m(t),ǫ]. Using unitarity of the time-dependentS-matrix (as elaborated in Appendix C), ex-\npand it to lowest order in the pumping frequency, and\ninsert it into (A28) leads to39\nI(s)\nout=i\n2π/summationdisplay\nβ/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftbigg∂Sfr\n∂tS†\nfrσ/bracerightbigg\n.\n(A29)\nWe evaluate the energy pumping by expanding (A27)\nto second order in the pumping frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\n−ifS∂S†\n∂t−(∂ǫf)1\n2∂S\n∂t∂S†\n∂t/bracerightbigg\n.\n(A30)\nAs a consequence of unitarity of the S-matrix (see Ap-\npendix C), the first term vanishes to second order in the\nprecession frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftBigg\n∂Sfr\n∂t∂S†\nfr\n∂t/bracerightBigg\n,(A31)\nwhere,at this point , we may insert the frozen scattering\nmatrix since the current expression is already propor-\ntional to the square of the pumping frequency. Further-\nmore, since there is no net pumped charge current in\none cycle (and we are assuming reservoirs in a common\nequilibrium), the pumped heat current is identical to the\npumped energy current, I(q)\nout=I(e)\nout.\nOur expression for the pumped energy current (A31)\nagrees with that derived in Ref. 27 at zero temperature.\nOur result (A31) differs from Ref. 40 at finite tempera-\ntures. The discrepancy can be explained as follows. In-\ntegration by parts over time tin Eq. (A27), using\n/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\ni/planckover2pi1∂S\n∂t/bracketrightbigg\nS†= 2/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−2/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†,(A32)\nand the unitarity condition from Appendix C,\n/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†=/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫǫf(ǫ), (A33)13\nthe DC pumped energy current can be rewritten as\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n. (A34)\nNext, we expand this to the second order in the pumping frequency and find\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\nǫf(ǫ)/parenleftbig\nSS†−1/parenrightbig\n−ǫ(∂ǫf)i/planckover2pi1\n2∂S\n∂tS†−ǫ(∂2\nǫf)/planckover2pi12\n8∂2S\n∂t2S†/bracerightbigg\n. (A35)\nThis form of the pumped energy current, Eq. (A35),\nagrees with Eq. (10) in Ref. 40 if one ( incorrectly ) as-\nsumesSS†= 1. Although for the frozen scattering ma-\ntrix,SfrS†\nfr= 1, unitarity does not hold for the Wigner\nrepresentation of the scattering matrix to the second or-\nder in the pumping frequency. ( SS†−1) therefore does\nnot vanish but contributes to leading order in the fre-\nquency to the pumped current, which may not be ne-\nglected at finite temperatures. Only when this term is\nincluded our new result Eq. (A31) is recovered.\nAppendix B: Fourier transform and Wigner\nrepresentation\nThere is a long tradition in quantum theory to trans-\nform the two-time dependence of two-operator correla-\ntion functions such as scattering matrices by a mixed\n(Wigner)representationconsistingofaFouriertransform\nover the time difference and an average time, which has\ndistinct advantages when the scattering potential varies\nslowlyintime.44Inordertoestablishconventionsandno-\ntations, we present here a short exposure how this works\nin our case.\nThe Fourier transform of the time dependent annihi-\nlation operators are defined in Eqs. (A11) and (A12).Consider a function Athat depends on two times t1\nandt2,A=A(t1,t2). The Wigner representation with\nt= (t1+t2)/2 andt′=t1−t2is defined as:\nA(t1,t2) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫA(t,ǫ)e−iǫ(t1−t2)//planckover2pi1,(B1)\nA(t,ǫ) =/integraldisplay∞\n−∞dt′A/parenleftbigg\nt+t′\n2,t−t′\n2/parenrightbigg\neiǫt′//planckover2pi1.(B2)\nWe also need the Wigner representation of convolutions,\n(A⊗B)(t1,t2) =/integraldisplay∞\n−∞dt′A(t1,t′)B(t′,t2).(B3)\nBy a series expansion, this can be expressed as44\n(A⊗B)(t,ǫ) =e−i(∂A\nt∂B\nǫ−∂B\nt∂A\nǫ)/2A(t,ǫ)B(t,ǫ) (B4)\nwhich we use in the following section.\nAppendix C: Properties of S-matrix\nHere we discuss some general properties of the two-\npoint time-dependent scattering matrix. Current conser-\nvation is reflected by the unitarity of the S-matrix which\ncan be expressed as\n/summationdisplay\nβn′s′/integraldisplay\ndt′S(α1β)\nn1s1,n′s′(t1,t′)S(α2β)∗\nn2s2,n′s′(t′,t2) =δn1n2δs1s2δα1α2δ(t1−t2). (C1)\nPhysically, this means that a particle entering the scattering region from a lead αat some time tis bound to exit the\nscattering region in some lead βat another (later) time t′. Using Wigner representation (B1) and integrating over\nthe local time variable, this implies (using Eq. (B4))\n1 =/parenleftbig\nS⊗S†/parenrightbig\n(t,ǫ) =e−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ), (C2)\nwhere 1 is a unit matrix in the space spanned by the wave guide modes ( labelled by spin sand orbital quantum\nnumbern). Similary, we find\n1 =/parenleftbig\nS†⊗S/parenrightbig\n(t,ǫ) =e+i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S†(t,ǫ)S(t,ǫ). (C3)\nTo second order in the precession frequency, by respectively sub tracting and summing Eqs. (C2) and (C3) give\nTr/braceleftbigg∂S\n∂t∂S†\n∂ǫ−∂S\n∂ǫ∂S†\n∂t/bracerightbigg\n= 0 (C4)14\nand\nTr/braceleftbig\nSS†−1/bracerightbig\n= Tr/braceleftbigg∂2S\n∂t2∂2S†\n∂ǫ2−2∂2S\n∂t∂ǫ∂2S†\n∂t∂ǫ+∂2S\n∂ǫ2∂2S†\n∂t2/bracerightbigg\n. (C5)\nFurthermore, foranyenergydependent function Z(ǫ)andarbitrarymatrixin thespacespannedbyspinandtransverse\nwaveguide modes Y, Eq. (C2) implies\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫZ(ǫ)Tr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ)−1/bracketrightbigg\nY/bracerightbigg\n= 0. (C6)\nIntegration by parts with respect to tandǫgives\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S\nt∂ZS†\nǫ/parenrightBig\n/2S(t,ǫ)Z(ǫ)S†(t,ǫ)−Z(ǫ)/bracketrightbigg\nY/bracerightbigg\n= 0, (C7)\nwhich can be simplified to\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg/bracketleftbigg\nZ/parenleftbigg\nǫ+i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\nS†(t,ǫ)−Z(ǫ)/parenrightbigg\nY/bracerightbigg\n= 0. (C8)\nSimilarly from (C3), we find\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nS†(t,ǫ)/bracketleftbigg\nZ/parenleftbigg\nǫ−i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\n−1/parenrightbigg\nY/bracerightbigg\n= 0. (C9)\nUsing this result for Y= 1 andZ(ǫ) =f(ǫ) in the\nexpression for the DC particle current (A25a), we see\nthat unitarity indeed implies particle current conserva-\ntion,/summationtext\nαI(c)\nα= 0 for a time-periodic potential. 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Grundler6,\u0003\n1Physik Department E10, Technische Universit at M unchen, D-85748 Garching, Germany\n2Physik Department E51, Technische Universit at M unchen, D-85748 Garching, Germany\n3Institute for Theoretical Physics, Universit at zu K oln, D-50937 K oln, Germany\n4Institut de Physique de la Mati\u0012 ere Complexe, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, 1015 Lausanne, Switzerland\n5Institut f ur Theoretische Physik, Technische Universit at Dresden, D-01062 Dresden, Germany\n6Institute of Materials and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN),\n\u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), Station 12, 1015 Lausanne, Switzerland\n(Dated: October 2, 2018)\nChiral magnets with topologically nontrivial spin order such as Skyrmions have generated enor-\nmous interest in both fundamental and applied sciences. We report broadband microwave spec-\ntroscopy performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetiza-\ntion dynamics we \fnd a remarkably small Gilbert damping parameter of about 1 \u000210\u00004at 5 K. This\nvalue is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium\niron garnet. We detect a series of sharp resonances and attribute them to con\fned spin waves in\nthe mm-sized samples. Considering the small damping, insulating chiral magnets turn out to be\npromising candidates when exploring non-collinear spin structures for high frequency applications.\nPACS numbers: 76.50.+g, 74.25.Ha, 4.40.Az, 41.20.Jb\nThe development of future devices for microwave ap-\nplications, spintronics and magnonics [1{3] requires ma-\nterials with a low spin wave (magnon) damping. In-\nsulating compounds are advantageous over metals for\nhigh-frequency applications as they avoid damping via\nspin wave scattering at free charge carriers and eddy\ncurrents [4, 5]. Indeed, the ferrimagnetic insulator yt-\ntrium iron garnet (YIG) holds the benchmark with a\nGilbert damping parameter \u000bintr= 3\u000210\u00005at room\ntemperature [6, 7]. During the last years chiral mag-\nnets have attracted a lot of attention in fundamental\nresearch and stimulated new concepts for information\ntechnology [8, 9]. This material class hosts non-collinear\nspin structures such as spin helices and Skyrmions be-\nlow the critical temperature Tcand critical \feld Hc2\n[10{12]. Additionally, Dzyaloshinskii-Moriya interaction\n(DMI) is present that induces both the Skyrmion lattice\nphase and nonreciprocal microwave characteristics [13].\nLow damping magnets o\u000bering DMI would generate new\nprospects by particularly combining complex spin order\nwith long-distance magnon transport in high-frequency\napplications and magnonics [14, 15]. At low tempera-\ntures, they would further enrich the physics in magnon-\nphoton cavities that call for materials with small \u000bintrto\nachieve high-cooperative magnon-to-photon coupling in\nthe quantum limit [16{19].\nIn this work, we investigate the Gilbert damping in\nCu2OSeO 3, a prototypical insulator hosting Skyrmions\n[20{23]. This material is a local-moment ferrimagnet\nwithTc= 58 K and magnetoelectric coupling [24] that\ngives rise to dichroism for microwaves [25{27]. The\nmagnetization dynamics in Cu 2OSeO 3has already been\nexplored [13, 28, 29]. A detailed investigation on thedamping which is a key quality for magnonics and spin-\ntronics has not yet been presented however. To eval-\nuate\u000bintrwe explore the \feld polarized state (FP)\nwhere the two spin sublattices attain the ferrimagnetic\narrangement[21]. Using spectra obtained by two di\u000ber-\nent coplanar waveguides (CPWs), we extract a minimum\n\u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K, i.e. only about four times\nhigher than in YIG. We resolve numerous sharp reso-\nnances in our spectra and attribute them to modes that\nare con\fned modes across the macroscopic sample and\nallowed for by the low damping. Our \fndings substanti-\nate the relevance of insulating chiral magnets for future\napplications in magnonics and spintronics.\nFrom single crystals of Cu 2OSeO 3we prepared two\nbar-shaped samples exhibiting di\u000berent crystallographic\norientations. The samples had lateral dimensions of\n2:3\u00020:4\u00020:3 mm3. They were positioned on CPWs that\nprovided us with a dynamic magnetic \feld hinduced by\na sinusoidal current applied to the signal surrounded by\ntwo ground lines. We used two di\u000berent CPWs with ei-\nther a broad [30] or narrow signal line width of ws= 1 mm\nor 20\u0016m, respectively [31]. The central long axis of the\nrectangular Cu 2OSeO 3rods was positioned on the central\naxis of the CPWs. The static magnetic \feld Hwas ap-\nplied perpendicular to the substrate with Hkh100iand\nHkh111ifor sample S1 and S2, respectively. The direc-\ntion ofHde\fned the z-direction. The dynamic \feld com-\nponent h?Hprovided the relevant torque for excita-\ntion. Components hkHdid not induce precessional mo-\ntion in the FP state of Cu 2OSeO 3. We recorded spectra\nby a vector network analyzer using the magnitude of the\nscattering parameter S12. We subtracted a background\nspectrum recorded at 1 T to enhance the signal-to-noisearXiv:1705.03416v1  [cond-mat.str-el]  9 May 20172\nratio (SNR) yielding the displayed \u0001 jS12j. In Ref. [7],\nKlingler et al. have investigated the damping of the in-\nsulating ferrimagnet YIG and found that Gilbert param-\neters\u000bintrevaluated from both the uniform precessional\nmode and standing spin waves con\fned in the macro-\nscopic sample provided the same values. For Cu 2OSeO 3\nwe evaluated \u000bin two ways[32]. When extracting the\nlinewidth \u0001 Hfor di\u000berent resonance frequencies fr, the\nGilbert damping parameter \u000bintrwas assumed to vary\naccording to [33, 34]\n\u00160\r\u0001\u0001H= 4\u0019\u000bintr\u0001fr+\u00160\r\u0001\u0001H0; (1)\nwhere\ris the gyromagnetic factor and \u0001 H0the contri-\nbution due to inhomogeneous broadening. Equation (1)\nis valid when viscous Gilbert damping dominates over\nscattering within the magnetic subsystem [35]. When\nperforming frequency-swept measurements at di\u000berent\n\feldsH, the obtained linewidth \u0001 fwas considered to\nscale linearly with the resonance frequency as [36]\n\u0001f= 2\u000bintr\u0001fr+ \u0001f0; (2)\nwith the inhomogeneous broadening \u0001 f0. The conver-\nsion from Eq. (1) to Eq. (2) is valid when frscales linearly\nwithHandHis applied along a magnetic easy or hard\naxis of the material [37, 38]. In Fig. 1 (a) to (d) we show\nspectra recorded in the FP state of the material using the\ntwo di\u000berent CPWs. For the same applied \feld Hwe ob-\nserve peaks residing at higher frequency fforHkh100i\ncompared to Hkh111i. From the resonance frequencies,\nwe extract the cubic magnetocrystalline anisotropy con-\nstantK= (\u00000:6\u00060:1)\u0001103J/m3for Cu 2OSeO 3[31].\nThe magnetic anisotropy energy is found to be extremal\nforh100iandh111ire\recting easy and hard axes, respec-\ntively [31]. The saturation magnetization of Cu 2OSeO 3\namounted to \u00160Ms= 0:13 T at 5 K[22].\nFigure 1 summarizes spectra taken with two di\u000ber-\nent CPWs on two di\u000berent Cu 2OSeO 3crystals exhibit-\ning di\u000berent crystallographic orientation in the \feld H.\nFor the narrow CPW [Fig. 1 (a) and (c)], we observed a\nbroad peak superimposed by a series of resonances that\nall shifted to higher frequencies with increasing H. The\n\feld dependence excluded them from being noise or arti-\nfacts of the setup. Their number and relative intensities\nvaried from sample to sample and also upon remounting\nthe same sample in the cryostat (not shown). They disap-\npeared with increasing temperature Tbut the broad peak\nremained. For the broad CPW [Fig. 1 (b) and (d)], we\nmeasured pronounced peaks whose linewidths were sig-\nni\fcantly smaller compared to the broad peak detected\nwith the narrow CPW. We resolved resonances below\nthe large peaks [arrows in Fig. 1 (b)] that shifted with\nHand exhibited an almost \feld-independent frequency\no\u000bset from the main peaks that we will discuss later. It\nis instructive to \frst follow the orthodox approach and\nanalyze damping parameters from modes re\recting the\n69121518-0.4-0.20.0(d)Δ  |S12|f\n (GHz)H\n || 〈111〉 \n69121518-6-30(c)Δ |S12| (10-2)f\n (GHz)\n-0.6-0.4-0.20.0H\n || 〈100〉 (b)broad CPWΔ |S12|\n-0.3-0.2-0.10.00\n.35 T(a)narrow CPWΔ |S12|0\n.25 T0\n.45 T0.55 TFIG. 1. (Color online) Spectra \u0001 jS12jobtained at T = 5 K\nfor di\u000berent Husing (a) a narrow and (b) broad CPW when\nHjjh100ion sample S1. Corresponding spectra taken on sam-\nple S2 for Hjjh111iare shown in (c) and (d), respectively.\nNote the strong and sharp resonances in (b) and (d) when us-\ning the broad CPW that provides a much more homogeneous\nexcitation \feld h. Arrows mark resonances that have a \feld-\nindependent o\u000bset with the corresponding main peaks and are\nattributed to standing spin waves. An exemplary Lorentz \ft\ncurve is shown in blue color in (b).\nexcitation characteristics of the CPW [29]. Second, we\nfollow Ref. [7] and analyze con\fned modes.\nLorentz curves (blue) were \ftted to the spectra\nrecorded with the broad CPW to determine resonance\nfrequencies and linewidths. Note that the corresponding\nlinewidths were larger by a factor ofp\n3 compared to the\nlinewidth \u0001 fthat is conventionally extracted from the\nimaginary part of the scattering parameters [39]. The\nextracted linewidths \u0001 fwere found to follow linear \fts\nbased on Eq. (2) at di\u000berent temperatures (details are\nshown in Ref. [31]). In Fig. 2 (a) we show a resonance\ncurve that was obtained as a function of Htaken with\nthe narrow CPW at 15 GHz. The curve does not show\nsharp features as Hwas varied in \fnite steps (symbols).\nThe linewidth \u0001 H(symbols) is plotted in Fig. 2 (b) for\ndi\u000berent resonance frequencies and temperatures. The\ndata are well described by linear \fts (lines) based on\nEq. (1). Note that the resonance peaks measured with\nthe broad CPW were extremely sharp. The sharpness\ndid not allow us to analyze the resonances as a function\nofH. We refrained from \ftting the broad peaks of Fig. 1\n(a) and (c) (narrow CPW) as they showed a clear asym-\nmetry attributed to the overlap of subresonances at \fnite\nwavevector k, as will be discussed below.\nIn Fig. 3 (a) and (b) we compare the parameter \u000bintr\nobtained from both di\u000berent CPWs (circles vs. stars) and\nthe two evaluation routes [40]. For Hkh100i[Fig. 3 (a)],\nbetween 5 and 20 K the lowest value for \u000bintramounts to\n(3.7\u00060.4)\u000210\u00003. This value is three times lower com-\npared to preliminary data presented in Ref. [29]. Beyond3\nFIG. 2. (Color online) (a) Lorentz curve (magenta line) \ftted\nto a resonance (symbols) measured at f= 15 GHz as a func-\ntion ofHat 5 K. (b) Frequency dependencies of linewidths\n\u0001H(symbols) for four di\u000berent T. We performed thep\n3-\ncorrection. The slopes of linear \fts (straight lines) following\nEq. 1 are considered to re\rect the intrinsic damping parame-\nters\u000bintr.\n04812H || 〈100〉 αintr (10-3)Δ H  narrow CPWΔ\n f    broad   CPW\nH || 〈111〉 \n1020304050T\n (K)\n10203040500.00.20.40.60.8Δf0 (GHz)T\n (K)(b)( a)(\nd)( c)\nFIG. 3. (Color online) (a) and (b) Intrinsic damping param-\neters\u000bintrand inhomogeneous broadening \u0001 f0for two di\u000ber-\nent \feld directions (see labels) obtained from the slopes and\nintercepts at fr= 0 of linear \fts to the linewidth data (see\nFig. 2 (b) and Ref. [31]). Dashed lines are guides to the eyes.\n20 K the damping is found to increase. For Hkh111i\n[Fig. 3 (b)] we extract (0.6 \u00060.6)\u000210\u00003as the smallest\nvalue. Note that these values for \u000bintrstill contain an ex-\ntrinsic contribution and thus represent upper bounds for\nCu2OSeO 3, as we will show later. For the inhomogeneous\nbroadening \u0001 f0in Fig. 3 (c) and (d) the datasets are\nconsistent (we have used the relation \u0001 f0=\r\u0001H0=2\u0019\nto convert \u0001 H0into \u0001f0). We see that \u0001 f0increases\nwithTand is small for the broad CPW, independent\nof the crystallographic direction of H. For the narrow\nCPW the inhomogeneous broadening is largest at small\nTand then decreases by about 40 % up to about 50\nK. Note that a CPW broader than the sample is as-\nsumed to excite homogeneously at fFMR [41] transfer-\nring a wave vector k= 0 to the sample. Accordinglywe ascribe the intense resonances of Fig. 1 (b) and (d) to\nfFMR. UsingfFMR= 6 GHz and \u000bintr= 3:7\u000210\u00003at 5\nK [Fig. 3 (a)], we estimate a minimum relaxation time of\n\u001c= [2\u0019\u000bintrfr]\u00001= 6:6 ns.\nIn the following, we examine in detail the additional\nsharp resonances that we observed in spectra of Fig. 1.\nIn Fig. 1 (b) taken with the broad CPW for Hkh100i,\nwe identify sharp resonances that exhibit a characteris-\ntic frequency o\u000bset \u000efwith the main resonance at all\n\felds (black arrows). We illustrate this in Fig. 4(a) in\nthat we shift spectra of Fig. 1 (b) so that the positions of\ntheir main resonances overlap. The additional small res-\nonances (arrows) in Fig. 1 (b) are well below the uniform\nmode. This is characteristic for backward volume magne-\ntostatic spin waves (BVMSWs). Standing waves of such\nkind can develop if they are re\rected at least once at the\nbottom and top surfaces of the sample. The resulting\nstanding waves exhibit a wave vector k=n\u0019=d , with\norder number nand sample thickness d= 0:3 mm. The\nBVMSW dispersion relation f(k) of Ref. [13] provides a\ngroup velocity vg=\u0000300 km/s at k=\u0019=d[triangles in\nFig. 4 (b)]. Hence, the decay length ld=vg\u001camounts\nto 2 mm considering \u001c= 6:6 ns. This is larger than\ntwice the relevant lateral sizes, thereby allowing stand-\ning spin wave modes to form in the sample. Based on\nthe dispersion relation of Ref. [13], we calculated the fre-\nquency splitting \u000ef=fFMR\u0000f(n\u0019=d ) [open diamonds\nin Fig. 4 (b)] assuming n= 1 andt= 0:4 mm for the\nsample width tde\fned in Ref. [13]. Experimental val-\nues (\flled symbols) agree with the calculated ones (open\nsymbols) within about 60 MHz. In case of the narrow\nCPW, we observe even more sharp resonances [Fig. 1 (a)\nand (c)]. A set of resonances was reported previously\nin the \feld-polarized phase of Cu 2OSeO 3[26, 28, 42, 43].\nMaisuradze et al. assigned secondary peaks in thin plates\nof Cu 2OSeO 3to di\u000berent standing spin-wave modes [43]\nin agreement with our analysis outlined above.\n0.30.40.51.101.151.201.25-\n500-300-100100δf (GHz)(b)/s61549\n0H (T)v\ng (km/s)\n-10 -0.8-0.6-0.4-0.20.0f\n - f (0)  (GHz)H || 〈100〉 (a)b\nroad CPWΔ |S12|δ\nf\nFIG. 4. (Color online) (a) Spectra of Fig. 1 (b) replotted as\nf\u0000fFMR(H) for di\u000berent Hsuch that all main peaks are at\nzero frequency and the \feld-independent frequency splitting\n\u000efbecomes visible. The numerous oscillations seen particu-\nlarly on the bottom most curve are artefacts from the cali-\nbration routine. (b) Experimentally evaluated (\flled circles)\nand theoretically predicted (diamonds) splitting \u000efusing dis-\npersion relations for a platelet. Calculated group velocity vg\natk=\u0019=(0:3 mm). Dashed lines are guides to the eyes.4\nThe inhomogeneous dynamic \feld hof the narrow\nCPW provides a much broader distribution of kcom-\npared to the broad CPW. This is consistent with the\nfact that the inhomogeneous broadening \u0001 f0is found to\nbe larger for the narrow CPW compared to the broad\none [Fig. 3 (c) and (c)]. Under these circumstances, the\nexcitation of more standing waves is expected. We at-\ntribute the series of sharp resonances in Fig. 1 (a) and\n(c) to such spin waves. In Fig. 5 (a) and (b) we highlight\nprominent and particularly narrow resonances with #1,\n#2 and #3 recorded with the narrow CPW. We trace\ntheir frequencies fras a function of HforHkh100iand\nHkh111i, respectively. They depend linearly on Hsug-\ngesting a Land\u0013 e factor g= 2:14 at 5 K.\nWe now concentrate on mode #1 for Hk h100iat\n5 K that is best resolved. We \ft a Lorentzian line-\nshape as shown in Fig. 5(c) for 0.85 T, and summarize\nthe corresponding linewidths \u0001 fin Fig. 5(d). The inset\nof Fig. 5(d) shows the e\u000bective damping \u000be\u000b= \u0001f=(2fr)\nevaluated directly from the linewidth as suggested in Ref.\n[29]. We \fnd that \u000be\u000bapproaches a value of about 3.5\n\u000210\u00004with increasing frequency. This value includes\nboth the intrinsic damping and inhomogeneous broad-\nening but is already a factor of 10 smaller compared to\n\u000bintrextracted from Fig. 3 (a). Note that Cu 2OSeO 3\nexhibiting 3.5\u000210\u00004outperforms the best metallic thin-\n\flm magnet [44]. To correct for inhomogeneous broad-\nening and determine the intrinsic Gilbert-type damping,\nwe apply a linear \ft to the linewidths \u0001 fin Fig. 5(d) at\nfr>10:6 GHz and obtain (9.9 \u00064.1)\u000210\u00005. Forfr\u0014\n10.6 GHz the resonance amplitudes of mode #1 were\nsmall reducing the con\fdence of the \ftting procedure.\nFurthermore, at low frequencies, we expect anisotropy to\nmodify the extracted damping, similar to the results in\nRef. [45]. For these reasons, the two points at low frwere\nleft out for the linear \ft providing (9.9 \u00064.1)\u000210\u00005.\nWe \fnd \u0001fand the damping parameters of Fig. 3 to\nincrease with T. It does not scale linearly for Hkh100i\n[31]. A deviation from linear scaling was reported for\nYIG single crystals as well and accounted for by the con-\n\ruence of a low- kmagnon with a phonon or thermally\nexcited magnon [5]. In the case of Hkh111i(cf. Fig. 3\n(b)) we obtain a clear discrepancy between results from\nthe two evaluation routes and CPWs used. We relate\nthis observation to a misalignment of Hwith the hard\naxish111i. The misalignment motivates a \feld-dragging\ncontribution [38] that can explain the discrepancy. For\nthis reason, we concentrated our standing wave analysis\non the case Hkh100i. We now comment on our spectra\ntaken with the broad CPW that do not show the very\nsmall linewidth attributed to the con\fned spin waves.\nThe sharp mode #1 yields \u0001 f= 15:3 MHz near 16 GHz\n[Fig. 5 (d)]. At 5 K the dominant peak measured at 0.55 T\nwith the broad CPW provides however \u0001 f= 129 MHz.\n\u0001fobtained by the broad CPW is thus increased by a\nfactor of eight and explains the relatively large Gilbert\nFIG. 5. (Color online) (a)-(b) Resonance frequency as a func-\ntion of \feld Hof selected sharp modes labelled #1 to #3 (see\ninsets) for Hkh100iandHkh111iat T = 5 K. (c) Exemplary\nLorentz \ft of sharp mode #1 for Hkh100iat 0.85 T. (d) Ex-\ntracted linewidth \u0001f as a function of resonance frequency fr\nalong with the linear \ft performed to determine the intrinsic\ndamping\u000bintrin Cu 2OSeO 3. Inset: Comparison among the\nextrinsic and intrinsic damping contribution. The red dotted\nlines mark the error margins of \u000bintr= (9:9\u00064:1)\u000210\u00005.\ndamping parameter in Fig. 3 (a) and (b). We con\frmed\nthis larger value on a third sample with Hkh100iand ob-\ntained (3.1\u00060.3)\u000210\u00003[31] using the broad CPW. The\ndiscrepancy with the damping parameter extracted from\nthe sharp modes of Fig. 5 might be due to the remaining\ninhomogeneity of hover the thickness of the sample lead-\ning to an uncertainty in the wave vector in z-direction.\nFor a standing spin wave such an inhomogeneity does\nnot play a role as the boundary conditions discretize k.\nAccordingly, Klingler et al. extract the smallest damp-\ning parameter of 2 :7(5)\u000210\u00005reported so far for the\nferrimagnet YIG when analyzing con\fned magnetostatic\nmodes [7].\nTo summarize, we investigated the spin dynamics in\nthe \feld-polarized phase of the insulating chiral mag-\nnet Cu 2OSeO 3. We detected numerous sharp reso-\nnances that we attribute to standing spin waves. Their\ne\u000bective damping parameter is small and amounts to\n3:5\u000210\u00004. A quantitative estimate of the intrinsic\nGilbert damping parameter extracted from the con\fned\nmodes provides even \u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K. The\nsmall damping makes an insulating ferrimagnet exhibit-\ning Dzyaloshinskii-Moriya interaction a promising can-\ndidate for exploitation of complex spin structures and\nrelated nonreciprocity in magnonics and spintronics.\nWe thank S. Mayr for assistance with sample prepa-\nration. Financial support through DFG TRR80, DFG\n1143, DFG FOR960, and ERC Advanced Grant 291079\n(TOPFIT) is gratefully acknowledged.5\n\u0003Electronic mail: dirk.grundler@ep\r.ch\n[1] I. Zutic and H. Dery, Nat. Mater. 10, 647 (2011).\n[2] M. Krawczyk and D. Grundler, J. Phys.: Condens. Mat-\nter26, 123202 (2014).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. 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Phys. 85(1999)."    },    {        "title": "1309.5523v4.Patterns_formation_in_axially_symmetric_Landau_Lifshitz_Gilbert_Slonczewski_equations.pdf",        "content": "arXiv:1309.5523v4  [math.AP]  30 Mar 2017Pattern formation in axially symmetric\nLandau-Lifshitz-Gilbert-Slonczewski equations\nOctober 2, 2018\nC. Melcher1& J.D.M. Rademacher2\nAbstract\nThe Landau-Lifshitz-Gilbert-Slonczewski equation describes mag netization dynamics\nin the presence of an applied field and a spin polarized current. In the case of axial sym-\nmetry and with focus on one space dimension, we investigate the eme rgence of space-time\npatterns in the form of wavetrainsand coherent structures, wh ose local wavenumbervaries\nin space. A major part of this study concerns existence and stabilit y of wavetrains and of\nfront- and domain wall-type coherent structures whose profiles a symptote to wavetrains\nor the constant up-/down-magnetizations. For certain polarizat ion the Slonczewski term\ncan be removed which allows for a more complete charaterization, inc luding soliton-type\nsolutions. Decisive for the solution structure is the polarization par ameteras well as size of\nanisotropy compared with the difference of field intensity and curre nt intensity normalized\nby the damping.\n1 Introduction\nThis paper concerns the analysis of spatio-temporal patter n formation for the axially sym-\nmetric Landau-Lifshitz-Gilbert-Slonczewski equation fo r which the applied magnetic field and\ncurrent are aligned with or orthogonal to the material aniso tropy. In one space dimension we\nthus consider\n∂tm=m×/bracketleftBig\nα∂tm−∂2\nxm+(µm3−h)ˆe3+β\n1+ccpm3m׈e3/bracketrightBig\n(1)\nasamodelforthemagnetization dynamics m=m(x,t)∈S2(i.e.misadirectionfield)driven\nby an external field h=hˆe3and current j=β\n1+ccpm3ˆe3with polarization parameter ccp∈\n(−1,1). The parameters α >0 andµ∈Rare the Gilbert damping factor and the anisotropy\nconstant, respectively. A brief overview of thephysical ba ckground and interpretation of terms\nis given below in Section 2.\nThe constant up- or down-magnetization states m=±ˆe3are always steady states of (1)\nand magnetic domain walls spatially separating these state s are of major interest. While\nthe combination of field and current excitations gives rise t o a variety of pattern formation\nphenomena, see e.g. [5, 14, 15, 19, 23], not much mathematica lly rigorous work is available so\nfar, in particular for the dissipative case α >0 that we consider. The case of axial symmetry\nis not only particularly convenient from a technical perspe ctive. It offers at the same time\nvaluable insight in the emergence of space-time patterns an d displays strong similarities to\nbetter studied dynamical systems such as real and complex Gi nzburg-Landau equations. In\n1Lehrstuhl I f¨ ur Mathematik and JARA-FIT, RWTH Aachen Unive rsity, 52056 Aachen, Germany,\nmelcher@rwth-aachen.de\n2Fachbereich 3 – Mathematik, Universit¨ at Bremen, Postfach 33 04 40, 28359 Bremen, Germany,\nrademach@math.uni-bremen.de\n1m1m2m3\n(a) (b)\nFigure 1: Illustration of a wavetrain profile m(ϕ) (a) in the 2-sphere showing their constant\naltitude and (b) as a space-time plot of, e.g., m2. In (a) the thick arrow represents m=\n(m,m3), the thin line its trajectory as a function of ϕ=kx−ωt.\nthis framework we examine the existence and stability of wav etrain solutions of (1), i.e.,\nsolutions of the form\nm(x,t) =ei(kx−ωt)m0,\nwhere the complex exponential acts on m0∈S2by rotation about the ˆe3-axis (cf. Figure 1\nfor an illustration). In the special case ccp= 0 it turns out that (1) can be transformed to the\nvariational LLG-equation with β= 0: in a rotating frame about the m3-axis with frequency\n−β/αthe current dependent term vanishes and hchanges to h−β/α; see§2.1. This allows\nfor a complete characterization of wavetrains and their L2-stability.\nWe also investigate the existence of coherent structure sol utions which are locally in space\nof wavetrain form\nm(x,t) =eiϕ(x,t)m0(x−st) where ϕ(x,t) =φ(x−st)+Ωt (2)\nsuch that m0(ξ) = (sin θ(ξ),0,cosθ(ξ)). Samples are plotted in Figures 2, 3, 4. In the\nvariational case ccp= 0 we completely characterize the existence of small amplit ude coherent\nstructuresandstationary ( s= 0) ones, whichinfactcorrespondtostandingwaves intheab ove\nrotating frame. Throughthecoherent structureviewpoint w e recover a family of ‘homogenous’\ndomain wall type solutions of arbitrary velocity, having no azimuthal profile, i.e., constant φ\nand thus vanishing local wavenumber d ϕ/dx. For general ccpand large speeds, |s| ≫1,\nwe prove existence of a family of more general front-type coh erent structures with nontrivial\nlocal wavenumbers, which can also form a spatial interface b etween±ˆe3and wavetrains. The\nanalysis of these kinds of solutions is inspired by and bears similarities with that of the real\nand complex Ginzburg-Landau equations.\nMore specifically, the parameter space for existence of wave trains and coherent structures\nis largely organized by the stability of the equilibrium sta tesm=±ˆe3. The nature of\nbifurcations that we find motivates the following notions to organize the parameter space of\n(1): We refer to parameters as being\n•‘supercritical’ if ±ˆe3are both unstable\n•‘subcritical’ if ±ˆe3have different stability\n•‘subsubcritical’ if ±ˆe3are both stable.\n20102030405060/Minus0.50.00.5\nxm2\n050100150200/Minus0.8/Minus0.400.4\nxm2\n/Minus0.8/Minus0.4 0 0.4/Minus0.50.00.5\nm1m2\n/Minus0.8/Minus0.6/Minus0.4/Minus0.20.00.2/Minus0.8/Minus0.400.4\nm1m2\n(a) (b)\nFigure 2: Plots of coherent structure profiles for supercrit ical anisotropy with µ= 7,h−Ω =\n−1,Ω =β/α,ccp= 0, and first integral C= 1, cf. (43). (a) A quasi-periodic solution that\nmaps to a solution with period ≈16 in the reduced equations (44). (b) The same solution\ntype with period 200 closer to a soliton-type solution with w avetrain as its asymptotic state.\nFrom a physical viewpoint µandαare material specific, while h,βare control parameters. In\nour exposition we choose µas a primary parameter and speak of super-, sub- or subsubcri tical\nanisotropy; one may also choose βorhat the price of less convenient conditions.\nOur results may be summarized somewhat informally as follow s.\nThe up- and down-magnetization equilibria ±ˆe3.(Lemma 1) Let β±:=β/(1±ccp).\nThe constant state m=ˆe3is strictly stable if and only if µ < h−β+/αandm=−ˆe3\nif and only if µ <−(h−β−/α). Instabilities are of Hopf-type for the essential spectru m\nwith onset via spatially homogenous modes of frequency β/α. In other words, stability of\n±ˆe3changes when the difference of signed anisotropy ±µand the force balance h−β±\nα, of\nmagnetic field strength minus the ratio of current-polariza tion intensity and damping factor,\nchanges sign. This corresponds to the well known instabilit y threshold in the more broadly\nstudied ODE for solutions that are homogeneous in space. Not ably, for ccp= 0 the anisotropy\nis subsubcritical precisely for −µ >|h−β/α|(‘easy-axis’), and supercritical precisely for\nµ >|h−β/α|(‘easy-plane’).\nFast and small amplitude coherent structure. (Theorems 6, 7 and corollaries) For each\nsufficiently large speed there exists a family of front-type c oherent structures parametrized by\ntheazimuthal frequency. Theirprofilesconnect ±ˆe3witheach otheror, if therearewavetrains,\nthere are fronts connecting these and/or ±ˆe3in the order of altitudes. An example is plotted\nin Figure 3. Small amplitude coherent structures are of fron t type and, for ccp= 0, exist only\nfor super- and subcritical anistropy.\n3−1 −0.5 0 0.5 1−101\nm1m3\n−1 −0.5 0 0.5 1−101\nm1m2\n0 100 200 30000.511.522.53\nxθ, q\n(a) (b)\nFigure 3: Profile of a ‘fast’ front connecting the wavetrain a nd the unstable −ˆe3computed\nwith the coherent structure ODE guided by the asymptotic pre diction of equation (35). Here\nµ= 1,h= 0.5,s= 5,Ω = 2,ccp= 0 and the asymptotic wavetrain on the left has wavenumber\nk= Ω/s= 0.4, and is spectrally stable.\nWavetrains. In the case ccp= 0 (Theorems 2, 4) for each wavenumber k∈Rat most one\nwavetrain exists, and moreover:\n1.Supercritical anisotropy: Wavetrains exist precisely for kwith|k|>/radicalbig\nµ+|h−β/α|or\n0≤ |k|</radicalbig\nµ−|h−β/α|. There is (explicitly known) k∗∈(0,/radicalbig\nµ−|h−β/α|such\nthat all wavetrains with |k|< k∗are stable and sideband unstable for |k|> k∗.\n2.Subcritical anisotropy: Wavetrain exists precisely for kwith|k|>/radicalbig\nµ+|h−β/α|, but\nare all unstable.\n3.Subsubcritical anisotropy: Wavetrains exist for all k, but are all unstable.\nThe overall picture for wavetrains of (1) with ccp= 0 can be viewed as a combination of\nthose in a supercritical and a subcritical real Ginzburg-La ndau equation; see Figure 11.\nFor general ccp∈(−1,1) additional effects are (1) a nontrivial nonlinear dispersi onrelation\nω(k) with nonzero group velocitiesd\ndkω(k), (2) the occurrence of ‘hyperbolic’ and ‘elliptic’\nbifurcation points of wavetrains and (3) coexistence of sta ble wavetrains and stable ±ˆe3.\nWavetrains for k2> µare always unstable (Theorems 2, 3), but for ccp/ne}ationslash= 0 wavetrains are\npotentially convectively but not absolutely unstable, tho ugh we do not investigate this here.\nDomain walls for ccp= 0.(Theorem 5) For any µ <0 there exists a family of fronts whose\nspatial profiles connect ±ˆe3withθ′=√−µsin(θ), and that are ‘homogeneous’ in the sense\nthatq≡0 so there is no azimuthal profile. They corresponds to well kn own domains walls of\nthe LLG-equation3. Here we readily locate these within the coherent structure framework.\nStationary coherent structures for ccp= 0.(Theorems 8, 9)\n1.Supercritical anisotropy: For fixed parameters there exist various stationary coheren t\nstructures ( s= 0) including ‘homogeneous’ ones (cf. Figure 4). An interes ting case of\n3After acceptance of the present manuscript for publication , we found these were also obtained in [12].\n4−15−10−5051015−1−0.500.51\nxm1\n−15−10−5051015−1−0.500.51\nxm1\n−15−10−5051015−1−0.500.51\nxm1\n−15−10−5051015−1−0.500.51\nxm3\n−15−10−5051015−1−0.500.51\nxm3\n−15−10−5051015−1−0.500.51\nxm3\n(a) (b) (c)\nFigure 4: Snapshots of sample homogeneous coherent structu res with spatially periodic pro-\nfiles, having d φ/dξ=q= 0. Compare Figure 12. Here ccp= 0,α= 1 and Ω = βis arbitrary.\n(a) Near a pair of domains walls ( µ=−1,h= 10−4−β). (b) Near an upward ‘phase slip\nsoliton’ with plateaus near the oscillation at m3=h−β(µ= 1,h= 0.8−β), and (c) near an\nupward-downward pair of such solitons ( µ= 1,h= 0.8−β).\nthe latter is a symmetric pair of ‘phase slip’ soliton-type c oherent structures, whose\nspatially asymptotic states are the same spatially homogen eous oscillation ( k= 0), but\nthe intermediate profile crosses either ˆe3or−ˆe3, so that the asymptotic states differ\nazimuthally by 180◦. There also exists a non-homogeneous soliton-type solutio n with\nasymptotic state being a wavetrain (cf. Figure 2).\n2.Sub- and subsubcritical anisotropy: All stationary coherent structures have periodic\nprofiles except a homogeneous phase slip soliton with spatia lly asymptotic state ±ˆe3for\nsgn(h−β/α) =±1.\nHigher space dimensions. The model for Nspace dimensions has the second derivative\nwith respect to xin (1) replaced by a Laplace operator/summationtextN\nj=1∂2\nxj. Wavetrain type solutions\nare then of the form\nm(x,t) =m∗(k·x−ωt),\nwherek= (k1,...,kN). Notably, for kj= 0, 2≤j≤Nthese are solutions from one space\ndimension extended trivially (constant) in the additional directions.\nConveniently, the rotation symmetry (gauge invariance in t he Ginzburg-Landau context),\nmeans that the analyses of ±ˆe3and these wavetrains is already covered by that of the one-\ndimensional case: the linearization is space-independent and therefore there is no symmetry\nbreaking due to different kj. Indeed, all relevant quantities are rotation symmetric, d epending\nonly onk2=/summationtextN\nj=1k2\njorℓ2=/summationtextN\nj=1ℓ2\nj, whereℓ= (ℓ1,...,ℓN) is the Fourier wavenumber vec-\ntor of the linearization. In particular, the instabilities occur simultaneously for all directions.\n5Concerning coherent structures, in higher space dimension the defining equation (see (33)\nbelow) turns into an elliptic PDE in general. The analysis in this paper only covers the trivial\nconstant extension into higher dimensions.\nThis paper is organized as follows. In Section 2, the terms in the model equation (1) and\nits well-posedness are discussed. Section 3 concerns the st ability of the trivial steady states\n±ˆe3and in§4 existence and stability of wavetrains are analyzed. Secti on 5 is devoted to\ncoherent structures.\nAcknowledgement. JR has been supported in part by the NDNS+ cluster of the Dutch\nScience Fund (NWO). We thank the anonymous reviewers for sug gestions that helped improve\nthe manuscript, and Lars Siemer as well as Ivan Ovsyannikov f or their critical reading.\n2 Review of Landau-Lifshitz-Gilbert-Slonczewski equatio ns\nThe classical equation of dissipative magnetization dynam ics, the Landau-Lifshitz-Gilbert\nequation [11, 22] for unit vector fields m=m(x,t)∈S2,\n∂tm=m×(α∂tm−γheff).\nfeatures a damped precession of maround the effective field heff=−δE(m), i.e., minus the\nvariational derivative of the interaction energy E=E(m). The gyromagnetic ratio γ >0 is\na parameter which appears as the typical precession frequen cy. By rescaling time, one can\nalways assume γ= 1. The Gilbert damping factor α >0 is a constant that can be interpreted\ndynamically as the inverse of the typical relaxation time. I t is useful to take into account that\nthere are several equivalent forms of LLG. Elementary algeb raic manipulations taking into\naccount that −m×m×ξ=ξ−(m·ξ)myield the so-called Landau-Lifshitz form\n(1+α2)∂tm=−m×(αm×heff+heff), (3)\nintroduced in the original work [22]. In case α >0, the energy E(m) is not conserved but is\na Lyapunov functional, i.e., more precisely (recall heff=−δE(m))\nd\ndtE(m(t)) =−α/bardbl∂tm(t)/bardbl2or equivalentlyd\ndtE(m(t)) =−α\n1+α2/bardblm×heff/bardbl2.\nGilbert damping enables the magnetization to approach (spi ral down to) a steady state, i.e.\nsatisfying m×heff= 0 (Browns equation), as t→ ∞.\nSpin-torque interaction. The system can be driven out of equilibrium conventionally b y\nan external magnetic field hwhich appears as part of the effective field. In modern spin-\ntronic applications, magnetic systems are excited by spin p olarized currents (with direction of\npolarization ˆep∈S2) giving rise to a spin torque\nm×m×jwherej=βˆep\n1+ccpm·ˆep, (4)\nwhich has been introduced in [2, 34]. Here, the parameters β >0 andccp∈(−1,1) depend\non the intensity of the current and ratio of polarization [4] . Typically we have ˆep=ˆe3. Ac-\ncordingly, the modified Landau-Lifshitz-Gilbert equation , also called Landau-Lifshitz-Gilbert-\nSlonczewski equation (LLGS), reads\n∂tm=m×(α∂tm−heff+m×j). (5)\n6One may extend the notion of effective field to include current i nteraction by letting\nHeff=heff−m×j,\nwhere the second term is usually called Slonczewski term. In this framework (5) can also be\nwritten in the form (3) with heffreplaced by Heff. Observe, however, that the Slonczewski\nterm(andhence Heff)isingeneralnon-variational andthattheenergyisnolong eraLyapunov\nfunctional. Introducing the potential Ψ( m) =β\nccpln(1+ccpm·ˆep) ofj(forccp/ne}ationslash= 0) reveals\ntheskew variational structure\nm×[α∂tm+δE(m)] =−m×m×[∂tm+δΨ(m)],\nsee[6]. Inthemicromagnetic model theunderlyinginteract ion energies areintegral functionals\ninmcontaining in particular exchange (Dirichlet) interactio n, dipolar stray-field interaction,\ncrystal anisotropy and Zeeman interaction with external ma gnetic field, see e.g. [16]. In this\npaper we shall mainly focus on the spatially one-dimensiona l situation and consider energies\nof the form\nE(m) =1\n2/integraldisplay/parenleftbig\n|∂xm|2+µm2\n3/parenrightbig\ndx−/integraldisplay\nh·mdx. (6)\nHere,h∈R3is a constant applied magnetic field. The parameter µ∈Rfeatures easy plane\nanisotropy for µ >0andeasy axis anisotropyfor µ <0, respectively, accordingtoenergetically\npreferred subspaces. This term comprises crystalline and s hape anisotropy effects. Shape\nanisotropy typically arises from stray-field interactions which prefer magnetizations tangential\nto the sample boundaries. Hence µ >0 corresponds to a thin-film perpendicular to the ˆe3-\naxis whereas µ <0 corresponds to a thin wire parallel to the ˆe3-axis. The effective field\ncorresponding to (6) reads\nheff=∂2\nxm−µm3ˆe3+h. (7)\nWith the choices h=hˆe3and ˆep=ˆe3, the Landau-Lifshitz-Gilbert-Slonczewski equation\n(5) exhibits the aforemented rotation symmetry about the ˆe3-axis. The presence of a spin\ntorquem×m×jexerted by a constant current may induce switching between m agnetization\nstates or magnetization oscillation [3, 4, 7]. For the latte r effect, the energy supply due to\nthe electric current compensates the energy dissipation du e to damping enabling a stable\noscillation, called precessional states . In applications the typical frequency is in the range of\nGHz, so that a precessional state would basically act as a mic rowave generator. In the class\nof spatially homogeneous states, precessional states are p eriodic orbits with m3=const.and\nof constant angular velocity β/αwhenccp= 0. It is more subtle, however, to understand the\noccurrence and stability of spatially non-homogeneous pre cessional states. This is the theme\nof this paper.\nExtensions and related work. There is a wealth of literature studying the dynamics of\nrelated Landau-Lifschitz models with and without damping a nd axial symmetry and including\neffects other than spin-torque interaction and as general ref erence we mention the book [6] as\nwell as the review article [21]. More specifically, spatiall y non-trivial states and their stabililty\nhave been considered in [18], where the spin-torque part of t he effective field is replaced by\na demagnetization term solving Maxwell’s equation. Also co upled nano-oscillators of LLGS\ntype have been considered widely, e.g. recently in [33, 35]. Recently, for a situation without\naxial symmetry, Turing patterns of spin states have been num erically found in [23].\n7Non-symmetric variants of our equation (1) have been used e. g. in the description of\nthe field driven motion of a flat domain wall connecting antipo dal steady states m3=±1\nasx1→ ±∞. A prototypical situation is the field driven motion of a flat B loch wall in an\nuniaxial the bulk magnet governed by\n∂tm=m×/parenleftbig\nα∂tm−∂2\nxm+µ1m1ˆe1+(µ3m3−h)ˆe3/parenrightbig\n. (8)\nIn this case µ1>0> µ3, whereµ1corresponds to stray-field and µ3to crystalline anisotropy.\nExplicit traveling wave solutions were obtained in unpubli shed work by Walker, see e.g. [16],\nand reveal interesting effects such as the existence of a termi nal velocity (called Walker ve-\nlocity) and the notion of an effective wall mass. A mathematica l account on Walker’s explicit\nsolutions and investigations on their stability, possible extensions to finite layers and curved\nwalls can be found e.g. in [9, 25, 30]. Observe that our axiall y symmetric model is obtained\nin the limit µ1ց0. On the other hand, the singular limit µ3→+∞leads to trajectories\nconfined to the {m3= 0}plane (equator map), and can be interpreted as a thin-film lim it.\nIn suitable parameter regimes it can be shown that the limit e quation is a dissipative wave\nequation governing the motion of N´ eel walls [8, 24, 26].\nWell-posedness of LLGS. It is well-known that Landau-Lifshitz-Gilbert equations a nd its\nvariants have the structure of quasilinear parabolic syste ms. In the specific case of (1), one\nhas the extended effective field Heff=heff−m×j, more precisely\nHeff=∂2\nxm−f(m) where f(m) = (µm3−h)ˆe3+β\n1+ccpm3m׈e3.(9)\nHence the corresponding Landau-Lifshitz form (3) of (1) rea ds\n(1+α2)∂tm=−m×/bracketleftBig\n∂2\nxm−f(m)/bracketrightBig\n−αm×m×/bracketleftBig\n∂2\nxm−f(m)/bracketrightBig\n. (10)\nTaking into account\nm×∂2\nxm=∂x(m×∂xm) and −m×m×∂2\nxm=∂2\nxm+|∂xm|2m,(11)\nvalid for msufficiently smooth and |m|= 1, one sees that (10) has the form\n∂tm=∂x(A(m)∂xm)+B(m,∂xm) (12)\nwith analytic functions A:R3→R3×3andB:R3×R3→R3such that A(m) is uniformly\nelliptic for α >0, in fact\nξ·A(m)ξ=α\n1+α2|ξ|2for allξ∈R3.\nWell-posedness results for α >0 can now be deduced from techniques based on higher\norder energy estimates as in [27, 28] or maximal regularity a nd interpolation as in [29]. In\nparticular, weshallrelyonresultsconcerningperturbati onsofwavetrains, travelingwaves, and\nsteady states. Suppose m∗=m∗(x,t) is a smooth solution of (1) with bounded derivatives\nup to all high orders (only sufficiently many are needed) and m0:R→S2is such that\nm0−m∗(·,0)∈H2(R). Then there exist T >0 and a smooth solution m:R×(0,T)→S2\nof (1) such that m−m∗∈C0([0,T);H2(R))∩C1([0,T);L2(R)) with\nlim\ntց0/bardblm(t)−m0/bardblH2= 0 and lim\ntրT/bardblm(t)−m∗(·,t)/bardblH2=∞ifT <∞.\n8The solution is unique in its class and the flow map depends smo othly on initial conditions\nand parameters.\nGiven the smoothness of solutions, we may compute pointwise ∂t|m|2= 2m·∂tm, so that\nfor|m|= 1 the cross product form of the right hand side of (10) gives ∂t|m|2= 0. Hence, the\nset of unit vector fields, {|m|= 1}, is an invariant manifold of (12) consisting of the solution s\nto (1) that we are interested in.\nIn addition to well-posedness, also stability and spectral theory for (12), see, e.g., [29],\ncarry over to (1). In particular, the computations of L2-spectra in the following sections are\njustified and yield nonlinear stability for strictly stable spectrum and nonlinear instability for\nthe unstable (essential) spectrum.\nLandau-Lifshitz-Gilbert-Slonczewski versus complex Ginzburg-Landau equations.\nStereographic projection of (1) yields\n(α+i)ζt=∂2\nxζ−2¯ζ(∂xζ)2\n1+|ζ|2+µ(1−|ζ|2)ζ\n1+|ζ|2−(h+iβ)ζwhereζ=m1+im2\n1+m3,\nvalid for magnetizations avoiding the south pole.\nStudying LLG-type equations via stereographic projection has a long history and has\nbeen employed in several of the aforementioned references, see, e.g., the review [21] and the\nreferences therein.mThere is also a global connection betw een LLG and CGL in the spirit\nof the classical Hasimoto transformation [13], which turns the (undamped) Landau-Lifshitz\nequation in one space dimension ( heff=∂2\nx) into the focussing cubic Schr¨ odinger equation\n[20, 36]. The idea is to disregard the customary coordinates representation and to introduce\ninstead a pull-back frame on the tangent bundle along m. In the case of µ=β=h= 0, i.e.\nheff=∂xm, this leads to\n(α+i)Dtu=D2\nxu (13)\nwhereu=u(x,t) is the complex coordinate of ∂xmin the moving frame representation,\nandDxandDtare covariant derivatives in space and time giving rise to cu bic and quintic\nnonlinearities, see [27, 28] for details.\n2.1 Symmetry and the variational structure for ccp= 0\nThe aforementioned rotation symmetry of (1) about the m3-axis of all terms manifests as\nan equivariance of the right hand side of (12) with respect to any such rotation Rϕ: let\nm=Rϕ/tildewiderm, then\n∂x(A(m)∂xm)+B(m,∂xm) =Rϕ(∂x(A(/tildewiderm)∂x/tildewiderm)+B(/tildewiderm,∂x/tildewiderm)).\nFor a rotating frame m=eiΩtm, write the rotation about the ˆe3-axis asRΩtand note\nthat time derivatives become ∂tm=RΩt(ΩR∗\nΩtR′\nΩt/tildewiderm+∂t/tildewiderm), where R∗\nΩtR′\nΩtm=m׈e3.\nTherefore, having ccp= 0, (1) is also an equation for /tildewidermwith the parameter βchanged to\nβ+αΩ(from∂tmwithin the brackets) and htoh+Ω(from∂tmon the left hand side).\nIn other words, for ccp= 0, changing spin torque current has the same effect as changin g the\nmagnetic field.\n9Choosing Ω=−β/αyieldsβ= 0 in (1), which is therefore variational with respect to\nthe energy (6) as discussed above. This has strong structura l consequences for the coherent\nstructures (2) and allows for a (largely) complete characte rization. In particular, it turns\nout that the existence of coherent structures that are stati onary (s= 0), but not necessarily\ntime independent, requires their superimposed azimuthal f requency Ω to satisfy Ω = β/α; see\n§5.3. The reduction to β= 0 thus implies Ω = 0 and therefore turns the stationary coher ent\nstructures into standing waves and hence to time-independe nt solutions.\n3 Hopf instabilities of the steady states m=±ˆe3\nAs a starting point and to motivate the subsequent analysis o f more complex patterns, let\nus consider the stability of the constant magnetizations ±ˆe3. It is well-known that a Hopf\nbifurcation of these states occurs in the ODE associated to ( 1) in the absence of diffusion,\nthat is, for spatially constant solutions. In the following , we account in addition for spatial\ndependence.\nUse the shorthand β±=β/(1±ccp). Substituting m=±ˆe3+δn+o(δ), where n=\n(n,0)∈TmS2, into (1) gives, at order δ, the linear equation\n∂tn= (±µ−h)n׈e3±ˆe3×(α∂tn−∂2\nxn+β±n׈e3),\nwhich may be written in complex form as\n∂tn±β±n=i/parenleftbig\nα∂tn−∂2\nxn−(h∓µ)n/parenrightbig\n.\nIts eigenvalue problem diagonalizes in Fourier space (for x) and yields the matrix eigenvalue\nproblem/vextendsingle/vextendsingle/vextendsingle/vextendsingle±β±−λΛ\n−Λ±β±−λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,\nwhere Λ = ±µ−h∓αλ∓ℓ2withℓthe Fourier wave number. The determinant reads\n(±β±−λ)2=−Λ2⇔ ±β±−λ=σiΛ, σ∈ {±1}.\nConsidering real and imaginary parts this leads to\n(1+α2)Re(λ) =±β±−α(ℓ2±h−µ) =α(µ∓(h−β±/α)−ℓ2)\nIm(λ) =σ(∓Re(λ)+β±/α),\nso that the maximal real part has ℓ= 0. At criticality, where Re( λ) = 0 the imaginary\nparts are ±β±/α, which (if nonzero) corresponds to a so-called Hopf-instab ility of the (purely\nessential) spectrum and we expect the emergence of oscillat ing solutions whose frequency at\nonset isβ±/α[31]. Since the critical mode has ℓ= 0, the onset of instability coincides with\nthe aforementioned Hopf-bifurcation of the ODE associated with diffusionless (1).\nThe formulas for real- and imaginary parts immediately give the results mentioned in §1\nand\nLemma 1 The constant state m=ˆe3is (strictly) L2-stable if and only if µ < µ+:=h−β+/α\nandm=−ˆe3if and only if µ < µ−:=−(h−β−/α). Instabilities are of Hopf-type for the\nessential spectrum and have frequency β/α.\nForccp= 0the anisotropy is subsubcritical precisely for −µ >|h−β/α|, and supercritical\nprecisely for µ >|h−β/α|.\n104 Wavetrains\nTo exploit the rotation symmetry about the ˆe3-axis, we change to polar coordinates in the\nplanar components m= (m1,m2) of the magnetization m= (m,m3). With m=rexp(iϕ)\nequation (1) changes to\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbiggr2∂tϕ\n∂tm3/parenrightbigg\n=/parenleftbigg∂x(r2∂xϕ)\n∂2\nxm3+|∂xm|2m3/parenrightbigg\n+r2/parenleftbiggβ/(1+ccpm3)\nh−µm3/parenrightbigg\n,(14)\nwhere\n|∂xm|2= (∂xr)2+r2(∂xϕ)2+(∂xm3)2andr2+m2\n3= 1.\nThis can be seen as follows. In view of (3), with heffreplaced by the extended effective field\nHeff=heff−m×jas in (9) and taking into account (11), (1) reads\nα∂tm+m×∂tm=∂2\nxm+(h−µm3)ˆe3+/parenleftbig\n|∂xm|2+µm2\n3−hm3/parenrightbig\nm−β\n1+ccpm3m׈e3.\nThe third component of the above equation is the second compo nent of (14), whereas the first\ncomponent of (14) is obtained upon inner multiplication by m⊥= (m⊥,0) = (ieiϕ,0) and\ntaking into account that m׈e3=−m⊥.\nTherotation symmetryhasturnedintotheshiftsymmetry ϕ/ma√sto→ϕ+const. Infullspherical\ncoordinates m=/parenleftbigeiϕsinθ\ncosθ/parenrightbig\n, (14) further simplifies to\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbiggsinθ∂tϕ\n−∂tθ/parenrightbigg\n=/parenleftbigg2cosθ∂xθ∂xϕ+sinθ∂2\nxϕ\n−∂2\nxθ+sinθcosθ(∂xϕ)2/parenrightbigg\n+sinθ/parenleftbiggβ/(1+ccpcos(θ))\nh−µcosθ/parenrightbigg\n.(15)\n4.1 Existence of wavetrains\nWavetrains are solutions of the form m(x,t) =m∗(kx−ωt), where kis referred to as the\nwavenumber and ωas the frequency. A natural type of wavetrains are relative e quilibria with\nrespect tothephaseshiftsymmetry forwhich ϕ=kx−ωtandm3,rareconstant. SeeFigure1\nfor an illustration.\nTheorem 1 Wavetrains with frequency ωand wavenumber kare in one-to-one correspon-\ndence to solutions of\nΓ(ω,k) :=ccpαω(ω+h)−(β+αω)(k2−µ) = 0,\nunder the constraint |(ω+h)/(k2−µ)| ≤1. In particular, for each kthere are at most two\nvalues of ωthat yield a wavetrain, and for each ω/ne}ationslash=−β/αthere is at most one value of k2\nthat gives a wavetrain, unless ccpαω(ω+h) = 0forω=−β/α. Moreover, for |ccp|<1,\n(a)ω/ne}ationslash= 0,k2/ne}ationslash=µ,sgn(ω) =−sgn(β)andω∈[min{−β±/α},max{−β±/α}].\n(b) As|k| → ∞we have ω→ −β/αandm3→0.\n(c) Bifurcations of k∼0fromk= 0for fixed ωare unfolded for increasing µ.\n11/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ1.55\n/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ1.57\nFigure 5: Plots of wavetrain locations in the ( k,ω)-plane when parameters pass through\nan elliptic bifurcation point. Labels are the values of µ. Other parameters are fixed at\nccp= 0.5,h= 2,α= 1,β= 2.1sothat ω±=−β±,β+= 1.4,β−= 4.2andµ+= 0.6,µ−= 2.2.\nTherefore, µ= 1.55,1.57 are both subcritical with ˆe3stable and −ˆe3unstable. Shaded regions\nhave|m3|>1. The elliptic point lies at µsn≈1.558,ωsn=−2.558.\n(d) Bifurcations for m/ne}ationslash=±ˆe3occur atω=ωsn:=β±√\nβ(β−4αh)\n2αwith|m3|<1if|1\nccp−h\nµ|<2\nand are either:\n... a hyperbolic point for k/ne}ationslash= 0, ifh=β/αand then ω=−h,k2=−ccph+µ,\n... a hyperbolic point for k= 0, ifsgn((ccph−µ)(ccph+µ)) = 1,\n... an elliptic point for k= 0, ifsgn((ccph−µ)(ccph+µ)) =−1.\nIn the following we discuss the existence problem and thereb y prove each statement of the\ntheorem.\nThe terms elliptic and hyperbolic refers to the use in [32] an d will be explained below.\nNote that the latter two bifurcation types do not occur for ccp= 0. That case is considered\nin detail in §4.2. We note that Γ depends on konly through k2−µso thatµ≥0 is the same\nasµ= 0 up to change in wavenumber and solutions for fixed ωcan increase |k|only through\nincreasing µ.\nSubstituting the wavetrain ansatz into (15) yields the alge braic equations\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbigg−sin(θ)ω\n0/parenrightbigg\n=/parenleftbigg0\nsin(θ)cos(θ)k2/parenrightbigg\n+sin(θ)/parenleftbiggβ/(1+ccpcos(θ))\nh−µcos(θ)/parenrightbigg\n.\nThus either θ≡0 modπor (recall |ccp|<1)\n−αω=β\n1+ccpm3,−ω= (k2−µ)m3+h. (16)\nIn the first case we have r= 0, which corresponds to the constant upward or downward\nmagnetizations, ( r,m3) = (0,±1) with unspecified kandω. In the second case, we notice\naside that absence of dissipation ( α= 0) requires absence of current ( β= 0) and there is a\ntwo-parameter set of wavetrains given by the second equatio n. The case we are interested in\nisα >0 and then β= 0 requires ω= 0, and this falls into the special case ccp= 0.\n12/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ0.55\n/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ0.64\n/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ0.65\nFigure 6: Analogue of Figure 5 with fixed parameters as there, whenµpasses through a\nhyperbolic bifurcation point at k= 0 with µsn≈0.641,ωsn=−1.641. Note that between\nµ= 0.55 andµ= 0.64 the upper branch enters the region |m3| ≤1 at a bifurcation of in this\ncaseˆe3atµ=µ+= 0.6. Therefore, µ= 0.64,0.65 are supercritical with ±ˆe3both unstable.\n/Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5\nkΩ1.99\n/Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5\nkΩ2\n/Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5\nkΩ2.001\nFigure 7: Analogue of Figure 5 when parameters pass through h yperbolic bifurcation points\nwithk/ne}ationslash= 0 so that h=β/α. Notably this cannot be unfolded by variation of µ. Labels are\nthe values of βand other parameters are fixed at ccp= 0.5,h= 2,α= 1,µ= 1.2. This is\nsubcritical with ˆe3unstable since µ+= 2(1−β/3),µ−= 2(β−1) and 1.2∈(µ+,µ−).\nIn the generic case β,ccp/ne}ationslash= 0, the first equation implies that ω≈0 is not possible for\n|m3| ≤1 and we obtain\nm3=−1\nccp/parenleftbiggβ\nαω+1/parenrightbigg\n, m 3=−ω+h\nk2−µ(17)\nwhere for k2=µwe have ω=−hand the first equation holds.\nEliminating m3and rearranging terms gives the existence condition in term s ofωandk\nas zeros of Γ( ω,k) as in the theorem. For ω/ne}ationslash=−β/αthis gives k2as a quadratic function\nofωinverse to the nonlinear dispersion relation ω(k). The exceptional ω=−β/αoccurs\nprecisely when h=β/αand implies m3= 0, i.e. a solution on the equator (or ccp= 0).\nThe strict monotonicity of Γ in kaway from k= 0 also means that upon parameter change\nnew solution branches can emerge only through local extrema of Γ atk= 0, i.e., an ‘elliptic’\npoint. Specifically, this occurs if at a critical point ∂2\nkΓ(ω,0) = 2(β+αω) has the same sign\n13as∂2\nωΓ(ω,0) = 2αccpand, e.g. Γ(0 ,0) =βµvaries.\nIn particular, Γ( ω,k) =∂ωΓ(ω,k) = 0 occurs at\nαω2+β(2ω+h) = 0 (18)\nand is a fold point of wavetrains (in the form of homogeneous o scillations) with fixed k. For\nk= 0 these have at frequency and parameters (recall ccp/ne}ationslash= 0)\nωsn=−ccph+µ\n2ccp,4βµccp=α(µ+ccph)2. (19)\nNote that |m3|<1 fork= 0 is by (17) equivalent to |ωsn+h|<|µ|which yields |1\nccp−h\nµ|<2.\nAt such critical point we also have\n∂2\nkΓ(ω,0) =−α\nccp(ccph−µ)(ccph+µ), (20)\nso that the relative size of ccphandµdetermines whether such a bifurcation point is elliptic\nor hyperbolic in the language of [31]. In terms of critical pa rameters, substituting ω=ωsn+˜ω\nand, for instance µ=µsn+ ˜µto unfold with µand other parameters fixed we obtain\n˜µ=k2−ccpα\nβ+αωsn˜ω2(21)\nwhich gives the options of hyperbola or ellipses for level se ts. Hyperbolic points are saddle\npoints of Γ and at such points the connectivity of existing br anches changes. For k/ne}ationslash= 0 this\noccurs in particular, if h=β/αwhen the two branches of Γ = 0 are the line ω=−hfor any\nkandω=1\nccp(k2−µ).\nWe plot examples of these situations in Figures 5, 6, 7.\nMore globally, since Γ is quadratic in ωthere are at most two solutions for each kand by\nstrict monotonicity in k2, away from h=β/α, there is at most one solution for each ωor the\nwhole line ω=−β/α. The only complication is the constraint |m3| ≤1 – dispersion curves\ntouch the boundary m3= 1 at bifurcations of ±ˆe3, which were studied in §3. The figures\nillustrate the essential scenarios.\n4.2 Existence in the case ccp= 0\nIn this case the existence conditions can be conveniently wr itten as\nω=−β\nα(22)\ncos(θ) =h−β/α\nµ−k2,(µ/ne}ationslash=k2). (23)\nAs expected from the variational structure in rotating coor dinates discussed in §2.1, all wave-\ntrains oscillate with frequency given by the ratio of applie d current and dissipation. In par-\nticular, in this case the natural representation of wavetra ins is that ( θ,k)-plane rather than\n(ω,k) as above.\nAn involution symmetry involving parameters is\n(h−β/α,θ)→(β/α−h,θ+π), (24)\n14/Minus4/Minus20240Π2Π\nkΘ\n/Minus4/Minus20240Π2Π\nkΘ\n/Minus4/Minus20240Π2Π\nkΘ\n(a) (b) (c)\nFigure 8: Plots of equilibrium locations in the ( k,m3)-plane including the trivial equilibria\n±ˆe3plotted with thick line if stable (when not intersecting wav etrain parameters). Compare\nFigure 9. See (23). (a) supercritical (easy plane) anisotro py (µ= 1,h−β/α= 1/2), (b)\nsubcritical anisotropy ( µ= 1,h−β/α= 2), where no homogeneous oscillations ( k= 0) exist,\nand (c) subsubcritical (easy axis) anisotropy µ=−1,h−β/α= 0.9.\nsupercritical\nsubcritical\nsubsubcritical−|h−β/α||h−β/α|\nσˆe3stable±ˆe3unstable\n±ˆe3stableµ\nk\nFigure 9: Sketch of existence region (shaded) in the ( k,µ)-plane, with boundary given by (25)\nforσ:= sgn(h−β/α)/ne}ationslash= 0. The sign of σdetermines which of ±ˆe3is stable in the subcritical\nrange.\nso that the sign of h−β/αis irrelevant for the qualitative picture.\nSolvability of (23) requires that |µ−k2|>|h−β/α|(unlessr= 0), so that only for super-\nand subsubcritical anisotropy,\n|µ|>|h−β/α|, (25)\nthere exist wavetrains with wavenumber in an interval aroun dk= 0. In other words, non-\ntrivial spatially nearly homogeneous oscillations requir e sufficiently small (in absolute value)\ndifferencebetweenappliedmagneticfieldandoscillationfre quency(ratioofappliedcurrentand\ndissipation). Thetransitionintothisregimegoesviathe‘ Hopf’instability from §3. Combining\n(25) with Lemma 1 and straightforward analysis of (23) gives the following lemma. The three\ntypes of solution sets are plotted in Figure 8. Clearly, the s olution sets are symmetric with\nrespect to the signs of kandθ, respectively.\nTheorem 2 There are three types of wavetrain parameter sets solving (23):\n151. For supercritical anisotropy there is one connected comp onent of wavetrain parameters\nincluding k= 0, and two connected components with unbounded |k|, each with constant\nsign ofk.\n2. For subcritical anisotropy there are two connected compo nents with unbounded |k|, each\nwith constant sign of k.\n3. For subsubcritical anisotropy there are two connected co mponents, each a graph over the\nk-axis.\nThe Hopf-type instabilities of ±ˆe3noted in §3 at the transition from sub- to supercritical\nanisotropy is a supercritical bifurcation in the sense that solutions emerge at the loss of stability\nof the basic solution, here ±ˆe3, while that from sub- to subsubcritical is subcritical in th e sense\nthat solutions emerge at the gain of stability.\nProof. Let us consider the existence region of wavetrains in wavenu mber-parameter space.\nFrom (23), r= 0 atθ≡0 modπgives the boundary for nontrivial amplitude,\nµ=k2±(h−β/α), (26)\nas a pair quadratic parabolas in ( k,µ)-space. The solution set in this projection is sketched in\nFigure9. Remark that this set is non-empty for any parameter setα,β,h∈Rof (1). However,\nas in the general case not all wavenumbers are possible due to the geometric constraint.\nNotably, the existence region consists of two disjoint sets , one contained in {µ >0}with\nconvex boundary and one extending into {µ <0}with concave boundary.\n4.3 Stability of wavetrains\nIn this section we discuss spectral stability of wavetrains and in summary we obtain the\nfollowing result.\nTheorem 3\n(a) Wavetrains bifurcating from ±ˆe3atk= 0are stable if µ±>max{0,ccpβ±/α}, which\nimplies supercritical bifurcation, i.e., for increasing µ. They are unstable if µ±<\nmax{0,ccpβ±/α}, which means subcritical bifurcation, i.e., decreasing µ.\n(b) Forµ >0there isk∗>0such that precisely the wavetrains with wavenumber |k|< k∗\nandµ > k2+αω2ccp/βare sideband stable. These are fully spectrally stable if βccp≥0\norα≥0is sufficiently small. A wavetrain and ˆe3or−ˆe3can be simultaneously stable\nonly for ccp/ne}ationslash= 0.\n(c) For each kat most one wavetrain can be stable. Wavetrains with r∼0andk/ne}ationslash= 0are\nunstable. All wavetrains with k2> µare unstable.\n(d) Near the hyperbolic or elliptic bifurcation points at k= 0the sideband stable wavetrains\nlie in a sector that is to leading order bounded by |ω−ωsn|=S|k|whose opening angle\nless than πand which includes ωwith a selected sign of ω−ωsn.\n16/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0\nkΩ1.57\n/Minus1.0/Minus0.50.00.51.0/Minus2.1/Minus2.0/Minus1.9/Minus1.8/Minus1.7/Minus1.6/Minus1.5/Minus1.4\nkΩ0.64\n/Minus1.0/Minus0.50.00.51.0/Minus2.4/Minus2.2/Minus2.0/Minus1.8/Minus1.6\nkΩ1.99\n/Minus1.0/Minus0.50.00.51.0/Minus2.10/Minus2.05/Minus2.00/Minus1.95/Minus1.90\nkΩ2.001\nFigure 10: Dispersion curves and stability regions. Wavetr ains on the dispersion curves in the\ngreen shaded region are unstable due to unstable eigenvalue ofA(0,0) using (29), and in the\nred shaded region due to unstable sideband only, using ˜λ′′\n0(0) with k2from Γ = 0. Parameters\nare as in (a): Figure 5; (b): Figure 6; (c),(d): Figure 7. Only the wavetrains near k= 0 are\nstable (except for the lower branch in (b)).\nRemark 1 Item (a) should be compared with the Hopf instabilities discu ssed in§3. The case\nccp= 0is simplest: the constraint simply means that wavetrains bi furcating at the subsub- to\nsubcritical transition are unstable. For general ccpthe condition also accounts for interaction\nwith folds.\nConcerning item (b), notably in the easy axis case µ <0all wavetrains are unstable. The\ncondition ccpβ≥0is not sharp. However, we do not know whether a ‘Hopf’ instabi lity can\noccur for otherwise stable wavetrains if ccpβ <0is sufficiently large negative. Item (d) is an\nanalog to the results in [32].\nIn order to study spectral stability of wavetrains we consid er the comoving frame y=\nx−cphtwithcph=ω/kthe wavespeed so that the wavetrain is an equilibrium of (14) . For\nconvenience, time is rescaled to t= (1+α2)˜t. The explicit formulation of (14) then reads\n∂˜t/parenleftbiggϕ\nm3/parenrightbigg\n=/parenleftbiggα(∂y(r2∂yϕ)/r2+˜β(m3))+(∂2\nym3+|∂ym|2m3)/r2+h−µm3+cph∂yϕ\nα(∂2\nym3+|∂ym|2m3+r2(h−µm3))−∂y(r2∂yϕ)−r2˜β(m3)+cph∂ym3/parenrightbigg\n,\n(27)\nwherer2= 1−m2\n3and˜β(m3) :=β/(1+ccpm3).\nLetF= (F1,F2)tdenote the right hand side of (27). Wavetrains have constant randm3\nso that quadratic terms in their derivatives can be discarde d for the linearization LofFin a\nwavetrain, and from |∂ym|2onlyr2(∂yϕ)2is relevant. The components of Lare\n∂ϕF1=α∂2\ny+cph∂y+2km3∂y\n∂m3F1=r−2∂2\ny+k2−µ−2αkm3r−2∂y+α˜β′(m3)\n∂ϕF2= 2αkm3r2∂y−r2∂2\ny\n∂m3F2=α∂2\ny+αr2(k2−µ)+cph∂y−2αm3(m3k2+h−µm3)+2m3(k∂y+˜β(m3))\n−r2˜β′(m3) =α∂2\ny+αr2(k2−µ)+cph∂y+2m3k∂y−r2˜β′(m3),\nwhere the last equation is due to (17) and ˜β′(m3) =−ccpβ/(1+ccpm3)2. Since all coefficients\nare constant, the eigenvalue problem\nLu=λu\n17is solved by the characteristic equation arising from an exp onential ansatz u= exp(νy)u0,\nwhich yields the matrix\nA(ν,cph) :=/parenleftbiggαν2+(cph+2km3)ν−r−2ν(−ν+2αkm3)+k2−µ+α˜β′(m3)\nr2ν(−ν+2αkm3)αν2+(cph+2km3)ν+αr2(k2−µ)−r2˜β′(m3)/parenrightbigg\n.\nThe characteristic equation then reads\ndcph(λ,ν) :=|A(ν,cph)−λ|=|A(ν,0)−(λ−νcph)|=d0(λ−cphν,ν)\nd0(λ,ν) =λ2−t(ν)λ+d(ν) (28)\nwith trace and determinant of A(ν,0)\nt(ν) := trA(ν,0) = 2ν(αν+2km3)+αr2(k2−µ)−r2˜β′(m3)\nd(ν) := detA(ν,0) = (1+ α2)ν/parenleftBig\nν(ν2+4k2m2\n3+r2(k2−µ))−2r2˜β′(m3)km3/parenrightBig\n= (1+α2)ν/parenleftBigg\nν/parenleftBigg\nν2+(3k2+µ)(˜β(m3)/α−h)2\n(k2−µ)2+k2−µ/parenrightBigg\n−2r2˜β′(m3)km3/parenrightBigg\n.\nIn the last equation (23) was used.\nThe characteristic equation is also referred to as the complex (linear) dispersion relation .\nThespectrum of L, for instance in L2(R), consists of solutions for ν= iℓand is purely essential\nspectrum (in the sense that λ−Lis not a Fredholm operator with index zero). Indeed, setting\nν= iℓcorresponds to Fourier transforming in ywith Fouriermode ℓ. Note that the solution\nd(0,0) = 0 stems from spatial translation symmetry in y. For the same reason the real part\nof solutions λof (28) for any given ν∈iRdoes not depend on cph, which means that spectral\nstability is independent of cphand is therefore completely determined by d0(λ,iℓ) = 0.\nAs a first observation concerning stability we note that at r= 0,|m3|= 1 the solutions λ±\ntod0(λ,iℓ) = 0 have Re( λ±) =−ℓ(ℓ±2k) whose maximum is k2. Hence, wavetrains at (and\nthus near) r= 0 are unstable for k/ne}ationslash= 0.\nFold stability. First note the eigenvalues A(0,0) are 0 and\nτ(k2) :=t(0) =r2(α(k2−µ)−˜β′(m3))\nso that a ‘fold instability’ occurs precisely for r= 0 (compare §3) orα(k2−µ) =˜β′(m3).\nWe readily compute that the latter corresponds to the critic al points in (18). Note that\n˜β′(m3) =−α2ω2ccp/β. In particular, for the loops of wavetrains emerging from an elliptic\npoint, the upper and lower ω-values have opposite signs of τ(0); specifically the lower is stable\nifβccp>0, cf. Figure 5.\nAt the bifurcation points of ±ˆe3(see§3) where r= 0, we have µ=µ±andω=−β±/α\nwhich yields Theorem 3(a) except for the super-/subcritica lity. To see this recall that ±ˆe3\ndestabilize always through increasing µ. Whether wavetrains with k= 0 emerge from ±ˆe3\ndepends on the sign of ∂µm3. From (16) we find µ=h\nm3−β\nαm3(1+ccpm3)and compute\n∂m3µ(±1) =±τ(0)\nαr2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm3=±1.\n18Hence, fold-stability implies ∂µm3(µ+)<0 or∂µm3(µ−)>0, respectively, so that increasing\nµyields|m3|<1 and thus emergence of solutions.\nMore generally, sign changes of τ(k2) correspond to fold points and a curve of spectrum\ncrosses the origin. We plot some fold stability boundaries i n Figure 10. Using τ(k2) we have\nthat wavetrains with µ < k2+ccpα\nβω2are unstable. In particular, for ccpβ≥0 all wavetrains\nwithk2> µare unstable. Using the existence condition we may also writ e this condition\nindependent of kas\nω(β(2ω+h)+αω2)\nβ(β+αω)<0, (29)\nwhich explains the changes in the fold stability indicator a tω=−β/αin the figure.\nComing back to k= 0, at the bifurcation points of ±ˆe3(see§3) where r= 0, we have\nµ=µ±. Fold-stability is then µ > αc cpω2/βwhich holds if\n±/parenleftbigg\nh−β±\nα/parenrightbigg\n>βccp\nα(1±ccp)2⇔ ±h >β\nα2ccp±1\n(1±ccp)2,\nas noted in Theorem 3 item (a).\nWe next check the other possible marginal stability configur ations case by case.\nSideband instability. A sideband instability occurs when the curvature of the curv e of\nessential spectrum attached to the origin changes sign so th at the essential spectrum extends\ninto positive real parts. Let ˜λ0(ℓ) denote the curve of spectrum of A(iℓ,0) attached to the\norigin, that is ˜λ0(0) = 0, and let′denote the differentiation with respect to ℓ. Derivatives of\n˜λ0can be computed by implicit differentiation of d0(λ,iℓ) =λ2−t(iℓ)+d(iℓ) = 0.\nThis gives ˜λ′\n0(0) = id′(0)\nt(0)=−i˜β′(m3)2(1+α2)km3\nα(k2−µ)−˜β′(m3)therefore\n˜λ′′\n0(0) =−2d′(d′−t′t)+d′′t2\nt3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nℓ=0=: Λ(k2)2(1+α2)r4\nτ3(k2),\nwhere the ω-dependence is suppressed. Some calculations yield Λ as a cu bic polynomial in\nK=k2given by Λ = a3K3+a2K2+a1K+a0with\na3=−α2(4−3r2),\na2=α(2˜β′r2+αµ(8−5r2)),\na1=−α2µ2(4−r2)−4(1−r2)α2˜β′(m3)2−(˜β′(m3)2+4α˜β′(m3)µ)r2,\na0=µ(˜β′(m3)+αµ)2r2.\nNotably, a0=µτ(0)2and also a3<0 sincer∈(0,1). A wavetrain with wavenumber kis\ntherefore (strictly) sideband stable precisely when Λ( k2)τ(k2)<0 andsideband unstable for\nΛ(k2)τ(k2)>0. As mentioned above, τ(k2)>0 for large enough k2so that all such (already\nunstable) wavetrains are also sideband unstable as a3<0 holds always.\nSinceτ(k2)>0 is the unstable fold case, we next assume τ(k2)<0 so that sideband\nstability is precisely Λ( k2)>0.\n19Homogeneous oscillations. Fork= 0 we obtain ˜λ′′\n0(0) = Λ(0)2(1+α2)r4\nτ3(0)=−2(1+α2)µ\nτ(0),\nso thatµ >0 is required for sideband stability (given fold stability τ(0)<0).\nLet us study this situation near the Hopf instability of ±ˆe3, whereµ≈µ±=±(h−β±/α)\nwithµ > µ±andω=β±/α. Hence, sideband stability of the emerging wavetrains requ ires\nh > β+/αorh < β−/α, that is, µ±>0. Compare Theorem 3(a).\nNear homogeneous. To leading order Λ( k2) = 0 isa1k2+a0=O(k4) and via (21) we have\na1=a1(˜ω,k2),a0=a0(˜ω,k2), where at bifurcation a0(0) =∂ωa0(0) = 0. Upon expanding we\ntherefore find sideband instabilities to leading order at\n˜ω2=a1(0)−∂k2a0(0)\n∂˜ω2a0(0)k2+O(|k|3+|˜ω|3),\nwhere∂˜ω2a0(0) =µsnr2\nsn/parenleftBig\n2\nβα2ωsnccp/parenrightBig2\nand∂k2a0(0) =αµsnr2\nsnwithrsn= 1−/parenleftBig\nωsn+h\nµ/parenrightBig2\nther-coordinate of the wavetrain at the bifurcation point. Some algebra yields a1(0) =\n−4α(1+α2)(h+ωsn)2.\nIn accordance with the results in [31], this means that wavet rains in a sector in the ( ω,k)-\nplane near the fold point are sideband stable, while wavetra ins outside this sector are sideband\nunstable. We expect that the opening angle of this sector can be changed while keeping\nthe dispersion curves essentially fixed. Here we do not pursu e this further, but note that\nsince sideband instabilities do occur and the stable region cannot include the fold points,\nthe prefactor of k2is positive and ˜ ωhas a selected sign. The sideband boundaries for some\nexamples are plotted in Figure 10.\nGeneral wavetrains. Concerning the sign of Λ in general, Λ(0) = a0has the sign of\nµand thus all wavetrains for µ <0 are sideband unstable. For µ >0 we have Λ( µ) =\n−4α2(˜β′(m3))2m2\n3µ <0, so a sign change occurs at some Ksb∈(0,µ), which implies sideband\ninstability for k2≥µ. Moreover, µ >0 andccpβ≤0 imply a1<0 anda2>0. Since\na3<0 this means both roots of Λ′, (2a2±/radicalbig\n4(a2\n2−3a3a1))/(6a3) lie at negative Kand so Λ\nis monotone decreasing for all K >0. Therefore Ksbis the unique sideband instability in this\ncase.\nOn the other hand, for a1>0 the roots of Λ′have opposite signs so that due to the sign\nchange in the interval (0 ,µ) the positive one must be a local maximum so that also in this\ncaseKsbis the unique sideband instability. Moreover, in all cases w avetrains with k2> µare\nunstable since Λ <0 in this range.\nHopf instability. A Hopf instability occurs when the essential spectrum touch es the imag-\ninary axis at nonzero values. In particular, there is γ/ne}ationslash= 0 so that d0(iγ,iℓ) = 0. At k= 0\nwe have Im( d0(iγ,iℓ)) =γ(2αℓ2−τ(0)) so that in the fold stable case τ(0)<0 there is real ℓ\nforγ/ne}ationslash= 0. Recall τ(0) =−(˜β′(m3)+αµ)r2. Therefore, there is no (relevant) Hopf instability\nneark= 0.\nMore generally, solving Im( d0(iγ,iℓ)) forγand substituting the result into Re( d0(iγ,iℓ))\nwe obtain up to a factor (1+ α2)ℓ2\nℓ2+(µ−k2)r2+4α2m2\n3k2G(ℓ2), G(L) =−4L2−4L(k2−µ)r2+(˜β′(m3)2+(k2−µ)2)r4\n(2αL−τ(k2))2\n20/Minus4/Minus20240Π2Π\nkΘsupercritical\nsubcritical\nsubsubcriticalµ\nk\n(a) (b)\nFigure 11: Analogues of Figures 8(a) and 9 with stable range o f wavetrains in (a) bold line,\nand in (b) the dark shaded region.\nwhere all terms except possibly G(L) are positive in the interesting range µ > k2. Note that\nfor sufficiently small α >0 there is no root besides ℓ= 0 and thus no Hopf and in fact no\nsideband instability. But there seems to be no satisfying ex plicit bound. (While the same\nseems to occur for m3∼0, such wavetrains have k2> µand are thus unstable.)\nHowever, Gis nondecreasing for ˜β′(m3)≤0, i.e.,ccpβ≥0, since then\nG′(L) =−4˜β′(m3)r22L+(µ−k2−α˜β′(m3))r4\n(2αL−τ(k2))3≥0,\nin the fold stable case τ(k2)≤0. Thus, besides ℓ=γ= 0, there is at most one solution\nd0(iγ,iℓ) = 0 for ccpβ≥0, which rules out a Hopf instability as this requires two suc h\nsolutions. We do not know whether or not Hopf instabilities c an occur for general αand\nccpβ <0.\nTuring instability. A Turing instability occurs when the spectrum touches the or igin for\nnonzero ℓ, that is, there is ℓ/ne}ationslash= 0 so that d0(0,iℓ) = 0, which means det A(iℓ,0) = 0. Since\nIm(detA(iℓ,0)) =−2(1 +α2)˜β′(m3)ℓkm3r2and our previous considerations already cover\nzeros of this, such instabilities do not occur.\nThis exhausts the list of possible marginal stability.\n4.4 Stability of wavetrains for ccp= 0\nForccp= 0 the wavetrain frequency ωis independent of the wavenumber kin (22) so that\nthe phase velocity ω/kof all wavetrains is −β/(αk) and the group velocity d ω/dk, which\ndescribes the motion of perturbations by localized wave pac kets, vanishes for all wavetrains.\nDue to Theorem 3 destabilizations of stable wavetrains can o nly occur through a unique\nsideband instability |k|=k∗>0. This value can be explicitly determined since d′(0) = 0 and\n21d′′(0) =−2(1+α2)((3K+µ)r2−4K) so that ˜λ′′\n0(0) = 0 at\nk2\n∗=µr2\n4−3r2.\nTaking into account that k2=µfor a wavetrain can occur only if h=β/αwe thus have\nTheorem 4 Consider ccp= 0. All wavetrains whose wavenumber ksatisfies |k|> k∗are\nunstable. For µ >0wavetrains with wavenumber |k|< k∗are spectrally stable, while those\nwith|k|> k∗are unstable. In case h/ne}ationslash=β/αa sideband instability occurs at k=±k∗. There\nis no secondary instability for k2< µ. Nontrivial spectrally stable wavetrains exist only for\nsupercritical anisotropy, |h−β/α|< µ.\nThe overall picture for wavetrains of (1) with ccp= 0 is thus a combination of the scenarios\nfrom a supercritical and a subcritical real Ginzburg-Landa u equation ∂tA=∂2\nxA+ ˜µA∓\nA|A|2, A(x,t)∈C, which describes the dynamics near pattern forming Turing i nstabilities\nand possesses the gauge-symmetry A→eiϕA. The interested reader is referred to the review\n[1] and the references therein.\n5 Coherent structures\nThe coexistence of wavetrains and constant magnetizations raises the question how these\ninteract. In this section we study solutions that have spati ally varying local wave number. In\nparticular, we consider solutions that spatially connect w avetrains or ±ˆe3in a coherent way.\nIn order to locate such solutions induced by symmetry we make the ansatz\nξ=x−st\nϕ=φ(ξ)+Ωt\nθ=θ(ξ),(30)\nwith constant s,Ω. Solutions of (14) of this form are generalized travelling waves to (1)\nwith speed sthat have a superimposed oscillation about ˆe3with frequency Ω. This ansatz\nis completely analogous to that used in the aforementioned s tudies of the real and complex\nGinzburg-Landau equations [1].\nLet¯β(θ) :=˜β(cos(θ)) =β/(1 +ccpcos(θ)) denote the ( β,ccp)-dependent term of (15).\nSubstituting ansatz (30) into (15) with′= d/dξandq=φ′gives, after division by sin( θ), the\nODEs\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbiggsin(θ)(Ω−sq)\nsθ′/parenrightbigg\n=/parenleftbigg2cos(θ)θ′q+sin(θ)q′\n−θ′′+sin(θ)cos(θ)q2/parenrightbigg\n+sin(θ)/parenleftbigg¯β(θ)\nh−µcos(θ)/parenrightbigg\n,(31)\non the cylinder ( θ,q)∈S1×R, which is the same as {(m3,r,q)∈R3:m2\n3+r2= 1}.\nWavetrains. Steady states with vanishing ξ-derivative of θandqhaveϕ=q(x−st)+Ωt\nand thus correspond to the wavetrains discussed in §4 with wavenumber k=qand frequency\nω=sq−Ω. Hence, the ansatz (30) removes all wavetrains whose waven umber and frequency\ndo not lie on the line {ω=sk−Ω}in (ω,k)-space.\n22We may visualize this by drawing the line ω=sk−Ω into the wavetrain existence and\nstability plots in the ( ω,k)-plane such as Figure 10. Specifically, for s/ne}ationslash= 0, steady states of\n(31) with m3/ne}ationslash=±1 are wavetrains with wavenumber qfor which there exists θsuch that\nq=Ω−¯β(θ)/α\ns. (32)\nIn particular, for ccp= 0 we have constant ¯β(θ) =βso that equilibria of (31) (other than\n±ˆe3) have uniquely selected wavenumber q. Hence, heteroclinic solutions to (31) for ccp= 0\ncan only be domain walls connecting ±ˆe3or connect one of ±ˆe3to a wavetrain.\nCoherent structure ODEs. Writing (31) as an explicit ODE gives\nθ′=p\np′= sin(θ)/parenleftbig\nh+(q2−µ)cos(θ)−(Ω−sq)/parenrightbig\n−αsp\nq′=α(Ω−sq)−β/(1+ccpcos(θ))−s+2cos(θ)q\nsin(θ)p,(33)\nwhose study is the subject of the following sections. For lat er use we also note the ‘desingu-\nlarization’ by the (singular) coordinate change p= sin(θ)˜pso that ˜p′=p′/sin(θ)−˜p2cos(θ),\nwhich gives\nθ′= sin(θ)˜p\n˜p′=h+(q2−µ)cos(θ)−(Ω−sq)−αs˜p−cos(θ)˜p2\nq′=α(Ω−sq)−β/(1+ccpcos(θ))−(s+2cos(θ)q)˜p.(34)\nHence, (33) is equivalent to (34) except at zeros of sin( θ). Inparticular, for ˜ p= 0the equilibria\nof (34) with sin( θ)/ne}ationslash= 0 are those of (33), but θ=nπ,n∈Zare invariant subspaces which\nmay contain equilibria with ˜ p/ne}ationslash= 0.\nNext, we first consider various moving heteroclinic coheren t structures with s/ne}ationslash= 0 and for\nccp= 0 give a complete analysis of stationary coherent structur es (s= 0). Coherent structures\nalso emerge near the elliptic and hyperbolic wavetrain bifu rcations that arise from fold points\nforccp/ne}ationslash= 0. However, the detailed analysis of this case is beyond the scope of this paper.\n5.1 Homogeneous domain walls\nClassical domain walls connect antipodal equilibria at x=±∞. For the model equation (8)\nexplicit (Walker) solutions are known to exist below a criti cal fieldh. These solutions exhibit a\ntilting of the azimuthal angle ϕ=const.in order to balance precessional forces. An analogue\nsituation arises in our context when q=q′= 0. In this case we have ϕ= Ωtand therefore no\nspatially varying azimuthal profile.4\nTheorem 5 Non-equilibrium coherent structure solutions with q≡0fors/ne}ationslash= 0and|ccp|<1\nexist for µ <0andccp= 0orβ= 0only, and have Ω =h+αβ\n1+α2,s2=−(β−αh)2\nµ(1+α2)2. They\nare oscillating heteroclinic fronts connecting ±ˆe3that solve θ′=σ√−µsin(θ)whereσ=\nsgn(s(αh−β)). The family of such fronts is smooth and extends to s= 0, whereh= Ω =β/α,\nand fronts exist for both signs of σ.\n4After acceptance of the present manuscript for publication , we found that the sufficiency of ccp= 0 for\nexistence of such domains walls in Theorem 5 is contained in [ 12].\n23Proof.Recall¯β(θ) =β/(1+ccpcos(θ)) sod\ndξ¯β(θ) =ccpβsin(θ)θ′/(1+ccpcos(θ))2.\nSuppose a solution ( θ,p,q) to (33) has q≡0, so also q′≡0. Then the third equation of\n(33) fors/ne}ationslash= 0 yields, using the first equation, θ′=αΩ−¯β(θ)\nssin(θ). Differentiation gives\nθ′′=αΩ−¯β(θ)\nscos(θ)θ′−ccpβsin2(θ)\ns(1+ccpcos(θ))2θ′\n= sin(θ)αΩ−¯β(θ)\ns2/parenleftbigg\n(αΩ−¯β(θ))cos(θ)−ccpβ(1−cos2(θ))\n(1+ccpcos(θ))2)/parenrightbigg\n.\nOn the other hand, the second equation of (33) requires\nθ′′= sin(θ)(h−µcos(θ)−Ω−α(αΩ−¯β(θ))).\nFirst consider ccp= 0 orβ= 0 so that ¯β(θ) =βand these two right hand sides for θ′′simplify.\nEquating them and comparing the coefficients of cos( θ)j,j= 0,1, yields h= Ω+α(αΩ−β)\nandµ=−/parenleftBig\nαΩ−β\ns/parenrightBig2\n, which means ( αΩ−β)/s=σ√−µforσ= sgn(s(αΩ−β)). Taken\ntogether, the parameter conditions can be equivalently cas t as the equations for Ω, s2andσ\nin the theorem statement.\nHence, for ccp= 0 orβ= 0 these parameter choices and θ′=σ√−µsin(θ) are necessary\nconditions for q≡0. As a scalar equation, the only non-trivial and bounded sol utions are\nheteroclinic orbits between equilibria. Taking Ω = β/α+s˜µfor some ˜ µ/ne}ationslash= 0 gives a smooth\nparametrization up to s= 0 and σ= sgn(˜µ).\nConversely, for ccp= 0 orβ= 0 and these choices of parameters, any ( θ,p,q)(ξ) where\nθ(ξ) satisfies θ′=σ√−µsin(θ),p=θ′andq≡0 is a solution to (33).\nIt remains to show that if β/ne}ationslash= 0 then ccp= 0 is necessary for a non-trivial solution\nwithq≡0: Subtracting the two right hand sides for θ′′from above and multiplication with\n(1+ccpcos(θ))3gives a polynomial in cos( θ) of degree four. A straightforward computation\nshows that the 4th order term to vanish requires µ=−α2Ω2/s2. Using this the coefficients\najof cos(θ)j,j= 0,1,2,3, in this polynomial can be computed as\na0=−h+(1+α2)Ω−αβ+(β−αΩ)βccp/s2,\na1= 3ccp((1+α2)Ω−h)+β2/s2−αβ(2ccp+(2+c2\ncp)Ω/s2),\na2=c2\ncp(3((1+α2)Ω−h)−αβ)−3αβΩccp/s2,\na3=c3\ncp((1+α2)Ω−h)−αβΩc2\ncp/s2.\nWe first solve a0= 0 trivially for hand proceed with somewhat tedious, but straightforward\ncalculations: substituting this hintoa1= 0 (which is linear in Ω) we solve for Ω, which uses\n|ccp|<1. Substituting the resulting h,Ω gives\na2=ccpβ\n2s2/parenleftbig\n3β(c2\ncp−1)+αccps2/parenrightbig\n,\nso that for a2= 0 either ccp= 0 (since β/ne}ationslash= 0; note that then also a3= 0) ors2= 3βc2\ncp−1\nαccp. In\nthe latter case, substituting the previous h,Ω and this s2intoa3would give a3=αβc3\ncp/3/ne}ationslash= 0.\nHence,ccp= 0 is necessary as claimed.\n24Remark 2\n1. Fors= 0further coherent structure solutions exist, but not as doma in walls. See\nTheorem 8 below.\n2. In§5.2.1 we find fast domain walls and fronts that have non-trivi alq.\n3. Numerical simulations suggest that these domain walls ar e dynamically stable solutions\nin the subsubcritical case. They are unstable in the subcriti cal case |h−β/α|>−µ,\nwhich occurs for large |s|, since then either ˆe3or−ˆe3is unstable.\n4. The profile of these domain walls depends only on the paramet erµ. In particular, the\nsubfamily parameterized by hhas arbitrarily large speed but constant shape, though the\noscillation frequency Ωdepends on h.\n5.2 Moving front-type coherent structures\nUsing the desingularized system (34), we prove existence of some non-stationary coherent\nstructures of front-type, spatially connecting wavetrain s or±ˆe3.\n5.2.1 Near the fast limit |s| ≫1\nTheorem 6 For any bounded set of (α,β,µ,Ω0,Ω1)andccp∈(−1,1)there exists s0>0and\nneighborhood UofM0:={θ∈[0,π],˜p=q−Ω1= 0}such that for all |s| ≥s0the following\nholds for (34)withΩ = Ω 0+Ω1s. The heteroclinic orbits of (34)inUform a smooth family\nin the parameter s−1for each sign of s, which reverses their orientation. These heteroclinics\nand are in one-to-one correspondence with those of the ODE\nd\ndηθ=−α\n1+α2sin(θ)/parenleftbigg¯β(θ)\nα−h+(Ω2\n1−µ)cos(θ)/parenrightbigg\n, (35)\non the spatial scale η=ξ/s, which also gives the θ-profile to leading order in s−1. Moreover,\nfor such a heteroclinic orbit (θh,˜ph,qh)(ξ)withθσ:= lim ξ→σ∞θh(ξ)∈ {0,π}forσ= 1or\nσ=−1, theq-limit is/are\nlim\nξ→σ∞qh(ξ) = Ω1−1\ns/parenleftbiggh+α¯β(θσ)−σµ\n1+α2−Ω0/parenrightbigg\n+O(s−2). (36)\nIn particular, for Ω1/ne}ationslash= 0or(1+α2)Ω0/ne}ationslash=h+α¯β(θσ)−σµlocal wavenumbers are nontrivial:\nqh/ne}ationslash≡0.\nBefore proving the theorem we note the consequences of this f or coherent structures and\ndomains walls in in (33) and (1).\nCorollary 1 The heteroclinic solutions of Theorem 6 are in one-to-one corr espondence with\nheteroclinic solutions to (33)and thus heteroclinic coherent structures in (1)that lie in Uand\nconnectθ= 0,πor a wavetrain with r/ne}ationslash= 0. Forθ∈(0,π)all properties carry over to (33)\nwith the bijection given by p= sin(θ)˜p.\n25Proof.Recall that (34) and (33) are equivalent for θ∈(0,π). Since the limit of the vector\nfield of (33) along such a heteroclinic from (34) is zero by con struction in all cases. Hence, for\neach of the heteroclinic orbits in (34) of Theorem 6, there ex ist a heteroclinic orbit in (33) in\nthe sense of the corollary statement.\nCorollary 2 For any ‘bandgap’ parameter set of (33)such that there exist no wavetrains\nsatisfying (32)for any|s|> s1, for some s1>0, there exist fast domain wall type coherent\nstructures spatially connecting ±ˆe3for all sufficiently large velocity |s|.\nProof. Choosing Ω 1=kthere are by assumption no equilibria in (35) besides ±ˆe3, which\nare therefore connected by a heteroclinic orbit. Theorem 6 t hen implies the claim.\nSuch ‘bandgaps’ occur in particular if µ >0 for Ω2\n1∼µ.\nRemark 3\n1. Concerning stability, Lemma 1 and Theorem 4 imply that for ccp= 0the domain walls\nconnecting ±ˆe3might be stable in the subsubcritical case only since otherw ise one of the\nasymptotic states is unstable: the unique wavetrain with θ∈(0,π)in the subsubcritical\ncase and ˆe3or−ˆe3in the sub- and supercritical cases. However, it may be that s ome\nfronts are stable in a suitable weighted sense as invasion fr onts into an unstable state.\n2. For increasing speeds these solutions are decreasingly l ocalized, hence far from a sharp\ntransition.\n3. The uniqueness statement in the corollaries is limited, si nce in the (θ,p,q)-coordinates\nthe neighborhood Ufrom the theorem is ‘pinched’ near θ= 0,π: a uniform neighborhood\nin(θ,˜p)has a sinus-shaped boundary in (θ,p).\nRemark 4 Part of the family homogeneous domains walls from Theorem 5, w hereccp= 0, is\na continuation to smaller |s|of homogeneous ( q≡0) fronts in the family of Theorem 6. The\nlatter are decreasingly localized, which requires in the fo rmer that√µ=O(s). Specifically,\nµ=−(β−αh)2\ns2(1+α2)2andΩ0=h+αβ\n1+α2,Ω1= 0in the heteroclinics of Theorem 6. Then µ→0as\ns2→ ∞so thatµ= 0in the leading order equation (35)and in(36)µis removed from the\norders−1term. Since σs√−µ=−(β−αΩ0)andβ−αΩ =−α\n1+α2/parenleftBig\nh−β\nα/parenrightBig\nindeed(35)equals\nthe equation in Theorem 5. In particular, (36)is consistent with q≡0.\nFinally, remark that the ODE (35) is the spatial variant of th e temporal heteroclinic\nconnection in (15): setting all space derivatives to zero, t heθ-equation of (15) reads\n−∂tθ=α\n1+α2sin(θ)/parenleftbigg\nh−¯β(θ)\nα−µcos(θ)/parenrightbigg\n,\nwhich is (35) with µreplaced by Ω2\n1−µand up to possible direction reversal. Since Ω 1=q\non the slow manifold M0(i.e. at leading order), the reduced flow equilibria reprodu ce the\nwavetrain existence condition (23). This kind of relation b etween temporal dynamics and fast\ntravelling waves holds formally (but in general not rigorou sly) for any evolution equation in\none space dimension. Here the symmetry makes the temporal OD E scalar.\n26Proof (Theorem 6) Let us set s=ε−1so that the limit to consider is ε→0. Since we will\nrescale space with εandε−1this means sign changes of sreflect the directionality of solutions.\nThe existence proof relies on a geometric singular perturba tion argument and we shall use\nthe terminology from this theory, see [10, 17], and also some times suppress the ε-dependence\nofθ,˜p,q.\nUpon multiplying the ˜ p- andq-equations of (34) by εwe obtain the, for ε/ne}ationslash= 0 equivalent,\n‘slow’ system\nθ′= sin(θ)˜p\nε˜p′=−α˜p+q−Ω1+ε(h+(q2−µ)cos(θ)−Ω0−cos(θ)˜p2)\nεq′=−˜p−α(q−Ω1)+ε(αΩ0−¯β(θ)−2cos(θ)q˜p).(37)\nSettingε= 0 gives the algebraic equations\nA/parenleftbigg˜p\nq/parenrightbigg\n=−Ω1/parenleftbigg−1\nα/parenrightbigg\n,whereA=−/parenleftbiggα−1\n1α/parenrightbigg\n.\nSince det A= 1+α2>0 the unique solution is ˜ p=q−Ω1= 0 and thus the ‘slow manifold’\nisM0as defined in the theorem, with ‘slow flow’ given by\nθ′= sin(θ)˜p.\nSince ˜p= 0 atε= 0,M0is a manifold (a curve) of equilibria at ε= 0, so that the slow\nflow is in fact ‘superslow’ and will be considered explicitly below. Since the slow manifold is\none-dimensional (and persists for ε >0 as shown below) it suffices to consider equilibria for\nε >0. These lie on the one hand at θ=θ0∈ {0,π}, if\nA/parenleftbigg˜p\nq/parenrightbigg\n+Ω1/parenleftbigg−1\nα/parenrightbigg\n+εF(˜p,q) = 0, F(˜p,q) :=/parenleftbiggh+σ(q2−µ)−Ω0−σ˜p2\nαΩ0−¯β(θ0)−2σq˜p/parenrightbigg\n,\nwhereσ= cos(θ0)∈ {−1,1}. Since det A=−(1 +α2)<0 the implicit function theorem\nprovides a locally unique curve of equilibria (˜ pε,qε) for sufficiently small ε, where\nd\ndε/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nε=0/parenleftbigg˜pε\nqε/parenrightbigg\n=−A−1F(0,0) =−A−1/parenleftbiggh−σµ−Ω0\nαΩ0−β±/parenrightbigg\n.\nThis proves the claimed location of asymptotic states.\nOn the other hand, for θ/ne}ationslash= 0 system (34) is equivalent to (33). From the previous consi d-\nerations of equilibria (=wavetrains) we infer that the uniq ue equilibria in an ε-neighborhood\nofM0are those at θ=θ0, (˜p,q) = (˜pε,qε) together with the possible additional θ-values of\nwavetrains, where kis now replaced by q=ε(Ω0−¯β(θ)/α).\nTowards the persistence of M0as a perturbed invariant manifold for |ε|>0, let us switch\nto the ‘fast’ system by rescaling the time-like variable to ζ=ξ/ε. With˙θ= dθ/dζetc., this\ngives\n˙θ=εsin(θ)˜p\n˙˜p=−α˜p+q−Ω1+ε(h+(q2−µ)cos(θ)−Ω0−cos(θ)˜p2)\n˙q=−˜p−α(q−Ω1)+ε(αΩ0−¯β(θ)−2cos(θ)q˜p).(38)\n27Note that M0is (also) a manifold of equilibria at ε= 0 in this system and the linearization of\n(38) inM0for transverse directions to M0is given by A. Since the eigenvalues of A,−α±i,\nare away from the imaginary axis, M0is normally hyperbolic and therefore persists as an\nε-close invariant one-dimensional manifold Mε, smooth in εand unique in a neighborhood of\nM0. See [10]. The aforementioned at least two and at most three e quilibria lie in Mε, and,\nMεbeing one-dimensional, these must be connected by heterocl inic orbits.\nFor the connectivity details it is convenient to derive an ex plicit expression of the leading\norder flow. We thus switch to the superslow time scale η=εξand setp= ˜p/ε,q= (q−Ω1)/ε,\nwhich changes (37) to (subdindex ηmeans d/dη)\nθη= sin(θ)p\nεpη=−αp+q+h+(Ω2\n1−µ)cos(θ)−Ω0+εcos(θ)(ε(q2−p2)+2qΩ1)\nεqη=−p−αq+αΩ0−¯β(θ)−2εcos(θ)(εqp+Ω1p).(39)\nAtε= 0, solving the algebraic equations for ( p,q) gives\n/parenleftbiggp\nq/parenrightbigg\n=−A−1/parenleftbiggh+(Ω2\n1−µ)cos(θ)−Ω0\nαΩ0−¯β(θ)/parenrightbigg\n=1\n1+α2/parenleftbiggαh+α(Ω2\n1−µ)cos(θ)−¯β(θ)\n(1+α2)Ω0−h−(Ω2\n1−µ)cos(θ)−α¯β(θ)/parenrightbigg\nwhose first component gives pso that the leading order superslow flow on the invariant man-\nifold is indeed given by (35).\n5.2.2 The case of small amplitudes\nIn this section we consider small amplitude coherent struct ures, which means qmust lie near\na bifurcation point of wavetrains. Here we focus on the inter section points of the solution\ncurves from (16) with θ=θ0= 0,π, which gives\ncos(θ0)/parenleftbig\nq2−µ/parenrightbig\n=¯β(θ0)\nα−h. (40)\nForccp= 0 this is possible for super- and subcritical anisotropy on ly, compare Figure 8.\nWe show that these intersection points are pitchfork-type b ifurcations in (34) that give\nrise to front-type coherent structures. As in the previous s ection, we locate such solutions in\n(33) from an analysis of (34).\nIt is convenient to write (16) in terms of m3= cos(θ) so equilibria of (34) with ˜ p= 0 solve\n˜Γ(m3) :=˜β(m3)\nα−/parenleftbig\nq2(m3)−µ/parenrightbig\nm3−h= 0,\nwhereq(m3) is the selected qfrom (32). Recall ˜β(m3) =β\n1+ccpm3.\nTheorem 7 Consider θ=θ0∈ {0,π}and setm0\n3:= cos(θ0). Suppose that parameters of (34)\nare such that s/ne}ationslash= 0,˜Γ(m0\n3) = 0and∂m3˜Γ(m0\n3)/ne}ationslash= 0. Then the equilibrium point (θ0,0,q(m0\n3))\nof(34)undergoes a pitchfork bifurcation upon any perturbation of horµ.\nMore precisely, let Sε= (αε,βε,hε,µε,Ωε,ccp(ε),sε),ε∈(−ε0,ε0)for some ε0>0, be\na curve in the parameter space of (34)with|ccp(ε)|<1,αε>0,sε/ne}ationslash= 0and such that S0\nsatisfies ˜Γ(m0\n3) = 0and˜γ:=∂m3˜Γ(m0\n3)/ne}ationslash= 0.\n28Then(34)has a curve of equilibria (θ0,˜pε,qε), with possibly adjusted ε0, such that ˜p0=\n0,q0=q(m0\n3)and two equilibria with θ/ne}ationslash=θ0bifurcate from (θ0,0,q(m0\n3))for increasing ε\nif, with parameters Sε,m0\n3˜γ∂ε˜Γ(m0\n3)|ε=0>0. The bifurcating equilibria are connected to\n(θ0,˜pε,qε)by heteroclinic orbits which converge to (θ0,˜pε,qε)asℓξ→ ∞forℓ=−m0\n3˜pε.\nSpecifically, this occurs if hε=h0−m0\n3˜γεorµε=µ0+˜γεandℓ= ˜γs, or ifβε=β0+m0\n3˜γε\nandℓ=−(2q(m0\n3)m0\n3+s)˜γ, with all other parameters fixed in each case.\nAnalogously to Corollary 1 we have\nCorollary 3 The heteroclinic solutions of Theorem 7 are in one-to-one corr espondence with\nheteroclinic solutions to (33), connecting to θ≡0orθ=πinU. Bounded solutions for\nθ/ne}ationslash∈ {0,π}are also in one-to-one correspondence.\nProof (Theorem 7) Note that ˜Γ(m3) = 0 with |m3|<1 is equivalent to (and if |m3|= 1\nsufficient for) the existence of an equilibrium of (34) at θwith cos( θ) =m3,q=q(m3) from\n(40) and ˜ p= 0. Assuming ˜ γ=∂m3˜Γ(m0\n3)/ne}ationslash= 0 and ∂ν˜Γ(m0\n3)/ne}ationslash= 0 forν=horν=µimplies\nexistence of a locally unique curve of equilibria m3(ν) that transversely crosses m0\n3. The case\nof parameters Sεis analogous with a curve m3(ε), where ∂εm3(0) =−∂ε˜Γ(m0\n3)/∂m3˜Γ(m0\n3)|ε=0\nhaving the sign of −m0\n3means bifurcation of two equilibria for ε >0.\nItremainstoshowthatthecentermanifoldassociatedtothe bifurcationisone-dimensional,\nand to obtain the directionality of heteroclinics.\nFor the former it suffices to show that the linearization at the bifurcation point has only\na simple eigenvalue on the imaginary axis, namely at zero. Th e linearization of (34) in any\npoint with ˜ p= 0,θ=θ0gives the 3 ×3 matrix\n˜A=\n00 0\n0\n0B\n, B=/parenleftbigg−αs 2qm0\n3+s\n−(s+2qm0\n3)−αs/parenrightbigg\n,\nwhich has a kernel with eigenvector (1 ,0,0)t. The remaining eigenvalues are those of B,\nwhich are −αs±(s+m0\n32q)i. Since these lie off the imaginary axis for s/ne}ationslash= 0 there is indeed\nat most one simple zero eigenvalue on the imaginary axis. Thi s implies the existence of a\none-dimensional center manifold which includes all equili bria and heteroclinic connections\nnear (θ0,0,q(m0\n3)) for nearby parameters. Equilibria in the symmetry plane {m3=m0\n3}are\nsolutions of (˜ p′,q′) =:G(˜p,q) = 0 with Ggiven by (34). Since DG(0,q(m0\n3)) =Bis invertible\nwe obtain a curve ( θ0,˜pε,qε) for parameters at Sεas claimed.\nTheuniquenessofbifurcatingequilibriaoneithersideoft hesymmetryplaneandinvariance\nof the one-dimensional center manifold implies existence a nd local uniqueness of heteroclinic\nconnections for sgn( ε) = sgn( m0\n3˜γ∂ε˜Γ(m0\n3)|ε=0). In order to determine the directionality of\nthese, a perturbation in the kernel gives θ′= sin(θ0+δ)˜p=m0\n3δ˜p+O(δ2) so that for m0\n3˜pε<0\nthe equilibrium at ( θ0,˜pε,qε) is stable in the center manifold, and unstable for reversed sign.\nSince existence of heteroclinics requires sgn( ε) = sgn(m0\n3˜γ∂ε˜Γ(m0\n3)|ε=0) this implies stability\nifℓ:=−˜γ∂ε˜Γ(m0\n3)∂ε˜p|ε=0<0 and thus convergence to ( θ0,˜pε,qε) asℓξ→ ∞.\nForhε=h0−m0\n3˜γεwith otherwise fixed parameters ∂ε˜Γ(m0\n3) =m0\n3˜γso heteroclinics\nexist for ε >0. On the other hand, ∂ε˜pε|ε=0is the first component −m0\n3˜γαsdet(B) of\n−B−1∂hG(0,q(m0\n3))(−m0\n3˜γ), where αdet(B)>0. Hence, ℓ=s˜γas claimed. The cases\nε=µ,βare determined analogously using ∂µ˜Γ(m0\n3) =m0\n3,∂µG(0,q(m0\n3) = (−m0\n3,0) and\n∂β˜Γ(m0\n3) = ((1+ ccpm0\n3)α)−1>0,∂βG(0,q(m0\n3) = (0,−((1+ccpm0\n3)α)−1).\n290Π 2Π/Minus2/Minus1012\nΘΘ'supercritical\n0Π 2Π/Minus2/Minus1012\nΘΘ'subcritical\n0Π 2Π/Minus2/Minus1012\nΘΘ'subsubcritical\n0Π 2Π/Minus2/Minus1012\nΘΘ'domainwalls\n(a) (b) (c) (d)\nFigure 12: Phase plane streamplots of (42) with Mathematica . (a)-(c) have h−Ω = 1/2.\n(a) supercritical anisotropy (here µ= 1), (b) subcritical (here µ= 0), (c) subsubcritical (here\nµ=−1), (d) subsubcritical case that allows for standing domain walls,h= Ω,µ=−1.\n5.3 Stationary coherent structures for ccp= 0\nIn this section we consider the case s= 0 (which does not imply time-independence) and\nccp= 0 (which will imply integrability), so that equations (33) reduce to\nθ′′= sin(θ)/parenleftbig\nh−Ω+(q2−µ)cos(θ)/parenrightbig\nq′=αΩ−β−2cot(θ)θ′q.(41)\nIn case Ω /ne}ationslash=β/αthere are no equilibria and it will be shown at the end of this s ection\nthat there are no coherent structure-type solutions in that case. Recall from §2.1 that for\nccp= 0 we may choose coordinates so that β= 0, which means Ω = 0 and thus stationary\ncoherent structures are turned into standing waves. Howeve r, we choose not to remove the\nparameter βin order to emphasize the typically oscillatory nature of so lutions to (1) and for\nconsistency in parameter relations. Nevertheless, the sym metries and integrals that we will\nfind are consequences of this reducibility.\nFor Ω = β/α, system (41) is invariant under the reflection q→ −qso that{q= 0}is an\ninvariant plane which separates the three dimensional phas e space. In particular, there cannot\nbe connections between equilibria (=wavetrains) with oppo site signs of q, that is, sign reversed\nspatial wavenumbers .\n5.3.1 Homogeneous solutions ( q= 0)\nSolutions in the invariant set {q= 0}have the form m(ξ) =r(ξ)exp(itΩ) and (31) turns into\na second order ODE on the circle {m2\n3+r2= 1}. The ODE for θfrom (41) is given by the\nnonlinear pendulum equation\nθ′′= sin(θ)(h−Ω−µcos(θ)), (42)\nwhich is invariant under θ→ −θand is Hamiltonian with potential energy\nP0(θ) = cos(θ)(h−Ω−µ\n2cos(θ)).\nThe symmetry (24) applies and we therefore assume in the foll owing that Ω = β/α < h.\n30We plot the qualitatively different vector fields of (42) in Fig ure 12 and some profiles\nin Figure 4. Coherent structure solutions are completely ch aracterized via the figure, which\nwe formulate next explicitly for the original PDE with m= (m,m3),m=reiϕ,r= sin(θ),\nm3= cos(θ). Homoclinic profiles may beinterpreted as (dissipative) s olitons. Theheteroclinic\nconnections in item 2(a) can be viewed as (dissipative) soli tons with ‘phase slip’.\nTheorem 8 Lets= 0andΩ =β/αand consider solutions to (14)of the form (30)withϕ\nconstant in ξ, i.e.,q= 0. These oscillate in time pointwise about the ˆe3-axis with frequency\nΩ =β/α. Assume without loss of generality, due to (24), thath >Ω.\n1. Subcritical anisotropy h−Ω>|µ|>0. There exist no nontrivial wavetrains with q= 0,\nand the coherent structure solutions with q= 0are a pair of homoclinic profiles to ˆe3, and\nthree one-parameter families of periodic profiles, one boun ded and two semi-unbounded.\nThe homoclinic profiles each cross once through −ˆe3in opposite θ-directions. The limit\npoints of the bounded curve of periodic profiles are −ˆe3and the union of homoclinic\nprofiles. Each of the homoclinics is the limit point of one of t he semi-unbounded families,\neach of which has unbounded θ-derivatives. The profiles from the bounded family each\ncross−ˆe3once during a half-period, the profiles of the unbounded fami ly cross both ±ˆe3\nduring one half-period.\n2. Suppose super- or subsubcritical anisotropy |µ|> h−Ω. There exists a wavetrain with\nk= 0, which is stable in the supercritical case ( µ >0) and unstable in the subsubcritical\ncase (µ <0). In(42)this appears in the form of two equilibria being the symmetri c pair\nof intersection points of the wavetrain orbit and a meridian on the sphere, phase shifted\nbyπinϕ-direction. Details of the following can be read off Figure 12 analogous to item\n1.\n(a) In the supercritical case ( µ > h−Ω) the coherent structure solutions with q= 0\nare two pairs of heteroclinic connections between the wavet rain and its phase shift,\nand four curves of periodic profiles; two bounded and two semi -unbounded.\n(b) In the subsubcritical case ( µ < h−Ω) the coherent structure solutions with q= 0are\ntwo pairs of homoclinic connections to ±ˆe3, respectively, and five curves of periodic\nprofiles, three bounded and two semi-unbounded.\n3. The degenerate case h= Ω,µ <0is the only possibility for profiles of stationary coherent\nstructures to connect between ±ˆe3, which then come in a pair as in the corresponding\npanel of Figure 12. The remaining coherent structures with q= 0are analogous to the\nsupercritical case with ±ˆe3and the pair of wavetrain and its phase shift interchanged.\nProof.As for wavetrains discussedin §4, thecondition cos( θ) = (h−Ω)/µyields theexistence\ncriterion |h−Ω|<|µ|for an equilibrium to (42) in (0 ,π). The derivative of the right hand\nside of (42) at θ= 0 ish−Ω−µ, which dictates the type of all equilibria and only saddles\ngenerate heteroclinic or homoclinic solutions.\nIt remains to study the connectivity of stable and unstable m anifolds of saddles, which is\ngiven by the difference in potential energy P0(θ). Since P0(0)−P0(π) = 2(h−Ω) the claims\nfollow.\n31/Minus4/Minus20240Π2Π\nkΘ0Π 2Π/Minus0.8/Minus0.400.4\nΘ\n0Π 2Π/Minus0.400.4\nΘ\n(a) (b)\nFigure 13: (a) Figure 11(a) with solutions to (43) for θ∈(0,π) andC= 0.1 (thick solid line),\nandC= 0.4 (thick dashed line). (b) Upper panel: potential P(θ) for parameters as in (a) and\nC= 0.1. Lower panel: same with C= 0.4.\n5.3.2 Non-homogeneous solutions ( q/ne}ationslash= 0)\nIn order to study (41) for q/ne}ationslash= 0, we note the following first integral. Since Ω = β/α, the\nequation for qcan be written as\n(log|q|)′=−2(log|sin(θ)|)′,\nand therefore explicitly integrated. With integration con stantC= sin(θ(0))2|q(0)|this gives\nq=C\nsin(θ)2. (43)\nSubstituting this into the equation for θyields the nonlinear pendulum\nθ′′= sin(θ)(h−Ω−µcos(θ))+C2cos(θ)\nsin(θ)3, (44)\nwith singular potential energy\nP(θ) =P0(θ)+1\n2C2cot(θ)2.\nThe energy introduces barriers at multiples of πso that solutions for q/ne}ationslash= 0 cannot pass ±ˆe3,\nand asCincreases the energy landscape becomes qualitatively inde pendent of Ω ,h,µ.\nAsanimmediateconsequenceoftheenergybarriersandthefa ctthatonlystablewavetrains\nare saddle points we get\nTheorem 9 Lets= 0andΩ =β/α. Consider solutions to (14)of the form (30). Intersec-\ntions in(q,θ)-space of the curve Cgiven by (43)with the wavetrain existence curves Wfrom\n(23)(withkreplaced by q) are in one-to-one correspondence with equilibria of (41).\n32Consider supercritical anisotropy 0≤h−Ω< µand assume Cis such that Ctransversely\nintersects the component of Wwhich intersects {q= 0}. Then the intersection point with\nsmallerq-value corresponds to a spectrally stable wavetrain, and in (41)there is a pair of\nhomoclinic solutions to this wavetrain. All other intersec tion points are unstable wavetrains.\nAll other non-equilibrium solutions of the form (30)withs= 0are periodic in ξ, and this\nis also the case for all other parameter settings.\nThe homoclinic orbit is a soliton-type solution to (1) with a symptotic state a wavetrain\n(cf. Figure 2).\nNotably, the tangential intersection of CandWis at the sideband instability.\nIn Figure 13 we plot an illustration in case of supercritical anisotropy µ >Ω−h >0. In\nthe upper panel of (b) the local maxima each generate a pair of homoclinic solutions to the\nstable wavetrain it represents. The values of qthat it visits lie on the bold curve in (a), whose\nintersections with the curve of equilibria are the local max ima and minima. The lower panel\nin (b) has C= 0.4 and the local maxima disappeared. All solutions are period ic and lie on\nthe thick dashed curve in (a).\n5.3.3 Absence of wavetrains ( Ω/ne}ationslash=β/α)\nIn this case the first integral Q= log(|q|sin(θ)2) is monotone,\nQ′=Ω−β/α\nq/ne}ationslash= 0,\nand therefore |q|is unbounded as ξ→ ∞orξ→ −∞. In view of (44), we also infer that θ\noscillates so that there are no relevant solutions of the for m (30).\nReferences\n[1]Aranson, I., and Kramer, L. The world of the complex Ginzburg-Landau equation.\nRev. Mod. Phys. 74 (2002), 99–143.\n[2]Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current.\nPhys. Rev. 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Bose National Centre \nfor Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India \n \n*E-mail: abarman@bose.res.in \n \n \nKeywords: (Thin Film Heterostructures, Interface Properties, Spin Pumping, Spin \nTransparency, Spin-Mixing Conductance, Gilbert Damping, Time-resolved Magneto-optical \nKerr Effect) \n \n \nAbstract \nPure spin current has transfigured the energy-efficient spintronic devices and it has the salient \ncharacteristic of transport of the spin angular momentum. Spin pumping is a potent method to \ngenerate pure spin current and for its increased efficiency high effective spin-mixing \nconductance ( Geff) and interfacial spin transparency ( T) are essential. Here, a giant T is reported \nin Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) heterostructures in beta-tungsten (β-W) phase by \nemploying all-optical time-resolved magneto-optical Kerr effect technique. From the variation \nof Gilbert damping with W and CoFeB thicknesses, the spin diffusion length of W and spin-\nmixing conductances are extracted. Subsequently, T is derived as 0.81 ± 0.03 for the β-\nW/CoFeB interface. A sharp variation of Geff and T with W thickness is observed in consonance \nwith the thickness-dependent structural phase transition and resistivity of W. The spin memory \nloss and two-magnon scattering effects are found to have negligible contributions to damping \nmodulation as opposed to spin pumping effect which is reconfirmed from the invariance of \ndamping with Cu spacer layer thickness inserted between W and CoFeB. The observation of \ngiant interfacial spin transparency and its strong dependence on crystal structures of W will be \nimportant for pure spin current based spin-orbitronic devices. \n    \n2 \n 1. Introduction \nThe rapid emergence of spintronics has promised a new paradigm of electronics based on the \nspin degree of freedom either associated with the charge or by itself.[1-3]  This has potential \nadvantages of non-volatility, reduced electrical power consumption, increased data processing \nspeed, and increased integration densities as opposed to its semiconductor counterpart.[4] A \nmajor objective of modern spintronics is to harness pure spin current, which comprises of flow \nof spins without any net flow of charge current.[5, 6] This has the inherent benefit of reduced \nJoule heating and Oersted fields together with the ability to manipulate magnetization . Three \nmajor aspects of spin current are its generation, transport, and functionalization. Pure spin \ncurrent can be generated by spin-Hall effect,[7,8] Rashba-Edelstein effect,[9,10] spin pumping,[11-\n13] electrical injection in a lateral spin valve using a non-local geometry,[14,15] and spin \ncaloritronic effects.[16,17] Among these, spin pumping is an efficient and extensively used \nmethod of spin injection from ferromagnet (FM) into normal metal (NM) where the precessing \nspins from FM transfer spin angular momentum to the conduction electrons of adjacent NM \nlayer in NM/FM heterostructure, which gets dissipated by spin-flip scattering. The efficiency \nof spin pumping is characterized by spin-mixing conductance and spin diffusion length. The \ndissipation of spin current into the NM layer results in loss of spin angular momentum in the \nFM layer leading to an increase in its effective Gilbert damping parameter ( αeff). Thus, spin \npumping controls the magnetization dynamics in NM/FM heterostructures, which is crucial for \ndetermining the switching efficiency of spin-torque based spintronic devices. The enhancement \nin αeff is more prominent in heavy metals (HM) with high spin-orbit coupling (SOC) due to \nstronger interaction between electron spin and lattice. Intense research in the field of spin-\norbitronics has revealed that interface dependent spin transport is highly influenced by the spin \ntransparency, which essentially determines the extent of spin current diffused through the \nNM/FM interface.[18,19]     \n3 \n The highly resistive β-W, which shows a distorted tetragonal phase commonly referred to as \nA15 structure, is well known for exhibiting large spin Hall angle (SHA) (up to ~0.50) [20]  as \ncompared to other transition metal elements such as Pt (0.08) [21]  and β-Ta (0.12).[7] Besides, \nin W/FM heterostructures, W leads to highly stable perpendicular magnetic anisotropy[22]  and \ninterfacial Dzyaloshinskii-Moriya interaction.[23] Another important characteristic associated \nwith W is that it shows a thickness-dependent phase transition in the sub-10 nm thickness \nregime.[24,25] In general, sputter-deposited W films with thickness well below 10 nm are found \nto have β phase with high resistivity, whereas the films with thickness above 10 nm possess \npredominantly α phase (bcc structure) with low resistivity. A small to moderate SHA has been \nreported for the α and mixed (α + β) phase (<0.2) of W.[24] As SHA and effective spin-mixing \nconductance ( Geff) are correlated, one would expect that interfacial spin transparency ( T), which \nis also a function of Geff, should depend on the structural phase of W thin films. Furthermore, \nthe magnitude of the spin-orbit torque (SOT) depends on the efficiency of spin current \ntransmission (i.e. T) across the NM/FM interface. It is worth mentioning that due to high SOC \nstrength, W is a good spin-sink material and also cost-effective in comparison with the widely \nused NM like Pt. On the other hand, CoFeB due to its notable properties like high spin \npolarization, large tunnel magnetoresistance, and low intrinsic Gilbert damping, is used as FM \nelectrode in magnetic tunnel junctions. The presence of Boron at the NM/CoFeB interface \nmakes this system intriguing as some recent studies suggest that a small amount of boron helps \nin achieving a sharp interface and increases the spin polarization, although an excess of it causes \ncontamination of the interface. To this end, determination of T of the technologically important \nW/CoFeB interface and its dependence on the W-crystal phase are extremely important but still \nabsent in the literature.  \nBesides spin pumping, there are different mechanisms like spin memory loss (SML),[26] Rashba \neffect,[10] two-magnon scattering (TMS),[27] and interfacial band hybridization[28] which may \nalso cause loss of spin angular momentum at NM/FM interface, resulting in increase of αeff and   \n4 \n decrease of the spin transmission probability. However, for improved energy efficiency, the \nNM/FM interface in such engineered heterostructures must possess high spin transmission \nprobability. Consequently, it is imperative to get a deeper insight into all the mechanisms \ninvolved in generation and transfer of spin current for optimizing its efficiency. Here, we \ninvestigate the effects of spin pumping on the Gilbert damping in W/CoFeB bilayer system as \na function of W-layer thickness using recently developed all-optical technique, which is free \nfrom delicate micro-fabrication and electrical excitation and detection.[29] This is a local and \nnon-invasive method based on time-resolved magneto-optical Kerr effect (TR-MOKE) \nmagnetometry. Here, the damping is directly extracted from the decaying amplitude of time-\nresolved magnetization precession, which is free from experimental artifacts stemming from \nmultimodal oscillation, sample inhomogeneity, and defects. From the modulation of damping \nwith W layer thickness, we have extracted the intrinsic spin-mixing conductance ( G↑↓) of the \nW/CoFeB interface which excludes the backflow of spin angular momentum and spin diffusion \nlength(𝜆௦ௗ) of W. Furthermore, we have modeled the spin transport using both the ballistic \ntransport model[30, 31] and the model based on spin diffusion theory[32,33]. Subsequently, Geff, \nwhich includes the backflow of spin angular momentum, is estimated from the dependence of \ndamping on the CoFeB layer thicknesses. By using both the spin Hall magnetoresistance \nmodel[34] and spin transfer torque based model utilizing the drift-diffusion approximation[35], \nwe have calculated the T of W/CoFeB interface. The spin Hall magnetoresistance model gives \nlower value of T than the drift-diffusion model, but the former is considered more reliable as \nthe latter ignores the spin backflow. We found a giant value of T exceeding 0.8 in the β phase \nof W, which exhibits a sharp decrease to about 0.6 in the mixed (α+β) phase using spin Hall \nmagnetoresistance model. We have further investigated the other possible interface effects in \nour W/CoFeB system, by incorporating a thin Cu spacer layer of varying thickness between the \nW and CoFeB layers. Negligible modulation of damping with Cu thickness confirms the   \n5 \n dominance of spin pumping generated pure spin current and its transport in the modulation of \ndamping in our system.   \n \n2. Results and Discussion \nFigure 1 (a) shows the grazing incidence x-ray diffraction (GIXRD) patterns of \nSub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures at the glancing angle of 2o. In these \nplots, the peaks corresponding to α and β phase of W are marked. The high-intensity GIXRD \npeak at ∼44.5° and low intensity peak at ∼64° correspond primarily to the β phase (A15 \nstructure) of W (211) and W(222) orientation, respectively. Interestingly, we find these peaks \nto be present for all thicknesses of W, but when t > 5 nm, then an additional peak at ∼40.1° \ncorresponding to α-W with (110) crystal orientation appears. Consequently, we understand that \nfor t ≤ 5 nm, W is primarily in β-phase, while for t > 5 nm a fraction of the α phase appears, \nwhich we refer to as the mixed (α+β) phase of W. These findings are consistent with some \nexisting literature.[24,25]  Some other studies claimed that this transition thickness can be tuned \nby carefully tuning the deposition conditions of the W thin films.[36] The average lattice \nconstants obtained from the β-W peak at 44.5o and α-W peak at 40.1o correspond to about 4.93 \nand 3.15 Å, respectively. By using the Debye-Scherrer formula, we find the average crystallite \nsize in β and α phase of W to be about 14 and 7 nm, respectively. \nIt is well known that the formation of β-W films is characterized by large resistivity due to its \nA-15 structure which is associated with strong electron-phonon scattering, while the α-W \nexhibits comparatively lower resistivity due to weak electron-phonon scattering. We measured \nthe variation of resistivity of W with its thickness across the two different phases, using the \nfour-probe method. The inverse of sheet resistance ( Rs) of the film stack as a function of W \nthickness is plotted in Figure 1(b). A change of the slope is observed beyond 5 nm, which \nindicates a change in the W resistivity. The data have been fitted using the parallel resistors \nmodel[24] (shown in Figure S1  of the Supporting Information). [37] We estimate the average   \n6 \n resistivity of W ( ρW) in β and mixed (α+β) phase to be about 287 ± 19 and 112 ± 14 µΩ.cm, \nrespectively, while the resistivity of CoFeB (ρCoFeB) is found to be 139 ± 16 μΩ.cm. Thus, the \nresistivity results corroborate well with those of the XRD measurement. \nThe AFM image of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) (t = 1, 5 and 10 nm) samples in \nFigure 1(c) revealed the surface topography. We have used WSxM software to process the \nimages.[38] The variation in the average surface roughness of the films with W thickness is listed \nin Table 1. The roughness varies very little when measured at various regions of space of the \nsame sample. The surface roughness in all samples is found to be small irrespective of the \ncrystal phase of W. Due to the small thicknesses of various layers in the heterostructures, the \ninterfacial roughness is expected to show its imprint on the measured topographical roughness. \nWe thus understand that the interfacial roughness in these heterostructures is very small and \nsimilar in all studied samples. Details of AFM characterization is shown in Figure S2  of the \nSupporting Information.[37] \n2.1. Principles behind the modulation of Gilbert damping with layer thickness: \nIn an NM/FM bilayer magnetic damping can have various additional contributions, namely \ntwo-magnon scattering, eddy current, and spin pumping in addition to intrinsic Gilbert damping. \nAmong these, the spin pumping effect is a non-local effect, in which an external excitation \ninduces magnetization precession in the FM layer. The magnetization precession causes a spin \naccumulation at the NM/FM interface. These accumulated spins carry angular momentum to \nthe adjacent NM layer, which acts as a spin sink by absorbing the spin current by spin-flip \nscattering, leading to an enhancement of the Gilbert damping parameter of FM. In 2002, \nTserkovnyak and Brataas theoretically demonstrated the spin pumping induced enhancement \nin Gilbert damping in NM/FM heterostructures using time-dependent adiabatic scattering \ntheory where magnetization dynamics in the presence of spin pumping can be described by a \nmodified Landau-Lifshitz-Gilbert (LLG) equation as: [11-13]   \n7 \n ௗ𝒎\nௗ௧= −𝛾(𝒎×𝑯eff)+𝛼0(𝒎×ௗ𝒎\nௗ௧)+ఊ\nVMೞ𝑰௦                          (1) \nwhere γ is the gyromagnetic ratio,  Is is the total spin current, Heff is the effective magnetic field, \nα0 is intrinsic Gilbert damping constant, V is the volume of ferromagnet and Ms is saturation \nmagnetization of the ferromagnet. As shown in equation (2), Is generally consists of a direct \ncurrent contribution 𝑰𝒔𝟎 which is nonexistent in our case as we do not apply any charge current, \n𝑰𝒔𝒑𝒖𝒎𝒑, i.e. spin current due to pumped spins from the FM to NM and 𝑰𝒔𝒃𝒂𝒄𝒌, i.e. a spin current \nbackflow to the FM reflecting from the NM/substrate interface which is assumed to be a perfect \nreflector. \n𝑰𝒔=𝑰𝒔𝟎+𝑰𝒔pump+𝑰𝒔back                                                    (2) \nHere, 𝑰𝒔𝒃𝒂𝒄𝒌 is determined by the spin diffusion length of the NM layer. Its contribution to \nGilbert damping for most metals with a low impurity concentration is parametrized by a \nbackflow factor β which can be expressed as:[39] \n𝛽=൭2𝜋𝐺↑↓ටఌ\nଷtanhቀ௧\nఒೞቁ൱ିଵ\n                                      (3) \nwhere ε is the material-dependent spin-flip probability, which is the ratio of the spin-conserved \nto spin-flip scattering time. It can be expressed as: [40]                                                                \n 𝜀= (𝜆𝜆௦ௗ⁄)ଶ3⁄                                                                                     (4) \nwhere λel and λsd are the electronic mean free path and spin diffusion length of NM, respectively.  \nThe spin transport through NM/FM interface directly depends on the spin-mixing conductance, \nwhich is of two types: (a) G↑↓, which ignores the contribution of backflow of spin angular \nmomentum, and (b) Geff, which includes the backflow contribution. Spin-mixing conductance  \ndescribes the conductance property of spin channels at the interface between NM and FM. Also, \nspin transport across the interface affects the damping parameter giving rise to αeff of the system   \n8 \n that can be modeled by both ballistic and diffusive transport theory. In the ballistic transport \nmodel, the αeff is fitted with the following simple exponential function:[30,31,39] \n𝐺eff=𝐺↑↓൬1−𝑒ିమ\nഊೞ൰=ସగdM\nఓಳ(𝛼eff−𝛼)                            (5) \n𝛥𝛼=𝛼eff−𝛼=ఓಳீ↑↓൭ଵିషమ\nഊೞ൱\nସdMeff                                         (6) \nHere, the exponential term signifies backflow spin current contribution and a factor of 2 in the \nexponent signifies the distance traversed by the spins inside the NM layer due to reflection from \nthe NM/substrate interface.  \nIn the ballistic approach, the resistivity of NM is not considered while the NM thickness is \nassumed to be less than the mean free path. To include the effect of the charge properties of \nNM on spin transport, the model based on spin diffusion theory is used to describe αeff (t). \nWithin this model, the additional damping due to spin pumping is described as:[32,33,36]  \n            𝐺eff=ீ↑↓\nቆଵାమഐഊೞಸ↑↓\nୡ୭୲୦ቀ௧ఒೞൗቁቇ=ସగdM\nఓಳ(𝛼eff−𝛼)                                         (7)                                                                                       \n                                                                                        \n           ∆𝛼=𝛼eff−𝛼=ఓಳீ↑↓\nସగdMቆଵା మഐഊೞಸ↑↓\nୡ୭୲୦ቀ௧ఒೞൗቁቇ (8)                                                                                            \n     \nwhere ρ is the electrical resistivity of the W layer. Here the term  మఘఒೞீ↑↓\ncothቀ𝑡𝜆௦ௗൗቁ account \nfor the back-flow of pumped spin current into the ferromagnetic layer. \nThe reduction of spin transmission probability implies a lack of electronic band matching, \nintermixing, and disorder at the interface. The spin transparency, T of an NM/FM interface \ntakes into account all such effects that lead to the electrons being reflected from the interface \ninstead of being transmitted during transport. Further, T depends on both intrinsic and extrinsic \ninterfacial factors, such as band-structure mismatch, Fermi velocity, interface imperfections, \netc.[19,39] According to the spin Hall magnetoresistance model, the spin current density that   \n9 \n diffuses into the NM layer is smaller than the actual spin current density generated via the spin \npumping in the FM layer. This model linked T with 𝐺eff  by the following relation:[34,39] \n𝑇=ீeff  tanh൬\nమഊೞ൰\nீeff  coth൬\nഊೞ൰ା\nమഊೞమഐ                              (9) \nThe interfacial spin transparency was also calculated by Pai et al. in the light of damping-like \nand field-like torques utilizing the drift-diffusion approximation. Here, the effects of spin \nbackflow are neglected as it causes a reduction in the spin torque efficiencies. Assuming t ≫ λ \nand a very high value of d, T can be expressed as:[35] \n𝑇=ଶீ↑↓ீಿಾ⁄\nଵାଶீ↑↓ீಿಾ⁄                                                                                                               (10) \nwhere, 𝐺ேெ=\nఘఒೞమ  is the spin conductance of the NM layer. \nIn an NM/FM heterostructure, other than spin pumping, there is a finite probability to have \nsome losses of spin angular momentum due to interfacial depolarization and surface \ninhomogeneities, known as SML and TMS, respectively. In SML, loss of spin angular \nmomentum occurs when the atomic lattice at the interface acts as a spin sink due to the magnetic \nproximity effect or due to the interfacial spin-orbit scattering which could transfer spin \npolarization to the atomic lattice.[26] The TMS arises when a uniform FMR mode is destroyed \nand a degenerate magnon of different wave vector is created.[27] The momentum non-\nconservation is accounted for by considering a pseudo-momentum derived from internal field \ninhomogeneities or secondary scattering. SML and TMS may contribute to the enhancement of \nthe Gilbert damping parameter considerably. Recently TMS is found to be the dominant \ncontribution to damping for Pt-FM heterostructures.[41] In the presence of TMS and SML \neffective Gilbert damping can be approximated as:[41] \n    αeff = α0 + αSP + αSML + αTMS    \n      ∆𝛼=𝛼eff−𝛼= 𝑔𝜇ீeff  ା ீೄಾಽ\nସdM+𝛽்ெௌ𝑑ିଶ                                                      (11) \n   \n10 \n where 𝐺ௌெ is the “effective SML conductance”, and βTMS is a “coefficient of TMS” that \ndepends on both interfacial perpendicular magnetic anisotropy field and the density of magnetic \ndefects at the FM surfaces.  \n2.2. All-optical measurement of magnetization dynamics: \nA schematic of the spin pumping mechanism along with the experimental geometry is shown \nin Figure 2(a). A typical time-resolved Kerr rotation data for the Sub/Co 20Fe60B20(3 nm)/SiO 2(2 \nnm) sample at a bias magnetic field, H = 2.30 kOe is shown in Figure 2(b) which consists of \nthree different temporal regimes. The first regime is called ultrafast demagnetization, where a \nsharp drop in the Kerr rotation (magnetization) of the sample is observed immediately after \nfemtosecond laser excitation. The second regime corresponds to the fast remagnetization where \nmagnetization recovers to equilibrium by spin-lattice interaction. The last regime consists of \nslower relaxation due to heat diffusion from the lattice to the surrounding (substrate) superposed \nwith damped magnetization precession. The red line in Figure 2(b) denotes the bi-exponential \nbackground present in the precessional data. We are mainly interested here in the extraction of \ndecay time from the damped sinusoidal oscillation about an effective magnetic field and its \nmodulation with the thickness of FM and NM layers. We fit the time-resolved precessional data \nusing a damped sinusoidal function given by: \n𝑀(𝑡)=𝑀(0)𝑒ିቀ\nഓቁsin(2π𝑓𝑡+𝜑)                               (12) \nwhere τ is the decay time, φ is the initial phase of oscillation and f  is the precessional frequency. \nThe bias field dependence of precessional frequency can be fitted using the Kittel formula given \nbelow to find the effective saturation magnetization ( Meff): \n𝑓=ఊ\nଶ(𝐻(𝐻+4π𝑀eff))ଵ/ଶ                                                              (13)                       \nwhere γ = gµB/ħ, g is the Landé g-factor and ћ is the reduced Planck’s constant. From the fit, \nMeff and g are determined as fitting parameters. For these film stacks, we obtained effective   \n11 \n magnetization, Meff ≈ 1200 ± 100 emu/cc, and g = 2.0 ± 0.1. The comparison between Meff \nobtained from the magnetization dynamics measurement and Ms from VSM measurement for \nvarious thickness series are presented systematically in Figures S3-S5  of the Supporting \nInformation.[37] For almost all the film stacks investigated in this work, Meff  is found to be close \nto Ms, which indicates that the interface anisotropy is small in these heterostructures. We \nestimate αeff using the expression: [42]  \n𝛼eff=1\nγτ(𝐻+2π𝑀eff)                                           (14) \nwhere τ is the decay time obtained from the fit of the precessional oscillation with equation (12). \nWe have plotted the variation of time-resolved precessional oscillation with the bias magnetic \nfield and the corresponding fast Fourier transform (FFT) power spectra in Figure S6  of the \nSupporting Information.[37] The extracted values of αeff are found to be independent of the \nprecession frequency f. Recent studies show that in presence of extrinsic damping contributions \nlike TMS, αeff should increase with f, while in presence of inhomogeneous anisotropy in the \nsystem αeff should decrease with f.[43] Thus, frequency-independent αeff rules out any such \nextrinsic contributions to damping in our system. \n2.3. Modulation of the Gilbert damping parameter: \nIn Figure 3 (a) we have presented time-resolved precessional dynamics for \nSub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) samples with 0 ≤ t ≤ 15 nm at H = 2.30 kOe. The \nvalue of α0 for the 3-nm-thick CoFeB layer without the W underlayer is found to be 0.006 ± \n0.0005. The presence of W underlayer causes αeff to vary non monotonically over the whole \nthickness regime as shown by the αeff vs. t plot in Figure 3(b). In the lower thickness regime, \ni.e.  0 ≤ t ≤ 3 nm, Δ α increases sharply by about 90% due to spin pumping but it saturates for t \n≥ 3 nm. However, for t > 5 nm, Δ α drops by about 30% which is most likely related to due to \nthe thickness-dependent phase transition of W.  At first, we have fitted our result for t ≤ 5 nm \nwith equation (6) of the ballistic transport model and determined G↑↓ = (1.46 ± 0.01) × 1015 cm-  \n12 \n 2 and λsd = 1.71 ± 0.10 nm as fitting parameters. Next, we have also fitted our results with \nequation (8) based on spin diffusion theory, where we have obtained G↑↓ = (2.19 ± 0.02) × 1015 \ncm-2 and λsd = 1.78 ± 0.10 nm. The value of G↑↓ using spin diffusion theory is about 28% higher \nthan that of ballistic model while the value of λsd is nearly same in both models. Using values \nfor λel (about 0.45 nm for W) from the literature[44]  and λsd derived from our experimental data, \nwe have determined the spin-flip probability parameter, ε = 2.30 × 10−2 from equation (4). To \nbe considered as an efficient spin sink, a nonmagnetic metal must have ε  ≥ 1.0 × 10-2 and hence \nwe can infer that the W layer acts as an efficient spin sink here.[13] The backflow factor β can \nbe extracted from equation (3). We have quantified the modulation of the backflow factor (Δ β) \nto be about 68% within the experimental thickness regime. \nTo determine the value of 𝐺eff  directly from the experiment, we have measured the time-\nresolved precessional dynamics for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) samples with 1 \nnm ≤ d ≤ 10 nm at H = 2.30 kOe as shown in Figure 4(a). The αeff  is found to increase with the \ninverse of FM layer thickness ( Figure 4(b)). We have fitted our results first with equation (5), \nfrom which we have obtained 𝐺eff  and 𝛼 to be (1.44 ± 0.01) × 1015 cm-2 and 0.006 ± 0.0005, \nrespectively.  \nBy modelling the W thickness dependent modulation of damping of Figure 3(b) using equation \n(5), we have obtained 𝐺eff  of W/CoFeB in β-phase (where ∆𝛼 ≈  0.006) and α+β-mixed phase \n(where ∆𝛼  ≈ 0.004) of W to be (1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2, \nrespectively. From these, we conclude that β-phase of W has higher conductance of spin \nchannels in comparison to the α+β-mixed phase. The variation of 𝐺eff  with W layer thickness \nis presented in Figure 5(a), which shows that 𝐺eff  increases non monotonically and nearly \nsaturates for t ≥ 3 nm. For t > 5 nm, 𝐺eff  shows a sharp decrease in consonance with the variation \nof αeff.  \nWe have further fitted the variation of αeff  with the inverse of FM layer thickness ( Figure 4(b)) \nusing with equation (11) to isolate the contributions from SML, TMS and spin pumping (SP).   \n13 \n The values of  𝐺ௌெ , and βTMS are found to be (2.45 ± 0.05) × 1013 cm-2 and (1.09 ± 0.02) × 10-\n18 cm2, respectively. 𝐺ௌெ is negligible in comparison with 𝐺eff  which confirms the absence of \nSML contribution in damping. Contribution of TMS to damping modulation ( 𝛽்ெௌ𝑑ଶ) is also \nbelow 2% for all the FM thicknesses. The relative contributions are plotted in Figure 5(b). It is \nclear that spin pumping contribution is highly dominant over the SML and TMS for our studied \nsamples. The value of our 𝐺eff  in β-W/CoFeB is found to be much higher than that obtained for \nβ-Ta/CoFeB[39] measured by all-optical TRMOKE technique as well as various other NM/FM \nheterostructures measured by conventional techniques as listed in Table 2. This provides \nanother confirmation of W being a good spin sink material giving rise to strong spin pumping \neffect. \nWe subsequently investigate the value of T for W/CoFeB interface, which is associated with \nthe spin-mixing conductances of interface, spin diffusion length, and resistivity of NM as \ndenoted in equations (9) and (10). T is an electronic property of a material that depends upon \nelectronic band matching of the two materials on either side of the interface. After determining \nthe resistivity, spin diffusion length and spin-mixing conductances experimentally, we have \ndetermined the value of T which depends strongly on the structural phase of W. Using equation \n(9) based on the spin-Hall magnetoresistance model, Tβ-W and T(α+β)-W are found to be 0.81 ± \n0.03 and 0.60 ± 0.02, respectively. On the other hand, equation (10) of spin transfer torque \nbased model utilizing the drift-diffusion approximation gives Tβ-W and T(α+β)-W to be 0.85 ± 0.03 \nand 0.63 ± 0.02, respectively, which are slightly higher than the values obtained from spin-Hall \nmagnetoresistance model. However, we consider the values of T obtained from the spin-Hall \nmagnetoresistance model to be more accurate as it includes the mandatory contribution of spin \ncurrent backflow from W layer into the CoFeB layer. Nevertheless, our study clearly \ndemonstrates that the value of spin transparency of the W/CoFeB interface is the highest \nreported among the NM/FM heterostructures as listed in Table 2. This high value of T, \ncombined with the high spin Hall angle of β-W makes it an extremely useful material for pure   \n14 \n spin current based spintronic and spin-orbitronic devices. The structural phase dependence of \nT for W also provides a particularly important guideline for choosing the correct thickness and \nphase of W for application in the above devices.  \nFinally, to directly examine the additional possible interfacial effects present in the W/CoFeB \nsystem, we have introduced a copper spacer layer of a few different thicknesses between the W \nand CoFeB layers. Copper has very small SOC and spin-flip scattering parameters and it shows \na very high spin diffusion length. Thus, a thin copper spacer layer should not affect the damping \nof the FM layer due to the spin pumping effect but can influence the other possible interface \neffects. Thus, if other interface effects are substantial in our samples, the introduction of the \ncopper spacer layer would cause a notable modulation of damping with the increase of copper \nspacer layer thickness ( c).[19,39] The time-resolved Kerr rotation data for the Sub/W(4 \nnm)/Cu(c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures with 0 ≤ c ≤ 1 nm are presented in \nFigure 6(a) at H = 2.30 kOe and Figure 6(b) shows the plot of αeff as a function of c. The \ninvariance of αeff with c confirms that the interface of Cu/CoFeB is transparent for spin transport \nand possible additional interfacial contribution to damping is negligible, which is in agreement \nwith our modelling as shown in Figure 5(b).  \n \n3. Conclusion \nIn summary, we have systematically investigated the effects of thickness-dependent structural \nphase transition of W in W( t)/CoFeB( d) thin film heterostructures and spin pumping induced \nmodulation of Gilbert damping by using an all-optical time-resolved magneto-optical Kerr \neffect magnetometer. The W film has exhibited structural phase transition from a pure β phase \nto a mixed (α + β) phase for t > 5 nm. Subsequently, β-W phase leads to larger modulation in \neffective damping ( αeff) than (α+β)-W. The spin diffusion length of W is found to be 1.71 ± \n0.10 nm, while the spin pumping induced effective spin-mixing conductance  𝐺eff  is found to be  \n(1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2 for β and mixed (α+β) phase of W,   \n15 \n respectively. This large difference in 𝐺eff  is attributed to different interface qualities leading \ntowards different interfacial spin-orbit coupling. Furthermore, by analyzing the variation of αeff \nwith CoFeB thickness in W (4 nm)/CoFeB  (d)/SiO2 (2 nm), we have isolated the contributions \nof spin memory loss and two-magnon scattering from spin pumping, which divulges that spin \npumping is the dominant contributor to damping. By modeling our results with the spin Hall \nmagnetoresistance model, we have extracted the interfacial spin transparency ( T) of β-\nW/CoFeB and (α + β)-W/CoFeB as 0.81 ± 0.03 and 0.60 ± 0.02, respectively. This structural \nphase-dependent T value will offer important guidelines for the selection of material phase for \nspintronic applications. Within the framework of ballistic and diffusive spin transport models, \nthe intrinsic spin-mixing conductance ( G↑↓) and spin-diffusion length ( λsd) of β-W are also \ncalculated by studying the enhancement of αeff as a function of β-W thickness. Irrespective of \nthe used model, the value of T for W/CoFeB interface is found to be highest among the NM/FM \ninterfaces, including the popularly used Pt/FM heterostructures. The other possible interface \neffects on the modulation of Gilbert damping are found to be negligible as compared to the spin \npumping effect. Thus, our study helps in developing a deep understanding of the role of W thin \nfilms in NM/FM heterostructures and the ensuing spin-orbit effects. The low intrinsic Gilbert \ndamping parameter, high effective spin-mixing conductance combined with very high interface \nspin transparency and spin Hall angle can make the W/CoFeB system a key material for spin-\norbit torque-based magnetization switching, spin logic and spin-wave devices.  \n \n4. Experimental Section/Methods \n4.1. Sample Preparation \nThin films of Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) were deposited by using RF/DC magnetron \nsputtering system on Si (100) wafers coated with 285 nm-thick SiO 2. We varied the W layer \nthickness as t = 0, 0.5, 1, 1.5, 2, 3, 4, 5, 8, 10 and 15 nm and CoFeB layer thickness as d = 1, 2, \n3, 5 and 10 nm. The depositions were performed at an average base pressure of 1.8  × 10-7 Torr   \n16 \n and argon pressure of about 0.5 mTorr at a deposition rate of 0.2 Å/s. Very slow deposition \nrates were chosen for achieving a uniform thickness of the films even at a very thin regime \ndown to sub-nm. The W and CoFeB layers were deposited using average DC voltages of 320 \nand 370 V, respectively, while SiO 2 was deposited using average RF power of 55 watts. All \nother deposition conditions were carefully optimized and kept almost identical for all samples. \nIn another set of samples, we introduced a thin Cu spacer layer in between the CoFeB and W \nlayers and varied its thickness from 0 nm to 1 nm. The Cu layer was deposited at a DC voltage \nof 350 V, argon pressure of 0.5 mTorr and deposition rate of 0.2 Ǻ/s. \n4.2. Characterization \nAtomic force microscopy (AFM) was used to investigate the surface topography and vibrating \nsample magnetometry (VSM) was used to characterize the static magnetic properties of these \nheterostructures. Using a standard four-probe technique the resistivity of the W films was \ndetermined and grazing incidence x-ray diffraction (GIXRD) was used for investigating the \nstructural phase of W. To study the magnetization dynamics, we used a custom-built TR-\nMOKE magnetometer based on a two-color, collinear optical pump-probe technique. Here, the \nsecond harmonic laser pulse (λ = 400 nm, repetition rate = 1 kHz, pulse width >40 fs) of an \namplified femtosecond laser, obtained using a regenerative amplifier system (Libra, Coherent) \nwas used to excite the magnetization dynamics, while the fundamental laser pulse (λ = 800 nm, \nrepetition rate = 1 kHz, pulse width ~40 fs) was used to probe the time-varying polar Kerr \nrotation from the samples. The pump laser beam was slightly defocused to a spot size of about \n300 µm and was obliquely (approximately 30° to the normal on the sample plane) incident on \nthe sample. The probe beam having a spot size of about 100 µm was normally incident on the \nsample, maintaining an excellent spatial overlap with the pump spot to avoid any spurious \ncontribution to the Gilbert damping due to the dissipation of energy of uniform precessional \nmode flowing out of the probed area. A large enough magnetic field was first applied at an \nangle of about 25° to the sample plane to saturate its magnetization. This was followed by a   \n17 \n reduction of the magnetic field to the bias field value ( H = in-plane component of the bias field) \nto ensure that the magnetization remained saturated along the bias field direction. The tilt of \nmagnetization from the sample plane ensured a finite demagnetizing field along the direction \nof the pump pulse, which was modified by the pump pulse to induce a precessional \nmagnetization dynamics in the sample. The pump beam was chopped at 373 Hz frequency and \nthe dynamic Kerr signal in the probe pulse was detected using a lock-in amplifier in a phase-\nsensitive manner. The pump and probe fluences were kept constant at 10 mJ/cm2 and 2 mJ/cm2, \nrespectively, during the measurement. All the experiments were performed under ambient \nconditions at room temperature. \n \nAcknowledgements \n \nAB gratefully acknowledges the financial assistance from the S. N. Bose National Centre for \nBasic Sciences (SNBNCBS), India under Project No. SNB/AB/18-19/211.  SNP, SM and SC \nacknowledge SNBNCBS for senior research fellowship. ArB acknowledges SNBNCBS for \npostdoctoral research associateship. 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Barwal, N. Behera, S. Akansel, P. Svedlindh, S. Chaudhary, \nPhys. Rev. B , 2018, 97, 064420. \n[52] G. Wu, Y. Ren, X. He, Y. Zhang, H. Xue, Z. Ji, Q. Y. Jin, Z. Zhang, Phys. Rev. Appl ., \n2020,  13, 024027. \n[53] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, F. Y. Yang, Phys. Rev. Lett ., \n2014, 112, 197201. \n \n \n    \n22 \n  \n  \n \nFigure 1.  (a) X-ray diffraction patterns measured at 2° grazing angle incidence for different W \nthickness. (b) Variation of inverse sheet resistance with W thickness. (c) AFM images of the \nsamples showing the surface topography.  \n \n  \n23 \n   \nFigure 2. (a) Schematic of experimental geometry and (b) typical TR-MOKE data from \nCo20Fe60B20(3 nm)/SiO 2(2 nm) heterostructure at an applied bias magnetic field of 2.30 kOe. \nThe three important temporal regimes are indicated in the graph. The solid red line shows a \nbiexponential fit to the decaying background of the time-resolved Kerr rotation data. \n  \n24 \n   \nFigure 3. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of W thickness at an \napplied bias magnetic field of 2.30 kOe. (b) Experimental result of variation damping with t \n(symbol) fitted with theoretical models (solid and dashed lines) of spin pumping. Two different \nregions corresponding to W crystal phase, namely β and α+β are shown.  \n \n \n \n \n  \n25 \n  \n  \nFigure 4. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) as function of Co 20Fe60B20 thickness \nd at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation of damping \nvs 1/d (symbol) fitted with theoretical models (solid and dashed lines).  \n \n  \n26 \n   \nFigure 5.  (a) Variation of effective spin-mixing conductance( 𝐺eff ) with W layer thickness t \n(symbol). The solid line is guide to the eye. (b) Contributions of SP, SML and TMS to the \nmodulation of damping for different Co 20Fe60B20 layer thickness d (symbol). The solid line is \nguide to the eye. 0 2 4 6 8 10039095100 \n  SP\n TMS\n SML\n  Damping (%) \n                d  (nm) 0 2 4 8 12 160.00.51.01.5 \n \n  Geff (1015 cm-2)\nt (nm)(a)\n(b)  \n27 \n   \nFigure 6. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W(4 nm)/Cu( c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of Cu layer \nthickness c at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation \nof damping vs c. The dotted line is guide to the eye, showing very little dependence of damping \non Cu layer thickness. \n  \n28 \n Table 1. The average surface roughness values of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) \nsamples obtained using AFM. \n \nTable 2. Comparison of the effective spin-mixing conductance and interfacial spin \ntransparency of the W/CoFeB samples studied here with the important NM/FM interfaces taken \nfrom the literature. \nMaterial  \nInterface Effective Spin-Mixing \nConductance  (×1015 cm-2) Interfacial Spin \nTransparency  \nPt/Py 1.52 [19] 0.25 [19] \nPt/Co 3.96 [19] 0.65 [19] \nPd/CoFe 1.07 [31] N.A. \nPt/FM 0.6-1.2 [35] 0.34-0.67 [35] \nβ-Ta/CoFeB  0.69 [39] 0.50 [39] \nβ-Ta/ CFA 2.90 [40] 0.68 [40] \nPd0.25Pt0.75/Co 9.11 [41] N.A. \nAu0.25Pt0.75/Co 10.73 [41] N.A. \nPd/Co 4.03 [41] N.A. \nPd0.25Pt0.75/FeCoB 3.35 [41] N.A. \nAu0.25Pt0.75/ FeCoB 3.64 [41] N.A. \nGr/Py 5.26 [45] N.A. \nRu/Py 0.24 [46] N.A. \nPt/YIG 0.3-1.2 [47] N.A. \nMoS2/CFA 1.49 [48] 0.46 [48] \nPd/Fe 0.49-1.17 [49] 0.04-0.33 [49] \nPd/Py 1.40 [50] N.A. \nMo/CFA 1.56 [51] N.A. \nMoS2/CoFeB 16.11 [52] N.A. \nTa/YIG 0.54 [53] N.A. \nW/YIG 0.45 [53] N.A. \nCu/YIG 0.16 [53] N.A. \nAg/YIG 0.05 [53] N.A. \nAu/YIG 0.27 [53] N.A. \nβ-W/CoFeB  1.44 (This work)  0.81 (This work)  \nMixed(α+β)-W/CoFeB  1.07 (This work)  0.60 (This work)  \n  \n((N.A. =  Not available)) \n \n t (nm) 0 0.5 1.0 1.5 2 3 5 8 10 15 \nRoughness \n(nm) 0.23 0.21 0.32 0.28 0.25 0.21 0.19 0.29 0.28 0.22 \n   \n29 \n Supporting Information \n \n \nStructural Phase Dependent Giant Interfacial Spin Transparency in W/CoFeB Thin \nFilm Heterostructure \n \nSurya Narayan Panda, Sudip Majumder, Arpan Bhattacharyya, Soma Dutta, Samiran \nChoudhury and Anjan Barman* \n \nDepartment of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre \nfor Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India \n \nE-mail: abarman@bose.res.in \n \n \nThis file includes: \n1. Determination of resistivity of W and Co 20Fe60B20 layers. \n2. Measurement of surface roughness of the sample using AFM. \n3. Determination of saturation magnetization of the samples from static and dynamic \nmeasurements.  \n4. Variation of effective damping with precessional frequency. \n \n \n \n1. Determination of resistivity of W and CoFeB layers :  \n \nThe variation of sheet resistance ( Rs) of the W( t)/Co20Fe60B20(3 nm) film stack with W layer \nthickness, t is shown in Figure S1 . The data is fitted with a parallel resistor model (Ref. 24 of \nthe article) by the formula given in the inset of the figure. This yields the resistivity of W in its \nβ and (α+β) phase as: 287 ± 19 µΩ.cm and 112 ± 14 µΩ.cm, respectively. On the other hand, \nthe resistivity of Co 20Fe60B20 is found to be 139 ± 16 µΩ.cm. \n   \n30 \n   \n \nFigure S1.  Variation of sheet resistance ( Rs) of the W ( t)/ Co20Fe60B20(3 nm) film stack vs. W \nthickness t used for the determination of resistivity of the W and Co 20Fe60B20 layers. \n \n2. Measurement of surface roughness of the sample using AFM:  \nWe have measured the surface topography of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) thin \nfilms by atomic force microscopy (AFM) in dynamic tapping mode by taking scan over 10 μm \n× 10 μm area. We have analyzed the AFM images using WSxM software. Figures S2 (a) and \nS2(d) show two-dimensional planar AFM images for t = 1 nm and 10 nm, respectively. Figures \nS2(b) and S2(e) show the corresponding three-dimensional AFM images for t = 1 nm and 10 \nnm, respectively. The dotted black lines on both images show the position of the line scans to \nobtain the height variation. Figures S2 (c) and S2(f) show the surface roughness profile along \nthat dotted lines, from which the average roughness ( Ra) is measured as 0.32 ± 0.10  nm and \n0.28 ± 0.12 nm for t = 1 nm and 10 nm, respectively. Topographical roughness is small and \nconstant within the error bar in all samples irrespective of the crystal phase of W. Furthermore, \nsurface roughness varies very little when measured at different regions of same sample. The \ninterfacial roughness is expected to show its imprint on the measured topographical roughness 0 3 10 150200400\n  Rs(Ω)\nt(nm)(ρW)β= 287 µΩ.cm\n(ρW)α+β= 112 µΩ.cm\n= 139 µΩ.cm  \n31 \n due to the small thickness of our thin films. Small and constant surface roughness in these \nheterostructures proves the high quality of the thin films.  \n \n  \nFigure S2.  (a) The two-dimensional AFM image, (b) the three-dimensional AFM image, and \n(c) the line scan profile along the black dotted line for W(1 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 \nnm) sample. (d) The two-dimensional AFM image, (e) the three-dimensional AFM image, and \n(f) the line scan profile along the black dotted line for W(10 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 \nnm) sample. \n \n3. Determination of saturation magnetization of the samples from static and dynamic \nmagnetic measurements :  \nWe have measured the in-plane saturation magnetization ( Ms) of all the W( t)/ \nCo20Fe60B20(d)/SiO2(2 nm) samples using vibrating sample magnetometry (VSM). Typical \nmagnetic hysteresis loops (magnetization vs. magnetic field) for W( t)/ Co20Fe60B20(3 \nnm)/SiO 2(2 nm), W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm) and W(4 nm)/Cu( c)/ Co20Fe60B20(3 \n  \n32 \n nm)/SiO 2(2 nm) series are plotted in Figures S3 (a), S4(a) and S5(a), respectively. Here, Ms is \ncalculated from the measured magnetic moment divided by the total volume of the Co 20Fe60B20 \nlayer. These films have very small coercive field (~5 Oe).  The effective magnetization Meff of \nthe samples are obtained by fitting the bias magnetic field ( H) dependent precessional frequency \n(f) obtained from the TR-MOKE measurements, with the Kittel formula (equation (13) of the \narticle) (see Figures S3 (b), S4(b) and S5(b)). We have finally plotted the variation of Meff and \nMs with W, Co 20Fe60B20, and Cu thickness in Figures S3 (c), S4(c), and S5(c), respectively. The \nMeff and Ms values are found to be in close proximity with each other, indicating that the \ninterfacial anisotropy is small for all these samples. Since these films were not annealed post-\ndeposition, the interfacial anisotropy stays small and plays only a minor role in modifying the \nmagnetization dynamics for these heterostructures.  \n \n \nFigure S3.     (a) VSM loops for W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit (solid line) \nto experimental data (symbol) of precessional frequency vs. magnetic field for W( t)/ \nCo20Fe60B20(3 nm)/SiO 2( 2 nm) samples. (c) Comparison of variation of Ms from VSM and  Meff \nfrom TR-MOKE as a function of W layer thickness. \n \n \n 0 4 8 12 16500100015002000\n  \nt (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)-100001000\n-100001000\n-0.4 0.0 0.4-100001000  \n  \nt =1 nm\nt = 8 nm\n  \n  \nH (kOe)t = 15 nm\n  \n  M (emu/cc)141618\n141618\n1.5 2.0 2.5141618  \n  \nt = 1 nm\nt = 8 nm\nt = 15 nm  \n  f (GHz)\nH (kOe)  \n  (a) (b)\n(c)  \n33 \n  \n \n \nFigure S4.     (a) VSM loops for W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm). (b) Kittel fit (solid line) \nto experimental data (symbol) of precessional frequency vs. magnetic field for W(4 nm)/ \nCo20Fe60B20(d)/SiO2(2 nm) samples. (c) Comparison between variation of Ms from VSM and  \nMeff from TR-MOKE as a function of Co 20Fe60B20 layer thickness. \n \n \nFigure S5.     (a) VSM loops for W(4 nm)/Cu( c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit \n(solid line) to experimental data (symbol) of precessional frequency vs. magnetic field for W(4 \nnm)/ Cu(c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) samples. (c) Comparison between variation of Ms \nfrom VSM and  Meff from TR-MOKE as a function of Cu layer thickness.  \n \n 0 3 6 9500100015002000\n  \nd (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)141618\n141618\n1.5 2.0 2.514161820  \n  \n  \n  \n  \n  f  (GHz)\nH (kOe)d = 10 nmd = 5 nmd = 2 nm\n-100001000\n-100001000\n-0.3 0.0 0.3-100001000  \n  \nd = 5 nmd = 2 nm\nd = 10 nmM (emu/cc)\n  \n  \nH (kOe)  \n  (a) (b)\n(c)\n0.00 0.25 0.50 0.75 1.00500100015002000\n  \nc (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)141618\n141618\n1.5 2.0 2.5141618  \n  \n  \n  \n  \n  f (GHz)\nH (kOe)c = 1 nmc = 0.5 nmc = 0 nm\n-100001000\n-100001000\n-0.4 0.0 0.4-100001000\nH (kOe)M (emu/cc)  \n  \nc = 0.5 nmc = 0 nm \nc = 1 nm  \n  \n  \n  (a) (b)\n(c)  \n34 \n 4. Variation of effective damping with precessional frequency:   \nFor all the sample series the time-resolved precessional oscillations have been recorded at \ndifferent bias magnetic field strength. The precessional frequency has been extracted by taking \nthe fast Fourier transform (FFT) of the background-subtracted time-resolved Kerr rotation. \nSubsequently, the time-resolved precessional oscillations have also been fitted with a damped \nsinusoidal function given by equation (12) of the article to extract the decay time τ. The value \nof effective Gilbert damping parameter ( αeff) have then been extracted using equation (14). \nVariation of this αeff with precessional frequency ( f) is plotted to examine the nature of the \ndamping. Here, we have plotted the time-resolved precessional oscillations ( Figure S6( a)), FFT \npower spectra ( Figure S6( b)) and αeff vs. f (Figure S6( c)) for Sub/W(0.5 nm)/Co 20Fe60B20(3 \nnm)/SiO 2(2 nm) sample.  It is clear from this data that damping is frequency independent, which \nrules out the contribution of various extrinsic factors such as two-magnon scattering, \ninhomogeneous anisotropy, eddy current in the damping for our samples.  \n \n \nFigure S6.     (a) Background subtracted time-resolved precessional oscillations at different bias \nmagnetic fields for Sub/W(0.5 nm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample, where symbols \nrepresent the experimental data points and solid lines represent fits using equation (12) of the \narticle. (b) The FFT power spectra of the time-resolved precessional oscillations showing the 0.0 0.3 0.6 0.9 1.2 1.5  \n  \n  \n  \n  \n  \nH = 1.50 kOeH = 1.80 kOeH = 2.10 kOe\n  \n  Kerr Rotation (arb. units)\nTime (ns)H = 2.30 kOe\n0 10 20 30  \n  \n  \n  \n  \n  \n  \n  Power (arb. units)\nf (GHz)12 14 16 180.0000.0080.016\n  \n \nf (GHz)eff\n(c)\n(b) (a)  \n35 \n precessional frequency. (c) Variation of effective damping with precessional frequency is \nshown by symbol and the dotted line is guide to the eye. "    },    {        "title": "1111.1219v1.Tunable_magnetization_relaxation_in_spin_valves.pdf",        "content": "arXiv:1111.1219v1  [cond-mat.mes-hall]  4 Nov 2011Tunable magnetization relaxation in spin valves\nXuhui Wang∗and Aurelien Manchon\nPhysical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia\n(Dated: June 24, 2018)\nIn spin values the damping parameters of the free layer are de termined non-locally by the entire\nmagnetic configuration. In a dual spin valve structure that c omprises a free layer embedded be-\ntween two pinned layers, the spin pumping mechanism, in comb ination with the angular momentum\nconservation, renders the tensor-like damping parameters tunable by varying the interfacial and dif-\nfusive properties. Simulations based on the Landau-Lifshi tz-Gilbert phenomenology for a macrospin\nmodel are performed with the tensor-like damping and the rel axation time of the free layer mag-\nnetization is found to be largely dependent on while tunable through the magnetic configuration of\nthe source-drain magnetization.\nPACS numbers: 75.70.Ak, 72.25.Ba, 75.60.Jk, 72.25.Rb\nA thorough knowledge of magnetization relaxation\nholds the key to understand magnetization dynamics in\nresponse to applied fields1and spin-transfer torques.2,3\nIn the framework of Landau-Lifshitz-Gilbert (LLG) phe-\nnomenology, relaxation is well captured by the Gilbert\ndamping parameterthat is usuallycited asa scalarquan-\ntity. As pointed out by Brown half a century ago,4the\nGilbert damping for a single domain magnetic particle is\nin general a tensor.\nWhen a ferromagnetic thin film is deposited on a nor-\nmal metal substrate, an enhanced damping has been ob-\nserved ferromagnetic resonance experiments.5This ob-\nservation is successfully explained by spin pumping:6,7\nThe slow precession of the magnetization pumps spin\ncurrent into the adjacent normal metal where the dis-\nsipation of spin current provides a non-local mechanism\nto the damping. The damping enhancement is found\nto be proportional to spin mixing conductance, a quan-\ntity playing key roles in the magneto-electronic circuit\ntheory.7,8\nThepumped spincurrent Ip∝M×˙Misalwaysin the\nplaneformedbythefreelayermagnetizationdirection M\nand the instantaneous axis about which the magnetiza-\ntion precesses. Therefore, in a single spin valve, when M\nis precessingaround the source(drain) magnetization m,\nthe pumpingcurrentisalwaysinthe planeof mandM.9\nLet us assume an azimuth angle θbetween mandM. In\nsuch anin-plane configuration, the pumping current Ip\nhas a component Ipsinθthat is parallel to m. The spin\ntransfer torque acting on the source (drain) ferromagnet\nmis the component of spin current that is in the plane\nand perpendicular to m. To simplify the discussion, we\nconsider it to be completely absorbed by m. The lon-\ngitudinal (to m) component experiences multiple reflec-\ntion at the source (drain) contact, and cancels the damp-\ning torque by an amount proportional to Ipsin2θbut is\nstill aligned along the direction of M×˙M. Therefore\nthe total damping parameter has an angle θdependence\nbut still picks up a scalar (isotropic) form. This is the\nwell-known dynamic stiffness explained by Tserkovnyak\net al.9In the most general case, when the precessing axis\nof the free layer is mis-aligned with m, there is always anout-of-plane pumping torque perpendicular to the plane.\nIn the paradigmof Slonczewski, this out-of-plane compo-\nnent is not absorbed at the interface of the source (drain)\nferromagnetic nodes, while the conservation of angular\nmomentum manifests it as a damping enhancement that\nshows the tensor form when installed in the LLG equa-\ntion.\nStudies in lateral spin-flip transistors have suggested\na tensor form for the enhanced damping parameters.10\nIn spin valves, works based on general scattering the-\nory have discussed the damping in the framework of\nfluctuation-dissipation theorem11and shown that the\nGilbert dampingtensorcanbe expressedusingscattering\nmatrices,12thus enabling first-principle investigation.13\nBut explicit analytical expressions of the damping ten-\nsor, its dependence on the magnetic configuration as well\nas the material properties and particularly its impact on\nthe magnetization relaxation are largely missing.\nIn this paper, we investigate the Gilbert damping pa-\nrameters of the free layer in the so-called dual spin valve\n(DSV).14–16We analyze the origin of the damping tensor\nand derive explicit analytical expressions of its non-local\ndependence on the magnetic configuration and materials\nproperties. A generalization of our damping tensor to a\ncontinuous magnetic texture agrees well with the results\nin earlier works. Particularly, we show, in numeric sim-\nulations, that by tuning the magnetic configurations of\nthe entire DSV, the relaxation time of the free layer can\nbe increased or decreased.\nmLM mR\nReservoir Reservoir\nFIG. 1: A dual spin valve consists of a free layer (with magne-\ntization direction M) sandwiched by two fixed ferromagnetic\nlayers (with magnetization directions mLandmR) through\ntwo normal metal spacers. The fixed layer are attached to\nreservoirs.2\nTo analyze the spin and charge currents in a DSV, we\nemploy the magneto-electronic circuit theory and spin\npumping,7,8in combination with diffusion equations.17\nPillar-shaped metallic spin valves usually consist of\nnormal-metal ( N) spacers much shorter than its spin-flip\nrelaxation length, see for example Ref.[3,15]. To a good\napproximation, in the Nnodes, a spatially homogeneous\nspin accumulation is justified and the spin current ( Ii)\nconservation dictates/summationtext\niIi= 0 (where subscript iindi-\ncates the source of spin current).\nA charge chemical potential ( µ) and a spin accumula-\ntion (s) are assigned to every ForNnode. In a transi-\ntion metalferromagnet,astrongexchangefieldalignsthe\nspin accumulation to the magnetization direction. At ev-\neryF|Ninterface, the charge and spin currents on the N\nside are determined by the contact conductance and the\ncharge and spin distributions on both sides of the con-\ntact. For example, at the contact between the left lead\nferromagnet to the left normal metal N1, called L|N1\nthereafter, the currents are8\nIL=e\n2hGL[(µ1−µL)+PL(s1−sL)·mL],\nIL=−GL\n8π[2PL(µ1−µL)mL+(s1−sL)·mLmL\n+ηL(s1−s1·mLmL)]. (1)\nWe have used the notation G=g↑+g↓is the sum of\nthe spin- σinterface conductance gσ. The contact polar-\nisationP= (g↑−g↓)/(g↑+g↓). The ratio η= 2g↑↓/G\nis between the real part of the spin-mixing conductance\ng↑↓and the total conductance G. The imaginary part\nofg↑↓is usually much smaller than its real part, thus\ndiscarded.18The spin-coherence length in a transition\nmetalferromagnetisusuallymuchshorterthanthethick-\nness of the thin film,19which renders the mixing trans-\nmission negligible.7The precession of the free layer mag-\nnetization Mpumps a spin current Ip= (/planckover2pi1/4π)g↑↓\nFM×\n˙Minto the adjacent normal nodes N1andN2, which\nis given by the mixing conductance g↑↓\nFat theF|N1(2)\ninterface (normal metals spacers are considered identical\non both sides of the free layer).\nA back flow spin current at the F|N1interface reads\nI1=−GF\n8π[2PF(µ1−µF)M+(s1−sF)·MM\n+ηF(s1−s1·MM)] (2)\non theN1side. Therefore, a weak spin-flip scattering\ninN1demands IL+I1+Ip= 0, which is dictated by\nangularmomentum conservation. The sameconservation\nlaw rules in N2, whereIR+I2+Ip= 0.\nFor the ferromagnetic ( F) nodes made of transition\nmetals, the spin diffusion is taken into account properly.9\nIn a strong ferromagnet, any transverse components de-\ncay quickly due to the large exchange field, thus the\nlongitudinal spin accumulation sν=sνmν(withν=\nL,R,F) diffuses and decays exponentially at a length\nscale given by spin diffusion length ( λsd) as∇2\nxsν=sν/λsd. The difference in spin-dependent conductivty\nof majority and minority carriers is taken into account\nby enforcing the continuity of longitudinal spin current\nmν·Iν=−(D↑\nν∇xs↑\nν−D↓\nν∇xs↓\nν) at the every F|Nin-\nterface. We assume vanishing spin currents at the outer\ninterfaces to reservoirs.\nThe diffusion equations and current conservation de-\ntermine, self-consistently, the spin accumulations and\nspin currents in both NandFnodes . We are mainly\nconcerned with the exchange torque9T=−M×(IL+\nIR)×Macting on M. A general analytical formula is\nattainable but lengthy. In the following, we focus on two\nscenarios that are mostly relevant to the state-of-the-art\nexperiments in spin valves and spin pumping: (1) The\nfree layer has a strong spin flip (short λsd) and the thick-\nnessdF≥λsd, for which the permalloy (Py) film is an\nideal candidate;15(2) The free layer is a half metal, such\nas Co2MnSi studied in a recent experiment.20\nStrong spin flip in free layer. We assume a strong spin\nflip scattering in the free layer i.e., dF≥λsd. We leave\nthe diffusivity properties in the lead Fnodes arbitrary.\nThe total exchange torque is partitioned into two parts:\nAnisotropic part that is parallel to the direction of the\nGilbert damping M×˙Mand ananisotropic part that is\nperpendicular to the plane spanned by mL(R)andM(or\nthe projection of M×˙Mto the direction mL(R)×M),\ni.e.,\nT=/planckover2pi1g↑↓\nF\n4π(DL\nis+DR\nis)/parenleftBig\nM×˙M/parenrightBig\n+/planckover2pi1g↑↓\nF\n4πM×/bracketleftBig\n(DL\nanˆAL,an+DR\nanˆAR,an)˙M/bracketrightBig\n,(3)\nwhere the material-dependent parameters DL(R)\nisand\nDL(R)\nanare detailed in the Appendix A.\nMost interest is in the anisotropic damping described\nby a symmetric tensor with elements\nˆAij\nan=−mimj (4)\nwherei,j=x,y,z(we have omitted the lead index Lor\nR). The elements of ˆAanare given in Cartesian coordi-\nnates of the source-drain magnetization direction. The\nanisotropic damping appears as M׈Aan˙Mthat is al-\nways perpendicular to the free layer magnetization direc-\ntion, thus keeping the length of Mconstant.11It is not\ndifficult to show that when Mis precessing around m,\nthe anisotropic part vanishes due to ˆAan˙M= 0.\nWe generalizeEq.(4) toa continuousmagnetictexture.\nConsider here only one-dimensional spatial dependence\nand the extension to higher dimensions is straightfor-\nward. The Cartesian component of vector U≡M×\nˆAan˙MisUi=−εijkMjmkml˙Ml(whereεijkis the Levi-\nCivita tensor and repeated indices are summed). We as-\nsume the fixed layer and the free layer differ in space by\na lattice constant a0, which allows mk≈Mk(x+a0). A\nTaylor expansion in space leads to U=−a2\n0M×(ˆD˙M),\nwherethematrixelements ˆDkl= (∂xM)k(∂xM)landwe3\nhave assumed that the magnetization direction is always\nperpendicular to ∂xM. In this case, three vectors ∂xM,\nM×∂xMandMare perpendicular to each other. A\nrotation around Mbyπ/2 leavesMand˙Munchanged\nwhile interchanging ∂xMwithM×∂xM, we have\nˆDkl= (M×∂xM)k(M×∂xM)l, (5)\nwhich agrees with the so-called differential damping ten-\nsor Eq.(11) in Ref.[21].\nEq.(3) suggests that the total exchange torque on the\nfree layer is a linear combination of two independent ex-\nchange torques arsing from coupling to the left and the\nrightFnodes. This form arises due to a strong spin-\nflip scattering in the free layer that suppresses the ex-\nchange between two spin accumulations s1ands2in the\nNnodes. In the pursuit of a concise notation for the\nGilbert form, the exchange torque can be expressed as\nT=M×← →α˙Mwith a total damping tensor given by\n← →α=/planckover2pi1g↑↓\nF\n4π/parenleftBig\nDL\nis+DR\nis+DL\nanˆAL,an+DR\nanˆAR,an/parenrightBig\n.(6)\nThe damping tensor← →αis determined by the entire mag-\nnetic configuration of the DSV and particularly by the\nconductance of F|Ncontacts and the diffusive proper-\nties theFnodes.\nHalf metallic free layer . This special while experimen-\ntally relevant20case means PF= 1. Half-metallicity\nin combination with the charge conservation enforces a\nlongitudinal back flow that is determined solely by the\nbias current: The spin accumulations in Nnodes do\nnot contribute to the spin accumulation inside the free\nlayer, thus an independent contribution due to left and\nright leads is foreseen. We summarize the material spe-\ncific parameters in the Appendix A. When spin flip is\nweak in the source-drain ferromagnets, ξL≈0 leads to\nDis≈0. In this configuration, by taking a (parallel or\nanti-parallel)source-drainmagnetization direction as the\nprecessingaxis,thetotaldampingenhancementvanishes,\nwhich reduces to the scenario of ν= 1 in Ref.[9].\nMagnetization relaxation . To appreciate the impact of\nan anisotropic damping tensor on the magnetization re-\nlaxation, we perform a simulation, for the free layermag-\nnetization, using Landau-Lifshitz-Gilbert (LLG) equa-\ntion augmented by the tensor damping, i.e.,\ndM\ndt=−γM×Heff+α0M×dM\ndt\n+γ\nµ0MsVM×← →αdM\ndt.(7)\nα0is the (dimensionless) intrinsic Gilbert damping pa-\nrameter. Symbol γis the gyromagnetic ratio, Msis the\nsaturation magnetization, and Vis the volume of the\nfree layer. µ0stands for the vacuum permeability. The\ndynamics under the bias-driven spin transfer torque is\nnot the topic in this paper, but can be included in a\nstraightforward way.22We give in the Appendix B the\nexpressions of the bias-driven spin torques.We are mostly interested in the relaxation of the mag-\nnetization, instead of particular magnetization trajecto-\nries, in the presence of a tensor damping. The follow-\ning simulation is performed for the scenario where the\nfree layer has a strong spin flip, i.e., Case (1). We em-\nploy the pillar structure from Ref.[15] while consider-\ning the free layer (Py) to be 8nm thick (a thicker free\nlayer favors a better thermal stability.15) The source-\ndrain ferromagnets are cobalt (Co) and we expect the\nresults are valid for a larger range of materials selec-\ntions. The Py film is elliptic with three axes given by\n2a= 90 nm, 2 b= 35 nm,15andc= 8 nm. The de-\nmagnetizing factors Dx,y,zin the shape anisotropy en-\nergyEdem= (1/2)µ0M2\nsV/summationtext\ni=x,y,zDiM2\niareDx= 0.50,\nDy= 0.37 and Dz= 0.13. An external field Haleads\nto a Zeeman splitting EZee=−Vµ0MsHa·M. For Py\nfilms, we neglect the uniaxial anisotropy. The total free\nenergyET=EZee+Edemgives rise to an effective field\nHeff=−(1/VMsµ0)∂ET/∂M.\nThe spin-dependent conductivities in the bulk of Co\nandthe spin diffusion length λCo≈60nm aretaken from\nthe experimental data.24For Py, we take λPy≈4 nm.25\nTo have direct connection with experiments, the above\nmentioned bare conductance has to be renormalized by\nthe Sharvin conductance.26For Py/Cu the mixing con-\nductance, we take the value g↑↓\nFS−1≈15 nm−2,26which\ngivesRL(R)F≈1.0.\n56789100.950.960.970.980.991(a) Bz = 50 G; I.P. y−axis.\n  \n(y,y)(y,x)(x,x)(x,z)(y,z)(z,z)369(b) Relaxation time\n(x,x)\n(y,y)\n(z,z)\nFIG. 2: (Color online) Mzas a function of time (in ns) in\npresence of differentsource-drain magnetic configurations and\napplied fields. (a) The external magnetic field Bz= 50 Gauss\nis applied along z-axis. The blue (dashed), red (solid) and\nblack (dotted dash) curves correspond to source-drain magn e-\ntization in configurations ( y,y), (x,x), and (z,z) respectively.\n(b) Magnetization relaxation times (in the unit of ns)versu s\nsource-drain magnetic configurations at different applied fi eld\nalongz-axis.:Bz= 10 G (red /square),Bz= 50 G (blue /circlecopyrt),\nBz= 200 G (green ▽),Bz= 800 G (black ♦). Lines are a\nguide for the eyes. The initial position (I.P.) of the free la yer\nis taken along y-axis.\nThe relaxation time τris extracted from the sim-\nulations by demanding at a specific moment τrthe\n|Mz−1.0|<10−3, i.e., reaches the easy axis. In the\nabsence of bias, panel (a) of Fig.2 shows the late stage\nof magnetization relaxation from an initial position ( y-4\naxis) in the presence of an tensor damping, under various\nsource-drain (SD) magnetic configurations. The results\nare striking: Under the same field, switching the SD con-\nfigurations increases or decreases τr. In panel (b), the\nextracted relaxation times τrversus SD configurations\nunder various fields are shown. At low field Bz= 10 G\n(red/square), when switching from ( z,z) to (y,y),τris im-\nproved from 8.0 ns to 6.3 ns, about 21%. At a higher\nfieldBz= 800 G (black ♦), the improvement is larger\nfrom 5.2 at ( z,z) to 3.6 at ( y,y), nearly 31%. To a large\ntrend, the relaxation time improvement is more signifi-\ncant at higher applied fields.\nIn conclusion, combining conservation laws and\nmagneto-electronic circuit theory, we have analyzed the\nGilbert damping tensor of the free layer in a dual spin\nvalve. Analytical results of the damping tensor as func-\ntions of the entire magnetic configuration and material\nproperties are obtained. Numerical simulations on LLG\nequation augmented by the tensor damping reveal a tun-\nable magnetization relaxation time by a strategic selec-\ntion of source-drain magnetization configurations. Re-\nsults presented in this paper open a new venue to the\ndesign and control of magnetization dynamics in spin-\ntronic applications.\nX.Wang is indebted to G. E. W. Bauer, who has\nbrought the problem to his attention and offered invalu-\nable comments.\nAppendix A: Material dependent parameters\nIn this paper, RL(R)F≡g↑↓\nL(R)/g↑↓\nFis the mixing con-\nductance ratio and χL(R)≡mL(R)·M. The diffusivity\nparameter ξL(R)=φL(R)(1−P2\nL(R))/ηL(R), where for the\nleftFnode\nφL=1\n1+(σ↑\nL+σ↓\nL)λLe2\n4hSσ↑\nLσ↓\nLtanh(dL/λL)GL(1−P2\nL)(A1)\nwherehthe Planck constant, Sthe area of the thin film,\nethe elementary charge, λLthe spin diffusion length, dL\nthe thickness of the film, and σ↑(↓)the spin-dependent\nconductivity. φRisobtainedbysubstituting all LbyRin\nEq.(A1). Parameter ξFis given by ξF= (1−P2\nF)φF/ηF\nwith\nφF=1\n1+(σ↑\nF+σ↓\nF)λFe2\n4hSσ↑\nFσ↓\nFGF(1−P2\nF).(A2)\nThe material dependent parameters as appearing in the\ndamping tensor Eq.(6) are: (1) In the case of a strongspin flip in free layer,\nDL(R)\nis=RL(R)F\nLL(R)F/bracketleftBig\nξL(R)RL(R)F+ξL(R)ξF(1−χ2\nL(R))\n+ξF(1−χ2\nL(R))χ2\nL(R)/bracketrightBig\n,\nDL(R)\nan=RL(R)F\nLL(R)F(ξL(R)−1)[ξF(1−χ2\nL(R))+RL(R)F]\n1+RL(R)F,\nLL(R)F=(1+RL(R)Fχ2\nL)ξF(1−χ2\nL(R))\n+RL(R)F/bracketleftBig\n(1−χ2\nL(R))(1+ξL(R)ξF)\n+ξL(R)RL(R)F+ξFχ2\nL(R)/bracketrightBig\n; (A3)\n(2)In the case of a half metallic free layer\nDL(R)\nis=RL(R)FξL(R)\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F),\nDL(R)\nan=RL(R)F\n1+RL(R)F\n×ξL(R)−1\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F).(A4)\nAppendix B: Bias dependent spin torques\nThe full analytical expression of bias dependent spin\ntorques are rather lengthy. We give here the expres-\nsions, under a bias current I, for symmetric SD fer-\nromagnets (i.e., φL=φR=φthusξL=ξR=ξ)\nwith parallelor anti-parallelmagnetization direction. (1)\nWith a strong spin flip in the free layer, the parallel\nSD magnetization leads to vanishing bias-driven torque\nT(b)\n⇑⇑= 0; WhentheSDmagnetizationsareanti-parallelly\n(i.e.,mL=−mR≡m),\nT(b)\n⇑⇓=I/planckover2pi1Pφ\ne(1+R)L/bracketleftbig\n(ξF+RξFχ2+R)(1−χ2)\n+R(R+ξF(1−χ2)+χ2)/bracketrightbig\nmF×(m×mF).\n(B1)\n(2) When the free layer is half metallic, for symmetric\nSD ferromagnets , T(b)\n⇑⇑= 0 and\nT(b)\n⇑⇓=I/planckover2pi1\neφP\n(1−ξ)(1−χ2)+ξ(χ2+R)mF×(m×mF).\n(B2)\n∗Electronic address: xuhui.wang@kaust.edu.sa\n1L. D. Landau and E. M. Lifshitz, Statistical Physics ,Part\n2(Pergamon, Oxford, 1980); T. L. Gilbert, IEEE. Trans.\nMag.40, 2443 (2004).2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B 54, 9353 (1996).\n3E. B. Myers, et al., Science 285, 867 (1999); J. A. Katine,\net al., Phys. Rev. Lett. 84, 3149 (2000); S. I. Kiselev, et5\nal., Nature (London) 425, 380 (2003).\n4W. F. Brown, Phys. Rev. 130, 1677 (1963).\n5Mizukami et al., Jpn. J. Appl. Phys. 40, 580 (2001); J.\nMagn. Mater. Magn. 226, 1640 (2001).\n6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002); Phys. Rev. B 66, 224403\n(2002).\n7Y. Tserkovnyak, et al., Rev. Mod. Phys. 77, 1375 (2005).\n8A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2005).\n9Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n10X. Wang, G. E. W. Bauer, and A. Hoffmann, Phys. Rev.\nB73, 054436 (2006).\n11J. Foros, et al., Phys. Rev. B 78, 140402(R) (2008); Phys.\nRev. B79, 214407 (2009).\n12A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008); arXiv:1104.1625.\n13A.A.Starikov, et al., Phys.Rev.Lett. 105, 236601 (2010).\n14L. Berger, J. Appl. Phys. 93, 7693 (2003).\n15G. D. Fuchs, et al., Appl. Phys. Lett. 86, 152509 (2005).16P. Bal´ aˇ z, M. Gmitra, and J. Barna´ s, Phys. Rev. B 80,\n174404 (2009); P. Yan, Z. Z. Sun, and X. R. Wang, Phys.\nRev. B83, 174430 (2011).\n17T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993); A. A.\nKovalev, A. Brataas, and G. E. W. Bauer Phys. Rev. B\n66, 224424 (2002).\n18K. Xia,et al., Phys. Rev. B 65, 220401(R) (2002).\n19M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n20H. Chudo, et al., J. Appl. Phys. 109, 073915 (2011).\n21S. Zhang and S. -L. Zhang, Phys. Rev. Lett. 102, 086601\n(2010).\n22J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72,\n014446 (2005).\n23J. Osborn, Phys. Rev. 67, 351 (1945).\n24J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274\n(1999).\n25A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 338\n(1999).\n26G. E. W. Bauer, et al., Phys. Rev. B 67, 094421 (2003)."    },    {        "title": "1807.11808v3.Comparative_study_of_methodologies_to_compute_the_intrinsic_Gilbert_damping__interrelations__validity_and_physical_consequences.pdf",        "content": "Comparative study of methodologies to compute the intrinsic Gilbert damping:\ninterrelations, validity and physical consequences\nFilipe S. M. Guimar~ aes,\u0003J. R. Suckert, Jonathan Chico, Juba Bouaziz, Manuel dos Santos Dias, and Samir Lounis\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n(Dated: December 20, 2018)\nRelaxation e\u000bects are of primary importance in the description of magnetic excitations, leading\nto a myriad of methods addressing the phenomenological damping parameters. In this work, we\nconsider several well-established forms of calculating the intrinsic Gilbert damping within a uni\fed\ntheoretical framework, mapping out their connections and the approximations required to derive\neach formula. This scheme enables a direct comparison of the di\u000berent methods on the same footing\nand a consistent evaluation of their range of validity. Most methods lead to very similar results for\nthe bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence\nof the spin pumping mechanism. The e\u000bects of inhomogeneities, temperature and other sources of\n\fnite electronic lifetime are often accounted for by an empirical broadening of the electronic energy\nlevels. We show that the contribution to the damping introduced by this broadening is additive, and\nso can be extracted by comparing the results of the calculations performed with and without spin-\norbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying\nelectronic structure, we unambiguously demonstrate that the damping parameter obtained within\nthe constant broadening approximation diverges for three-dimensional bulk magnets in the clean\nlimit, while it remains \fnite for monolayers. Our work puts into perspective the several methods\navailable to describe and compute the Gilbert damping, building a solid foundation for future\ninvestigations of magnetic relaxation e\u000bects in any kind of material.\nI. INTRODUCTION\nDynamical processes lie at the core of magnetic manip-\nulation. From the torques acting on the magnetic mo-\nments to how fast they relax back to their equilibrium\norientations, a material-speci\fc time-dependent theory\nis essential to describe and predict their behavior. In\nmost cases, the description of the time evolution of the\nmagnetization is done via micromagnetics1or atomistic\nspin dynamics (ASD)2,3approaches, in which the mag-\nnetization is considered either as a classical continuous\nvector \feld or as individual 3D vectors on a discrete\nlattice, respectively. They have been successfully used\nto describe a plethora of magnetic phenomena, ranging\nfrom spin waves in low dimensional magnets4, domain\nwalls5and skyrmion6dynamics to thermal stability of\nmagnetic textures7. These approaches model the mag-\nnetization dynamics via a phenomenological equation of\nmotion that contains both precessional and relaxation\nterms.\nA \frst attempt to address these processes was per-\nformed by Landau and Lifshitz (LL), by considering\na Larmor-like precessional torque and adding to it a\n(weaker) damping term of relativistic origin8. Since its\nphenomenological inception in 1935, the precise nature\nof the relaxation processes has been a source of intense\ndebate. In particular, the original LL formulation was\nfound to not properly describe situations in which the\ndamping was large. This problem was addressed by\nGilbert, who introduced a Rayleigh-like dissipation term\ninto the magnetic Lagrangian, thus obtaining the now-ubiquitous Landau-Lifshitz-Gilbert (LLG) equation9,\ndM\ndt=\u0000\rM\u0002B+\u000b\nMM\u0002dM\ndt\n=\u0000e\rM\u0002B\u0000\u000be\r\nMM\u0002(M\u0002B):(1)\nwhere\r >0 is the gyromagnetic factor, Mis the (spin)\nmagnetic moment, Bis the time-dependent e\u000bective\nmagnetic \feld acting on M, and\u000bis the scalar damping\nparameter named after Gilbert. The upper form of the\nLLG equation is due to Gilbert, and the lower one shows\nthat it is equivalent to a LL equation with a renormalized\ngyromagnetic factor, e\r=\r=(1 +\u000b2). The \frst term in\nthe right-hand side of Eq. (1) describes the precession of\nthe magnetic moments around the e\u000bective \feld, while\nthe second term is the Gilbert damping one, that de-\nscribes the relaxation of the magnetic moments towards\nB. This equation corrects the previously mentioned issue\nfor large values of \u000b, for which the original LL equation\nis expected to fail10,11.\nThe ferromagnetic resonance (FMR) technique is one\nof the most common procedures to probe magnetiza-\ntion dynamics12, in which the damping parameter is re-\nlated to the linewidth of the obtained spectra13. Al-\nthough many measurements have been carried out in bulk\nmaterials12,14{18, their description at low temperatures is\nstill controversial19{22. This can be attributed to the dif-\nferent intrinsic and extrinsic mechanisms that can con-\ntribute to the relaxation processes23{36. When varying\nthe temperature, two distinct regimes could be identi-\n\fed in the measured relaxation parameters37. For high\ntemperatures, a proportionality between the linewidth\nand the temperature was observed in most of the exper-arXiv:1807.11808v3  [cond-mat.mes-hall]  19 Dec 20182\niments. It was called resistivity-like, due to the simi-\nlarity with the temperature dependence of this quantity.\nA conductivity-like regime (linewidth inversely propor-\ntional to the temperature) was identi\fed at low temper-\natures for certain materials such as Ni15,17, but not for\nFe18,38. It was also seen that di\u000berent concentrations\nof impurities a\u000bected this low-temperature regime, even\nsuppressing it altogether16.\nFrom the theoretical point-of-view, the calculation of\nthe Gilbert parameter is a challenging problem due to\nthe many di\u000berent mechanisms that might be at play for\na given material39,40. Perhaps this is why most of the\ntheoretical approaches have focused on contributions to\nthe damping from electronic origin. The ultimate goal\nthen becomes the development of a predictive theory of\nthe Gilbert damping parameter, based on the knowledge\nof a realistic electronic structure of the target magnetic\nmaterial. The ongoing e\u000borts to complete this quest\nhave resulted in the development of a myriad of tech-\nniques21,22,37,41{43. Comparisons between a few of these\napproaches are available44,45, including experimental val-\nidation of some methods24,46, but a complete picture is\nstill lacking.\nWe clarify this subject by addressing most of the well-\nestablished methods to calculate the Gilbert damping\nfrom \frst principles. First, we connect the many dif-\nferent formulas, highlighting the approximations made\nin each step of their derivations, determining what con-\ntributions to the damping they contain, and establish-\ning their range of validity. These are schematically illus-\ntrated in Fig. 1. Second, we select a few approaches and\nevaluate the Gilbert damping within a uni\fed and con-\nsistent framework, making use of a multi-orbital tight-\nbinding theory based on \frst-principles electronic struc-\nture calculations. FMR simulations and the mapping of\nthe slope of the inverse susceptibility are used to bench-\nmark the torque correlation methods based on the ex-\nchange and spin-orbit torques. We apply these di\u000berent\ntechniques to bulk and monolayers of transition metals\n(Fe, Ni and Co), for which the spin pumping mecha-\nnism is not present and only the spin-orbit interaction\n(SOI) contributes to the relaxation. Disorder and tem-\nperature e\u000bects are included by an empirical broadening\nof the electronic energy levels37,43,47,48. Third, we engage\na longstanding question regarding the behavior of the\ndamping in the low-temperature and low-disorder limits:\nshould the intrinsic contribution to the Gilbert damping\ndiverge for clean systems? Our results using the con-\nstant broadening model demonstrate that the divergence\nis present in the clean limit of 3D systems but not of\nthe 2D ones49, which we attest by eliminating the pos-\nsibility of them being caused by numerical convergence\nissues or di\u000berent anisotropy \felds. Our results also in-\ndicate that the limit !!0 is not responsible for the\ndivergence of the intrinsic damping, as it is commonly\nattributed19,37,43,50. Finally, we propose a new way to\nobtain the spin-orbit contribution that excludes the \fc-\ntitious temperature/disorder contribution caused by thearti\fcial broadening51,52: they can be discounted by sub-\ntracting the values of damping calculated without SOI.\nFor bulk systems, this yields the total damping, while in\nlayered materials this method should also discount part\nof the spin pumping contribution. In Ref. 20, where tem-\nperature and disorder are included via a CPA analogy, a\nsimilar arti\fcial increase of \u000bfor high temperatures was\nremoved by including vertex corrections.\nThis work is organized as follows. We start, in Sec. II,\nwith a brief overview of the di\u000berent methods proposed\nin the literature. In Sec. III, we explain the theory used\nto calculate the response functions. We then turn to\nthe distinct theoretical forms of calculating the damping:\nIn Sec. IV, we analyze the di\u000berent approaches related\nto the spin-spin responses, while in Sec. V, the torque\nmethods are explored. We then discuss the obtained re-\nsults and conclude in Sec. VII. The Hamiltonian used in\nthe microscopic theory is given in Appendix A, while the\nanisotropy \felds for the 3D and 2D systems together with\nthe transverse dynamical magnetic susceptibility given\nby the LLG equation are given in Appendix B.\nII. OVERVIEW OF METHODS ADDRESSING\nINTRINSIC GILBERT DAMPING\nWe now focus on the di\u000berent methods to describe\nthe microscopic contributions to the Gilbert parameter,\nwhich encompasses e\u000bects that transfers energy and an-\ngular momentum out of the magnetic system. Within\nthese mechanisms, the relativistic SOI comes to the fore.\nThis is often referred to as the intrinsic contribution to\nthe damping, and was \frst identi\fed by Landau and Lif-\nshitz8. The origin of this damping mechanism lies in\nthe non-hermiticity of the relativistic corrections to the\nspin Hamiltonian when the magnetization precesses26,27.\nThe elementary magnetic excitations, called magnons,\ncan also be damped via Stoner excitations (electron-hole\npairs with opposite spins)33,34,53. Alternatively, the con-\nduction electrons can carry spin angular momentum even\nin absence of the SOI. This leads to damping via the spin-\npumping mechanism32,54{56.\nEarly models proposed to describe these processes al-\nready argued that the interaction between the magnetic\nmoments and the conduction electrons is a key ingre-\ndient57. This led to the so-called breathing Fermi sur-\nface model, where the shape of the Fermi surface de-\npends on the orientation of the magnetization through\nthe SOI41. This approach, however, could only capture\nthe conductivity-like regime, which diverges at low tem-\nperatures. The decay of magnons into Stoner excitations\nwas also considered early on39, describing well the exper-\nimental behavior of Ni but also missing the increase at\nlarger temperatures of other materials.\nAn important progress was made by Kambersky us-\ning the spin-orbit torque correlation function to calculate\nthe damping parameter37. This approach captures both\nconductivity- and resistivity-like behaviors, which were3\n↵\u0000↵noSOI\n<latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit><latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit><latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit><latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit>\nInversion\nInversion for  low frequencies + sum ruleDyson equation of susceptibilityEquation of motion of susceptibility\nSpectral representation  at T=0KSpectral representation at T=0KLow SOI\nLow spin pumpingNo spin pumpingComputational costsFull SOI Spin pumpingFMR linewidthSlope of inverse susceptibility\nSlope of inverse mean-field susceptibility\nExchange torque correlation at Fermi surfaceSlope of mean-field  SO-torque susceptibility with SOIProduct of spectral functions of opposite spinsSlope of mean-field susceptibilitySpin correlation  at Fermi surfaceSO-torque correlation at Fermi surface with SOISpin response methodsTorque response methods\nEquation of motion of susceptibility + perturbation theory\nDyson equation for Green function + Orbital quenchingSlope of SO-torque susceptibility  without SOI\nSpectral representation at T=0KSlope of mean-field  SO-torque susceptibility without SOISO-torque correlation at Fermi surface  without SOILarge broadening\nFigure 1. Diagram exhibiting the di\u000berent methods investigated in this work, their connections and range of validity. Two\ngroups are identi\fed: one related to the spin susceptibility (spin response methods), including the ferromagnetic resonance\nand the slope of the inverse susceptibility that involves a direct mapping of this quantity to the LLG equation; and the other\nassociated with torque responses, for which approximations need to be taken. The steps indicated by solid lines represent\nexact connections, while dashed arrows involve some kind of approximation. The arrow on the left points from the methods\nthat require less computational power (lower part) to the more demanding ones (upper part). Boxes are hyper-linked with the\nrespective equations and sections.\nshown to originate from the intra- and interband transi-\ntions, respectively58. Recently, this so-called torque cor-\nrelation method was re-obtained using a di\u000berent per-\nturbative approach19, spurring discussions about the va-\nlidity of the obtained results, specially the divergence\ncaused by the intraband transitions22. A similar method\nalso based on torque correlation functions was developed\nusing a scattering theory approach42involving the ex-\nchange torque operator instead of the spin-orbit torque\none. Results obtained in this way also present diverg-\ning behavior in the clean limit of 3D structures20. A\nsimilar scattering framework was used to explain the\nenhancement of the Gilbert damping due to the spin\npumping in thin \flms32. Yet another method relating\nthe Gilbert damping to the spin-spin response was pro-\nposed and related to the existing spin-orbit torque cor-\nrelation method43. It also presented diverging intraband\ncontributions when the parameter used to broaden the\ndelta functions (which mimics the e\u000bect of disorder or\ntemperature) was taken to zero59. The vertex correc-\ntions proposed in Ref. 59 did not remove this diver-\ngence. More recently, Costa and Muniz21showed that\nthe damping parameters of layered structures remain \f-nite in the zero broadening limit, when extracted directly\nfrom the linewidth of the dynamical magnetic suscepti-\nbility (within the random phase approximation).\nSeveral of these methods have been implemented\nfor material-speci\fc calculations20,47,49,58,60{62, and some\napproaches were compared and related43{45,63. In this\nwork, we start our analysis with the uniform frequency-\ndependent spin-spin susceptibility, which is measured ex-\nperimentally in FMR setups, to derive the other expres-\nsions for the damping parameter based on the spin- and\ntorque-correlation methods.\nIII. MICROSCOPIC THEORY\nWe begin by setting the grounds of the theory we use\nto evaluate the di\u000berent formulas of the Gilbert damping\non equal footing. The electronic structure of the system\nis described by the mean-\feld Hamiltonian\n^H=^H0+^Hxc+^HSOI+^Hext: (2)\nThe paramagnetic band structure is described by ^H0\nwithin a multi-orbital tight-binding parametrization. An4\ne\u000bective local electron-electron interaction within the\nmean-\feld approximation is included in ^Hxc, which is re-\nsponsible for ferromagnetism. We also account for spin-\norbit interaction through ^HSOI, and the interaction with\nexternal static magnetic \felds via ^Hext. The explicit\nforms of all the terms are given in Appendix A.\nIn this work, we investigate the di\u000berent methods to\ncompute the intrinsic Gilbert damping utilizing the pro-\ntotypical bulk magnets Fe (bcc), Co (fcc) and Ni (fcc),\nand also square lattices corresponding to the (001) planes\nof those materials, with the same nearest-neighbor dis-\ntances as in its bulk forms.\nFor simplicity, we consider the spin-orbit interaction\nand the local e\u000bective Coulomb interaction only on the\ndorbitals, with U= 1 eV64{66for all systems, and the\nspin-orbit strengths \u0015Fe\nSOI= 54 meV67,\u0015Co\nSOI= 70 meV68,\nand\u0015Ni\nSOI= 133 meV68. The magnetic ground state is\nfound by self-consistently enforcing charge neutrality for\nthe bulk materials69. For the monolayer cases, the total\nnumber of electrons in the atomic plane is decreased to\nn= 7:3 (Fe),n= 8:1 (Co) and n= 9:0 (Ni), as the re-\nmaining charge spills into the vacuum (which we are not\nexplicitly taking into account within the model). The\nground-state properties (spin moment M, orbital mo-\nmentM`and magnetic anisotropy energy K) obtained\nwithin this framework are listed in Table I. The easy axis\nfor all the bulk systems and the monolayers were found\nto be along the (001) direction. We emphasize that our\ngoal is not to achieve the most realistic description of the\nelectronic structure of these materials, but rather to de-\n\fne a concrete set of cases that allow us to compare the\ndi\u000berent methods to compute the Gilbert damping.\nThe magnetic excitations are described using linear re-\nsponse theory, where the transverse magnetic response\n\u000eM(t) due to an oscillatory magnetic \feld \u000eB(t) is given\nby70\n\u000eM\u000b(t) =Z\ndt0\u001f\u000b\f(t\u0000t0)\u000eB\f(t0); (3)\nwhere the convention to sum over repeated indices of\nthe components \f=fx;y;zgis used. This approach\ncaptures the orbitally-averaged part of the response. The\nbulk monolayer\nbcc Fe fcc Co fcc Ni Fe Co Ni\nM(\u0016B) 2.32 1.48 0.43 2.90 1.90 0.96\nM`(\u0016B) 0.072 0.079 0.055 0.28 0.22 0.20\nK(meV) 0.19 0.26 0.084 1.7 1.8 1.9\nTable I. Ground state properties of the investigated systems.\nMandM`denotes the spin and orbital magnetic moments,\nrespectively. Values obtained for \u0011= 1:36 meV. The mag-\nnetic anisotropy constant Kis obtained from the anisotropy\n\felds given by Eq. (B3).magnetic susceptibility is given by\n\u001f\u000b\f(t\u0000t0) =\u00004\n\n^S\u000b(t);^S\f(t0)\u000b\u000b\n= 4i\n\u0002^S\u000b(t);^S\f(t0)\u0003\u000b; (4)\nin atomic units. ^S\u000b(t) is the\u000b-component of the spin op-\nerator. In the \frst line of the equation above, we reprise\nthe double-bracket notation of Zubarev for the spin-spin\nretarded Green function71. This notation is convenient\nfor the derivations of Sec. V.\nFor the crystal symmetries of the systems we are in-\nterested in, it is convenient to work in the circular ba-\nsis^S\u0006=^Sx\u0006i^Sy, which diagonalizes the susceptibil-\nity matrix with components \u001f\u0000+(t) and\u001f+\u0000(t). The\nfrequency- and wave vector-dependent transverse suscep-\ntibility\u001f\u0000+(q;!) is obtained within the random phase\napproximation (RPA), which captures the collective spin\nwave modes21,72, as well as the possible decay into\nparticle-hole excitations (Stoner modes) described by the\nsingle-particle response function \u001f\u0000+\n0(!). Considering\nmatrices that take into account the orbital dependency,\nthe two susceptibilities are related by\n[\u001f\u0000+]\u00001= [\u001f\u0000+\n0]\u00001\u00001\n4U: (5)\nHere,U\u0016\u0017=U\u000e\u0016\u0017is a matrix with the e\u000bective lo-\ncal Coulomb interaction strength within the dorbitals.\nIt plays a similar role to the exchange-correlation ker-\nnel in the adiabatic local-density approximation of time-\ndependent DFT calculations73. We de\fne the transverse\nmagnetic response of the system by summing the suscep-\ntibility matrix over all the dorbitals.\nThe uniform single particle transverse susceptibility\n\u001f\u0000+\n0(!) =\u001f\u0000+\n0(q= 0;!), obtained within the mean-\n\feld approximation, is expressed in terms of the single-\nparticle Green functions as\n\u001f\u0000+\n0;\u0016\u0017(!) =1\n\u0019NX\nkZ\u000fF\nd\"\b\nG\"\"\n\u0016\u0017(k;\"+!) ImG##\n\u0017\u0016(k;\")\n+ ImG\"\"\n\u0016\u0017(k;\")\u0002\nG##\n\u0016\u0017(k;\"\u0000!)\u0003\u0003o\n:\n(6)\nThe sum is over the wave vectors in the \frst Brillouin\nzone, with Ntheir number. The indices \u0016;\u0017represent\norbitals and \u000fFis the Fermi level.\nIn the spirit of many preceding works37,43,47,48, the\ne\u000bect of temperature and disorder is modeled by in-\ntroducing a constant band broadening \u0011on the en-\nergy levels, such that G(!)!G(!+ i\u0011). The imag-\ninary part of the Green function is then de\fned as\nImG\u0016\u0017(!) =1\n2ifG\u0016\u0017(!+ i\u0011)\u0000G\u0016\u0017(!\u0000i\u0011)g. This ap-\nproach attempts to capture all the intrinsic e\u000bects origi-\nnated from the electronic structure of the system.\nThe imaginary part of the susceptibility is related\nto the energy dissipation of the system74, encoding\nthe relaxation mechanism of the magnetization towards\nequilibrium. The damping parameter is then obtained5\nby mapping the transverse magnetic susceptibility ob-\ntained from the quantum mechanical calculation de-\nscribed above to the phenomenological form provided by\nthe LLG, Eq. (1). On the following sections, we present\ndi\u000berent mapping procedures involving several approx-\nimations and explore their range of validity when the\nbroadening \u0011is taken to zero (clean limit).\nIV. SPIN RESPONSE METHODS\nA. Ferromagnetic resonance\nMagnetic excitations can be investigated by applying\ntime-dependent perturbations. This is done in FMR ex-\nperiments where the magnetic sample is subjected to a\nstatic magnetic \feld and an oscillatory radio-frequency\none. By varying either the strength of the static compo-\nnent or the frequency of the oscillatory \feld, the system\ncan be driven through magnetic resonance. This setup\nyields the uniform mode of the transverse magnetic sus-\nceptibility. As the Gilbert parameter describes the relax-\nation mechanisms of the magnetization, it is related to\nthe linewidth of the resonance peak21,75.\nWe simulate this kind of experiments by calculating\nthe transverse magnetic response relying on the linear\nresponse theory discussed in Sec. III, and mapping the\nimaginary part of the susceptibility into the result ob-\ntained from the LLG equation (see Appendix B),\nIm\u001f\u0000+(!) =\u00002\u000b\r!M\n[!\u0000\r(Bext+Ban)]2+ (\u000b!)2:(7)\nWhen \fxing the frequency and varying Bext;z, this func-\ntion presents a resonance at Bres= (!\u0000\rBan;z)=\r\nwith linewidth given by the full width at half maxi-\nmum \u0001B= 2\u000b!=\r . On the other hand, when the\n\feld is kept \fxed and the frequency is varied, the res-\nonance is located at !res=\r(Bext;z+Ban;z)=p\n1 +\u000b2\nwith full width at half maximum approximately given by\n\u0001!\u00192\u000b\rjBext;z+Ban;zj, in the limit \u000b\u001c175.\nThe Gilbert parameter can then be obtained either by\n\ftting Eq. (7) or through the ratio between the linewidth\nand the resonance position. In this sense, a divergence of\nthe damping when \u0011!0 seems counter-intuitive, since\nthis would imply that either the resonance position ( Bres\nor!res) goes to zero or that the corresponding linewidth\nincreases drastically. In the presence of SOI, the SU(2)\nrotational symmetry is broken and the anisotropy \feld\nBan;zshifts the resonance position to a \fnite value | it\ncosts a \fnite amount of energy to set the magnetization\ninto precession76. Therefore, the divergence of the damp-\ning parameter can only happen if the linewidth increases\nand goes to in\fnity.\nTo verify this claim, we simulate FMR experiments\nin fcc Co bulk by calculating the imaginary part of the\ntransverse magnetic susceptibility as a function of the\nfrequency!, in the presence of the spin-orbit interac-\ntion. In Fig. 2a, we present the obtained spectra fordi\u000berent values of the broadening \u0011. When a relatively\nlarge value of the broadening is used, \u0011= 13:6 meV (solid\ncurve), the spectra displays a broad resonance peak,\nwhich can be characterized by a value \u000b= 1:3\u000210\u00002,\nobtained by \ftting the linear response data with Eq. (7).\nWhen the broadening of the energy levels is decreased\nto\u0011= 4:1 meV (dashed curve), the peak shifts and be-\ncomes sharper ( \u000b= 3:8\u000210\u00003), as one intuitively ex-\npects when disorder and/or temperature decreases. No-\ntice that most of the change in \u000bis due to the change\nin the peak width, while the resonance shift is relatively\nsmall. This can be viewed as a consequence of the smaller\nenergy overlap between the bands, which decrease possi-\nble interband transitions58. Surprisingly, by further de-\ncreasing the broadening to \u0011= 0:41 meV (dotted curve),\nthe peak becomes broader when compared to the pre-\nvious case, with \u000b= 5:6\u000210\u00003. This counter-intuitive\nresult represents a shorter lifetime of the magnetic excita-\ntion when the electronic lifetime (mean time between two\nsuccessive scattering events) \u001c=\u0011\u00001becomes longer.\nObtaining the damping from the FMR curves is com-\nputationally demanding, though. The response function\nmust be calculated for many frequencies (or magnetic\n\felds) to resolve the peak. For the case of low broaden-\nings that require many k-points in the Brillouin zone for\na converged result, this task becomes prohibitive. In the\nnext section, we provide alternative methods to obtain\nthe Gilbert parameter based on the static limit of the\nsusceptibility, and compare their outcomes with the ones\nobtained using the resonance approach.\nB. Inverse Susceptibility Method\nWe proceed now to investigate a di\u000berent mapping\nof the microscopic transverse susceptibility to the LLG\nequation and possible approximations to simplify the cal-\nculation of the Gilbert damping. From Eq. (B4), one can\nsee that\u000bde\fnes the slope of the imaginary part of the\ninverse susceptibility43, i.e.,\n\u000b= 2\rMlim\n!!0Im[\u001f\u0000+(!)]\u00001\n!: (8)\nWe will refer to this as the inverse susceptibility method\n(ISM). The mapping to the LLG model of the slope at\nsmall frequencies has a great advantage over the FMR\none since it only requires a single frequency-point calcula-\ntion, instead of a full sweep over frequencies or magnetic\n\felds for the \ftting procedure.\nIn Fig. 2b, we display the damping parameter for bcc\nFe, fcc Co and fcc Ni bulk systems calculated as a func-\ntion of the electronic energy broadening. We also include\nthe results obtained from the FMR approach (solid sym-\nbols), which compare well with the ISM given in Eq. (8).\nNote that although Eq. (8) has an explicit linear depen-\ndence on the spin moment M, the susceptibility implic-\nitly depends on its value. The obtained curves are in-\nversely related to M: highest for Ni ( M\u00180:45\u0016B), low-6\n0.5 0.55 0.6 0.6502468·105\nFrequencyω(meV)−Imχ−+(ω) (states/eV)η1= 13.6 meV→α= 1.3·10−2\nη2= 4.1 meV→α= 3.8·10−3\nη3= 0.41 meV→α= 5.6·10−3\n1 10 10010−210−1100101102\nBroadening η(meV)Gilbert damping αXC-TCM ISM\nFe\nFe 5·λSOI\nFe 10·λSOI\n1 10 10010−310−210−1100101\nBroadening η(meV)Gilbert damping αISM Fe\nISM Co\nISM Co no SOI\nFMR Co\nISM Ni\n1 10 10010−310−210−1100101\nBroadening η(meV)Gilbert damping αFe monolayer\nCo monolayer\nNi monolayer10 100 1,000Temperature (K)\n10 100 1,000Temperature (K)\n10 100 1,000Temperature (K)(a)\n(b)(c)\n(d)\nFigure 2. Characteristics of the Gilbert damping in 3D and 2D systems in presence and absence of SOI. (a) Ferromagnetic\nresonance spectra for fcc Co, in presence of spin-orbit interaction and no external \feld, calculated for three di\u000berent decreasing\nbroadenings \u00111= 13:6 meV (solid), \u00112= 4:1 meV (dashed) and \u00113= 0:41 meV (dotted). The values of the Gilbert damping\ngiven in the legend box, obtained by \ftting to Eq. (7), decrease from the \frst case to the second, but increases again when \u0011is\nfurther decreased. (b) Gilbert damping in presence of spin-orbit interaction for bcc Fe (blue triangles), fcc Co (red circles, solid\nline) and fcc Ni (green squares) as a function of the broadening, obtained from the slope of the inverse susceptibility, Eq. (8).\nAll values were computed with 108k-points in the full Brillouin zone. Solid red circles are the values obtained from the FMR\nspectra in (a), while the open red circles connected by dashed lines represent the damping parameter for fcc Co when SOI is\nnot included in the calculations. (c) Damping parameter for bcc Fe for di\u000berent SOI strenghts: \u0015SOI= 54:4 meV, 5 \u0002\u0015SOI,\nand 10 \u0002\u0015SOI. (d) Gilbert damping of Fe, Co and Ni monolayers in the presence of SOI. No increase in the Gilbert damping\nis seen when the broadening \u0011is decreased.\nest for Fe (M\u00182:3\u0016B) and Co in-between ( M\u00181:5\u0016B).\nThis trend is con\frmed by setting the SOI strength \u0015SOI\nto the same values for all the elements (not shown). The\nposition of the minimum value of \u000bis connected with\n\u0015SOI, which determines when the intraband or interband\ntransitions become more important58. To substantiate\nthis claim, we employed the technique of arti\fcially scal-\ning the\u0015SOI, as previously done in connection to the\nmagnetic anisotropy energy77. The results are shown in\nFig. 2c, where the SOI strength \u0015SOIof Fe bulk is mag-\nni\fed by factors of 5 and 10. Indeed, the minimum can\nclearly be seen to shift to larger values of \u0011.An important aspect to be considered is the conver-\ngence of Eq. (6) | failing to achieve numerical precision\nmay give rise to spurious results49,78. This can be partly\nsolved using sophisticated schemes to perform those cal-\nculations79,80. When the broadening is lowered, the con-\nvergence of the wave vector summation is a\u000bected by\nthe increasingly dominant role of the poles of the Green\nfunctions in the vicinity of the Fermi energy. For that\nreason, to capture the intricacies of the electronic states\n| in particular, the important contributions from the\nsmall gaps opened by the weak SOI |, we calculated\nthe slope of the response function using a very \fne in-7\ntegration mesh on the Brillouin zone reaching up to 109\nk-points. The results in Fig. 2c also demonstrate that the\ndivergence is not an issue of numerical convergence, since\nthis behavior is shifted to larger values of broadenings,\nfor which the convergence is more easily achieved.\nNevertheless, such diverging e\u000bect only occurs in the\npresence of spin-orbit interaction. In Fig. 2b we also dis-\nplay the values of \u000bfor Co fcc obtained using the ISM\nwhen the SOI is not included in the calculations (cir-\ncles connected by dashed lines). In this case, \u000bnoSOI lin-\nearly goes to zero when the broadening is decreased21.\nThe non-vanishing damping when SOI is not present\ncan be interpreted as originating from the \fnite elec-\ntronic lifetimes introduced by the constant broadening\nparameter. As it stands, \u0011represents the coupling to a\n\fctitious reservoir51,52providing dissipation mechanisms\nthat physically should originate from disorder or temper-\nature, for example.\nObtaining the damping from the FMR spectra when\nSOI is not present requires an applied magnetic \feld,\nsuch that the resonance frequency becomes \fnite and\navoiding an in\fnite response at zero frequency (repre-\nsenting no cost of energy due to the rotational symmetry,\ni.e., the Goldstone mode). Nevertheless, the results pre-\nsented in Fig. 2d were obtained using the ISM without\nany applied \feld. Calculations with an applied magnetic\n\feld shifting the peak to the original anisotropy energy\nwere indistinguishable from those values (with variations\nsmaller than 3%). This is accordance to the phenomeno-\nlogical expectations expressed through Eq. (B4), where\nthe slope is independent of the magnetic \feld.\nOne can put our results for bulk ferromagnets into\nperspective by comparing with low dimensional systems.\nWe investigated this case within our linear response ap-\nproach, using monolayers of Fe, Co and Ni. The calcu-\nlations follow the same procedure, except that the sum\noverkvectors in Eq. (6) is restricted to the 2D Brillouin\nzone. The results are presented as triangles (Fe), cir-\ncles (Co) and squares (Ni) connected by dotted lines in\nFig. 2d, and once again exhibit a monotonous decay with\nthe decrease of \u0011. We note that previous calculations of\nthe damping parameter in thin \flms also did not \fnd it\nto increase rapidly for decreasing broadening21,49.\nBesides the dimensionality, another main di\u000berence\nfrom the bulk to the layered case is the larger anisotropy\n\felds of the latter (see Table I). Nevertheless, this can-\nnot explain the non-diverging behavior in the monolay-\ners. We have already shown that by arti\fcially increas-\ning the SOI strength of the bulk | and, consequently, its\nanisotropy \feld |, the conductivity-like behavior of the\ndamping occurs at even larger broadenings (see Fig. 2c).\nOn the other hand, to rule out a possible divergence hap-\npening at lower broadenings ( \u0011<0:1 meV, not reachable\nin our calculations), we have also scaled up \u0015SOIof the\nmonolayers by one order of magnitude. This resulted in\nlarger dampings, nonetheless, the same decreasing be-\nhaviour with \u0011!0 was observed (not shown). There-\nfore, the divergence can only be attributed to the three-dimensionality of the ferromagnet.\nC. Approximate static limit methods\nWe now look back to Fig. 1 and proceed to perform\napproximations on Eq. (8) in order to simplify the calcu-\nlations of the damping parameter. Here we follow Ref. 43.\nFirst, we use Eq. (5) that relates the RPA susceptibility\nmatrix to the mean-\feld response matrix \u001f0, such that\nIm\u001f\u00001\u0019Im\u001f\u00001\n0. Although Uis a real matrix, the sum\nover orbitals ( \u001f=P\n\u0016\u0017\u001f\u0016\u0017) ends up mixing the real\nand imaginary parts of the matrix elements. Only when\nRe\u001f\u00001\n0=U=4 the relation above becomes an equality.\nThis means that, within our model with Uacting only on\nthedorbitals,\u001fmust also be de\fned by summing over\nthose orbitals only. Under the previous assumption, we\nobtain\n\u000b\u00192\rMlim\n!!0Im[\u001f\u0000+\n0(!)]\u00001\n!: (9)\nThis relation is only valid when \u001f\u0000+\n0is decoupled from\nthe other types of susceptibilities (transverse and longi-\ntudinal), as in the systems we investigate in this work.\nThe damping parameter can therefore be obtained from\nthe single-particle transverse susceptibility \u001f0.\nFor frequencies !in the meV range (where the col-\nlective spin excitations are located), \u001f\u0000+\n0has a simple\n!-dependence81:\n\u001f\u0000+\n0(!)\u0019Re\u001f0(0) + i!Im\u001f0\n0(0): (10)\nwhere\u001f0\n0(0) =d\u001f\u0000+\n0\nd!\f\f\f\n!=0. These results are valid also in\nthe presence of spin-orbit coupling. Using Eq. (10), the\nGilbert damping can be written as\n\u000b\u0019\u00002\rM\u0002\nRe\u001f\u0000+\n0(0)\u0003\u00002lim\n!!0Im\u001f\u0000+\n0(!)\n!:(11)\nAlthough the expansion of the susceptibility for low fre-\nquencies was used, no extra approximation is employed,\nsince Eq. (9) is calculated in the limit !!0. Re\u001f\u0000+\n0(0)\ncan be obtained using the sum rule that relates the\nstatic susceptibility with the magnetic moments76. For\n3dtransition metals, the external and the spin-orbit\n\felds are three orders of magnitude smaller than U, and\nso the static susceptibility of the bulk systems reads\nRe\u001f\u0000+\n0(0)\u00194=U. Thus,\n\u000b\u0019\u0000\rMU2\n8lim\n!!0Im\u001f\u0000+\n0(!)\n!: (12)\nFinally, from Eq. (6) it is possible to show that Eq. (12)8\nsimpli\fes as\n\u000b\u0019\rMU2\n2\u0019NX\nk;\u0016\u0017TrfImG\u0017\u0016(k;\u000fF)^S\u0000ImG\u0016\u0017(k;\u000fF)^S+g\n=\r\n2M\u0019NX\nk;\u0016\u0017TrfImG\u0017\u0016(k;\u000fF)^T\u0000\nxcImG\u0016\u0017(k;\u000fF)^T+\nxcg\n=\rMU2\u0019\n8NX\nk;\u0016\u0017n#\n\u0017\u0016(k;\u000fF)n\"\n\u0016\u0017(k;\u000fF):\n(13)\nwheren\u001b\n\u0016\u0017(k;\u000fF) =\u00001\n\u0019ImG\u001b\u001b\n\u0016\u0017(k;\u000fF) is the matrix el-\nement of the spectral function of spin \u001bcalculated at\nkand\u000fF. The second equation is written in terms\nof the \\exchange-correlation torque operator\", T\u0006\nxc=\n\u0000i\u0002^S\u0006;^Hxc\u0003\n=\u0007iUM^S\u0006. This result is equivalent\nto the one obtained in Ref. 42, which we reference as\ntheexchange torque correlation method (XC-TCM) |\nalthough, in reality, it relates \u000bwith the spin-spin re-\nsponse. The last step in Eq. (13) connects the damping\nwith the product of spectral functions of opposite spins\nat the Fermi level, as shown theoretically in Ref. 81 and\ncon\frmed experimentally in Ref. 46.\nIn Fig. 2c, we compare the results obtained with this\napproximated method with the ISM described before, for\nthe di\u000berent values of SOI scalings. For the bulk tran-\nsition metals we investigate, the approximation is very\ngood, since the SOI is relatively small. In fact, even\nwhen the SOI is scaled one order of magnitude higher,\nthe results of the XC-TCM are still very good.\nThe formulas in Eq. (13) show that we have arrived\nat the bottom of the triangle in Fig. 1. These forms\ndo not involve an integral over energy, which simpli\fes\nsubstantially the calculation of \u000b. For that reason, they\nare suitable for \frst-principles approaches (e.g., Refs. 20\nand 62). This concludes our investigations of the spin\nresponse methods. In the next section, we take a di\u000berent\npath to calculate the Gilbert damping.\nV. TORQUE RESPONSE METHODS\nDespite the simplicity of the methods based on the spin\nsusceptibility discussed in the previous section, seminal\nwork was based on a di\u000berent type of response function.\nThe main idea, \frst proposed by Kambersky37, is to di-\nrectly relate \u000bto the spin-orbit interaction. Here, our\naim is twofold. First, we connect the spin susceptibility\nwith the spin-orbit torque response via the equation of\nmotion, clarifying the damping mechanisms captured by\nthis formalism. Second, we compare the results obtained\nwith both types of methods.\nWe start with the equation of motion for the spin-spin\nsusceptibility. Its time-Fourier transform can be written\nas19\n!\n\n^S\u0000;^S+\u000b\u000b\n!=M+\n\n\u0002^S\u0000;^H\u0003\n;^S+\u000b\u000b\n!; (14)whereM=\u00002\n^Sz\u000b\n. From the Hamiltonian given in\nEq. (2), the commutator [ ^S\u0000;^H\u0003\nhas four contributions:\nkinetic (spin currents, from ^H0), exchange torque (from\n^Hxc), external torque (from ^Hext) and spin-orbit torque\n(from ^HSOI). In presence of SOI, the total spin magnetic\nmoment is not a conserved quantity and spin angular\nmomentum can be transferred to the orbital degrees of\nfreedom. For bulk systems subjected to static external\n\felds and in the present approximation for the electron-\nelectron interaction, the only two non-vanishing torques\nare due to the external \feld and the spin-orbit interac-\ntion. It also follows from these assumptions that the\nmechanisms that contribute to the relaxation arises then\nfrom the spin-orbit torques ^T\u0006\nSOI=\u0000i\u0002^S\u0006;^HSOI\u0003\nand\nfrom the broadening of the energy levels \u0011.\nIt can be shown19that the inverse of the susceptibility\n\u001f\u0000+(!) =\n\n^S\u0000;^S+\u000b\u000b\n!is given by\n\u0002\n\u001f\u0000+(!)\u0003\u00001=\u0002\n\u001f\u0000+\nnoSOI (!)\u0003\u00001\u0002\n1 +\u001f\u0000+\nnoSOI (!) \u0000(!)\u0003\u00001\n\u0019\u0002\n\u001f\u0000+\nnoSOI (!)\u0003\u00001\u0000\u0000(!):\n(15)\nHere,\u001f\u0000+\nnoSOI (!) is the susceptibility calculated excluding\nthe SOI contribution to the Hamiltonian. The connection\nbetween the two susceptibilities in Eq. (15) is provided\nby the quantity\nM2\u0000(!) = i\n\u0002^T\u0000\nSOI;^S+\u0003\u000b\n+\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!:(16)\nUsing Eq. (8), and noticing that the \frst term on the\nright-hand side of the equation above does not contribute\nto the imaginary part, we \fnd\n\u000b=\u000bnoSOI\u00002\r\nMlim\n!!0Im\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!\n!: (17)\n\u000bnoSOI is the contribution obtained by inputting\n\u001f\u0000+\nnoSOI (!) into Eq. (8), which is \fnite due to the broad-\nening\u0011.\nKambersky37\frst obtained this same result following a\ndi\u000berent approach. In our framework, this would involve\nstarting from Eq. (5) and exploiting the consequences of\nthe fact that the collective spin excitations ( !\u0018meV)\nhave low frequencies when compared to the exchange en-\nergy (U\u0018eV). On the other hand, Hankiewicz et al.19\ndescribed the same expansion for low SOI, and justi\fed\nits use for !.\rBext. Finally, Edwards22shows that\nthis formula is equivalent to a perturbation theory cor-\nrect to\u00152\nSOI(compared to \rBext\u0000!). For that rea-\nson, he suggests that the states used in the calculation\nof\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!should not include SOI, since the op-\nerator ^T\u0000\nSOI/\u0015SOI. Due to the orbital quenching in the\nstates without SOI, this leads to the absence of intra-\nband contributions and, consequently, of the divergent\nbehavior for \u0011!082.\nIn this approach, temperature and disorder e\u000bects are\nincluded in \u000bnoSOI (shown in Fig. 2d for Co), while the9\nspin-orbit intrinsic broadening is calculated by the sec-\nond term in Eq. (17), which can also be obtained as\n\u000b\u0000\u000bnoSOI . An extra advantage of calculating the damp-\ning as the aforementioned di\u000berence is that one explic-\nitly subtracts the contributions introduced by \u0011, provid-\ning similar results to those obtained with vertex correc-\ntions20. Considering the torque-torque response within\nthe mean-\feld approximation (an exact result in the per-\nturbative approach22), we obtain, similarly to Eq. (13),\n\u000b\u0000\u000bnoSOI =\n2\r\nM\u0019NX\nkTrfImG(k;\u000fF)^T\u0000\nSOIImG(k;\u000fF)^T+\nSOIg:\n(18)\nIn this formula, the involved quantities are matrices in\nspin and orbital indices and the trace runs over both.\nThis is known as Kambersy's formula, commonly used in\nthe literature43,44,47,49,58, which we refer to as spin-orbit\ntorque correlation method (SO-TCM). As in Eq. (13), it\nrelates the damping to Fermi level quantities only. When\nthe SOI is not included in the calculation of the Green\nfunctionsG(k;\u000fF) and enters only through the torque\noperators, we name it perturbative SO-TCM22. These\nmethods are placed at the bottom right of Fig. 1, with\nthe main approximations required indicated by the long\ndashed arrows.\nWe now proceed to compare these approaches with the\nISM explained in Sec. IV B. Fig. 3 presents the calcula-\ntions of the SOI contribution to the damping parameter\nof bulk Fe (a), Co (b) and Ni (c) using the SO-TCM ob-\ntained in Eq. (18), when no external \feld is applied. Both\napproaches, including SOI (red curve with squares) in\nthe Green functions or not (green curve with triangles),\nare shown. For a meaningful comparison, we compute\n\u000b\u0000\u000bnoSOI within the ISM.\nWe \frst note that the perturbative approach suggested\nby Edwards22describes reasonably well the large broad-\nening range (i.e., mostly given by the interband transi-\ntions), but deviates from the other approaches for low \u0011.\nThis is an expected behaviour since it does not include\nthe intraband transitions that display the \u0011\u00001behav-\nior within the constant broadening model. In the clean\nlimit, the Gilbert damping computed from the pertur-\nbative SO-TCM approaches zero for all elements, in a\nvery monotonic way for Co and Ni, but not for Fe. This\nmethod is thus found to be in agreement with the other\nones only when \u0015SOI\u001c\u0011. The SO-TCM formula in-\ncluding the SOI in the states (i.e., Kambersky's formula)\nmatches very well \u000bobtained within ISM in the whole\nrange of broadenings.\nFinally, after demonstrating that the SO-TCM pro-\nvides very similar results to the ISM, we can use it to\nresolve the wave-vector-dependent contributions to the\nGilbert parameter by planes in the reciprocal space, as\n\u000b(kmax\nz) =kmax\nzX\njkzj\u000b(kz); (19)\n1 10 10010−310−2Gilbert damping α(a)Fe\nISM ( α−αnoSOI )\nSO-TCM\nperturbative SO-TCM10 100 1,000Temperature [K]\n1 10 10010−410−310−2Gilbert damping α(b)Co\n10 100 1,000\n1 10 10010−410−310−210−1100\nBroadening η[meV]Gilbert damping α(c)Ni\n10 100 1,000Figure 3. Comparison between \u000b\u0000\u000bnoSOI for (a) Fe bcc, (b)\nCo fcc and (c) Ni fcc, obtained using the inverse susceptibility\nmethod (ISM) with the spin-orbit-torque correlation method\n(SO-TCM) with and without SOI in the states (perturbative\nSO-TCM). All the points were computed with 108k-points in\nthe full Brillouin zone.\nwhere\u000b(kz) is given by the right-hand side of Eq. (18)\nsummed over kx;ky. The result, displayed in Fig. 4, uses\n100 million k-points for all curves and shows the expected\ndivergence in presence of SOI and a decrease with \u0011when\nthis interaction is absent. In every case, most of the con-\ntribution arises from the \frst half ( kmax\nz<0:4). Note\nthat when the broadening of the energy levels is low, the\nintegrated alpha without SOI (Fig. 4b) displays step-\nlike contributions, while when SOI is present, they are\nsmoother. This is a consequence of the damping being10\n01234Gilbert damping α(·10−2)(a) With SOI\nη(meV):\n0.14\n0.41\n0.68\n1.1\n1.4\n4.1\n6.8\n10.9\n13.6\n40.8\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\nkmax\nzGilbert damping α(·10−3)(b) Without SOIDecreasingη\nDecreasingη\nFigure 4. Integrated Gilbert damping for fcc Co as a function\nofkzplotted against the maximum value, kmax\nz(see Eq. (19)),\nwith SOI (a) and without SOI (b). The curves were obtained\nusing the SO-TCM given in Eq. (18). Colors represent di\u000ber-\nent values of the broadening \u0011(in units of meV). The value of\n\u000bforkmax\nz= 0 (i.e., a single value of kzin the sum) represents\na two-dimensional system, whilst for kmax\nz= 1 the sum covers\nthe whole 3D Brillouin zone. In the latter case, the damping\ndecreases when \u0011is decreased without SOI, while it increases\ndrastically when SOI is present. For 2D systems, the\ncaused by interband transitions in the former and intra-\nband in the latter.\nThe convergence of the previous results for the smallest\n\u0011including SOI were tested with respect to the total\nnumber of k-points in the Brillouin zone in Fig. 5. By\ngoing from 10 million to 10 billion k-points, the results\nvary\u001820%. However, compared with the result shown\nin Fig. 4a, the damping gets even larger, corroborating\nonce more the divergent results.\nVI. DISCUSSIONS\nIn this section, we make a few \fnal remarks on the pre-\nviously obtained results and we go beyond bulk systems\nto comment on the approximations taken and additional\nphysical mechanisms that may come into play in other\n0 0.2 0.4 0.6 0.8 1024\nkmax\nzGilbert damping α(·10−2)\nk-points:\n107\n108\n109\n1010Figure 5. Integrated Gilbert damping for fcc Co as a func-\ntion ofkzplotted against the maximum value, kmax\nz, for\n\u0011= 0:14 meV and di\u000berent amount of k-points (up to 10\nbillion) in the Brillouin zone.\nmaterials. We also provide a new analytical explanation\nfor the divergence of the damping parameter within the\nconstant broadening model.\nOur \frst comment regards the application of static\nmagnetic \felds B. As described in Refs. 19 and 22, the\napproximations done in Eq. (15) to derive an expression\nfor\u000binvolves comparisons between the excitation en-\nergy andB. However, all the results we have presented\nhere were obtained in absence of static \felds. We also\nperformed calculations including external magnetic \felds\nup toB\u00187 T, and the computed damping parameter is\nweakly in\ruenced by their presence. We conclude that\nthe validity of the SO-TCM formula given in Eq. (18)\ndoes not hinge on having a magnetic \feld, supporting\nthe arguments already given in Ref. 19.\nA further remark concerns the approximations made\nto obtain the mean-\feld result in Eq. (12). We assumed\nthat SOI is weak when using the magnetic sum rule.\nThis approximation may break down when this is not\nthe case. The spin pumping also a\u000bects the magnetic\nsum rule, which may worsen the agreement with the ISM\nresults. Although this contribution is not present in the\ninvestigated (bulk-like) systems, it plays an important\nrole in magnetic multilayers. This e\u000bect enhances the\ndamping factor32,54,55. Furthermore, the SO-TCM ex-\nplicitly excludes spin pumping, as this is described by\n^I\u0000\nS=\u0000i\u0002^S\u0000;^H0\u0003\n, dropped from the equation of mo-\ntion. These validity conditions are indicated in Fig. 1 by\nthe large blue rectangle (low SOI), red triangle (low spin\npumping) and green rectangle (no spin pumping).\nAnother mechanism that opens new spin relaxation\nchannels is the coupling between transverse and longi-\ntudinal excitations induced by the SOI. This was one of\nthe reasons raised in Ref. 21 to explain the divergence of\nthe damping parameter. However, this is absent not only11\nwhen the system has full spin rotational symmetry83, but\nalso when rotational symmetry is broken by the SOI in\n2D and 3D systems for the symmetries and materials we\ninvestigated. Even though the damping is \fnite in the\n\frst two cases (as shown in Fig. 2d), the divergence is\nstill present in the latter (Fig. 2b).\nWe can also recognize that the mathematical expres-\nsion for\u000bin terms of the mean-\feld susceptibility given\nin Eq. (12) is similar to the conductivity one (i.e., the\nslope of a response function)84| which leads to the same\nissues when approaching the clean limit ( \u0011!0). How-\never, the physical meaning is the exact opposite: While\nthe divergence of the conductivity represents an in\fnite\nacceleration of an ideal clean system, in\fnite damping\ndenotes a magnetic moment that is instantly relaxed in\nwhichever direction it points (as d M=dt!0 for\u000b!1 )\n| i.e., no dynamics10,11. This means that a clean 3D\nspin system is in\fnitely viscous. Within the constant\nbroadening model, the divergence of the Gilbert damp-\ning can also be seen analytically by comparing Eq. (12)\nwith the calculations of the torkance done in Ref. 48.\nBy replacing the torque operator and the current density\nby the spin lowering and raising operators, respectively,\nthe even contribution (in the magnetization) to the re-\nsponse function vanishes and only the odd one remains.\nIn this approximation, it is also seen that only the Fermi\nsurface quantities are left, while the Fermi sea does not\ncontribute85. In the limit of low broadenings, this con-\ntribution is shown to diverge as \u0011\u00001. This divergence\narises from intraband transitions which are still present\nin the clean limit, and originate from the \fnite electronic\nlifetimes introduced by the constant broadening approx-\nimation.\nThe static limit ( !!0) is another reason that many\nauthors considered to be behind the divergent damping\nbehavior19,37,43,50. This limit is taken in Eq. (8) in or-\nder to eliminate the contribution of terms nonlinear in\nfrequency from the inverse susceptibility (e.g., inertia ef-\nfects68,86). They can be present in the full microscopic\ncalculation of the susceptibility but are not included in\nthe phenomenological model discussed in Appendix B.\nAdding the quadratic term in frequency leads to an in-\nverse susceptibility given by\nIm[\u001f\u0000+(!)]\u00001=\u0000!\n2\rM(\u000b\u0000!I)\nwhereIis the o\u000b-diagonal element of the moment of iner-\ntia tensor86. The \ft to the expression linear in frequency\nthen yields an e\u000bective \u000be\u000b(!). In the vicinity of the res-\nonance frequency, \u000be\u000b(!res) =\u000b\u0000!resI, which is clearly\nreduced in comparison to the one obtained in the static\nlimit,\u000be\u000b(0) =\u000b. According to Ref. 68, I\u0018\u000b=\u0011, which\nexplains the discrepancy between the FMR and the ISM\nseen in Fig. 2b as \u0011!0. We can then conclude that the\nstatic limit is not the culprit behind the divergence of \u000b\nin the clean limit.VII. CONCLUSIONS\nIn this work, we presented a study of di\u000berent meth-\nods to calculate the intrinsic Gilbert damping \u000b, o\u000bering\na panorama of how the approaches are related and their\nrange of validity (see Fig. 1). They can be grouped into\nthree main categories: the methods that directly employ\nthe results of full microscopic calculations of the dynam-\nical magnetic susceptibility \u001f(!) (FMR and ISM); the\nexchange-torque method (XC-TCM), which is also based\non\u001f(!) but making use of the mean-\feld approximation;\nand the spin-orbit torque-correlation method (SO-TCM),\nobtained from the (spin-orbit) torque-torque response via\nan equation of motion for \u001f(!). While the FMR, ISM\nand XC-TCM include all the contributions to the mag-\nnetic relaxation, the SO-TCM provides only the intrinsic\ncontribution due to the angular momentum transfer to\nthe orbital degrees of freedom (not including, for exam-\nple, the spin pumping mechanism). The XC- and SO-\nTCM, given by Eqs. (13) and (18), are predominant in\nthe literature due to their simplicity in obtaining \u000bin\nterms of Fermi level quantities. It is important to note,\nhowever, that they rely on approximations that may not\nalways be full\flled21.\nIn order to implement and compare the di\u000berent meth-\nods, we constructed a uni\fed underlying framework\nbased on a multi-orbital tight-binding Hamiltonian using\nas case studies the prototypical bulk 3D systems: bcc Fe,\nfcc Co and fcc Ni. For this set of materials, the di\u000berent\nmethods lead to similar results for \u000b, showing that the\ncorresponding approximations are well-founded. Even\nwhen the SOI strength is scaled up by one order of mag-\nnitude, this excellent agreement remains, as we explic-\nitly veri\fed for bcc Fe. We found one method that falls\nout-of-line with the others in the clean limit, namely the\nperturbative form of the SO-TCM formula22,82. In this\ncase, although the equation is identical to the well-known\nKambersky formula, Eq. (18), the electronic states used\nto evaluate it do not include SOI. By comparison with\nthe other methods, we conclude that the results obtained\nby the perturbative SO-TCM are only valid in the large\nbroadening regime (compared to the SOI strength). Cen-\ntral to our analysis was a careful study of the convergence\nof our results with respect to the number of k-points,\nreaching up to 1010k-points in the full Brillouin zone.\nThe behavior of \u000bis intimately connected with the con-\nstant broadening approximation for the electronic life-\ntimes. For high temperatures, the Gilbert damping in-\ncreases with increasing temperature ( \u000b\u0018\u0011), while for\nlow temperatures it diverges for 3D ferromagnets ( \u000b\u0018\n1=\u0011), but not for 2D (ferromagnetic monolayers). Our\ncalculations revealed that the high temperature values\nof\u000barise mostly from the broadening of the electronic\nstates. In Ref. 20, the strongly increasing behaviour of\n\u000bfor high temperatures was found to be spurious, and\ncured employing a more realistic treatment of disorder\nand temperature, and the so-called vertex corrections.\nWe found that the contribution of the intrinsic SOI to \u000b12\nis additive to the one arising from the broadening, and\ncan be easily extracted by performing a calculation of\n\u000bwithout SOI and subtracting this result from the SOI\none,\u000b\u0000\u000bnoSOI . Combined with the ISM, this provides\na relatively simple and accurate way to obtain the in-\ntrinsic damping, which discounts contributions from the\nadditional broadening \u0011. This establishes an alternative\nway of accessing the high temperature regime of \u000b.\nThe low-temperature divergence of \u000bwhen approach-\ning the clean limit for 3D ferromagnets has also been the\nsubject of much discussion. The \frst di\u000eculty is in es-\ntablishing numerically whether this quantity actually di-\nverges or not. Our results consistently show an increase\nof\u000bwith decreasing \u0011, down to the smallest achievable\nvalue of\u0011= 0:14 meV (Fig. 5), with no hints of a plateau\nbeing reached, but only when accounting for SOI. This\ndivergence arises from the intraband contributions to \u000b,\nas discussed in Ref. 58. Refs. 22 and 82 used pertur-\nbation theory arguments to claim that such intraband\ncontributions should be excluded. However, as we dis-\ncussed in Sec. IV B, adapting the formalism of Ref. 48 to\nthe calculation of \u000bshows that these intraband terms are\nenabled by the constant broadening approximation, and\nso should be included in the calculations. Contrary to\nthe high temperature regime, works that employ a more\nrealistic treatment of disorder and temperature still \fnd\nthe diverging behavior of \u000b20,52.\nIn real experiments, any kind of material disturbance\nsuch as disorder or temperature e\u000bects leads to a \fnite\nvalue of the damping. Besides that, a non-uniform com-\nponent of the oscillatory magnetic \feld (either from the\napparatus itself or due to its limited penetration into the\nsample) induces excitations with \fnite wave vectors and\n\fnite linewidths39,87. A di\u000berent way to determine the\ndamping parameter is using the time-resolved Magneto-\nOptic Kerr E\u000bect (TR-MOKE)40,88. It has the advan-\ntage that, as it accesses a smaller length scale ( \u00181µm)\nthan FMR experiments (which probe the whole magnetic\nvolume), the measured magnetic properties are more ho-\nmogeneous and thus the e\u000bect of linewidth broadening\nmay be weaker. The magnetic excitations in nanomag-\nnets can also be probed by recent re\fnements of FMR\nexperimental setups89,90.\nAlthough the methods we described here are gen-\neral, we did not explicitly addressed non-local sources\nof damping such as the spin-pumping32. As a future\nproject, we plan to ascertain whether our conclusions\nhave to be modi\fed for systems where this mechanism\nis present. Systems that combine strong magnetic el-\nements with heavy ones possessing strong SOI are ex-\npected to have anisotropic properties, as well-known for\nthe magnetic interactions91. It is then natural to explore\nwhen the Gilbert damping can also display signi\fcant\nanisotropy, becoming a tensor instead of a scalar quan-\ntity47,78. Indeed, this has been observed experimentally\nin magnetic thin \flms92,93. As the SOI, magnetic non-\ncollinearity can also lead to other forms of damping in do-\nmain walls and skyrmions50,94{98. From the microscopicpoint of view, the potential coupling between transverse\nand longitudinal degrees of freedom allowed by the non-\ncollinear alignment should also be considered. Lastly,\nhigher order terms in frequency, such as the moment of\ninertia68,86,99{101, might also become important in the\ndynamical magnetic susceptibility for large frequencies\nor for antiferromagnets, for instance.\nThe description of magnetization dynamics of real ma-\nterials helps to design new spintronic devices able to con-\ntrol the \row of information. Our work sheds light on fun-\ndamental questions about the main relaxation descrip-\ntions used in the literature and sets ground for future\ntheoretical predictions.\nAppendix A: Ground-state Hamiltonian\nIn this Appendix, we give the explicit forms of the\nterms in the Hamiltonian written in Eq. 2. As the inves-\ntigated systems only have one atom in the unit cell, the\nsite indices are omitted.\nThe electronic hoppings in the lattice are described by\n^H0=1\nNX\nk\u001bX\n\u0016\u0017t\u0016\u0017(k)cy\n\u0016\u001b(k)c\u0017\u001b(k); (A1)\nwithcy\n\u0016\u001b(k) andc\u0017\u001b(k) being the creation and annihila-\ntion operators of electrons with spin \u001band wave vector\nkin the orbitals \u0016and\u0017, respectively. The tight-binding\nparameters t\u0016\u0017(k) were obtained by \ftting paramagnetic\nband structures from \frst-principles calculations up to\nsecond nearest neighbors102, within the two-center ap-\nproximation103.\nThe electron-electron interaction is characterized by\na local Hubbard-like104interaction within the Lowde-\nWindsor approximation105, resulting in the mean-\feld\nexchange-correlation term\n^Hxc=\u0000X\n\u00162d\n\u001bU\n2(\nM\u000b\u001b\u000b\n\u001b\u001b0+X\n\u00172d\u000en\u0017(2\u000e\u001b\u001b0\u000e\u0016\u0017\u0000\u000e\u001b\u001b0))\ncy\n\u0016\u001b(k)c\u0016\u001b0(k):\n(A2)\nHere,Uis the local e\u000bective Coulomb interaction, M\u000b\nand\u001b\u000bare the\u000b-component of the magnetic moment\nvector (summed over the dorbitals) and of the Pauli\nmatrix, respectively. \u000en\u0016is the change in the occupation\nof orbital\u0016compared to the DFT calculations included\nin Eq. A1. M\u000band\u000en\u0016are determined self-consistently.\nThe atomic SOI is described by\n^HSOI=\u0015X\n\u0016\u0017\n\u001b\u001b0^L\u000b\n\u0016\u0017^S\u000b\n\u001b\u001b0cy\n\u0016\u001b(k)c\u0017\u001b0(k);(A3)\nwhereL\u000bandS\u000bare the\u000bcomponents of the orbital and\nspin vector operators, respectively. The strength of the\nSOI,\u0015, is also obtained from \frst-principles calculations.13\nThe interaction with a static magnetic \feld Bextis\ndescribed by\n^Hext=B\u000b\nextX\n\u0016\u0017\n\u001b\u001b0(^L\u000b\n\u0016\u0017\u000e\u001b\u001b0+\u001b\u001b\u001b0\u000e\u0016\u0017)cy\n\u0016\u001b(k)c\u0016\u001b0(k);\n(A4)\nwhere\u0016Bis absorbed to B\u000b\nextand we used gL= 1 and\ngS= 2 as the Land\u0013 e factors for the orbital and spin\nangular momentum.\nAppendix B: Phenomenology of FMR\nThe semi-classical description of the magnetization is\nobtained using the Landau-Lifshitz-Gilbert (LLG) equa-\ntion (1)9. The e\u000bective \feld acting on the magnetic mo-\nment is obtained from the energy functional of the system\nasBe\u000b(t) =\u0000@E=@M. For the symmetries we investi-\ngate, the model energy106for the 3D cubic cases77can\nbe written as\nE3D(M) =K4\nM4(M2\nxM2\ny+M2\nyM2\nz+M2\nxM2\nz)\u0000M\u0001Bext;\n(B1)\nwhile for 2D systems,\nE2D(M) =\u0000K2\nM2M2\nz\u0000M\u0001Bext: (B2)\nPositive values of K4andK2yield easy magnetization\ndirection along the (001) direction.\nWe consider magnetic moments pointing along the easy\naxis, which de\fnes the ^ zdirection. Static magnetic \felds\nare applied along the same orientation. The magnetic\nmoment is set into small angle precession, M=M^ z+\u000eMx(t)^ x+\u000eMy(t)^ y, by an oscillatory \feld in the trans-\nverse plane, i.e., Bext(t) =Bext^ z+\u000eBext(t). In this form,\nthe e\u000bective \feld (linear in the transverse components of\nthe magnetization) is given by Be\u000b(t) =Ban(t)+Bext(t),\nwith\nB3D\nan(t) =\u00002K4\nM2(\u000eMx^ x+\u000eMy^ y) , and B2D\nan=2K2\nM^ z\n(B3)\nbeing the anisotropy \felds for 3D and 2D systems, respec-\ntively. In the following expressions, K4andK2appear\nin the same way, so they are denoted by K.\nThe Fourier transform of the linearized equation of mo-\ntion can be written using the circular components \u000eM\u0006=\n\u000eMx\u0006i\u000eMy. Within this convention, \u000eM\u0000=\u000eB\u0000=\n\u001f\u0000+=2 and\n\u001f\u0000+(!) =\u00002\rM\n[!\u0000\r(Bext+Ban)]\u0000i\u000b!; (B4)\nwhereBan= 2K=M .\nACKNOWLEDGMENTS\nWe are very grateful to R. B. Muniz, A. T. Costa\nand D. M. Edwards for fruitful discussions. 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Press, Oxford, 2008)."    },    {        "title": "1809.09429v1.Theory_of_damping_in_magnetization_dynamics__dispelling_a_myth_and_pointing_a_way_forward.pdf",        "content": "arXiv:1809.09429v1  [cond-mat.mtrl-sci]  25 Sep 2018Theory of damping in magnetization dynamics, dispelling a m yth and pointing a way\nforward\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nThere is a widely-held belief amongst theoreticians that th e Gilbert damping parameter αin\nmagnetization dynamics is infinite for a pure metal at T=0. Th e basic error leading to this belief\nis pointed out explicitly and the various methods of calcula tion used are viewed in a unified way\nbased on the Lorentzian lineshape of ferromagnetic resonan ce spectra. A general torque formula\nforαis proposed as a good starting-point for treating inhomogen eous materials such as alloys,\ncompounds and layered structures. Local spin density funct ional theory provides a simple physical\npicture, in terms of a non-uniform precessional cone angle i n ferromagnetic resonance, of how such\ninhomogeneity contributes to the damping. In acomplementa ry many-bodytheory this contribution\nis given by a vertex correction to the torque-torque respons e function.\nThe damping of magnetization dynamics in ferromagnetic metals and a lloys is of critical importance in spintronic\ndevices. Damping largely controls the speed at which a device can ope rate and its energy requirement. In device\nphysics damping is usually treated phenomenologically by means of a Gilb ert term in the Landau-Lifshitz-Gilbert\nequation [1, 2] and many quantum-mechanical calculations of the Gilb ert parameter have been made for specific\nmaterials [3–10]. A reliable treatment of damping in transition metals an d alloys would be an invaluable guide in\nthe search for materials with very low damping [11, 12], as required fo r the future development of devices such\nas magnetic random access memory(MRAM). Most recent work in th is direction is concerned with the important\nintrinsic contribution arising from spin-orbit coupling(SOC) and it is th is which concerns us here. A satisfactory\ntheory should work in the limit of a pure metal but almost all existing ca lculations predict that the Gilbert damping\nparameter αdiverges to infinity for a pure metal at T=0. This would mean that in th e pure metals Fe, Co and Ni\nat low temperature the linewidth in a ferromagnetic resonance (FMR ) experiment would be much too large for the\nresonance to be observed. The prediction or acceptance of infinit e damping has been made by so many authors [3–\n10, 13, 14] over the last forty years that it has acquired the stat us of a myth. A very recent paper [15] repeats it\nonce again. It is noteworthy that no experimentalist seems to have troubled to investigate the problem by work\non high purity metals and dilute alloys at low temperature. The aim of th is article is not only to dispel the myth\nbut to formulate a firm starting-point for future calculations of αin technically important materials such as alloys,\ncompounds and layered structures.\nThe most direct method to investigate damping,both experimentally a nd theoretically, is to study the ferromagnetic\nresonance (FMR) linewidth. In FMR a uniform static magnetic field His applied and the absorption of a transverse\nmicrowave field of angular frequency ωpeaks around the frequency bex//planckover2pi1wherebex= 2µBHis the Zeeman energy.\nForHin the z direction the absorption is determined by the imaginary part o f the dynamical transverse susceptibility\nχ−+(ω). This susceptibility, which must include the effect of SOC, can be calc ulated by standard many-body theory\nusing the Kubo formula or by time-dependent spin-density function al theory (SDFT). In practice the many-body\nmethod is usually based on a tight-binding approximation and employs t he random phase approximation (RPA)\nwith a short-range screened Coulomb interaction. This is then equiv alent to a time-dependent Hartee-Fock mean-\nfield theory. The long-range interaction can also be included if care is taken that it does not enter the exchange\nterms [16, 17]. SDFT is approximated similarly as a time-dependent mea n-field theory in the local spin density\napproximation (LSDA) and the long-range Coulomb interaction pres ents no problem since it is effectively screened\nin the exchange-correlation functional. It is useful to consider bo th these methods in parallel. In a system with\nvarying direction of magnetization SDFT is based on a density matrix o f order 2 [18] rather than just spin and\nparticle densities. χ−+(ω) is then coupled to fifteen other response functions which determ ine the longitudinal spin\nsusceptibility as well as the charge response and mixed charge-spin responses [19, 20]. These last relate to phenomena\nlike the spin-Hall effect. Some of these response functions, includin g the longitudinal spin susceptibility, involve the\nlong-range Coulomb interaction importantly even in the absence of S OC [16, 17, 20]. Costa and Muniz [22], following\nan earlier paper [23], show how SOC produces mode coupling in the RPA m any-body approach. However the long-\nrange Coulomb interaction is left untreated. Their paper is particula rly important for being the first to challenge the\nmyth of infinite damping.\nWe firstdiscussthe caseofaBravaislattice whichisappropriatefor puremetalswith acubic structurelikeFe andNi\nat T=0. In both the approaches described above the dynamical su sceptibility is related to mean-field susceptibilities\nof the general form\nχ0(ω) =N−1/summationdisplay\nkmnMmn(k)fkn−fkm\nEkm−Ekn−/planckover2pi1ω+iη. (1)2\nHereEkmis the energy of the one-electron state with wave-vector kin bandm, calculated in the presence of SOC,\nfkmis the corresponding occupation number, Mmn(k) is a product of matrix elements and ηis a small positive\nconstant which ultimately tends to zero. As in usual time-dependen t perturbation theory equation (1) represents the\nresponse to a perturbing field of angular frequency ωin which transitions occur between occupied and unoccupied\nstates. ”Intraband transitions” with m=nclearly do not occur for ω/negationslash= 0 owing to the cancellation of the Fermi\nfunctions. These transitions between identical states, which are not really transitions at all, can play no role in a\ndynamical process. Hankiewicz et al have made a similar point [24]. How ever, in nearly all calculations of the Gilbert\ndamping parameter α, intraband transitions appear and lead to the infinite damping discus sed above.\nTo dispel a myth effectively it is necessary to see how it has arisen. It is instructive to review, in a unified way, some\nmethods which have been used to calculate α. We start from the Lorentzian form of the FMR lineshape which is\nwell-established experimentally [21] and theoretically [22]. Near the re sonance the dynamical transverse susceptibility\nis dominated by a pole at /planckover2pi1ω=bex+/planckover2pi1∆ωwhere ∆ω∼ξ2,ξbeing the SOC parameter, so that\nχ−+(ω) =−2/angbracketleftSz/angbracketright/N\n/planckover2pi1(ω−∆ω)−bex. (2)\nHereSzis thezcomponent of total spin and Nis the number of atoms in the crystal. Near the resonance the FMR\nabsorption is determined by\nℑ(χ−+(ω)) =−2(/angbracketleftSz/angbracketright/N)ℑ(/planckover2pi1∆ω)\n(/planckover2pi1ω−ℜ(/planckover2pi1∆ω)−bex)2+(ℑ(/planckover2pi1∆ω))2. (3)\nℜ(/planckover2pi1∆ω) corresponds to a shift in the resonance frequency and ℑ(/planckover2pi1∆ω) determines the linewidth, both due to SOC.\nThe Gilbert damping factor αis given by ℑ(/planckover2pi1∆ω)/bex(e.g. [25]). The most direct way to calculate αis a brute-force\nnumerical RPA calculation of ℑ(χ−+(ω)), with SOC included, as a function of ωaround the resonance. Costa and\nMuniz [22] performed such calculations using the tight-binding appro ximation and found perfect Lorentzians from\nwhich they deduced α. Taking a monnolayer of Co as an example they found no tendency fo rαto diverge in the pure\nlimit of sharp electronic states. This method of calculating αis very computer intensive and more economic methods\nexist if one assumes a Lorentzian curve from the outset.\nIt follows immediately from (2) that\nα=ℑ(/planckover2pi1∆ω)/bex=2/angbracketleftSz/angbracketright\nNbexℑ(1\nχ−+(bex//planckover2pi1)). (4)\nThis new formula for αmay be regarded as exact. A full treatment of the transverse su sceptibility includes coupling\nto other modes and leads to a rather complex expression in terms of sixteen mean-field susceptibilities of the form (1)\nwith different sets of matrix elements [20]. There is an enormous simplifi cation in the case of a Bravais lattice if we\ncalculate αonly to second order in the SOC parameter ξ. Following the arguments of [20] it is readily found that\ncoupling of the transverse susceptibility to other modes is then elimin ated and that χ−+in (4) may be replaced by\nthe mean-field susceptibility χ0\n−+. This elimination depends on inversion symmetry, which is a property o f a Bravais\nlattice. Without this symmetry, coupling of the transverse suscep tibility to other modes occurs in general even to\norderξ2, as discussed later. It follows further that to order ξ2\nα= (N∆2/2/angbracketleftSz/angbracketrightbex)ℑ(χ0\n−+(bex//planckover2pi1)) (5)\nwhere ∆ is the exchange splitting in the band structure. It is usually s ufficient to calculate the last factor to first\norder in bexso we may take the unphysical limit bex→0, but with due care as discussed below. Then\nα= (N∆2/2/angbracketleftSz/angbracketright)[∂ωℑ(χ0\n−+(ω)]ω=0 (6)\nwhere the electronic state energiesand matrixelements in ℑ(χ0\n−+(ω) are calculatedwith bex= 0. Beforeproceedingto\nthe static ω→0 limit it is essential not to include contributions from ”intraband tran sitions”, as pointed out after (1).\nThis precaution was not taken in [14], where a similar formula was obtain ed, so the spurious infinite damping for a\npure metal appeared. Sometimes it is preferable to keep the physic al non-zero Zeeman field to remove all danger of\nincluding intraband transitions. This also gives the option of calculatin g the frequency-swept FMR linewidth as a\nfunction of Zeeman field. This has been measured [21] and can be con verted to a frequency dependence of α. Such\na dependence has been discussed by Costa and Muniz [22]. However the low-field limit is usually sufficient and here\nwe take the limit bex→0, with the precaution mentioned above, to compare with other the oretical work. Following\n[14], but excluding intraband terms, we find the following two express ions forαat T=0:\nα= (π∆2/2/angbracketleftSz/angbracketright)/summationdisplay\nk/summationdisplay′\nmn|/angbracketleftkm|S−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF)\n= (πξ2/2/angbracketleftSz/angbracketright)/summationdisplay\nk/summationdisplay′\nmn|/angbracketleftkm|T−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF).(7)3\nHereSSS= (Sx,Sy,Sz) is the total spin operator, S−=Sx−iSy,ξhsois the total spin-orbit interaction, T−= [S−,hso]\nis a torque operator and EFis the Fermi energy. The prime on the sum over bands means m/negationslash=nand the sum over k\nis to be carried out as an integral over the Brillouin zone as usual. As p ointed out these expressions are only correct\nto orderξ2so that in the second expression we must evaluate the electronic st ates and energies with ξ= 0. The prime\non the summation sign may then be omitted since the m=nterms are zero owing to inversion symmetry [10]. The\nresulting expression, which can now be written in terms of one-part icle Green functions if desired, is just the version\nof Kambersky’s torque formula [13] for a Bravais lattice derived in tw o ways by Edwards [20]. It is the mean-field\napproximation to a much more general formula [20], valid for an orde red or disordered system,\nα=−(ξ2/2bex/angbracketleftSz/angbracketright)ℑ[χξ=0\nT(bex//planckover2pi1)]. (8)\nWe shall refer to this as the general torque formula. It is exact to orderξ2and we have left open the option of taking\nthe limit bex→0. Here the torque-torque response function is given by the Four ier transform of a retarded Green\nfunction using the Kubo formula\nχT(ω) =/integraldisplay\n/angbracketleft/angbracketleftT−(t),T+/angbracketright/angbracketrighte−iωtdt. (9)\nThe wide application of (8) is discussed later and we recall that the se cond expression in (7), corresponding to the\nmean-field approximation to χT, is only valid for a Bravais lattice. To evaluate the integral over kin the formula (7)\nnumerically it is usual to replace the delta-functions by Lorentzians of width proportional to an inverse relaxation\ntime parameter τ−1. This broadening of the electron states may be regarded as a crud e representation of the effect\nof impurity and/or phonon scattering. The limit τ−1→0 of a perfect crystal at T=0 leads to a finite value of αbut\nis quite tricky to perform numerically [26]. If we wrongly retain SOC in ca lculating the electron states in the second\nexpression of (7) the diagonal matrix elements are non-zero and le ad to the notorious infinite damping parameter\nα. The only work which deals correctly with αin pure metals is reported in fig.1 of [10] and in [22, 26].(In [26] the\ncaption of fig.1 should read ”with and without SOC included in calculating electronic states”).\nWenowturntothetaskofestablishingafirmbasisforcalculatingthe dampingparameter αintechnicallyimportant\nmaterials, which are typically random alloys or layered structures. T his task is greatly simplified if we are satisfied\nwith calculating αto second order in the SOC parameter ξ. This should be sufficient in nearly all systems of interest.\nAt room temperature the ξ2dependence of αis well-established experimentally in several alloy systems, including\nsome containing Pt with its large SOC [27, 28]. The general torque for mula (8) is a very convenient starting-point. Its\nderivation in Appendix A of [20] is for a completely general ferromag netic material, either ordered or disordered, and\nagain relies only on the universal FMR Lorentzian lineshape. The deriv ation proceeds by comparing an exact relation\nbetween χ−+(ω) andχT(ω) with an expansion of (2) in the limit /planckover2pi1∆ω/(/planckover2pi1ω−bex)→0 followed by /planckover2pi1ω→bex. This\norder of limits is essential and results in the form (8) where χTis evaluated in the absence of SOC. A similar formula\nwas derived by Kambersky [13] in another way where crucially the pre scription ξ= 0 did not become apparent. The\nformula is remarkable for describing the essence of a phenomenon a rising solely from SOC without the need to include\nSOC in the calculation.\nThe calculation of χTin a disordered system is still a very demanding problem. It may be app roached using the\nRPA of standard many-body theory or, less obviously, using time-d ependent LSDA. A diagrammatic RPA treatment\nofχTinvolves a sum of ladder diagrams and the first term, without an inter action line, corresponds to the mean-field\napproximation χ0\nT. The remaining terms constitute a vertex correction and we have s hown above that in a monatomic\nBravais lattice this vanishes. In a disordered system like an alloy, or a metal at finite temperature in a frozen phonon\npicture, this is not the case. However this great simplification persis ts if, in a very crude approximation, the system is\nreplaced, at the outset, by an effective medium with the full transla tional symmetry of the lattice but finite electron\nlifetime. We are then led to the Kambersky-like formula (7) for αwith a Lorentzian broadening of the delta-functions\ndetermined by relaxation times which may be dependent on spin and te mperature. This is the background to a\nrecent calculation of αin bulk Ni at room temperature [26] which is in reasonable agreement w ith experiment. A\nproper treatment of χTin a disordered material must deal simultaneously with the RPA verte x correction and any\nvertex corrections which arise in connection with methods of taking a configurational average, such as the coherent\npotential approximation (CPA). There is a small literature on this pr oblem as applied to χ−+, notχT, in a one-band\nmodel [29, 30]. Santos and Costa [30] find that for dilute non-magne tic impurities the RPA vertex correction is\nparticularly important. However as yet the many-body approach is far from being able to provide reliable results for\nαin real disordered materials. The time-dependent LSDA method see ms more promising. As shown below, it gives\na clear physical picture of the RPA vertex correction and separat es it from the configurational averaging problem.\nIn a FMR experiment the local magnetization vector sweeps out a co ne as it precesses around the Zeeman field\ndirection and in the presence of SOC the cone angle θ(rrr) is a function of position. In the time-dependent LSDA θ(rrr)\nsatisfies an integral equation whose solution is avoided in [14] by tak ing a spatially-independent averaged cone angle4\nθ(rrr). This approximation enforces a uniform precession, as occurs in t he absence of SOC, and removes the possibility\nof coupling between transverse and longitudinal susceptibilities. It is very reasonable for a monatomic Bravais lattice\nwhere the variation of θ(rrr) within a unit cell is largely an artificial consequence of the local appr oximation. In the\ntight-binding framework of [20] it would not be an approximation at all for a monatomic Bravais lattice. However, in\ncompounds, alloys and layered structures, variation of the cone a ngle between different types of atom and different\nlayers may be very important. In the many-body approach the ver tex correction in χTis the difference between\nthe fullχTand the mean-field approximation χ0\nT. Since we have seen that the mean-field approximation works well\nfor a homogeneous system, like a monatomic Bravais lattice, we conc lude that the vertex correction corresponds to\nthe effect of the spatial variation of the cone angle, which can be st udied with the LSDA approach. This will be\ndemonstrated explicitly in a forthcoming publication. This productive interplay between standard many-body theory\nand density-functional theory is quite unusual.\nI would like to acknowledge a useful exchange of e-mails with Filipe Guima res on the subject-matter of this paper.\n[1] L.D. Landau, E.M. Lifshitz and L.P. Pitaevski, Statistical Physics, Part 2 (Oxford: Pergamon 1980).\n[2] T.L. Gilbert, Phys. 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Commun. 8234 (2017).\n[13] V. Kambersky, Czech. J. Phys. B 261366 (1976).\n[14] I. Garate and A. MacDonald, Phys. Rev. B 79, 064403 (2009).\n[15] F.S.M. Guimares, J.R. Suckert, J. Chico, J. Bouaziz, M. dos Santos Dias and S. Lounis, arXiv:1807.11808v1.\n[16] D.J. Kim, H.C. Praddaude and B.B. Schwartz, Phys. Rev. L ett.23419 (1969).\n[17] D.J. Kim, B.B. Schwartz and H.C. Praddaude, Phys. Rev. B 7205 (1973).\n[18] U. von Barth and L. Hedin, J. Phys. C 51629 (1972).\n[19] A.R .Williams and U. von Barth, in Theory of the Inhomogeneous Electron Gas ed. S. Lundqvist and N.H. March (Plenum\n1983)\n[20] D.M. Edwards, J. Phys. Condens. Mater. 28086004 (2016).\n[21] S.S. Kalarickai, P. Krivosik, M. Wu, C.E. Patton, M.L. S chneider, P. Kabos, T.J. Silva and J.P. Nibarger, J. Appl. Ph ys.\n99093909 (2006).\n[22] A.T. Costa and R.B. Muniz, Phys. Rev. B 92014419 (2015).\n[23] A.T. Costa, R.B. Muniz, S. Lounis, A.B. Klautau and D.L. Mills, Phys. Rev. B 82014428 (2010).\n[24] E.M. Hankievicz, G. Vignale and Y. Tserkovnyak, Phys. R ev. B75174434 (2007).\n[25] D.M. Edwards and O. Wessely, J. Phys. Condens. Matter 21146002 (2009).\n[26] A. Umerski and D.M. Edwards, IOP Conf. Series: Journal o f Physics: Conf. Series 903012056 (2017).\n[27] C. Scheck, L. Cheng, I. Barsukov, Z. Frait and W.E. Baile y, Phys. Rev. Lett. 98117601 (2007).\n[28] P. He, X. Ma, J.W. Zhang, H.B. Zhao, G. L¨ upke, Z. Shi and S .M. Zhou, Phys. Rev. Lett. 110077203 (2013).\n[29] H. Yamada and M. Shimizu, J. Phys. Soc. Japan 311344 (1971).\n[30] E.B. Santos and A.T. Costa, J. Mag. Magn. Mat. 46969 (2019)."    },    {        "title": "2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf",        "content": "Dynamical Majorana Ising spin response in a topological superconductor-magnet\nhybrid by microwave irradiation\nYuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan.\n3Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: March 20, 2024)\nWe study a dynamical spin response of surface Majorana modes in a topological superconductor-\nmagnet hybrid under microwave irradiation. We find a method to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and\nthe microwave frequency. This reflects the topological nature of the Majorana modes, enhancing\nthe Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising\nspins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative\nspin devices.\nIntroduction.— The quest for Majoranas within matter\nstands as one of the principal challenges in the study of\ncondensed matter physics, more so in the field of quan-\ntum many-body systems [1]. The self-conjugate nature\nof Majoranas leads to peculiar electrical characteristics\nthat have been the subject of intensive research, both\ntheoretical and experimental [2]. In contrast, the focus of\nthis paper lies on the magnetic properties of Majoranas,\nspecifically the Majorana Ising spin [3–8]. A distinctive\ncharacteristic of Majorana modes, appearing as a surface\nstate in topological superconductors (TSC), is its exceed-\ningly strong anisotropy, which makes it behave as an Ising\nspin. In particular, this paper proposes a method to ex-\nplore the dynamical response of the Majorana Ising spin\nthrough the exchange interaction at the magnetic inter-\nface, achieved by coupling the TSC to a ferromagnet with\nferromagnetic resonance (FMR) (as shown in Fig.1 (a)).\nFMR modulation in a magnetic hybrid system has at-\ntracted much attention as a method to analyze spin ex-\ncitations in thin-film materials attached to magnetic ma-\nterials [9, 10]. Irradiating a magnetic material with mi-\ncrowaves induces dynamics of localized spin in magnetic\nmaterials, which can excite spins in adjacent thin-film\nmaterials via the magnetic proximity effect. This setup\nis called spin pumping, and has been studied intensively\nin the field of spintronics as a method of injecting spins\nthrough interfaces [11, 12]. Recent studies have theoret-\nically proposed that spin excitation can be characterized\nby FMR in hybrid systems of superconducting thin films\nand magnetic materials [13–18]. Therefore, it is expected\nto be possible to analyze the dynamics of surface Majo-\nrana Ising spins using FMR in hybrid systems.\nIn this work, we consider a TSC-ferromagnetic insula-\ntor (FI) hybrid system as shown in Fig. 1 (a). The FMR\nis induced by microwave irradiation on the FI. At the\ninterface between the TSC and the FI, the surface Ma-\n(b)\n(c)(a)\nFI~~\n~~Microwave\nϑS\nY, yX\nxZhdcHex\nTSC\n(d)\nhdchdc+δhα+δα\nHz\nFIG. 1. (a) The TSC-FI hybrid schematic reveals how,\nunder resonance frequency microwave irradiation, localized\nspins commence precessional motion, consequently initiating\nthe dynamical Majorana Ising spin response at the TSC inter-\nface. (b) In the TSC context, the liaison between a spin-up\nelectron and a spin-down hole with the surrounding sea of\nspin-triplet Cooper pairs drastically modulate their proper-\nties; notably, a spin-down hole can engage with a spin-triplet\nCooper pair, thereby inheriting a negative charge. (c) No-\ntably, spin-triplet Cooper pairs amass around holes and scat-\nter around electrons, thereby eroding the rigid distinction be-\ntween the two. (d) The interplay between the Majorana mode\nand the localized spin manipulates the FMR spectrum, trig-\ngering a frequency shift and linewidth broadening.\njorana modes interact with the localized spins in the FI.\nAs a result, the localized spin dynamics leads to the dy-\nnamical Majorana Ising spin response (DMISR), which\nmeans the Majorana Ising spin density is dynamically in-\nduced, and it is possible to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjust-\ning the external magnetic field angle and the microwave\nfrequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2  [cond-mat.mes-hall]  19 Mar 20242\nspin dynamics due to the interface interaction leads to a\nfrequency shift and a linewidth broadening, which reflect\nthe properties of the Majorana Ising spin dynamics. This\nwork proposes a setup for detecting Majorana modes and\npaves the way for the development of quantum comput-\ning and spin devices using Majoranas.\nModel.— We introduce a model Hamiltonian Hconsist-\ning of three terms\nH=HM+HFI+Hex. (1)\nThe first, second, and third terms respectively describe\nthe surface Majorana modes on the TSC surface, the bulk\nFI, and the proximity-induced exchange coupling. Our\nfocus is on energy regions significantly smaller than the\nbulk superconducting gap. This focus allows the spin ex-\ncitation in the TSC to be well described using the surface\nMajorana modes. The subsequent paragraphs provide\ndetailed explanations of each of these three terms.\nThe first terms HMdescribes the surface Majorana\nmodes,\nHM=1\n2Z\ndrψT(r)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r),(2)\nwhere r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant\nvelocity, and σ= (σx, σy, σz) are the Pauli matrices.\nThe two component Majorana field operator is given by\nψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization\naxis along the xaxis. The Majorana field operators sat-\nisfy the Majorana condition ψσ(r) =ψ†\nσ(r) and the an-\nticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′)\nwhere σ, σ′=→,←. We can derive HMby using surface-\nlocalized solutions of the BdG equation based on the bulk\nTSC Hamiltonian. The details of the derivation of HM\nare provided in the Supplemental Material [19].\nA notable feature of the surface Majorana modes is\nthat the spin density is Ising like, which we call the Majo-\nrana Ising spin [3–8]. The feature follows naturally from\nthe Majorana condition and the anticommutation rela-\ntion. The Majorana Ising spin density operator is given\nbys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r))\n(See the Supplemental Material for details [19]). The\nanisotropy of the Majorana Ising spin is the hallmark of\nthe surface Majorana modes on the TSC surface.\nThe second term HFIdescries the bulk FI and is given\nby the ferromagnetic Heisenberg model,\nHFI=− JX\n⟨n,m⟩Sn·Sm−ℏγhdcX\nnSZ\nn, (3)\nwhere J>0 is the exchange coupling constant, Snis the\nlocalized spin at site n,⟨n, m⟩means summation for near-\nest neighbors, γis the electron gyromagnetic ratio, and\nhdcis the static external magnetic field. We consider the\nspin dynamics of the localized spin under microwave irra-\ndiation, applying the spin-wave approximation. This al-\nlows the spin excitation to be described by a free bosonic\noperator, known as a magnon [20].The third term Hexrepresents the proximity exchange\ncoupling at the interface between the TSC and the FI,\nHex=−Z\ndrX\nnJ(r,rn)s(r)·Sn=HZ+HT,(4)\nHZ=−cosϑZ\ndrX\nnJ(r,rn)sz(r)SZ\nn, (5)\nHT=−sinϑZ\ndrX\nnJ(r,rn)sz(r)SX\nn, (6)\nwhere the angle ϑis shown in Fig. 1 (a). HZis the\ncoupling along the precession axis and HTis the coupling\nperpendicular to the precession axis. In our setup, HZ\nleads to gap opening of the energy spectrum of the surface\nMajorana modes and HTgives the DMISR under the\nmicrowave irradiation.\nDynamical Majorana Ising spin response.— We con-\nsider the microwave irradiation on the FI. The coupling\nbetween the localized spins and the microwave is given\nby\nV(t) =−ℏγhacX\nn\u0000\nSX\nncosωt−SY\nnsinωt\u0001\n,(7)\nwhere hacis the microwave amplitude, and ωis the mi-\ncrowave frequency. The microwave irradiation leads to\nthe precessional motion of the localized spin. When the\nfrequency of the precessional motion and the microwave\ncoincide, the FMR occurs. The FMR leads to the DMISR\nvia the exchange interaction. The DMISR is character-\nized by the dynamic spin susceptibility of the Majorana\nmodes, ˜ χzz(q, ω), defined as\n˜χzz(q, ω) :=Z\ndre−iq·rZ\ndtei(ω+i0)tχzz(r, t),(8)\nwhere χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩\nwith the interface area L2and the spin den-\nsity operator in the interaction picture, sz(r, t) =\nei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou-\npling, we consider configuration average and assume\n⟨P\nnJ(r,rn)⟩ave=J1, which means that HZis treated\nas a uniform Zeeman like interaction and the interface\nis specular [21]. Using eigenstates of Eq. (2) and after a\nstraightforward calculation, the uniform spin susceptibil-\nity is given by\n˜χzz(0, ω)\n=−X\nk,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ)\n2Ek,λ+ℏω+i0,\n→ −Z\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω+i0, (9)\nwhere |k, λ⟩is an eigenstate of HMwith eigenenergy\nEk,λ=λp\n(ℏvk)2+M2, (λ=±).M=J1Scosϑis\nthe Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3\ndistribution function, and D(E) is the density of states\ngiven by\nD(E) =L2\n2π(ℏv)2|E|θ(|E| − |M|), (10)\nwith the Heaviside step function θ(x). It is important to\nnote that the behavior of the uniform spin susceptibil-\nity is determined by the interband contribution, which is\nproportional to the Fermi distribution function, i.e., the\ncontribution of the occupied states. This mechanism is\nsimilar to the Van Vleck paramagnetism [22]. The con-\ntribution of the occupied states often plays a crucial role\nin topological responses [23].\nReplacing the localized spin operators with their statis-\ntical average values, we find the induced Majorana Ising\nspin density, to the first order of J1S, is given by\nZ\ndr⟨sz(r, t)⟩= ˜χzz\n0(0,0)J1Scosϑ\n+ Re[˜ χzz\n0(0, ω)]hac\nαhdcJ1Ssinϑsinωt, (11)\nwhere ˜ χzz\n0(0,0) is the spin susceptibility for M= 0. The\nfirst term originates from HZand gives a static spin den-\nsity, while the second term originates from HTand gives\na dynamic spin density. Figure 2 shows the induced Ising\nspin density as a function of time at several angles. As\nshown in Eq. (11), the Ising spin density consists of the\nstatic and dynamic components. The dynamic compo-\nnent is induced by the precessional motion of the local-\nized spin, which means one can induce the DMISR using\nthe dynamics of the localized spin.\nThe inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of\nϑat a fixed frequency. When the frequency ℏωis smaller\nthan the Majorana gap, Im˜ χzz(0, ω) is zero. Once the\nfrequency overcomes the Majorana gap, Im˜ χzz(0, ω) be-\ncomes finite. The implications of these behaviors are that\nif the magnon energy is smaller than the Majorana gap,\nthere is no energy dissipation due to the DMISR. How-\never, once the magnon energy exceeds the Majorana gap,\nfinite energy dissipation associated with the DMISR oc-\ncurs at the surface of the TSC. Therefore, one can toggle\nbetween dissipative and non-dissipative Majorana Ising\nspin dynamics by adjusting the precession axis angle and\nthe microwave frequency.\nFMR modulation.— The retarded component of the\nmagnon Green’s function is given by GR(rn, t) =\n−(i/ℏ)θ(t)⟨[S+\nn(t), S−\n0(0)]⟩with the interaction picture\nS±\nn(t) =eiHFIt/ℏS±\nne−iHFIt/ℏ. The FMR signal is char-\nacterized by the spectral function defined as\nA(q, ω) :=−1\nπIm\"X\nne−iq·rnZ\ndtei(ω+i0)tGR(rn, t)#\n.\n(12)\nSSImχzz(0, ω) ˜⟨s   z⟩\n2\n1ωtϑ\nFInon-dissipativenon-dissipativedissipativedissipativeTSC\nFITSC000.00.51.0\nπ/4\nπ/2\n0 π/4 π/20\nϑ2π\nπFIG. 2. The induced Ising spin density, with a unit\n˜χzz\n0(0,0)J1S, is presented as a function of ωtandϑ. The\nfrequency and temperature are set to ℏω/J1S= 1.5 and\nkBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is\nset to 0 .3. The static Majorana Ising spin density arises\nfrom HZ. When the precession axis deviates from the di-\nrection perpendicular to the interface, the precessional mo-\ntion of the localized spins results in the dynamical Majorana\nIsing spin response (DMISR). Energy dissipation due to the\nDMISR is zero for small angles ϑas the Majorana gap ex-\nceeds the magnon energy. However, once the magnon energy\novercomes the Majorana gap, the energy dissipation becomes\nfinite. Therefore, one can toggle between dissipative and non-\ndissipative DMISR by adjusting ϑ.\nFor uniform external force, the spectral function is given\nby\nA(0, ω) =2S\nℏ1\nπ(α+δα)ω\n[ω−γ(hdc+δh)]2+ [(α+δα)ω]2.\n(13)\nThe peak position and width of the FMR signal is given\nbyhdc+δhandα+δα, respectively. hdcandαcorre-\nspond to the peak position and the linewidth of the FMR\nsignal of the FI alone. δhandδαare the FMR modu-\nlations due to the exchange interaction HT. We treat\nHM+HFI+HZas an unperturbed Hamiltonian and HT\nas a perturbation. In this work, we assume the specular\ninterface, where the coupling J(r,rn) is approximated\nasDP\nn,n′J(r,rn)J(r′,rn′)E\nave=J2\n1. The dynamics\nof the localized spins in the FI is modulated due to the\ninteraction between the localized spins and the Majo-\nrana Ising spins. In our setup, the peak position and the\nlinewidth of the FMR signal are modulated and the FMR4\nmodulation is given by\nδh= sin2ϑSJ2\n1\n2NγℏRe˜χzz(0, ω), (14)\nδα= sin2ϑSJ2\n1\n2NℏωIm˜χzz(0, ω), (15)\nwhere Nis the total number of sites in the FI. These for-\nmulas were derived in the study of the FMR in magnetic\nmultilayer systems including superconductors. One can\nextract the spin property of the Majorana mode from the\ndata on δhandδα. Because of the Ising spin anisotropy,\nthe FMR modulation exhibits strong anisotropy, where\nthe FMR modulation is proportional to sin2ϑ.\nFigure 3 shows the FMR modulations (a) δαand (b)\nδh. The FMR modulation at a fixed frequency increases\nwith angle ϑand reaches a maximum at π/2, as can be\nread from Eqs. (14) and (15). When the angle ϑis fixed\nand the frequency ωis increased, δαbecomes finite above\na certain frequency at which the energy of the magnon\ncoincides with the Majorana gap. When ϑ < π/ 2 and\nℏω≈2M,δαlinearly increases as a function of ωjust\nabove the Majorana gap. The localized spin damping is\nenhanced when the magnon energy exceeds the Majorana\ngap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes\nandδαis proportional to ω/T. In the high frequency\nregion ℏω/J 1S≫1,δαconverges to its upper threshold.\nThe frequency shift δhis almost independent of ωand\nhas a finite value even in the Majorana gap. This behav-\nior is analogous to the interband contribution to the spin\nsusceptibility in strongly spin-orbit coupled band insula-\ntors, and is due to the fact that the effective Hamiltonian\nof the Majorana modes includes spin operators. It is im-\nportant to emphasize that although the Majorana modes\nhave spin degrees of freedom, only the zcomponent of the\nspin density operator is well defined. This is a hallmark\nof Majorana modes, which differs significantly from elec-\ntrons in ordinary solids. Note that δhis proportional to\nthe energy cutoff, which is introduced to converge energy\nintegral for Re˜ χzz(0, ω). The energy cutoff corresponds\nto the bulk superconducting gap, which is estimated as\n∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap-\nplicable in the frequency region below ℏω∼0.1[meV]\n(∼30[GHz]). In addition, we assume that Majorana gap\nis estimated to be J1S∼0.01[meV] ( ∼0.1[K]).\nDiscussion.— Comparing the present results with spin\npumping (SP) in a conventional metal-ferromagnet hy-\nbrid, the qualitative behaviors are quite different. In con-\nventional metals, spin accumulation occurs due to FMR.\nIn contrast, in the present system, no corresponding spin\naccumulation occurs due to the Ising anisotropy. Also, in\nthe present calculations, the proximity-induced exchange\ncoupling is assumed to be an isotropic Heisenberg-like\ncoupling. However, in general, the interface interaction\ncan also be anisotropic. Even in such a case, it is no qual-\nitative change in the case of ordinary metals, although a\n0.00.5\n(a) (b)\nϑℏω/J1S 0\nπ/4\nπ/2024\nϑℏω/J1S 0\nπ/4\nπ/2024δ  α δ  h10\n0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a)\nThe damping modulation δαonly becomes finite when the\nmagnon energy exceeds the Majorana gap; otherwise, it van-\nishes. This behavior corresponds to the energy dissipation of\nthe Majorana Ising spin. (b) The peak shift is finite, except\nforϑ= 0, and is almost independent of ω. This behavior\nresembles the spin response observed in strongly spin-orbit\ncoupled band insulators, where the interband contribution to\nspin susceptibility results in a finite spin response, even within\nthe energy gap.\ncorrection term due to anisotropy is added [24]. There-\nfore, the Ising anisotropy discussed in the present work\nis a property unique to the Majorana modes and can\ncharacterize the Majorana excitations.\nLet us comment on the universal nature of the toggling\nbetween non-dissipative and dissipative dynamical spin\nresponses observed in our study. Indeed, such toggling\nbecomes universally feasible when the microwave fre-\nquency and the energy gap are comparable, and when the\nHamiltonian and spin operators are non-commutative,\nindicating that spin is not a conserved quantity. The\nnon-commutativity can be attributed to the presence of\nspin-orbit couplings [25–27], and spin-triplet pair corre-\nlations [28].\nMicrowave irradiation leads to heating within the FI,\nso that thermally excited magnons due to the heating\ncould influence the DMISR. Phenomena resulting from\nthe heating, which can affect interface spin dynamics, in-\nclude the spin Seebeck effect (SSE) [29], where a spin\ncurrent is generated at the interface due to a tempera-\nture difference. In hybrid systems of normal metal and\nFI, methods to separate the inverse spin Hall voltage due\nto SP from other signals caused by heating have been\nwell studied [30]. Especially, it has been theoretically\nproposed that SP and SSE signals can be separated us-\ning a spin current noise measurement [24]. Moreover, SP\ncoherently excites specific modes, which qualitatively dif-\nfers from SSE induced by thermally excited magnons [14].\nTherefore, even if heating occurs in the FI in our setup,\nthe properties of Majorana Ising spins are expected to\nbe captured. Details of the heating effect on the DMISR\nwill be examined in the near future.\nWe also mention the experimental feasibility of our the-\noretical proposals. As we have already explained, the\nFMR modulation is a very sensitive spin probe. Indeed,\nthe FMR modulation by surface states of 3D topological5\ninsulators [31] and graphene [32–36] has been reported\nexperimentally. Therefore, we expect that the enhanced\nGilbert damping due to Majorana Ising spin can be ob-\nservable in our setup when the thickness of the ferromag-\nnetic insulator is sufficiently thin.\nFinally, it is pertinent to mention the potential candi-\ndate materials where surface Majorana Ising spins could\nbe detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39],\nSrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p-\nwave superconducting state and theoretically can host\nsurface Majorana Ising spins. Recent NMR measure-\nments indicate that UTe 2could be a bulk p-wave su-\nperconductor in the Balian-Werthamer state [41], which\nhosts the surface Majorana Ising spins with the per-\npendicular Ising anisotropy, as considered in this work.\nAxBi2Se3(A= Cu, Sr, Nb) is considered to possess in-\nplane Ising anisotropy [8], differing from the perpendic-\nular Ising anisotropy explored in this work. Therefore,\nwe expect that it exhibits anisotropy different from that\ndemonstrated in this work.\nConclusion.— We present herein a study of the spin\ndynamics in a topological superconductor (TSC)-magnet\nhybrid. Ferromagnetic resonance under microwave irra-\ndiation leads to the dynamically induced Majorana Ising\nspin density on the TSC surface. One can toggle between\ndissipative and non-dissipative Majorana Ising spin dy-\nnamics by adjusting the external magnetic field angle and\nthe microwave frequency. Therefore, our setup provides\na platform to detect and control Majorana excitations.\nWe expect that our results provide insights toward the\ndevelopment of future quantum computing and spintron-\nics devices using Majorana excitations.\nAcknowledgments.— The authors are grateful to R.\nShindou for valuable discussions. This work is partially\nsupported by the Priority Program of Chinese Academy\nof Sciences, Grant No. XDB28000000. We acknowl-\nedge JSPS KAKENHI for Grants (Nos. JP20K03835,\nJP21H01800, JP21H04565, and JP23H01839).\nSUPPLEMENTAL MATERIAL\nSurface Majorana modes\nIn this section, we describe the procedure for deriv-\ning the effective Hamiltonian of the surface Majorana\nmodes. We start with the bulk Hamiltonian of a three-\ndimensional topological superconductor. Based on the\nbulk Hamiltonian, we solve the BdG equation to demon-\nstrate the existence of a surface-localized solution. Us-\ning this solution, we expand the field operator and show\nthat it satisfies the Majorana condition when the bulk\nexcitations are neglected. As a result, on energy scales\nmuch smaller than the bulk superconducting gap, the\nlow-energy excitations are described by surface-localized\nMajorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three-\ndimensional coordinates and r∥for two-dimensional ones\nin the Supplemental Material.\nWe start with the mean-field Hamiltonian given by\nHSC=1\n2Z\ndrΨ†\nBdG(r)HBdGΨBdG(r), (16)\nwithr= (x, y, z ). We consider the Balian-Werthamer\n(BW) state, in which the pair potential is given by\n∆ˆk=∆\nkF\u0010\nˆk·σ\u0011\niσywith the bulk superconducting gap\n∆. Here, we do not discuss the microscopic origin of the\npair correlation leading to the BW state. As a result, the\nBdG Hamiltonian HBdGis given by\nHBdG=\nεˆk−EF 0 −∆\nkFˆk−∆\nkFˆkx\n0 εˆk−EF∆\nkFˆkx∆\nkFˆk+\n−∆\nkFˆk+∆\nkFˆkx−εˆk+EF 0\n∆\nkFˆkx∆\nkFˆk− 0 −εˆk+EF\n,\n(17)\nwith ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2\n2m. The four\ncomponent Nambu spinor ΨBdG(r) is given by\nΨBdG(r) :=\nΨ→(r)\nΨ←(r)\nΨ†\n→(r)\nΨ†\n←(r)\n, (18)\nwith the spin quantization axis along the xaxis. The\nmatrices of the spin operators are represented as\nσx=\u00121 0\n0−1\u0013\n, (19)\nσy=\u0012\n0 1\n1 0\u0013\n, (20)\nσz=\u00120−i\ni0\u0013\n. (21)\nThe fermion field operators satisfy the anticommutation\nrelations\n{Ψσ(r),Ψσ′(r′)}= 0, (22)\n{Ψσ(r),Ψ†\nσ′(r′)}=δσσ′δ(r−r′), (23)\nwith the spin indices σ, σ′=→,←.\nTo diagonalize the BdG Hamiltonian, we solve the BdG\nequation given by\nHBdGΦ(r) =EΦ(r). (24)\nWe assume that a solution is written as\nΦ(r) =eik∥·r∥f(z)\nu→\nu←\nv→\nv←\n, (25)6\nwithk∥= (kx, ky) and r∥= (x, y). If we set the four\ncomponents vector to satisfy the following equation (Ma-\njorana condition)\n\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n\nu→\nu←\nv→\nv←\n=±\nu→\nu←\nv→\nv←\n, (26)\nwe can obtain a surface-localized solution. If we take a\npositive (negative) sign, we obtain a solution localized\non the top surface (bottom surface). As we will consider\nsolutions localized on the bottom surface below, we take\na negative sign. Finally, we obtain the normalized eigen-\nvectors of the BdG equation given by\nΦλ,k∥(r) =eik∥·r∥\n√\nL2fk∥(z)uλ,k∥, (27)\nwith\nfk∥(z) =Nk∥sin(k⊥z)e−κz, (28)\nNk∥=s\n4κ(k2\n⊥+κ2)\nk2\n⊥, (29)\nκ=m∆\nℏ2kF, (30)\nk⊥=q\nk2\nF−k2\n∥−κ2, (31)\nand\nu+,k∥=\nu+,→k∥\nu+,←k∥\nv+,→k∥\nv+,←k∥\n=1√\n2\nsinϕk∥+π/2\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\n,(32)\nu−,k∥=\nu−,→k∥\nu−,←k∥\nv−,→k∥\nv−,←k∥\n=1√\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\nsinϕk∥+π/2\n2\n.(33)\nThe eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can\nshow that the eigenvectors satisfy\nu−,−k∥=u+,k∥. (34)\nConsequently, the field operator is expanded as\nΨBdG(r) =X\nk∥\u0012\nγk∥eik∥·r∥\n√\nL2+γ†\nk∥e−ik∥·r∥\n√\nL2\u0013\n×fk∥(z)u+,k∥+ (bulk modes) ,(35)\nwhere γk∥(γ†\nk∥) is the quasiparticle creation (annihila-\ntion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk\nmodes and performing the integration in the z-direction,\nwe obtain the effective Hamiltonian for the surface states\nHM=1\n2Z\ndr∥ψT(r∥)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r∥),(36)\nwhere v= ∆/ℏkFand we introduced the two component\nMajorana field operator\nψ(r∥) =\u0012ψ→(r∥)\nψ←(r∥)\u0013\n, (37)\nsatisfying the Majorana condition\nψσ(r∥) =ψ†\nσ(r∥), (38)\nand the anticommutation relation\nn\nψσ(r∥), ψσ′(r′\n∥)o\n=δσσ′δ(r∥−r′\n∥). (39)\nThe spin density operator of the Majorana mode is\ngiven by\ns(r∥) =ψ†(r∥)σ\n2ψ(r∥). (40)\nThexcomponent is given by\nsx(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00121/2 0\n0−1/2\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ→(r∥)−ψ†\n←(r∥)ψ←(r∥)\u0003\n=1\n2\u0002\nψ2\n→(r∥)−ψ2\n←(r∥)\u0003\n= 0. (41)\nIn a similar manner, the yandzcomponents are given\nby\nsy(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120 1/2\n1/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ←(r∥) +ψ†\n←(r∥)ψ→(r∥)\u0003\n=1\n2\b\nψ→(r∥), ψ←(r∥)\t\n= 0, (42)\nand\nsz(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120−i/2\ni/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=−i\n2\u0000\nψ†\n→(r∥)ψ←(r∥)−ψ†\n←(r∥)ψ→(r∥)\u0001\n=−iψ→(r∥)ψ←(r∥), (43)\nrespectively. As a result, the spin density operator is\ngiven by\ns(r∥) =\u0000\n0,0,−iψ→(r∥)ψ←(r∥)\u0001\n. (44)\nOne can see that the spin density of the Majorana mode\nis Ising like.7\nMajorana Ising spin dynamics\nIn this section, we calculate the Ising spin density in-\nduced on the TSC surface by the proximity coupling Hex.\nHexconsists of two terms, HZandHT.HZleads to the\nstatic spin density and HTleads to the dynamic spin\ndensity. First, we calculate the static spin density. Next,\nwe calculate the dynamic spin density.\nThe total spin density operator is given by\nsz\ntot=Z\ndr∥sz(r∥). (45)\nThe statistical average of the static spin density is calcu-\nlated as\n⟨sz\ntot⟩=−X\nk∥M\n2Ek∥\u0002\nf(Ek∥)−f(−Ek∥)\u0003\n→ −\u0012L\n2πℏv\u00132Z∆\nMEdEZ2π\n0dϕM\n2E[f(E)−f(−E)]\n=−Z∆\n0dED (E)f(E)−f(−E)\n2EM. (46)\nAt the zero temperature limit T→0, the static spin\ndensity is given by\n⟨sz\ntot⟩=1\n2L2\n2π(ℏv)2(∆−M)M≈˜χzz\n0(0,0)M, (47)\nwhere ˜ χzz\n0(0,0) = D(∆)/2 and we used ∆ ≫M.\nThe dynamic spin density is given by the perturbative\nforce\nHT(t) =Z\ndr∥sz(r∥)F(r∥, t), (48)\nwhere F(r∥, t) is given by\nF(r∥, t) =−sinϑX\nnJ(r∥,rn)\nSX\nn(t)\u000b\n≈ −sinϑJ1Sγhacp\n(ω−γhdc)2+α2ω2cosωt\n=:Fcosωt. (49)\nThe time dependent statistical average of the Ising spin\ndensity, to the first order of J1S, is given by\nZ\ndr∥\nsz(r∥, t)\u000b\n=Z\ndr∥Z\ndr′\n∥Z\ndt′χzz(r∥−r′\n∥, t′)F(r′\n∥, t−t′)\n= Re\u0002\n˜χzz(0, ω)Fe−iωt\u0003\n≈Re[˜χzz\n0(0, ω)]Fcosωt, (50)\nwhere we used Re˜ χzz\n0(0, ω)≫Im˜χzz\n0(0, ω). The real part\nof ˜χzz(0, ω) is given by\nRe˜χzz(0, ω) =−PZ\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω,\n(51)where Pmeans the principal value. When the integrand\nis expanded with respect to ω, the lowest order correc-\ntion term becomes quadratic in ω. In the frequency range\nconsidered in this work, this correction term is signifi-\ncantly smaller compared to the static spin susceptibility\nRe˜χzz(0,0). Therefore, the spin susceptibility exhibits\nalmost no frequency dependence and remains constant\nas a function of ω. The imaginary part of ˜ χzz(0, ω) is\ngiven by\nIm˜χzz(0, ω)\n=πD(ℏω/2)(ℏω/2)2−M2\n2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)].\n(52)\nFMR modulation due to the proximity exchange\ncoupling\nIn this section, we provide a brief explanation for the\nderivation of the FMR modulations δhandδα. The FMR\nmodulations can be determined from the retarded com-\nponent of the magnon Green’s function, which is given\nby\n˜GR(k, ω) =2S/ℏ\nω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53)\nwhere we introduce the Gilbert damping constant αphe-\nnomenologically. In the second-order perturbation calcu-\nlation with respect to HT, the self-energy is given by\nΣR(k, ω) =−\u0012sinϑ\n2\u00132X\nq∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54)\nwhere ˜J(q∥,0) is given by\n˜J(q∥,k) =1\nL2√\nNZ\ndr∥X\nnJ(r∥,rn)ei(q∥·r∥+k·rn)\n(55)\nThe pole of ˜GR(k, ω) signifies the FMR modulations,\nincluding both the frequency shift and the enhanced\nGilbert damping. These are given by\nδh=2S\nγℏReΣR(0, ω), δα =−2S\nℏωImΣR(0, ω).(56)\nFrom the above equations and Eq. (54), it is apparent\nthat FMR modulations provide information regarding\nboth the properties of the interface coupling and the dy-\nnamic spin susceptibility of the Majorana modes.\nThe form of matrix element ˜J(q∥,0) depends on the\ndetails of the interface. In this work, we assume the\nspecular interface. |˜J(q∥,0)|2is given by\n|˜J(q∥,0)|2=J2\n1\nNδq∥,0. (57)8\nUsing Eq. (57), the self-energy for the uniform magnon\nmode is given by\nΣR(0, ω) =−\u0012sinϑ\n2\u00132J2\n1\nN˜χzz(0, ω). (58)\n[1] F. Wilczek, Nat. 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Jpn. 92, 063701 (2023)."    },    {        "title": "1209.3120v1.Skyrmion_Dynamics_in_Multiferroic_Insulator.pdf",        "content": "arXiv:1209.3120v1  [cond-mat.str-el]  14 Sep 2012Skyrmion Dynamics in Multiferroic Insulator\nYe-Hua Liu,1You-Quan Li,1and Jung Hoon Han2,3,∗\n1Zhejiang Institute of Modern Physics and Department of Physi cs,\nZhejiang University, Hangzhou 310027, People’s Republic of China\n2Department of Physics and BK21 Physics Research Division,\nSungkyunkwan University, Suwon 440-746, Korea\n3Asia Pacific Center for Theoretical Physics, Pohang, Gyeong buk 790-784, Korea\n(Dated: October 31, 2018)\nRecent discovery of Skyrmion crystal phase in insulating mu ltiferroic compound Cu 2OSeO 3calls\nfor new ways and ideas to manipulate the Skyrmions in the abse nce of spin transfer torque from\nthe conduction electrons. It is shown here that the position -dependent electric field, pointed along\nthe direction of the average induced dipole moment of the Sky rmion, can induce the Hall motion\nof Skyrmion with its velocity orthogonal to the field gradien t. Finite Gilbert damping produces\nlongitudinal motion. We find a rich variety of resonance mode s excited by a.c. electric field.\nPACS numbers: 75.85.+t, 75.70.Kw, 76.50.+g\nSkyrmions are increasingly becoming commonplace\nsightings among spiral magnets including the metallic\nB20 compounds[1–5] and most recently, in a multiferroic\ninsulator Cu 2OSeO3[6]. Both species of compounds dis-\nplaysimilarthickness-dependentphasediagrams[5,6] de-\nspitetheircompletelydifferentelectricalproperties,high-\nlighting the generality of the Skyrmion phase in spiral\nmagnets. Along with the ubiquity of Skyrmion matter\ncomes the challenge of finding means to control and ma-\nnipulatethem, inadevice-orientedmannerakintoefforts\nin spintronics community to control the domain wall and\nvortex motion by electrical current. Spin transfer torque\n(STT) is a powerful means to induce fast domain wall\nmotion in metallic magnets[7, 8]. Indeed, current-driven\nSkyrmion rotation[9] and collective drift[10], originating\nfrom STT, have been demonstrated in the case of spiral\nmagnets. Theory of current-induced Skyrmion dynam-\nics has been worked out in Refs. [11, 12]. In insulating\ncompounds such as Cu 2OSeO3, however, the STT-driven\nmechanism does not work due to the lack of conduction\nelectrons.\nAs with other magnetically driven multiferroic\ncompounds[13], spiral magnetic order in Cu 2OSeO3is\naccompanied by finite electric dipole moment. Recent\nwork by Seki et al.[14] further confirmed the mecha-\nnism of electric dipole moment induction in Cu 2OSeO3\nto be the so-called pd-hybridization[15–17]. In short, the\npd-hybridization mechanism claims the dipole moment\nPijfor every oxygen-TM(transition metal) bond propor-\ntional to ( Si·ˆeij)2ˆeijwhereiandjstand for TM and\noxygen sites, respectively, and ˆ eijis the unit vector con-\nnecting them. Carefully summing up the contributions\nof such terms over a unit cell consisting of many TM-\nO bonds, Seki et al.were able to deduce the dipole\nmoment distribution associated with a given Skyrmionic\nspin configuration[14]. It is interesting to note that the\nnumerical procedure performed by Seki et al.is pre-\ncisely the coarse-graining procedure which, in the text-book sense of statistical mechanics, is tantamount to the\nGinzburg-Landau theory of order parameters. Indeed we\ncan show that Seki et al.’s result for the dipole moment\ndistribution is faithfully reproduced by the assumption\nthat the local dipole moment Piis related to the local\nmagnetization Siby\nPi=λ(Sy\niSz\ni,Sz\niSx\ni,Sx\niSy\ni) (1)\nwith some coupling λ. A similar expression was pro-\nposed earlier in Refs. [18, 19] as the GL theory\nof Ba 2CoGe2O7[20], another known pd-hybridization-\noriginated multiferroic material with cubic crystal struc-\nture. Each site icorresponds to one cubic unit cell of\nCu2OSeO3with linear dimension a∼8.9˚A, and we have\nnormalized Sito have unit magnitude. The dimension of\nthe coupling constant is therefore [ λ] = C·m.\nHaving obtained the proper coupling between dipole\nmoment and the magnetizaiton vector in Cu 2OSeO3one\ncan readily proceed to study the spin dynamics by solv-\ning Landau-Lifshitz-Gilbert (LLG) equation. Very small\nvalues of Gilbert damping parameter are assumed in the\nsimulation as we are dealing with an insulating magnet.\nA new, critical element in the simulation is the term aris-\ning from the dipolar coupling\nHME=−/summationdisplay\niPi·Ei=−λ\n2/summationdisplay\niSi\n0Ez\niEy\ni\nEz\ni0Ex\ni\nEy\niEx\ni0\nSi,(2)\nwhere we have used the magneto-electric coupling ex-\npression in Eq. (1). In essence this is a field-dependent\n(voltage-dependent) magnetic anisotropy term. The to-\ntal Hamiltonian for spin is given by H=HHDM+\nHME, whereHHDMconsists of the Heisenberg and the\nDzyaloshinskii-Moriya (DM) exchange and a Zeeman\nfield term. Earlier theoretical studies showed HHDMto\nstabilize the Skyrmion phase[1, 21–24].2\nTwo field orientations can be chosen independently in\nexperiments performed on insulating magnets. First, the\ndirection of magnetic field Bdetermines the plane, or-\nthogonal to B, in which Skyrmions form. Second, the\nelectric field Ecan be applied to couple to the induced\ndipole moment of the Skyrmion and used as a “knob” to\nmove it around. Three field directions used in Ref. [14]\nand the induced dipole moment in eachcase areclassified\nas (I)B/bardbl[001],P= 0, (II) B/bardbl[110],P/bardbl[001], and\n(III)B/bardbl[111],P/bardbl[111]. One can rotate the spin axis\nappearing in Eq. (1) accordingly so that the z-direction\ncoincides with the magnetic field orientation in a given\nsetup and the x-direction with the crystallographic[ 110].\nIn each of the cases listed above we obtain the magneto-\nelectric coupling, after the rotation,\nH(I)\nME=−λ\n2/summationdisplay\niEi([Sy\ni]2−[Sx\ni]2),\nH(II)\nME=−λ\n2/summationdisplay\niEi([Sz\ni]2−[Sx\ni]2),\nH(III)\nME=−λ\n2√\n3/summationdisplay\niEi(3[Sz\ni]2−1).(3)\nIn cases (II) and (III) the E-field is chosen parallel to the\ninduced dipole moment P,Ei=EiˆP, to maximize the\neffect of dipolar coupling. In case (I) where there is no\nnet dipole moment for Skyrmions we chose E/bardbl[001] to\narrive at a simple magneto-electric coupling form shown\nabove.\nSuppose now that the E-field variation is sufficiently\nslow on the scale of the lattice constant ato allow the\nwriting of the continuum energy,\nHME=−λd\na/integraldisplay\nd2rE(r)ρD(r). (4)\nIt is assumed that all variables behave identically along\nthe thickness direction, oflength d. The “dipolarcharge”\ndensityρD(r)couplestotheelectricfield E(r)inthesame\nwayasthe conventionalelectric chargedoes tothe poten-\ntial field in electromagnetism. The analogy is also useful\nin thinking about the Skyrmion dynamics under the spa-\ntially varying E-field as we will show. The continuum\nform of dipolar charge density in Eq. 4 is\nρ(I)\nD(ri) =1\n2a2([Sy\ni]2−[Sx\ni]2),\nρ(II)\nD(ri) =1\n2a2([Sz\ni]2−[Sx\ni]2−1),\nρ(III)\nD(ri) =√\n3\n2a2([Sz\ni]2−1). (5)\nDivision by the unit cell area a2ensures that ρD(r) has\nthe dimension of areal density. Values for the ferromag-\nnetic case, Sz\ni= 1, is subtracted in writing down the def-\ninition (5) in order to isolate the motion of the Skyrmion\nFIG. 1: (color online) (a) Skyrmion configuration and (b)-(d )\nthe corresponding distribution of dipolar charge density f or\nthree magnetic field orientations as in Ref. 14. (b) B/bardbl[001]\n(c)B/bardbl[110] (d) B/bardbl[111]. For each case, electric field is\nchosen as E/bardbl[001],E/bardbl[001] and E/bardbl[111], respectively. See\ntext for the definition of dipolar charge density. As schemat i-\ncally depicted in (a), the Skyrmion executes a Hall motion in\nresponse to electric field gradient.\nrelative to the ferromagnetic background. Due to the\nsubtraction, the dipolar charge is no longer equivalent to\nthe dipole moment of the Skyrmion. The distribution of\ndipolar charge density for the Skyrmion spin configura-\ntion in the three cases are plotted in Fig. 1. In case (I)\nthe total dipolar charge is zero. In cases (II) and (III)\nthe net dipolar charges are both negative with the re-\nlation,Q(II)\nD/Q(III)\nD=√\n3/2, where QD, of order unity,\nis obtained by integrating ρD(r) over the space of one\nSkyrmion and divide the result by the number of spins\nNSkinside the Skyrmion. If the field variation is slow on\nthe scale of the Skyrmion, then the point-particle limit is\nreached by writing ρD(r) =QDNSk/summationtext\njδ(r−rj) whererj\nspans the Skyrmion positions, and identical charge QDis\nassumed for all the Skyrmions. We arrive at the “poten-\ntial energy” of the collection of Skyrmion particles,\nHME=−λQDNSkd\na/summationdisplay\njE(rj). (6)\nA force acting on the Skyrmion will be given as the gra-\ndientFi=−∇iHME. Inter-Skyrmion interaction is ig-\nnored.\nThe response of Skyrmions to a given force, on the\nother hand, is that of an electric charge in strong\nmagnetic field, embodied in the Berry phase action3\n(−2πS¯hQSkd/a3)/summationtext\nj/integraltext\ndt(rj×˙rj)·ˆz, whereQSkis the\nquantized Skyrmion charge[12, 25], and Sis the size of\nspin. Equation of motion follows from the combination\nof the Berry phase action and Eq. (6),\nvj=λ\n4πS¯ha2NSkQD\nQSkˆz×∇jE(rj), (7)\nwherevjis thej-th Skyrmion velocity. Typical Hall ve-\nlocity can be estimated by replacing |∇E|with ∆E/lSk,\nwhere ∆Eis the difference in the field strength between\nthe left and the right edge of the Skyrmion and lSkis its\ndiameter. Taking a2NSk∼l2\nSkwe find the velocity\nλlSk\n4πS¯h∆E∼10−6∆E[m2/V·s], (8)\nwhich gives the estimated drift velocity of 1 mm/s for the\nfield strength difference of 103V/m across the Skyrmion.\nExperimental input parameters of lSk= 10−7m, and\nλ= 10−32C·m were taken from Ref. [14] in arriving at\nthe estimation, as well as the dipolar and the Skyrmion\nchargesQD≈ −1 andQSk=−1. We may estimate the\nmaximum allowed drift velocity by equating the dipolar\nenergy difference λ∆Eacross the Skyrmion to the ex-\nchange energy J, also corresponding to the formation en-\nergy of one Skyrmion[24]. The maximum expected veloc-\nity thus obtained is enormous, ∼104m/s forJ∼1meV,\nimplying that with the right engineering one can achieve\nrather high Hall velocity of the Skyrmion. In an encour-\naging step forward, electric field control of the Skyrmion\nlattice orientation in the Cu 2OSeO3crystal was recently\ndemonstrated[26].\nResults of LLG simulation is discussed next. To start,\na sinusoidal field configuration Ei=E0sin(2πxi/Lx) is\nimposed on a rectangular Lx×Lysimulation lattice with\nLxmuch larger than the Skyrmion size. In the absence\nof Gilbert damping, a single Skyrmion placed in such\nan environment moved along the “equi-potential line” in\nthey-direction as expected from the guiding-center dy-\nnamics of Eq. (7). In cases (II) and (III) where the\ndipolar charges are nonzero the velocity of the Skyrmion\ndrift is found to be proportional to their respective dipo-\nlar charges QDas shown in Fig. 2. The drift velocity\ndecreased continuously as we reduced the field gradient,\nobeying the relation (7) down to the zero velocity limit.\nThe dipolar charge is zero in case (I), and indeed the\nSkyrmion remains stationary for sufficiently smooth E-\nfield gradient. Even for this case, Skyrmions can move\nfor field gradient modulations taking place on the length\nscale comparable to the Skyrmion radius, for the reason\nthat the forces acting on the positive dipolar charge den-\nsity blobs (red in Fig. 1(a)) are not completely canceled\nby those on the negative dipolar charge density blobs\n(blue in Fig. 1(a)) for sufficiently rapid variations of thefield strength gradient. A small but non-zero drift veloc-\nityensues, asshowninFig. 2. Longitudinalmotionalong\nthe field gradient begins to develop with finite Gilbert\ndamping, driving the Skyrmion center to the position of\nlowest “potential energy” E(r). For the Skyrmion lat-\ntice, imposing a uniform field gradient across the whole\nlattice may be too demanding experimentally, unless the\nmagnetic crystal is cut in the form of a narrow strip the\nwidth of which is comparable to a few Skyrmion radii.\nIn this case we indeed observe the constant drift of the\nSkyrmions along the length of the strip in response to\nthe field gradient across it. The drift speed is still pro-\nportional to the field gradient, but about an order of\nmagnitude less than that of an isolated Skyrmion under\nthe same field gradient. We observed the excitation of\nbreathing modes of Skyrmions when subject to a field\ngradient, and speculate that such breathing mode may\ninterfere with the drift motion as the Skyrmions become\nclosed-packed.\n0 500 1000 1500 2000−30−25−20−15−10−505\nty(ab. unit)\n  \nv(I)\nHall=−7×10−4\nv(II)\nHall=−1.32×10−2\nv(III)\nHall=−1.49×10−2\nv(II)\nHall\nv(III)\nHall≈Q(II)\nD\nQ(III)\nD=√\n3\n2case (I)  : B||[001]\ncase (II) : B||[110]\ncase (III): B||[111]\nFIG. 2: (color online) Skyrmion position versus time for cas es\n(I) through (III) for sinusoidal electric field modulation ( see\ntext) with the Skyrmion center placed at the maximum field\ngradient position. The average Hall velocities (in arbitra ry\nunits) in cases (II) and (III) indicated in the figure are ap-\nproximately proportional to the respective dipolar charge s, in\nagreement with Eq. (7). A small velocity remains in case\n(I) due to imperfect cancelation of forces across the dipola r\ncharge profile.\nSeveral movie files are included in the supplementary\ninformation. II.gif and III.gif give Skyrmion motion for\nEi=E0sin(2πxi/Lx) onLx×Ly= 66×66 lattice for\nmagneto-electric couplings (II) and (III) in Eq. (3). III-\nGilbert.gif gives the same E-field as III.gif, with finite\nGilbert damping α= 0.2. I.gif describes the case (I)\nwhere the average dipolar charge is zero, with a rapidly\nvarying electric field Ei=E0sin(2πxi/λx) andλxcom-\nparable to the Skyrmion radius. The case of a narrow\nstrip with the field gradient across is shown in strip.gif.\nMochizuki’s recent simulation[27] revealed that inter-4\nnal motion of Skyrmions can be excited with the uniform\na.c. magnetic field. Some of his predictions were con-\nfirmed by the recent microwave measurement[29]. Here\nwe show that uniform a.c. electric field can also ex-\ncite several internal modes due to the magneto-electric\ncoupling. Time-localized electric field pulse was applied\nin the LLG simulation and the temporal response χ(t)\nwas Fourier analyzed, where the response function χ(t)\nrefersto(1 /2)/summationtext\ni([Sy\ni(t)]2−[Sx\ni(t)]2), (1/2)/summationtext\ni([Sz\ni(t)]2−\n[Sx\ni(t)]2), and (√\n3/2)/summationtext\ni[Sz\ni(t)]2for cases (I) through\n(III), respectively. (In Mochizuki’s work, the response\nfunction was the component of total spin along the a.c.\nmagnetic field direction.)\nIn case (I), the uniform electric field perturbs the ini-\ntial cylindrical symmetry ofSkyrmion spin profileso that/summationtext\ni([Sx\ni(t)]2−[Sy\ni(t)]2) becomes non-zero and the over-\nall shape becomes elliptical. The axes of the ellipse then\nrotates counter-clockwise about the Skyrmion center of\nmass as illustrated in supplementary figure, E-mode.gif.\nThere are two additional modes of higher energies with\nbroken cylindrical symmetry in case (I), labeled X1 and\nX2 in Fig. 3 and included as X1-mode.gif and X2-\nmode.gif in the supplementary. The rotational direction\nof the X1-mode is the same as in E-mode, while it is the\nopposite for X2-mode.\nAs in Ref. [27], we find sharply defined breathing\nmodesin cases(II) and(III) atthe appropriateresonance\nfrequency ω, in fact the same frequency at which the a.c.\nmagnetic field excites the breathing mode. The verti-\ncal dashed line in Fig. 3 indicates the common breath-\ning mode frequency. Movie file B1-mode.gif shows the\nbreathing mode in case (III). Additional, higher energy\nB2-mode (B2-mode.gif) was found in cases (II) and (III),\nwhich is the radial mode with one node, whereas the B1\nmode is nodeless.\nIn addition to the two breathing modes, E-mode and\nthe two X-modes are excited in case (II) as well due\nto the partly in-plane nature of the spin perturbation,\n−(λE(t)/2)/summationtext\ni([Sz\ni(t)]2−[Sx\ni(t)]2). In contrast, case\n(III), where the perturbation −(√\n3λE(t)/2)/summationtext\ni[Sz\ni(t)]2\nis purely out-of-plane, one only finds the B-modes. As\na result, case (I) and (III) have no common resonance\nmodes or peaks, while case (II) has all the peaks (though\nX1 and X2 peaks are small). Compared to the magnetic\nfield-induced resonances, a richer variety of modes are\nexcited by a.c. electric field. In particular, the E-mode\nhas lower excitation energy than the B-mode and has a\nsharp resonance feature, which should make its detection\na relatively straightforward task. Full analytic solution\nof the excited modes[28] will be given later.\nIn summary, motivated by the recent discov-\nery of magneto-electric material Cu 2OSeO3exhibiting\nSkyrmion lattice phase, we have outlined the theory of\nSkyrmion dynamics in such materials. Electric field gra-\ndient is identified as the source of Skyrmion Hall motion.00.20.40.60.81Imχ(ω) (ab. unit)\n  \nB1\nR1\nR2(a)B||z,Bω||x,y\nB||z,Bω||z\n00.05 0.10.15 0.20.25 0.30.3500.20.40.60.81\nωImχ(ω) (ab. unit)\n  \nE\nX1X2B1\nB2(b)case (I)  : B||[001],Eω||[001]\ncase (II) : B||[110],Eω||[001]\ncase (III): B||[111],Eω||[111]\nFIG. 3: (color online) (a) Absorption spectra for a.c. unifo rm\nmagnetic field as in Mochizuki’s work (reproduced here for\ncomparison). (b) Absorption spectra for a.c. uniform elect ric\nfield in cases (I) through (III). In case (I) where there is no\nnet dipolar charge we find three low-energy modes E, X1, and\nX2. For case (III) where the dipolar charge is finite we find\nB1 and B2 radial modes excited. Case (II) exhibits all five\nmodes. Detailed description of each mode is given in the text .\nSeveralresonantexcitationsbya.c. electricfieldareiden-\ntified.\nJ. H. H. is supported by NRF grant (No. 2010-\n0008529, 2011-0015631). Y. Q. L. is supported by NSFC\n(Grant No. 11074216). J. H. H. acknowledges earlier\ncollaboration with N. Nagaosa, Youngbin Tchoe, and J.\nZang on a related model and informative discussion with\nY. Tokura.\n∗Electronic address: hanjh@skku.edu\n[1] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[2] W. M¨ unzer, A. Neubauer, T. Adams, S. M¨ uhlbauer, C.\nFranz, F. Jonietz, R. Georgii, P. B¨ oni, B. Pedersen, M.\nSchmidt, A. Rosch, and C. Pfleiderer, Phys. Rev. B 81,\n041203(R) (2010).\n[3] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).\n[4] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nature\nMat.10, 106 (2011).\n[5] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K.\nOhoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and\nY. Tokura, Phys. Rev. Lett. 106, 156603 (2011).\n[6] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science5\n336, 198 (2012).\n[7] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[8] D. C. Ralph and M. Stiles, J. Magn. Mag. Mat. 320,\n1190 (2008).\n[9] F. Jonietz, S. M¨ uhlbauer, C. Pfleiderer, A. Neubauer, W.\nM¨ unzer, A. Bauer, T. Adams, R. Georgii, P. B¨ oni, R. A.\nDuine, K. Everschor, M. Garst, and A. Rosch, Science\n330, 1648 (2010).\n[10] T. Schulz, R. Ritz, A. Bauer, M. Halder, M.Wagner, C.\nFranz, C. Pfleiderer, K. Everschor, M. Garst, and A.\nRosch, Nat. Phys. 8, 301 (2012).\n[11] K. Everschor, M. Garst, R. A. Duine, and A. Rosch,\nPhys. Rev. B 84, 064401 (2011).\n[12] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.\nRev. Lett. 107, 136804 (2011).\n[13] Y. Tokura and S. Seki, Adv. Mat. 21, 1 (2009).\n[14] S. Seki, S. Ishiwata, and Y. Tokura, arXiv:1206.4404v1\n(2012).\n[15] Chenglong Jia, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 74, 224444 (2006).\n[16] Chenglong Jia, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 76, 144424 (2007).\n[17] Taka-hisa Arima, J. Phys. Soc. Jpn. 76, 073702 (2008).\n[18] Judit Romh´ anyi, Mikl´ os Lajk´ o, and Karlo Penc, Phys.Rev. B84, 224419 (2011).\n[19] J. H. Han, unpublished note.\n[20] H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and\nY. Tokura, Phys. Rev. Lett. 105, 137202 (2010).\n[21] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101 (1989); A. Bogdanov and A. Hubert, J. Magn.\nMagn. Mater. 138, 255 (1994).\n[22] U. K. Roßler, A. N. Bogdanov, and C. Pfleiderer, Nature\n442, 797 (2006).\n[23] Su Do Yi, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 80, 054416 (2009).\n[24] Jung Hoon Han, Jiadong Zang, Zhihua Yang, Jin-Hong\nPark, and Naoto Nagaosa, Phys. Rev. B 82, 094429\n(2010).\n[25] Michael Stone, Phys. Rev. B 53, 16573 (1996).\n[26] J. S. White, et al. arXiv:1208.1146 (2012).\n[27] Masahito Mochizuki, Phys. Rev. Lett. 108, 017601\n(2012).\n[28] Olga Petrova and Oleg Tchernyshyov, Phys. Rev. B 84,\n214433 (2011); Imam Makhfudz, Benjamin Kr¨ uger, and\nOleg Tchernyshyov, arXiv:1208.3123 (2012).\n[29] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y.\nTokura, Phys. Rev. Lett. 109, 037603 (2012)."    },    {        "title": "2211.13486v1.Influence_of_non_local_damping_on_magnon_properties_of_ferromagnets.pdf",        "content": "In\ruence of non-local damping on magnon properties of ferromagnets\nZhiwei Lu,1,\u0003I. P. Miranda,2,\u0003Simon Streib,2Manuel Pereiro,2Erik Sj oqvist,2\nOlle Eriksson,2, 3Anders Bergman,2Danny Thonig,3, 2and Anna Delin1, 4\n1Department of Applied Physics, School of Engineering Sciences, KTH Royal\nInstitute of Technology, AlbaNova University Center, SE-10691 Stockholm, Swedeny\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden\n3School of Science and Technology, Orebro University, SE-701 82, Orebro, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n(Dated: November 28, 2022)\nWe study the in\ruence of non-local damping on magnon properties of Fe, Co, Ni and Fe 1\u0000xCox\n(x= 30%;50%) alloys. The Gilbert damping parameter is typically considered as a local scalar\nboth in experiment and in theoretical modelling. However, recent works have revealed that Gilbert\ndamping is a non-local quantity that allows for energy dissipation between atomic sites. With\nthe Gilbert damping parameters calculated from a state-of-the-art real-space electronic structure\nmethod, magnon lifetimes are evaluated from spin dynamics and linear response, where a good\nagreement is found between these two methods. It is found that non-local damping a\u000bects the\nmagnon lifetimes in di\u000berent ways depending on the system. Speci\fcally, we \fnd that in Fe, Co,\nand Ni the non-local damping decreases the magnon lifetimes, while in Fe 70Co30and Fe 50Co50an\nopposite, non-local damping e\u000bect is observed, and our data show that it is much stronger in the\nformer.\nINTRODUCTION\nIn recent years, there has been a growing interest in\nmagnonics, which uses quasi-particle excitations in mag-\nnetically ordered materials to perform information trans-\nport and processing on the nanoscale. Comparing to the\nconventional information device, the magnonics device\nexhibits lower energy consumption, easier integrability\nwith complementary metal-oxide semiconductor (CMOS)\nstructure, anisotropic properties, and e\u000ecient tunability\nby various external stimuli to name a few [1{10]. Yttrium\niron garnet (YIG) [11] as well as other iron garnets with\nrare-earth elements (Tm, Tb, Dy, Ho, Er) [12] are very\npromising candidates for magnonics device applications\ndue to their low energy dissipation properties and, thus,\nlong spin wave propagation distances up to tens of \u0016m.\nContrary, the damping of other materials for magnonics,\nlike CoFeB, is typically two orders of magnitude higher\ncompared to YIG [12], leading to much shorter spin wave\npropagation distances. A clear distinction can be made\nbetween materials with an ultra-low damping parame-\nter, like in YIG, and those with a sign\fciantly larger,\nbut still small, damping parameter. Materials like YIG\nare insulating, which hinders many of the microscopic\nmechanisms for damping, resulting in the low observed\ndamping parameter. In contrast, materials like CoFeB\nare metallic. In research projects that utilize low damp-\ning materials, YIG and similar non-metallic low damping\nsystems are typically favored. However, metallic systems\nhave an advantage, since magnetic textures can easily by\nin\ruenced by electrical currents. Hence, there is good\n\u0003These two authors contributed equally\nyCorresponding author: zhiweil@kth.sereason to consider metallic systems for low damping ap-\nplications, even though their damping typically is larger\nthan in YIG. One can conclude that Gilbert damping\nis one of the major bottlenecks for the choice of mate-\nrial in magnonics applications and a detailed experimen-\ntal as well as theoretical characterisation is fundamen-\ntal for this \feld of research, especially for metallic sys-\ntems. Thus, a more advanced and detailed understand-\ning of Gilbert damping is called for, in order to overcome\nthis obstacle for further development of magnonics-based\ntechnology.\nWhereas most studies consider chemical modi\fcations\nof the materials in order to tune damping [13, 14], only a\nfew focus on the fundamental physical properties as well\nas dependencies of the Gilbert damping. Often Gilbert\ndamping is considered as a phenomenological scalar pa-\nrameter in the equation of motion of localized atom-\nistic magnetic moments, i.e. the Landau-Lifshitz-Gilbert\n(LLG) equation [15]. However, from using the general\nRayleigh dissipation function in the derivation proposed\nby Gilbert [16], it was theoretically found that the Gilbert\ndamping should be anisotropic, a tensor, and non-local.\nFurthermore, it depends on the temperature and, thus,\non underlying magnon as well as phonon con\fgurations\n[17{20]. This is naturally built into the multiple theoret-\nical methods developed to predict the damping parame-\nter, including breathing Fermi surface model [21], torque\ncorrelation model [22], and linear response formulation\n[23]. For instance, the general Gilbert damping tensor\nas a function of the non-collinear spin con\fguration has\nbeen proposed in Ref. 24.\nNonetheless, an experimental veri\fcation is still miss-\ning due to lacking insights into the impact of the gen-\neralised damping on experimental observables. In a re-\ncent experiment, however, the anisotropic behavior of the\ndamping has been con\frmed for Co 50Fe50thin \flms andarXiv:2211.13486v1  [cond-mat.mtrl-sci]  24 Nov 20222\nwas measured to be of the order of 400% [25], with respect\nto changing the magnetization direction. Changes of\nGilbert damping in a magnetic domain wall and, thus, its\ndependency on the magnetic con\fguration was measured\nin Ref. [26] and \ftted to the Landau-Lifshitz-Baryakhtar\n(LLBar) equation, which includes non-locality of the\ndamping by an additional dissipation term proportional\nto the gradient of the magnetisation [27{29]. However,\nthe pair-wise non-local damping \u000bijhas not yet been\nmeasured.\nThe most common experimental techniques of evaluat-\ning damping are ferromagnetic resonance (FMR) [30] and\ntime-resolved magneto-optical Kerr e\u000bect (TR-MOKE)\n[31]. In these experiments, Gilbert damping is related\nto the relaxation rate when (i)slightly perturbing the\ncoherent magnetic moment out of equilibrium by an ex-\nternal magnetic \feld [32] or (ii)when disordered mag-\nnetic moments remagnetise after pumping by an ultrafast\nlaser pulse [33]. Normally, in case (i)the non-locality is\nsuppressed due to the coherent precession of the atomic\nmagnetic moments. However, this coherence can be per-\nturbed by temperature, making non-locality in principle\nmeasurable. One possible other path to link non-local\ndamping with experiment is magnon lifetimes. Theoret-\nically, the magnon properties as well as the impact of\ndamping on these properties can be assessed from the\ndynamical structure factor, and atomistic spin-dynamics\nsimulations have been demonstrated to yield magnon dis-\npersion relations that are in good agreement with exper-\niment [34]. In experiment, neutron scattering [35] and\nelectron scattering [36] are the most common methods for\nprobing magnon excitations, where the linewidth broad-\nening of magnon excitations is related to damping and\nprovides a way to evaluate the magnon lifetimes [37]. It is\nfound in ferromagnets that the magnon lifetimes is wave\nvector (magnon energy) dependent [38{40]. It has been\nreported that the magnon energy in Co \flms is nearly\ntwice as large as in Fe \flms, but they have similar magnon\nlifetimes, which is related to the intrinsic damping mech-\nanism of materials [41]. However, this collective e\u000bect of\ndamping and magnon energy on magnon lifetimes is still\nan open question. The study of this collective e\u000bect is of\ngreat interest for both theory and device applications.\nHere, we report an implementation for solving the\nstochastic Landau-Lifshitz-Gilbert (SLLG) equation in-\ncorporating the non-local damping. With the dynamical\nstructure factor extracted from the spin dynamics sim-\nulations, we investigate the collective e\u000bect of non-local\ndamping and magnon energy on the magnon lifetimes.\nWe propose an e\u000ecient method to evaluate magnon life-\ntimes from linear response theory and verify its validity.\nThe paper is organized as follows. In Sec. I, we give\nthe simulation details of the spin dynamics, the adiabatic\nmagnon spectra and dynamical structure factor, and the\nmethodology of DFT calculations and linear response.\nSec. II presents the non-local damping in real-space, non-\nlocal damping e\u000bects on the spin dynamics and magnon\nproperties including magnon lifetimes of pure ferromag-nets (Fe, Co, Ni), and Fe 1\u0000xCox(x= 30%;50%) alloys.\nIn Sec. III, we give a summary and an outlook.\nI. THEORY\nA. Non-local damping in atomistic spin dynamics\nThe dynamical properties of magnetic materials at \f-\nnite temperature have been so far simulated from atom-\nistic spin dynamics by means of the stochastic Landau-\nLifshitz-Gilbert equation with scalar local energy dissipa-\ntion. Here, the time evolution of the magnetic moments\nmi=mieiat atom site iis well described by:\n@mi\n@t=mi\u0002\u0012\n\u0000\r[Bi+bi(t)] +\u000b\nmi@mi\n@t\u0013\n;(1)\nwhere\ris the gyromagnetic ratio. The e\u000bective \feld Bi\nacting on each magnetic moment is obtained from:\nBi=\u0000@H\n@mi: (2)\nThe here considered spin-Hamiltonian Hconsists of a\nHeisenberg spin-spin exchange:\nH=\u0000X\ni6=jJijei\u0001ej: (3)\nHere,Jij{ the Heisenberg exchange parameter { cou-\nples the spin at site iwith the spin at site jand is cal-\nculated from \frst principles (see Section I C). Further-\nmore,\u000bis the scalar phenomenological Gilbert damp-\ning parameter. Finite temperature Tis included in\nEq. (1) via the \ructuating \feld bi(t), which is modeled\nby uncorrelated Gaussian white noise: hbi(t)i= 0 and\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u000eij\u000e\u0016\u0017\u000e(t\u0000t0), where\u000eis the Kro-\nnecker delta, i;jare site and \u0016;\u0017=fx;y;zgCartesian\nindices. Furthermore, the \ructuation-dissipation theo-\nrem givesD=\u000bkBT\n\rmi[42], with the Boltzman constant\nkB.\nA more generalized form of the SLLG equation that\nincludes non-local tensorial damping has been reported\nin previous studies [20, 43, 44] and is:\n@mi\n@t=mi\u00020\n@\u0000\r[Bi+bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA;(4)\nwhich can be derived from Rayleigh dissipation func-\ntional in the Lagrange formalism used by Gilbert [16].\nIn the presence of non-local damping, the Gaussian \ruc-\ntuating \feld ful\flls [43, 45, 46]\n\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u0016\u0017\nij\u000e(t\u0000t0); (5)\nwithD\u0016\u0017\nij=\u000b\u0016\u0017\nijkBT\n\rmi. The damping tensor \u000b\u0016\u0017\nijmust be\npositive de\fnite in order to be physically-de\fned. Along3\nwith spatial non-locality, the damping can also be non-\nlocal in time, as discussed in Ref. [47]. To prove the\n\ructuation-dissipation theorem in Eq. (5), the Fokker-\nPlanck equation has to be analysed in the presence of\nnon-local damping, similar to Ref. [15]. This is, however,\nnot the purpose of this paper. Instead, we will use the\napproximation \u000b\u0016\u0017\nij=1\n3Trf\u000biig\u000eij\u000e\u0016\u0017within the di\u000busion\nconstantD. Such an approximation is strictly valid only\nin the low temperature limit.\nTo solve this SLLG equation incorporating the non-\nlocal damping, we have implemented an implicit mid-\npoint solver in the UppASD code [48]. This iterative\n\fx-point scheme converges within an error of 10\u000010\u0016B,\nwhich is typically equivalent to 6 iteration steps. More\ndetails of this solver are provided in Appendix A. The\ninitial spin con\fguration in the typical N= 20\u000220\u000220\nsupercell with periodic boundary conditions starts from\ntotally random state. The spin-spin exchange interac-\ntions and non-local damping parameters are included up\nto at least 30 shells of neighbors, in order to guarantee\nthe convergence with respect to the spatial expansion of\nthese parameters (a discussion about the convergence is\ngiven in Section II A). Observables from our simulations\nare typically the average magnetisation M=1\nNPN\nimi\nas well as the magnon dispersion.\nB. Magnon dispersion\nTwo methods to simulate the magnon spectrum are\napplied in this paper: i)the dynamical structure factor\nandii)frozen magnon approach.\nFor the dynamical structure factor S(q;!) at \fnite\ntemperature and damping [34, 49], the spatial and time\ncorrelation function between two magnetic moments iat\npositionrandjat positionr0as well as di\u000berent time 0\nandtis expressed as:\nC\u0016(r\u0000r0;t) =hm\u0016\nr(t)m\u0016\nr0(0)i\u0000hm\u0016\nr(t)ihm\u0016\nr0(0)i:(6)\nHereh\u0001idenotes the ensemble average and \u0016are Carte-\nsian components. The dynamical structure factor can be\nobtained from the time and space Fourier transform of\nthe correlation function, namely:\nS\u0016(q;!) =1p\n2\u0019NX\nr;r0eiq\u0001(r\u0000r0)Z1\n\u00001ei!tC\u0016(r\u0000r0;t)dt:\n(7)\nThe magnon dispersion is obtained from the peak\npositions of S(q;!) along di\u000berent magnon wave vectors\nqin the Brillouin zone and magnon energies !. It\nshould be noted that S(q;!) is related to the scattering\nintensity in inelastic neutron scattering experiments [50].\nThe broadening of the magnon spectrum correlates to\nthe lifetime of spin waves mediated by Gilbert damping\nas well as intrinsic magnon-magnon scattering processes.\nGood agreement between S(q;!) and experiment hasbeen found previously [34].\nThe second method { the frozen magnon approach\n{ determines the magnon spectrum directly from the\nFourier transform of the spin-spin exchange parameters\nJij[51, 52] and non-local damping \u000bij. At zero tempera-\nture, a time-dependent external magnetic \feld is consid-\nered,\nB\u0006\ni(t) =1\nNX\nqB\u0006\nqeiq\u0001Ri\u0000i!t; (8)\nwhereNis the total number of lattice sites and B\u0006\nq=\nBx\nq\u0006iBy\nq. The linear response to this \feld is then given\nby\nM\u0006\nq=\u001f\u0006(q;!)B\u0006\nq: (9)\nWe obtain for the transverse dynamic magnetic suscep-\ntibility [53, 54]\n\u001f\u0006(q;!) =\u0006\rMs\n!\u0006!q\u0007i!\u000bq; (10)\nwith saturation magnetization Ms, spin-wave frequency\n!q=E(q)=~and damping\n\u000bq=X\nj\u000b0je\u0000iq\u0001(R0\u0000Rj): (11)\nWe can extract the spin-wave spectrum from the imagi-\nnary part of the susceptibility,\nIm\u001f\u0006(q;!) =\rMs\u000bq!\n[!\u0006!q]2+\u000b2q!2; (12)\nwhich is equivalent to the correlation function S\u0006(q;!)\ndue to the \ructuation-dissipation theorem [55]. We\n\fnd that the spin-wave lifetime \u001cqis determined by the\nFourier transform of the non-local damping (for \u000bq\u001c1),\n\u001cq=\u0019\n\u000bq!q: (13)\nThe requirement of positive de\fniteness of the damping\nmatrix\u000bijdirectly implies \u000bq>0, since\u000bijis diago-\nnalized by Fourier transformation due to translational\ninvariance. Hence, \u000bq>0 is a criterion to evaluate\nwhether the damping quantity in real-space is physically\nconsistent and whether \frst-principles calculations are\nwell converged. If \u000bq<0 for some wave vector q, energy\nis pumped into the spin system through the correspon-\ndent magnon mode, preventing the system to fully reach\nthe saturation magnetization at su\u000eciently low temper-\natures.\nThe e\u000bective damping \u000b0of the FMR mode at q=\n0 is determined by the sum over all components of the\ndamping matrix, following Eqn.11,\n\u000btot\u0011\u000b0=X\nj\u000b0j: (14)\nTherefore, an e\u000bective local damping should be based\non\u000btotif the full non-local damping is not taken into\naccount.4\nC. Details of the DFT calculations\nThe electronic structure calculations, in the framework\nof density functional theory (DFT), were performed us-\ning the fully self-consistent real-space linear mu\u000en-tin\norbital in the atomic sphere approximation (RS-LMTO-\nASA) [56, 57]. The RS-LMTO-ASA uses the Haydock\nrecursion method [58] to solve the eigenvalue problem\nbased on a Green's functions methodology directly in\nreal-space. In the recursion method, the continued frac-\ntions have been truncated using the Beer-Pettifor termi-\nnator [59], after a number LLof recursion levels. The\nLMTO-ASA [60] is a linear method which gives precise\nresults around an energy E\u0017, usually taken as the center\nof thes,panddbands. Therefore, as we calculate \fne\nquantities as the non-local damping parameters, we here\nconsider an expression accurate to ( E\u0000E\u0017)2starting\nfrom the orthogonal representation of the LMTO-ASA\nformalism [61].\nFor bcc FeCo alloys and bcc Fe we considered LL= 31,\nwhile for fcc Co and fcc Ni much higher LLvalues (51 and\n47, respectively), needed to better describe the density of\nstates and Green's functions at the Fermi level.\nThe spin-orbit coupling (SOC) is included as a l\u0001s\n[60] term computed in each variational step [62]. All\ncalculations were performed within the local spin den-\nsity approximation (LSDA) exchange-functional (XC) by\nvon Barth and Hedin [63], as it gives general magnetic\ninformation with equal or better quality as, e.g., the\ngeneralized gradient approximation (GGA). Indeed, the\nchoice of XC between LSDA and GGA [64] have a mi-\nnor impact on the onsite damping and the shape of the\n\u000bqcurves, when considering the same lattice parame-\nters (data not shown). No orbital polarization [65] was\nconsidered here. Each bulk system was modelled by a\nbig cluster containing \u001855000 (bcc) and\u0018696000 (fcc)\natoms located in the perfect crystal positions with the re-\nspective lattice parameters of a= 2:87\u0017A (bcc Fe and bcc\nFe1\u0000xCox, su\u000eciently close to experimental observations\n[66]),a= 3:54\u0017A (fcc Co [20, 67]), and a= 3:52\u0017A (fcc\nNi [68]). To account for the chemical disorder in the\nFe70Co30and Fe 50Co50bulks, the electronic structure\ncalculated within the simple virtual crystal approxima-\ntion (VCA), which has shown to work well for the fer-\nromagnetic transition metals alloys (particularly for el-\nements next to each other in the Periodic Table, such\nas FeCo and CoNi) [69{76], and also describe in a good\nagreement the damping trends in both FeCo and CoNi\n(see Appendix C).\nAs reported in Ref. [77], the total damping of site\ni, in\ruenced by the interaction with neighbors j, can\nbe decomposed in two main contributions: the onsite\n(fori=j), and the non-local (for i6=j). Both can be\ncalculated, in the collinear framework, by the followingexpression,\n\u000b\u0016\u0017\nij=\u000bCZ1\n\u00001\u0011(\u000f)Tr\u0010\n^T\u0016\ni^Aij(^T\u0017\nj)y^Aji\u0011\nd\u000fT!0K\u0000\u0000\u0000\u0000!\n\u000bCTr\u0010\n^T\u0016\ni^Aij(\u000fF+i\u000e)(^T\u0017\nj)y^Aji(\u000fF+i\u000e)\u0011\n;\n(15)\nwhere we de\fne ^Aij(\u000f+i\u000e) =1\n2i(^Gij(\u000f+i\u000e)\u0000^Gy\nji(\u000f+i\u000e))\nthe anti-Hermitian part of the retarded physical Green's\nfunctions in the LMTO formalism, and \u000bC=g\nmti\u0019a\npre-factor related to the i-th site magnetization. The\nimaginary part, \u000e, is obtained from the terminated con-\ntinued fractions. Also in Eq. 15, ^T\u0016\ni= [\u001b\u0016\ni;Hso] is the\nso-called torque operator [20] evaluated in each Cartesian\ndirection\u0016;\u0017=fx;y;zgand at site i,\u0011(\u000f) =\u0000@f(\u000f)\n@\u000fis\nthe derivative of the Fermi-Dirac distribution f(\u000f) with\nrespect to the energy \u000f,g= 2\u0010\n1 +morb\nmspin\u0011\ntheg-factor\n(not considering here the spin-mixing parameter [78]),\n\u001b\u0016are the Pauli matrices, and mtiis the total magnetic\nmoment of site i(mti=morbi+mspini). This results\nin a 3\u00023 tensor with terms \u000b\u0016\u0017\nij. In the real-space bulk\ncalculations performed in the present work, the \u000bij(with\ni6=j) matrices contain o\u000b-diagonal terms which are can-\ncelled by the summation of the contributions of all neigh-\nbors within a given shell, resulting in a purely diagonal\ndamping tensor, as expected for symmetry reasons [15].\nTherefore, as in the DFT calculations the spin quanti-\nzation axis is considered to be in the z([001]) direction\n(collinear model), we can ascribe a scalar damping value\n\u000bijas the average \u000bij=1\n2(\u000bxx\nij+\u000byy\nij) =\u000bxx\nijfor the\nsystems investigated here. This scalar \u000bijis, then, used\nin the SLLG equation (Eq. 1).\nThe exchange parameters Jijin the Heisenberg\nmodel were calculated by the Liechtenstein-Katsnelson-\nAntropov-Gubanov (LKAG) formalism [79], according to\nthe implementation in the RS-LMTO-ASA method [61].\nHence all parameters needed for the atomistic LLG equa-\ntion have been evaluated from ab-initio electronic struc-\nture theory.\nII. RESULTS\nA. Onsite and non-local dampings\nTable I shows the relevant ab-initio magnetic prop-\nerties of each material; the TCvalues refer to the Curie\ntemperature calculated within the random-phase approx-\nimation (RPA) [80], based on the computed Jijset. De-\nspite the systematic \u000btotvalues found in the lower limit\nof available experimental results (in similar case with,\ne.g., Ref. [81]), in part explained by the fact that we\nanalyze only the intrinsic damping, a good agreement\nbetween theory and experiment can be seen. When the\nwhole VCA Fe 1\u0000xCoxseries is considered (from x= 0%\ntox= 60%), the expected Slater-Pauling behavior of5\nthe total magnetic moment [73, 82] is obtained (data not\nshown).\nFor all systems studied here, the dissipation is domi-\nnated by the onsite ( \u000bii) term, while the non-local pa-\nrameters (\u000bij,i6=j) exhibit values at least one order of\nmagnitude lower; however, as it will be demonstrated in\nthe next sections, these smaller terms still cause a non-\nnegligible impact on the relaxation of the average magne-\ntization as well as magnon lifetimes. Figure 1 shows the\nnon-local damping parameters for the investigated ferro-\nmagnets as a function of the ( i;j) pairwise distance rij=a,\ntogether with the correspondent Fourier transforms \u000bq\nover the \frst Brillouin Zone (BZ). The \frst point to no-\ntice is the overall strong dependence of \u000bon the wave\nvectorq. The second point is the fact that, as also re-\nported in Ref. [20], \u000bijcan be an anisotropic quantity\nwith respect to the same shell of neighbors, due to the\nbroken symmetry imposed by a preferred spin quantiza-\ntion axis. This means that, in the collinear model and for\na given neighboring shell, \u000bijis isotropic only for equiva-\nlent sites around the magnetization as a symmetry axis.\nAnother important feature that can be seen in Fig. 1\nis the presence of negative \u000bijvalues. Real-space neg-\native non-local damping parameters have been reported\npreviously [20, 77, 97]. They are related to the decrease\nof damping at the \u0000-point, but may also increase \u000bqfrom\nthe onsite value in speci\fc qpoints inside the BZ; there-\nfore, they cannot be seen as ad hoc anti-dissipative con-\ntributions. In the ground-state, these negative non-local\ndampings originate from the overlap between the anti-\nHermitian parts of the two Green's functions at the Fermi\nlevel, each associated with a spin-dependent phase factor\n\b\u001b(\u001b=\";#) [20, 80].\nFinally, as shown in the insets of Fig. 1, a long-range\nconvergence can be seen for all cases investigated. An\nillustrative example is the bcc Fe 50Co50bulk, for which\nthe e\u000bective damping can be \u001860% higher than the con-\nverged\u000btotif only the \frst 7 shells of neighbors are con-\nsidered in Eq. 14. The non-local damping of each neigh-\nboring shell is found to follow a1\nr2\nijtrend, as previously\nargued by Thonig et al. [20] and Umetsu et al. [97].\nExplicitly,\n\u000bij/sin(k\"\u0001rij+ \b\") sin(k#\u0001rij+ \b#)\njrijj2; (16)\nwhich also qualitatively justi\fes the existence of negative\n\u000bij's. Thus, the convergence in real-space is typically\nslower than other magnetic quantities, such as exchange\ninteractions ( Jij/1\njrijj3) [80], and also depends on the\nimaginary part \u000e(see Eq. 15) [20]. The di\u000berence in the\nasymptotic behaviour of the damping and the Heisenberg\nexchange is distinctive; the \frst scales with the inverse of\nthe square of the distance while the latter as the inverse\nof the cube of the distance. Although this asymptotic\nbehaviour can be derived from similar arguments, both\nusing the Greens function of the free electron gas, the\nresults are di\u000berent. The reason for this di\u000berence issimply that the damping parameter is governed by states\nclose to the Fermi surface, while the exchange parameter\ninvolves an integral over all occupied states [20, 79].\nFrom bcc Fe to bcc Fe 50Co50(Fig. 1(a-f)), with in-\ncreasing Co content, the average \frst neighbors \u000bijde-\ncreases to a negative value, while the next-nearest neigh-\nbors contributions reach a minimum, and then increase\nagain. Similar oscillations can be found in further shells.\nAmong the interesting features in the Fe 1\u0000xCoxsystems\n(x= 0%;30%;50%), we highlight the low \u000bqaround the\nhigh-symmetry point H, along the H\u0000PandH\u0000N\ndirections, consistently lower than the FMR damping.\nBoth\u000bvalues are strongly in\ruenced by non-local con-\ntributions &5 NN. Also consistent is the high \u000bqob-\ntained forq=H. For long wavelengths in bcc Fe, some\n\u000bqanisotropy is observed around \u0000, which resembles the\nsame trait obtained for the corresponding magnon dis-\npersion curves [80]. This anisotropy changes to a more\nisotropic behavior by FeCo alloying.\nFar from the more noticeable high-symmetry points,\n\u000bqpresents an oscillatory behavior along BZ, around the\nonsite value. It is noteworthy, however, that these oscil-\nlatory\u000bqparameters exhibit variations up to \u00182 times\n\u000bii, thus showing a pronounced non-local in\ruence in\nspeci\fcqpoints.\nIn turn, for fcc Co (Fig. 1(g,h)) the \frst values are\ncharacterized by an oscillatory behavior around zero,\nwhich also re\rects on the damping of the FMR mode,\n\u000bq=0. In full agreement with Ref. [20], we compute a\npeak of\u000bijcontribution at rij\u00183:46a, which shows\nthe long-range character that non-local damping can ex-\nhibit for speci\fc materials. Despite the relatively small\nmagnitude of \u000bij, the multiplicity of the nearest neigh-\nbors shells drives a converged \u000bqdispersion with non-\nnegligible variations from the onsite value along the BZ,\nspecially driven by the negative third neighbors. The\nmaximum damping is found to be in the region around\nthe high-symmetry point X, where thus the lifetime of\nmagnon excitations are expected to be reduced. Simi-\nlar situation is found for fcc Ni (Fig. 1(i,j)), where the\n\frst neighbors \u000bijare found to be highly negative, con-\nsequently resulting in a spectrum in which \u000bq> \u000bq=0\nfor everyq6= 0. In contrast with fcc Co, however, no\nnotable peak contributions are found.\nB. Remagnetization\nGilbert damping in magnetic materials determines the\nrate of energy that dissipates from the magnetic to other\nreservoirs, like phonons or electron correlations. To ex-\nplore what impact non-local damping has on the energy\ndissipation process, we performed atomistic spin dynam-\nics (ASD) simulations for the aforementioned ferromag-\nnets: bcc Fe 1\u0000xCox(x= 0%;30%;50%), fcc Co, and\nfcc Ni, for the (i)fully non-local \u000bijand (ii)e\u000bective\n\u000btot(de\fned in 14) dissipative case. We note that, al-\nthough widely considered in ASD calculations, the adop-6\nTABLE I. Spin ( mspin) and orbital ( morb) magnetic moments, onsite ( \u000bii) damping, total ( \u000btot) damping, and Curie temper-\nature (TC) of the investigated systems. The theoretical TCvalue is calculated within the RPA. In turn, mtdenotes the total\nmoments for experimental results of Ref. [82].\nmspin(\u0016B)morb(\u0016B)\u000bii(\u000210\u00003) \u000btot(\u000210\u00003) TC(K)\nbcc Fe (theory) 2.23 0.05 2.4 2.1 919\nbcc Fe (expt.) 2.13 [68] 0 :08 [68] \u0000 1:9\u00007:2 [33, 83{89] 1044\nbcc Fe 70Co30(theory) 2.33 0.07 0.5 0.9 1667\nbcc Fe 70Co30(expt.) mt= 2:457 [82] \u0000 0:5\u00001:7a[33, 83, 90] 1258 [92]\nbcc Fe 50Co50(theory) 2.23 0.08 1.5 1.6 1782\nbcc Fe 50Co50(expt.) mt= 2:355 [82] \u0000 2:0\u00003:2b[25, 33, 83] 1242 [93]\nfcc Co (theory) 1.62 0 :08 7.4 1.4 1273\nfcc Co (expt.) 1 :68(6) [94] \u0000 \u0000 2:8(5) [33, 89] 1392\nfcc Ni (theory) 0 :61 0 :05 160.1 21.6 368\nfcc Ni (expt.) 0 :57 [68] 0 :05 [68] \u0000 23:6\u000064 [22, 83, 87{89, 95, 96] 631\naThe lower limit refers to polycrystalline Fe 75Co2510 nm-thick \flms from Ref. [33]. Lee et al. [90] also found a low Gilbert damping in\nan analogous system, where \u000btot<1:4\u000210\u00003. For the exact 30% of Co concentration, however, previous results [33, 84, 91] indicate\nthat we should expect a slightly higher damping than in Fe 75Co25.\nbThe upper limit refers to the approximate minimum intrinsic value for a 10 nm-thick \flm of Fe 50Co50jPt (easy magnetization axis).\ntion of a constant \u000btotvalue (case (ii)) is only a good ap-\nproximation for long wavelength magnons close to q= 0.\nFirst, we are interested on the role of non-local damp-\ning in the remagnetization processes as it was already\ndiscussed by Thonig et al. [20] and as it is important\nfor,e.g., ultrafast pump-probe experiments as well as all-\noptical switching. In the simulations presented here, the\nrelaxation starts from a totally random magnetic con-\n\fguration. The results of re-magnetization simulations\nare shown in Figure 2. The fully non-local damping (i)\nin the equation of motion enhances the energy dissipa-\ntion process compared to the case when only the e\u000bective\ndamping (ii)is used. This e\u000bect is found to be more pro-\nnounced in fcc Co and fcc Ni compared to bcc Fe and bcc\nFe50Co50. Thus, the remagnetization time to 90% of the\nsaturation magnetisation becomes \u00185\u00008 times faster\nfor case (i)compared to the case (ii). This is due to\nthe increase of \u000bqaway from the \u0000 point in the whole\nspectrum for Co and Ni (see Fig. 1), where in Fe and\nFe50Co50it typically oscillates around \u000btot.\nFor bcc Fe 70Co30, the e\u000bect of non-local damping on\nthe dynamics is opposite to the data in Fig. 2; the re-\nlaxation process is decelerated. In this case, almost the\nentire\u000bqspectrum is below \u000bq=0, which is an interest-\ning result given the fact that FMR measurements of the\ndamping parameter in this system is already considered\nan ultra-low value, when compared to other metallic fer-\nromagnets [33]. Thus, in the remagnetization process of\nFe70Co30, the majority of magnon modes lifetimes is un-\nderestimated when a constant \u000btotis considered in the\nspin dynamics simulations, which leads to a faster overall\nrelaxation rate.\nAlthough bcc Fe presents the highest Gilbert damp-ing obtained in the series of the Fe-Co alloys (see Table\nI) the remagnetization rate is found to be faster in bcc\nFe50Co50. This can be explained by the fact that the ex-\nchange interactions for this particular alloy are stronger\n(\u001880% higher for nearest-neighbors) than in pure bcc\nFe, leading to an enhanced Curie temperature (see Table\nI). In view of Eq. 13 and Fig. 1, the di\u000berence in the\nremagnetization time between bcc Fe 50Co50and elemen-\ntal bcc Fe arises from \u000bqvalues that are rather close,\nbut where the magnon spectrum of Fe 50Co50has much\nhigher frequencies, with corresponding faster dynamics\nand hence shorter remagnetization times.\nFrom our calculations we \fnd that the sum of non-local\ndamping\u0010P\ni6=j\u000bij\u0011\ncontributes with \u000013%,\u000081%,\n\u000087%, +80%, and +7% to the local damping in bcc Fe,\nfcc Co, fcc Ni, bcc Fe 70Co30, and bcc Fe 50Co50, respec-\ntively. The high positive ratio found in Fe 70Co30indi-\ncates that, in contrast to the other systems analyzed, the\nnon-local contributions act like an anti-damping torque,\ndiminishing the local damping torque. A similar anti-\ndamping e\u000bect in antiferromagnetic (AFM) materials\nhave been reported in theoretical and experimental in-\nvestigations ( e.g., [98, 99]), induced by electrical current.\nHere we \fnd that an anti-damping torque e\u000bect can have\nan intrinsic origin.\nTo provide a deeper understanding of the anti-damping\ne\u000bect caused by a positive non-local contribution, we an-\nalytically solved the equation of motion for a two spin\nmodel system, e.g. a dimer. In the particular case when\nthe onsite damping \u000b11is equal to the non-local con-\ntribution\u000b12, we observed that the system becomes un-\ndamped (see Appendix B). As demonstrated in Appendix\nB, ASD simulations of such a dimer corroborate the re-7\nFIG. 1. Non-local damping ( \u000bij) as a function of the nor-\nmalized real-space pairwise ( i;j) distance computed for each\nneighboring shell, and corresponding Fourier transform \u000bq\n(see Eq. 11) from the onsite value ( \u000bii) up to 136 shells of\nneighbors (136 NN) for: (a,b) bcc Fe; (c,d) bcc Fe 70Co30;\n(e,f) bcc Fe 50Co50in the virtual-crystal approximation; and\nup to 30 shells of neighbors (30 NN) for: (g,h) fcc Co; (i,j) fcc\nNi. The insets in sub\fgures (a,c,e,g,i) show the convergence\nof\u000btotin real-space. The obtained onside damping values are\nshown in Table I. In the insets of the left panel, green full\nlines are guides for the eyes.\nsult of undamped dynamics. It should be further noticed\nthat this proposed model system was used to analyse\nthe stability of the ASD solver, verifying whether it can\npreserve both the spin length and total energy. Full de-\ntail of the analytical solution and ASD simulation of a\nspin-dimer and the anti-damping e\u000bect are provided inAppendix B.\nFIG. 2. Remagnetization process simulated with ASD, con-\nsidering fully non-local Gilbert damping ( \u000bij, blue sold lines),\nand the e\u000bective damping ( \u000btot, red dashed lines), for: (a) fcc\nNi; (b) fcc Co; and (c) bcc Fe 1\u0000xCox(x= 0%;30%;50%).\nThe dashed gray lines indicate the stage of 90% of the satu-\nration magnetization.\nC. Magnon spectra\nIn order to demonstrate the in\ruence of damping on\nmagnon properties at \fnite temperatures, we have per-\nformed ASD simulations to obtain the excitation spectra\nfrom the dynamical structure factor introduced in Sec-\ntion I. Here, we consider 16 NN shells for S(q;!) calcula-\ntions both from simulations that include non-local damp-\ning as well as the e\u000bective total damping (see Appendix\nD for a focused discussion). In Fig. 3, the simulated\nmagnon spectra of the here investigated ferromagnets are\nshown. We note that a general good agreement can be\nobserved between our computed magnon spectra (both\nfrom the the frozen magnon approach as well as from the\ndynamical structure factor) and previous theoretical as\nwell as experimental results [34, 52, 80, 100{103], where\ndeviations from experiments is largest for fcc Ni. This\nexception, however, is well known and has already been\ndiscussed elsewhere [104].\nThe main feature that the non-local damping causes to\nthe magnon spectra in all systems investigated here, is in\nchanges of the full width at half maximum (FWHM) 4q\nofS(q;!). Usually,4qis determined from the super-\nposition of thermal \ructuations and damping processes.\nMore speci\fcally, the non-local damping broadens the\nFWHM compared to simulations based solely on an e\u000bec-\ntive damping, for most of the high-symmetry paths in all\nof the here analyzed ferromagnets, with the exception of\nFe70Co30. The most extreme case is for fcc Ni, as \u000bqex-\nceeds the 0:25 threshold for q=X, which is comparable\nto the damping of ultrathin magnetic \flms on high-SOC\nmetallic hosts [105]. As a comparison, the largest di\u000ber-\nence of FWHM between the non-local damping process\nand e\u000bective damping process in bcc Fe is \u00182 meV, while\nin fcc Ni the largest di\u000berence can reach \u0018258 meV. In\ncontrast, the di\u000berence is \u0018\u00001 meV in Fe 70Co30and the8\nlargest non-local damping e\u000bect occurs around q=N\nand in the H\u0000Pdirection, corroborating with the dis-\ncussion in Section II A. At the \u0000 point, which corresponds\nto the mode measured in FMR experiments, all spins in\nthe system have a coherent precession. This implies that\n@mj\n@tin Eq. 4 is the same for all moments and, thus, both\ndamping scenarios discussed here (e\u000becive local and the\none that also takes into account non-local contributions)\nmake no di\u000berence to the spin dynamics. As a conse-\nquence, only a tiny (negligible) di\u000berence of the FWHM\nis found between e\u000bective and non-local damping for the\nFMR mode at low temperatures.\nThe broadening of the FWHM on the magnon spec-\ntrum is temperature dependent. Thus, the e\u000bect of non-\nlocal damping to the width near \u0000 can be of great in-\nterest for experiments. More speci\fcally, taking bcc Fe\nas an example, the di\u000berence between width in e\u000bective\ndamping and non-local damping process increases with\ntemperature, where the di\u000berence can be enhanced up to\none order of magnitude from T= 0:1 K toT= 25 K.\nNote that this enhancement might be misleading due to\nthe limits of \fnite temperature assumption made here.\nThis temperature dependent damping e\u000bect on FWHM\nsuggests a path for the measurement of non-local damp-\ning in FMR experiments.\nWe have also compared the di\u000berence in the imaginary\npart of the transverse dynamical magnetic susceptibility\ncomputed from non-local and e\u000bective damping. De\fned\nby Eq. 12, the imaginary part of susceptibility is re-\nlated to the FWHM [15]. Similar to the magnon spectra\nshown in Fig. 3, the susceptibility di\u000berence is signi\f-\ncant at the BZ boundaries. Taking the example of fcc\nCo, Im\u001f\u0006(q;!) for e\u000bective damping processes can be\n11:8 times larger than in simulations that include non-\nlocal damping processes, which is consistent to the life-\ntime peak that occurs at high the symmetry point, X,\ndepicted in Fig. 4. In the Fe 1\u0000xCoxalloy, and Fe 70Co30,\nthe largest ratio is 1 :7 and 2:7 respectively. The intensity\nat \u0000 point is zero since \u000bqis independent on the coupling\nvector and equivalent in both damping modes. The ef-\nfect of non-local damping on susceptibility coincides well\nwith the magnon spectra from spin dynamics. Thus, this\nmethod allows us to evaluate the magnon properties in a\nmore e\u000ecient way.\nD. Magnon lifetimes\nBy \ftting the S(q;!) curve at each wave vector with\na Lorentzian curve, the FWHF and hence the magnon\nlifetimes,\u001cq, can be obtained from the simple relation\n[15]\n\u001cq=2\u0019\n4q: (17)\nFigure 4 shows the lifetimes computed in the high-\nsymmetry lines in the BZ for all ferromagnets here in-vestigated. As expected, \u001cqis much lower at the qvec-\ntors far away from the zone center, being of the order\nof 1 ps for the Fe 1\u0000xCoxalloys (x= 0%;30%;50%),\nand from\u00180:01\u00001 ps in fcc Co and Ni. In view of\nEq. 13, the magnon lifetime is inversely proportional to\nboth damping and magnon frequency. In the e\u000bective\ndamping process, \u000bqis a constant and independent of\nq; thus, the lifetime in the entire BZ is dictated only by\n!q. The situation becomes more complex in the non-\nlocal damping process, where the \u001cqis in\ruenced by the\ncombined e\u000bect of changing damping and magnon fre-\nquency. Taking Fe 70Co30as an example, even though\nthe\u000bqis higher around the \u0000, the low magnon frequency\ncompensates the damping e\u000bect, leading to an asymp-\ntotically divergent magnon lifetime as !q!0. However,\nthis divergence becomes \fnite when including e.g. mag-\nnetocrystalline anisotropy or an external magnetic \feld\nto the spin-Hamiltonian. In the H\u0000Npath, the magnon\nenergy of Fe 70Co30is large, but \u000bqreaches\u00184\u000210\u00004\natq=\u00001\n4;1\n4;1\n2\u0001\n, resulting in a magnon lifetime peak of\n\u001810 ps. This value is not found for the e\u000bective damping\nmodel.\nIn the elemental ferromagnets, as well as for Fe 50Co50,\nit is found that non-local damping decreases the magnon\nlifetimes. This non-local damping e\u000bect is signi\fcant in\nboth Co and Ni, where the magnon lifetimes from the \u000bij\nmodel di\u000ber by an order of magnitude from the e\u000bective\nmodel (see Fig. 4). In fact, considering \u001cqobtained from\nEq. 13, the e\u000bective model predicts a lifetime already\nhigher by more than 50% when the magnon frequencies\nare\u001833 meV and\u001814 meV in the K\u0000\u0000 path ( i.e.,\nnear \u0000) of Ni and Co, respectively. This di\u000berence mainly\narises, in real-space, from the strong negative contriu-\ntions of\u000bijin the close neighborhood around the refer-\nence site, namely the NN in Ni and third neighbors in Co.\nIn contrast, due to the \u000bqspectrum composed of almost\nall dampings lower than \u000btot, already discussed in Section\nII A, the opposite trend on \u001cqis observed for Fe 70Co30:\nthe positive overall non-local contribution guide an anti-\ndamping e\u000bect, and the lifetimes are enhanced in the\nnon-local model.\nAnother way to evaluate the magnon lifetimes is from\nthe linear response theory. As introduced in Section I B,\nwe have access to magnon lifetimes at low temperatures\nfrom the imaginary part of the susceptibility. The \u001cq\ncalculated from Eq. 13 is also displayed in Fig. 4. Here\nthe spin-wave frequency !qis from the frozen magnon\nmethod. The magnon lifetimes from linear response have\na very good agreement with the results from the dynam-\nical structure factor, showing the equivalence between\nboth methods. Part of the small discrepancies are re-\nlated to magnon-magnon scattering induced by the tem-\nperature e\u000bect in the dynamical structure factor method.\nWe also \fnd a good agreement on the magnon lifetimes\nof e\u000bective damping in pure Fe with previous studies\n[106]. They are in the similar order and decrease with\nthe increasing magnon energy. However, their results\nare more di\u000bused since the simulations are performed at9\nFIG. 3. Magnon spectra calculated with non-local Gilbert damping and e\u000bective Gilbert damping in: (a) bcc Fe; (b) bcc\nFe70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni. The black lines denote the adiabatic magnon spectra calculated from\nEq. 7. Full red and open blue points denote the peak positions of S(q;!) at each qvector for\u000btotand\u000bijcalculations,\nrespectively, at T= 0:1 K. The width of transparent red and blue areas corresponds to the full width half maximum (FWHM)\non the energy axis \ftted from a Lorentzian curve, following the same color scheme. To highlight the di\u000berence of FWHM\nbetween the two damping modes, the FWHMs shown in the magnon spectrum of Fe 1\u0000xCox, Co, and Ni are multiplied by 20,\n5, and1\n2times, in this order. The triangles represent experimental results: in (a), Fe at 10 K [102] (yellow up) and Fe with\n12% Si at room-temperature [101] (green down); in (d), Co(9 ML)/Cu(100) at room-temperature [103] (green down); in (e) Ni\nat room-temperature (green down) [100]. The standard deviation of the peaks are represented as error bars.\nroom-temperature.\nIII. CONCLUSION\nWe have presented the in\ruence of non-local damping\non spin dynamics and magnon properties of elemental fer-\nromagnets (bcc Fe, fcc Co, fcc Ni) and the bcc Fe 70Co30\nand bcc Fe 50Co50alloys in the virtual-crystal approxima-\ntion. It is found that the non-local damping has impor-\ntant e\u000bects on relaxation processes and magnon prop-\nerties. Regarding the relaxation process, the non-local\ndamping in Fe, Co, and Ni has a negative contribution\nto the local (onsite) part, which accelerates the remagne-\ntization. Contrarily, in\ruenced by the positive contribu-\ntion of\u000bij(i6=j), the magnon lifetimes of Fe 70Co30and\nFe50Co50are increased in the non-local model, typically\nat the boundaries of the BZ, decelerating the remagneti-\nzation.\nConcerning the magnon properties, the non-local\ndamping has a signi\fcant e\u000bect in Co and Ni. More\nspeci\fcally, the magnon lifetimes can be overestimated\nby an order of magnitude in the e\u000bective model for these\ntwo materials. In real-space, this di\u000berence arises as a\nresult of strong negative non-local contributions in theclose neighborhood around the reference atom, namely\nthe NN in Ni and the third neighbors in Co.\nAlthough the e\u000bect of non-local damping to the\nstochastic thermal \feld in spin dynamics is not included\nin this work, we still obtain coherent magnon lifetimes\ncomparing to the analytical solution from linear response\ntheory. Notably, it is predicted that the magnon lifetimes\nat certain wave vectors are higher for the non-local damp-\ning model in some materials. An example is Fe 70Co30, in\nwhich the lifetime can be \u00183 times higher in the H\u0000N\npath for the non-local model. On the other hand, we\nhave proposed a fast method based on linear response\nto evaluate these lifetimes, which can be used to high-\nthroughput computations of magnonic materials.\nFinally, our study provides a link on how non-local\ndamping can be measured in FMR and neutron scat-\ntering experiments. Even further, it gives insight into\noptimising excitation of magnon modes with possible\nlong lifetimes. This optimisation is important for any\nspintronics applications. As a natural consequence of\nany real-space ab-initio formalism, our methodology and\n\fndings also open routes for the investigation of other\nmaterials with preferably longer lifetimes caused by non-\nlocal energy dissipation at low excitation modes. Such\nmaterials research could also include tuning the local10\nFIG. 4. Magnon lifetimes \u001cqof: (a) bcc Fe; (b) bcc Fe 70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni as function of q,\nshown in logarithmic scale. The color scheme is the same of Fig. 3, where blue and red represents \u001cqcomputed in the e\u000bective\nand non-local damping models. The transparent lines and opaque points depict the lifetimes calculated with Eq. 13 and by\nthe FWHM of S(q;!) atT= 0:1 K (see Eq. 17). The lifetime asymptotically diverges around the \u0000-point due to the absence\nof anisotropy e\u000bects or external magnetic \feld in the spin-Hamiltonian.\nchemical environments by doping or defects.\nIV. ACKNOWLEDGMENTS\nFinancial support from Vetenskapsr\u0017 adet (grant num-\nbers VR 2016-05980 and VR 2019-05304), and the\nKnut and Alice Wallenberg foundation (grant number\n2018.0060) is acknowledged. Support from the Swedish\nResearch Council (VR), the Foundation for Strategic Re-search (SSF), the Swedish Energy Agency (Energimyn-\ndigheten), the European Research Council (854843-\nFASTCORR), eSSENCE and STandUP is acknowledged\nby O.E. . Support from the Swedish Research Coun-\ncil (VR) is acknowledged by D.T. and A.D. . The\nChina Scholarship Council (CSC) is acknowledged by\nZ.L.. The computations/data handling were enabled by\nresources provided by the Swedish National Infrastruc-\nture for Computing (SNIC) at the National Supercom-\nputing Centre (NSC, Tetralith cluster), partially funded\nby the Swedish Research Council through grant agree-\nment No. 2016-07213.\n[1] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye,\nM. Krawczyk, J. Gr afe, C. Adelmann, S. Cotofana,\nA. Naeemi, V. I. Vasyuchka, et al. , J. Phys. Condens.\nMatter 33, 413001 (2021).\n[2] P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nat. Rev. Mater 6, 1114 (2021).\n[3] B. Rana and Y. 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Sakuma, J. Appl. Phys. 117, 013912 (2015).13\nAppendix A: Numerical solver\nIn this Appendix, the numerical method to solve Eq.\n1 is described. In previous studies, several numerical\napproaches have been proposed to solve the local LLG\nequations, including HeunP method, implicit midpoint\nmethod, Depondt-Merten's method [107], semi-implicit\nA (SIA) and semi-implicit B (SIB) methods [42]. To solve\nthis non-local LLG equation, we use the \fxed-point iter-\nation midpoint method. We have done convergence tests\non this method and \fnd that it preserve the energy and\nspin length of the system, which is demonstrated in Fig.\n5 for the case of a dimer. With stable outputs, the solver\nallows for a relatively large time step size, typically of\nthe order of \u0001 t\u00180:1\u00001 fs.\nFollowing the philosophy of an implicit midpoint\nmethod, the implemented algorithm can be described as\nfollows. Let mt\nibe the magnetic moment of site iat a\ngiven time step t. Then we can de\fne the quantity mmidand the time derivative of mi, respectively, as\nmmid=mt+1\ni+mt\ni\n2;\n@mi\n@t=mt+1\ni\u0000mt\ni\n\u0001t:(A1)\nUsing this de\fnition in Eq. 4, the equation of motion\nof thei-th spin becomes:\n@mi\n@t=mmid\u00020\n@\u0000\r[Bi(mmid) +bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA:\n(A2)\nThus, with a \fxed-point scheme, we can do the follow-\ning iteration\nmt+1(k+1)\ni =mt\ni+ \u0001t0\n@ \nmt+1(k)\ni +mt\ni\n2!\n\u00020\n@\u0000\r\"\nBi \nmt+1(k)\ni +mt\ni\n2!\n+bi(t)#\n+X\nj\u000bij\nmjmt+1(k)\nj\u0000mt\nj\n\u0001t1\nA1\nA:\n(A3)\nIfmt+1(k+1)\ni\u0019mt+1(k)\ni , the self-consistency con-\nverges. Typically, about 6 iteration steps are needed.\nThis solver was implemented in the software package Up-\npASD [48] for this work.\nAppendix B: Analytical model of anti-damping in\ndimers\nIn the dimer model, there are two spins on site 1 and\nsite 2 denoted by m1andm2, which are here supposed\nto be related to the same element { so that, naturally,\n\u000b11=\u000b22>0. Also, let's consider a su\u000eciently low\ntemperature so that bi(t)!0, which is a reasonable\nassumption, given that damping has an intrinsic origin\n[108]. This simple system allows us to provide explicit\nexpressions for the Hamiltonian, the e\u000bective magnetic\n\felds and the damping term. From the analytical solu-\ntion, it is found that the dimer spin system becomes an\nundamped system when local damping is equal to non-\nlocal damping, i.e.the e\u000bective damping of the system\nis zero.\nFollowing the de\fnition given by Eq. 4 in the main\ntext, the equation of motion for spin 1 reads:\n@m1\n@t=m1\u0002\u0012\n\u0000\rB1+\u000b11\nm1@m1\n@t+\u000b12\nm2@m2\n@t\u0013\n;(B1)\nand an analogous expression can be written for spin 2.\nFor sake of simplicity, the Zeeman term is zero and thee\u000bective \feld only includes the contribution from Heisen-\nberg exchange interactions. Thus, we have B1= 2J12m2\nandB2= 2J21m1. Withj\u000bijj\u001c 1, we can take the\nLL form@mi\n@t=\u0000\rmi\u0002Bito approximate the time-\nderivative on the right-hand side of the LLG equation.\nLetm1=m2and\u000b12=\u0015\u000b11. SinceJ12=J21and\nm1\u0002m2=\u0000m2\u0002m1, then we have\n@m1\n@t=\u00002\rJ12m1\u0002\u0014\nm2+ (1\u0000\u0015)\u000b11\nm1(m1\u0002m2)\u0015\n:\n(B2)\nTherefore, when \u000b12=\u000b21=\u000b11(i.e.,\u0015= 1), Eq. B1\nis reduced to:\n@m1\n@t=\u00002\rJ12m1\u0002m2; (B3)\nand the system becomes undamped. It is however\nstraightforward that, for the opposite case of a strong\nnegative non-local damping ( \u0015=\u00001), Eq. B2 describes\na common damped dynamics. A side (and related) con-\nsequence of Eq. B2, but important for the discussion in\nSection II B, is the fact that the e\u000bective onsite damp-\ning term\u000b\u0003\n11= (1\u0000\u0015)\u000b11becomes less relevant to the\ndynamics as the positive non-local damping increases\n(\u0015!1), or, in other words, as \u000btot= (\u000b11+\u000b12) strictly\nincreases due to the non-local contribution. Exactly the\nsame reasoning can be made for a trimer, for instance,\ncomposed by atoms with equal moments and exchange\ninteractions ( m1=m2=m3,J12=J13=J23), and\nsame non-local dampings ( \u000b13=\u000b12=\u0015\u000b11).14\nThe undamped behavior can be directly observed from\nASD simulations of a dimer with \u000b12=\u000b11, as shown in\nFig. 5. Here the magnetic moment and the exchange are\ntaken the same of an Fe dimer, m1= 2:23\u0016BandJ12=\n1:34 mRy. Nevertheless, obviously the overall behavior\ndepicted in Fig. 5 is not dependent on the choice of\nm1andJ12. Thezcomponent is constant, while the x\nandycomponents of m1oscillate in time, indicating a\nprecessing movement.\nIn a broader picture, this simple dimer case exempli\fes\nthe connection between the eigenvalues of the damping\nmatrix\u000b= (\u000bij) and the damping behavior. The occur-\nrence of such undamped dynamics has been recently dis-\ncussed in Ref. [109], where it is shown that a dissipation-\nfree mode can occur in a system composed of two sub-\nsystems coupled to the same bath.\n0.00 0.02 0.04 0.06 0.08 0.10\nt(ps)0.2\n0.00.20.40.60.81.0Magnetization\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\nEnergy(mRy)mxmymzmEnergy\nFIG. 5. Spin dynamics at T= 0 K of an undamped dimer\nin which\u000b12=\u000b21=\u000b11(see text). The vector m1is\nnormalized and its Cartesian components are labeled in the\n\fgure asmx,myandmz. The black and grey lines indicate\nthe length of spin and energy (in mRy), respectively.\nAppendix C: E\u000bective and onsite damping in the\nFeCo and CoNi alloys\nAs mentioned in Section I, the simple VCA model al-\nlows us to account for the disorder in 3 d-transition-metal\nalloys in a crude but e\u000ecient way which avoids the use\nof large supercells with random chemical distributions.\nWith exactly the same purpose, the coherent potential\napproximation (CPA) [110] has also been employed to\nanalyze damping in alloys ( e.g., in Refs. [84, 111, 112]),\nshowing a very good output with respect to trends, when\ncompared to experiments [33, 81]. In Fig. 6 we show\nthe normalized calculated local (onsite, \u000bii) and e\u000bec-\ntive damping ( \u000btot) parameters for the zero-temperature\nVCA Fe 1\u0000xCoxalloy in the bcc structure, consistent with\na concentration up to x\u001960% of Co [33]. The computed\nvalues in this work (blue, representing \u000bii, and red points,\nrepresenting \u000btot) are compared to previous theoretical\nCPA results and room-temperature experimental data.\nThe trends with VCA are reproduced in a good agree-ment with respect to experiments and CPA calculations,\nshowing a minimal \u000btotwhen the Co concentration is\nx\u001930%. This behavior is well correlated with the local\ndensity of states (LDOS) at the Fermi level, as expected\nby the simpli\fed Kambersk\u0013 y equation [113], and the on-\nsite contribution. Despite the good agreement found, the\nvalues we have determined are subjected to a known error\nof the VCA with respect to the experimental results.\nThis discrepancy can be partially explained by three\nreasons: ( i) the signi\fcant in\ruence of local environ-\nments (local disorder and/or short-range order) to \u000btot\n[25, 77]; ( ii) the fact that the actual electronic lifetime\n(i.e., the mean time between two consecutive scattering\nevents) is subestimated by the VCA average for random-\nness in the FeCo alloy, which can have a non-negligible\nimpact in the damping parameter [22, 114]; and ( iii) the\nin\ruence on damping of noncollinear spin con\fgurations\nin \fnite temperature measurements [54, 115]. On top of\nthat, it is also notorious that damping is dependent on\nthe imaginary part of the energy (broadening) [22, 114],\n\u000e, which can be seen as an empirical quantity, and ac-\ncounts for part of the di\u000berences between theory and ex-\nperiments.\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035\n 0  10  20  30  40  50  60 0 5 10 15 20 25Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)onsite  (αii)\ntotal  (αtot)\nTurek et al.  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nMankovsky  et al.  [2018]  (αtot)\nSchoen et al.\nn(EF)\nFIG. 6. (Color online) Left scale : Computed Gilbert e\u000bec-\ntive (\u000btot, red circles) and onsite ( \u000bii, blue squares) damping\nparameters as a function of Co the concentration ( x) for bcc\nFe1\u0000xCoxbinary alloy in the virtual-crystal approximation.\nThe values are compared with previous theoretical results us-\ning CPA, from Ref. [84] (gray full triangles), Ref. [54] (black\nopen rhombus), Ref. [112] (yellow open triangles), and room-\ntemperature experimental data [33]. Right scale : The calcu-\nlated density of states (DOS) at the Fermi level as a function\nofx, represented by the black dashed line.\nIn the spirit of demonstrating the e\u000bectiveness of the\nsimple VCA to qualitatively (and also, to some extent,\nquantitatively) describe the properties of Gilbert damp-\ning in suitable magnetic alloys, we also show in Fig. 7 the\nresults obtained for Co xNi1\u0000xsystems. The CoNi alloys15\nare known to form in the fcc structure for a Ni concen-\ntration range of 10% \u0000100%. Therefore, here we mod-\neled CoxNi1\u0000xby a big fcc cluster containing \u0018530000\natoms in real-space with the equilibrium lattice parame-\nter ofa= 3:46\u0017A. The number of recursion levels consid-\nered isLL= 41. A good agreement with experimental\nresults and previous theoretical calculations can be no-\nticed. In particular, the qualitative comparison with the-\nory from Refs. [81, 84] indicates the equivalence between\nthe torque correlation and the spin correlation models\nfor calculating the damping parameter, which was also\ninvestigated by Sakuma [116]. The onsite contribution\nfor each Co concentration, \u000bii, is omitted from Fig. 7\ndue to an absolute value 2 \u00004 times higher than \u000btot,\nbut follows the same decreasing trend. Again, the over-\nall e\u000bective damping values are well correlated with the\nLDOS, and re\rect the variation of the quantity1\nmtwith\nCo concentration (see Eq. 15).\n 0 0.005 0.01 0.015 0.02 0.025\n 0  10  20  30  40  50  60  70 10 15 20 25 30Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)total  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nStarikov et al.  (αtot)\nSchoen [2017] et al.\nn(EF)\nFIG. 7. (Color online) Left scale : Computed Gilbert e\u000bective\n(\u000btot, red circles) damping parameters as a function of the Co\nconcentration ( x) for fcc Co xNi1\u0000xbinary alloy in the virtual-\ncrystal approximation. The values are compared with previ-\nous theoretical results using CPA, from Ref. [84] (gray full\ntriangles), Ref. [81] (gold full circles), and room-temperature\nexperimental data [89]. Right scale : The calculated density of\nstates (DOS) at the Fermi level as a function of x, represented\nby the black dashed line.\nAppendix D: E\u000bect of further neighbors in the\nmagnon lifetimes\nWhen larger cuto\u000b radii ( Rcut) of\u000bijparameters are\nincluded in ASD, Eq. A3 takes longer times to achieve a\nself-consistent convergence. In practical terms, to reach a\nsizeable computational time for the calculation of a given\nsystem,Rcutneeds to be chosen in order to preserve the\nmain features of the magnon properties as if Rcut!1 .\nA good quantity to rely on is the magnon lifetime \u001cq,as it consists of both magnon frequency and q-resolved\ndamping (Eq. 13). In Section II C, we have shown the\nequivalence between Eq. 13 and the inverse of FWHM\non the energy axis of S(q;!) for the ferromagnets inves-\ntigated here. Thus, the comparison of two \u001cqspectra for\ndi\u000berentRcutcan be done directly and in an easier way\nusing Eq. 13.\nFIG. 8. (Color online) Magnon lifetimes calculated using Eq.\n13 for: (a) bcc Fe; and (b) bcc Fe 50Co50, using a reduced set\nof 16 NN shells (opaque lines), and the full set of 136 NN\nshells (transparent lines).\nAn example is shown in Figure 8 for bcc Fe and bcc\nFe50Co50. Here we choose the \frst 16 NN ( Rcut\u00183:32a)\nand compare the results with the full calculated set of\n136 NN (Rcut= 10a). It is noticeable that the reduced\nset of neighbors can capture most of the features of the\n\u001cqspectrum for a full NN set. However, long-range in-\n\ruences of small magnitudes, such as extra oscillations\naround the point q=Hin Fe, can occur. In particu-\nlar, these extra oscillations arise mainly due to the pres-\nence of Kohn anomalies in the magnon spectrum of Fe,\nalready reported in previous works [52, 80]. In turn, for\nthe case of Fe 50Co50, the long-range \u000bijreduces\u000btot, and\ncauses the remagnetization times for non-local and e\u000bec-16\ntive dampings to be very similar (see Fig. 2). For the\nother ferromagnets considered in the present research,comparisons of the reduced Rcutwith analogous quality\nwere reached."    },    {        "title": "2010.00642v1.Modeling_coupled_spin_and_lattice_dynamics.pdf",        "content": "Modeling coupled spin and lattice dynamics\nMara Strungaru,1Matthew O A Ellis,2Sergiu Ruta,3Oksana Chubykalo-Fesenko,4Richard F L Evans,1and Roy W Chantrell1\n1Department of Physics, University of York, York, United Kingdom\n2Department of Computer Science, University of Sheffield, Sheffield, United Kingdom\n3Department of Physics,University of York, York, United Kingdom\n4Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain\nA unified model of molecular and atomistic spin dynamics is presented enabling simulations both in micro-\ncanonical and canonical ensembles without the necessity of additional phenomenological spin damping. Trans-\nfer of energy and angular momentum between the lattice and the spin systems is achieved by a coupling term\nbased upon the spin-orbit interaction. The characteristic spectra of the spin and phonon systems are analyzed\nfor different coupling strength and temperatures. The spin spectral density shows magnon modes together with\nthe uncorrelated noise induced by the coupling to the lattice. The effective damping parameter is investigated\nshowing an increase with both coupling strength and temperature. The model paves the way to understanding\nmagnetic relaxation processes beyond the phenomenological approach of the Gilbert damping and the dynamics\nof the energy transfer between lattice and spins.\nI. INTRODUCTION\nWith the emergent field of ultrafast magnetisation\ndynamics1understanding the flow of energy and angular mo-\nmentum between electrons, spins and phonons is crucial for\nthe interpretation of the wide range of observed phenom-\nena2–5. For example, phonons strongly pumped in the THz\nregime by laser excitation can modulate the exchange field\nand manipulate the magnetisation as shown for the magnetic\ninsulator YIG6or in Gd7. The excitation of THz phonons\nleads to a magnetic response with the same frequency in\nGd7, proving the necessity of considering the dynamics of\nboth lattice and spins. Phonon excitations can modulate both\nanisotropy and exchange which can successfully manipulate\n8–10or potentially switch the magnetisation11,12, ultimately\nleading to the development of low-dissipative memories.\nMagnetisation relaxation is typically modeled using the\nphenomenological description of damping proposed by Lan-\ndau and Lifshitz13and later Gilbert14, where the precessional\nequation of motion is augmented by a friction-like term, re-\nsulting in the Landau-Lifshitz-Gilbert (LLG) equation. This\nrepresents the coupling of the magnetic modes (given pri-\nmarily by the localised atomic spin) with the non-magnetic\nmodes (lattice vibrations and electron orbits). The LLG equa-\ntion and its generalisations can be deduced from the quantum-\nmechanical approaches assuming an equilibrium phonon bath\nand the weak coupling of the spin to the bath degrees of\nfreedom15–17. Thus the standard approach works on the sup-\nposition that the time scales between the environmental de-\ngrees of freedom and the magnetic degrees of freedom are\nwell separated and reducing the coupling between the mag-\nnetization and its environment to a single phenomenological\ndamping parameter18,19. In reality, the lattice and magneti-\nsation dynamics have comparable time-scales, where the in-\nteraction between the two subsystems represents a source of\ndamping, hence the necessity of treating spin and lattice dy-\nnamics in a self-consistent way.\nTo investigate these phenomena, and aiming at predictive\npower for the design of competitive ultrafast magnetic nano-\ndevices, advanced frameworks beyond conventional micro-magnetics and atomistic spin dynamics20are needed21. A\ncomplete description of magnetic systems includes the inter-\naction between several degrees of freedom, such as lattice,\nspins and electrons, modeled in a self-consistent simulation\nframework. The characteristic relaxation timescales of elec-\ntrons are much smaller ( \u0019fs) in comparison to spin and lattice\n(100fs\u0000ps), hence magnetisation relaxation processes can be\ndescribed via coupled spin and lattice dynamics, termed Spin-\nLattice Dynamics (SLD)22–29. SLD models can be crucial in\ndisentangling the interplay between these sub-systems.\nSLD models have so far considered either micro-canonical\n(NVE - constant particle number, volume and energy)27,28or\ncanonical (NVT - constant particle number, volume and tem-\nperature) ensembles with two Langevin thermostats connected\nto both lattice and spin subsystems23,30. Damping due to spin-\nlattice interactions only within the canonical ensemble (NVT)\nhas not yet been addressed, but is of interest in future mod-\nelling of magnetic insulators at finite temperature. Here we\nintroduce a SLD model capable of describing both ensem-\nbles. Specifically, our model (i) takes into account the transfer\nof angular momentum from spin to lattice and vice-versa, (ii)\nworks both in a micro-canonical ensemble (constant energy)\nand in a canonical ensemble (constant temperature), (iii) al-\nlows a fixed Curie temperature of the system independent of\nthe spin-lattice coupling strength, (iv) disables uniform trans-\nlational motion of the system and additional constant energy\ndrift, which can be produced by certain spin-lattice coupling\nforms. Furthermore, in this work, the characteristics of the in-\nduced spin-lattice noise, the magnon-phonon induced damp-\ning and the equilibrium properties of the magnetic system has\nbeen systematically investigated.\nThe paper is organised as follows. We start by describ-\ning the computational model of Spin-Lattice Dynamics and\nthe magnetic and mechanical energy terms used in this frame-\nwork (Section II). We then explore the equilibrium properties\nof the system for both microcanonical and canonical simula-\ntions, proving that our model is able to efficiently transfer both\nenergy and angular momentum between the spin and lattice\ndegrees of freedom. In Section III we compute the equilibrium\nmagnetisation as function of temperature for both a dynamic\nand static lattice and we show that the order parameter is notarXiv:2010.00642v1  [cond-mat.mtrl-sci]  1 Oct 20202\ndependent on the details of the thermostat used. In Section IV\nwe analyse the auto-correlation functions and spectral char-\nacteristics of magnon, phonons and the coupling term prov-\ning that the pseudo-dipolar coupling efficiently mediates the\ntransfer of energy from spins to the lattice and vice-versa. We\nthen calculate the temperature and coupling dependence of the\ninduced magnon-phonon damping and we conclude that the\nvalues agree well with damping measured in magnetic insula-\ntors, where the electronic contributions to the damping can be\nneglected ( Section V).\nII. COMPUTATIONAL MODEL\nIn order to model the effects of both lattice and spin dy-\nnamics in magnetic materials an atomistic system is adopted\nwith localised atomic magnetic moments at the atomic coor-\ndinates. Within this framework there are now nine degrees of\nfreedom; atomic magnetic moment (or spin) S, atomic posi-\ntionrand velocity v. The lattice and the magnetic system can\ndirectly interact with each other via the position and spin de-\npendent Hamiltonians. The total Hamiltonian of the system\nconsists of a lattice Hlatand magnetic Hmagparts:\nHtot=Hlat+Hmag: (1)\nThe lattice Hamiltonian includes the classical kinetic and\npairwise inter-atomic potential energies:\nHlat=å\nimiv2\ni\n2+1\n2å\ni;jU(ri j): (2)\nOur model considers a harmonic potential (HP) defined as:\nU(ri j) =(\nV0(ri j\u0000r0\ni j)2=a2\n0ri j<rc\n0 ri j>rc:(3)\nwhere V0has been parametrised for BCC Fe in27anda0=1˚A\nis a dimension scale factor. To be more specific we consider\nthe equilibrium distances r0\ni jcorresponding to a symmetric\nBCC structure. The interaction cut-off is rc=7:8˚A. The pa-\nrameters of the potential are given in Table II. The harmonic\npotential has been used for simplicity, however it can lead to\nrather stiff lattice for a large interaction cutoff.\nAnother choice of the potential used in our model is an an-\nharmonic Morse potential (MP) parameterised in31for BCC\nFe and defined as:\nU(ri j) =(\nD[e\u00002a(ri j\u0000r0)\u00002e\u0000a(ri j\u0000r0)]ri j<rc\n0 ri j>rc(4)\nThe Morse potential approximates well the experimental\nphonon dispersion observed experimentally for BCC Fe32as\nshown in33. The phonon spectra for the choices of potential\nused in this work are given in Section IV. Other nonlinear\nchoices of potential can be calculated via the embedded atom\nmethod34,35.The spin Hamiltonian ( Hmag) used in our simulations con-\nsists of contributions from the exchange interaction, Zeeman\nenergy and a spin-lattice coupling Hamiltonian, given by the\npseudo-dipolar coupling term ( Hc), which we will describe\nlater:\nHmag=\u00001\n2å\ni;jJ(ri j)(Si\u0001Sj)\u0000å\nimiSi\u0001Happ+Hc;(5)\nwhere miis the magnetic moment of atom i,Siis a unit\nvector describing its spin direction and Happis an external ap-\nplied magnetic field. The exchange interactions used in our\nsimulations depend on atomic separation J(ri j). They were\ncalculated from first principle methods for BCC Fe by Ma et\nal23and follow the dependence:\nJ(ri j) =J0\u0012\n1\u0000ri j\nrc\u00133\nQ(rc\u0000ri j); (6)\nwhere rcis the cutoff and Q(rc\u0000ri j)is the Heaviside step\nfunction, which implies no exchange coupling between spins\nsituated at larger distance than rc.\nSeveral previous SLD models suffered from the fact that\nthey did not allow angular momentum transfer between lattice\nand spin systems28. This happened for magnetisation dynam-\nics in the absence of spin thermostat, governed by symmetric\nexchange only, due to total angular momentum conservation.\nTo enable transfer of angular momentum, Perera et al26have\nproposed local anisotropy terms to mimic the spin-orbit cou-\npling phenomenon due to symmetry breaking of the local en-\nvironment. Their approach was successful in thermalising the\nsubsystems, however, single site anisotropy spin terms with\na position dependent coefficients as employed in26can induce\nan artificial collective translational motion of the sample while\nthe system is magnetically saturated, due to the force \u0000¶Htot\n¶riproportional to spin orientation. To avoid large collective mo-\ntion of the atoms in the magnetic saturated state, we consider\na two-site coupling term, commonly known as the pseudo-\ndipolar coupling, described by\nHc=\u0000å\ni;jf(ri j)\u0014\n(Si\u0001ˆri j)(Sj\u0001ˆri j)\u00001\n3Si\u0001Sj\u0015\n: (7)\nThe origin of this term still lies in the spin-orbit interaction,\nappearing from the dynamic crystal field that affects the elec-\ntronic orbitals and spin states. It has been employed previ-\nously in SLD simulations22,27. It was initially proposed by\nAkhiezer36, having the same structure of a dipolar interac-\ntion, however with a distance dependence that falls off rapidly,\nhence the name pseudo-dipolar interaction. The exchange-\nlike term\u00001\n3Si\u0001Sjis necessary in order to preserve the Curie\ntemperature of the system under different coupling strengths\nand to ensure no net anisotropy when the atoms form a sym-\nmetric cubic lattice. For the mechanical forces, the exchange\nlike term eliminates the anisotropic force that leads to a non-\nphysical uniform translation of the system when the mag-\nnetic system is saturated. The magnitude of the interactions3\nis assumed to decay as f(ri j) =CJ0(a0=ri j)4as presented\nin27with Ctaken as a constant, for simplicity measured rel-\native to the exchange interactions and a0=1˚A is a dimen-\nsion scale factor. The constant Ccan be estimated from\nab-initio calculations26, approximated from magneto-elastic\ncoefficients27, or can be chosen to match the relaxation times\nand damping values, as in this work.\nSince the total Hamiltonian now depends on the coupled\nspin and lattice degrees of freedom ( vi,ri,Si), the following\nequations of motion (EOM) need to be solved concurrently to\nobtain the dynamics of our coupled system:\n¶ri\n¶t=vi; (8)\n¶vi\n¶t=\u0000hvi+Fi\nmi; (9)\n¶Si\n¶t=\u0000gSi\u0002Hi; (10)\nFi=\u0000¶Htot\n¶ri+Gi; (11)\nHi=\u00001\nmSm0¶Htot\n¶Si; (12)\nwhere FiandHirepresent the effective force and field, Girep-\nresents the fluctuation term (thermal force) and hrepresents\nthe friction term that controls the dissipation of energy from\nthe lattice into the external thermal reservoir. The strength of\nthe fluctuation term can be calculated by converting the dissi-\npation equations into a Fokker-Planck equation and then cal-\nculating the stationary solution. The thermal force has the\nform of a Gaussian noise:\nhGia(t)i=0; (13)\nhGia(t)Gjb(t0)i=2hkBT\nmidabdi jd(t\u0000t0): (14)\nTo prove that the complete interacting many-body spin-\nlattice framework presented in here is in agreement with the\nfluctuation-dissipation theorem, we have followed the ap-\nproach presented by Chubykalo et al37based on the Onsager\nrelations. Linearising the equation of motion for spins, we find\nthat the kinetic coefficients for the spin system are zero, due\nto the fact that the spin and internal field are thermodynamic\nconjugate variables. Hence, if the noise applied to the lattice\nobeys the fluctuation dissipation theory, the coupled system\nwill obey it as well, due to the precessional form of the equa-\ntion of motion for the spin.\nWe compare the SLD model presented here with other ex-\nisting model that do not take into account the lattice degrees\nof freedom (Atomistic Spin Dynamics - ASD). Particularly,\nin our case we assume a fixed lattice positions. The summary\nof the comparison is presented in Table I. Atomistic spin dy-\nnamics simulations (ASD)18,20,38,39have been widely used to\nstudy finite size effects, ultrafast magnetisation dynamics and\nnumerous other magnetic phenomena. Here the intrinsic spin\ndamping (the Gilbert damping - aG) is phenomenologically\nincluded. In our case since the lattice is fixed it is assumedModel Lattice Lattice Spin Intrinsic Spin\nthermostat thermostat damping\nSLD Dynamic On Off Phonon\ninduced\nASD Fixed Off On Electronic\nmainly\nTABLE I. Summary comparison of the SLD model developed here\nagainst other spin dynamics models.\nQuantity Symbol Value Units\nExchange23J0 0:904 eV\nrc 3:75 ˚A\nHarmonic potential27V0 0:15 eV\nrc 7:8 ˚A\nMorse potential31D 0:4174 eV\na 1:3885 ˚A\nr0 2:845 ˚A\nrc 7:8 ˚A\nMagnetic moment ms 2:22 mB\nCoupling constant C 0:5\nMass m 55:845 u\nLattice constant a 2:87 ˚A\nLattice damping h 0:6 s\u00001\nTABLE II. Parameters used in the spin-lattice model to simulate BCC\nFe.\nto come from electronic contributions. Consequently, only 3\nequations of motion per atom describing the spin dynamics\nare used:\n¶Si\n¶t=\u0000g\n(1+a2\nG)Si\u0002(Hi+aGSi\u0002Hi) (15)\nwith an additional field coming from the coupling to the\nfixed lattice positions. The temperature effects are introduced\nin spin variables by means of a Langevin thermostat. The spin\nthermostat is modeled by augmenting the effective fields by a\nthermal stochastic field ( Hi=xi\u0000¶H=¶Si) and its proper-\nties also follow the fluctuation-dissipation theorem:\nhxia(t)i=0; (16)\nhxia(t)xjb(t0)i=2aGkBT\ngmSda;bdi jd(t\u0000t0): (17)\nThe characteristics of the above presented models are sum-\nmarised in Table I.\nTo integrate the coupled spin and lattice equations of mo-\ntion we used a Suzuki-Trotter decomposition (STD) method40\nknown for its numerical accuracy and stability. The scheme\ncan integrate non-commuting operators, such as is the case of\nspin-lattice models and conserves the energy and space-phase\nvolume. The conservation of energy is necessary when deal-\ning with microcanonical simulations. Considering the gener-\nalized coordinate X=fr;v;Sgthe equations of motion can be4\nre-written using the Liouville operators:\n¶X\n¶t=ˆLX(t)= (ˆLr+ˆLv+ˆLS)X(t): (18)\nThe solution for the Liouville equation is X(t+Dt) =\neLDtX(t). Hence, following the form of this solution and ap-\nplying a Suzuki-Trotter decomposition as in Tsai’s work41,42,\nwe can write the solution as:\nX(t+Dt) =eˆLsDt\n2eˆLvDt\n2eˆLrDteˆLvDt\n2eˆLsDt\n2X(t)+O(Dt3);(19)\nwhere Ls;Lv;Lrare the Liouville operators for the spin, veloc-\nity and position. This update can be abbreviated as (s,v,r,v,s)\nupdate. The velocity and position are updated using a first\norder update, however the spin needs to be updated using a\nCayley transform43,44, due to the fact that the norm of each\nindividual spin needs to be conserved. Thus we have\neˆLvDtvi=vi+Dt\nmiFi; (20)\neˆLrDtri=ri+Dtvi; (21)\neˆLSDtSi=Si+DtHi\u0002Si+Dt2\n2\u0002\n(Hi\u0001Si)Hi\u00001\n2H2\niSi\u0003\n1+1\n4Dt2H2\ni:(22)\nThe spin equations of motions depend directly on the neigh-\nbouring spin orientations (through the effective field) hence\nindividual spins do not commute with each other. We need to\nfurther decompose the spin system ˆLs=åiˆLsi. The following\ndecomposition will be applied for the spin system:\neˆLs(Dt=2)=eˆLs1(Dt=4):::eˆLsN(Dt=2):::eˆLs1(Dt=4)+O(Dt3)(23)\nTests of the accuracy of the integration have been per-\nformed by checking the conservation of energy within the mi-\ncrocanonical ensemble. To ensure that the spin and lattice\nsub-systems have reached equilibrium, we calculate both the\nlattice temperature (from the Equipartition Theorem) and spin\ntemperature45. These are defined as:\nTL=2\n3NkBå\nip2\ni\n2m;TS=åi(Si\u0002Hi)2\n2kBåiSi\u0001Hi: (24)\nIII. SPIN-LATTICE THERMALISATION\nAs an initial test of our model we look at the thermalisation\nprocess within micro-canonical (NVE) and canonical (NVT)\nsimulations for a periodic BCC Fe system of 10\u000210\u000210 unit\ncells. No thermostat is applied directly to the spin system and\nits thermalisation occurs via transfer of energy and angular\nmomentum from the lattice, i.e. via the magnon-phonon inter-\naction. In the case of the NVE simulations, the energy is de-\nposited into the lattice by randomly displacing the atoms from\nan equilibrium BCC structure positions within a 0 :01˚A radius\nsphere and by initialising their velocities with a Boltzmann\nFIG. 1. NVE (top) and NVT (bottom) simulations for a 10 \u000210\u000210\nunit cell BCC Fe system. The spin system is randomly initialised with\na temperature of 1900 K, while the lattice velocities are initialised\nby a Boltzmann distribution at T=300 K. In both cases we obtain\nequilibration of the two subsystems on the ps timescale.\ndistribution at T=300 K. The spin system is initialised ran-\ndomly in the x\u0000yplane with a constant component of mag-\nnetisation of 0.5 in the out of plane ( z) direction. In the case\nof NVT simulations, the lattice is connected to a thermostat\nat a temperature of T=300 K. The parameters used in the\nsimulations are presented in Table II.\nFig. 1 shows the thermalisation process for the two types of\nsimulation. In both cases the spin system has an initial temper-\nature of T=1900 K due to the random initialisation. For the\nNVE simulations, the two subsystems are seen to equilibrate\nat a temperature of T=600 K, this temperature being depen-\ndent on the energy initially deposited into the system. In the\nNVT simulations, the lattice is thermalised at T=300 K fol-\nlowed by the relaxation of the spin towards the same temper-\nature. In both cases we observe that the relaxation of the spin\nsystem happens on a 100 ps timescale, corresponding to typi-\ncal values for spin-orbit relaxation. The corresponding change\nin the magnetisation is emphasized by the green lines in Fig.\n1 showing a transfer of angular momentum between the spin\nand lattice degrees of freedom.\nTo gain a better understanding of properties at thermal equi-\nlibrium within the Spin-Lattice Dynamics model, we have in-\nvestigated the temperature dependence of the magnetic order\nparameter in different frameworks that either enable or dis-\nable lattice dynamics, specifically: SLD or ASD. Tab. I il-\nlustrates the differences between the models. Since reaching\njoint thermal equilibrium depends strongly on the randomness\nalready present in the magnetic system this process is acceler-\nated by starting with a reduced magnetisation of M=MS=0:95\nFIG. 2. Magnetisation versus temperature curves for the SLD model\n(with different choices of lattice potential: MP-Morse Potential, HP-\nHarmonic Potential) and fixed lattice ASD model. The inset zooms\naround the ferromagnetic to paramagnetic phase transition tempera-\nture.\nforT>300 K.\nFig. 2 shows the comparison of the equilibrium magnetisa-\ntion using either the harmonic potential (HP), Morse potential\n(MP) or fixed lattice (ASD) simulations. The magnetisation\nis calculated by averaging for 200 ps after an initial equilibra-\ntion for 800 ps (for SLD type simulations) or 100 ps (for ASD)\nsimulations. We observe that even without a spin thermostat\n(in SLD model) the magnetisation reaches equilibrium via the\nthermal fluctuations of the lattice, proving that both energy\nand angular momentum can be successfully transferred be-\ntween the two sub-systems. Additionally, both the SLD and\nASD methods give the same equilibrium magnetisation over\nthe temperature range considered. This confirms that the equi-\nlibrium quantities are independent of the details of the thermo-\nstat used, in agreement with the fact that both SLD and ASD\nmodels obey the fluctuation-dissipation theorem.\nIn principle, since the strength of the exchange interaction\ndepends on the relative separation between the atoms, any\nthermal expansion of the lattice could potentially modify the\nCurie temperature. However, as highlighted in the inset of\nFig. 2, the same Curie temperature is observed in each model.\nWe attribute this to fact that the thermal lattice expansion is\nsmall in the temperature range considered due to two reasons:\ni) the Curie temperature of the system is well below the melt-\ning point of Fe (\u00191800K) and ii) we model a bulk, constant-\nvolume system with periodic boundary conditions that does\nnot present strong lattice displacements due to surfaces. We\nnote that Evans et al46found a reduction of TCin nanoparticles\ndue to an expansion of atomic separations at the surface that\nconsequently reduces the exchange interactions. For systems\nwith periodic boundary conditions we anticipate fluctuations\nin the exchange parameter due to changes in interatomic spac-\nings to be relatively small. Although the equilibrium proper-\nties are not dependent on the details of the thermostat or thepotential, the magnetisation dynamics could be strongly influ-\nenced by these choices.\nThe strength of the pseudo-dipolar coupling parameters C\ndetermines the timescale of the thermalisation process. Its\nvalue can be parametrised from magneto-elastic simulations\nvia calculations of the anisotropy energy as a function of\nstrain. The magneto-elastic Hamiltonian can be written for a\ncontinuous magnetisation Mand elastic strain tensor eas47,48:\nHm\u0000e=B1\nM2\nSå\niM2\nieii+B2\nM2så\niMiMjei j (25)\nwhere constants B1;B2can be measured experimentally49.\nThe pseudo-dipolar term acts as a local anisotropy, however,\nfor a lattice distorted randomly, this effective anisotropy is av-\neraged out to zero. At the same time under external strain\neffects, an effective anisotropy will arise due to the pseudo-\ndipolar coupling which is the origin of the magneto-elastic\neffects. To calculate the induced magnetic anisotropy energy\n(MAE), the BCC lattice is strained along the zdirection whilst\nfixed in the xyplane. The sample is then uniformly rotated and\nthe energy barrier is evaluated from the angular dependence of\nthe energy. Fig. 3 shows MAE for different strain values and\ncoupling strengths, with the magneto-elastic energy densities\nconstants B1obtained from the linear fit. The values of the ob-\ntained constants B1are larger than the typical values reported\nforBCC FeB1=\u00003:43 MJ m\u00003=\u00006:2415\u000210\u00006eV A\u0000349\nmeasured at T=300 K. Although the obtained magneto-\nelastic coupling constants for BCC Fe are larger than experi-\nmental values, it is important to stress that, as we will see later,\na large coupling is necessary in order to obtain damping pa-\nrameters comparable to the ones known for magnetic insula-\ntors where the main contribution comes from magnon-phonon\nscattering. In reality, in BCC Fe there is an important contribu-\ntion to the effective damping from electronic sources, which if\nconsidered, can lead to the smaller coupling strengths, consis-\ntent in magnitude with experimental magneto-elastic parame-\nters. Indeed, as we will show later, our finding suggests that\nphonon damping is a very small contribution in metallic sys-\ntems such as BCC Fe .\nIV . DYNAMIC PROPERTIES AT THERMAL\nEQUILIBRIUM\nSection III showed that the equilibrium magnetisation does\nnot depend on the details of the thermostat used and a success-\nful transfer of both energy and angular momentum is achieved\nbetween the spin and lattice sub-systems by the introduction\nof a pseudo-dipolar coupling term. In this section, we inves-\ntigate the properties of the magnons, phonons and the cou-\npling term that equilibrates the spin and phonon systems in\nthe absence of a phenomenological spin damping. Two types\nof simulations are presented here: i)magnon and phonon\nspectra calculated along the high symmetry path of a BCC\nlattice and ii)averaged temporal Fourier transform (FT) of\nindividual atoms datasets (spin, velocity, pseudo-dipolar cou-\npling field). The phonon - Fig. 4 and magnon - Fig. 5 spec-6\nFIG. 3. Magnetic anisotropy energy as function of strain for different\ncoupling strengths for T=0K.\ntra are calculated by initially equilibrating the system for 10\nps with a spin thermostat with aG=0:01 and a coupling of\nC=0:5, followed by 10 ps of equilibration in the absence\nof a spin thermostat. For the method i)the correlations are\ncomputed for a runtime of 20 ps after the above thermalisa-\ntion stage. For each point in k-space, the first three maxima\nof the auto-correlation function are plotted for better visual-\nisation. The auto-correlation function is projected onto the\nfrequency space and the average intensity is plotted for dif-\nferent frequencies. The phonon spectra are calculated from\nthe velocity auto-correlation function defined in Fourier space\nas33,50:\nAp(k;w) =Ztf\n0hvp\nk(t)vp\nk(t\u0000t0)ie\u0000iwtdt (26)\nwhere p=x;y;z,tfis the total time and vp\nk(t)is the spatial\nFourier Transform calculated numerically as a discrete Fourier\nTransform:\nvp\nk(t) =å\nivp\nie\u0000ik\u0001ri (27)\nThe same approach is applied for the magnon spectra, us-\ning the dynamical spin structure factor, which is given by\nthe space-time Fourier transform of the spin-spin correlation\nfunction defined as Cmn(r\u0000r0;t\u0000t0) =<Sm(r;t)Sn(r0;t0)>,\nwith m;ngiven by the x,y,z components51:\nSmn(k;w) =å\nr;r0eik\u0001(r\u0000r0)Ztf\n0Cmn(r\u0000r0;t\u0000t0)e\u0000iwtdt(28)\nThe second method ( ii)) to investigate the properties of the\nsystem involves calculating temporal Fourier transform of in-\ndividual atoms datasets, and averaging the Fourier response\nover 1000 atoms of the system. This response represents an\nintegrated response over the k-space. Hence, the projection of\nintensities on the frequency space presented by method i)has\nsimilar features as the spectra presented by method ii). For theresults presented in Fig. 6, a system of 10 \u000210\u000210BCC unit\ncells has been chosen. The system has been equilibrated for\na total time of 20 ps with the method presented in i)and the\nFast Fourier transform (FFT) is computed for the following\n100 ps.\nFig. 4 shows the phonon spectra for a SLD simulations at\nT=300K, C=0:5 for the Morse Potential - Fig. 4(a) and the\nHarmonic Potential - Fig. 4(b) calculated for the high sym-\nmetry path of a BCC system with respect to both energy and\nfrequency units. The interaction cutoff for both Morse and\nHarmonic potential is rc=7:8˚A. The Morse phonon spec-\ntrum agrees well with the spectrum observed experimentally32\nand with the results from33. The projection of the spectra onto\nthe frequency domain shows a peak close to 10.5 THz, due\nto the overlap of multiple phonon branches at that frequency\nand consequently a large spectral density with many k-points\nexcited at this frequency. Moving now to the harmonic poten-\ntial, parameterised as in Ref. 27, we first note that we observe\nthat some of the phonon branches overlap - Fig.4b). Secondly,\nthe projection of intensity onto the frequency domain shows a\nlarge peak at 8.6THz, due to a flat region in the phonon spec-\ntra producing even larger number of k-points in the spectrum\nwhich contribute to this frequency. Finally, the large cutoff\nmakes the Harmonic potential stiffer as all interactions are\ndefined by the same energy, V0, and their equilibrium posi-\ntions corresponding to a BCC structure. This is not the case\nfor the Morse potential which depends exponentially on the\ndifference between the inter-atomic distance and a constant\nequilibrium distance, r0. For a long interaction range, the har-\nmonic approximation will result in a more stiff lattice than the\nMorse parameterisation.\nIn principle, the harmonic potential with a decreased in-\nteraction cutoff and an increased strength could better repro-\nduce the full phonon spectra symmetry for BCC Fe. How-\never, in this work we preferred to use the parameterisation\nexisting in literature27and a large interaction cutoff for sta-\nbility purposes. Although the full symmetry of the BCC Fe\nphonon spectra is not reproduced by this harmonic potential,\nthe phonon energies/frequencies are comparable to the values\nobtained with the Morse potential. Nevertheless, we observed\nthe same equilibrium magnetisation and damping (discussed\nlater) for both potentials, hence the simple harmonic potential\nrepresents a suitable approximation, that has the advantage of\nbeing more computationally efficient.\nFig. 5 shows the magnon spectrum obtained within the SLD\nframework using the Morse potential together with its pro-\njection onto the frequency domain. The results agree very\nwell with previous calculations of magnon spectra28,52. For\nthe harmonic potential the magnon spectrum is found to be\nidentical to that for the Morse potential with only very small\nchanges regarding the projection of intensity onto the fre-\nquency domain. This is in line with our discussion in the pre-\nvious section where the choice of inter-atomic potential had\nlittle effect on the Curie temperature, which is closely linked\nto the magnonic properties. As the harmonic potential is more\ncomputationally efficient than the Morse, we next analyse the\nproperties of the system for a 10 \u000210\u000210 unit cells system\natT=300K with the harmonic potential.7\nFIG. 4. Phonon spectra calculated for a 32 \u000232\u000232 unit cell system at T=300K, C=0:5 for a) Morse potential, b)Harmonic potential. The\nspectra are calculated via method i).\nRight figure includes the projection of the intensity of the spectra onto the frequency domain. Solid lines are the experimental data of\nMinkiewicz et al32. For the Minkiewicz et al data there is only 1 datapoint for the N- Gpath for the second transverse mode which does not\nshow up on the line plots.\nFIG. 5. Magnon spectrum (x component) calculated for a 32 \u000232\u0002\n32 unit cell system at T=300K, C=0:5 for a Morse potential. The\nspectrum is calculated via method i).\nRight figure includes the projection of the intensity of the spectrum\nonto the frequency domain.\nThe power spectral density (auto-correlation in Fourier\nspace) of the magnon, phonons and coupling field at 300 K\nis shown in Fig. 6 computed using method iidetailed previ-\nously. The amplitude of the FFT spectra of velocities and\ncoupling field has been scaled by 0.12 and 0.05 respectively to\nallow for an easier comparison between these quantities. As\nshown in Fig. 6.a) the coupling term presents both magnon\nand phonon characteristics; demonstrating an efficient cou-\npling of the two sub-systems. The large peak observed at\na frequency of 8 :6 THz appears as a consequence of the flat\nphonon spectrum for a Harmonic potential, as observed in the\nspectrum and its projection onto the frequency domain in Fig.\n4.b). Additionally, Fig. 6.a) can give us an insight into the in-duced spin noise within the SLD framework. The background\nof the FFT of the coupling field is flat for the frequencies plot-\nted here, showing that the noise that acts on the spin is uncor-\nrelated. The inset shows a larger frequency domain where it\nis clear that there are no phonon modes for these frequencies,\nand only thermal noise decaying with frequency is visible. At\nthe same time an excitation of spin modes are visible for fre-\nquencies up to ca .100 THz.\nThe characteristics of the coupling field with respect to the\ncoupling strength for a dynamic (SLD) and fixed lattice simu-\nlations (ASD) are presented in Fig. 6(b). The only difference\nbetween the ASD and SLD simulations is given by the pres-\nence of phonons (lattice fluctuations) in the latter. Since the\nlarge peak at 8 :6 THz is due to the lattice vibrations, it is not\npresent in the ASD simulations. The smaller peaks are present\nin both models since they are proper magnonic modes. With\nincreasing coupling the width of the peaks increases suggest-\ning that the magnon-phonon damping has increased. Moving\ntowards the larger frequency regimes, Fig. 6.b) - (inset), we\nobserve that large coupling gives rise to a plateau in the spec-\ntra at around 150 THz, which is present as well for the fixed-\nlattice simulations (ASD). The plateau arises from a weak an-\ntiferromagnetic exchange that appears at large distances due to\nthe competition between the ferromagnetic exchange and the\nantiferromagnetic exchange-like term in the pseudo-dipolar\ncoupling.\nWe have also analysed the characteristics of the magnon\nand phonon spectra for different temperatures- Fig. 7. With8\nFIG. 6. The power spectral density of the auto-correlation function in the frequency domain for magnons, phonons and coupling field for a\nSLD simulations with a Harmonic lattice, calculated by method ii). Panel a) shows the power density of the auto-correlation function of the\nx component of the velocity vx, spin Sxand coupling field Hcx. Panel b) presents the power density of the auto-correlation function for the x\ncomponent of the coupling field for either static (ASD) or dynamic (SLD) lattice. The insets show the high-frequency spectra. For Panel a) the\nvelocity and the coupling field have been multiplied by a factor of 0.12 and 0.05 respectively for easier graphical comparison.\nFIG. 7. The power spectral density of the auto-correlation function in the frequency domain for magnons - Panel a) and phonons - Panel b) for\na SLD simulations with a Harmonic lattice, calculated by method ii), for three distinct temperatures and a coupling constant of C=0:5.\nincreasing temperature, the peaks corresponding to magnons\nshift to smaller frequencies. This is a typical situation known\nas a softening of low-frequency magnon modes due to the in-\nfluence of thermal population, see e.g.53- Panel a). The same\neffect can be observed by calculating the magnon spectra via\nmethod ifor various temperatures. In Panel b), the peak cor-\nresponding to phonons remains almost at the same frequency\nof about 8 :6 THz, as the phonon spectra is not largely affected\nby temperatures up to T=600K. The increase of the effec-\ntive damping (larger broadening) of each magnon mode with\ntemperature is clearly observed.\nV . MACROSOPIC MAGNETISATION DAMPING\nIn this section we evaluate the macroscopic damping pa-\nrameter experienced by magnetisation due to the magnon-\nphonon excitations for a periodic BCC system using our SLD\nmodel. This method for calculating the damping has been\npresented in54–56. The system is first thermalised at a non-\nzero temperature in an external field of Bext=50T applied in\nthezdirection, then the magnetisation is rotated coherently\nthrough an angle of 30\u000e. The system then relaxes back to\nequilibrium allowing the relaxation time to be extracted. The\naveraged zcomponent of magnetisation is then fitted to the\nfunction mz(t) =tanh(agBext(t+t0)=(1+a2))where arep-resents the macrosopic (LLG-like) damping, gthe gyromag-\nnetic ratio and t0a constant related to the initial conditions.\nThe model system consists of 10 \u000210\u000210 unit cells and the\ndamping value obtained from fitting of mz(t)is averaged over\n10 different simulations.\nFig. 8 shows the dependence of the average damping pa-\nrameter together with the values obtained from individual\nsimulations for different temperatures and coupling strengths\nfor two choices of mechanical potential. In our model, the\nspin system is thermalised by the phonon thermostat, hence\nno electronic damping is present. With increasing coupling,\nthe energy and angular momentum transfer is more efficient,\nhence the damping is enhanced. Since the observed value of\ninduced damping is small, calculating the damping at higher\ntemperature is challenging due to the strong thermal fluctua-\ntions that affect the accuracy of the results. Despite the low\ntemperatures simulated here, the obtained damping values (at\nT=50K, a=4:9\u000210\u00005) are of the same order as reported\nfor magnetic insulators such as YIG (1 \u000210\u00004to 1\u000210\u0000657,58\n) as well as in different SLD simulations (3 \u000210\u00005,27). Gener-\nally, the induced damping value depends on the phonon char-\nacteristics and the coupling term, that allows transfer of both\nenergy and angular momentum between the two subsystems.\nFig. 8(a) and (b) compare the calculated damping for the\nMorse and Harmonic potential for two values of the coupling\nstrength. We observe that the values are not greatly affected9\nFIG. 8. Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic\nPotential (green open squares) and MP-Morse Potential (black open circles) as function of temperature Fig. a), b) and as function of the\ncoupling strength Fig. c), d); Fig a) and b) are calculated for a constant coupling strength of C=0:3,C=0:5 respectively. Fig c) and d)\nare calculated for temperatures of T=100K,T=300Krespectively. The black and green lines represents the average damping parameter\nobtained from the simulations using the Morse and the Harmonic Potentials, respectively.\nby the choice of potential. This arises due to the fact that only\nthe spin modes around Gpoint are excited and for this low k-\nvectors modes the inter-atomic distances between neighbour-\ning atoms do not vary significantly. The extracted damping\nparameter as a function of coupling strength for 100 K and\n300 K is presented in Fig. 8(c) and (d) respectively. The func-\ntional form of the variation is quadratic, in accordance with\nthe form of the coupling term. Measurements of damping in\nmagnetic insulators, such as YIG, show a linear increase in the\ndamping with temperature,58which agrees with the relaxation\nrates calculated by Kasuya and LeCraw59and the relaxation\nrates calculated in the NVE SLD simulations in Ref. 27. How-\never, Kasuya and LeCraw suggest that the relaxation rate can\nvary as Tn, where n=1\u00002 with n=2 corresponding to larger\ntemperature regimes. Nevertheless, the difference between\nthe quadratic temperature variation of the damping observed\nin our simulations and the linear one observed in experiments\nfor YIG can be attributed to the difference in complexity be-\ntween the BCC Fe model and YIG. The difference between\nthe trends may as well suggest that the spin-orbit coupling in\nYIG could be described better by a linear phenomenological\ncoupling term, such as the one used in Refs. 26 and 29, but\nwe note that such forms can lead to a uniform force in the di-\nrection of the magnetisation and so might need further adap-\ntation before being suitable. To test an alternate form of the\ncoupling we have changed the pseudo-dipolar coupling to an\non-site form, specifically Hc=\u0000åi;jf(ri j)((Si\u0001ˆri j)2\u00001\n3S2\ni)\ni.e a N ´eel-like anisotropy term. This leads to much smallerdamping as shown in Fig. 9 ( T=300 K, a=3:3\u000210\u00005,\naveraged over 5 realisations) making it difficult to accurately\ncalculate the temperature dependence of the damping, espe-\ncially for large temperatures. The magnon-phonon damping\ncan clearly have complex behavior depending on the proper-\nties of the system, especially the coupling term, hence no uni-\nversal behaviour of damping as function of temperature can\nbe deduced for spin-lattice models.\nNeglecting the lattice contribution, the temperature depen-\ndence of the macrosopic damping can be mapped onto the\nLandau-Lifshitz-Bloch formalism (LLB)54and theory17and\nASD simulations60have shown it to vary inversely with the\nequilibrium magnetisation. The LLB theory shows that the\nmacrosopic damping is enhanced for large temperatures due\nto thermal spin fluctuations. Using the equilibrium magneti-\nsation it is possible to approximate the variation of damping\nwith temperature produced due to thermal fluctuations within\nthe LLB model. From 100K to 400K the damping calculated\nvia the LLB model increases within the order of 10\u00005, which\nis considerably smaller than the results obtained via the SLD\nmodel. This shows that within the SLD model the temperature\nincrease of the damping parameter is predominantly due to\nmagnon-phonon interaction, and not due to thermal magnon\nscattering, as this process is predominant at larger tempera-\ntures.10\nFIG. 9. Temperature variation of the damping parameter for N ´eel-\nlike on-site coupling, Hc=\u0000åi;jf(ri j)((Si\u0001ˆri j)2\u00001\n3S2\ni). The val-\nues are extracted from mz(t)fittings for 10 realisations;\nVI. CONCLUSIONS AND OUTLOOK\nTo summarise, we have developed a SLD model that is able\nto transfer energy and angular momentum efficiently from\nthe spin to lattice sub-systems and vice-versa via a pseudo-\ndipolar coupling term. Our approached takes the best fea-\ntures from several previously suggested models and general-\nize them which allows modelling in both canonical and mi-\ncrocanonical ensembles. With only the lattice coupled to\na thermal reservoir and not the spin system, we reproduce\nthe temperature dependence of the equilibrium magnetisation,\nwhich agrees with the fact that the spin-lattice model obeys\nthe fluctuation-dissipation theorem. We are able to study the\ndynamic properties such as phonon and spin spectrum and\nmacrosopic damping, showing that the magnetic damping isnot greatly influenced by the choice of potential, however it\nis influenced by the form of the coupling term. This enables\nthe possibility of tailoring the form of the coupling term so it\ncan reproduce experimental dependencies of damping for dif-\nferent materials. In future, the addition of quantum statistics\nfor Spin Lattice Dynamics models61,62may also yield better\nagreement with experimental data.\nThe SLD model developed here opens the possibility of the\ninvestigation of ultrafast dynamics experiments and theoret-\nically studies of the mechanism through which angular mo-\nmentum can be transferred from spin to the lattice at ultrafast\ntimescales. As we have demonstrated that the model works\nwell in the absence of an phenomenological Gilbert damping,\nwhich consists mainly of electronic contributions, the SLD\nmodel can be employed to study magnetic insulators, such\nas YIG, where the principal contribution to damping is via\nmagnon-phonon interactions. Future application of this model\nincludes controlling the magnetisation via THz phonons7\nwhich can lead to non-dissipative switching of the magnetisa-\ntion11,12. With the increased volume of data stored, field-free,\nheat-free switching of magnetic bits could represent the future\nof energy efficient recording media applications. Another pos-\nsible application is more advanced modelling of the ultrafast\nEinstein-de-Haas effect2or phonon-spin transport63.\nVII. ACKNOWLEDGEMENTS\nWe are grateful to Dr. Pui-Wai Ma and Prof. Matt\nProbert for helpful discussions. Financial support of the Ad-\nvanced Storage Research Consortium is gratefully acknowl-\nedged. MOAE gratefully acknowledges support in part from\nEPSRC through grant EP/S009647/1. The spin-lattice simula-\ntions were undertaken on the VIKING cluster, which is a high\nperformance compute facility provided by the University of\nYork. 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The highly Co-doped ZnO is paramagentic at room temperature and becomes\nan uncompensated antiferromagnet at low temperatures, showing a narrowly opened hysteresis and\na vertical exchange bias shift even in the absence of any ferromagnetic layer. At low temperatures in\ncombination with Permalloy an exchange bias is found causing a horizontal as well as vertical shift\nof the hysteresis of the heterostructure together with an increase in coercive \feld. Furthermore,\nan increase in the Gilbert damping parameter at room temperature was found by multifrequency\nFMR evidencing spin pumping. Temperature dependent FMR shows a maximum in magnetic\ndamping close to the magnetic phase transition. These measurements also evidence the exchange\nbias interaction of Permalloy and long-range ordered Co-O-Co structures in ZnO, that are barely\ndetectable by SQUID due to the shorter probing times in FMR.\nI. Introduction\nIn spintronics a variety of concepts have been devel-\noped over the past years to generate and manipulate spin\ncurrents [1, 2]. Amongst them are the spin Hall e\u000bect\n(SHE), which originates from the spin orbit coupling [3],\nspin caloritronics [4] utilizing the spin seebeck e\u000bect [5]\nor spin transfer torque (current induced torque) due to\nangular momentum conservation [6] as examples. Spin\npumping [7], where a precessing magnetization transfers\nangular momentum to an adjacent layer, proved to be a\nvery versatile method since it has been reported for di\u000ber-\nent types of magnetic orders [8{11] or electrical properties\n[12{14] of materials. Furthermore it could also be veri-\n\fed in trilayer systems where the precessing ferromagnet\nand the spin sink, into which the angular momentum is\ntransferred, are separated by a non-magnetic spacer [15{\n18]. This is strongly dependent on the material, while\nfor Cu [15], Au [16], or Al [17] pumping through a few\nnanometers is possible an MgO barrier of 1 nm is enough\nto completely suppress spin pumping [18].\nSpintronic devices are usually based on a ferromagnet\n(FM) although antiferromagnetic spintronics [19] holds\nthe advantages of faster dynamics, less perturbation by\nexternal magnetic \felds and no stray \felds. The latter\ntwo are caused by the zero net magnetization of an an-\ntiferromagnet (AFM), which on the other hand makes\nthem harder to manipulate. One way to control an\nAFM is by using an adjacent FM layer and exploiting\nthe exchange-bias (EB) e\u000bect [20, 21]. Measuring spin-\ntransfer torque in FM/AFM bilayer structures, is possi-\n\u0003Electronic address: martin.buchner@jku.at; Phone: +43-732-\n2468-9651; FAX: -9696\nyCurrent address: NanoSpin, Department of Applied Physics,\nAalto University School of Science, P.O. Box 15100, FI-00076\nAalto, Finlandble [22, 23], but challenging due to Joule heating [24{26]\nor possible unstable antiferromagnetic orders [27]. Anti-\nferromagnets can be used either as spin source [28] or as\nspin sink [11, 29] in a spin pumping experiment. Thereby\nthe spin mixing conductance, a measure for the absorp-\ntion of angular (spin) momentum at the interface [7],\nis described by intersublattice scattering at an antiferro-\nmagnetic interface [30]. Linear response theory predicted\nan enhancement of spin pumping near magnetic phase\ntransitions [31], which could recently also be veri\fed ex-\nperimentally [29].\nIn this work we investigate the behavior of the uncom-\npensated, antiferromagnetic Co xZn1-xO with x2f0.3,\n0.5, 0.6g(in the following 30 %, 50 % and 60 % Co:ZnO)\nin contact to ferromagnetic permalloy (Py). While\nweakly paramagnetic at room temperature, Co:ZnO\nmakes a phase transition to an antiferromagnetic state at\na N\u0013 eel temperature ( TN) dependent on the Co concentra-\ntion [32]. This resulting antiferromagnetism is not fully\ncompensated which is evidenced by a narrow hysteresis\nand a non saturating magnetization up to 17 T [33]. Fur-\nthermore, Co:ZnO \flms exhibit a vertical EB in complete\nabsence of a FM layer [34]. This vertical exchange shift is\ndependent on the Co concentration [32], temperature and\ncooling \feld [35] and the \feld imprinted magnetization\npredominantly shows orbital character [36]. Note that\nbelow the coalesence limit of 20 % the vertical EB van-\nishes. Co:ZnO therefore o\u000bers to study magnetic inter-\nactions between an uncompensated AFM and a FM Py\nlayer. Static coupling, visible as EB, is investigated using\nsuper conducting quantum interference device (SQUID)\nmagnetometry. The dynamic coupling across the inter-\nface is measured using ferromagnetic resonance (FMR)\nat room temperature and around the magnetic transi-\ntion temperatures determined from M(T) SQUID mea-\nsurements. Element selective XMCD studies are carried\nout to disentangle the individual magnetic contributions.\nFinally heterostructures with an Al spacer were investi-\ngated to rule out intermixing at the interface as sourcearXiv:1909.04362v3  [cond-mat.mtrl-sci]  14 Oct 20192\nfor the coupling e\u000bect.\nII. Experimental Details\nHeterostructures consisting of Co:ZnO, Py and Al, as\nshown in Fig. 1 were fabricated on c-plane sapphire sub-\nstrates using reactive magnetron sputtering (RMS) and\npulsed laser deposition (PLD) at a process pressure of 4\n\u000210-3mbar. The di\u000berent layers of a heterostructure are\nall grown in the same UHV chamber with a base pressure\nof 2\u000210-9mbar in order to ensure an uncontaminated\ninterface. While Py and Co:ZnO are grown by magnetron\nsputtering, the Al spacer and capping layers are grown\nby PLD. Al and Py are fabricated at room temperature\nusing 10 standard cubic centimeters per minute (sccm)\nAr as a process gas.\nFor the heterostructures containing a Co:ZnO layer,\nsamples with three di\u000berent Co concentrations of 30 %,\n50 % and 60 % are grown utilizing preparation conditions\nthat yield the best crystalline quality known for Co:ZnO\nsingle layers [32, 33, 36]. For 30 % and 50 % Co:ZnO\nmetallic sputter targets of Co and Zn are used at an\nAr:O 2ratio of 10 : 1 sccm, while for 60 % Co:ZnO no oxy-\ngen and a ceramic composite target of ZnO and Co 3O4\nwith a 3 : 2 ratio is used. The optimized growth temper-\natures are 450\u000eC, 294\u000eC and 525\u000eC. Between Co:ZnO\ngrowth and the next layer a cool-down period is required,\nto minimize inter-di\u000busion between Py and Co:ZnO.\nThe static magnetic properties are investigated by\nSQUID magnetometry. M(H) curves are recorded at\n300 K and 2 K in in-plane geometry with a maximum\nmagnetic \feld of \u00065 T. During cool-down either a mag-\nnetic \feld of\u00065 T or zero magnetic \feld is applied to dif-\nferentiate between plus-\feld-cooled (pFC), minus-\feld-\ncooled (mFC) or zero-\feld-cooled (ZFC) measurements.\nAll measurements shown in this work have been corrected\nby the diamagnetic background of the sapphire substrate\nand care was taken to avoid well-known artifacts [37, 38].\nFor probing the element selective magnetic properties\nX-ray absorption (XAS) measurements were conducted\nat the XTreme beamline [39] at the Swiss Synchrotron\nLightsource (SLS). From the XAS the X-ray magnetic\ncircular dichroism (XMCD) is obtained by taking the\ndirect di\u000berence between XAS with left and right cir-\ncular polarization. The measurements were conducted\nwith total \ruoresence yield under 20\u000egrazing incidence.\nThereby, the maximum magnetic \feld of 6.8 T was ap-\nplied. Both, external magnetic \feld and photon helic-\nity have been reversed to minimize measurement arte-\nfacts. Again pFC, mFC and ZFC measurements were\nconducted applying either zero or the maximum \feld in\nthe respective direction.\nThe dynamic magnetic properties were measured us-\ning multi-frequency and temperature dependent FMR.\nMulti-frequency FMR is exclusively measured at room\ntemperature from 3 GHz to 10 GHz using a short cir-\ncuited semi-rigid cable [40]. Temperature dependentmeasurements are conducted using an X-band resonator\nat 9.5 GHz. Starting at 4 K the temperature is increased\nto 50 K in order to be above the N\u0013 eel-temperature of the\nCo:ZnO samples [32, 35]. At both FMR setups the mea-\nsurements were done in in-plane direction.\nThe measured raw data for SQUID, FMR, XAS and\nXMCD can be found in a following data repository [41].\nIII. Experimental results & Discussion\nFIG. 1: (a) shows the schematic setup of the samples. For the\nCo:ZnO layer three di\u000berent Co concentrations of 30 %, 50 %\nand 60 % are used. The cross section TEM image of the 60 %\nCo:ZnO/Py sample as well as the electron di\u000braction pattern\nof the Co:ZnO layer (b) and a magni\fcation on the interface\nbetween Co:ZnO and Py (c) are shown.\nFigure 1(a) displays the four di\u000berent types of samples:3\nCo:ZnO layers, with Co concentrations of 30 %, 50 % and\n60 %, are grown with a nominal thickness of 100 nm and\nPy with 10 nm. To prevent surface oxidation a capping\nlayer of 5 nm Al is used. For single 60 % Co:ZnO \flms\nthe vertical-exchange bias e\u000bect was largest compared to\nlower Co concentrations. Therefore, for 60 % Co:ZnO\nsamples with an additional Al layer as spacer between\nCo:ZnO and Py have been fabricated. The thickness of\nthe Al spacer (1 nm, 1.5 nm and 2 nm) is in a range where\nthe Al is reported not to suppress spin pumping e\u000bects\nitself [17].\nTEM\nTo get information about the interface between Py\nand Co:ZnO high resolution cross section transmission\nelectron microscopy (TEM) was done. In Fig. 1(b) the\ncross section TEM image of 60 % Co:ZnO/Py with the\nelectron di\u000braction pattern of the Co:ZnO is shown. A\nmagni\fcation of the interface between Co:ZnO and Py is\nshown in Fig. 1(c). From XRD measurements [32] it is\nobvious that the quality of the wurtzite crystal slightly\ndecreases for higher Co doping in ZnO. A similar be-\nhavior is observed in TEM cross section images. While\n35 % Co:ZnO shows the typical only slightly misoriented\ncolumnar grain growth [32] it is obvious from Fig. 1(b)\nthat the crystalline nanocolumns are less well ordered for\n60 % Co:ZnO. Although the electron di\u000braction pattern\ncon\frms a well ordered wurtzite structure, the misorien-\ntation of lattice plains is stronger than for 35 % Co:ZnO\n[32], even resulting in faint Moir\u0013 e fringes which stem from\ntilted lattice plains along the electron path. This cor-\nroborates previous \fndings of !-rocking curves in XRD\n[32, 36] where the increase in the full width at half maxi-\nmum also evidences a higher tilting of the crystallites, i.e.\nan increased mosaicity. The interface to the Py layer is\nsmooth, although it is not completely free of dislocations.\nAlso the interface seems to be rather abrupt within one\natomic layer, i.e. free of intermixing. A similar behavior\nis found for the interface between 50 % Co:ZnO and Py\n(not shown).\nXAS and XMCD\nFigure 2 shows XAS and XMCD spectra recorded at\n3 K and a magnetic \feld of 6.8 T at the Ni L 3/2and\nCo L 3/2edges of 60 % Co:ZnO/Py after pFC, mFC or\nZFC. For all three cooling conditions the Ni L 3/2edges\n(Fig. 2(a)) show a metallic character of the Ni XAS with-\nout any additional \fne structure characteristics for NiO\nand thus no sign of oxidation of the Py. Further, no dif-\nferences in the XAS or the XMCD of the Ni edges of\ndi\u000berent cooling conditions are found. The same is ob-\nserved for the Fe L 3/2edges, however, they are a\u000bected\ngreatly by self-absorption processes in total \ruorescence\nyield (not shown).\n/s56/s52/s48 /s56/s53/s48 /s56/s54/s48 /s56/s55/s48 /s56/s56/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s40/s97/s41\n/s78/s105/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84\n/s32/s88/s65/s83/s32/s112/s70/s67\n/s32/s88/s65/s83/s32/s90/s70/s67\n/s32/s88/s65/s83/s32/s109/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s112/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s90/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s109/s70/s67/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s40/s98/s41/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s88/s77/s67/s68/s32/s40/s37/s41\n/s55/s55/s48 /s55/s56/s48 /s55/s57/s48 /s56/s48/s48 /s56/s49/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s67/s111/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84/s32/s88/s65/s83/s32/s112/s70/s67\n/s32/s88/s65/s83/s32/s90/s70/s67\n/s32/s88/s65/s83/s32/s109/s70/s67\n/s32/s88/s77/s67/s68/s32/s112/s70/s67\n/s32/s88/s77/s67/s68/s32/s90/s70/s67\n/s32/s88/s77/s67/s68/s32/s109/s70/s67\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s52/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s88/s77/s67/s68/s32/s40/s37/s41FIG. 2: In (a) the XMCD at the Ni L 3/2edges after pFC,\nmFC and ZFC for 60 % Co:ZnO/Py are shown. (b) shows\nthe same for the Co L 3/2edges.\nThe Co L 3/2edges in Fig. 2(b) are also greatly af-\nfected by the self absorption of the total \ruorescence\nyield, since it is buried below 10 nm of Py and 5 nm of\nAl. In contrast to Ni the XAS and XMCD at the Co\nL3/2edges (Fig. 2(b)) are not metallic and evidence the\nincorporation of Co as Co2+in the wurtzite structure\nof ZnO [32, 36]. The overall intensity of the Co XMCD\nis strongly reduced indicating a small magnetic moment\nper Co atom well below metallic Co. This small e\u000bective\nCo moment in 60 % Co:ZnO can be understood by the\ndegree of antiferromagnetic compensation that increases\nwith higher Co doping concentrations [32]. Furthermore,\nno indications of metallic Co precipitates are visible in\nthe XAS and XMCD of the heterostructure as it would\nbe expected for a strong intermixing at the interface to\nthe Py.\nNo changes between the pFC, mFC and ZFC measure-\nments are visible also for the Co edges either in XAS or\nXMCD indicating that the spin system of the Co dopants\nis not altered in the exchange bias state. This corrob-\norates measurements conducted at the Co K-edge [36].\nAfter \feld cooling the XMCD at the Co main absorption\nincreased compared to the ZFC conditions. At the Co\nK-edge the main absorption stems from the orbital mo-\nment. The spin system is only measured indirectly at the\npre-edge feature which remained una\u000bected by the cool-\ning \feld conditions. The data of K- and L-edges com-\nbined evidences that the imprinted magnetization after\n\feld cooling is composed predominantly of orbital mo-4\nment, which is in good agreement with other EB systems\n[42, 43]\nSQUID\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s49/s48/s49\n/s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s49/s48/s49\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s45/s49/s54/s48/s45/s49/s50/s48/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48/s51/s48/s48/s75\n/s40/s98/s41/s32/s77/s47/s77/s91/s49/s48/s109/s84/s93\n/s48/s72/s32/s40/s109/s84/s41/s32/s80/s121/s32\n/s32/s51/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121/s40/s97/s41\n/s32\n/s32/s51/s48/s48/s75\n/s32/s50/s75/s77/s32/s40 /s101/s109/s117/s41\n/s48/s72/s32/s40/s109/s84/s41/s32/s32/s109/s70/s67\n/s32/s32/s112/s70/s67\n/s32/s32/s90/s70/s67/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s50/s75/s32/s97/s102/s116/s101/s114/s58\nFIG. 3: At 300 K the M(H) curves of the single Py \flm al-\nmost overlaps with the M(H) curves of the heterostructures\nwith all three Co:ZnO concentrations (a). In the inset it can\nbe seen that there is no di\u000berence in coercive \feld for Py at\n300 K and 2 K. Measuring the 60 % Co:ZnO/Py heterostruc-\nture after plus, minus and zero \feld cooling, horizontal and\nvertical exchange bias shifts are visible, as well as an increase\nin the coercive \feld (b).\nThe static coupling in the heterostructures was investi-\ngated by integral SQUID magnetometry. Measurements\ndone at 300 K, as shown in Fig. 3(a), do not reveal a sig-\nni\fcant in\ruence of the Co:ZnO on the M(H) curve of\nPy. Just a slight increase in coercive \feld from 0.1 mT\nto 0.4 mT is determined. Some of the M(H) curves\nin Fig. 3(a) are more rounded than the others. This\ncan be attributed to slight variations in the aspect ra-\ntio of the SQUID pieces and thus variations in the shape\nanisotropy. The inset of Fig. 3(a) shows the hysteresis of\nthe single Py \flm at 300 K and 2 K, where no di\u000berence\nin coercivity is visible. Please note that up to now mea-\nsurements were conducted only in a \feld range of \u000610 mT\nand directly after a magnet reset. This is done to avoid\nin\ruences of the o\u000bset \feld of the SQUID [38]. At lowtemperatures, to determine the full in\ruence of Co:ZnO,\nhigh \felds need to be applied, as it has been shown in [35].\nTherefore, coercive \felds obtained from low temperature\nmeasurements are corrected by the known o\u000bset \feld of\n1.5 mT of the SQUID [38].\nSince the paramagnetic signal of Co:ZnO is close to the\ndetection limit of the SQUID and thus, orders of mag-\nnitude lower than the Py signal it has no in\ruence on\nthe room temperature M(H) curve. However, with an\nadditional Co:ZnO layer a broadening of the hysteresis,\na horizontal and a small vertical shift are measured at\n2 K as can be seen exemplary for 60 % Co:ZnO/Py in\nFig. 3(b). Similar to single Co:ZnO \flms where an open-\ning of theM(H) curve is already visible in ZFC mea-\nsurements [32, 34{36] also in the heterostructure no \feld\ncooling is needed to increase the coercive \feld.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50\n/s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s40/s98/s41\n/s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s54/s48/s37 /s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s97/s102/s116/s101/s114/s32/s90/s70/s67/s40/s97/s41\n/s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s112/s70/s67\n/s32/s109/s70/s67\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s45/s49/s50/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s32/s112/s70/s67\n/s32/s109/s70/s67/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s58\n/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s58\nFIG. 4: (a) At 2 K the coercivity increases with Co concen-\ntration in the heterostructure. In the inset the temperature\ndependence of the coercivity of the 60 % Co:ZnO/Py het-\nerostructure is given. (b) The vertical shift (circles) and the\nhorizontal shift (squares) depend on the Co concentration.\nBoth shifts reverse the direction when the measurement is\nchanged from pFC to mFC.\nEarlier works [32, 34] demonstrated that the hystere-\nsis opening and vertical shift in Co:ZnO are strongly de-\npendent on the Co concentration and increase with in-\ncreasing Co doping level. Furthermore, the EB e\u000bects\nare observed in the in-plane and out-of-plane direction,\nwith a greater vertical shift in the plane. Therefore,\nthe heterostructers with Py are measured with the mag-\nnetic \feld in in-plane direction. Figure 4(a) provides an\noverview of the coercive \feld after ZFC for the di\u000ber-5\nent Co concentrations. The coercive \feld increases from\n0.1 mT for single Py to 20.6 mT for 60 % Co:ZnO/Py.\nAdditionally, in the inset the temperature dependence of\nthe coercive \feld of the 60 % Co:ZnO/Py heterostructure\nis shown, since it shows the strongest increase in coercive\n\feld. From the 20.6 mT at 2 K it \frst increases slightly\nwhen warming up to 5 K. That the maximum coercivity\nis not at 2 K is in good agreement with measurements at\nsingle 60 % Co:ZnO \flms where a maximum hysteresis\nopening at 7 K was determined [35]. Afterwards the co-\nercive \feld decreases. At the N\u0013 eel temperature of 20 K a\ncoercive \feld of 11.6 mT is measured. Above T Nit de-\ncreases even further but the coercivity is still 3.65 mT at\n50 K. A coupling above T Ncould stem from long range\nmagnetic ordered structures in Co:ZnO where \frst in-\ndications are visible already in single Co:ZnO \flms [32].\nHowever, for single layers they are barely detectable with\nthe SQUID.\nThe vertical (circles) and horizontal (squares) hystere-\nsis shifts after pFC and mFC are shown in Fig. 4(b) for\nthe Py samples with Co:ZnO layers. Similar to single\nCo:ZnO \flms the vertical shift increases with rising Co\nconcentration. The shift is given in percent of the magne-\ntization at 5 T to compensate for di\u000berent sample sizes.\nDue to the overall higher magnetization at 5 T in combi-\nnation with Py this percentage for the heterostructures\nis lower than the vertical shift for single Co:ZnO \flms.\nWith increasing Co concentration the degree of antiferro-\nmagnetic compensation increases [32, 35], which in turn\nshould lead to a stronger EB coupling. This can be\nseen in the horizontal shift and thus EB \feld which is\nstrongest for 60 % Co:ZnO/Py and nearly gone for 30 %\nCo:ZnO/Py. For both kinds of shift the pFC and mFC\nmeasurements behave similar, except the change of di-\nrection of the shifts.\nMultifrequency FMR\nThe dynamic coupling between the two layers has been\ninvestigated by multifrequency FMR measured at room\ntemperature. The frequency dependence of the resonance\nposition between 3 GHz and 10 GHz of the heterostruc-\ntures is shown in Fig. 5(a). The resonance position of Py\nyields no change regardless of the Co concentration in\nthe Co:ZnO layer or its complete absence. Also in 2 nm\nAl/Py and 60 % Co:ZnO/2 nm Al/Py the resonance po-\nsition stays unchanged. The resonance position of a thin\n\flm is given by Kittel formula [44]:\nf=\r\n2\u0019p\nBres(Bres+\u00160M) (1)\nwith the gyromagnetic ratio \r=g\u0016B\n\u0016hand magnetiza-\ntionM. However, any additional anisotropy adds to Bres\nand therefore alters eq. (1) [44]. The fact that all samples\nshow the identical frequency dependence of the resonance\nposition evidences that neither the gyromagnetic ratio \rand thus the Py g-factor are in\ruenced nor any addi-\ntional anisotropy BAniso is introduced by the Co:ZnO.\nBy \ftting the frequency dependence of the resonance po-\nsition using the Kittel equation with the g-factor of 2.11\n[45] all the samples are in the range of (700 \u000615) kA/m,\nwhich within error bars is in good agreement with the\nsaturation magnetization of (670 \u000650) kA/m determined\nfrom SQUID.\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48/s50/s52/s54/s56/s49/s48\n/s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s48/s49\n/s40/s98/s41\n/s32/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s66\n/s114/s101/s115/s32/s40/s109/s84/s41/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121 \n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s65/s108/s47/s80/s121 \n/s32/s80/s121 /s40/s97/s41\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s32/s32\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s32/s32/s110/s111/s114/s109/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32/s40/s109/s84/s41/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s32/s32/s32/s32/s102/s114/s101/s113/s46/s32/s61/s32/s54/s46/s53/s56/s71/s72/s122\n/s32/s76/s111/s114/s101/s110/s116/s105/s97/s110/s32/s102/s105/s116\nFIG. 5: The resonance \felds determined at room temperature\nwith multifrequency FMR are seen in (a). In the inset an ex-\nemplary FMR spectrum for of 50 % Co:ZnO/Py at 6.58 GHz is\nshown with the corresponding Lorentian \ft. For the linewidth\n(b) and the associated damping parameter \u000b(inset) an in-\ncrease is visible for the heterostructures with higher Co con-\ncentration in the Co:ZnO. The lines are linear \fts to the data.\nEven though the Co:ZnO layer does not in\ruence the\nresonance position of the FMR measurement the het-\nerostructures exhibit an increase in linewidth. This cor-\nresponds to a change of the damping in the system. The\nfrequency dependence of the linewidth can be used to sep-\narate the inhomogeneous from the homogeneous (Gilbert\nlike) contributions, from which the Gilbert damping pa-\nrameter\u000bcan be determined.\n\u0001B= \u0001Bhom+ \u0001Binhom (2)6\nwhere\n\u0001Bhom=4\u0019\u000b\n\rf (3)\nNo di\u000berence in linewidth between Al/Py (open stars)\nand Py (full stars) is found, as can be seen in Fig. 4(b)\nwhere the peak to peak linewidth B ppis plotted over the\nmeasured frequency range for all the heterostructures.\nWhile the heterostructure with 30 % Co:ZnO/Py (green\ntriangles) lies atop the single Py and the Al/Py \flm,\nthe linewidth increases stronger with frequency for 50 %\nCo:ZnO/Py (blue circles). The broadest FMR lines are\nmeasured for the 60 % Co:ZnO/Py heterostructure (red\nsqaures).\nUsing the Py g-factor of 2.11 [45], \u000bcan be calcu-\nlated from the slopes of the frequency dependence ex-\ntracted from the linewidths seen in Fig. 5(b): the result-\ning\u000bare shown in the inset. For the single Py layer \u000bPy\n= (5.7\u00060.3)\u000210-3which compares well to previously re-\nported values [7]. This increases to \u000b50= (8.0\u00060.3)\u000210-3\nfor 50 % Co:ZnO/Py and even \u000b60= (9.4\u00060.3)\u000210-3for\n60 % Co:ZnO/Py. So the damping increases by a factor\nof 1.64 resulting in a spin pumping contribution \u0001 \u000b=\n(3.7\u00060.5)\u000210-3that stems from the angular momentum\ntransfer at the interface of Py and Co:ZnO. By insertion\nof a 2 nm Al spacer layer \u0001 \u000breduces to (0.8\u00060.5)\u000210-3.\nDependence on the Al spacer thickness\nTo obtain information about the lengthscale of the\nstatic and dynamic coupling, heterostructures with Al\nspacer layers of di\u000berent thickness (1 nm, 1.5 nm and\n2 nm thick) between Py and the material beneath (sap-\nphire substrate or 60 % Co:ZnO) were fabricated. With-\nout a Co:ZnO layer the spacer underlying the Py layer\ndoes not exhibit any changes in either SQUID (not\nshown) or FMR (see Fig 5 (a) and (b)). The results ob-\ntained for the 60 % Co:ZnO/Al/Py heterostructure for\nthe coercive \feld, vertical and horizontal shift extracted\nfromM(H) curves are shown in Fig. 6(a), whereas the\ndamping parameter \u000bfrom room temperature multifre-\nquency FMR measurements, analogues to Fig. 5(b), are\ndepicted in Fig. 6(b).\nThe horizontal shift and the increased coercive \feld\nare caused by the coupling of FM and AFM moments in\nrange of a few \u0017Angstrom to the interface [46{48]. There-\nfore, both e\u000bects show a similar decrease by the insertion\nof an Al spacer. While the horizontal shift and coer-\ncive \feld are reduced signi\fcantly already at a spacer\nthickness of 1 nm, the vertical shift (inset of Fig. 6(a))\nis nearly independent of the Al spacer. Comparing with\nthe XMCD spectra of Fig. 2 it can be concluded that\nthe vertical shift in the uncompensated AFM/FM sys-\ntem Co:ZnO/Py stems solely from the increased orbital\nmoment of pinned uncompensated moments in Co:ZnO\nand is independent of the FM moments at the interface.Furthermore, the FM moments do not exhibit any ver-\ntical shift and the exchange between the two layers only\nresults in the horizontal shift.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100\n/s32/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41\n/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116\n/s32/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s40/s98/s41\n/s32/s32\n/s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s65/s108/s32/s115/s112/s97/s99/s101/s114/s47/s80/s121/s40/s97/s41\n/s65/s108/s47/s80/s121\nFIG. 6: When an Al spacer is inserted between the Py and\nthe Co:ZnO layer horizontal shift and coercive \feld show a\nstrong decrease already at 1 nm spacer thickness (a) while the\nvertical shift (inset) is not dependent on the spacer thickness.\n(b) shows the e\u000bect of the Al spacer on the Gilbert damping\nparameter\u000b, which also decreases if the spacer gets thicker\nthan 1 nm. As shaded region the Gilbert damping parameter\nof a Al/Py \flm is indicated within error bars.\nFor the FMR measurments after inserting an Al spacer\nno e\u000bect on the resonance position is found, as was shown\nalready in Fig. 5(a). For a 1 nm thick Al spacer the damp-\ning results in \u000b= (8.8\u00060.3)\u000210-3, which gives a \u0001 \u000b=\n(3.1\u00060.5)\u000210-3. This is only a slight decrease compared\nto the sample without Al spacer. By increasing the spacer\nthickness\u000breduces to values just above the damping ob-\ntained for pure Py or Al/Py, shown as shaded region in\nFig. 6(b). The 1 nm thick Al layer is thick enough to sup-\npress intermixing between the Co:ZnO and the Py layer\nas can be seen in Fig. 1(b). Together with the unchanged\nbehavior of Al/Py without Co:ZnO damping e\u000bects due\nto intermixing between Al and Py can be excluded. Also,\na change in two magnon scattering can be ruled out, since\nit would account for non-linear e\u000bects on the linewidth\nand contribute to \u0001 Binhom [49]. Therefore, the increase\nin Gilbert damping can be attributed to a dynamic cou-\npling, e.g. spin pumping from Py into Co:ZnO. Further-\nmore, the dynamic coupling mechanism is extends over a\nlonger range than the static coupling. With 1 nm spacer\nthe dynamic coupling is only slightly reduced whereas\nthe static coupling is already completely suppressed.7\nTemperature dependent FMR\nIn vicinity to the magnetic phase transition temper-\nature the spin pumping e\u000eciency should be at a max-\nimum [29, 31]. Therefore, the samples are measured\ninside a resonator based FMR setup, as a function of\ntemperature. During the cooldown no magnetic \feld is\napplied and the results shown in Fig. 7 are ZFC mea-\nsurements. For 50 % Co:ZnO/Py the resonance posi-\ntions shifts of Py to lower magnetic \felds as the tem-\nperature decreases as can be seen in Fig. 7(a). Not only\nthe resonance position is shifting, but also the linewidth\nis changing with temperature as shown in Fig. 7(b). The\nlinewidth has a maximum at a temperature of 15 K which\ncorresponds well to T Ndetermined by M(T) SQUID\nmeasurements for a 50 % Co:ZnO layer [32]. This max-\nium of the linewidth in the vicinity of T Nis also ob-\nserved for 60 % Co:ZnO/Py and even 30 % Co:ZnO/Py,\nas shown in Fig. 7(c). The measured maximum of 30 %\nCo:ZnO/Py and 60 % Co:ZnO/Py are at 10.7 K, 19.7 K\nrespectively and are marked with an open symbol in\nFig. 7(c). For comparison the N\u0013 eel temperatures de-\ntermined from M(T) measurements [32] are plotted as\ndashed line. Py on the other hand shows only a slight\nincrease in linewidth with decreasing temperature. The\nobserved e\u000bects at low temperatures vanish for the 60 %\nCo:ZnO/2 nm Al/Py heterostructure.\nFigure 7(d) shows the temperature dependence of the\nresonance \feld for all samples. For Py Bresonly decreases\nslightly whereas for 50 % and 60 % Co:ZnO a strong shift\nofBrescan be observed. This shift evidences a magnetic\ncoupling between the Py and the Co:ZnO layer. Even\nin the heterostructure with 30 % Co:ZnO/Py a clear de-\ncrease in resonance position below 10 K (the previously\ndetermined T N[32]) is visible. This shift of the resonance\nposition is only observed at low temperatures. At room\ntemperature no shift of the resonance position at 9.5 GHz\nhas been observed as shown in Fig. 5(a). From the low-\ntemperature behavior of the single Py layer and eq. 1 it\nis obvious that the gyromagnetic ratio is not changing\nstrongly with temperature, therefore shift of the reso-\nnance position in the heterostructure can be attributed\nto a change in anisotropy. From the SQUID measure-\nments at 2 K, see Fig. 3(b) and Fig. 4(b) EB between the\ntwo layers has been determined, which acts as additional\nanisotropy [20] and therefore causes the shift of the reso-\nnance position. Both the shift of the resonance position\nand the maximum in FMR linewidth vanish if the Py is\nseparated from 60 % Co:ZnO by a 2 nm Al spacer layer.\nSo, also at low temperatures the static EB coupling and\nthe dynamic coupling can be suppressed by an Al spacer\nlayer.\nM(T) measurements indicated a more robust long-\nrange magnetic order in 60 % Co:ZnO by a weak sepa-\nration of the \feld heated and ZFC curves lasting up to\n200 K [32]. Additionally, the coercive \feld measurements\non the 60 % Co:ZnO/Py hetersotructure revealed a weak\ncoupling above T N. However, this has not been observedfor lower Co concentrations. In the heterostructure with\n30 % Co:ZnO the FMR resonance position and linewidth\nreturn quickly to the room temperature value for temper-\natures above the T Nof 10 K. For both 50 % Co:ZnO/Py\nand 60 % Co:ZnO/Py the resonance positions are still de-\ncreased and the linewidths are increased above their re-\nspective N\u0013 eel temperatures and are only slowly approach-\ning the room temperature value. In the 60 % Co:ZnO/Py\nheterostructure measurements between 100 K and 200 K\nrevealed that a reduced EB is still present. It is known for\nthe blocking temperatures of superparamagnetic struc-\ntures that in FMR a higher blocking temperature com-\npared to SQUID is obtained due to much shorter probing\ntimes in FMR of the order of nanoseconds compared to\nseconds in SQUID [50]. Hence, large dopant con\fgura-\ntions in Co:ZnO still appear to be blocked blocked on\ntimescales of the FMR whereas they already appear un-\nblocked on timescales of the SQUID measurements.\nV. Conclusion\nThe static and dynamic magnetic coupling of Co:ZnO,\nwhich is weakly paramagnetic at room temperature and\nan uncompensated AFM at low temperatures, with ferro-\nmagnetic Py was investigated by means of SQUID mag-\nnetometry and FMR. At room temperature no static in-\nteraction is observed in the M(H) curves. After cooling\nto 2 K an EB between the two layers is found resulting\nin an increase of coercive \feld and a horizontal shift.\nAdditionally, a vertical shift is present caused by the un-\ncompensated moments in the Co:ZnO. While this vertical\nshift is nearly una\u000bected by the insertion of an Al spacer\nlayer between Co:ZnO and Py the EB vanishes already\nat a spacer thickness of 1 nm.\nThe FMR measurements at room temperature re-\nveal an increase of the Gilbert damping parameter for\n50 % Co:ZnO/Py and 60 % Co:ZnO/Py, whereas 30 %\nCo:ZnO/Py is in the range of an individual Py \flm. At\nroom temperature the resonance position is not a\u000bected\nfor all the heterostructures. For the 60 % Co doped sam-\nple \u0001\u000b= 3.7\u000210-3, which is equivalent to an increase\nby a factor of 1.64. In contrast to the static magnetic\ncoupling e\u000bects, an increased linewidth is still observed\nin the heterostructure containing a 1 nm Al spacer layer.\nAt lower temperatures the resonance position shifts\nof the heterostructures to lower resonance \felds, due to\nthe additional EB anisotropy. The temperature depen-\ndence of the linewidth shows a maximum at tempera-\ntures, which by comparison with M(T) measurements\ncorrespond well to T Nof single Co:ZnO layers and thus\ncorroborate the increase of the damping parameter and\nthus spin pumping e\u000eciency in vicinity to the magnetic\nphase transition. Furthermore, the shift of the resonance\nposition has been observed at temperatures well above\nTNfor 50 % Co:ZnO/Py and 60 % Co:ZnO/Py. Up to\nnow only indications for a long range AFM order in 60 %\nCo:ZnO/Py had been found by static M(T) measure-8\n/s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41\n/s40/s99/s41\n/s32/s32/s110/s111/s114/s109/s46/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32/s40/s109/s84/s41/s32/s84/s32/s61/s32/s32/s32/s52/s46/s48/s75\n/s32/s84/s32/s61/s32/s49/s52/s46/s57/s75\n/s32/s84/s32/s61/s32/s51/s49/s46/s52/s75\n/s32/s84/s32/s61/s32/s53/s48/s46/s50/s75/s40/s97/s41\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s84\n/s78/s32/s100/s101/s116/s101/s114/s109/s105/s110/s101/s100/s32\n/s102/s114/s111/s109/s32/s77/s40/s84/s41/s32/s83/s81/s85/s73/s68/s32/s91/s51/s50/s93\n/s32/s32/s66\n/s114/s101/s115/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121\n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121/s84\n/s78/s32/s54/s48/s37/s84\n/s78/s32/s53/s48/s37/s84\n/s78/s32/s51/s48/s37\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121\n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121\nFIG. 7: By decreasing the temperature the resonance position of 50 % Co:ZnO/Py shifts to lower resonance \felds (a) and\nthe linewidth increases, showing a maxium at the T N(b). A similar behavior is observed for the heterostructures with 30 %\nand 60 % Co doping while a single Py \flm does not exhibit a maximum when cooling (c). The maximum is marked as open\nsymbol in the temperature dependence, while the T Ndetermined from M(T) [32] are shown as dashed lines. Furthermore, the\nresonance position of the heterostructures with Co:ZnO shifts at low temperatures (d).\nments. The dynamic coupling, however, is sensitive to\nthose interactions due to the higher time resolution in\nFMR resulting in a shift of the resonance position above\nthe T Ndetermined from M(T) SQUID.\nAcknowledgment\nThe authors gratefully acknowledge funding by the\nAustrian Science Fund (FWF) - Project No. P26164-N20 and Project No. ORD49-VO. All the mea-\nsured raw data can be found in the repository at\nhttp://doi.org/10.17616/R3C78N. 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Gerhard, C. Schumacher, C. Gould, L.W. Molenkamp1\nPhysikalisches Institut (EP3), Universit¨ at W¨ urzburg,\nAm Hubland, D-97074 W¨ urzburg, Germany\n(Dated: 10 April 2018)\nNiMnSb is a ferromagnetic half-metal which, because of its rich aniso tropy and very\nlow Gilbert damping, is a promising candidate for applications in informat ion tech-\nnologies. Wehaveinvestigatedthein-planeanisotropypropertieso fthin, MBE-grown\nNiMnSbfilms asafunctionoftheir Mnconcentration. Using ferromag neticresonance\n(FMR)todetermine theuniaxialandfour-foldanisotropyfields,2KU\nMsand2K1\nMs, wefind\nthat a variationin composition can change the strength of the four -fold anisotropy by\nmore than an order of magnitude and cause a complete 90◦rotation of the uniaxial\nanisotropy. This provides valuable flexibility in designing new device geo metries.\n1INTRODUCTION\nNiMnSb is a half-metallic ferromagnetic material offering 100% spin pola rization in its\nbulk1, and was therefore long considered a very promising material for s pintronic appli-\ncations such as spin injection. Experience has shown however that preserving sufficiently\nhigh translation symmetry to maintain this perfect polarization at su rfaces and interface\nis a major practical challenge, reducing its atractiveness for spin in jection. The material\nnevertheless continues to be very promising for use in other spintr onic applications; in par-\nticular in spin torque devices such as spin-transfer-torque (STT) controlled spin valves and\nspin torque oscillators (STO). This promise is based on its very low Gilbe rt damping, of\norder 10−3or lower2which should enhance device efficiency, as well as on its rich and stron g\nmagnetic anisotropy which allows for great flexibility in device engineer ing.\nFor example, it has been shown that STO oscillators formed from two layers of orthogo-\nnal anisotropy can yield significantly higher signal than those with co -linear magnetic easy\naxis3–6. Being able to tune the magnetic anisotropy of individual layers is clea rly useful for\nthe production of such devices.\nPrevious results have shown a dependence of the anisotropy of NiM nSb on film thickness7,\nwhich offers some control possibilities when device geometries allow fo r appropriate layer\nthicknesses, but that is not always possible due to other design or lit hography limitations.\nHere we show how the anisotropy of layers of a given range of thickn ess can effectively be\ntuned by slight changes in layer composition, achieved by adjusting t he Mn flux.\nEXPERIMENTAL\nThe NiMnSb layers are grown epitaxial by molecular beam epitaxy (MBE ) on top of\na 200 nm thick (In,Ga)As buffer on InP (001) substrates. All sample s have a protective\nnon-magnetic metal cap (Ru or Cu) deposited by magnetron sputt ering before the sample\nis taken out of the UHV environment, in order to avoid oxidation and/ or relaxation of the\nNiMnSb8. The flux ratio Mn/Ni, and thus the composition, is varied between sa mples by\nadjusting the Mn cell temperature while the flux ratio Ni/Sb is kept co nstant. The thickness\nof most of the studied NiMnSb layers is 38 ±2 nm. Two samples have a slightly larger film\nthickness (45 nm, marked with ( ) in Fig. 3a), caused by the change in growth rate due\n2/s49/s52/s46/s53 /s49/s53/s46/s48 /s49/s53/s46/s53/s49/s48/s48/s49/s48/s50/s49/s48/s52/s49/s48/s54/s49/s48/s56/s49/s48/s49/s48/s49/s48/s49/s50/s49/s48/s49/s52/s49/s48/s49/s54\n/s49/s52/s46/s56/s52 /s49/s52/s46/s56/s56 /s49/s52/s46/s57/s50/s49/s48/s49/s49/s48/s50/s49/s48/s51\n/s66\n/s67\n/s32 /s32/s40/s176/s41/s32/s108/s111/s119/s101/s115/s116/s32/s77/s110/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s115/s97/s109/s112/s108/s101/s32/s65/s41\n/s32/s109/s101/s100/s105/s117/s109/s32/s77/s110/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s115/s97/s109/s112/s108/s101/s32/s66/s41\n/s32/s104/s105/s103/s104/s101/s115/s116/s32/s77/s110/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s115/s97/s109/s112/s108/s101/s32/s67/s41\n/s32/s32/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s41/s65/s32/s32/s32\n/s32 /s32/s40/s176/s41\nFIG. 1. HRXRD ω-2θ-scans of 3 NiMnSb samples with various Mn concentrations. T he curves are\nvertically offset for clarity. Inlet: ω-scans showing high crystal quality.\nto the change in Mn flux. We verified that there is no correlation betw een anisotropy and\nsample thickness in this range.\nHigh Resolution X-Ray Diffraction (HRXRD) measurements of the (00 2) Bragg reflection\nare used to determine the vertical lattice constant of each sample . Fig.1 shows standard\nω-2θ-scans of the (002) Bragg reflection on layers with the lowest and h ighest Mn concen-\ntrations used in the study, as well as a scan for a sample with medium M n concentration.\nThe sample with the lowest Mn content has a vertical lattice constan t of 5.939 ˚A (sample A)\nand that with the highest Mn content (sample C) has a vertical lattic e constant of 6.092 ˚A.\nTo get an estimate of the vertical lattice constant of stoichiometr ic NiMnSb in our layer\nstacks, we used an XRD measurement of a stoichiometric, relaxed s ample9. We determine\na relaxed lattice constant of arel= (5.926 ±0.007)˚A. Together with the lattice constant\nof our InP/(In,Ga)As substrate, 5.8688 ˚A, and an estimated Poisson ratio of 0.3 ±0.03,\nwe get the minimal and maximal values for the vertical lattice consta nt of stoichiometric\nNiMnSb: a⊥,max= 5.999˚A,a⊥,min= 5.957˚A. The vertical lattice constant of the sample\nwith medium Mn concentration (sample B, 5.968 ˚A) lies in this range. We conclude that\n3the composition of sample B is approximately stoichiometric.\nIn Ref. 10 and 11, the effects of off-stoichiomteric defects in NiMnS b are discussed. Among\nthe possible defects related to Mn, Mn Ni(Mn substituting Ni) is most likely (it has lowest\nformation energy) and the predicted decrease of the saturation magnetization is consistent\nwith our observation (see Fig. 3b). Furthermore, an increase of t he lattice constant with\nincreasing concentration of this kind of defect is predicted theore tically and observed ex-\nperimentally. Thus, we can use the (vertical) lattice constant as a m easure for the Mn\nconcentration in our samples.\nThe crystal quality is also assessed by the HRXRD measurements. T he inset in Fig. 1 shows\ntheω-scans of the same three NiMnSb layers. The ω-scans of both the low and medium Mn\nconcentration sample are extremely narrow with a full width half-ma ximum (FWHM) of 15\nand 14 arcsec, respectively. A broadening for the sample with highe st Mn concentration\ncan be seen (FWHM of 35 arcsec). Reasons for the broadening can be partial relaxation of\nthe layer due to the increased lattice mismatch with the (In,Ga)As Bu ffer, and/or defects\nrelated to the surplus of Mn.\nUsing the experimental data of the lattice constant in Ref. 11, we c an estimate a difference\nin Mn concentration between sample A and C (extreme samples) of ab out 40%. For sample\nC (extreme high Mn concentration), we determine a saturation mag netization of 3.4 µBohr\n(see Fig. 3b). According to Ref. 11, this corresponds to a crysta l where about 20% of Ni is\nreplaced by Mn. It should be noted that we investigated the effect o f extreme surplus/deficit\nof Mn within the limits of acceptable crystal quality. As can be seen in F ig. 3a, already a\nmuch smaller change incomposition canchange the strength andorie ntationof the magnetic\nanisotropy significantly.\nTo map out the in-plane anisotropy of our samples, we use frequenc y-domain ferromagnetic\nresonance (FMR) measurements at a frequency of 12.5 GHz. The r esonance fields are de-\ntermined as a function of an external magnetic field applied at fixed a ngles ranging from 0◦\n(defined as the [100] crystal direction) to 180◦. Fig. 2 shows results of these measurements\nfor four different samples with four distinct types of anisotropy: S ample A and D both ex-\nhibit large uniaxial anisotropies with anadditional four-foldcompone nt, however of opposite\nsign. The hard axis of sample A is along the [1 ¯10] crystal direction, where for sample D the\nhard axis is along the [110] crystal direction. Sample B and C both sho w mainly uniaxial\nanisotropies, again with opposite signs.\n4/s50/s46/s48/s50/s46/s50/s50/s46/s52/s50/s46/s54\n/s50/s46/s48/s50/s46/s50/s50/s46/s52/s50/s46/s54\n/s48 /s52/s53 /s57/s48 /s49/s51/s53 /s49/s56/s48/s49/s46/s56/s56/s49/s46/s57/s50/s49/s46/s57/s54/s50/s46/s48/s48/s50/s46/s48/s52\n/s48 /s52/s53 /s57/s48 /s49/s51/s53 /s49/s56/s48/s50/s46/s48/s50/s46/s49/s50/s46/s50/s50/s46/s51/s32/s32\n/s32/s114/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41/s32/s65\n/s32/s32\n/s32\n/s32/s32/s67\n/s32/s32 /s32/s32\n/s32/s66\n/s32/s32\n/s32/s91/s49/s48/s48/s93/s32/s32/s32/s91/s49/s49/s48/s93/s32/s32/s91/s48/s49/s48/s93/s32/s32/s32/s91/s49/s49 /s48/s93/s32/s32/s91/s49/s48/s48/s93/s32/s32/s32/s91/s49/s49/s48/s93/s32/s32/s91/s48/s49/s48/s93/s32/s32/s32/s91/s49/s49 /s48/s93/s32/s32\n/s32/s68\n/s97/s110/s103/s108/s101/s32/s98/s101/s116/s119/s101/s101/s110/s32/s101/s120/s116/s101/s114/s110/s97/s108/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s97/s110/s100/s32/s99/s114/s121/s115/s116/s97/s108/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110/s32/s91/s49/s48/s48/s93/s32/s40/s176/s41\nFIG. 2. FMR measurements and simulation for four different sam ples. The symbols are measure-\nments of the resonance frequency for magnetic fields along sp ecific crystal directions, where 0◦lies\nalong [100]. The lines are simulations (see below) and also s erve as a guide to the eye. Sample A,\nB and C correspond to the samples with lowest, medium and high est Mn concentration shown in\nFig. 1. Sample D completes the various kinds of anisotropy ob served in NiMnSb.\nTheFMRdatacanbesimulated withasimple phenomenological magneto staticmodel toex-\ntract the anisotropy components (derivation taken from Ref. 12 ). The free energy equation\nfor thin films of cubic materials is given by:\nǫc=−K/bardbl\n1\n2(α4\nx+α4\ny)−K⊥\n1\n2α4\nz−Kuα2\nz (1)\nwhereαx,αyandαzdescribe the magnetization with respect to the crystal directions [100],\n[010] and [001]. K/bardbl\n1is the four-fold in-plane anisotropy constant, KuandK⊥\n1represent\nthe perpendicular uniaxial anisotropy (second and fourth order r espectively). In our in-\nplane FMR geometry, the fourth order perpendicular anisotropy t ermK⊥\n1can be neglected.\nInstead, an additional uniaxial in-plane anisotropy term is added:\nǫu=−K/bardbl\nu(ˆn·ˆM)2\nM2s\nwith the unit vector ˆ nalong the uniaxial anisotropy and the saturation magnetization Ms,\nˆM. The Zeeman term coupling to the external field H0and a demagnetization term origi-\nnating from the thinness of the sample, are defined as\nǫZ=−M·H0, ǫdemag=−4πDM2\n⊥\n2(2)\n5and added as well to the free energy. The effective magnetic field\nHeff=−∂ǫtotal\n∂M(3)\nwith\nǫtotal=ǫc+ǫu+ǫZ+ǫdemag (4)\nis used to solve the Landau-Lifshitz-Gilbert-Equation (LLG):\n−1\nγ∂M\n∂t= [M×Heff]−G\nγ2M2s[M×∂M\n∂t] (5)\nwith the gyromagnetic ratio γ=gµB\n/planckover2pi1and the Gilbert damping constant G. The resonance\ncondition can be found by calculating the susceptibility13,χ=∂M\n∂H:\n(ω\nγ)2=BeffH∗\neff (6)\nIn the following, we neglect the damping contribution sinceG\nγMsin our samples is of the\norder of 10−3or lower. Thus, BeffandH∗\neffin our case can be found to be:\nH∗\neff=H0cos[φM−φH]+2K/bardbl\n1\nMscos[4(φM−φF)]\n+2K/bardbl\nU\nMscos[2(φM−φU)] (7)\nBeff=H0cos[φM−φH]+K/bardbl\n1\n2Ms(3+cos[4(φM−φF)])\n+4πDMs−2K⊥\nU\nMs+K/bardbl\nU\nMs(1+cos[2(φM−φU])) (8)\nHere,φM,φHandφUdefine the angles of the magnetization, external magnetic field and in-\nplane easy axis of the uniaxial anisotropy, respectively, with respe ct to the crystal direction\n[100].φFaccounts for the angle of the four-fold anisotropy. At the magne tic fields used in\nthese studies, it is safe to assume φM=φH14. In equation (8), 4 πDMs−2K⊥\nU\nMScan be defined\nas an effective magnetization 4 πMeff, containing the out-of-plane anisotropy. It is used as\na constant in our simulation.\nFor each sample, we extract2K1\nMsand2KU\nMs, the four-fold and uniaxial in-plane anisotropy\nfield, from the simulation and plot them versus the vertical lattice co nstant (Fig. 3a). The\nvertical, dotted lines mark the range where stoichiometric NiMnSb is e xpected. For vertical\nlattice constants in the range from 5.96 to 6.00 ˚A, both anisotropy fields are relatively small.\nThe four-fold contribution increases for samples with decreasing v ertical lattice constant\n6/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s53/s46/s57/s50 /s53/s46/s57/s54 /s54/s46/s48/s48 /s54/s46/s48/s52 /s54/s46/s48/s56 /s54/s46/s49/s50/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s32/s32\n/s32\n/s67/s66/s65\n/s65/s66\n/s68/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s75\n/s85\n/s32\n/s40/s98/s41/s97/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s115/s32\n/s50/s75\n/s49/s47/s77\n/s83/s32/s44/s32 /s50/s75\n/s117/s47/s77\n/s83/s32/s40/s79/s101/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s75\n/s85\n/s32\n/s40/s97/s41/s50/s75\n/s49/s47/s77\n/s115\n/s50/s75\n/s85/s47/s77\n/s115\n/s67/s40/s32/s32/s41\n/s32/s32/s40/s32/s32/s41/s115/s97/s116/s46/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s77\n/s115/s32/s40/s105/s110/s32/s181/s66/s111/s104/s114/s41\n/s118/s101/s114/s116/s105/s99/s97/s108/s32/s108/s97/s116/s116/s105/s99/s101/s32/s99/s111/s110/s115/s116/s97/s110/s116/s32/s40 /s197 /s41\nFIG. 3. (a) Uniaxial anisotropy field2KU\nMsand four-fold anisotropy field2K1\nMsfor NiMnSb layers with\nvarious Mn concentrations. The vertical lattice constant i s used as a gauge of the Mn content.\nSamples with a rotated RHEED pattern (see last section) are i ndicated by open symbols. The\ndotted lines mark the range where stoichiometric NiMnSb is e xpected. The samples of lowest,\nmedium and highest Mn concentration (A, B and C) together wit h sample D are marked. The\ntwo samples marked with ( ) exhibit slightly higher film thick ness than the other samples. (b)\nSaturation magnetization Msdepending on the vertical lattice constant.\n(lower Mn concentration) but remains small for larger vertical latt ice constant (increasing\nMn concentration). The uniaxial anisotropy gets more strongly ne gative with increasing\nvertical lattice constant, whereas in samples with lower vertical lat tice constants, the uni-\naxial field can be either positive or negative while its absolute value gro ws significantly with\ndecreasing vertical lattice constant. The change in sign of the unia xial anisotropy field at a\nvertical lattice constant of about 5.99 ˚A corresponds to a rotation of the easy axis from the\n[110] direction (positive anisotropy fields) to the [1 ¯10] direction. One can see that already\na small change of the vertical lattice constant (small change in com position) is sufficient to\nrotate the uniaxial anisotropy as well as to induce a significant four -fold anisotropy.\nThe fitting accuracy of the extracted anisotropy fields is ∼5%, giving error bars smaller than\n7the symbols in Fig. 3a. It should be noted that in order to exactly ext ract the anisotropy\nconstants K1andKUfrom the anisotropy fields, the saturation magnetization Msof each\nsample is needed. This can be determined by SQUID measurements. W e have performed\nsuch measurements on a representative fraction of the samples ( Fig. 3b). Samples with\nmedium Mn concentration show saturation magnetizations which, to experimental accuracy\nof about 8% are consistent with the theoretically expected 4.0 µBohrper unit formula for\nstoichiometric NiMnSb15. The estimated measurement accuracy of 8% accounts for un-\ncertainty in the sample thickness extracted from the HRXRD data o f about 5%, as well\nas errors in determining the exact sample area, SQUID calibration an d SQUID response\ndue to finite sample size. Our samples with highest and lowest magnetiz ation show a slight\ndecrease in saturation magnetization, of order 12%. This change is sufficiently small to be\nneglected in the overall assesment of the anisotropy vs. vertical lattice constant of Fig. 3a.\nIn an attempt to understand the effect of higher or lower Mn conce ntration on the crystal\nstructure in our samples, we consider the possible non-stoichiomet ric defects which can exist\nin NiMnSb, asdiscussed in Ref. 10. Formationenergies, magnetic mom ent change and effect\non the half-metallic character are presented there for each type of defect. Mn-related defects\nare a) Mn substituting Ni or Sb (Mn Ni, MnSb), b) Mn on a vacancy position (Mn I), c) Ni or\nSb substituting Mn (Ni Mn, SbMn) or d) a vacancy position at the Mn site (vac Mn). With a\nsurplus of Mn, both Mn substituting Ni or Sb and Mn incorporated on the vacancy position\nseem plausible. However, the formation energy of Mn Sbis more than three times larger than\nfor the other defects, suggesting it should be very rare. On the o ther hand, in the case of a\nMn deficiency, either Ni or Sb could substitute Mn or vacancies can b e built into the crystal.\nAll those three defects have similar formation energies, making the m equally possible.\nExcept for Mn Iand Mn Sb, all of these possible defects reduce the magnetic moment per\nformula unit. Our observations of a lower magnetic moment for samp les with either high or\nlow Mn flux, are thus consistent with the defects Mn Ni, NiMn, SbMnand vac Mn. The posi-\ntive contribution of Mn Ito the magnetic moment is however some 5 times smaller than the\ndecrease induced by the other defects, so some fraction of defe cts of the Mn Ivariety could\nalso be present in the samples. A detailed discussion on the transition from stoichiometric\nNiMnSb towards off-stoichiometric Ni 1-xMn1+xSb is given in Ref. 11. It is shown that the\nlattice constant of off-stoichiometric NiMnSb increases for increas ing substitution of Ni by\nMn. This behavior is clearly seen in our samples for increasing Mn conce ntration and we\n8FIG. 4. Typical RHEED reconstruction of the NiMnSb surface i llustrating the two reconstructions\ndiscussed in the text.\nconclude that this kind of defect is most prominent in our samples. An explanation for a\ndecreasing lattice constant for decreasing Mn concentration is ye t to be found.\nAfurtherobservationwhichmayprovideinsightintotheobserveda nisotropybehaviorcomes\nfrom Reflective High Energy Electron Diffraction (RHEED), which is us ed to monitor the\nsurface of the sample in-situ during the growth. RHEED provides inf ormation about the\nsurface reconstruction, which turns out to be sensitive to the Mn content. In all samples,\nat the beginning of the growth (after approximately one minute), t he surface reconstruction\nexhibits a clear 2 ×1 pattern, meaning a d/2 reconstruction in the [110] crystal direction and\nad/1 reconstruction along [1 ¯10] direction (see Fig.4). How this pattern then evolves during\ngrowth depends on the Mn flux. For ideal Mn flux, the pattern is sta ble throughout the\nentire 2 hour growth time corresponding to a 40 nm layer. A reduced Mn flux results in a\nmore blurry RHEED pattern, but does not lead to any change in the s urface reconstruction.\nA higher Mn flux, on the other hand, causes a change of the recons truction such that the\nd/2 pattern also becomes visible along the [1 ¯10] direction and fades over time in the [110]\ndirection until a 90◦rotation of the original pattern has been completed. The length of\ntime (and thus the thickness) required for this rotation depends s trongly on the Mn flux.\nA slightly enhanced Mn flux causes a very slow rotation of the recons truction that can last\nthe entire growth time, whereas a significant increase of the Mn flux (sample with vertical\nlattice constants above 6.05 ˚A) will cause a rotation of the reconstruction within a few min-\nutes of growth start, corresponding to a thickness of only very f ew monolayers. Based on\nthese observations, our samples can be split into two categories: s amples with a stable 2 ×1\n9reconstruction and those with a 2 ×1 reconstruction that rotates during growth. In Fig.\n3a, samples with a stable RHEED pattern are indicated with filled symbo ls while empty\nsymbols show samples with a rotated RHEED reconstruction. It is int eresting to note that\nall samples with a rotated reconstruction exhibit a very low four-fo ld anisotropy field. In\naddition, the sooner the rotation of the RHEED pattern occurs, t he stronger the uniaxial\nanisotropy is.\nSUMMARY\nWe have shown that the anisotropy of NiMnSb strongly depends on t he composition\nof the material. A variation of the Mn flux results in different (vertica l) lattice constants\n(measured by HRXRD) that can be used for a measure of the Mn con centration. RHEED\nobservations (in-situ) during thegrowthalreadygive anindication o fhighorlowMn concen-\ntration. The anisotropy shows a clear trend for increasing Mn cont ent. Using this together\nwith the RHEED observations, NiMnSb layers with high crystal quality and anisotropies\nas-requested can be grown. The microscopic origin of this behavior remains to be under-\nstood, and it is hoped that this paper will stimulate further efforts in this direction. The\nphenomenology itself is nevertheless of practical significance in tha t it provides interesting\ndesign opportunitiesfordevices such asspin-valves thatcouldbem adeoftwo NiMnSb layers\nwith mutually parallel or orthogonal magnetic easy axes as desired.\nACKNOWLEDGEMENTS\nWe thank T. Naydenova for assistance with the SQUID measuremen ts. This work was\nsupported by the European Commission FP7 contract ICT-257159 “MACALO”.\nREFERENCES\n1R. A. de Groot, F. M. Mueller, P. G. v. Engen, and K. H. J. Buschow, Phys. Rev. Lett.\n50, 2024 (1983).\n2A.Riegler, Ferromagneticresonancestudyof the Half-Heusler alloyNi MnSb: Thebenefitof\nusing NiMnSb as a ferromagnetic layer in pseudo spin-valve b ased spin-torque oscillators ,\nPh.D. thesis, Universitaet Wuerzburg (2011).\n103T. Devolder, A. Meftah, K. Ito, J. A. Katine, P. Crozat, and C. Ch appert, Journal of\nApplied Physics 101, 063916 (2007).\n4D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq, I. Firastrau , F. Ponthenier,\nM. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C. Cyrille , O. Redon, and\nB. Dieny, Nature Materials 6, 447 (2007).\n5G. Consolo, L. Lopez-Diaz, L. Torres, G. Finocchio, A. Romeo, and B. Azzerboni, Applied\nPhysics Letters 91, 162506 (2007).\n6S. M. Mohseni, S. R. Sani, J. Persson, T. N. Anh Nguyen, S. Chung, Y. Pogoryelov,\nand J.˚Akerman, physica status solidi (RRL) Rapid Research Letters 5, 432 (2011) and\nreferences therein\n7A. Koveshnikov, G. Woltersdorf, J. Q. Liu, B. Kardasz, O. Mosend z, B. Heinrich, K. L.\nKavanagh, P. Bach, A. S. Bader, C. Schumacher, C. R¨ uster, C. Gould, G. Schmidt, L. W.\nMolenkamp, and C. Kumpf, Journal of Applied Physics 97, 073906 (2005).\n8C. Kumpf, A. Stahl, I. Gierz, C. Schumacher, S. Mahapatra, F. Lo chner, K. Brunner,\nG. Schmidt, L. W. Molenkamp, and E. Umbach, physica status solidi ( c)4, 3150 (2007).\n9W. Van Roy and M. W´ ojcik, inHalf-metallic Alloys , Lecture Notes in Physics, Vol. 676,\nedited by I. Galanakis and P. Dederichs (Springer Berlin Heidelberg, 2 005) pp. 153–185.\n10B. Alling, S. Shallcross, and I. A. Abrikosov, Phys. Rev. B 73, 064418 (2006).\n11M. Ekholm, P. Larsson, B. Alling, U. Helmersson, and I. A. Abrikosov , Journal of Applied\nPhysics108, 093712 (2010).\n12B. Heinrich and J. A. C. Bland, eds., Ultrathin Magnetic Structures II (Springer Berlin\nHeidelberg, 1994).\n13C. Kittel, Phys. Rev. 73, 155 (1948).\n14We have confirmed from frequency dependent measurements tha t this assumption leads\nto errors smaller than the size of the symbols in Fig. 3a.\n15T. Graf, C. Felser, and S. S. Parkin, Progress in Solid State Chemist ry39, 1 (2011).\n11"    },    {        "title": "1703.09444v2.Temperature_dependent_magnetic_damping_of_yttrium_iron_garnet_spheres.pdf",        "content": "Temperature dependent magnetic damping of yttrium iron garnet spheres\nH. Maier-Flaig,1, 2,\u0003S. Klingler,1, 2C. Dubs,3O. Surzhenko,3R.\nGross,1, 2, 4M. Weiler,1, 2H. Huebl,1, 2, 4and S. T. B. Goennenwein1, 2, 4, 5, 6\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany\n4Nanosystems Initiative Munich, 80799 M unchen, Germany\n5Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden, Germany\n6Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n(Dated: June 5, 2017)\nWe investigate the temperature dependent microwave absorption spectrum of an yttrium iron\ngarnet sphere as a function of temperature (5 K to 300 K) and frequency (3 GHz to 43 :5 GHz). At\ntemperatures above 100 K, the magnetic resonance linewidth increases linearly with temperature\nand shows a Gilbert-like linear frequency dependence. At lower temperatures, the temperature\ndependence of the resonance linewidth at constant external magnetic \felds exhibits a characteristic\npeak which coincides with a non-Gilbert-like frequency dependence. The complete temperature and\nfrequency evolution of the linewidth can be modeled by the phenomenology of slowly relaxing rare-\nearth impurities and either the Kasuya-LeCraw mechanism or the scattering with optical magnons.\nFurthermore, we extract the temperature dependence of the saturation magnetization, the magnetic\nanisotropy and the g-factor.\nI. INTRODUCTION\nThe magnetization dynamics of the ferrimagnetic insu-\nlator yttrium iron garnet (YIG) recently gained renewed\ninterest as YIG is considered an ideal candidate for spin-\ntronic applications as well as spin-based quantum infor-\nmation storage and processing1{4due to the exception-\nally low damping of magnetic excitations as well as its\nmagneto-optical properties5{7. In particular, consider-\nable progress has been made in implementing schemes\nsuch as coupling the magnetic moments of multiple YIG\nspheres2,8or interfacing superconducting quantum bits\nwith the magnetic moment of a YIG sphere3,9.\nMagnetization dynamics in YIG have been investi-\ngated in a large number of studies in the 1960s.10{12\nHowever, a detailed broadband study of the magnetiza-\ntion dynamics in particular for low temperatures is still\nmissing for bulk YIG. Nevertheless, these parameters are\nessential for the design and optimization of spintronic\nand quantum devices. Two recent studies13,14consider\nthe temperature dependent damping of YIG thin \flms.\nHaidar et al.13report a large Gilbert-like damping of\nunknown origin, while the low damping thin \flms inves-\ntigated by Jermain et al.14show a similar behavior as\nreported here.\nOur systematic experiments thus provide an impor-\ntant link between the more recent broadband studies on\nYIG thin \flms and the mostly single-frequency studies\nfrom the 1960s: We investigate the magnetostatic spin\nwave modes measured in a YIG sphere using broadband\nmagnetic resonance up to 43.5 GHz in the temperature\nrange from 5 to 300K. We extract the temperature de-\npendent magnetization, the g-factor and the magnetic\nanisotropy of YIG. Additionally, we focus our analysis on\nthe temperature dependent damping properties of YIGand identify the phenomenology of slowly relaxing rare-\nearth impurities and either the Kasuya-LeCraw mecha-\nnism or the scattering with optical magnons as the mi-\ncroscopic damping mechanism.\nThe paper is organized as follows. We \frst give a short\nintroduction into the experimental techniques followed by\na brief review of the magnetization damping mechanisms\nreported for YIG. Finally, we present the measured data\nand compare the evolution of the linewidth with temper-\nature and frequency with the discussed damping models.\nThe complete set of raw data and the evaluation routines\nare publicly available.15\nII. EXPERIMENTAL DETAILS AND\nFERROMAGNETIC RESONANCE THEORY\nThe experimental setup for the investigation of the\ntemperature dependent broadband ferromagnetic reso-\nnance (bbFMR) is shown schematically in Fig. 1. It\nconsists of a coplanar wave guide (CPW) onto which\na 300 µm diameter YIG sphere is mounted above the\n300µm wide center conductor. The [111] direction of\nthe single crystalline sphere is aligned along the CPW\nsurface normal as con\frmed using Laue di\u000braction (not\nshown). We mount a pressed diphenylpicrylhydrazyl\n(DPPH) powder sample in a distance of approximately\n5 mm from the sphere. The identical sample with the\nsame alignment has been used in Ref. 16. This assembly\nis mounted on a dip stick in order to place the YIG sphere\nin the center of a superconducting magnet (Helmholtz\ncon\fguration) in a Helium gas-\row cryostat. End-launch\nconnectors are attached to the CPW and connected to\nthe two ports of a vector network analyzer (VNA) mea-\nsuring the phase sensitive transmission of the setup uparXiv:1703.09444v2  [cond-mat.mtrl-sci]  2 Jun 20172\nliquid Helium\nside view\nCPWP1\nP2H0hMW hMWVNAP1P2\nside viewside view5mm\nDPPH\nYIG\ncom ±1‰ ±1‰\nFIG. 1. The coplanar waveguide (CPW), on which the YIG\nsphere and a DPPH marker are mounted (right), is inserted\ninto a magnet cryostat (left). The microwave transmission\nthrough the setup is measured phase sensitively using a vec-\ntor network analyzer (VNA). As the Oersted \feld hMW(red)\naround the center conductor of the CPW extends into the\nYIG sphere and the DPPH, we can measure the microwave\nresponse spectra of both samples. A superconducting magnet\nprovides the static external magnetic \feld H0(orange) at the\nlocation of the sample. Also shown are the lines correspond-\ning to the speci\fed 1 ‰homogeneity of the \feld for a on-axis\ndeviation from the center of magnet (com).\nto 43:5 GHz.\nThe sphere is placed within the microwave Oersted\n\feldhMWof the CPW's center conductor which is excited\nwith a continuous wave microwave of variable frequency.\nWe apply a static external magnetic \feld H0perpen-\ndicular to the CPW surface and thus hMWis oriented\nprimarily perpendicular to H0. The microwave Oersted\n\feld can therefore excite magnetization precession at fre-\nquencies that allow a resonant drive. The magnetization\nprecession is detected by electromagnetic induction via\nthe same center conductor.17This induction voltage in\ncombination with the purely transmitted microwave sig-\nnal is measured phase sensitively as the complex scatter-\ning parameter Sraw\n21(!) at port 2 of the VNA.\nThe frequency-dependent background is eliminated as\nfollows: A static external magnetic \feld su\u000eciently large\nthat no resonances are expected in the given microwave\nfrequency range is applied and the transmission at this\n\feld is recorded as the background reference SBG\n21. Then,\nthe external \feld is set to the value at which we ex-\npect resonances of YIG in the given frequency range and\nrecord the transmission Sraw\n21. We \fnally divide Sraw\n21by\nSBG\n21. This corrects for the frequency dependent attenu-\nation and the electrical length of the setup. We choose\nthis background removal method over a microwave cal-\nibration because it additionally eliminates the \feld andtemperature dependence of S21that arises from the ther-\nmal contraction and movement of the setup and magnetic\nmaterials in the microwave connectors. In the following\nwe always display S21=Sraw\n21=SBG\n21.\nFor the evaluation of the magnetization dynamics, we\n\ft the transmission data to S21=\u0000ifZ\u001f +A1+A2f\nfor eachH0. Here,A1;2describe a complex-valued back-\nground and\n\u001f(f;H 0) =\u00160Ms\r\n2\u0019\u0000\r\n2\u0019\u00160H0\u0000i\u0001f\u0001\nf2res\u0000f2\u0000if\u0001f(1)\nis the ferromagnetic high-frequency susceptibility.17,18\nThe free parameters of the \ft are the resonance frequency\nfres, the full width at half maximum (FWHM) linewidth\n\u0001fas well as the complex scaling parameter Z, which\nis proportional to the strength of the inductive coupling\nbetween the speci\fc magnetic resonance mode and the\nCPW. For a given \fxed magnetic \feld the \ft parame-\nters\r\n2\u0019(gyromagnetic ratio) and Ms(saturation magne-\ntization) are completely correlated with Zand are thus\n\fxed. They are later determined from \ftting the disper-\nsion curves.19\nIn spheres various so-called magnetostatic modes\n(MSM) arise due to the electromagnetic boundary\nconditions.20These modes can be derived from the\nLandau-Lifshitz equation in the magnetostatic limit ( ~r\u0002\n~H= 0) for insulators.21The lineshape of all modes is\ngiven by Eq. 1. Due to the di\u000berent spatial mode pro\fles\nand the inhomogeneous microwave \feld, the inductive\ncoupling and thus Zis mode dependent.22A detailed re-\nview of possible modes, their distribution and dispersion\nis given in R oschmann and D otsch20. We will only dis-\ncuss the modes (110) and (440) in detail in the following\nas all the relevant characteristics of all other modes can\nbe related to these two modes. Their linear dispersions\nare given by20\nf110\nres=\r\n2\u0019\u00160(H0+Hani) (2)\nf440\nres=\r\n2\u0019\u00160\u0012\nH0+Hani+Ms\n9\u0013\n(3)\nwhereHaniis the magnetic anisotropy \feld and\r\n2\u0019is\nthe gyromagnetic ratio which relates to the g-factor by\n\r\n2\u0019=\u0016B\nhg. It is thus generally assumed that gis the\nsame for all modes. We note that the apparent g-factor\nmay still vary in between modes if the modes experience\na di\u000berent anisotropy.23,24Such an anisotropy contribu-\ntion can be caused by surface pit scattering as it a\u000bects\nmodes that are localized at the surface stronger than bulk\nlike modes25. In our experiment, no such variation in g\ncoinciding with a change in anisotropy was observed and\nwe use a mode number independent gin the following.\nKnowledge of the dispersion relations of the two modes\nallows to determine the saturation magnetization from\n\u00160Ms(T) = 92\u0019\n\r\u0001fM= 92\u0019\n\r\u0000\nf440\nres\u0000f110\nres\u0001\n:(4)3\nThe anisotropy \feld is extracted by extrapolating the\ndispersion relations in Eqs. (2) and (3) to H0= 0.\nThe temperature dependent linewidth \u0001 fof the modes\nis the central result of this work. For a short review of\nthe relevant relaxation processes we refer to the dedicated\nSec. III.\nIn this work, we investigate the T-dependence of Ms,\nHani,gand \u0001f. Accurate determination of the g-factor\nand the anisotropy Hanirequires accurate knowledge of\nH0. In order to control the temperature of the YIG\nsphere and CPW, they are placed in a gas-\row cryo-\nstat as displayed schematically in Fig. 1. The challenge\nin this type of setup is the exact and independent deter-\nmination of the static magnetic \feld and its spatial in-\nhomogeneity. Lacking an independent measure of H0,26\nwe only report the relative change of gandHanifrom\ntheir respective room temperature values which were de-\ntermined separately using the same YIG sphere.16Note\nthat we determine the resonance frequencies directly in\nfrequency space. Our results on linewidth and magneti-\nzation are hence independent of a potential uncertainty\nin the absolute magnitude of H0and its inhomogeneity.\nIII. RELAXATION THEORY\nWhen relaxation properties of ferromagnets are dis-\ncussed today, the most widely applied model is the so-\ncalled Gilbert type damping. This purely phenomeno-\nlogical model is expressed in a damping term of the form\n\u000bM\u0002dM\ndtin the Landau-Lifshitz equation. It describes\na viscous damping, i.e. a resonance linewidth that de-\npends linearly on the frequency. A linear frequency de-\npendence is often found in experiments and the Gilbert\ndamping parameter \u000bserves as a \fgure of merit of the\nferromagnetic damping that allows to compare samples\nand materials. It contains, however, no insight into the\nunderlying physical mechanisms.\nIn order to understand the underlying microscopic re-\nlaxation processes of YIG, extensive work has been car-\nried out. Improvements on both the experimental side\n(low temperatures27, temperature dependence10,28,29,\nseparate measurements of MzandMxy11) and on the\nsample preparation (varying the surface pit size25, pu-\nrifying Yttrium10, doping YIG with silicon30and rare-\nearth elements30{33) led to a better understanding of\nthese mechanisms.\nHowever, despite these e\u000borts the microscopic origin\nof the dominant relaxation mechanism for bulk YIG at\nroom temperature is still under debate. It has been\ndescribed by a two-magnon process by Kasuya and\nLeCraw28(1961). In this process, a uniformly-precessing\nmagnon (k= 0) relaxes under absorption of a phonon\nto ak6= 0 magnon. If the thermal energy kBTis\nmuch larger than the energy of the involved magnons and\nphonons (T > 100 K) and low enough that no higher-\norder processes such as four-magnon scattering play a\nrole (T < 350 K), the Kasuya-LeCraw process yields alinewidth that is linear in frequency and temperature:\n\u0001fKL/T;f.28,30This microscopic process is therefore\nconsidered to be the physical process that explains the\nphenomenological Gilbert damping for low-damping bulk\nYIG. More recently, Cherepanov et al.34pointed out\nthat the calculations by Kasuya and LeCraw28assume\na quadratic magnon dispersion in k-space which is only\ncorrect for very small wave numbers k. Taking into ac-\ncount a more realistic magnon dispersion (quadratic at\nlowk, linear to higher k), the Kasuya-LeCraw mechanism\ngives a value for the relaxation rate that is not in line\nwith the experimental results. Cherepanov therefore de-\nveloped an alternative model that traces back the linear\nfrequency and temperature dependence at high tempera-\ntures (150 K to 300 K) to the interaction of the uniform-\nprecession mode with optical magnons of high frequency.\nRecently, atomistic calculations by Barker and Bauer35\ncon\frmed the assumptions on the magnon spectrum that\nare necessary for the quantitative agreement of the latter\ntheory with experiment.\nBoth theories, the Kasuya-LeCraw theory and the\nCherepanov theory, aim to describe the microscopic ori-\ngin of the intrinsic damping. They deviate in their\nprediction only in the low-temperature ( T < 100 K)\nbehavior.30At these temperatures, however, impurities\ntypically dominate the relaxation and mask the contri-\nbution of the intrinsic damping process. Therefore, the\ndominant microscopic origin of the YIG damping at tem-\nperatures above 150K has not been unambiguously de-\ntermined to date.\nIf rare-earth impurities with large orbital momentum\nexist in the crystal lattice, their exchange coupling with\nthe iron ions introduces an additional relaxation chan-\nnel for the uniform precession mode of YIG. Depend-\ning on the relaxation rate of the rare-earth impurities\nwith respect to the magneto{dynamics of YIG, they are\nclassi\fed into slowly and fast relaxing rare-earth impu-\nrities. This is an important distinction as the e\u000eciency\nof the relaxation of the fundamental mode of YIG via\nthe rare-earth ion to the lattice at a given frequency de-\npends on the relaxation rate of the rare-earth ion and the\nstrength of the exchange coupling. In both the slow and\nthe fast relaxor case, a characteristic peak-like maximum\nis observed in the linewidth vs. temperature dependence\nat a characteristic, frequency-dependent temperature12.\nThe frequency dependence of this peak temperature al-\nlows to distinguish fast and slowly relaxing rare-earth\nions: The model of a fast relaxing impurity predicts that\nthe peak temperature is constant, while in the case of\nslowly relaxing rare-earth ions the peak temperature is\nexperted to increase with increasing magnetic \feld (or\nfrequency). The relaxation rate of rare-earths \u001cREis typ-\nically modeled by a direct magnon to phonon relaxation,\nan Orbach processes36,37that involves two phonons, or\na combination of both. The inverse relaxation rate of\nan Orbach process is described by1\n\u001cOrbach =B\ne\u0001=(kBT)\u00001\nwith the crystal \feld splitting \u0001 and a proportionality\nfactorB. A direct process leads to an inverse relax-4\nation rate of1\n\u001cdirect =1\n\u001c0coth\u000e\n2kBTwith\u001c0, the relaxation\ntime atT= 0 K. It has been found experimentally that\nmost rare-earth impurities are to be classi\fed as slow\nrelaxors.30The sample investigated here is not intention-\nally doped with a certain rare-earth element and the peak\nfrequency and temperature dependence indicates a slow\nrelaxor. We therefore focus on the slow relaxing rare-\nearth impurity model in the following.\nDeriving the theory of the slowly relaxing impurities\nhas been performed comprehensively elsewhere.30The\nlinewidth contribution caused by a slowly relaxing rare-\nearth impurity is given by31:\n\u0001fSR=C\n2\u0019f\u001cRE\n1 + (f\u001cRE)2(5)\nwithC/1\nkBTsech\u0010\n\u000ea\n2kBT\u0011\n. Therein, \u000eais the splitting\nof the rare-earth Kramers doublet which is given by the\ntemperature independent exchange interaction between\nthe iron ions and the rare-earth ions.\nAlso Fe2+impurities in YIG give rise to a process\nthat leads to a linewidth peak at a certain temperature.\nThe physical origin of this so-called valence exchange or\ncharge-transfer linewidth broadening is electron hopping\nbetween the iron ions.30Simpli\fed, it can be viewed as\na two level system just like a rare-earth ion and thus re-\nsults in the same characteristic linewidth maximum as\na slowly relaxing rare-earth ion. For valence exchange,\nthe energy barrier \u0001 hopthat needs to be overcome for\nhopping determines the time scale of the process. The\ntwo processes, valence exchange and rare-earth impurity\nrelaxation, can therefore typically not be told apart from\nFMR measurements only. In the following, we use the\nslow relaxor mechanism exclusively. This model consis-\ntently describes our measurement data and the resulting\nmodel parameters are in good agreement with literature.\nWe would like to emphasize, however, that the valence\nexchange mechanism as the relevant microscopic process\nresulting for magnetization damping can not be ruled out\nfrom our measurements.\nIV. EXPERIMENTAL RESULTS AND\nDISCUSSION\nTwo exemplary S21broadband spectra recorded at two\ndistinct temperatures are shown in Fig. 2. The color-\ncoded magnitude jS21jis a measure for the absorbed mi-\ncrowave power. High absorption (bright color) indicates\nthe resonant excitation of a MSM in the YIG sphere or\nthe excitation of the electron paramagnetic resonance of\nthe DPPH. In the color plot the color scale is truncated\nin order to improve visibility of small amplitude reso-\nnances. In addition, the frequency axis is shifted relative\nto the resonance frequency of a linear dispersion with\ng= 2:0054 (fg=2:0054\nres =g\u0016B\nh\u00160H) for each \feld. In this\nway, modes with g= 2:0054 appear as vertical lines. A\n0.20.40.60.81.01.21.4¹0H0 (T)290 K\nminmax\nAbsorption\n−1.0 −0.5 0.0 0.5 1.0\nf¡f(g=2:0054)\nres  (GHz)0.20.40.60.81.01.21.4¹0H0 (T)20 KΔfAΔfM\nf (GHz)10.05 10.06 10.07 10.08\nIm(S21)\n2\n046810\n0\n-4-224Re(S21)\nȴffres(a)\n(b)FIG. 2. Eigenmode spectra of the YIG sphere at (a) 290 K\nand (b) 20 K. The (110) and (440) MSM are marked with red\ndashed lines. The change in their slope gives the change of the\ng-factor of YIG. Their splitting (\u0001 fM, red arrow) depends lin-\nearly on the YIG magnetization. The increase in Msto lower\ntemperatures is already apparent from the increased splitting\n\u0001fM. Marked in orange is the o\u000bset of the resonance fre-\nquency \u0001fAextrapolated to H0= 0 resulting from anisotropy\n\feldsHanipresent in the sphere. The green marker denotes\nthe position of the DPPH resonance line which increases in\namplitude considerably to lower temperatures. Inset: S21pa-\nrameter (data points) and \ft (lines) at \u00160H= 321 mT and\nT= 20 K.\ndeviatingg-factor is therefore easily visible as a di\u000ber-\nent slope. Comparing the spectra at 290 K [Fig. 2 (a)]\nto the spectra at 20 K [Fig. 2 (b)], an increase of the g-\nfactor is observed for all resonance modes upon reducing\nthe temperature. The rich mode spectrum makes it nec-\nessary to carefully identify the modes and assign mode\nnumbers. Note that the occurrence of a particular mode\nin the spectrum depends on the position of the sphere\nwith respect to the CPW due to its inhomogeneous exci-\ntation \feld. We employ the same method of identifying\nthe modes as used in Ref. 16 and \fnd consistent mode\nspectra. As mentioned before, we do not use the DPPH\nresonance (green arrow in Fig. 2) but the (110) YIG\nmode as \feld reference. For this \feld reference, we take\ng(290 K) = 2 :0054 and\r\n2\u0019\u00160Hani(290 K) = 68 :5 MHz de-\ntermined for the same YIG sphere at room temperature\nin an electromagnet with more accurate knowledge of the\napplied external magnetic \feld.16The discrepancy of the\nDPPHg-value from the literature values of g= 2:0036 is5\nattributed to the non-optimal location of the DPPH spec-\nimen in the homogeneous region of the superconducting\nmagnet coils.\nIn Fig. 2, the \ftted dispersion of the (110) and (440)\nmodes are shown as dashed red lines. As noted previ-\nously, we only analyze these two modes in detail as all\nparameters can be extracted from just two modes. The\n(110) and (440) mode can be easily and unambiguously\nidenti\fed by simply comparing the spectra with the ones\nfound in Ref 16. Furthermore, at high \felds, both modes\nare clearly separated from other modes. This is necessary\nas modes can start hybridizing when their (unperturbed)\nresonance frequencies are very similar (cf. low-\feld re-\ngion of Fig. 2 (b)) which makes a reliable determination of\nthe linewidth and resonance frequency impossible. These\nattributes make the (110) and the (440) mode the ideal\nchoice for the analysis.\nAs described in Sec. II, we simultaneously \ft the (110)\nand the (440) dispersions with the same g-factor in or-\nder to extract Ms,Haniandg. In the \ft, we only take\nthe high-\feld dispersion of the modes into account where\nno other modes intersect the dispersion of the (110) and\n(440) modes. The results are shown in Fig. 3. Note that\nthe statistical uncertainty from the \ft is not visible on the\nscale of any of the parameters Ms,Haniandg. Following\nthe work of Solt38, we model the resulting temperature\ndependence of the magnetization (Fig. 3 (a)) with the\nBloch-law taking only the \frst order correction into ac-\ncount:\nMs=M0\u0010\n1\u0000aT3\n2\u0000bT5\n2\u0011\n: (6)\nThe best \ft is obtained for \u00160M0 =\n249:5(5) mT, a = (23\u00063)\u000210\u00006K\u00003=2and\nb= (1:08\u00060:11)\u000210\u00007K\u00005=2. The obtained \ft\nparameters depend strongly on the temperature window\nin which the data is \ftted. Hence, the underlying\nphysics determining the constants aandbcannot be\nresolved.39Nevertheless, the temperature dependence\nofMsis in good agreement with the results determined\nusing a vibrating sample magnetometer.40\nIn particular, also the room temperature saturation\nmagnetization of \u00160Ms(300 K) = (180 :0\u00060:8) mT is in\nperfect agreement with values reported in literature.41,42\nNote that the splitting of the modes is purely in frequency\nspace and thus errors in the \feld do not add to the uncer-\ntainty. We detect a small non-linearity of the (110) and\n(440) mode dispersions that is most likely due to devia-\ntions from an ideal spherical shape or strain due to the\nYIG mounting. This results in a systematic, temperature\nindependent residual of the linear \fts to these disper-\nsions. This resulting systematic error of the magnetiza-\ntion is incorporated in the uncertainty given above. How-\never, a deviation from the ideal spherical shape, strain in\nthe holder or a misalignment of the static magnetic \feld\ncan also modify the splitting of the modes and hence re-\nsult in a di\u000berent Ms.43This fact may explain the small\ndiscrepancy of the value determined here and the valuedetermined for the same sphere in a di\u000berent setup at\nroom temperature.16\nFrom the same \ft that we use to determine the mag-\nnetization, we can deduce the temperature dependence\nof the anisotropy \feld \u00160Hani[Fig. 3 (b)]. Most notably,\nHanichanges sign at 200 K which has not been observed\nin literature before and can be an indication that the\nsample is slightly strained in the holder. The resonance\nfrequency of DPPH extrapolated to \u00160H0= 0 (\u0001fani, red\nsquares) con\frms that the error in the determined value\nHaniis indeed temperature independent and very close\nto zero. Thus, the extracted value for the anisotropy is\nnot merely given by an o\u000bset in the static magnetic \feld.\nThe evolution of the g-factor with temperature is\nshown in Fig. 3 (c). It changes from 2 :005 at room tem-\nperature to 2 :010 at 10 K where it seems to approach a\nconstant value. As mentioned before, the modes' disper-\nsion is slightly non-linear giving rise to a systematic, tem-\nperature independent uncertainty in the determination of\ngof\u00060:0008. Theg-factor of YIG has been determined\nusing the MSMs of a sphere for a few selected tempera-\ntures before.12Comparing our data to these results, one\n\fnds that the trend of the temperature dependence of g\nagrees. However, the absolute value of gand the mag-\nnitude of the variation di\u000ber. At the same time, we \fnd\na change of the g-factor of DPPH that is on the scale\nof 0:0012. This may be attributed to a movement of\nthe sample slightly away from the center of magnet with\nchanging temperature due to thermal contraction of the\ndip stick. In this case, the YIG g-factor has to be cor-\nrected by this change. The magnitude of this e\u000bect on\nthe YIGg-factor can not be estimated reliably from the\nchange of the DPPH g-factor alone. Furthermore, the\ntemperature dependence of the DPPH g-factor has not\nbeen investigated with the required accuracy in literature\nto date to allow excluding a temperature dependence of\ntheg-factor of DPPH. We therefore do not present the\ncorrected data but conclude that we observe a change in\nthe YIGg-factor from room temperature to 10 K of at\nleast 0.2 %.\nNext, we turn to the analysis of the damping properties\nof YIG. We will almost exclusively discuss the damping of\nthe (110) mode in the following but the results also hold\nquantitatively and qualitatively for the other modes.16\nVarying the applied microwave excitation power P(not\nshown) con\frms that no nonlinear e\u000bects such as a power\nbroadening of the modes are observed with P= 0:1 mW.\nNote that due to the microwave attenuation in the mi-\ncrowave cabling, the microwave \feld at the sample loca-\ntion decreases with increasing frequency for the constant\nexcitation power.\nFirst, we evaluate the frequency dependent linewidth\nfor several selected temperatures [Fig. 4 (a)]. At temper-\natures above 100 K, a linear dependence of the linewidth\nwith the resonance frequency is observed. This depen-\ndence is the usual so-called Gilbert-like damping and\nthe slope is described by the Gilbert damping param-\neter\u000b. A linear frequency dependence of the damping6\n180200220240260¹0M (mT)\n(a)model\n−20246¹0Hani (mT)\n(b)YIG\nDPPH\n0 50 100 150 200 250 300\nTemperature (K)2.0042.0082.0122.0162.020g-factor (unitless)\n0.08%\n0.26%\n(c)\nFIG. 3. (a)YIG magnetization as function of tempera-\nture extracted from the (110) and (440) mode dispersions us-\ning Eq. 4. The purple line shows the \ft to a Bloch model\n(cf. parameters in the main text). (b)YIG anisotropy \feld\n\u00160Hani(T) =2\u0019\n\r\u0001fani. Red squares: Same procedure applied\nto the DPPH dispersion as reference. (c)YIGg-factor (blue\ncircles). For reference, the extracted DPPH g-factor is also\nshown (red squares). The gray numbers indicate the rela-\ntive change of the g-factors from the lowest to the highest\nmeasured temperature (gray horizontal lines). As we use the\nYIG (110) mode as the magnetic \feld reference, the extracted\nvalue ofgandHaniat 300 K are \fxed to the values determined\nin the room temperature setup.16\nin bulk YIG has been described by the theory developed\nby Kasuya and LeCraw28and the theory developed by\nCherepanov et al.34(cf. Sec. III). We extract \u000bfrom\na global \ft of a linear model to the (110) and (440)\nlinewidth with separate parameters for the inhomoge-\nneous linewidths \u0001 f110\n0and \u0001f440\n0and a shared Gilbert\ndamping parameter \u000bfor all modes:16\n\u0001f= 2\u000bf+ \u0001f110;440\n0 (7)\nThe \ft is shown exemplarily for the 290 K (red) data in\nFig. 4 (a).\nThe Gilbert damping parameter \u000bextracted using this\n\ftting routine for each temperature is shown in Fig. 5 (a).\nConsistently with both theories, \u000bincreases with increas-\ning temperature. The error bars in the \fgure correspond\nto the maximal deviation of \u000bextracted from separate\n\fts for each mode. They therefore give a measure of\nhow\u000bscatters in between modes. The statistical error\nof the \ft (typically \u00060:00001) is not visible on this scale.\nThe Gilbert damping parameter \u000blinearly extrapolatedto zero temperature vanishes. Note that this is consis-\ntent with the magnon-phonon process described by Ka-\nsuya and LeCraw28but not with the theory developed\nby Cherepanov et al.34. For room temperature, we ex-\ntract a Gilbert damping of 4 \u000210\u00005which is in excel-\nlent agreement with the literature value.16,44From the\n\ft, we also extract the inhomogeneous linewidth \u0001 f0,\nwhich we primarily associate with surface pit scattering\n(Sec. III, Ref 16). In the data, a slight increase of \u0001 f0\ntowards lower temperatures is present [Fig. 5 (b)]. Such a\nchange in the inhomogeneous linewidth can be caused by\na change in the surface pit scattering contribution when\nthe spin-wave manifold changes with Ms.16,25\nNote that according to Fig. 5 (b) \u0001 f0is higher for the\n(440) mode than for the (110) mode. This is in agreement\nwith the theoretical expectation that surface pit scatter-\ning has a higher impact on \u0001 f0for modes that are more\nlocalized at the surface of the sphere like the (440) mode\ncompared to the more bulk-like modes such as the (110)\nmode25.45\nTurning back to Fig. 4 (a), for low temperatures (20 K,\nblue data points), a Gilbert-like damping model is obvi-\nously not appropriate as the linewidth increases consider-\nably towards lower frequencies instead of increasing lin-\nearly with increasing frequency. Typically, one assumes\nthat the damping at low frequencies is dominated by so-\ncalled low \feld losses that may arise due to domain for-\nmation. The usual approach is then to \ft a linear trend\nto the high-frequency behavior only. Note, however, that\neven though the frequency range we use is already larger\nthan usually reported13,14,46, this approach yields an un-\nphysical, negative damping. We conclude that the model\nof a Gilbert-like damping is only valid for temperatures\nexceeding 100 K (Fig. 5) for the employed \feld and fre-\nquency range.\nThe linewidth data available in literature are typi-\ncally taken at a \fxed frequency and the linewidth is dis-\nplayed as a function of temperature12,29,30. We can ap-\nproximately reproduce these results by plotting the mea-\nsured linewidth at \fxed H0as a function of temperature\n[Fig. 4 (b)].47A peak-like maximum of the linewidth be-\nlow 100 K is clearly visible. For increasing magnetic \feld\n(frequency), the peak position shifts to higher tempera-\ntures. This is the signature of a slowly relaxing rare-earth\nimpurity (Sec. III). A fast relaxing impurity is expected\nto result in a \feld-independent linewidth vs. temperature\npeak and can thus be ruled out. At the peak position,\nthe linewidth shows an increase by 2 :5 MHz which trans-\nlates with the gyromagnetic ratio to a \feld linewidth in-\ncrease of 0:08 mT. For 0.1 at. % Terbium doped YIG, a\nlinewidth increase of 80 mT has been observed48. Con-\nsidering that the linewidth broadening is proportional to\nthe impurity concentration and taking the speci\fed pu-\nrity of the source material of 99.9999% used to grow the\nYIG sphere investigated here, we estimate an increase of\nthe linewidth of 0 :08 mT, in excellent agreement with the\nobserved value.\nModeling the linewidth data is more challenging: The7\n0 5 10 15 20 25 30 35 40 45\nf110\nres (GHz)0¢f023456¢f110 (MHz)(a) 20 K\n290 K\n0 50 100 150 200\nTemperature (K)0.51.01.52.02.53.03.5¢f110 (MHz)\n(b) 341 mT, 9.6 GHz\n1007 mT, 28.3 GHzTmax\n0:3TTmax\n1:0T\nFIG. 4. (a)Full width at half maximum (FWHM) linewidth \u0001 f110\nresof the (110) mode as a function of frequency for di\u000berent\ntemperatures. A linear Gilbert-like interpretation is justi\fed in the high- Tcase (T > 100 K) only. Below 100 K, the slope of\n\u0001f110(f110\nres) is not linear so that a Gilbert type interpretation is no longer applicable. (b)FWHM linewidth as a function of\ntemperature for two di\u000berent \fxed external magnetic \felds. The linewidth peaks at a magnetic \feld dependent temperature\nthat can be modeled using the phenomenology of rare-earth impurities resulting in Tmax(vertical dotted lines).\n01234® (unitless)£10−5\n(a)\n0 50 100 150 200 250 300\nTemperature (K)0123¢f0 (MHz) (b)110\n440\nFIG. 5. (a)Gilbert damping parameter \u000bdetermined from the slope of a linear \ft to the \u0001 f(f;T) data for frequencies\nabove 20 GHz. The red line shows the linear dependence of the linewidth with temperature expected from the Kasuya-LeCraw\nprocess. (b)Inhomogeneous linewidth \u0001 f0(intersect of the aforementioned \ft with f110\nres= 0) as a function of temperature.\nThe inhomogeneous linewidth shows a slight increase with decreasing temperature down to 100 K. In the region where the\nslow relaxor dominates the linewidth (gray shaded area, cf. Fig. 4), the linear \ft is not applicable and unphysical damping\nparameters and inhomogeneous linewidths are extracted.\nmodel of a slowly relaxing rare-earth ion contains the\nexchange coupling of the rare-earth ion and the iron\nsublattice, and its temperature dependent relaxation fre-\nquency as parameters. As noted before, typically a di-\nrect and an Orbach process model the relaxation rate,\nand both of these processes have two free parameters.\nUnless these parameters are known from other experi-\nments for the speci\fc impurity and its concentration in\nthe sample, \ftting the model to the temperature behav-\nior of the linewidth at just one \fxed frequency gives am-\nbiguous parameters. In principle, frequency resolved ex-\nperiments as presented in this work make the determina-\ntion of the parameters more robust as the mechanism re-\nsponsible for the rare-earth relaxation is expected not to\nvary as a function of frequency. The complete frequency\nand \feld dependence of the linewidth is shown in Fig. 6.At temperatures above approx 100 K, the linewidth in-\ncreases monotonically with \feld, in agreement with a\ndominantly Gilbert-like damping mechanisms, which be-\ncomes stronger for higher temperatures. On the same\nlinear color scale, the linewidth peak below 100 K and its\nfrequency evolution is apparent. Fig. 4 (b) corresponds\nto horizontal cuts of the data in Fig. 6 at \u00160H0= 341\nand 1007 mT.\nFor typical YIG spheres, that are not speci\fcally en-\nriched with only one rare-earth element, the composition\nof the impurities is unknown. Di\u000berent rare-earth ions\ncontribute almost additively to the linewidth and have\ntheir own characteristic temperature dependent relax-\nation frequency respectively peak position. This is most\nprobably the case for the YIG sphere of this study. The\nconstant magnitude of the peak above 0 :3 T and the con-8\nstant peak width indicates that fast relaxing rare-earth\nions play a minor role. The evolution of the linewidth\nwithH0andfcan therefore not be \ftted to one set\nof parameters. We thus take a di\u000berent approach and\nmodel just the shift of the peak position in frequency\nand temperature as originating from a single slowly re-\nlaxing rare-earth impurity. For this, we use a value\nfor the exchange coupling energy between the rare-earth\nions and the iron sublattice in a range compatible with\nliterature30of\u000ea= 2:50 meV. To model the rare-earth\nrelaxation rate as a function of temperature, we use the\nvalues determined by Clarke et al.32for Neodymium\ndoped YIG: \u001c0= 2:5\u000210\u000011s for the direct process and\n\u0001 = 10:54 meV and B= 9\u00021011s\u00001for the Orbach pro-\ncess. The model result, i.e. the peak position, is shown as\ndashed line in Fig. 6 and shows good agreement with the\ndata. This indicates that, even though valence exchange\nand other types of impurities cannot be rigourously ex-\ncluded, rare-earth ions are indeed the dominant source\nfor the linewidth peak at low temperatures.\nV. CONCLUSIONS\nWe determined the ferromagnetic dispersion and\nlinewidth of the (110) magnetostatic mode of a polished\nYIG sphere as a function of temperature and frequency.\nFrom this data, we extract the Gilbert damping param-\neter for temperatures above 100 K and \fnd that it varies\nlinearly with temperature as expected according to the\ntwo competing theories of Kasuya and LeCraw28andCherepanov et al.34. At low temperatures, the temper-\nature dependence of the linewidth measured at constant\nmagnetic \feld shows a peak that shifts to higher tempera-\ntures with increasing frequency. This indicates slowly re-\nlaxing impurities as the dominant relaxation mechanism\nfor the magnetostatic modes below 100 K. We model\nthe shift of the peak position with temperature and fre-\nquency with values reported for Neodymium impurities32\nin combination with a typical value for the impurity-ion\nto iron-ion exchange coupling. We \fnd that these param-\neters can be used to describe the position of the linewidth\npeak. We thus directly show the implications of (rare\nearth) impurities as typically present in YIG samples\non the dynamic magnetic properties of the ferrimagnetic\ngarnet material. Furthermore, we extract the temper-\nature dependence of the saturation magnetization, the\nanisotropy \feld and the g-factor.\nACKNOWLEDGEMENTS\nThe authors thank M. S. Brandt for helping out with\nthe microwave equipment. C.D. and S.O. would like to\nacknowledge R. Meyer, M. Reich, and B. Wenzel (IN-\nNOVENT e.V.) for technical assistance in the YIG crys-\ntal growth and sphere preparation. We gratefully ac-\nknowledge funding via the priority program Spin Caloric\nTransport (spinCAT), (Projects GO 944/4 and GR\n1132/18), the priority program SPP 1601 (HU 1896/2-\n1) and the collaborative research center SFB 631 of the\nDeutsche Forschungsgemeinschaft.\n\u0003hannes.maier-\raig@wmi.badw.de\n1X. Zhang, C.-l. Zou, L. Jiang, and H. X. Tang, Physical\nReview Letters 113, 156401 (2014).\n2X. Zhang, C.-l. Zou, N. Zhu, F. Marquardt, L. Jiang, and\nH. X. Tang, Nature Communications 6, 8914 (2015).\n3Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami,\nand Y. Nakamura, Physical Review Letters 113, 083603\n(2014).\n4L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and\nC.-M. Hu, Physical Review Letters 114, 227201 (2015).\n5S. Klingler, H. Maier-Flaig, R. Gross, C.-M. Hu, H. Huebl,\nS. T. B. Goennenwein, and M. Weiler, Applied Physics\nLetters 109, 072402 (2016).\n6S. Viola Kusminskiy, H. X. Tang, and F. Marquardt, Phys-\nical Review A 94, 033821 (2016).\n7R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa,\nA. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura,\nPhysical Review B 93, 174427 (2016).\n8N. J. Lambert, J. A. Haigh, S. Langenfeld, A. C. Doherty,\nand A. J. Ferguson, Physical Review A 93, 021803 (2016).\n9Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. Usami, and Y. Nakamura, Science 349, 405\n(2015).\n10E. G. Spencer, R. C. Lecraw, and A. M. Clogston, Physical\nReview Letters 3, 32 (1959).11M. Sparks and C. Kittel, Physical Review Letters 4, 232\n(1960).\n12K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet\nPhysics JETP 12, 1074 (1961).\n13M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas,\nS. Khartsev, and J. \u0017Akerman, Journal of Applied Physics\n117, 17D119 (2015).\n14C. L. Jermain, S. V. Aradhya, J. T. Brangham, M. R. Page,\nN. D. Reynolds, P. C. Hammel, R. A. Buhrman, F. Y.\nYang, and D. C. Ralph, arXiv preprint arXiv:1612.01954\n(2016).\n15H. Maier-Flaig, \\Temperature dependent damp-\ning of yttrium iron garnet spheres { Measure-\nment data and analysis programs,\" (2017),\nhttps://dx.doi.org/10.17605/OSF.IO/7URPT.\n16S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko,\nR. Gross, H. Huebl, S. T. B. Goennenwein, and M. Weiler,\nApplied Physics Letters 110, 092409 (2017).\n17S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Jour-\nnal of Applied Physics 99, 093909 (2006).\n18M. L. Schneider, J. M. Shaw, A. B. Kos, T. Gerrits, T. J.\nSilva, and R. D. McMichael, Journal of Applied Physics\n102, 103909 (2007).9\n50 100 150 200 250\nTemperature (K)0.20.40.60.81.01.21.4¹0H0 (T)\n123456\n¢f110 (MHz)\n5152535\nf110\nres (GHz)\nFIG. 6. Full map of the FWHM linewidth of the (110) mode as function of temperature and \feld resp. resonance frequency\nf110\nres. At low temperatures, only the slow relaxor peak is visible while at high temperatures the Gilbert-like damping becomes\ndominant. The position of the peak in the linewidth modeled by a slow relaxor is shown as dashed orange line. The model\nparameters are taken from Clarke31and taking \u000ea= 2:50 meV. The dotted lines indicate the deviation of the model for 0 :5\u000ea\n(lowerTmax) and 2\u000ea(higherTmax).\n19H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress, Physical Review\nB84, 054424 (2011).\n20P. R oschmann and H. D otsch, physica status solidi (b) 82,\n11 (1977).\n21L. R. Walker, Physical Review 105, 390 (1957).\n22Due to the comparable dimensions of center conductor\nwidth and sphere diameter, we expect that the sphere ex-\nperiences an inhomogeneous microwave magnetic \feld with\nits main component parallel to the surface of the CPW and\nperpendicular to its center conductor. As the microwave\nmagnetic \feld is su\u000eciently small to not cause any non-\nlinear e\u000bects, a mode dependent excitation e\u000eciency is the\nonly e\u000bect of the microwave magnetic \feld inhomogeneity.\n23C. Kittel, Physical Review 76, 743 (1949).\n24P. Bruno, Physical Review B 39, 865 (1989).\n25J. Nemarich, Physical Review 136, A1657 (1964).\n26The resonance frequency of the DPPH sample that has\nbeen measured simultaneously was intended as a \feld cal-\nibration but can not be utilized due to the magnetic \feld\ninhomogeneity. In particular, since the homogeneity of our\nsuperconducting magnet system is speci\fed to 1 ‰for an\no\u000b-axis deviation of 2 :5 mm, the spatial separation of 5 mm\nof the DPPH and the YIG sphere already falsi\fes DPPH\nas an independent magnetic \feld standard. Placing DPPH\nand YIG in closer proximity is problematic as the stray\n\feld of the YIG sphere will a\u000bect the resonance frequency\nof the DPPH. Note further that we are not aware of any re-\nports showing the temperature independence of the DPPH\ng-factor with the required accuracy.\n27J. F. Dillon, Physical Review 111, 1476 (1958).\n28T. Kasuya and R. C. LeCraw, Physical Review Letters 67,\n223 (1961).\n29E. G. Spencer, R. C. Lecraw, and J. Linares, Physical\nReview 123, 1937 (1961).\n30M. Sparks, Ferromagnetic-Relaxation Theory , edited by\nW. A. Nierenberg (McGraw-Hill, 1964).\n31B. H. Clarke, Physical Review 139, A1944 (1965).\n32B. H. Clarke, K. Tweedale, and R. W. Teale, Physical\nReview 139, A1933 (1965).33M. Sparks, Journal of Applied Physics 38, 1031 (1967).\n34V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re-\nports 229, 81 (1993).\n35J. Barker and G. E. W. Bauer, Physical Review Letters\n117, 217201 (2016).\n36R. Orbach, Proceedings of the Physical Society 77, 821\n(1961).\n37B. H. Clarke, Journal of Applied Physics 36, 1211 (1965).\n38I. H. Solt, Journal of Applied Physics 33, 1189 (1962).\n39Note that we failed to reproduce the \ft of Ref. 38 using\nthe data provided in this paper and that the reasonable\nagreement with the there-reported \ft parameters might\nbe coincidence.\n40E. E. Anderson, Physical Review 134, A1581 (1964).\n41P. Hansen, Journal of Applied Physics 45, 3638 (1974).\n42G. Winkler, Magnetic Garnets , Tracts in pure and applied\nphysics; Vol. 5 (Vieweg, 1981).\n43R. L. White, Journal of Applied Physics 31, S86 (1960).\n44P. R oschmann and W. Tolksdorf, Materials Research Bul-\nletin18, 449 (1983).\n45In comparison to Klingler et al.16, here, we do not see\nan increased inhomogeneous linewidth of the (110) mode\nand no secondary mode that is almost degenerate with the\n(110) mode. The di\u000berence can be explained by the ori-\nentation of the sphere which is very di\u000ecult to reproduce\nvery accurately ( <1°) between the experimental setups:\nThe change in orientation either separates the mode that\nis almost degenerate to the (110) mode or makes the degen-\neracy more perfect in our setup. The di\u000berent placement of\nthe sphere on the CPW can also lead to a situation where\nthe degenerate mode is not excited and therefore does not\ninterfere with the \ft.\n46Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz,\nW. Schneider, M. Wu, H. Schultheiss, and A. Ho\u000bmann,\nApplied Physics Letters 101, 152405 (2012).\n47Naturally, the resonance frequency varies slightly\n(\u00060:9 GHz) between the data points because the magneti-\nzation and the anisotropy changes with temperature.\n48J. F. Dillon and J. W. Nielsen, Physical Review Letters 3,\n30 (1959)."    },    {        "title": "0706.3160v3.Spin_pumping_by_a_field_driven_domain_wall.pdf",        "content": "arXiv:0706.3160v3  [cond-mat.mes-hall]  9 Jan 2008Spin pumping by a field-driven domain wall\nR.A. Duine\nInstitute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n(Dated: October 26, 2018)\nWe present the theory of spin pumping by a field-driven domain wall for the situation that spin\nis not fully conserved. We calculate the pumped current in a m etallic ferromagnet to first order\nin the time derivative of the magnetization direction. Irre spective of the microscopic details, the\nresult can be expressed in terms of the conductivities of the majority and minority electrons and\nthe dissipative spin transfer torque parameter β. The general expression is evaluated for the specific\ncase of a field-driven domain wall and for that case depends st rongly on the ratio of βand the\nGilbert damping constant. These results may provide an expe rimental method to determine this\nratio, which plays a crucial role for current-driven domain -wall motion.\nPACS numbers: 72.25.Pn, 72.15.Gd\nI. INTRODUCTION\nAdiabatic quantum pumping of electrons in quantum\ndots1,2has recently been demonstrated experimentally\nforboth charge3and spin4. Currently, the activityin this\nfieldismostlyconcentratedontheeffectsofinteractions5,\ndissipation6, and non-adiabaticity7. Complementary to\nthese developments, the emission of spin current by a\nprecessing ferromagnet — called spin pumping — has\nbeen studied theoretically and experimentally in single-\ndomain magnetic nanostructures8,9,10. One of the differ-\nences between spin pumping in single-domain ferromag-\nnets and quantum pumping in quantum dots is that in\nthelatterthehamiltonianoftheelectronicquasi-particles\nis manipulated directly, usually by varying the gate volt-\nage of the dot. In the case of ferromagnets, however,\nit is the order parameter — the magnetization direction\n— that is driven by an external (magnetic) field. The\ncoupling between the order parameter and the current-\ncarrying electrons in turn pumps the spin current11. The\nopposite effect, i.e., the manipulation of magnetization\nwith spin current, is called spin transfer12,13,14,15.\nRecently, the possibility of manipulating with cur-\nrent the position of a magnetic domain wall via\nspin transfer torques has attracted a great deal of\ntheoretical16,17,18,19,20,21,22,23,24,25,26,27,28,29,30and\nexperimental31,32,33,34,35,36,37,38interest. Although the\nsubject is still controversial18,21, it is by now established\nthat in the long-wavelength limit the equation of motion\nfor the magnetization direction Ω, which in the absence\nof current describes damped precession around the\neffective field −δEMM[Ω]/(/planckover2pi1δΩ), is given by\n/parenleftbigg∂\n∂t+vs·∇/parenrightbigg\nΩ−Ω×/parenleftbigg\n−δEMM[Ω]\n/planckover2pi1δΩ/parenrightbigg\n=−Ω×/parenleftbigg\nαG∂\n∂t+βvs·∇/parenrightbigg\nΩ, (1)\nand contains, to lowest order in spatial derivatives of the\nmagnetization direction, two contributions due the pres-\nence of electric current.The first is the reactivespin transfertorque16,17, which\ncorresponds to the term proportional to ∇Ωon the left-\nhand side of the above equation. It is characterized by\nthe velocity vsthat is linear in the curent and related to\nthe external electric field Eby\nvs=(σ↓−σ↑)E\n|e|ρs, (2)\nwhereσ↑andσ↓denote the conductivities of the major-\nity and minority electrons, respectively, and ρsis their\ndensity difference. (The elementary charge is denoted by\n|e|.) The second term in Eq. (1) due to the current is the\ndissipative spin transfer torque39that is proportional to\nβ19,20,21. Both this parameter, and the Gilbert damping\nparameter αG, have their microscopic origin in processes\nin the hamiltonian that break conservation of spin, such\nas spin-orbit interactions.\nIt turns out that the phenomenology of current-driven\ndomain-wallmotion depends crucially on the value of the\nratioβ/αG. For example, for β= 0 the domain wall\nis intrinsically pinned18, meaning that there is a criti-\ncal current even in the absence of inhomogeneities. For\nβ/αG= 1ontheotherhand, thedomainwallmoveswith\nvelocity vs. Although theoretical studies indicate that\ngenerically β/ne}ationslash=αG26,27,28,30, it is not well-understood\nwhat the relative importance of spin-dependent disorder\nand spin-orbit effects in the bandstructure is, and a pre-\ncise theoretical prediction of β/αGfor a specific mate-\nrial has not been attempted yet. Moreover, the determi-\nnation of the ratio β/αGfrom experiments on current-\ndriven domain wall motion has turned out to be hard\nbecause of extrinsic pinning of the domain and nonzero-\ntemperature29,38effects.\nIn this paper we present the theory of the current\npumped by a field-driven domain wall for the situation\nthat spin is not conserved. In particular, we show that\na field-driven domain wall in a metallic ferromagnet gen-\nerates a charge current that depends strongly on the ra-\ntioβ/αG. This charge current arises from the fact that\na time-dependent magnetization generates a spin cur-\nrent, similar to the spin-pumping mechanism proposed2\nby Tserkovnyak et al.8for nanostructures containing fer-\nromagnetic elements. Since the symmetry between ma-\njority and minority electrons is by definition broken in\na ferromagnet, this spin current necessarily implies a\ncharge current. In view of this, we prefer to use the term\n“spin pumping” also for the case that spin is not fully\nconserved, and defining the spin current as a conserved\ncurrent is no longer possible.\nThe generation of spin and charge currents by a mov-\ning domain wall via electromotiveforces is discussed very\nrecently by Barnes and Maekawa40. We note here also\nthe work by Ohe et al.41, who consider the case of the\nRashba model, and the very recent work by Saslow42,\nYanget al.43, and Tserkovnyak and Mecklenburg44. In\naddition to these recent papers, we mention the much\nearlier work by Berger, which discusses the current in-\nduced by a domain wall in terms of an analogue of the\nJosephson effect45.\nBarnes and Maekawa40consider the case that spin\nis fully conserved. In this situation it is convenient to\nperform a time and position dependent rotation in spin\nspace, such that the spin quantization axis is locally par-\nallel to the magnetization direction. As a result of spin\nconservation, the hamiltonian in this rotated frame con-\ntains nowonly time-independent scalarand exchangepo-\ntential terms. The kinetic-energy term of the hamilto-\nnian, however, will acquire additional contributions that\nhave the form of a covariant derivative. Perturbation\ntheory in these terms then amounts to performing a gra-\ndient expansion in the magnetization direction17. Hence,\nthe fact that Barnes and Maekawa consider the case\nthat spin is fully conserved is demonstrated mathemat-\nically by noting that in Eq. (5) of Ref. [40] there are\nno time-dependent potential-energy terms. Generaliz-\ning this approach to the case of spin-dependent disorder\nor spin-orbit coupling turns out to be difficult. Never-\ntheless, Kohno and Shibata were able to determine the\nGilbert damping and dissipative spin transfer torques us-\ning the above-mentioned method46. Since Barnes and\nMaekawa40consider the situation that spin is fully con-\nserved, they effectively are dealing with the case that\nαG=β= 0. This is because both the Gilbert damping\nparameter αGandthedissipativespintransfertorquepa-\nrameterβarise from processes in the microscopic hamil-\ntonian that do not conserve spin26,27,28,30. Hence, for the\ncase that αG=β= 0 our results agree with the results\nof Barnes and Maekawa40.\nThe remainder of this paper is organized as follows. In\nSec. II we derive a general expression for the electric cur-\nrent induced by a time-dependent magnetizationtexture.\nThis general expression is then evaluated in Sec. III for\na simple model of field-driven domain wall motion. We\nend in Sec. IV with a short discussion, and present our\nconclusions and outlook.II. ELECTRIC CURRENT\nQuite generally, the expectation value of the charge\ncurrent density, defined by j=−cδH/δAwithcthe\nspeed oflight, Hthe hamiltonian, and Athe electromag-\nnetic vector potential, is given as a functional derivative\nof the effective action\n/an}b∇acketle{tj(x,τ)/an}b∇acket∇i}ht=cδSeff\nδA(x,τ), (3)\nwithτthe imaginary-time variable that runs from 0 to\n/planckover2pi1/(kBT). (Planck’s constant is denoted by /planckover2pi1andkBTis\nthe thermal energy.) First, we assume that spin is con-\nserved meaning that the hamiltonian is invariant under\nrotations in spin space. The part of the effective action\nfor the magnetization direction that depends on the elec-\ntromagnetic vector potential is then given by17\nSeff=/integraldisplay\ndτ/integraldisplay\ndx/angbracketleftbig\njz\ns,α(x,τ)/angbracketrightbig˜Aα′(Ω(x,τ))∇αΩβ(x,τ),\n(4)\nwhere a summation over Cartesian indices α,α′,α′′∈\n{x,y,z}is implied throughout this paper. In this ex-\npression,\njα\ns,α′(x,τ) =/planckover2pi12\n4mi/bracketleftbig\nφ†(x,τ)τα∇α′φ(x,τ)\n−/parenleftbig\n∇α′φ†(x,τ)/parenrightbig\nταφ(x,τ)/bracketrightbig\n+|e|/planckover2pi1\n2mcAα′φ†(x,τ)ταφ(x,τ),(5)\nis the spin current, given here in terms of the Grassman\ncoherent state spinor φ†= (φ∗\n↑,φ∗\n↓). Furthermore, ταare\nthe Pauli matrices, and mis the electron mass. (Note\nthat since we are, for the moment, considering the situa-\ntion that spin is conserved there are no problems regard-\ning the definition of the spin current.) The expectation\nvalue/an}b∇acketle{t···/an}b∇acket∇i}htis taken with respect to the current-carrying\ncollinear state of the ferromagnet. Finally, ˜Aα(Ω) is the\nvector potential of a magnetic monopole in spin space\n[not to be confused with the electromagnetic vector po-\ntentialA(x,τ)] that obeys ǫα,α′,α′′∂˜Aα′/∂Ωα′′= Ωαand\nis well-known from the path-integral formulation for spin\nsystems47. Eq. (4) is most easily understood as arising\nfrom the Berry phase picked up by the spin of the elec-\ntrons as they drift adiabatically through a non-collinear\nmagnetization texture16,17. Variation of this term with\nrespect to the magnetization direction gives the reactive\nspin transfer torque in Eq. (1).\nThe expectation value of the spin current is given by\n/angbracketleftbig\njz\ns,α(x,τ)/angbracketrightbig\n=/integraldisplay\ndτ′/integraldisplay\ndx′Πz\nα,α′(x−x′;τ−τ′)Aα′(x′,τ′)\n/planckover2pi1c.\n(6)\nThe zero-momentum low-frequency part of the response\nfunction Πz\nα,α′(x−x′;τ−τ′)≡/angbracketleftbig\njz\ns,α(x,τ)jα′(x′,τ′)/angbracketrightbig\n0,\nwith/an}b∇acketle{t···/an}b∇acket∇i}ht0the equilibrium expectation value, is deter-\nmined by noting that for the vector potential A(x,τ) =3\n−cEe−iωτ/ωthe above equation [Eq. (6)] should in\nthe zero-frequency limit reduce to Ohm’s law /an}b∇acketle{tjz\ns/an}b∇acket∇i}ht0=\n−/planckover2pi1(σ↑−σ↓)E/(2|e|). Using this result together with\nEqs. (3-6), we find, after a Wick rotation τ→itto\nreal time, that\n/an}b∇acketle{tjα/an}b∇acket∇i}ht=−/planckover2pi1\n2|e|V(σ↑−σ↓)∂\n∂t/integraldisplay\ndx˜Aα′(Ω(x,t))∇αΩα′(x,t),\n(7)\nwithVthe volume of the system. We note that the time-\nderivative of the Berry phase term is also encountered\nby Barnes and Maekawa in discussing the electromotive\nforce in a ferromagnet40. Such Berry phase terms are\nknown to occur in adiabatic quantum pumping48.\nWe now generalize this result to the situation where\nspin is no longer conserved, for example due to spin-\norbit interactions or spin-dependent impurity scatter-\ning. Linearizing around the collinear state by means of\nΩ≃(δΩx,δΩy,1−δΩ2\nx/2−δΩ2\ny/2) we find that the part\nof the effective action that contains the electromagnetic\nvector potential reads30\nSeff=/integraldisplay\ndτ/integraldisplay\ndx/integraldisplay\ndτ′/integraldisplay\ndx′/integraldisplay\ndτ′′/integraldisplay\ndx′′[δΩa(x,τ)\n×Kab(x,x′,x′′;τ,τ′,τ′′)·A(x′′,τ′′)δΩb(x′,τ′)],(8)\nwhere a summation over transverse indices a,b∈ {x,y}\nis implied. The spin-wave photon interaction vertex\nKab(x,x′,x′′;τ,τ′,τ′′) =\n∆2\n8/planckover2pi1c/an}b∇acketle{tφ†(x,τ)τaφ(x,τ)φ†(x′,τ′)τbφ(x′,τ′)j(x′′,τ′′)/an}b∇acket∇i}ht0,\n(9)\ngiven in terms of the exchange splitting ∆, is also en-\ncountered in a microscopic treatment of spin transfer\ntorques30. The reactive part of this interaction vertex\ndetermines the reactive spin transfer torque and, via\nEqs. (3) and (8), reproduces Eq. (7). The zero-frequency\nlong-wavelength limit of the dissipative part of the spin-\nwave photon interaction vertex determines the dissipa-\ntive spin transfer torque. (Note that in this approach\nthe definition of the spin current does not enter in deter-\nmining the spin transfer torques.) Although Eq. (9) may\nbe evaluated for a given microscopic model within some\napproximation scheme30, we need here only that varia-\ntion of the action in Eq. (8) reproduces both the reactive\nand dissipative spin torques in Eq. (1). The final result\nfor the electric current density is then given by\n/an}b∇acketle{tjα/an}b∇acket∇i}ht=−/planckover2pi1\n2|e|V(σ↑−σ↓)/bracketleftbigg\nβ/integraldisplay\ndx∂Ω(x,t)\n∂t·∇αΩ(x,t)\n+∂\n∂t/integraldisplay\ndx˜Aα′(Ω(x,t))∇αΩα′(x,t)/bracketrightbigg\n.(10)\nThe above equation is essentially the result of a linear-\nresponse calculation in ∂Ω/∂t, and is the central result\nof this paper. We emphasize that the way in which thetransport coefficients σ↑andσ↓and theβ-parameter en-\nter does not rely on the specific details of the underlying\nmicroscopic model. Note that the above result reduces\nto that of Barnes and Maekawa (Eq.(9) of Ref. [40]) if we\ntakeβ= 0.\nIII. FIELD-DRIVEN DOMAIN WALL MOTION\nTo bring out the qualitative physics, we evaluate\nthe result in Eq. (10) using a simple model for field-\ndriven domain wall motion in a magnetic wire of length\nL. In polar coordinates θandφ, defined by Ω=\n(sinθcosφ,sinθsinφ,cosθ), we choose the micromag-\nnetic energy functional\nEMM[θ,φ] =ρs/integraldisplay\ndx/braceleftbiggJ\n2/bracketleftBig\n(∇θ)2+sin2θ(∇φ)2/bracketrightBig\n+K⊥\n2sin2θsin2φ−Kz\n2cos2θ+gBcosθ/bracerightbigg\n,(11)\nwhereJis the spin stiffness, and K⊥andKzare\nanisotropy constants larger than zero. The external field\nin the negative z-direction leads to an energy splitting\n2gB >0. We solve the equation of motion in Eq. (1)\nwithin the variational ansatz18,49\nθ(x,t) =θ0(x,t)≡2tan−1/bracketleftBig\ne−(rdw(t)−x)/λ/bracketrightBig\n,(12)\ntogether with φ(x,t) =φ0(t), that describes a rigid do-\nmain wall with width λ=/radicalbig\nJ/Kzat position rdw(t).\nThe chirality of the domain wall is determined by the\nangleφ0(t) and the magnetization direction is assumed\nto depend only on xwhich is taken in the long direction\nof the wire.\nThe equations ofmotion for the variationalparameters\nare given by18,29,49\n˙φ0(t)+αG/parenleftbigg˙rdw(t)\nλ/parenrightbigg\n=gB\n/planckover2pi1;\n/parenleftbigg˙rdw(t)\nλ/parenrightbigg\n−αG˙φ0(t) =K⊥\n2/planckover2pi1sin2φ0(t).(13)\nNote that the velocity vsis absent from these equations\nsince we consider the generation of electric current by\na field-driven domain wall. The above equations pro-\nvide a description of the field-driven domain wall and,\nin particular, of Walker breakdown49. That is, for an\nexternal field smaller than the Walker breakdown field\nBw≡αGK⊥/(2g) the domain wall moves with a con-\nstant velocity. For fields B > B wthe domain wall under-\ngoesoscillatorymotion, whichinitially makestheaverage\nvelocity smaller.\nSolving the equations of motion results in\n˙φ0=1\n(1+α2\nG)Re\n/radicalBigg/parenleftbigggB\n/planckover2pi1/parenrightbigg2\n−/parenleftbiggαGK⊥\n2/planckover2pi1/parenrightbigg2\n;\n˙rdw\nλ=gB\nαG/planckover2pi1−˙φ0\nαG, (14)4\n 0 0.5 1 1.5 2\n 0  0.5  1  1.5  2 |j/j0|\nB/Bwβ=0.015\nβ=0.01\nβ=0.005\nβ=0\nFIG. 1: Current generated by a field-driven domain wall in\nunits of j0= 2L/[|e|(σ↑−σ↓)αGK⊥], forαG= 0.01 and\nvarious values of β. The result is plotted as a function of\nmagnetic field in units of the Walker breakdown field Bw≡\nαGK⊥/(2g).\nwhere the ···indicates taking the time-averaged value.\nInsertingthe variational ansatzintoEq. (10) leads in first\ninstance to\n/an}b∇acketle{tjx/an}b∇acket∇i}ht=−/planckover2pi1\n|e|L(σ↑−σ↓)/bracketleftbiggβ˙rdw(t)\nλ+˙φ0(t)/bracketrightbigg\n,(15)\nwhich, using Eq. (14), becomes\n/an}b∇acketle{tjx/an}b∇acket∇i}ht=−/planckover2pi1\n|e|L(σ↑−σ↓)\n\nβgB\nαG/planckover2pi1\n+/parenleftBigg\n1−β\nαG\n1+α2\nG/parenrightBigg\nRe\n/radicalBigg/parenleftbigggB\n/planckover2pi1/parenrightbigg2\n−/parenleftbiggαGK⊥\n2/planckover2pi1/parenrightbigg2\n\n\n.(16)\nAs shown in Fig. 1, this result depends strongly on the\nratioβ/αG. In particular, for β > α Ga local maximum\nappears in the current as a function of magnetic field.\nSinceαGis determined independently from ferromag-\nnetic resonanceexperiments, measurementofthe slopeof\nthecurrentforsmallmagneticfieldsenablesexperimental\ndetermination of β. We note that within the present ap-\nproximation the current does not depend on the domain\nwall width λ. Furthermore, in the limit of zero Gilbert\ndampingand β, thedissipationlesslimit, wehavethatthe\ncurrent density is equal to /an}b∇acketle{tjx/an}b∇acket∇i}ht= (σ↓−σ↑)gB/(|e|L).\nThis is the result of Barnes and Maekawa40that corre-\nsponds to the situation that αG=β= 0, as discussed\nin the Introduction. We point out that, within our ap-\nproximation for the description of domain-wall motion,\nputtingβ=αGin Eq. (16) gives the same result as us-\ning Eqs. (13) and (15) with αG=β= 0. That the\nsituation discussed by Barnes and Maekawa40is indeed\nthat ofαG=β= 0 is seen by comparing their result\n[Eqs. (8) and (9) of Ref. [40], and the paragraph follow-\ning Eq. (9)] with our results in Eqs. (10) and (13).IV. DISCUSSION AND CONCLUSIONS\nOur result in Eq. (16) is a simple expression for the\npumped current as a function of magnetic field for a\nfield-driven domain wall. A possible disadvantage in us-\ning Eq. (16), however, is that in deriving this result we\nassumed a specific model to describe the motion of the\ndomain wall. This model does in first instance not in-\nclude extrinsic pinning and nonzero temperature. Both\nextrinsic pinning18and nonzerotemperature29can be in-\ncluded in the rigid-domain wall description. However,\nit is in some circumstances perhaps more convenient to\ndirectly use the result in Eq. (15) together with the ex-\nperimental determination of ˙ rdw(t). Since the only way\nin which the parameter βenters this equation is as a\nprefactor of ˙ rdw(t), this should be sufficient to determine\nits value from experiment. We note, however, that the\nprecision with which the ratio β/αGcan be determined\ndepends on how accurately the magnetization dynamics,\nand, in particular, the motion of the domain wall, is im-\naged experimentally. With respect to this, we note that\nthe various curves in Fig. 1 are qualitatively different for\ndifferent values of β/αG. In particular, the results for\nβ/αG>1 andβ/αG<1 differ substantially, and could\nmost likely be experimentally distinguished. In view of\nthis discussion, future research will in part be directed\ntowards evaluating Eq. (10) for more complicated mod-\nels of field-driven domain-wall motion, which will benefit\nthe experimental determination of β/αG.\nA typical current density is estimated as follows. For\nthe experiments of Beach et al.50we have that L∼20\nµm, andλ∼20 nm. The domain velocities measured\nin this experiment are ˙ rdw∼40−100 m/s. Taking as\na typical conductivity σ↑∼106Ω−1m−1we find, using\nequation Eq. (15) with β∼0.01, typical electric current\ndensities of the order of /an}b∇acketle{tjx/an}b∇acket∇i}ht ∼103−104A m−2. This re-\nsult depends somewhat on the polarization of the electric\ncurrent in the ferromagnetic metal, which we have taken\nequal to 50% −100% in this rough estimate. 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L.\nErskine, Nature Materials 4, 741 (2005).\n51F. Colaiori, G. Durin, and S. Zapperi, arXiv:0706.2122v1.\n52See for example Eq. (27) of Ref. [26] for possible higher-\norder terms in the presence of current."    },    {        "title": "2008.12221v3.Nutation_Resonance_in_Ferromagnets.pdf",        "content": "1 \n Nutation  Resonance in Ferromagnets  \nMikhail Cherkasskii1,*, Michael Farle2,3, and Anna Semisalova2 \n1 Department of General Physics 1 , St. Petersburg State University , St. Petersburg , 199034, Russia  \n2 Faculty of Physics and Center of Nanointegration (CENIDE), University of Duisburg -Essen, Duisburg, 47057, Germany  \n3 Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Russia  \n \n* Corresponding author: macherkasskii@hotmail.co m \n \n \nThe inertial  dynamic s of magnetization in a ferromagnet is investigated  theoreticall y. The analytically derived dynamic \nresponse upon microwave excitation shows two pea ks: ferromagnetic and nutation resonances. The exact analytical expressions \nof frequency and linewidth of  the magnetic  nutation resonance are deduced from the frequency dependent susceptibility \ndetermined by the i nertial Landau -Lifshitz -Gilbert equation.  The study  shows that the dependence of nutation linewidth on the \nGilbert precess ion damping  has a minimum , which becomes more expressive with increas e of the applied magnetic field.  \n \nPACS numbers: 76.50.+g, 78.47.jp, 75.50. -y \n \nI. INTRODUCTION  \nRecently, the effects of inertia in the spin dynamics of \nferromag nets were reported to cause nutation resonance  [1-\n12] at frequencies  higher  than the conventional ferromagnetic \nresonance . It was shown  that inertia  is responsible for the \nnutation , and that this type of motion should be considered  \ntogether with magnetization precession in the applied \nmagnetic field . Nutation in ferromagnets was confirmed \nexperimentally only recently  [2], since  nutation and \nprecession operate at substantially different time scales , and \nconventi onal microwave ferromagnetic resonance (FMR) \nspectroscopy techniques do not easily reach the high -\nfrequency (sub-Terahertz) regime re quired to observe the \ninertia effect which in addition yields a much weaker signal . \nSimilar to any other oscillatory system, t he magnetiza tion \nin a ferromagnet has resonant frequencies  usually studied by  \nferromagnetic resonance  [13,14]. The resonant \neigenfrequency is determined by  the magnetic parameters  of \nthe material  and applied magnetic field . Including inertia of \nthe magnetization in th e model description shows that  nutation \nand precession are complementary to each other  and several \nresonances can be generated . In this Letter , we concentrate on \nthe investigation of the resonance characteristics of nutation.  \nThe investigation of nutation is connected to the progress \nmade  in studies of the  spin dynamics at ultrashort  time \nscales  [15,16] . These successes led to the rapi d development \nof a new scientific field , the so -called ultrafast magnetism  [17-\n25]. The experimental as well as theoretical investigation of \nthe inertial  spin dynamics is at the very beginning , although it \nmight be of significance for future high speed spintronics \napplications  including ultrafast magnetic switching .  Besides nutation driven by magnetization inertia, several  \nother origins  of nutation have been reported . Transient \nnutations (Rabi oscillations) have been widely investigate d in \nnuclear magnetic resonance  [26] and electron spin \nresonance  [27-29], they were  recently addressed in \nferromagnets  [30]. A complex dynamics and Josephson \nnutation of a local spin \n1/ 2s  as well as large spin cluster \nembedded in the tunnel junction between ferromagnetic leads \nwas shown to occur  due to  a coupling to Josephson \ncurrent  [31-33]. Low-frequency nutation was observ ed in \nnanomagnets exhibiting a non -linear FMR  with the large -\nangle precession of magnetization where the onset of spin \nwave instabilities can be delay ed due to geometric \nconfinement  [34]. Nutation dynamics due to inertia of \nmagnetization in ferromagnetic thin films was observed for \nthe first time by Neeraj et al.  [2]. \nThe microscopic derivation of the magnetization  inertia \nwas performed in  ref. [3-7]. A relation between the Gilbert \ndamping constant  and the inertia l regime  characteristic time  \nwas elaborated in ref. [3]. The exchange interaction, damping, \nand moment of inertia can be calculated from first principles \nas shown in  [7]. The study of inertia spin dynamics  with a \nquantum approach  in metallic ferromagnets  was performed \nin [8]. In addition, nutation  was theoretically analyzed as a \npart of magnetization dynamics in ferromagnetic \nnanostructure  [9,10] and nanoparticles  [11]. Despite these \nadvances, exact analytical expressions for the high-frequency \nsusceptibility including inertia had not been derived yet.  \nIn [35], the inertial regime  was introduced in the \nframework of the mesoscopic nonequilibrium \nthermodynamics theory , and it was shown to be responsible \nfor the nutation superimposed on the precession of \nmagnetization . Wegrowe and Ciornei  [1] discussed  the  \n2 \n equivalence between the inertia in the dynamics of uniform \nprecession and a spinning top within the framework of the  \nLandau –Lifshitz –Gilbert equation generalized to the inertial \nregime. This equation was  studied  analytical ly and \nnumerical ly [12,36]. Although the se reports provide \nnumerical tools for obtaining resonance characteristics, the \ncomplexity of the numerical solution of differential equations \ndid not allow to estimate the nutation frequency and linewidth \naccurately . Also in a recent  remarkable paper  [37] a novel \ncollective excitation  – the nutation wave  – was reported,  and \nthe dispersion characteristics  were derived wit hout discussion \nof the nutation  resonance lineshapes  and intensities.  \nThus, at present, there is a necessity to study  the resonance \nproperties of nutation in ferromagnet s, and this paper is \ndevoted to this study. We performed the investigation based \non the Landau -Lifshitz -Gilbert equation with  the addition al \ninertia term  and provide an  analytical solution.   \nIt is well known that the Landau -Lifshitz -Gilbert equation \nallow s finding the susceptibility as the ratio between  the time-\nvarying magnetization and the time-varying driving magnetic \nfield (see for exampl e [38,39] and references therein). This \nsusceptibility describes  well the magnetic response of a \nferromagnet in the linear regime, that is  a small cone angle of \nthe precession . In this description , the ferromagnet usually is \nplaced in a  magnetic field  big enough to align all  atomic \nmagnetic moments along the field , i.e., the ferromagnet is in \nthe saturated state and the  magnetization  precess es. The \napplied  driving magnetic field  allows one to obser ve FMR  as \nsoon as the driving field frequency coincides with  \neigenfrequency of precession. Using the expression for \nsusceptibility, one can elaborate  such resonance \ncharacteristics as eigenfrequency and linewidth.  We will \npresent similar expressions for the dynamic susceptibility, \ntaking nutation into account.  \nII. SUSCEPTIBILITY  \nThe ferromagnet  is subjected to a uniform bias magnetic \nfield \n0H  acting along the z -axis and being strong enough to \ninitiate  the magnetic saturation  state. The small time -varying \nmagnetic field \nh  is superimposed on the bias field. The \ncoupling between impact and response, taking into account \nprecession, damping, and nutation, is given by the Inertial  \nLandau -Lifshitz -Gilbert ( ILLG)  equation  \n \n2\n2\n0,effd d d\ndt M dt dt       M M MMH   (1) \nwhere \n  is the gyromagnetic ratio, \nM  the magnetization  \nvector , \n0M  the magnetization at saturation, \neffH  the vector \nsum of all magnetic fields, external and internal, acting upon \nthe magnetization , \n the Gilbert damping , and \n  the inertial \nrelaxatio n time. For simplicity, we assume that the \nferromagnet is infinite, i.e. there is no demagnetization  correction , with negligible magnetocrystalline anisotropy , and \nonly the  externally applied field s contribute  to the total  field. \nThus, the bias magnetic field \n0H  and signal field \nh  are \nincluded in \neffH . We assume that the signal is small \n0,hH\n hence the magnetization is directed along \n0.H  \nOur interest is to study the correlated  dynamics of nutation \nand precession  simultaneously;  therefore we write the \nmagnetization  and magnetic field  in the general ized form  \nusing  the Fourier transformation  \n \n 01ˆ  ,\n2itt M z d e\n\n\n Mm   (2) \n \n 01ˆ  ,\n2it\nefft H z d e\n\n\n Hh   (3) \nwhere \nˆz  is the unit vector along the z -axis. If we  substitute  \nthese expressions  in the ILLG equation  and neglect the small \nterms, it leads to  \n \n\n \n 00\n211 \n22\nˆˆ\nˆˆ .i t i td i e d e\nM z H z\ni z z   \n\n   \n   \n\n     \n    \n        m\nhm\nmm   (4) \nBy performing the Fourier transform and changing the order \nof integration , equation  (4) becomes   \n \n\n\n \n00\n21 2\n1 2\nˆˆ\nˆˆ ,it\nitd dt i e\nd dt e\nM z H z\ni z z\n  \n\n   \n    \n\n \n\n\n   \n \n     \n       \nm\nhm\nmm   (5) \nwhere the integral representation of the Dirac delta  function  \ncan be found. With the delta function,  the equation (5)\nsimplifies to  \n \n  \n 00\n2ˆˆ\nˆˆ .i M z H z\ni z z     \n      \n   m h m\nmm   (6) \nBy projecting to Cartesian coordinates and  introducing the \ncircular variables  for positive and negative circular \npolarization  \n,xy m m im  \n,xy h h ih one obtains  \n \n  \n 2\n20,\n0,HM\nHMm m i m m h\nm m i m m h   \n\n        \n          \n        (7) \nwhere \n0 H H  is the precession frequency and \n0.M M\n The small -signal susceptibility follows  from \nthese equations :  \n3 \n  \n2\n2,\n,\n.M\nH\nM\nHim\nih\n\n   \n\n  \n   \n\n  \n  \n\n   (8) \nIt is seen that the susceptibility (8) is identical with  the \nsusceptibility for LLG equation , if one drops  the inertial term , \nthat is  \n0.  \nLet us separate dispersive and d issipative  parts  of the \nsusceptibility \n,i       \n \n \n 2\n2,\n,\n,\n,MH\nM\nMH\nMD\nD\nD\nD\n\n   \n\n   \n\n\n\n\n\n\n  \n\n\n   (9) \n \n 2 2 4 3\n2 2 22\n22 , 1   \n     \n    H H HD   (10) \n \n 2 2 4 3\n2 2 22\n22 . 1   \n     \n    H H HD   (11) \n \nThe frequency dependence of the dissipative parts of \nsusceptibilit ies \n and \n  is shown in the Fig. 1. The plus \nand minus subscripts correspond to right -hand and left -hand \ndirection of rotation. Since the denominators \nD  and \nD  are \nquartic polynomials, four critical points  for either  \n or \n \ncan be expected . Two of them  that are extrema with  a clear \nphysical meaning are plotted.  In Fig. 1(a) the extremum , \ncorresponding  to FMR  at \n0 H H  is shown . Due to the \ncontribution  of nutation , the frequency and linewidth of this \nresonance are slightly different from the ones of usual FMR . \nThe resonance occurs  for right -hand precession, i.e. positive \npolarization.   \nIn Fig. 1(b) the nutation resonance  possessing negative \npolarization is presented.  Note  that the polarizations of \nferromagnetic and nutation resonances are reverse d. \nIII. APPROXI MAT ION FOR  NUTATION \nFREQUENCY  \nLet us  turn to the description  of an approximation  of the \nnutation resonance frequency. If we equate the denominator \nD\n to zero, solve the resulting equation, we obtain the \napproximation from the real part of  the roots. This is reasonable , since the numerator of \n  is the linear function of \n\n , and we  are interested in \n1.  Indeed , the equation  \n \n 2 2 4 3 2 2\n202 2 1\n2H\nHH      \n     \n    (12) \nhas four roots that are complex conjugate in pairs  \n \n1,221 1 4 2,2H\nFMRiiw   \n       (13) \n \n1,221 1 4 2.2H\nNiiw   \n       (14) \n \nFIG. 1. (Color online) (a) The FMR peak with nutation. (b) \nThe nutation resonance. The calculation was performed for \n1/ 2 28  GHz T ,\n \n00 1 T, M  \n00 100 mT, H  \n0.0065\n and \n1110  s.  \n \nOne should choose  the same sign from the \n  symbol  in each \nformula , simultaneously . The real part of expression  (13) \ngives the approximate frequency for FMR , but in negative \nnumbers, so the sign should be inversed . The approximate \nfrequency of FMR in positive numbers can be derived from \nequation \n0. D  The approximate nutation frequency  is \nobtained  by the real part of  the expression (14). One takes half \nthe sum of two conjugate roots  \n1,2,Nw neglect s the high \n  \nterms , and obtains  the nutation resonance frequency  \n \n1 1 2\n.2NH\nw \n\n   (15) \nNote  that the expression of \nNw  is close  to the approximation  \ngiven in  [36] at \n1/ , H    namely  \n \nweak\nnu1\n.H \n\n   (16) \nThe similarity of both approximations b ecomes clear , if we \nperform a Taylor series expansion  and return to the notation \n,H\n \n\n2\n2 2 3 3 3\n2\nweak\nnu\n2 2 3 3 31 1 2 1\n2 2 4\n1,4\n1 1\n2\n1.18\n6H HH\nN\nH\nH HH\nHw\nO\nO  \n \n    \n   \n       \n\n   \n\n \n \n4 \n IV. PRECISE  EXPRESSIONS FOR  FREQUENCY \nAND LINEWIDTH OF NUT ATION  \nThe analytical approach proposed in this Letter yields  \nprecise values of the frequency  of nutation resonance  and the \nfull width at half maximum (FWHM) of the peak . The \nfrequency is found by extremum, when the derivative of the \ndissipative part of susceptibilities (9) is zero \n \n0.\n   (17) \nIt is enough to determine  zeros of the n umerator of th e \nderivative , that are given by  \n \n 2 2 4 3 2 2 23 4 2 1 0.HH                (18) \nLet us use Ferrari's solution  for this quartic equation  and \nintroduce the notation:  \n \n22\n2\n2\n2\n2\n3\n23\n24\n343\n4,\n2 1,  \n3,8\n,28,\n3.16 25,\n6rH\nrH\nrr\nr r\nr r r\nrr\nr r r r\nr\nr rrr\nr\nr\nrC\nE\nC\nC\nCEcA\nB\nBaA A\nBBbAA\nBB\nA AA\n\n \n\n\n\n\n\n\n  \n\n   (19) \nIn Ferrari's method , one should determine a root of the nested \ndepressed cubic equation . In the investigated case , the root is \nwritten  \n \n5,6r\nr r ray U V     (20) \nwhere  \n \n32\n3\n2\n32,27 4 2\n,3\n12\n1,\n.3 108 8r r r\nr\nr\nr\nr\nr\nrr\nrr\nr r rP Q QU\nPVU\nPc\nQaa\nabc   \n\n \n     (21) \nThus, the precise value of the nutation frequency is given by  \n \n2\n42\n2 13 2 .2 2rr r\nN\nr\nr\nrr\nrry\nA\nbaa\nay\nyB  \n   \n   (22) \nThe performed analysis shows that approximate value of \nnutation resonance frequency is close to precise value.  The linewidth of the nutation resonance is found at  a half \npeak height. If one denotes the maximum by \n,N X \n the equation  which determines \nfrequencies at half magnitude  is \n \n 2 2 4 3 2 2\n212 2 12\n2 0.H\nH H MX      \n      \n\n   (23) \nWe repeat the procedure for finding solu tions with Ferrari's \nmethod  introducing the  new notations  \n \n 22\n2\n2\n2\n2\n3\n23\n24\n3 4 21\n2\n,\n12 1 ,2\n1\n2\n3,8\n,28\n3.16 2,\n56,\n4lw\nlw\nlw\nlw\nlw\nlw lw\nlw\nlw lw\nlw lw lw\nlw\nlw lw\nlw lw lw lw lw lw\nlw\nlw lwH\nHM\nH\nlw\nlw\nlw lwA\nB\nBaA A\nBBbAA\nB B B D\nA AX\nX\nCX\nDX\nEX\nC\nCD\nA\nCEcA A\n\n \n \n\n\n\n\n\n\n  \n\n\n\n  \n  \n\n\n   (24) \nA root of the nested depressed cubic equation \nlwy  must be \nfound in the same way as provided in (20) with the \ncorresponding replacement of variables, i.e. subscript r is \nreplaced by lw. The difference between two adjacent roots \ngives the nutation linewidth  \n \n23 2 .\n2lw\nN lw lw\nlw lwbay\na y   \n   (25) \nThe explicit expression for the linewidth can be written using \nthe equations  (19)-(25). \n \n \nFIG. 2. (Color online) The dependence of the nutation \nlinewidth on the inertial relaxat ion time for \n00 100 mT, H\n \n00 1 T, M  and \n0.0065.  \n \n5 \n  \nThe effect of the inertial relaxation time on the nutation \nlinewidth is shown in Fig.  2. One can see that increasing \ninertial relaxation time leads to narrowing of the linewidth. \nThis behavior is expected and is consistent with the traditional \nview that decreasing of losses results in narrowing of \nlinewidth.  \n \n \nFIG. 3. (Color online) The dependence of nutation \nresonance linewidth on precession Gilbert damping \nparameter at different magnetic fields \n0H  for \n00 1 T M\nand \n1110  s.  \n \nSince the investig ated oscillatory system implemen ts \nsimultaneous two types of  motions , it is of interest to study the \ninfluence of the Gilbert precession  damping parameter \n  on \nthe nutation  resonance linewidth. The result is presented in \nFig. 3 and is valid for ferromagnets with vanishing anisotropy.  \nOne sees that the dependence of \nN  on \n  shows a \nminimum that becomes more expressive with increasing  bias \nmagnetic field.  In other words, t he linewidth is parametrized \nby the magnitude of field. This non-trivial  behavior of \nlinewidth relates with the nature of th is oscillatory system, \nwhich performs  two coupled  motions.  \nTo elucidate the non-trivial behavior , one can consider  the \nsusceptibility  (9) in the same way as it is usually performed \nfor the forced harmonic oscillator with damping  [40]. For this \noscillator , the linewidth can be direct ly calculated from the \ndenominator of the response expression once the driving \nfrequency is equal to eigenfreq uency. In the investigated case \nof magnetization with inertia , the response expression is (9) \nwith denominator s (10) and (11) written as  \n \n 2 2 4 3\n2 2 22\n21 . 2H H HD   \n      \n   \n   (26) \nSince the applied magnetic field is included in this expression  \nas \n0,H H the linewi dth depends on the field.  \nThe obtained result can be generalized to a fin ite sample \nwith magnetocrystalline anisotropy with method of effective \ndemagnetizing factors  [41,42] . In this case the bias magnetic field \n0H  denotes an external field and in the final expressions \nthis field should be replaced by \n 0 0 0ˆˆ ,i a d NN  H H M  \nwhere \nˆ\naN  is the anisotropy demagnetizing tensor and \nˆ\ndN  is \nthe shape demagnetizing tensor.  \nV. CONCLUSION  \nIn summary, we derived a general analytical expression for \nthe linewidth and f requency of nutation resonance in \nferromagnets, depending on magnetization, the Gilbert \ndamping, the inertial relaxation time and applied magnetic \nfield.  We show the nutation linewidth can be tuned by the \napplied magnetic field , and this tunability breaks the direct \nrelation between losses and the linewidth. This for example  \nleads to the appearance of a minimum  in the nutation \nresonance linewidth for the damping parameter  around  \n0.15.\n The obtained results are valid for ferromagnets with \nvanishing anisotropy . \nACKNOWLEDGEMENTS  \nWe thank  Benjamin Zingsem for helpful discussions . 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A 64, 968 (1951).  \n "    },    {        "title": "1011.5054v1.Ultra_fast_magnetisation_rates_within_the_Landau_Lifshitz_Bloch_model.pdf",        "content": "arXiv:1011.5054v1  [cond-mat.mtrl-sci]  23 Nov 2010Ultra-fastmagnetisationrates withintheLandau-Lifshit z-Bloch model.\nU. Atxitia and O. Chubykalo-Fesenko\nInstituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco, 28049 Madrid, Spain\nThe ultra-fast magnetisation relaxation rates during the l aser-induced magnetisation process are analyzed\nin terms of the Landau-Lifshitz-Bloch (LLB) equation for di fferent values of spin S. The LLB equation is\nequivalentinthelimit S→∞totheatomisticLandau-Lifshitz-Gilbert(LLG)Langevind ynamicsandfor S=1/2\nto the M3TM model [B. Koopmans, et al.Nature Mat. 9(2010) 259]. Within the LLB model the ultra-fast\ndemagnetisation time( τM)and the transverse damping ( α⊥) are parameterizedby theintrinsic coupling-to-the-\nbath parameter λ, defined by microscopic spin-flip rate. We show that for the ph onon-mediated Elliott-Yafet\nmechanism, λis proportional to the ratio between the non-equilibrium ph onon and electron temperatures. We\ninvestigate the influence of the finite spin number and the sca ttering rate parameter λon the magnetisation\nrelaxation rates. The relation between the fs demagnetisat ion rate and the LLG damping, provided by the LLB\ntheory, is checked basing on the available experimental dat a. A good agreement is obtained for Ni, Co and\nGd favoring the idea that the same intrinsic scattering proc ess is acting on the femtosecond and nanosecond\ntimescale.\nPACS numbers: 75.40Gb,78.47.+p, 75.70.-i\nI. INTRODUCTION\nMagnetisation precession and the spin-phonon relaxation\nratesat picosecondtimescale wereconsideredto bethe limi t-\ningfactorforthespeedofthemagnetisationswitching1,2, un-\ntil using optical excitation with fs pulsed lasers the possi bil-\nity to influence the magnetisation on femtosecond timescale\nwasdemonstrated3–6. Theultra-fastlaser-induceddemagneti-\nsation immediately became a hot topic of solid state physics\ndue to an appealing possibility to push further the limits of\noperation of magnetic devices. This ultra-fast process has\nnow been shown to proceed with several important charac-\nteristictimescales6: (i)thefemtoseconddemagnetisationwith\ntimescale τM(ii) the picosecond recovery with timescale τE\nand (iii) the hundredpicoseconds-nanosecondmagnetisati on\nprecession, traditionally characterized by the ferromagn etic\nresonance frequency ωFMRand the Landau-Lifshitz-Gilbert\ndampingparameter αLLG(see Fig.1).\nThephysicsofthemagnetisationchangesonfemto-second\ntimescales is obviously not-trivial and will require novel the-\norieswithintherelativisticquantumelectrodynamicsofm any\nelectronsystems. From theoretical pointof view, the exist ing\nmodels try to answer an open question of the role of differ-\nentsubsystems(photons,phonons,electronsandspins)int he\nultra-fast angular momentumtransfer7. This common goal is\nstimulated by experimental findings provided by the XMCD\nmeasurements showing the important role of the spin-orbit\ninteractions8,9. For the present state of art quantummechani-\ncal descriptions8,10–13of ultra-fast demagnetisationprocesses\ninvolveunavoidablesimplificationsandsometimesevensom e\nad-hoc assumptions necessary to explain experimental find-\nings, such as reduced exchange interactions, enhanced spin -\norbit coupling or a Gaussian distribution of occupied state s\naroundthe Fermi level. While some degreeof agreementhas\nbeen achieved in modelling of the ultra-fast demagnetisati on\n(τM)scale14,themodellingofallthreeultra-fastdemagnetisa-\ntionrateswithinthesameapproachisoutsidethepossibili ties\nofthequantummechanicalapproaches.\nThe three-temperature (3T) phenomenological model in-volves the rate equations for the electron, phonon and spin\ntemperatures (energies)10,15–17. Recently, it has been shown\nthattheintroductionofthespintemperatureisnotadequat e18\nsince the spin system is not in the equilibrium on the fem-\ntosecond timescale. It has been suggested to couple the spin\ndynamics to the two-temperature (2T) model for phonon and\nelectrontemperatures18–22. Thesemodelsarebasedontheen-\nergy flow picture and leave unidentified the angular momen-\ntum transfer mechanism and the underlying quantum mech-\nanism responsible for the spin flip22. They essentially inter-\nprettheultra-fastdemagnetisationas\"thermal\"processe s,un-\nderstanding the temperature as energy input from photon to\nelectron and then to the spin system. By using these mod-\nels the important role of the linear reversal path in the femt o-\nseconddemagnetisationhasbeenidentified23,24. Thecompar-\nison with experiment seems to indicate that in order to have\nmagnetisation switching in the ultra-fast timescale, a com -\nbinedactionof\"heat\"andlargefieldcomingfromtheinverse\nFaradayeffectisnecessary24.\nThe most successful recent phenomenological models de-\nscribing the ultra-fast magnetisation dynamics are (i) the\nLangevin dynamics based on the Landau-Lifshitz-Gilbert\n(LLG) equationand classical HeisenbergHamiltonian for lo -\ncalized atomic spin moments18,19, (ii) the Landau-Lifshitz-\nBloch (LLB) micromagnetics21,22and (iii) the Koopmans’s\nmagnetisation dynamics model (M3TM)25. The spin dy-\nnamicscouldbe coupledtotheelectrontemperaturefromthe\n2T model, underlying the electronic origin of the spin-flip\nprocess18,19,21,22,24or to both electron and phonon temper-\natures, underlying the Elliott-Yafet mechanism mediated b y\nphonons25. When the 2T model was carefully parameterised\nfromthemeasuredreflectivity,it gaveanexcellentagreeme nt\nwith the experiment in Ni22using the former approach or in\nNi,Co andGdusingthe latterapproach25.\nIn the classical derivation of the LLB equation the ther-\nmal averaging has been performed analytically within the\nmean field (MFA) approximation26. Thus, the LLB equa-\ntion for classical spins ( S→∞) is equivalent to an ensem-\nbleofexchange-coupledatomisticspinsmodelledbystocha s-2\ntic LLG equations20,27. At the same time, in some cases the\nLLBequationmaybepreferablewith respectto theatomistic\nHeisenbergmodel,sincebeingmicromagneticit canincorpo -\nrate quantum nature of magnetism and the quantum deriva-\ntion of LLB also exists28. In particular the limits of validity\nforthestatisticalmechanicsbasedontheclassicalHeisen berg\nmodel for the description of materials with delocalized mag -\nnetism of d-electrons in transition metals or magnetism of f-\nelectronsin rare earthsare not clear. An alternativestati stical\nsimplified descriptionof d-metalsconsists of a two level sys-\ntem with spin-up and spin-down bands (i.e. S=±1/2), as\nhas been done by B. Koopmans et al.25. Their model, as we\nshow in the present article, is also equivalent to the quantu m\nLLB equation with spin S=1/2. An additional advantage\nin the use of the LLB equation is the possibility to model\nlarger spatial scales20,21. Therefore the LLB micromagnet-\nicsisanimportantparadigmwithinthemultiscalemagnetis a-\ntiondynamicsdescription. TheLLBequationhasbeenshown\nto describe correctly the three stages of the ultra-fast dem ag-\nnetisationprocesses: thesub-picoseconddemagnetisatio n,the\npicosecondmagnetisationrecoveryand the nanosecondmag-\nnetisationprecession20–22,see Fig.1.\nThe intrinsic quantum mechanical mechanisms responsi-\nble for the ultra-fast demagnetisation in the LLB model\nare included in the intrinsic coupling-to-the-bath parame -\nterλ22,28. The coupling process is defined by the rate of\nthe spin flip. Several possible underlying quantum mech-\nanisms are currently under debate: the Elliott-Yafet (EY)\nelectron scattering mediated by phonons or impurities13,25,\nor other electrons14and electron-electron inelastic exchange\nscattering29,30. By combining the macroscopic demagnetisa-\ntion equation (M3TM model) with the rate of spin flip calcu-\nlatedonthebasisoffullHamiltonian,Koopmans etal.25have\nbeenabletorelatetheultra-fastdemagnetisationtime τMwith\nthespinfliprateofthephonon-mediatedElliott-Yafetscat ter-\ning. The authors fitted experimentaldemagnetisation rates in\nNi,Co,GdtothephenomenologicalM3TMmodelandfound\nthemtobeconsistentwiththevaluesestimatedonthebasiso f\nab-initiotheory13. Thecoupling-to-the-bathparameter λ(mi-\ncroscopicdampingparameterinatomisticLLGmodel)should\nbe distinguished from that of the macroscopic damping αLLG\n(α⊥in the LLB model), a more complicated quantity which\nincludesthe magnon-magnonprocesses.\nThefirst attempttorelatethesub-picoseconddemagnetisa-\ntion time with the macroscopicdampingprocesses was given\nbyKoopmans etal.6whosuggestedtherelation τM∼1/αLLG.\nSubsequently and with the aim to check this relation several\nexperiments in doped permalloy were performed32–34. The\npermalloythinfilmsweredopedwithrareearthimpurities,a l-\nlowingtoincreaseinacontrolledwaythedampingparameter\nαLLG. The effect on the demagnetisation time τMwas shown\nto be opposite34or null32, in contrast to the above relation.\nHowever, it should be noted that the analysis leading to this\nexpression was performed in terms of the Landau-Lifshitz-\nGilbert equation, relating the ultra-fast demagnetisatio n time\nτMtothetransversedampingwithouttakingintoaccounttheir\ntemperature dependence. Moreover, one should take into ac-\ncount that the rare-earth impurities may introduce a differ ent\nFigure 1. Characteristic time scales in ultrafast laser-in duced mag-\nnetisation dynamics experiments. The curve is obtained by t he in-\ntegration of the Landau-Lifshitz-Bloch equation coupled t o the two-\ntemperature model with the parameters from Ref.21. For the m od-\nelling of precession the applied field Hap=1T at 30 degrees was\nused.\nscatteringmechanismwith aslowertimescale33.\nPartially basing on the above mentioned experimental re-\nsults andfroma generalpointof view,the longitudinalrela x-\nation (the ultra-fast demagnetisation rate τM) and the trans-\nverse relaxation (the LLG damping αLLG) may be thought\nto be independent quantities. Indeed, different intrinsic and\nextrinsic mechanisms can contribute to the demagnetisatio n\nrates at different timescales. One can, for example, men-\ntionthatduringthefemtoseconddemagnetisationtheelect ron\ntemperature is often raised up to the Curie temperature22,24.\nAt this moment, the high frequency THz spinwaves35,36in-\ncludingtheStonerexcitations30contribute. At thesame time,\nthe transverse relaxation is related to the homogeneous pre -\ncessional mode. The LLBequationtakescare of thedifferent\nnaturesoflongitudinalandtransverserelaxation,arisin gfrom\nthe spin disordering. The LLB model calculates them inde-\npendently but basing on the same intrinsic scattering mecha -\nnismparameterizedbythe parameter λ. Theincrementof the\nnumber of scattering events is mimicked by the increases of\nthe electron temperature. Consequently, the relation betw een\nthe ultra-fast demagnetisation and precession remains val id\nbutwithatemperature-dependentcorrection. Ifthisrelat ionis\nconfirmedexperimentally,a uniqueintrinsic couplingpara m-\netermeansthatthesamemainmicroscopicmechanismisact-\ningonbothtimescales. Inthepresentarticlewewillshowth at\nthe analysis of the available experimental data seems to ind i-\ncate towards this possibility, at least in pure transition m etals\nsuch as Ni or Co and in rare earth metal Gd. We did not find\nvalidityofthecorrespondingrelationin Fe.\nUp to now only classical version ( S→∞) of the LLB\nequation was used to model the ultra-fast demagnetisation\nprocesses20,21,24. In the present article we show the impor-\ntant role of the choice of the quantum spin value, resulting\ninthedifferencesinthecorrespondinglongitudinalrelax ation\ntimes. The article is organized as follows. In section II we\npresentdifferentformulationsofthequantumLLBmodeland\nits main features for different spin values S. In section III3\nwe present results on the modelling of the demagnetisation\nprocesses within LLB models with different choices of the\nquantumspinsnumber Sandofthe intrinsic scatteringmech-\nanisms. In section IV we present our attempts to link the\nultra-fast demagnetisation rates in transition metals and Gd\nand comparison with available experimental data. Section\nV concludes the article. In the Appendix to the article we\ndemonstrate the equivalence of the LLB model with S=1/2\nandtheM3TMmodelbyB.Koopmans et al.25.\nII. THE LANDAU-LIFSHITZ-BLOCHMODELWITH\nQUANTUMSPINNUMBER S.\nTheLLBequationforaquantumspinwasderivedfromthe\ndensity matrix approach28. Although the model Hamiltonian\nwas rather the simplest form of the spin-phonon interaction ,\nthegeneralizationoftheapproachshouldbepossibleto mor e\ncomplex situations. The macroscopic equation for the mag-\nnetisationdynamics,validatalltemperatures,iswritten inthe\nfollowingform:\n˙n=γ[n×H]+γα/bardbl\nn²[n·Heff]n−γα⊥\nn2[n×[n×Heff]](1)\nwheren=M/Me(T) =m/meis the reduced magnetisation,\nnormalizedtotheequilibriumvalue Meatgiventemperature T\nandm=M/Me(T=0K). Theeffectivefield Heff,containsall\nusualmicromagneticcontributions,denotedby Hint(Zeeman,\nanisotropy,exchangeandmagnetostatic)andisaugmentedb y\nthecontributioncomingfromthetemperature\nHeff=Hint+me\n2/tildewideχ/bardbl/parenleftbig\n1−n2/parenrightbig\nn, (2)\nwhere/tildewideχ/bardbl(T) = (∂m/∂H)H→0is the longitudinal susceptibil-\nity . The LLB equation contains two relaxational parame-\nters: transverse α⊥andlongitudinal α/bardbl,relatedtotheintrinsic\ncoupling-to-the-bathparameter λ. Inthe quantumdescription\nthe couplingparameter λcontains the matrix elements repre-\nsenting the scattering events and, thus, is proportional to the\nspin-fliprateduetotheinteractionwiththeenvironment. T his\nparameter,inturn,couldbetemperaturedependentand,ino ur\nopinion, it is this microscopic parameter which should be re -\nlatedtotheGilbertparametercalculatedthroughab-initi ocal-\nculations as in Refs.38,39, since the contribution coming from\nthespindisorderingisnotproperlytakenintoaccountinth ese\nmodels. In the quantum case the temperature dependence of\nthe LLB damping parameters is given by the following ex-\npressions:\nα/bardbl=λ\nme2T\n3TC2qS\nsinh(2qS)=⇒\nS→∞λ\nme2T\n3TC, (3)\nα⊥=λ\nme/bracketleftbiggtanh(qS)\nqS−T\n3TC/bracketrightbigg\n=⇒\nS→∞λ\nme/bracketleftbigg\n1−T\n3TC/bracketrightbigg\n,(4)withqS=3TCme/[2(S+1)T], whereSis the quantum spin\nnumber and TCis the Curie temperature. In the case S→∞\nthedampingcoefficientshavetheformsusedinseveralprevi -\nouslypublishedworks40,suitableforthecomparisonwiththe\nLangevindynamicssimulationsbasedontheclassicalHeise n-\nbergHamiltonianandinagreementwiththem20,27.\nEq.(1) is singular for T>TC, in this case it is more con-\nvenient to use the LLB equation in terms of the variable\nm=M/Me(T=0K)27. The corresponding LLB equation is\nindistinguishable from Eq.(1) but with different relaxati onal\nparameters/tildewideα/bardbl=meα/bardbl,/tildewideα⊥=meα⊥and/tildewideα⊥=/tildewideα/bardblforT>TC,\nin this case the contribution of temperature to Heff[the sec-\nondterminEq.(2)]is (−1//tildewideχ/bardbl)[1−3Tcm2/5(T−Tc)m]m. Al-\nthough this formulation is more suitable for the modelling o f\nthe laser-induced demagnetisation process, during which t he\nelectronic temperature is usually raised higher than TC, it is\nthe expression (4) which should be compared with the trans-\nverse relaxation parameter αLLGdue to the similarity of the\nformulationoftheEq.(1)withthemacromagneticLLGequa-\ntion. In the classical case and far from the Curie temperatur e\nT≪TC,λ=α⊥=/tildewideα⊥(αLLG).\nS→∞S=7/2S=3/2S=1/2\nT/T CαLLG\n10.90.80.70.60.50.40.09\n0.06\n0.03\nS→∞S=7/2S=3/2S=1/2\nT/T Cτ/bardbl[ps]\n1.2 1 0.8 0.6 0.46\n4\n2\n0.1\nFigure 2. (Up) The transverse damping parameter α⊥(αLLG) as\na function of temperature within the LLB model for different spin\nvaluesS. The intrinsic coupling parameter was set to λ=0.03.\n(Down) The longitudinal relaxation time τ/bardblas a function of tem-\nperature within the LLB model for different spin values S. The\ntemperature-dependent magnetisation and the longitudina l suscepti-\nbility/tildewideχ/bardblwereevaluatedinbothcasesintheMFAapproachusingthe\nBrillouinfunction.\nIn the \"thermal\" model the nature of the longitudinal and4\nthe transverse relaxation differs from the point of view of\ncharacteristicspinwavefrequencies. Thetransverserela xation\n(knownastheLLGdamping)isbasicallytherelaxationofthe\nFMR mode. The contributionof other spinwave modes is re-\nducedtothethermalaveragingofthemicromagneticparame-\nters and the main effect comes fromthe decrease of the mag-\nnetisation at high temperature. Consequently, the transve rse\ndamping parameter increases with temperature (see Fig.2),\nconsistentwithatomisticmodellingresults27andwell-known\nFMRexperiments37,41.\nOn the contrary, the main contribution to the longitudinal\nrelaxation comes from the high-frequency spin waves. This\nprocessoccursinastrongexchangefield. Asaresult,thelon -\ngitudinal relaxation time (the inverse longitudinal relax ation)\nis much faster and increases with temperature,knownas crit -\nical slowing down, see Fig.2. This slowing down has been\nshown to be responsible for the slowing down of the femto-\nsecond demagnetisation time τMas a function of laser pump\nfluency18,22. The characteristic longitudinal timescale is not\nonly defined by the longitudinal damping parameter (3) but\nalso by the temperature-dependentlongitudinal susceptib ility\n/tildewideχ/bardbl(T)27, accordingtothefollowingequation:\nτ/bardbl(T)=/tildewideχ/bardbl(T)\nγ/tildewideα/bardbl(T). (5)\nAs it can be observed in Fig. 2 the transverse relaxation\nparameter α⊥(αLLG) and the longitudinal relaxation time τ/bardbl\nhaveastrongdependenceonthequantumspinnumber Scho-\nsen to describe system’s statistics. We conclude here about\nthe occurrence of quite different relaxation rates for the t wo\nextremecases S=1/2andS=∞.\nB. Koopmans et al.recently used a different equation\nto describe the ultrafast demagnetisation dynamics25, called\nM3TMmodel:\ndm\ndt=RmTp\nTC/parenleftbigg\n1−mcoth/parenleftbiggmTC\nTe/parenrightbigg/parenrightbigg\n. (6)\nEq.(6)hasbeenobtainedthroughthegeneralMasterequatio n\napproach for the dynamics of the populations of a two level\nsystem (spin S=1/2 was used) with the switching probabil-\nityevaluatedquantum-mechanicallyforthephonon-mediat ed\nEY spin-flips. Here TpandTeare phonon and electron tem-\nperatures,respectively,and Risa materialspecificparameter,\nrelatedtothespin-flipprobabilityinthephonon-mediated EY\nscatteringevents asf, as\nR=8asfGepµBkBVaT2\nC\nµatE2\nD, (7)\nwhereVaandµataretheatomicvolumeandmagneticmoment,\nrespectively, Gepistheelectron-phononcouplingconstant, kB\nistheBoltzmannconstant, µBistheBohrmagnetonand EDis\nthe Debye energy. This equation has allowed to fit the ultra-\nfast demagnetisation time ( τM) obtaining the values of Rin\nNi, Co and Gd25and relating them to the phonon-mediated\nEYscatteringrates asf.\nAs we show in the Appendix, the M3TM equation (6) cor-\nresponds to the longitudinal part of the LLB equation withthermal field only ( Hint=0) and with spin S=1/2, i.e. it is\nequivalentto\ndm\ndt=γ/tildewideα/bardblHeff. (8)\nThisgivesarelationbetweentheintrinsiccouplingparame ter\nλand the material specific parameter Rand finally with the\nphonon-mediatedEYspin-flipprobability asfviatheformula:\nλ=3R\n2γµat\nkBTCTp\nTe=λ0Tp\nTe. (9)\nThus the two approaches are reconciled, provided that the\ntemperature-dependent coupling rate (9) is used in the LLB\nequation,incontrasttootherworks18,21,22wherethecoupling\nλis considered to be temperature-independent. Combining\nexpressions (5) (7) and (9), one can immediately see that in\nthecaseofthephonon-mediatedEYprocess,thelongitudina l\nrelaxationtimeisdeterminedby\nτ/bardbl∝/tildewideχ/bardbl\nasfE2\nD\nGepVaTp. (10)\nInRef.25andbasingonthephonon-mediatedEYpicture,the\nclassification of materials on the basis of the \"magnetic in-\nteraction strength\" parameter µat/Jwas proposed, where Jis\nthematerialexchangeparameter. Accordingtotheexpressi on\nabove,thedemagnetisationratedependsonmoreparameters ,\namong which the important one is also the electron-phonon\ncoupling Gepdefining how fast the electron system can pass\ntheenergytothephononone. Anotherimportantparameteris\nthe microscopicspin-fliprate asf. Comparingto theB. Koop-\nmanset al.25materialsclassification,thelongitudinalsuscep-\ntibilityinEq.(10) isindeeddefinedbythevalueoftheatomi c\nmomentµatandbythefactthatthisfunctionrapidlyincreases\nwithtemperatureanddivergescloseto TC∝J. AtT≈TCone\nobtainsasimplelinearrelation27/tildewideχ/bardbl∝µat/J,thusshowingthe\ndependenceof the demagnetisation rate on this parameter, a s\nsuggestedinRef.25.\nIn the case of the phonon-mediated EY process the tem-\nperature dependence of the longitudinal relaxation is comi ng\nfrom the longitudinalsusceptibiliy only (cf. Eq. (10)), as op-\nposed to the case λ=const (cf. Eq.(5)). (We do not discuss\nherethepossibilitythatthephonon-mediatedEYspin-flipr ate\nasfmay be also temperature dependent.) However, the tem-\nperature dependence of the susceptibility is characterize d by\nitsexponentialdivergencecloseto TC. Inthesecircumstances\nan additional linear temperature dependence provided by th e\nlongitudinal damping is difficult to distinguish in the fitti ng\nprocedureofexperimentaldata.\nIII. MODELLINGOF THELASER-INDUCED\nULTRA-FASTDEMAGNETISATIONWITHINTHE LLB\nMODELS.\nInthespiritofRefs.18,20–22,25forthemodellingofultra-fast\ndemagnetisationdynamics,theLLBequationmaybecoupled5\nto the electron temperature Teonly, understanding the elec-\ntrons as the main source for the spin-flip mechanism18,20–22\nor to both phonon and electron temperatures in the spirit of\nthe phonon-mediatedElliott-Yafet process25. In both cases it\nis the electrontemperature T=Tewhich couplesto the mag-\nnetisation in the LLB formalism, since the phonon tempera-\nture could only enter into the temperature dependence of the\ncoupling-to-the bath parameter λvia Eq.(9) . Note that the\ntemperature Tis not the spin temperature, since the resulting\ndynamicsistakingplaceout-of-equilibrium.\nTheelectron Teandphonon Tptemperaturesaretakenfrom\nthe two-temperature (2T) model15,45,46. Within this model\ntheirdynamicsisdescribedbytwodifferentialequations:\nCedTe\ndt=−Gep(Te−Tp)+P(t),\nCpdTp\ndt=Gep(Te−Tp). (11)\nHereCe=γeTe(γe=const) and Cpare the specific heats of\ntheelectronsandthelattice. TheGaussiansourceterm P(t)is\na function which describes the laser power density absorbed\nin the material. The function P(t)is assumed to be propor-\ntional to the laser fluence Fwith the proportionality coeffi-\ncient which could be obtained from the long time scale de-\nmagnetization data (for which Te=Tp)22. The dynamics of\nthe electron temperature can be also measured directly in th e\ntime-resolvedphotoemissionexperiment47.\nThe first of Eqs.(11) may also include a diffusion term\n∇z(κ∇zTe)taking into account a final penetration depth of\nthe deposited energy into the film thickness25and a term,\nCe(Te−300K)/τthdescribing the heat diffusion to the exter-\nnal space22. In the present article, the parameters for the 2T-\nmodel were taken either from Koopmans et al.25or from U.\nAtxitiaet al.22(for Ni only), where they were carefully pa-\nrameterized through the reflectivity measurements. The Ni\n(Co, Gd etc) parameters, such as magnetisation as a function\nof temperature were taken assuming the Brilloiun (Langevin\nforS→∞)function.\nThe coupling of the 2T model to the LLB equation ade-\nquately describes all three stages of the ultra-fast demagn eti-\nsation rates: sub-ps demagnetisation, ps recovery and sub- ns\nprecession21,22, see Fig.1. As a consequence of the temper-\nature dependenceof both longitudinal dampingand suscepti -\nbility, and since the temperature is dynamically changed ac -\ncording to Eqs.(11), the longitudinal relaxation time is ti me-\ndependent via Eq.(5). It is also strongly dependent on the\nparameters of the 2T model and its dynamics is not simple.\nConsequently,thesub-psultra-fastdemagnetisationgene rally\nspeaking is not exponentialand cannot be described in terms\nof one relaxation time τM. Simple analytical expression is\npossible to obtain with the supposition of a square-shaped\ntemperaturepulse23. The two-exponentialfitting is also often\nused22,36. In our approachthe fs demagnetisation is fitted di-\nrectlytothesolutionoftheLLBequationwithoutassumptio n\nof the one- or two-exponential decay. However, to comply\nwith the existing approaches, we still discuss the demagnet i-\nsationratein termsofauniqueparameter τM.\nIn the experiment performedin the same material the onlyremainingfittingparameterfortheLLBmodelisthecoupling\nparameter λ. The choice of λtogether with the parameters\nof the 2T model defines all magnetisation rates. In Fig.3 we\npresent modelling of the ultra-fast demagnetisation and re -\nmagnetisationforvariousvaluesofthecouplingparameter λ,\nchosen to be independent on temperature, as in Ref. 22. If\nfor some reason the scattering channel was suppressed, this\nwouldleadtoasmallscatteringrateandconsequentlyasmal l\ndemagnetisation and a slow recovery. Indeed, the value of λ\nforGdwasfoundtobe60timessmallerthanforNi(seeTable\nI).Thissmallvalueof λassuresalargedelayinthemagnetis-\narionrelaxationtowardstheequilibriumelectrontempera ture.\nThus this parameter defines the diversity of the demagnetisa -\ntion rates in larger extend than the ratio µat/J, suggested in\nRef.25anddiscussedintheprevioussubsection.\nλ=0.001\nλ=0.01\nλ=0.1\nt[ps]∆m/m 0\n20 10 00\n-0.1\n-0.2\n-0.3\nFigure 3. The result of integration of the LLB model ( S→∞) with\ndifferent parameters λ(increasing from top to the bottom). In this\ncase the the 2T model parameters were taken from Ref.22withl aser\nfluenceF=30 mJ/cm2\nAnother parameter strongly influencing the demagnetisa-\ntion rates is the phonon-electron coupling Gepdefining the\nrate of the electron temperature equilibration time. This i s\nthe main parameter governing the magnetisation recovering\ntimeτE. Indeed, in Ref.25the phonon-electroncoupling Gep\nwas chosen to be 20 times smaller for Gd than for Ni. By\nadjustingthis parameter,the ultra-slowdemagnetisation rates\nobserved in TbFe alloy48, Gd49and in half-metals50as well\nas the two time-scales demagnetisation25,49are also well-\nreproduced (see, as an example, Fig.4). Within this model\nthetwo-timescaleprocessconsistsofarelativelyfastdem ag-\nnetisation (however much slower than in Ni), defined by the\nelectrontemperatureandsmallvalueof λ,followedbyamuch\nslowerprocessduetoaslowenergytransferfromtheelectro n\ntothelattice system.\nAs it was mentioned in the previous subsection, the\nphonon-mediatedEY mechanismpredictsthe couplingto the\nbath parameter λto be dependent on the ratio between the\nphonon and electron temperature through the relation (9). A\ndecrease of λup to two times at high fluencies is observed\nfor Ni and Co. The analysis of the data presented in Ref. 25\nand47forGdhasshownthatduringthedemagnetisationpro-\ncess the ratio Te/Tphas increased almost 6 times. In Fig.5\nwe present the magnetisation dynamics for Ni evaluated for6\nt[ps]400300200100m/m0\n32101\n0.8\n0.6\n0.4\nFigure 4. The result of integration of the LLB model ( S→∞) with\nconstantλ0=0.0015 (seeTableI).Inthiscase the2Tmodel param-\neters were takenfrom Ref.25corresponding toGd.\ntwo laser pulse fluencies, assumingvariousvaluesof the spi n\nSandtemperature-dependentandindependent λvalues. Note\nquite different demagnetisation rates at high fluency for tw o\nlimiting cases S=1/2, used in Ref.25 and S=∞, used in\nRef.22. The differences in the choice of λare pronounced\nat high pump fluency but are not seen at low fluency. One\ncan also hope that in the fitting procedure of experimental\ndata it would be possible to distinguish the two situations.\nUnfortunately, the fitting to experimental data procedure i s\ncomplicated and the changes coming from the two cases de-\nscribed above are competing with several different possibi l-\nities such as an additional temperature dependency in elec-\ntron or phonon specific heats51. Additionally, we would like\nto mention different electron-phonon coupling constants Gep\nused in Refs. 22 and 25. Fitting to experimental data from\nRef.25 for Ni for high fluence, we have found that the case\nof the temperature-dependent λ=λ0(Tp/Te)can be equally\nfitted with the temperature-independent λ≈λ0/2. To answer\ndefinitely which fitting is better, more experimental data pr o-\nmoting one or another intrinsic mechanism and varying laser\nfluencyisnecessary.\nIV. LINKING DIFFERENTTIMESCALES\nSince the longitudinal relaxation occurs under strong ex-\nchange field and the transverse relaxation - under external\napplied field, their characteristic timescales are quite di ffer-\nent. However, the LLB equation provides a relation be-\ntween the ultra-fast demagnetisation(longitudinalrelax ation)\nand the transverse relaxation (ordinaryLLG dampingparam-\neter) via the parameter λ0(λ=λ0orλ=λ0(Tp/Te)for\nTp=Te). The two demagnetisation rates could be measured\nindependently by means of the ultra-fast laser pump-probe\ntechnique52. It has been recently demonstrated53that the\ndamping of the laser-induced precession coincides with the\nmeasuredby FMR intransitionmetals. By separatemeasure-\nmentsofthetwomagnetisationrates,therelations(4)and( 5)\ngiven by the LLB theory could be checked. This can pro-\nvidethevalidationoftheLLBmodel,aswellastheanswerto\nthe question if the same microscopic mechanism is acting onM3TMLLB\nS→∞S→1/2S→∞S→1/2\nt[ps]∆m/m 0\n3 2 1 00.1\n0\n-0.1\n-0.2\n-0.3\n-0.4\n-0.5\nFigure 5. Magnetisation dynamics during laser-induced dem agneti-\nsation process calculated within the LLB model with differe nt spin\nnumbers and for two laser-fluencies F=10 mJ/cm2(upper curves)\nandF=40 mJ/cm2(bottom curves). Ni parameters from Ref.22\nwere used. The symbols are calculated with the LLB equation w ith\nthe intrinsic damping parameter using a constant λ0=0.003 value,\nand the solid lines with the LLB equation and the intrinsic co upling\nwiththe temperature dependent λ=λ0/parenleftbig\nTp/Te/parenrightbig\n.\nfemtosecond and picosecond timescales. Unfortunately, th e\ndamping problem in ferromagnetic materials is very compli-\ncated and the literature reveals the diversity of measured v al-\nuesinthesamematerial,dependingonthepreparationcondi -\ntions.\nThus, to have a definite answer the measurement on\nthe same sample is highly desired. The measurements of\nbothα⊥andτMare available for Ni22where an excel-\nlent agreement between ultra-fast magnetisation rates via a\nuniquetemperature-independentparameter λ=0.04hasbeen\nreported22. The resultsof thesystematic measurementsof τM\nare also available for Ni, Co, Gd in Ref. 25, as well as for\nFe55. The next problem which we encounter here is that the\ndemagnetisationratesstronglydependon the spin value S, as\nis indicatedin Figs. 2 and 5. The fitting of experimentaldata\nusing LLB model with different Svalues results in different\nvalues of the coupling parameter λ0. The use of S=1/2\nvalue25orS=∞value22is quite arbitrary and these values\ndo not coincide with the atomic spin numbers of Ni,Co, Gd.\nGenerally speaking, for metals the spin value is not a good\nquantum number. The measured temperature dependence of\nmagnetisation, however, is well fitted by the Brillouin func -\ntionwith S=1/2forNi andCoand S=7/2forGd54. These\narethevaluesof Swhichwe usein TableI.\nConsequently in Table I we present data for the coupling\nparameter λ0extractedfromRef.25. Differentlytothisarticle,\nfor Gd we corrected the value of the parameter Rto account\nforadifferentspinvaluebytheratioof thefactors,i.e. RS1=\n(fS2/fS1)RS2with\nfS=2qS\nsinh(2qS)1\nm2\ne,SχS\n/bardbl, (12)\nwhere the parameters are evaluated at 120 Kusing the MFA\nexpressions for each spin value S. The data are evaluated7\nMaterial S R25λ0α⊥ αLLG\nNi 1/2 17.2 0.0974 0 .032 0 .01942-0.02841\nCo 1/2 25.3 0.179 0.025 0.003641-0.00643-0.01144\nGd 7/2 0.009 0.0015 0.00036 0 .000533\nTable I. The data for ultra-fast demagnetisation rate param eters for\nthree different metals from ultrafast demagnetization rat es and from\nFMR mesurements. The third column presents the demagnetisa tion\nparameter Rfrom Ref. 25, corrected in the case of Gd for spin\nS=7/2. The fourth column presents the value of the λ0parame-\nter, as estimated from the M3TM model25and the formula Eq.(9).\nThe fifth column presents the data for α⊥estimated via the LLB\nmodel Eq.(4) and the λ0value from the third column, at room tem-\nperatureT=300Kfor Co and Ni and at T=120Kfor Gd . The\nlast column presents the experimentally measured Gilbert d amping\ncollectedfrom different references.\nfor the phonon-mediated EY process with the temperature-\ndependent parameter λvia the expression (9). The value of\ntheGilbertdampingparameter α⊥wasthenestimatedthrough\nformula(4)at300 K(forNiandCo)andat120 KforGd. Note\nthat for temperature-independent λ=λ0the resulting λ0and\nα⊥valuesareapproximatelytwotimessmallerforNiandCo.\nThelastcolumnpresentsexperimentalvaluesforthesamepa -\nrameterfoundinliteratureforcomparisonwiththeonesint he\nfifth column, estimated through measurements of the ultra-\nfast demagnetisation times τMand the relation provided by\ntheLLBequation.\nGiven the complexityof the problem, the results presented\ninTableIdemonstratequiteasatisfactoryagreementbetwe en\nthe values, extractedfrom the ultra-fast demagnetisation time\nτMand the Gilbert damping parameter α⊥via one unique\ncoupling-to-the-bath parameter λ. The agreement is particu-\nlarlygoodforNi,indicatingthatthesamespinflipmechanis m\nisactingonbothtimescales. Thisistrueforbothexperimen ts\nin Refs.22 and 25. For Co the value is some larger. For the\ntemperature-independent λ, the resulting value is two times\nsmaller and the agreement is again satisfactory. We would\nlike to note that no goodagreementwas obtainedforFe. The\nreporteddampingvalues41are5-10timessmallerasestimated\nfromthedemagnetisationratesmeasuredinRef. 55.\nV. CONCLUSIONS\nThe Landau-Lifshitz-Bloch(LLB) equation providesa mi-\ncromagnetic tool for the phenomenological modelling of the\nultra-fast demagnetisation processes. Within this model o ne\ncan describe the temperature-dependent magnetisation dy-\nnamics at arbitrary temperature, including close and above\nthe Curie temperature. The micromagnetic formulation can\ntake into account the quantum spin number. The LLB model\nincludes the dynamics governed by both the atomistic LLG\nmodel and the M3TM model by Koopmans et al.25. In\nthe future it represents a real possibility for the multisca le\nmodelling20.\nWe have shown that within this model the ultra-fast de-\nmagnetisation rates could be parameterized through a uniqu etemperature dependent or independent parameter λ, defined\nby the intrinsic spin-flip rate. The magnetisation dynamics\nis coupled to the electron temperature throughthis paramet er\nand is always delayed in time. The observed delay is higher\nforhigherelectrontemperature. Thisisinagreementwitht he\nexperimentalobservationthatdifferentmaterialsdemagn etize\natdifferentrates25,50andthattheprocessissloweddownwith\nthe increase of laser fluency. We have shown that for the\nphonon-mediatedEY mechanism the intrinsic parameter λis\ndependent on the ratio between phonon and electron temper-\natures and therefore is temperature dependent on the femto\nsecond-severalpicosecondtimescale. TheLLBequationcan\nreproduce slow demagnetizing rates observed in several ma-\nterials such as Gd, TbFe and half metals. This is in agree-\nment with both phonon-mediated EY picture since in Gd a\nlowerspin-flipprobabilitywaspredictedandalsowiththei n-\nelastic electron scattering picture, since the electron di ffusive\nprocesses are suppressed in insulators and half-metals31,50.\nHowever, we also stress the importance of other parameters\ndetermining the ultra-fast demagnetisation rates, such as the\nelectron-latticecoupling.\nThe macroscopic damping parameters (longitudinal and\ntransverse) have different natures in terms of the involved\nspinwaves and in terms of the timescales. Their temperature\ndependenceisdifferent,however,theyarerelatedbythesp in-\nflip rate. We have tried to check this relation in several tran -\nsition metals such as Ni, Co, Fe and the rare-earth metal Gd.\nA good agreement is obtained in Co and Gd and an excel-\nlent agreement in Ni. This indicates that on both timescales\nthe same main microscopic mechanism is acting. In Ni the\nagreement is good both within the assumptions λ=λ0and\nλ=λ0Tp/Te. InCotheagreementseemstobebetterwiththe\ntemperature-independentparameter λ=λ0whichdoesnotin-\ndicate towards the phonon-mediated EY mechanism. How-\never, given a small discrepancy and the complexity of the\ndamping problem, this conclusion cannot be considered defi-\nnite. Wecanneitherexcludeanadditionaltemperaturedepe n-\ndence of the intrinsic scattering probability (i.e. the par ame-\nterλ0)forbothphonon-mediatedEYandexchangescattering\nmechanismswhichwasnottakenintoaccount.\nAnopenquestionistheproblemofdopedpermalloywhere\nanattempttosystematicallychangethedampingparameterb y\ndopingwithrare-earthimpuritieswasundertaken33inorderto\nclarify the relation between the LLG damping and the ultra-\nfast demagnetisation rate32,34. The results are not in agree-\nment with the LLB model. However in this case we think\nthat the hypothesis of the slow relaxing impurities present ed\nin Ref.34 might be a plausible explanation. Indeed, if the\nrelaxation time of the rare earth impurities is high, the sta n-\ndardLLB modelis not valid since it assumes an uncorrelated\nthermal bath. The correlation time could be introduced in\ntheclassicalspindynamicsviatheLandau-Lifshitz-Miyas aki-\nSeki approach56. It has been shown that the correlation time\nof the order of 10 fs slows down the longitudinal relaxation\nindependentlyon the transverse relaxation. Thus in this ca se,\nthemodificationoftheoriginalLLBmodeltoaccountforthe\ncolorednoiseisnecessary.8\nVI. ACKNOWLEDGEMENT\nThis work was supported by the Spanish projects\nMAT2007-66719-C03-01,CS2008-023.\nAppendixA\nTo show the equivalence between the LLB model with\nS=1/2 and the M3TM model25, we compare the relaxation\nrates resulting fromboth equations. We start with the M3TM\nequation\ndm\ndt=−RTp\nTC/parenleftbigg\n1−mcoth/bracketleftbigg/parenleftbiggTC\nTe/parenrightbigg\nm/bracketrightbigg/parenrightbigg\nm(A1)\nwhereweidentifytheBrillouinfunctionforthecase S=1/2,\nB1/2=tanh(q)withq=q1/2=(TC/Te)m. Now, we use the\nidentityB1/2=2/B′\n1/2sinh(2q)towrite\ndm\ndt=−RTp\nTC/bracketleftbigg2\nsinh(2q)/bracketrightbigg/parenleftBigg\n1−B1/2\nm\nB′\n1/2/parenrightBigg\nm2(A2)\nwe multiplyanddivideby qµatβto obtain\ndm\ndt=−RTp\nTCµat\nkBTC/bracketleftbigg2q\nsinh(2q)/bracketrightbigg/parenleftBigg\n1−B1/2\nm\nµatβB′\n1/2/parenrightBigg\nm(A3)\nM3TMLLB\nt[ps]∆m/m 0\n6 4 2 00\n-0.1\n-0.2\n-0.3\nFigure 6. Longitudinal relaxation calculated with M3TM and LLB\n(S=1/2) models for Nickel parameters22andT/Tc=0.8.We expand around equilibrium me=B1/2(qe)the small\nquantity1 −B1/2/m\n1−B1/2(q)\nm∼=δm\nme/parenleftbigg\n1−/parenleftbiggTC\nTe/parenrightbigg\nB′\n1/2(qe)/parenrightbigg\n(A4)\nwhereδm=m−me. Next,weexpand maroundm2\ne\nm=me+1\n2(m2−m2\ne)\nme=⇒δm\nme=(m2−m2\ne)\n2m2e(A5)\nand,\n1−B1/2/m\nβµatB′\n1/2≈1\n2/tildewideχ/bardbl(m2−m2\ne)\nm2e(A6)\nFinally,collectingthe equations(A3)and(A6)altogether :\ndm\ndt=/parenleftbigg3R\n2µat\nkBTC/parenrightbigg2Tp\n3TC2q\nsinh(2q)/parenleftbigg1\n2/tildewideχ/bardbl(1−m2\nm2e)m/parenrightbigg\n(A7)\nComparing this to the LLB equation with longitudinal re-\nlaxation only and without anisotropy and external fields, we\ncanwriteEq. (A7)in termsof n:\ndn\ndt=γλ\nme2Te\n3TC2q\nsinh(2q)Heff=γα/bardblHeff(A8)\nwhereHeff=me\n2/tildewideχ/bardbl(1−n2)n,and\nα/bardbl=/bracketleftbigg3R\n2γµat\nkBTCTp\nTe/bracketrightbigg2Te\n3TC2q\nsinh(2q)(A9)\nThustheKoopmans’M3TMequationisequivalenttotheLLB\nequationwith S=1/2andwheretheprecessionalaspectsare\nnotconsidered. 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Nowak\nand A.Rebei, Phys.Rev. Lett. 102057203 (2009)."    },    {        "title": "1812.07244v2.Thermal_gradient_driven_domain_wall_dynamics.pdf",        "content": "arXiv:1812.07244v2  [cond-mat.mes-hall]  26 May 2019Thermal gradient driven domain wall dynamics\nM. T. Islam,1,2X. S. Wang,3,4and X. R. Wang1,5,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2Physics Discipline, Khulna University, Khulna, Banglades h\n3School of Electronic Science and Engineering and State Key L aboratory of Electronic Thin Film and Integrated Devices,\nUniversity of Electronic Science and Technology of China, C hengdu 610054, China\n4Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Tr ondheim, Norway\n5HKUST Shenzhen Research Institute, Shenzhen 518057, China\nThe issue of whether a thermal gradient acts like a magnetic fi eld or an electric current in the\ndomain wall (DW) dynamics is investigated. Broadly speakin g, magnetization control knobs can\nbe classified as energy-driving or angular-momentum drivin g forces. DW propagation driven by a\nstatic magnetic field is the best known example of the former i n which the DW speed is proportional\nto the energy dissipation rate, and the current-driven DW mo tion is an example of the latter. Here\nwe show that DW propagation speed driven by a thermal gradien t can be fully explained as the\nangular momentum transfer between thermally generated spi n current and DW. We found DW-\nplane rotation speed increases as DW width decreases. Both D W propagation speed along the wire\nand DW-plane rotation speed around the wire decrease with th e Gilbert damping. These facts\nare consistent with the angular momentum transfer mechanis m, but are distinct from the energy\ndissipation mechanism. We further show that magnonic spin- transfer torque (STT) generated by a\nthermal gradient has both damping-like and field-like compo nents. By analyzing DW propagation\nspeed and DW-plane rotational speed, the coefficient ( β) of the field-like STT arising from the non-\nadiabatic process, is obtained. It is found that βdoes not depend on the thermal gradient; increases\nwith uniaxial anisotropy K/bardbl(thinner DW); and decreases with the damping, in agreement w ith the\nphysical picture that a larger damping or a thicker DW leads t o a better alignment between the\nspin-current polarization and the local magnetization, or a better adiabaticity.\nI. INTRODUCTION\nManipulating domain walls (DW) in magnetic nanos-\ntructures has attracted much attention because of its po-\ntential applications in data storage technology [ 1] and\nlogic gates [ 2]. The traditional DW control knobs,\nnamely magnetic fields and spin-polarized currents, have\ncertain drawbacks in applications. In the magnetic-field-\ndriven DW motion, energy dissipation is the main cause\nofDWpropagationwhosespeedisproportionaltotheen-\nergy dissipation rate [ 3,4], and the magnetic field tends\ntodestroyunfavorabledomainsandDWs, insteadofdriv-\ning a series of DWs synchronously [ 5–7]. An electrical\ncurrent drives a DW to move mainly through the angu-\nlar momentum transfer so that it pushes multiple DWs\n[8–11] in the same direction. To achieve a useful DW\nspeed, it requires high electrical current densities that\nresult in a Joule heating problem [ 12–14]. To avoid these\nproblems, spin-wave spin current has been proposed as a\nmoreenergy-efficientcontrolparameter[ 15–18]. Thermal\ngradient, a way to generate spin-wave spin current, is an\nalternative control knob of the DW motion. The inves-\ntigation on thermal-gradient-driven domain wall motion\nis meaningful not only for conventional applications, but\nalso for the understanding of spin wave and domain wall\ndynamics [ 16,17,20–23], as well as for possible recycling\nof waste heat [ 19,24].\n∗[Corresponding author:]phxwan@ust.hkTo understand the mechanism behind thermal-\ngradient-drivenDWdynamics, therearemicroscopicthe-\nories [15–17,25,26] and macroscopic thermodynamic\ntheories [ 21,22]. Briefly speaking, the microscopic theo-\nries suggest that magnons populated in the hotter region\ndiffuses to the colder region to form a magnon spin cur-\nrent. The magnon spin currentpassesthrough a DWand\nexerts a torque on the DW by transferring spin angular\nmomentum to the DW. Thus, magnons drive the DW\npropagating toward the hotter region of the nanowire,\nopposite to the magnon current direction [ 15,16,18].\nThe thermodynamic theories anticipate that a thermal\ngradient generates an entropy force which always drives\nthe DW towards the hotter region in order to minimize\nthe system free energy. The macroscopic theories do not\nprovide any microscopic picture about DW dynamics al-\nthough a thermal gradient is often considered as an effec-\ntive magnetic field to estimate DW speed [ 21,22] from\nfield-driven DW theories. Thus, one interesting issue is\nwhether a thermal gradient in DW dynamics acts like a\nmagnetic field or an electric current. DW propagation\nspeed should be sensitive to both DW width and types\nof a DW (transverse DW) under an energy-driving force\nwhile the speed should be insensitive to the DW and DW\nstructure in the angular-momentum-driving force. This\nis the focus of the current work.\nIn this paper, we investigate DW motion along a uni-\naxial wire with the easy axis along the wire direction\nunder a thermal gradient. We found that the DW al-\nways propagates to the hotter region with an accom-\npanied DW-plane rotation. DW propagation speed and2\nz\nxy\nFIG. 1. Schematic diagram of a uniaxial magnetic nanowire\nwith a head-to-head DW at the center under a thermal gra-\ndient∇T. Black (white) color represents colder (hotter) end\nof the sample.\nDW-plane rotation speed increases as the magnetic easy-\naxis anisotropy and damping decreases. We show that\nDW motion can be attributed to the angular momen-\ntum transfer between magnonic spin current and the\nDW. Thus, we conclude that a thermal gradient in-\nteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Similar to an\nelectric current [ 27], a thermal gradient can generate\nboth damping-like (or adiabatic) STT and field-like (or\nnon-adiabatic) STT. From the damping-dependence and\nanisotropy-dependence of the average DW velocity and\nDW-plane rotation angular velocity, we extract field-like\nSTT coefficient ( β). It is found that βis independent\nof thermal gradient; is bigger for a thinner DW; and de-\ncreases with the damping coefficient. We also show that\nin the presence of a weak hard-axis anisotropy perpen-\ndicular to the wire, the DW still undergoes a rotating\nmotion. The DW propagation speed increases slightly\nwhile the DW-plane rotation speed decreases with the\nstrength of the hard-axis anisotropy.\nII. MODEL AND METHOD\nWe consider a uniaxial nanowire of length Lxand\ncross-section Ly×Lzalong the x-axis (easy axis) with\na head-to-head DW at the center, as shown in Fig. 1.\nLy,Lzis much smaller than the DW width ∆, and ∆\nis much smaller than Lx. A thermal gradient is applied\nalong the wire. The highest temperature is far below\nthe Curie temperature Tc. The magnetization dynam-\nics is governed by the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation [ 28,29],\ndm\ndt=−γm×(Heff+hth)+αm×∂m\n∂t,(1)\nwherem=M/MsandMsare respectively the magne-\ntization direction and the saturation magnetization. α\nis the Gilbert damping constant and γis the gyromag-netic ratio. Heff=2A\nµ0Ms/summationtext\nσ∂2m\n∂x2σ+2K/bardbl\nµ0Msmxˆx+hdipoleis\nthe effective field, where Ais the exchange constant, xσ\n(σ= 1,2,3) denote Cartesian coordinates x,y,z,K/bardblis\nthe easy-axis anisotropy, and hdipoleis the dipolar field.\nhthis the stochastic thermal field.\nThe stochastic LLG equation is solved numerically by\nMUMAX3 package [ 30] in which we use adaptive Heun\nsolver. To balance stability and efficiency, we choose the\ntime step 10−14s with the cell size (2 ×2×2) nm3. Mag-\nnetic charges at the two ends of the wire are removed to\navoid their attraction to the DW. The saturation mag-\nnetization Ms= 8×105A/m and exchange constant\nA= 13×10−12J/m are used to mimic permalloy in\nour simulations. The thermal field follows the Gaussian\nprocess characterized by following statistics [ 31]\n/angb∇acketlefthth,ip(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthth,ip(t)hth,jq(t+∆t)/angb∇acket∇ight=2kBTiαi\nγµ0Msa3δijδpqδ(∆t),(2)\nwhereiandjdenote the micromagnetic cells, and p,q\nrepresent the Cartesian components of the thermal field.\nTiandαiare respectively temperature and the Gilbert\ndamping at cell i, andais the cell size. kBis the Boltz-\nmann constant [ 28]. The numerical results presented in\nthis study are averaged over 15 random configurations\n(for DW velocity) and 4000-5000 random configurations\n(for spin current).\nUnderthethermalgradient ∇xT,magnetizationatdif-\nferent positions deviate from their equilibrium directions\ndifferently and small transverse components myandmz\nare generated. The transverse components vary spatial-\ntemporally and depend on the local temperature. This\nvariation generates a magnonic spin current [ 16]. This\nmagnonic spin current can interact with spin textures\nsuch as DWs. In the absence of damping (the thermal\nfield also vanishes), the spin currentalong the xdirection\ncan be defined from the spin continuity equation derived\nfrom Eq. ( 1) as follows [ 15],\n∂m\n∂t=−1\n1+α2m׈xmxK/bardbl−∂J\n∂x,(3)\nwhere\nJ(x) =2γA\nµ0Msm×∂m\n∂x, (4)\nis the spin current density along x-direction due to the\nexchangeinteraction. J(x) can be numerically calculated\n[15,23]. In the presence of damping as well as the ther-\nmal field, the contribution of the damping term and the\nthermal term is proportional to α, which is relatively\nsmall. More importantly, according to the fluctuation-\ndissipation theorem [ 28], the damping term and the ther-\nmal term should cancel each other after average over a\nlong time. Since the time scale of DW dynamics is much\nlonger than the thermal fluctuation, the combined con-\ntribution of damping and thermal terms should be very\nsmall.3\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s52/s56/s49/s50/s49/s54/s50/s48\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s50/s52/s54/s56/s49/s48/s49/s50/s45/s56/s48/s48 /s45/s52/s48/s48 /s48 /s52/s48/s48 /s56/s48/s48/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s32/s32\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41/s32/s32/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\n/s48/s46/s48/s55/s32 /s75/s47/s110/s109\n/s48/s46/s49 /s75/s47/s110/s109\n/s48/s46/s49/s53/s32 /s75/s47/s110/s109\n/s48/s46/s50/s32 /s75/s47/s110/s109\n/s48/s46/s50/s53/s32 /s75/s47/s110/s109\n/s48/s46/s51/s32 /s75/s47/s110/s109/s74\n/s116/s111/s116/s40/s120/s41/s40\n/s115/s41/s41\n/s120 /s32/s40/s110/s109/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41/s40/s97/s41\n/s32\n/s75 /s32/s40/s49/s48/s52\n/s32/s74/s47/s109/s51\n/s41/s118 /s32/s40/s109/s47/s115/s41/s40/s99/s41/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s51/s54/s51/s57/s52/s50\n/s32\nFIG. 2. (a) The spatial dependence of spin current densities\nJtot(x) for various ∇xT. The DW center is chosen as x=\n0. (b) Thermal gradient dependence of DW velocity vsimu\nfrom micromagnetic simulations (open squares) and vcurrent\ncomputedfrom total spin current(solid squares). (c)Therm al\ngradient dependence of DW-plane rotation angular velocity\n(squares). In (a)(b)(c) model parameters are Lx= 2048 nm,\nLy=Lz= 4 nm, α= 0.004 and K/bardbl= 5×105J/m3. (d)\nvsimu(solid squares) and dφ/dt(open squares) as a function\nofK⊥forLx= 1024 nm and ∇xT= 0.5 K/nm.\nIntegrating the x−component of Eq. ( 3) over a space\nenclosed the DW in the center and noticing the absence\nof the first term on the right, we have\nvcurrent=1\n2/integraldisplayLx/2\n−Lx/2∂mx\n∂tdx\n=−2γA\nµ0Ms/bracketleftbig1\n2(Jx|left−Jx|right)].(5)\nwhere we have assumed the fluctuations in the domains\nare small and the DW is not far from a symmetric one.\nJx|left,Jx|rightmean the x-components of the total spin\ncurrent on the left and right sides of the DW. The equa-\ntion clearly shows that the DW propagates opposite to\nthe spin current. This is the theoretical DW velocity un-\nder the assumption of angular momentum conservation,\nand it will be compared with the directly simulated DW\nvelocity below.\nIII. RESULTS\nA. Average spin current and DW velocity\nTosubstantiateourassertionthatDWpropagationun-\nder a thermal gradient is through angular-momentum ef-\nfect instead of energy effect, we would like to compare\nthe DW velocity obtained from micromagnetic simula-\ntions and that obtained from total spin current based on/s49 /s50 /s51 /s52 /s53 /s54 /s55/s56/s49/s50/s49/s54/s50/s48\n/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41/s32/s118 /s32/s40/s109/s47/s115/s41\n/s32/s76\n/s120/s61/s50/s48/s52/s56/s32/s110/s109\nFIG. 3. Damping αdependence of the DW dynamics: vsimu\n(Open squares); vcurrent(solid squares ); and dφ/dt(solid\ncircles). Model parameters are ∇xT= 0.2 K/nm, K/bardbl= 5×\n105J/m3,Lx= 2048 nm and Ly=Lz= 4 nm.\nEq. (5). Eq. (4) is used to calculate Jx(x). Fig.2(a) is\nspatial distribution of the ensemble averaged Jx(x) with\nDW atx= 0 for various thermal gradients. The sud-\nden sign change of Jx(x) at the DW center is a clear\nevidence of strong angular-momentum transfer from spin\ncurrent to the DW. Technically, magnetizationof the two\ndomains separated by the DW point to the opposite di-\nrections, thus the spin current polarization changes its\nsign. In calculating DW velocity vcurrentfrom Eq. ( 5),\nthe spin currents before entering DW and after passing\nDW are the averages of Jx(x) overx∈[−2∆,−∆] and\nx∈[∆,2∆], where ∆ is the DW width which is 16 nm\nin the current case. The thermal gradient dependence\nofvcurrentis shown in Fig. 2(b) (solid squares). vcurrent\ncompares well with the velocity vsimu(open squares) ob-\ntained directly from simulations by extracting the speed\nof the DW center along x-direction. The DW veloc-\nity is linearly proportional to the temperature gradient\nv=C∇xT, with the thermal mobility C= 6.66×10−8\nm2s−1K−1forvsimuorC= 6.59×10−8m2s−1K−1for\nvcurrent. It is noted that vcurrentalmost coincides with\nvsimuexcept a small discrepancy at very high thermal\ngradient when the nonlinear effects is strong. The small\ndiscrepancy may be attributed to the large fluctuations\nas well as the contribution from the damping, the dipo-\nlar and stochastic fields. These observations are consis-\ntent with magnonic STT [ 15,16,25,26]. It is observed\nthat the DW-plane rotates around the x-axis counter-\nclockwise for head-to-head DW and clockwise for tail-to-\ntail DW during DW propagation. DW rotation speed\ndφ/dt(squares) is shown in Fig. 2(c)) as a function of\n∇xT.4\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54\n/s52/s54/s56/s49/s48/s49/s50/s49/s52/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s100 /s47/s100/s116/s40/s100/s101/s103/s47/s110/s115/s41/s32\nFIG. 4. Anisotropy K/bardbldependence of the DW dynamics:\nvsimu(open squares); vcurrent(solid squares); and dφ/dt(solid\ncircles). Model parameters are Lx= 2048 nm, Ly=Lz=4 nm,\nα= 0.004 and ∇xT= 0.2 K/nm.\nB. Damping and anisotropy dependence of DW\ndynamics\nAn energy-effect and angular-momentum-effect\nhave different damping-dependence and anisotropy-\ndependence of DW dynamics. To distinguish the roles of\nenergy and the angular-momentum in thermal-gradient\ndriven DW dynamics, it would be useful to probe how\nthe DW dynamics depends on αandK/bardbl. Damping have\ntwo effects on the spin currents: one is the decay of\nspin current during its propagation so that the amount\nof spin angular momentum deposited on a DW should\ndecrease with the increase of the damping coefficient.\nAs a result, the DW propagation speed and DW-plane\nrotation speed should also be smaller for a larger α.\nIndeed, this is what we observed in our simulations\nas shown in Fig. 3(a) for DW speed and DW-plane\nrotation speed (open squares for vsimu, solid circles for\nvcurrent, and stars for dφ/dt). The model parameters are\nLx= 2048, Ly=Lz= 4 nm, ∇xT= 0.2 K/nm and\nK/bardbl= 5×105J/m3. The second damping effect is that\nthe larger αhelps the spin current polarization to align\nwith the local spin. This second effect enhances the\nadiabatic process that is important for non-adiabatic\nSTT or field-like torque discussed in the next subsection.\nTherefore, α−dependence of DW dynamics supports\nthe origin of thermal driven DW dynamics to be the\nangular-momentum effect, not the energy effect that\nwould lead to a larger vsimuanddφ/dtfor a larger α\n[3,4,33–35] instead of a decrease observed here.\nHere we would like to see how the DW dynamics de-\npendsonuniaxialanisotropy K/bardbl. Fig.4showsboth vsimu\n(open squares), vcurrent(filled squares) and dφ/dt(cir-\ncles) for Lx= 2048 nm, α= 0.004 and ∇xT= 0.2. The\nDW propagation speed, vsimudecreases with K/bardblwhileDW-plane rotational speed increases with K/bardbl. These re-\nsults seem follow partially the behavior of magnetic-field\ninduced DW motion, in which DW propagation speed\nis proportional to DW width (∆ ∼/radicalBigg\nA\nK/bardbl) or decrease\nwithK/bardbl, and partially electric current driven DW mo-\ntion, in which DW-plane rotational speed increases with\nK/bardbl. Thus, one may tend to conclude that a thermal gra-\ndient behaves more like a magnetic field rather than an\nelectric current from the DW width dependence of DW\npropagation speed, opposite to our claim of the angular-\nmomentum effects of the thermal gradient. It turns out,\nthis is not true. The reason is that magnon spectrum,\nωk=2γ\nµ0Ms/parenleftbig\nAk2+K/bardbl/parenrightbig\n, has a gap in a system with mag-\nnetic anisotropy. The larger K/bardblis, the bigger the energy\ngap will be. Thus, it becomes harder to thermally excite\nmagnon. As a result, the spin current decreasesas K/bardblin-\ncreases. To see whether the thermal-gradient driven DW\nmotion is due to the angular-momentum transfer or not,\none should compare whether vsimuandvcurrentmaintain\na good agreement with each other as K/bardblvaries. Indeed,\na good agreement between vsimuandvcurrentis shown in\nFig.4. This conclusion is also consistent with existing\nmagnonic STT theories [ 33–35].\nC. Separation of adiabatic and non-adiabatic\ntorques\nWe have already demonstrated that a thermal gradi-\nent interacts with DW through magnonic STT rather\nthan through energy dissipation. It is then interesting to\nknow what kind of STTs a thermal gradient can gen-\nerate. Specifically, whether a magnonic spin current\ngenerates damping-like (adiabatic), or field-like (Non-\nadiabatic) torques, or both just like an electric current\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s73/s32/s40/s49/s48/s49/s48\n/s32/s65/s47/s109/s50\n/s41\n/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\nFIG. 5. Model parameters are K/bardbl= 5×105J/m3,α=\n0.004,Lx= 1024 and Ly=Lz=4 nm. Effective electric current\ndensityI(open squares) and β(solid squares) are plotted as\nfunctions of ∇xT.5\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s49/s52/s48/s46/s50/s49/s48/s46/s50/s56\n/s40/s49/s48/s45/s51\n/s41\n/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s40/s97/s41\n/s40/s98/s41/s32/s32\nFIG. 6. Model parameters are ∇xT=0.5 K/nm, Lx= 1024\nnm and Ly=Lz=4 nm. (a) α-dependence of βforK/bardbl= 106\nand J/m3. (b)K/bardbl-dependence of βforα= 0.004.\n[27] does. To extract the STT generated from a thermal\ngradient, we approximate DW dynamics by the motion\nof its collective modes of DW center Xand the titled\nangleφof DW-plane. Subject to both damping-like and\nfield-like torques, using the travelling-wave ansatz [ 33–\n35], tan(θ/2) = exp[( x−X)/∆] where ∆ ∼/radicalbig\nA/K/bardbl, one\ncan derive the equations for X and φ,\nα\n∆dX\ndt+dφ\ndt=β\nαu,1\n∆dX\ndt−αdφ\ndt=u\nα.(6)\nFrom the above two equations, one can straightfor-\nwardly find DW propagating speed and DW-plane ro-\ntation speed,\nv=(1+αβ)\n(1+α2)u,˙φ=(β−α)\n(1+α2)u. (7)\nOne can extract βand equivalent electric current den-\nsityI= (2eMsu)/gµBPfromvanddφ/dtobtained in\nsimulations. For α= 0.004,K/bardbl= 106J/m3, theIandβ\nare obtained and plotted in Fig. 5as a function of ∇xT.\nIt is evident that Ilinearly increases with ∇xTandβ\nis independent of ∇xTas it should be. We then fixed\n∇xT= 0.5 K/nm, and repeat simulations and analysis\nmentioned above for various αandK/bardbl. Fig.6(a) and\n(b) shows βas a function of αandK/bardbl. From the figure,\nit is evident that βdecreases with α. This is because\nthe larger damping favors the alignment of spin current\npolarization with the local spin so that the non-adiabatic\neffect,β, becomes smaller. βincreases with K/bardblfor the\nsimilar reason: Larger K/bardblmeans a thinner DW so that\nit is much harder for the spin current polarization to re-\nverse its direction after passing through the thinner DW,\ni.e. a stronger non-adiabatic effect.\nIn some experiments, the temperature gradient is gen-\nerated by a laser spot[ 36]. The laser spot will induce a\nGaussian distribution of the temperature over the space\n[36,37]. In Fig. 7, weshowtheDWmotionin aGaussian\ntemperature profile T(x) =T0exp/parenleftBig\n−(x−xL)2\n2σ2/parenrightBig\nby plot-\nting the DW position against the time. Here we use the/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s32/s32/s68/s87/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s110/s109/s41\n/s116/s105/s109/s101/s32/s40/s110/s115/s41\nFIG. 7. Domain wall position versus time in a Gaussian tem-\nperature profile. The gray lines are raw data for different\nrandom seeds and the red line is the averaged result. The\ngreen dashed line is theoretical result using the thermal mo -\nbilityC= 6.66×10−8m2s−1K−1obtained from Fig. 2(b).\nsame parameters as those in Fig. 2(b), except a longer\nwireLx= 2048 nm, and T0= 400 K, σ= 200 nm,\nxL= 200 nm. Theoretically, if the instantaneous DW\nspeed under a Gaussian temperature is the same as that\nin the constant thermal-gradient case, we should expect\ndx\ndt=CdT\ndx, where the thermal mobility Cis the same as\nthat in Fig. 2(b). Using C= 6.66×10−8m2s−1K−1,\nthe above differential equation for x(t) can be numeri-\ncally solved with initial condition x(0) = 0. The result\nis plotted in Fig. 7in green dashed line. The simu-\nlated speed is smaller than this theoretical result. This\nis probably because, for the constant thermal-gradient,\nwe focus on the steady-state DW motion speed. In a\nGaussian temperature, the DW cannot immediately fol-\nlow the local temperature gradient. Before the DW can\nreach the steady-state speed corresponding to the local\ntemperature, it already moves to a position of smaller\ntemperature gradient. More details about DW motion in\nGaussian temperature profile may be an issue of future\nstudies.\nIV. DISCUSSION AND SUMMARY\nWe have studied the thermal gradient-driven DW dy-\nnamicsinanuniaxialnanowire. Inreality, thereisalways\ncertain hard anisotropy in a wire whose cross-section is\nnot a perfect ellipse. Thus, it is interesting to see how\nthe above results will change in a weak biaxial nanowire\nwith a small hard anisotropy K⊥= 1/2µ0M2\ns(Nz−Ny),\nsay along y-direction. Our simulations show that a DW\nstill propagatestowardsthe higher temperature region in\na similar way as that in a uniaxial wire. Interestingly, as\nshown in Fig. 2(d) for the K⊥-dependence of vsimu(solid\nsquares) and dφ/dt(open squares), DW speed increases6\nslightly with K⊥. This may be due to the increase of\ntorque along θ-direction [ 33] since Γ θis proportional to\n(Nz−Ny). This is also consistent with the early results\nforthe uniaxialwire that vsimu(which includes stochastic\nthermal field and demagnetisation fields) is always larger\nthanvcurrent(where the transverse fields are neglected).\nAt the meanwhile, dφ/dtdecreases with K⊥.\nThe main purpose of this paper is to study the\nmagnonic effects in thermal-gradient-driven domain wall\ndynamics. We consider the spin waves explicitly and\nall the material parameters (exchange constant A, crys-\ntalline anisotropy K, saturation magnetization Ms, and\nGilbert damping α) are assumed to be constant. Indeed,\nthe atomistic magnetic moments are independent of tem-\nperature. At the atomistic level, the exchange constant\nAoriginating from the Pauli exclusion principle and the\ncrystalline anisotropy Koriginating from the spin-orbit\ncoupling onlyweaklydepend on the temperature because\nof the vibration of atoms [ 39]. In micromagnetic models,\nbecause finite volumes that contains many magnetic mo-\nments are considered as unit cells, the parameters A,K,\nandMsdepend on the temperature. This is because the\nthermally excited spin waves with wavelengths shorter\nthan the length scale of the unit cells are included in\nthe effective A,K, andMsby doing an average [ 16,38].\nSince we use small mesh size 2 ×2×2 nm3, only spin\nwaves of very short wavelength affect the parameters A,\nK, andMsin our model. Those short-wavelength spin\nwavespossess high energyaswell as low density ofstates,\nso their contributions to the effective A,K, andMsare\nnot significant. The Gilbert damping αdepends on the\ntemperature non-monotonically [ 40–43]. The underlying\nmechanism is still under debate, but for many cases the\ndependence is not significant in a wide range of temper-\nature.\nIn summary, our results show that the uniform ther-\nmal gradient always drives a DW propagating towards\nthe hotter region and the DW-plane rotates around the\neasy axis. The DW velocity and DW-plane rotational\nspeed decrease with the damping coefficient. The DW\nvelocity obtained from simulation agrees with the veloc-\nity obtained from angular momentum conservation when\nthe magnon current density ( J(x)) from the simulation is\nusedtoestimatetheamountofangularmomentumtrans-\nferred from magnon current to the DW. All the above\nfindings lead to the conclusion that the thermal gradient\ninteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Furthermore,\nwe demonstrated that the magnonic STT generated by\na thermal gradient has both damping-like and field-like\ncomponents. The field-like STT coefficient βis deter-\nmined from DW speed and DW-plane rotation speed. β\ndoes not depend on the thermal gradient as expected,\nbut increases with a decrease of DW width. This behav-\nior can be understood from the expected strongmisalign-\nment of magnon spin polarization and the local spin so\nthat non-adiabatic torque (also called field-like torque)\nis larger. For the same reason, a larger Gilbert dampingresults in a better alignment between spin current polar-\nization and the local spin, thus βshould decrease with\nα. The thermal gradientcan be a veryinteresting control\nknob for nano spintronics devices, especially those made\nfrom magnetic insulators.\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No. 11774296) as well\nas Hong Kong RGC Grants Nos. 16300117, 16301518\nand 16301816. X.S.W acknowledges support from NSFC\n(GrantNo. 11804045),ChinaPostdoctoralScienceFoun-\ndation (Grant No. 2017M612932and 2018T110957),and\nthe Research Council of Norway through its Centres of\nExcellence funding scheme, Project No. 262633, “QuS-\npin.” M. T. I acknowledges the Hong Kong PhD fellow-\nship.7\n[1] Parkin S S P, Hayashi M and Thomas L 2008 Science\n320 190\n[2] Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit\nD and Cowburn R P 2005 Science309 1688\n[3] Wang X R, P Yan, Lu J and He C 2009 Ann. Phys. (N.\nY.)324 1815\n[4] Wang X R, Yan P and Lu J 2009 Europhys. Lett. 86\n67001\n[5] Atkinson D, Allwood D A, Xiong G, Cooke M D,\nFaulkner C C, and Cowburn R P 2003 Nat. Mater. 2\n85\n[6] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine\nJ L 2005 Nat. Mater. 4 741\n[7] Hayashi M, Thomas L, Bazaliy Ya B , Rettner C, Moriya\nR, Jiang X, and Parkin S S P 2006 Phys. Rev. Lett. 96\n197207\n[8] Berger L 1996 Phys. Rev. B 54 9353\n[9] Slonczewski J 1996 J. Magn. Magn. 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B84 014412\n[43] Maier-Flaig H, Klingler S , Dubs C, Surzhenko O, Gross\nR, Weiler M, Huebl H, and Goennenwein S T B 2017\nPhys. Rev. B 95 214423"    },    {        "title": "2201.11498v3.Effect_of_vertex_corrections_on_the_enhancement_of_Gilbert_damping_in_spin_pumping_into_a_two_dimensional_electron_gas.pdf",        "content": "E\u000bect of vertex corrections on the enhancement of Gilbert damping in spin pumping\ninto a two-dimensional electron gas\nM. Yama,1M. Matsuo,2;3;4;5T. Kato1,\n1Institute for Solid State Physics,\nThe University of Tokyo, Kashiwa, Japan\n2Kavli Institute for Theoretical Sciences,\nUniversity of Chinese Academy of Sciences, Beijing, China\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing, China\n4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai, Japan\n5RIKEN Center for Emergent Matter Science (CEMS),\nWako, Saitama, Japan\n(Dated: May 15, 2023)\nWe theoretically consider the e\u000bect of vertex correction on spin pumping from a ferromagnetic\ninsulator (FI) into a two-dimensional electron gas (2DEG) in which the Rashba and Dresselhaus\nspin-orbit interactions coexist. The Gilbert damping in the FI is enhanced by elastic spin-\ripping\nor magnon absorption. We show that the Gilbert damping due to elastic spin-\ripping is strongly\nenhanced by the vertex correction when the ratio of the two spin-orbit interactions is near a special\nvalue at which the spin relaxation time diverges while that due to magnon absorption shows only\nsmall modi\fcation. We also show that the shift in the resonant frequency due to elastic spin-\ripping\nis strongly enhanced in a similar way as the Gilbert damping.\nI. INTRODUCTION\nIn the \feld of spintronics1,2, spin pumping has long\nbeen used as a method of injecting spins into various\nmaterials3{5. Spin pumping was \frst employed to inject\nspins from a ferromagnetic metal into an adjacent normal\nmetal (NM)6{9. Subsequently, it was used on ferromag-\nnetic insulator (FI)/NM junctions10. Because spin injec-\ntion is generally related to the loss of the magnetization\nin ferromagnets, it a\u000bects the Gilbert damping measured\nin ferromagnetic resonance (FMR) experiments11. When\nwe employ spin injection from the FI, the modulation of\nthe Gilbert damping re\rects the properties of the spin ex-\ncitation in the adjacent materials, such as magnetic thin\n\flms12, magnetic impurities on metal surfaces13, and su-\nperconductors14{17. This is in clear contrast with the\nGilbert damping of a bulk FI, which re\rects properties\nof electrons and phonons18{20.\nAn attractive strategy is to combine spin pumping\nwith spin-related transport phenomena in semiconduc-\ntor microstructures1,21. A two-dimensional electron gas\n(2DEG) in a semiconductor heterostructure is an easily\ncontrolled physical system that has been used in spin-\ntronics devices22{25. A 2DEG system has two types of\nspin-orbit interaction, i.e., Rashba26,27and Dresselhaus\nspin-orbit interactions28,29.\nIn our previous work30, we theoretically studied spin\npumping into a 2DEG in semiconductor heterostructures\nwith both Rashba and Dresselhaus spin-orbit interac-\ntions, which can be regarded as a prototype for a 2DEG\nwith a complex spin-texture near the Fermi surface [see\nFig. 1 (a)]. In that study, we formulated the modu-\nlation of the Gilbert damping in the FI by using the\n2DEG\nmicrowave\nFI\n(a) (b)\ntotFIG. 1. (a) Schematic picture of junction composed of a fer-\nromagnetic insulator (FI) and a two-dimensional electron gas\n(2DEG) realized in a semiconductor heterostructure. Stotin-\ndicates the total spin of the FI. We consider a uniform spin\nprecession of the FI induced by microwave irradiation. (b)\nLaboratory coordinates ( x;y;z ) and the magnetization-\fxed\ncoordinates ( x0;y0;z0). The red arrow indicates the expec-\ntation value of the spontaneous spin polarization of the FI,\nhSi.\nsecond-order perturbation with respect to the interfa-\ncial coupling15,31{35and related it to the dynamic spin\nsusceptibility of the 2DEG. We further calculated the\nspin susceptibility and obtained characteristic features\nof the Gilbert damping modulation. This modulation\ncontains two contributions: elastic spin-\ripping, which\ndominates at low resonant frequencies, and magnon ab-\nsorption, which dominates at high resonant frequencies.\nIn addition, we clari\fed that these contributions have\ndi\u000berent dependence on the in-plane azimuth angle \u0012of\nthe ordered spin in the FI [see Fig. 1 (b)].\nWhen the Rashba and Dresselhaus spin-orbit interac-\ntions have almost equal magnitudes, spin relaxation byarXiv:2201.11498v3  [cond-mat.mes-hall]  12 May 20232\nnonmagnetic impurity scattering is strongly suppressed\nbecause the direction of the e\u000bective Zeeman \feld gen-\nerated by the spin-orbit interactions is unchanged along\nthe Fermi surface. Due to this substantial suppression of\nspin relaxation, there emerge characteristic physical phe-\nnomena such as the persistent spin helix state36{39. In\ngeneral, the vertex corrections have to be taken into ac-\ncount to treat various conservation laws, i.e., the charge,\nspin, momentum, and energy conservation laws in cal-\nculation of the response functions40{43. Therefore, for\nbetter description of realistic systems, we need to con-\nsider vertex correction, which captures e\u000bect of impurity\nmore accurately by re\recting conservation laws. How-\never, the vertex corrections were neglected in our pre-\nvious work30. This means that our previous calculation\nshould fail when the Rashba and Dresselhaus spin-orbit\ninteractions compete.\nIn this study, we consider the same setting, i.e., a\njunction composed of an FI and a 2DEG as shown in\nFig. 1 (a), and examine e\u000bect of the spin conservation law\nby taking the vertex correction into account. We theo-\nretically calculate the modulation of the Gilbert damping\nand the shift in the FMR frequency by solving the Bethe-\nSalpeter equation within the ladder approximation. We\nshow that the vertex correction substantially changes the\nresults, in particular, when the strengths of the Rashba-\nand Dresselhaus-type spin-orbit interactions are chosen\nto be almost equal but slightly di\u000berent; Speci\fcally,\nboth the Gilbert damping and the FMR frequency shift\nare largely enhanced at low resonant frequencies re\rect-\ning strong suppression of spin relaxation. This remark-\nable feature should be able to be observed experimen-\ntally. In contrast, the vertex correction changes their\nmagnitude only slightly at high resonant frequencies.\nBefore describing our calculation, we brie\ry comment\non study of the vertex corrections in a di\u000berent context.\nIn early studies of the spin Hall e\u000bect, there was a de-\nbate on the existence of intrinsic spin Hall e\u000bect44{46. By\nconsidering the vertex corrections, the spin Hall conduc-\ntivity, which is calculated from the correlation function\nbetween the current and spin current, vanishes in the\npresence of short-range disorder for simple models even if\nits strength is in\fnitesimally small47{49. This seemingly\ncontradictory result stimulated theoretical researches on\nrealistic modi\fed models50,51as well as de\fnition of the\nspin current52{56. However, we stress that the vertex\ncorrections for the dynamic spin susceptibility, which is\ncalculated from the spin-spin correlation function, have\nno such subtle problem57because it does not include the\nspin current.\nThe rest of this work is organized as follows. In Sec. II,\nwe brie\ry summarize our model of the FI/2DEG junc-\ntion and describe a general formulation for the magnon\nself-energy following Ref. 30. In Sec. III, we formulate\nthe vertex correction that corresponds to the self-energy\nin the Born approximation. We show the modulation of\nthe Gilbert damping and the shift in the FMR frequency\nin Secs. IV and V, respectively, and discuss the e\u000bect\nky\nkx\n(a) (b)\nFIG. 2. Schematic picture of the spin-splitting energy bands\nof 2DEG for (a) \f=\u000b= 0 and (b) \f=\u000b= 1. The red and blue\narrows represent spin polarization of each band. In the case\nof (b), the spin component in the direction of the azimuth\nangle 3\u0019=4 is conserved.\nof the vertex correction in detail. Finally, we summa-\nrize our results in Sec. VI. The six Appendices detail the\ncalculation in Sec. III.\nII. FORMULATION\nHere, we describe a model for the FI/2DEG junction\nshown in Fig. 1 (a) and formulate the spin relaxation\nrate in an FMR experiment. Because we have already\ngiven a detailed formulation on this model in our previous\npaper30, we will brie\ry summarize it here.\nA. Two-dimensional electron gas\nWe consider a 2DEG whose Hamiltonian is given as\nHNM=Hkin+Himp, whereHkinandHimpdescribe the\nkinetic energy and the impurity, respectively. The kinetic\nenergy is given as\nHkin=X\nk(cy\nk\"cy\nk#)^hk\u0010ck\"\nck#\u0011\n; (1)\n^hk=\u0018k^I\u0000he\u000b(k)\u0001\u001b; (2)\nwhereck\u001bis the annihilation operator of conduction elec-\ntrons with wave number k= (kx;ky) andzcomponent\nof the spin, \u001b(=\";#),^Iis a 2\u00022 identity matrix, \u001ba\n(a=x;y;z ) are the Pauli matrices, \u0018k=~2k2=2m\u0003\u0000\u0016\nis the kinetic energy measured from the chemical poten-\ntial, andm\u0003is an e\u000bective mass. Hereafter, we assume\nthat the Fermi energy is much larger than the other en-\nergy scales such as the spin-orbit interactions, the tem-\nperature, and the ferromagnetic resonance energy. Then,\nthe low-energy part of the spin susceptibility depends on\nthe chemical potential \u0016and the e\u000bective mass m\u0003only\nthrough the density of states at the Fermi energy, D(\u000fF).\nThe spin-orbit interaction is described by the e\u000bective3\nZeeman \feld,\nhe\u000b(k) =jkj(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0)\n'kF(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0); (3)\nwhere\u000band\frespectively denote the amplitudes of the\nRashba- and Dresselhaus-type spin-orbit interactions and\nthe electron wave number is expressed by polar coordi-\nnates as (kx;ky) = (jkjcos';jkjsin'). In the second\nequation of Eq. (3), we have approximated jkjwith the\nFermi wave number kFassuming that the spin-orbit in-\nteraction energies, kF\u000bandkF\f, are much smaller than\nthe Fermi energy58. When only the Rashba spin-orbit in-\nteraction exists ( \f= 0), the energy band is spin-splitted\nas shown in Fig. 2 (a). The spin polarization of each band\ndepends on the azimuth angle 'because it is determined\nby the e\u000bective Zeeman \feld he\u000bwhich is a function of\n'as seen in Eq. (3). In the special case of \f=\u000b= 1, the\nspin polarization always becomes parallel to the direction\nof the azimuth angle 3 \u0019=4 in thexyplane as shown in\nFig. 2 (b). Then, the spin component in this direction is\nconserved. This observation indicates that e\u000bect of the\nspin conservation may become important when the two\nspin-orbit interactions compete ( \u000b'\f).\nThe Hamiltonian of the impurity potential is given as\nHimp=uX\ni2impX\n\u001b\ty\n\u001b(ri)\t\u001b(ri); (4)\nwhere \t\u001b(r) =A\u00001=2P\nkck\u001beik\u0001r,Ais the area of the\njunction,uis the strength of the impurity potential, and\nriis the position of the impurity site.\nThe \fnite-temperature Green's function for the con-\nduction electrons is de\fned by a 2 \u00022 matrix ^g(k;i!m)\nwhose elements are\ng\u001b\u001b0(k;i!m) =Z~\f\n0d\u001cei!m\u001cg\u001b\u001b0(k;\u001c); (5)\ng\u001b\u001b0(k;\u001c) =\u0000~\u00001hck\u001b(\u001c)cy\nk\u001b0i; (6)\nwhereck\u001b(\u001c) =eHNM\u001c=~ck\u001be\u0000HNM\u001c=~,HNM=Hkin+\nHimp,!m=\u0019(2m+ 1)=~\fis the fermionic Matsubara\nfrequency, and \fis the inverse temperature. By em-\nploying the Born approximation, the \fnite-temperature\nGreen's function can be expressed as\n^g(k;i!m) =(i~!m\u0000\u0018k+i\u0000sgn(!m)=2)^I\u0000he\u000b\u0001\u001bQ\n\u0017=\u0006(i~!m\u0000E\u0017\nk+i\u0000sgn(!m)=2);\n(7)\nwhereE\u0006\nk=\u0018k\u0006jhe\u000b(')jis the spin-dependent electron\ndispersion,\n\u0000 = 2\u0019niu2D(\u000fF) (8)\nis level broadening, and niis the impurity concentration\n(see Appendix A and Ref. 30 for detailed derivation).\nAs already mentioned, the case of \f=\u000b = 1 is special\nbecause the spin component parallel to the direction ofthe azimuth angle 3 \u0019=4 in thexyplane is conserved (see\nFig. 2 (b)). By de\fning the spin component in this di-\nrection as\ns3\u0019=4\ntot\u00111\n2X\nk(cy\nk+ck+\u0000cy\nk\u0000ck\u0000); (9)\n\u0012\nck+\nck\u0000\u0013\n=\u0012\n1=p\n2e\u0000i3\u0019=4=p\n2\n\u0000ei3\u0019=4=p\n2 1=p\n2\u0013\u0012\nck\"\nck#\u0013\n;(10)\nwe can prove [ Hkin+Himp;s3\u0019=4\ntot] = 0. When the value of\n\f=\u000b is slightly shifted from 1, the spin conservation law\nis broken slightly and this leads to a slow spin relaxation.\nAs will be discussed in Secs. IV and V, this slow spin re-\nlaxation, which is a remnant of the spin conservation at\n\f=\u000b = 1, strongly a\u000bects the spin injection from the FI\ninto the 2DEG. To describe this feature, we need to con-\nsider the vertex correction to take the conservation law\ninto account in our calculation as explained in Sec. III.\nB. Ferromagnetic insulator\nWe consider the quantum Heisenberg model for the FI\nand employ the spin-wave approximation assuming that\nthe temperature is much lower than the magnetic tran-\nsition temperature and the magnitude of the localized\nspins,S0, is su\u000eciently large. We write the expectation\nvalue of the localized spins in the FI as hSi, whose direc-\ntion is (cos \u0012;sin\u0012;0) as shown in the Fig. 1 (b). Using\nthe Holstein-Primakov transformation, the Hamiltonian\nin the spin-wave approximation is obtained as\nHFI=X\nk~!kby\nkbk; (11)\nwherebkis the magnon annihilation operator with wave\nnumberk,~!k=Dk2+~\rhdcis the energy dispersion of\na magnon,Dis the spin sti\u000bness, \ris the gyromagnetic\nratio, andhdcis the externally applied DC magnetic \feld.\nWe note that the external DC magnetic \feld controls the\ndirection of the ordered spins. We introduce new coor-\ndinates (x0;y0;z0) \fxed on the ordered spins by rotating\nthe original coordinates ( x;y;z ) as shown in Fig. 1 (b).\nThen, the magnon annihilation operator is related to the\nspin ladder operator by the Holstein-Primakov transfor-\nmation asSx0+\nk\u0011Sy0\nk+iSz0\nk= (2S0)1=2bk. The spin\ncorrelation function is de\fned as\nG0(k;i!n) =Z~\f\n0d\u001cei!n\u001cG0(k;\u001c); (12)\nG0(k;\u001c) =\u00001\n~hSx0+\nk(\u001c)Sx0\u0000\nk(0)i; (13)\nwhere!n= 2n\u0019=~\fis the bosonic Matsubara fre-\nquency. The spin correlation function is calculated from\nthe Hamiltonian (11), as\nG0(k;i!n) =2S0=~\ni!n\u0000!k\u0000\u000bGj!nj; (14)4\nwhere\u000bG>0 is a phenomenological dimensionless pa-\nrameter that describes the strength of the Gilbert damp-\ning in the bulk FI.\nC. E\u000bect of the FI/2DEG interface\nThe coupling between the FI and 2DEG can be ac-\ncounted for by the Hamiltonian,\nHint=X\nk(TkSx0+\nksx0\u0000\nk+T\u0003\nksx0+\nkSx0\u0000\nk); (15)\nwhereTkis an exchange interaction at a clean interface,\nfor which the momentum of spin excitation is conserved.\nThe spin ladder operators for conduction electrons, sx0\u0006\nk,\nare obtained using a coordinate rotation as30\nsx0\u0006\nk=1\n2X\n\u001b;\u001b0X\nk0cy\nk0\u001b(^\u001bx0\u0006)\u001b\u001b0ck0\u0006k\u001b0; (16)\n^\u001bx0\u0006=\u0000sin\u0012\u001bx+ cos\u0012\u001by\u0006i\u001bz; (17)\nwhere ^\u001bx0\u0006\u0011^\u001by0\u0006i^\u001bz0and\n0\n@^\u001bx0\n^\u001by0\n^\u001bz01\nA=0\n@cos\u0012sin\u00120\n\u0000sin\u0012cos\u00120\n0 0 11\nA0\n@\u001bx\n\u001by\n\u001bz1\nA:\nAssuming that the interfacial exchange interaction is\nmuch smaller than the spin-orbit interactions, kF\u000band\nkF\f59,60, we perform a second-order perturbation theory\nwith respect to the interfacial exchange interaction Hint.\nAccordingly, the spin correlation function of the FI is\ncalculated as\nG(k;i!n) =1\n(G0(k;i!n))\u00001\u0000\u0006(k;i!n); (18)\n\u0006(k;i!n) =jTkj2A\u001f(k;i!n); (19)\nwhere \u0006(k;i!n) is the self-energy due to the interfacial\nexchange coupling and \u001f(k;i!n) is the spin susceptibility\nfor conduction electrons per unit area, de\fned as\n\u001f(k;i!n) =Z~\f\n0d\u001cei!n\u001c\u001f(k;\u001c); (20)\n\u001f(k;\u001c) =\u00001\n~Ahsx0+\nk(\u001c)sx0\u0000\nk(0)i; (21)\nwheresx0\u0006\nk(\u001c) =eHNM\u001c=~sx0\u0006\nke\u0000HNM\u001c=~. Within the\nsecond-order perturbation, we only need to calculate the\nspin susceptibility for pure 2DEG without considering\nthe junction because the interfacial coupling is already\ntaken into account in the prefactor of the self-energy in\nEq. (19). The uniform component of the retarded spin\ncorrelation function is obtained by analytic continuation\n(a)(b)\nFIG. 3. Feynman diagrams of (a) the uniform spin susceptibil-\nity and (b) the Bethe-Salpeter equation for the ladder-type\nvertex function derived from the Born approximation. The\ncross with two dashed lines indicates interaction between an\nelectron and an impurity.\ni!n!!+i\u000e, as\nGR(0;!) =2S0=~\n!\u0000(!0+\u000e!0) +i(\u000bG+\u000e\u000bG)!;(22)\n\u000e!0\n!0'2S0jT0j2A\n~!0Re\u001fR(0;!0); (23)\n\u000e\u000bG'\u00002S0jT0j2A\n~!0Im\u001fR(0;!0); (24)\nwhere the superscript Rindicates the retarded compo-\nnent,!0=!k=0(=\rhdc) is the FMR frequency, and\n\u000e!0and\u000e\u000bGare respectively the changes in the FMR\nfrequency and Gilbert damping due to the FI/2DEG in-\nterface. We note that in contrast with the bulk Gilbert\ndamping\u000bG, the increase of the Gilbert damping, \u000e\u000bG,\ncan be related directly to the spin susceptibility of 2DEG\nas shown by Eq. (24). In fact, measurement of \u000e\u000bG\nhas been utilized as a qualitative indicator of spin cur-\nrent through a junction61,62. In Eqs. (23) and (24), we\nmade an approximation by replacing !with the FMR\nfrequency!0by assuming that the FMR peak is su\u000e-\nciently sharp ( \u000bG+\u000e\u000bG\u001c1). Thus, both the FMR\nfrequency shift and the modulation of the Gilbert damp-\ning are determined by the uniform spin susceptibility of\nthe conduction electrons, \u001f(0;!). In what follows, we\ninclude the vertex correction for calculation of \u001f(0;!),\nwhich was not taken into account in our previous work30.\nIII. VERTEX CORRECTION\nWe calculate the spin susceptibility in the ladder\napproximation42,43that obeys the Ward-Takahashi re-\nlation with the self-energy in the Born approximation57.\nThe Feynman diagrams for the corresponding spin sus-\nceptibility and the Bethe-Salpeter equation for the vertex\nfunction are shown in Figs. 3 (a) and 3 (b), respectively.5\nThe spin susceptibility of 2DEG is written as\n\u001f(0;i!n) =1\n4\fAX\nk;i!mTrh\n^g(k;i!m)^\u0000(k;i!m;i!n)\n^g(k;i!m+i!n)^\u001bx0\u0000i\n; (25)\nwhere the vertex function ^\u0000(k;i!m;i!n) is a 2\u00022 matrix\nwhose components are determined by the Bethe-Salpeter\nequation [see Fig. 3 (b)],\n\u0000\u001b0\u001b(k;i!m;i!n)\n= (^\u001bx0+)\u001b0\u001b+u2ni\nAX\nqX\n\u001b1\u001b2g\u001b0\u001b2(q;i!m)\n\u0002\u0000\u001b2\u001b1(q;i!m;i!n)g\u001b1\u001b(q;i!m+i!n):(26)\nSince the right-hand side of this equation is indepen-\ndent ofk, the vertex function can simply be described\nas^\u0000(i!m;i!n). We express the vertex function with the\nPauli matrices as\n^\u0000(i!m;i!n)\u0011E^I+X^\u001bx0+Y^\u001by0+Z^\u001bz0; (27)\nwhereE,X,Y, andZwill be determined self-\nconsistently later. The Green's function for the conduc-\ntion electrons can be rewritten as\n^g(q;i!m) =A^I+B^\u001bx0+C^\u001by0\nD; (28)\nA(i!m) =i~!m\u0000\u0018q+i\u0000\n2sgn(!m); (29)\nB=\u0000he\u000bcos(\u001e\u0000\u0012); (30)\nC=\u0000he\u000bsin(\u001e\u0000\u0012); (31)\nD(i!m) =Y\n\u0017=\u0006[i~!m\u0000E\u0017\nq+i\u0000\n2sgn(!m)]; (32)\nwhere\u001eis the azimuth angle by which the e\u000bective Zee-\nman \feld is written as he\u000b= (he\u000bcos\u001e;he\u000bsin\u001e;0).\nThishe\u000bis written as he\u000b'kFp\n\u000b2+\f2+ 2\u000b\fsin 2'\nusing the Fermi wave number kF. By substituting\nEqs. (27) and (28) into the second term of Eq. (26) and\nby the algebra of Pauli matrices, we obtain\nu2ni\nAX\nq^g(q;i!m)^\u0000(q;i!m;i!n)^g(q;i!m+i!n)\n=E0^I+X0^\u001bx0+Y0^\u001by0+Z0^\u001bz0; (33)\nwhere\n0\nB@E0\nX0\nY0\nZ01\nCA=0\nB@\u00030+ \u00031 0 0 0\n0 \u0003 0+ \u00032 \u00033 0\n0 \u0003 3 \u00030\u0000\u00032 0\n0 0 0 \u0003 0\u0000\u000311\nCA0\nB@E\nX\nY\nZ1\nCA;\n(34)and \u0003j(i!m;i!n) (j= 0;1;2;3) are expressed as\n\u00030(i!m;i!n) =u2ni\nAX\nqAA0\nDD0; (35)\n\u00031(i!m;i!n) =u2ni\nAX\nqh2\ne\u000b\nDD0; (36)\n\u00032(i!m;i!n) =u2ni\nAX\nqh2\ne\u000bcos 2(\u001e\u0000\u0012)\nDD0; (37)\n\u00033(i!m;i!n) =u2ni\nAX\nqh2\ne\u000bsin 2(\u001e\u0000\u0012)\nDD0; (38)\nusing the abbreviated symbols, A=A(i!m),A0=\nA(i!m+i!n),D=D(i!m), andD0=D(i!m+i!n).\nHere, we have used the fact that the contributions of the\n\frst-order terms of BandCbecome zero after replacing\nthe sum with the integral with respect to qand perform-\ning the azimuth integration. We can solve for E,X,\nY, andZby combining Eq. (34) and the Bethe-Salpeter\nequation (26), which we rewrite as\nE^I+X^\u001bx0+Y^\u001by0+Z^\u001bz0\n= ^\u001bx0++E0^I+X0^\u001bx0+Y0^\u001by0+Z0^\u001bz0; (39)\nwith ^\u001bx0+= ^\u001by0+i^\u001bz0. The solution is\nE= 0; (40)\nX=\u00033\n(1\u0000\u00030)2\u0000\u00032\n2\u0000\u00032\n3; (41)\nY=1\u0000\u00030\u0000\u00032\n(1\u0000\u00030)2\u0000\u00032\n2\u0000\u00032\n3; (42)\nZ=i\n1\u0000\u00030+ \u00031: (43)\nBy replacing the sum with an integral as \u0018\u0011\u0018q,\n1\nAX\nq(\u0001\u0001\u0001)'D(\u000fF)Z1\n\u00001d\u0018Z2\u0019\n0d'\n2\u0019(\u0001\u0001\u0001); (44)\nEqs. (35)-(38) can be rewritten as\n\u0003j(i!m;i!n) =\u0012(\u0000!m)\u0012(!m+!n)~\u0003j(i!n); (45)\n~\u0003j(i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019\n\u0002X\n\u0017;\u00170=\u0006fj(\u0017;\u00170;')\ni~!n+ (\u0017\u0000\u00170)he\u000b(') +i\u0000;\n(46)\nwhere we have used Eq. (8), \u0012(x) is a step function, and\nf0(\u0017;\u00170;') = 1; (47)\nf1(\u0017;\u00170;') =\u0017\u00170; (48)\nf2(\u0017;\u00170;') =\u0017\u00170cos 2(\u001e(')\u0000\u0012); (49)\nf3(\u0017;\u00170;') =\u0017\u00170sin 2(\u001e(')\u0000\u0012): (50)6\nFor detailed derivation, see Appendix B. Substituting the\nGreen's function and the vertex function into Eq. (25),\nwe obtain\n\u001f(0;i!n) =1\n4\fAX\nk;i!m2\nDD0h\n2BCX\n+ (AA0\u0000B2+C2)Y\u0000i(AA0\u0000B2\u0000C2)Zi\n:(51)\nBy summing over kand!mand by analytical continu-\nation,i!n!!+i\u000e, the retarded spin susceptibility is\nobtained as63\n\u001fR(0;!)\n=D(\u000fF)~!\n2i\u0000\u0014~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n+~\u0003R\n0\u0000~\u0003R\n1\n1\u0000~\u0003R\n0+~\u0003R\n1\u0015\n\u0000D(\u000fF); (52)\nwhere\n~\u0003R\nj=~\u0003R\nj(!) =~\u0003j(i!n!!+i\u000e)\n=i\u0000\n4\u00010Z2\u0019\n0d'\n2\u0019\n\u0002X\n\u0017\u00170fj(\u0017;\u00170;')\n~!=\u00010+ (\u0017\u0000\u00170)he\u000b=\u00010+i\u0000=\u00010:(53)\nA detailed derivation is given in Appendix C. Here, we\nhave introduced a unit of energy, \u0001 0=kF\u000b, for the con-\nvenience of making the physical quantities dimensionless.\nUsing Eqs. (23) and (24), we \fnally obtain the shift in\nthe FMR frequency and the modulation of the Gilbert\ndamping as\n\u000e!0\n!0=\u000bG;0ReF(!0); (54)\n\u000e\u000bG=\u0000\u000bG;0ImF(!0); (55)\nF(!) =\u00010\n2\u0019i\u0000\u0014~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n+~\u0003R\n0\u0000~\u0003R\n1\n1\u0000~\u0003R\n0+~\u0003R\n1\u0015\n\u0000\u00010\n\u0019~!; (56)\nwhere\u000bG;0= 2\u0019S0jT0j2AD(\u000fF)=\u00010is a dimensionless\nparameter that describes the coupling strength at the\ninterface. This is our main result.\nThe spin susceptibility without the vertex correction\ncan be obtained by taking the \frst-order term with re-spect to ~\u0003R\nj:\n\u001fR(0;!)'~!D(\u000fF)\n2i\u0000\u0002\n2~\u0003R\n0\u0000~\u0003R\n1\u0000~\u0003R\n2\u0003\n\u0000D(\u000fF)\n=~!D(\u000fF)Zd'\n2\u0019h1\n~!+i\u00001\u0000cos2(\u001e(')\u0000\u0012)\n2\n+1\n~!\u00002he\u000b(') +i\u00001 + cos2(\u001e(')\u0000\u0012)\n4\n+1\n~!+ 2he\u000b(') +i\u00001 + cos2(\u001e(')\u0000\u0012)\n4i\n\u0000D(\u000fF):\n(57)\nThe imaginary part of \u001fR(0;!) reproduces the result of\nRef. 30. Using this expression, the shift in the FMR\nfrequency and the modulation of the Gilbert damping\nwithout the vertex correction are obtained as\n\u000e!nv\n0\n!0=\u000bG;0ReFnv(!0); (58)\n\u000e\u000bnv\nG=\u0000\u000bG;0ImFnv(!0); (59)\nFnv(!) =\u00010\n2\u0019i\u0000\u0014\n2~\u0003R\n0\u0000~\u0003R\n1\u0000~\u0003R\n2\u0015\n\u0000\u00010\n\u0019~!; (60)\nIV. MODULATION OF THE GILBERT\nDAMPING\nFirst, we show the result for the modulation of the\nGilbert damping, \u000e\u000bG, for\f=\u000b = 0, 1, and 3 and dis-\ncuss the e\u000bect of the vertex correction by comparing it\nwith the result without the vertex correction in Sec. IV A.\nNext, we discuss the strong enhancement of the Gilbert\ndamping near \f=\u000b= 1 in Sec. IV B.\nA. E\u000bect of vertex corrections\nFirst, let us discuss the case of \f=\u000b = 0, i.e., the\ncase when only the Rashba spin-orbit interaction exists64.\nFigure 4 (a) shows the e\u000bective Zeeman \feld he\u000balong\nthe Fermi surface. Figures 4 (b) and 4 (c) show the\nmodulations of the Gilbert damping without and with\nthe vertex correction. The horizontal axes of Figs. 4 (b)\nand 4 (c) denote the resonant frequency !0=\rhdcin\nthe FMR experiment. Note that the modulation of the\nGilbert damping, \u000e\u000bG, is independent of \u0012, i.e., the az-\nimuth angle ofhSi. The four curves in Figs. 4 (b) and\n4 (c) correspond to \u0000 =\u00010= 0:1, 0:2, 0:5, and 1:065. We\n\fnd that these two graphs have a common qualitative\nfeature; the modulation of the Gilbert damping has two\npeaks at!0= 0 and!0= 2\u0001 0and their widths become\nlarger as \u0000 increases. The peak at !0= 0 corresponds\nto elastic spin-\ripping of conduction electrons induced\nby the transverse magnetic \feld via the exchange bias\nof the FI, while the peak at ~!0= 2\u0001 0is induced by\nspin excitation of conduction electrons due to magnon7\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\n\u0001\nWithout vertex corrections With vertex corrections With vertex corrections\nFIG. 4. (Left panels) E\u000bective Zeeman \feld he\u000bon the Fermi surface. (Middle panels) Modulation of the Gilbert damping,\n\u000e\u000bnv\nG, without vertex correction. (Right panels) Modulation of the Gilbert damping with vertex correction, \u000e\u000bG. In the middle\nand right panels, the modulation of the Gilbert damping is plotted as a function of the FMR frequency, !0=\rhdc. The\nspin-orbit interactions are as follows. (a), (b), (c): \f=\u000b= 0. (d), (e), (f): \f=\u000b= 1. (g), (h), (i): \f=\u000b= 3. We note that (b),\n(e), (h) are essentially the same result as Ref. 30.\nabsorption30. In the case of \f=\u000b = 0, the vertex cor-\nrection changes the modulation of the Gilbert damping\nmoderately [compare Figs. 4 (c) with 4 (b)]. The widths\nof the two peaks at !0= 0 and!0= 2\u0001 0become nar-\nrower when the vertex correction is taken into account\n(see Appendix D for the analytic expressions).\nThe case of \f=\u000b = 1 is special because the e\u000bective\nZeeman \feld he\u000balways points in the direction of ( \u00001;1)\nor (1;\u00001), as shown in Fig. 4 (d). The amplitude of\nhe\u000bdepends on the angle of the wave number of the\nconduction electrons, ',\nhe\u000b(') = 2\u0001 0jsin('+\u0019=4)j; (61)\nand varies in the range of 0 \u00142he\u000b\u00144\u00010. Figures 4 (e)\nand 4 (f) show the modulation of the Gilbert dampingwithout and with the vertex correction for \u0000 =\u00010= 0:5.\nThe \fve curves correspond to \fve di\u000berent angles of hSi,\n\u0012=\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. The most remarkable\nfeature revealed by comparing Figs. 4 (f) with 4 (e) is that\nthe peak at !0= 0 disappears if the vertex correction\nis taken into account (see Appendix E for the analytic\nexpressions). In the subsequent section, we will show\nthat\u000e\u000bG(!0) has a\u000e-function-like singularity at !0= 0\nfor\f=\u000b = 1 due to the spin conservation law along the\ndirection ofhe\u000b.\nIn the case of \f=\u000b = 3, the direction of the e\u000bec-\ntive Zeeman \feld he\u000bvaries along the Fermi surface\n[Fig. 4 (g)]. Figures 4 (h) and 4 (i) show the modula-\ntion of the Gilbert damping without and with the ver-\ntex correction for \u0000 =\u00010= 0:5. For\f=\u000b = 3, a peak8\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\n\u0001\nWithout vertex corrections With vertex corrections With vertex corrections\nFIG. 5. Modulation of the Gilbert damping calculated for \f=\u000b= 1:1 (a) without the vertex correction and (b) with the vertex\ncorrection. The horizontal axis is the FMR frequency !0and the \fve curves correspond to \fve di\u000berent angles of hSi, i.e.,\n\u0012=\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. (c) Enlarged plot of the modulations of the Gilbert damping as a function of the FMR\nfrequency!0. The angle ofhSiis \fxed as\u0012=\u0019=4 and the three curves correspond to \f=\u000b = 1:03, 1:05, and 1:1. In all the\nplots, we have chosen \u0000 =\u00010= 0:5.\nat!0= 0 appears even when the vertex correction is\ntaken into account. The broad structure in the range of\n4\u00010\u0014~!0\u00148\u00010is caused by the magnon absorption\nprocess where its range re\rects the distribution of the\nspin-splitting energy 2 he\u000balong the Fermi surface. By\ncomparing Figs. 4 (h) and 4 (i), we \fnd that the vertex\ncorrection changes the result only moderately as in the\ncase of\f=\u000b = 0; the peak structure at !0= 0 becomes\nsharper when the vertex correction is taken into account\nwhile the broad structure is slightly enhanced.\nB. Strong enhancement of the Gilbert damping\nHere, we examine the strong enhancement of the\nGilbert damping for \f=\u000b'1. As explained in Sec. II A,\nthe spin component in the direction of the azimuth angle\n3\u0019=4 in thexyplane is exactly conserved at \f=\u000b = 1\n[see also Fig. 4 (d)]. When the value of \f=\u000b is shifted\nslightly from 1, the spin conservation law is broken but\nthe spin relaxation becomes remarkably slow. To see this\ne\u000bect, we show the modulation of the Gilbert damping\nwithout and with the vertex correction for \f=\u000b = 1:1\nin Figs. 5 (a) and 5 (b), respectively. The \fve curves\ncorrespond to \fve di\u000berent azimuth angles of hSi, and\nthe energy broadening is set as \u0000 =\u00010= 0:5. Figs. 5 (a)\nand 5 (b) indicate that the Gilbert damping is strongly\nenhanced at !0= 0 only when the vertex correction is\ntaken into account. This is the main result of our work.\nFigure 5 (c) plots the modulation of the Gilbert damp-\ning with the vertex correction for \u0000 =\u00010= 0:5 and\n\u0012=\u0019=4, the latter of which corresponds to the case of\nthe strongest enhancement at !0= 0. The three curves\ncorrespond to \f=\u000b= 1:03, 1:05, and 1:1. As the ratio of\n\f=\u000bapproaches 1, the peak height at !0= 0 gets larger.\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\nWithout vertex corrections With vertex corrections With vertex corrections\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001FIG. 6. Modulation of the Gilbert damping as a function\nof\f=\u000b. The \fve curves correspond to ~!0=\u00010= 0;0:005,\n0:01, 0:02, and 0:05. We have taken the vertex correction into\naccount and have chosen \u0000 =\u00010= 0:5. The inset illustrates\nmaximum values of the modulation of the Gilbert damping,\n\u000e\u000bG;max, in varying \f=\u000b for a \fxed value of ~!0=\u00010.\nFor\f=\u000b'1,\u000e\u000bGis calculated approximately as\n\u000e\u000bG\n\u000bG;0'\u00010\n2\u0019\u0000s\n(~!0)2+ \u00002ssin2\u0010\n\u0012+\u0019\n4\u0011\n; (62)\n\u0000s\u00112\n\u0000Z2\u0019\n0d'\n2\u0019(hx+hy)2\n1 + (2he\u000b=\u0000)2; (63)\nwhere \u0000sgives the peak width in Figs. 5 (b) and 5 (c) (see\nAppendix F for a detailed derivation). For \f=\u000b= 1 +\u000e\n(\u000e\u001c1), \u0000sis proportional to \u000e2and approaches zero in\nthe limit of \u000e!0. This indicates that \u0000 scorresponds\nto the spin relaxation rate due to a small breakdown of\nthe spin conservation law away from the special point of\n\f=\u000b = 1. Note that the peak height of \u000e\u000bGat!0= 0\ndiverges at \f=\u000b = 1. This indicates that for \f=\u000b =\n1,\u000e\u000bG(!0) has a\u000e-function-like singularity at !0= 0,\nwhich is not drawn in Fig. 4 (f).9\nFIG. 7. (Upper panels) Modulations of the Gilbert damping, \u000e\u000bG=\u000bG;0for (a)\f=\u000b= 0, (b)\f=\u000b= 1, and (c) \f=\u000b= 3. (Lower\npanels) Shifts in the FMR frequency, \u000e!0=(\u000bG;0!0), for (d)\f=\u000b = 0, (e)\f=\u000b = 1, and (f) \f=\u000b = 3. The horizontal axes are\nthe FMR frequency, !0=\rhdc, while the vertical axes show the azimuth angle of the spontaneous spin polarization, \u0012, in the\nFI. In all the plots, we have considered vertex corrections and have chosen \u0000 =\u00010= 0:5. In (a), (c), and (e) there are regions\nin which the values exceed the upper limits of the color bar located in the right side of each plot; the maximum value is about\n0:45 in (a), 0 :65 in (c), and about 10 in (e) (see also Fig. 8). In addition, (b) cannot express a \u000e-function-like singularity at\n!0= 0 (see the main text).\nFigure 6 plots the modulation of the Gilbert damping\nfor \u0000=\u00010= 0:5 and\u0012=\u0019=4 as a function of \f=\u000b. The\n\fve curves correspond to ~!0=\u00010= 0;0:005;0:01;0:02,\nand 0:05, respectively. This \fgure indicates that when\nwe \fx the resonant frequency !0and vary the ratio of\n\f=\u000b, the Gilbert damping is strongly enhanced when \f=\u000b\nis slightly smaller or larger than 1. We expect that this\nenhancement of the Gilbert damping is strong enough to\nbe observed experimentally. We note that \u000e\u000bG=\u000bG;0ap-\nproaches 0:378 (0:318) for\f=\u000b!0 (\f=\u000b!1 ). The\ninset in Fig. 6 plots maximum values of \u000e\u000bG=\u000bG;0when\n\f=\u000bis varied for a \fxed value of ~!0=\u00010. In other words,\nthe vertical axis of the inset corresponds to the peak\nheight in the main panel for each value of ~!0=\u00010. We\n\fnd that the maximum value of \u000e\u000bG=\u000bG;0diverges as !0\napproaches zero.\nV. SHIFT IN THE FMR FREQUENCY\nNext, we discuss the shift in the FMR frequency when\nthe vertex correction is taken into account. The den-\nsity plots in Figs. 7 (a), 7 (b), and 7 (c) for \f=\u000b= 0, 1,\nand 3 summarize the modulation of the Gilbert damping,\n\u000e\u000bG. These plots have the same features as in Figs. 4 (c),\n4 (f), and 4 (i). Figures. 7 (d), 7 (e), and 7 (f) plot the\nshift in the FMR frequency \u000e!0=!0with density plots\nfor\f=\u000b = 0, 1, and 3. By comparing Figs. 7 (a), 7 (b),\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\nWithout vertex corrections With vertex corrections With vertex corrections\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001FIG. 8. Shift in FMR frequency, \u000e!0=(\u000bG;0!0), as a func-\ntion of the resonance frequency !0for\f=\u000b = 1:1. The in-\nset shows the same quantities in the low-frequency range of\n0\u0014~!0=\u00010\u00140:05 with a larger scale on the vertical axis.\nWe have taken the vertex correction into account and have\nchosen \u0000=\u00010= 0:5.\nand 7 (c) with 7 (d), 7 (e), and 7 (f), we \fnd that some\nof the qualitative features of the FMR frequency shift\nare common to those of the modulation of the Gilbert\ndamping,\u000e\u000bG; (i) they depend on \u0012for\f=\u000b > 0, while\nthey do not depend on \u0012for\f=\u000b = 0, (ii) the structure10\nat!0= 0 due to elastic spin-\ripping appears, and (iii)\nthe structure within a \fnite range of frequencies due to\nmagnon absorption appears. We can also see a few dif-\nferences between \u000e\u000bGand\u000e!0=!0. For example, \u000e!0=!0\nhas a dip-and-peak structure at ~!0=\u00010= 2 where\u000e\u000bG\nhas only a peak. Related to this feature, \u000e!0=!0has a\ntail that decays more slowly than that for \u000e\u000bG. The most\nremarkable di\u000berence is that \u000e!0=!0diverges at !0= 0\nfor\f=\u000b = 1 except for \u0012= 3\u0019=4;7\u0019=4, re\recting the \u000e-\nfunction-like singularity of \u000e\u000bGat!0= 0. These features\nare reasonable because \u000e!0=!0and\u000e\u000bG, which are de-\ntermined by the real and imaginary parts of the retarded\nspin susceptibility, are related to each other through the\nKramers-Kronig conversion.\nThe main panel of Fig. 8 shows the frequency shift\n\u000e!0=!0for\f=\u000b = 1:1 as a function of the resonant\nfrequency!0. The \fve curves correspond to \u0012=\n\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. Although the frequency\nshift appears to diverge in the limit of !0!0 in the\nscale of the main panel, it actually grows to a \fnite value\nand then goes to zero as !0approaches zero (see the inset\nof Fig. 8). For \f=\u000b= 1 +\u000e(\u000e\u001c1), the frequency shift\nis calculated approximately as\n\u000e!0\n\u000bG;0!0'\u00010\n2\u0019~!0\n(~!0)2+ \u00002ssin2\u0010\n\u0012+\u0019\n4\u0011\n; (64)\nwhere \u0000sis the spin relaxation rate de\fned in Eq. (63)\n(see Appendix F for the detailed derivation). We expect\nthat this strong enhancement of the frequency shift near\n\f=\u000b= 1 can be observed experimentally.\nVI. SUMMARY\nWe theoretically investigated spin pumping into a two-\ndimensional electron gas (2DEG) with a textured e\u000bec-\ntive Zeeman \feld caused by Rashba- and Dresselhaus-\ntype spin-orbit interactions. We expressed the change\nin the peak position and the linewidth in a ferromag-\nnetic resonance (FMR) experiment that is induced by\nthe 2DEG within a second-order perturbation with re-\nspect to the interfacial exchange coupling by taking the\nvertex correction into account. The FMR frequency\nand linewidth are modulated by elastic spin-\ripping or\nmagnon absorption. We found that, for almost all of the\nparameters, the vertex correction modi\fes the modula-\ntion of the Gilbert damping only moderately and does not\nchange the qualitative features obtained in our previous\npaper30. However, we found that the Gilbert damping at\nlow frequencies, which is caused by elastic spin-\ripping,\nis strongly enhanced when the Rashba- and Dresselhaus-\ntype spin-orbit interactions are chosen to be almost equal\nbut slightly di\u000berent. Even in this situation, the Gilbert\ndamping at high frequencies, which is caused by magnon\nabsorption, shows small modi\fcation. This strong en-\nhancement of the Gilbert damping at low frequencies ap-\npears only when the vertex correction is taken into ac-\ncount and is considered to originate from the slow spinrelaxation related to the spin conservation law that holds\nwhen the two spin-orbit interactions completely match.\nA similar enhancement was found for the frequency shift\nof the FMR due to elastic spin-\ripping. We expect that\nthis remarkable enhancement can be observed experi-\nmentally.\nOur work provides a theoretical foundation for spin\npumping into two-dimensional electrons with a spin-\ntextured Zeeman \feld on the Fermi surface. Although\nwe have treated a speci\fc model for two-dimensional\nelectron systems with both the Rashba and Dresselhaus\nspin-orbit interactions, our formulation and results will\nbe helpful for describing spin pumping into general two-\ndimensional electron systems such as surface/interface\nstates66{68and atomic layer compounds69,70.\nACKNOWLEDGEMENTS\nThe authors thank Y. Suzuki, Y. Kato, and A. Shi-\ntade for helpful discussion. T. K. acknowledges sup-\nport from the Japan Society for the Promotion of Sci-\nence (JSPS KAKENHI Grant No. JP20K03831). M. M.\nis \fnancially supported by a Grant-in-Aid for Scienti\fc\nResearch B (Grants No. JP20H01863, No. JP21H04565,\nand No. JP21H01800) from MEXT, Japan. M. Y. is sup-\nported by JST SPRING (Grant No. JPMJSP2108).\nAppendix A: Calculation of Green's function\nIn our work, Green's function of conduction electrons\nis calculated by taking e\u000bect of impurity scattering into\naccount. In general, the \fnite-temperature Green's func-\ntion ^g(k;i!m) after the impurity average is described\nby the Dyson equation with the impurity self-energy\n^\u0000(k;!m) as\n^g(k;i!m) =1\n^g0(k;i!m)\u00001\u0000^\u0000(k;i!m); (A1)\nwhere ^g0(k;i!m)\u00001is Green's function of electrons in the\nabsence of impurities. In our work, we employ the Born\napproximation in which the self-energy is approximated\nby second-order perturbation with respect to an impurity\npotential. In the Born approximation, the self-energy is\ngiven as\n^\u0000(k;i!m) =niu2Zd2k\n(2\u0019)2^g0(k;i!m); (A2)\nwhereniis the impurity concentration. The correspond-\ning Feynman diagram of the Dyson equation is shown in\nFig. 9. By straightforward calculation, Eq. (7) can be\nderived. For a detailed derivation, see Ref. 30.11\nFIG. 9. The Feynman diagram for Green's function within\nthe Born approximation.\nAppendix B: Derivation of Equations. (45)-(50)\nEqs. (35)-(38) can be rewritten with \u0000 = 2 \u0019niu2D(\u000fF)\nas\n\u00030(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017;\u00170I\u0017\u00170; (B1)\n\u00031(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017;\u00170\u0017\u00170I\u0017\u00170; (B2)\n\u00032(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019cos 2('\u0000\u0012)X\n\u0017;\u00170\u0017\u00170I\u0017\u00170;\n(B3)\n\u00033(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019sin 2('\u0000\u0012)X\n\u0017;\u00170\u0017\u00170I\u0017\u00170;\n(B4)\nwhere\nI\u0017\u00170=Z1\n\u00001d\u0018\n2\u0019i1\ni~!m\u0000\u0018\u0000\u0017he\u000b+i(\u0000=2)sgn(!m)\n\u00021\ni~(!m+!n)\u0000\u0018\u0000\u00170he\u000b+i(\u0000=2)sgn(!m+!n):\n(B5)\nWe note that one needs to calculate this integral only for\n!n>0 to obtain the retarded component by analytic\ncontinuation. Then, we can easily prove by the residue\nintegral that I\u0017\u00170= 0 for!m>0 and!m+!n>0\n(!m<0 and!m+!n<0) because both of the two poles\nin the integrand are located only in the upper (lower) half\nof the complex plane of \u0018. For!m<0 and!m+!n>0,\nthe integral is evaluated by the residue integral as\nI\u0017\u00170=1\ni~!n+ (\u0017\u0000\u00170)he\u000b+i\u0000: (B6)\nBy combining these results, Eqs. (45)-(50) can be de-\nrived.\n(a) (b)\nFIG. 10. Schematic picture of the change in the contour in-\ntegral. (a) The original contour. (b) The modi\fed contour.\nAppendix C: Derivation of Eq. (52)\nIn this Appendix, we give a detailed derivation of\nEq. (52) from Eq. (51). First, we modify Eq. (51) as\n\u001f(0;i!n) =1\n8AX\nkX\n\u0017;\u00170\"\n\u0017\u00170sin 2(\u001e\u0000\u0012)I\u0017\u00170;1\n+n\n1\u0000\u0017\u00170cos 2(\u001e\u0000\u0012)o\nI\u0017\u00170;2\n\u0000i(1\u0000\u0017\u00170)I\u0017\u00170;3#\n; (C1)\nwhere\nI\u0017\u00170;j\u00111\n\fX\ni!mXj\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m)\n\u00021\ni~!m+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(!m+!n);(C2)\nand (X1;X2;X3) = (X;Y;Z ). A standard procedure\nbased on the residue integral enables us to express the\nsumI\u0017\u00170;jfor!n>0 as a complex integral on the con-\ntour C shown in Fig. 10 (a). This contour can be modi\fed\ninto a sum of the four contours, C l(l= 1;2;3;4), shown\nin Fig. 10 (b). Accordingly, I\u0017\u00170;jis written as\nI\u0017\u00170;j=4X\nl=1ICl\n\u0017\u00170;j; (C3)\nICl\n\u0017\u00170;j=\u0000Z\nCldz\n2\u0019if(z)Xj(z;i!n)\nz\u0000E\u0017\nk+i\u0000=2 sgn(Imz)\n\u00021\nz+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(Imz+!n);(C4)\nwheref(z) = 1=(e\fz+ 1) is the Fermi distribution func-\ntion. The sum of the contributions from the two contours,12\nC2and C 3, is calculated as\nIC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j\n=\u0000~Xj(i!n)ZdE\n2\u0019if(E)\n\u0002\"\n\u00001\nE\u0000E\u0017\nk\u0000i\u0000=21\nE+i~!n\u0000E\u00170\nk+i\u0000=2\n+1\nE\u0000i~!n\u0000E\u0017\nk\u0000i\u0000=21\nE\u0000E\u00170\nk+i\u0000=2#\n:(C5)\nHere, we have used the fact that Xj(z;i!n) is indepen-\ndent ofzfor 0<Imz < !nfrom Eq. (45) and have\nde\fned its value as ~Xj(i!n) (j= 1;2;3). From Eqs. (41)-\n(43), ~Xj(i!n) are calculated as\n~X1(i!n) =~\u00033(i!n)\n(1\u0000~\u00030(i!n))2\u0000~\u00032(i!n)2\u0000~\u00033(i!n)2;(C6)\n~X2(i!n) =1\u0000~\u00030(i!n)\u0000~\u00032(i!n)\n(1\u0000~\u00030(i!n))2\u0000~\u00032(i!n)2\u0000~\u00033(i!n)2;(C7)\n~X3(i!n) =i\n1\u0000~\u00030(i!n) +~\u00031(i!n): (C8)\nBy changing the integral variable to E0=E\u0000E\u0017\nkin the\n\frst term and to E0=\u0000(E\u0000E\u00170\nk) in the second term in\nEq. (C5), we obtain\nIC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j=\u0000~Xj(i!n)ZdE0\n2\u0019i1\nE0\u0000i\u0000=2\n\u0002\"\nf(\u0000E0+E\u00170\nk)\u0000f(E0+E\u0017\nk)\nE0+i~!n+E\u0017\nk\u0000E\u00170\nk+i\u0000=2#\n:(C9)\nUsing formula (44), we replace the sum over kin Eq. (C1)\nby the integral with respect to \u0018and'. We can perform\nthe\u0018-integral by using\n\u0000Z1\n\u00001d\u0018[f(E0+E\u0017\nk)\u0000f(\u0000E0+E\u00170\nk)]\n= 2E0+E\u0017\nk\u0000E\u00170\nk: (C10)\nThen, by performing the E0-integral, we obtain\nZ1\n\u00001d\u0018(IC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j) =i~!n~Xj(i!n)\nE\u0017\nk\u0000E\u00170\nk+i~!n+i\u0000:\n(C11)\nNext, let us consider the contribution from C 1and C 4.\nOn these two contours, Xj(z;i!n) is independent of z\nand its value is de\fned by ~X0\nj(i!n) (j= 1;2;3). Because\n\u0003j(z;i!n) (j= 0;1;2;3) becomes zero for Im z < 0 or\n!n<Imzfrom Eq. (45), ~X0\nj(i!n) are given as\n~X0\n1(i!n) = 0;~X0\n2(i!n) = 1;~X0\n3(i!n) =i:(C12)A similar calculation to that of C 2and C 3yields\nZ1\n\u00001d\u0018(IC1\n\u0017\u00170;j+IC4\n\u0017\u00170;j) =\u0000~X0\nj(i!n): (C13)\nBy substituting these results into Eq. (C1), we obtain\n\u001f(0;i!n) =D(\u000fF)\n8X\n\u0017;\u00170Z2\u0019\n0d'\n2\u0019\"\n\u0017\u00170sin 2(\u001e\u0000\u0012) ~x1(i!n)\n+ [1\u0000\u0017\u00170cos 2(\u001e\u0000\u0012)][\u00001 + ~x2(i!n)]\n\u0000i(1\u0000\u0017\u00170)[\u0000i+ ~x3(i!n)]#\n: (C14)\nwhere\n~xj(i!n) =i~!n~Xj(i!n)\nE\u0017\nk\u0000E\u00170\nk+i~!n+i\u0000: (C15)\nFinally, Eq. (52) is derived by substituting the expres-\nsions for ~Xj(i!n) and by analytic continuation i!n!\n!+i\u000e.\nAppendix D: Analytic Expression for \f=\u000b= 0\nIn this appendix, we derive analytic expressions of the\nmodulation of the Gilbert damping when \f=\u000b= 0, only\nthe Rashba spin-orbit interaction exists, to see the quan-\ntitative e\u000bect of taking the vertex correction into account.\nFor\f=\u000b= 0, the spin-splitting energy 2 he\u000b= 2\u0001 0is con-\nstant along the Fermi surface, and ~\u0003R\nj(!) (j= 0;1;2;3)\nis simpli\fed as\n~\u0003R\n0(!) =i\u0000\n4\u00010X\n\u0017\u001701\n~!=\u00010+ (\u0017\u0000\u00170) +i\u0000=\u00010;(D1)\n~\u0003R\n1(!) =i\u0000\n4\u00010X\n\u0017\u00170\u0017\u00170\n~!=\u00010+ (\u0017\u0000\u00170) +i\u0000=\u00010;(D2)\n~\u0003R\n2(!) =~\u0003R\n3(!) = 0: (D3)\nThen, we obtain the modulation of the Gilbert damping\nwith the vertex corrections,\n\u000e\u000bG\n\u000bG;0'\u00010\n2\u0019\u0000Re\"~\u0003R\n0(!0)\n1\u0000~\u0003R\n0(!0)+~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)\n1\u0000~\u0003R\n0(!0) +~\u0003R\n1(!0)#\n:\n(D4)\nThe modulation of the Gilbert damping without the ver-\ntex correction is obtained by considering only the \frst-\norder term with respect to ~\u0003R\nj(!0),\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n2\u0019\u0000Re\"\n2~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)#\n: (D5)\nWhen \u0000\u001c\u00010, the contribution of \u0017=\u00170is dominant\nfor the peak at !0= 0 and the modulation of the Gilbert13\ndamping can be analytically calculated as\n\u000e\u000bG\n\u000bG;0'\u00010\n4\u0019\u0001\u0000=2\n(~!0)2+ (\u0000=2)2; (D6)\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n4\u0019\u0001\u0000\n(~!0)2+ \u00002: (D7)\nThis indicates that the peak width is halved by taking\nthe vertex correction into account, which is consistent\nwith the results shown in Figs. 4 (b) and 4 (c).\nIn a similar way, we can evaluate the modulation of\nthe Gilbert damping near the peak at !0= 2\u0001 0=~as\n\u000e\u000bG\n\u000bG;0'\u00010\n4\u0019\u0001\u00141\n23\u0000=4\n(~!0\u00002\u00010)2+ (3\u0000=4)2\n+\u0000=2\n(~!0\u00002\u00010)2+ (\u0000=2)2\u0015\n; (D8)\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n4\u0019\u00013\u0000=2\n(~!0\u00002\u00010)2+ \u00002: (D9)\nAs well, for the peak at !0= 2\u0001 0=~, the peak width\nbecomes smaller when the vertex correction is taken into\naccount. This observation is consistent with the results\nshown in Figs. 4 (b) and 4 (c). For a \fnite value of \u0000, a\nsum of Eqs. (D6) and (D8) [Eqs. (D7) and (D9)] gives a\nbetter analytic form which \fts the numerical result with\n(without) the vertex correction. Note that \u000e\u000bGand\u000e\u000bnv\nG\ndepend on the impurity potential strength, u, and im-\npurity concentration, ni, through \u0000 = 2 \u0019niu2D(\u000fF) [see\nEq. (8)]. As shown in Eqs. (D6)-(D9), the peak widths\nof the Lorentzian functions in \u000e\u000bGand\u000e\u000bnv\nGare deter-\nmined by \u0000 [see Figs. 4 (b) and 4 (c)]. It is remarkable\nthat the peak width in \u000e\u000bGis reduced from \u0000 to \u0000 =2 by\ntaking the vertex correction into account. To summarize\nthe e\u000bect of the vertex correction, we show \u000e\u000bG\u0000\u000e\u000bnv\nG\nand\u000e!0\u0000\u000e!nv\n0in Figs. 11 (a) and 11 (d), respectively.\nWe \fnd that the vertex correction modi\fes mainly the\npeak width around ~!0=\u00010= 0 and 2, in consistent with\nthe above analytic expressions.\nFinally, we note that the same analytical expressions\nfor\u000e\u000bGand\u000e\u000bnv\nGcan be obtained for the case of \u000b=\f=\n0, i.e., when only the Dresselhaus spin-orbit interaction\nexists. We also note that for general values of \f=\u000b,\u000e\u000bG\nand\u000e\u000bnv\nGdepend on \u0000 in a more complicated way.\nAppendix E: Analytic Expression for \f=\u000b= 1\nIn this Appendix, we derive analytic expressions of the\nmodulation of the Gilbert damping when \f=\u000b = 1. In\nthis case, the e\u000bective Zeeman \feld is parallel to the\n(\u00001;1;0) direction and its amplitude is given as\nhe\u000b(') = 2\u0001 0jsin('+\u0019=4)j: (E1)Then, ~\u0003R\nj(!) (j= 0;1;2;3) becomes\n~\u0003R\n0(!) =i\u0000\n4\u00010X\n\u0017\u00170J\u0017\u00170 (E2)\n~\u0003R\n1(!) =i\u0000\n4\u00010X\n\u0017\u00170\u0017\u00170J\u0017\u00170; (E3)\n~\u0003R\n2(!) =\u0000sin 2\u0012~\u0003R\n1(!); (E4)\n~\u0003R\n3(!) =\u0000cos 2\u0012~\u0003R\n1(!) (E5)\nwhere\nJ\u0017\u00170(!)\u0011Z2\u0019\n0d'\n2\u0019\u00010\n~!+ (\u0017\u0000\u00170)he\u000b(') +i\u0000:(E6)\nIn the case of \u0012=\u0019=4, the modulation of the Gilbert\ndamping with the vertex correction is expressed as\n\u000e\u000bG\n\u000bG;0=\u00010\n2\u0019\u0000Re\"\n\u00002 +1\n1\u0000~\u0003R\n0(!0) +~\u0003R\n1(!0)\n+1\n1\u0000~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)#\n: (E7)\nThe third term of the above equation is calculated as\n1\n1\u0000~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)=1\n1\u0000i\u0000\n~!0+i\u0000=~!0+i\u0000\n~!0:(E8)\nThis indicates that the expansion with respect to ~\u0003R\njcan\nnot be allowed for !0\u001c\u0000. This is why the modula-\ntion without the vertex correction, which is obtained by\ntaking from the \frst-order term of ~\u0003R\njin Eq. (E7) as\n\u000e\u000bnv\nG\n\u000bG;0=\u00010\n2\u0019\u0000Re\u0014\n2~\u0003R\n0(!0)\u0015\n; (E9)\ngives a di\u000berent result near !0'0. Actually, for \u0012=\u0019=4,\n\u000e\u000bGand\u000e\u000bnv\nGare calculated as\n\u000e\u000bG\n\u000bG;0=\u00010\n2\u0019\u0000Re\"\ni\u0000\n2\u00010(J+\u0000+J\u0000+)\n1\u0000i\u0000\n2\u00010(J+\u0000+J\u0000+)#\n; (E10)\n\u000e\u000bnv\nG\n\u000bG;0=1\n4\u0019Re\"\ni\u0012\nJ+\u0000+J\u0000++J+++J\u0000\u0000\u0013#\n:(E11)\nNote that Eq.(E10) is not valid for !0= 0. As indicated\nfrom the absence of J++andJ\u0000\u0000, the graph of \u000e\u000bG(!0)\nhas no peak at zero frequency even though \u000e\u000bnv\nG(!0) has a\npeak there. This observation is consistent with Figs. 4 (e)\nand 4 (f).\nIn the case of \u0012=\u0000\u0019=4, the modulations of the Gilbert\ndamping with and without the vertex correction are\n\u000e\u000bG\n\u000bG;0=\u00010\n\u0019\u0000Re\"\ni\u0000\n2\u00010(J+\u0000+J\u0000+)\n1\u0000i\u0000\n2\u00010(J+\u0000+J\u0000+)#\n; (E12)\n\u000e\u000bnv\nG\n\u000bG;0=1\n2\u0019Re\"\ni(J+\u0000+J\u0000+)#\n: (E13)14\nFIG. 11. (Upper panels) Change of the Gilbert damping due to the vertex correction, \u000e\u000bG\u0000\u000e\u000bnv\nG, for (a)\f=\u000b= 0, (b)\f=\u000b= 1,\nand (c)\f=\u000b= 3. (Lower panels) Change of the FMR frequency due to the vertex correction, \u000e!0\u0000\u000e!nv\n0, for (d)\f=\u000b= 0, (e)\n\f=\u000b= 1, and (f) \f=\u000b= 3. The horizontal axes are the FMR frequency, !0=\rhdc, whereas the vertical axes are the azimuth\nangle of the spontaneous spin polarization, \u0012, in the FI. In all the plots, we have set \u0000 =\u00010= 0:5.\nNote that\u000e\u000bnv\nGis obtained by taking the \frst-order term\nin Eq. (E12). As indicated by the absence of the terms,\nJ++andJ\u0000\u0000, neither\u000e\u000bGnor\u000e\u000bnv\nGhas any structure\naround!0= 0. It can be checked that these two expres-\nsions give almost the same result when \u0000 .\u00010, which is\nconsistent with Figs. 4 (e) and 4 (f). Note as well that\n\u000e\u000bGis just doubled compared with the result for \u0012=\u0019=4\nin Eq. (E10).\nTo summarize the e\u000bect of the vertex correction, we\nshow\u000e\u000bG\u0000\u000e\u000bnv\nGand\u000e!0\u0000\u000e!nv\n0in Figs. 11 (b) and\n11 (e), respectively. We \fnd that the vertex correction\nmodi\fes mainly the peak width around ~!0=\u00010= 0.\nIn addition, the broad peak in the range of 0 <~!0<\n2\u00010is enhanced or suppressed depending on the azimuth\nangle of the ordered spin. These features are consistent\nwith the above analytic expressions. We note that similar\nfeatures are observed for \f=\u000b= 3 as seen in Figs. 11 (c)\nand 11 (f).\nAppendix F: Approximate Expressions near \f=\u000b= 1\nIn this Appendix, we derive the approximate expres-\nsions Eqs. (62) and (64) for \f=\u000b = 1 +\u000e(\u000e\u001c1) and\n!'0. For\f=\u000b= 1 +\u000e(\u000e\u001c1), we can use the approx-\nimation,\ncos 2(\u001e\u0000\u0012)'sin 2\u0012\u0010\n\u00001 +(hx+hy)2\nh2\ne\u000b\u0011\n; (F1)\nsin 2(\u001e\u0000\u0012)'cos 2\u0012\u0010\n\u00001 +(hx+hy)2\nh2\ne\u000b\u0011\n: (F2)Then, we obtain\n~\u0003R\n2'Xsin 2\u0012; (F3)\n~\u0003R\n3'Xcos 2\u0012; (F4)\nX\u0011i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017\u00170\u0017\u00170\u0010\n\u00001 +(hx+hy)2\nh2\neff\u0011\n~!+ (\u0017\u0000\u00170)he\u000b+i\u0000(F5)\nin the low-frequency region. Here, the contribution of\nthe second term of the bracket in Eq. (56) does not have\na singularity at !0= 0 because ~\u0003R\n0and ~\u0003R\n1do not de-\npend on the e\u000bective Zeeman \feld he\u000b. Therefore, the\nsingularity comes from the \frst term of the bracket in\nEq. 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Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nAn analysis of a single-domain magnetic needle (MN) in the pr esence of an external magnetic\nfieldBis carried out with the aim of achieving a high precision magn etometer. We determine the\nuncertainty ∆ Bof such a device due to Gilbert dissipation and the associate d internal magnetic\nfield fluctuations that give rise to diffusion of the MN axis dir ectionnand the needle orbital angular\nmomentum. The levitation of the MN in a magnetic trap and its s tability are also analyzed.\nA rigid single-domain magnet with large total spin,\ne.g.,S≃1012/planckover2pi1, can be used as a magnetic needle magne-\ntometer (MNM). Recently Kimball, Sushkov and Budker\n[1] predicted that the sensitivity of a precessing MNM\ncan surpass that of present state-of-the-art magnetome-\nters by orders of magnitude. This prediction motivates\nour present study of MNM dynamics in the presence of\nan external magnetic field B. Such analysis requires in-\nclusion of dissipation of spin components perpendicular\nto the easy magnetization axis (Gilbert damping). It is\ndue to interactions of the spin with internal degrees of\nfreedom such as lattice vibrations (phonons), spin waves\n(magnons), thermal electric currents, etc. [2, 3]. Once\nthere is dissipation, fluctuations are also present [6], and\nresult in a source of uncertainty that can affect the ac-\ncuracy of the magnetometer. Here we determine the un-\ncertainty in the measurement of the magnetic field by a\nMNM. We also analyze a related problem concerning the\ndynamics of the needle’s levitation in an inhomogeneous\nmagnetic field, e.g., a Ioffe-Pritchard trap [8].\nThe Hamiltonian for a MN, treated asa symmetric top\nwith body-fixed moments of inertia IX=IY≡ I ∝negationslash=IZ,\nsubject to a uniform magnetic field Bis,\nH=1\n2IˆL2+(1\n2IZ−1\n2I)ˆL2\nZ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nHR−(ω0//planckover2pi1)(ˆS·ˆn)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHA−ˆµ·B/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHB,\n(1)\nwhere a hat denotes quantum operator. In the rotational\nHamiltonain HR,ˆLis the orbital angular momentum op-\nerator and ˆLZ=ˆL·ˆZis its component along the body-\nfixed symmetry axis. ˆSis the needle spin angular mo-\nmentum operator, and ˆnis the operator for nthat is the\nunit vector in the direction of the easy magnetization\naxis. The frequency appearing in the anisotropy Hamil-\ntonianHA[4] isω0= 2γ2KS/V, whereKis the strengthofthe anisotropy, Vis the needle volume, and γ=gµB//planckover2pi1\nis the gyromagnetic ratio, in which µBis the Bohr mag-\nnetron, and gis theg-factor (taken to be a scalar for\nsimplicity). In the expression for the Zeeman Hamilto-\nnianHB,ˆµ=gµBˆSis the magnetic moment operator.\nThe Heisenberg equations of motion are\n˙ˆS=−gµBB׈S+2ω0\n/planckover2pi1(ˆS׈n)(ˆS·ˆn),(2)\n˙ˆL=-2ω0\n/planckover2pi1(ˆS׈n)(S·ˆn), (3)\n˙ˆJ=−gµBB׈S, (4)\n˙ˆn=I−1\n/planckover2pi1[ˆL׈n+i/planckover2pi1ˆn], (5)\nwhereˆJ=ˆL+ˆSis the total angularmomentum operator\nandIis the moment of inertia tensor.\nThe dynamics of a MN can be treated semiclassically\nbecause Sis very large. A mean–field approximation\n[9–11] is obtained by taking quantum expectation values\nof the operator equations and assuming that for a given\noperator ˆA, the inequality/radicalBig\n∝angbracketleftˆA2∝angbracketright−∝angbracketleftˆA∝angbracketright2≪ |∝angbracketleftˆA∝angbracketright|holds,\n(an assumption warranted for large S). Hence, the ex-\npectation values of a product of operators on the RHS\nof Eqs. (2)-(5) can be replaced by a product of expecta-\ntion values. The semiclassical equations are equivalent\nto those obtained in a classical Lagrangian formulation.\nDissipation is accounted for by adding the Gilbert term\n[2, 4]−αS×(˙S//planckover2pi1−Ω×S//planckover2pi1) to the RHS of the expecta-\ntion value of Eq. (2) and subtracting it from the RHS of\nEq. (3). Here αis the dimensionless friction parameter,\nand the term Ω×Stransforms from body fixed to space\nfixed frames. Note that Gilbert damping is due to inter-\nnalforces, hence Jis not affected and Eq. (4) remains\nintact.2\nIt is useful to recast the semiclassical dynamical equa-\ntions of motion in reduced units by defining dimension-\nless vectors: the unit spin m≡S/S, the orbital angu-\nlar momentum ℓ≡L/S, the total angular momentum,\nj=m+ℓand the unit vector in the direction of the\nmagnetic field b=B/B:\n˙m=ωBm×b+ω0(m×n)(m·n)−αm×(˙m−Ω×m),(6)\n˙ℓ=−ω0(m×n)(m·n)+αm×(˙m−Ω×m),(7)\n˙n=Ω×n, (8)\n˙j=ωBm×b, (9)\nwhere the angular velocity vector Ωis given by\nΩ= (ω3−ω1)(ℓ·n)n+ω1ℓ\n= (ω3−ω1)[(j−m)·n]n+ω1(j−m).(10)\nHereωB=γ|B|is the Larmor frequency, ω1=S/IX,\nandω3=S/IZ. Similar equations were obtained in\nRef. [5], albeit assuming that the deviations of n(t) and\nm(t) frombare small. We show below that the dynam-\nics can be more complicated than simply precession of\nthe needle about the magnetic field, particularly at high\nmagnetic fields where nutation can be significant.\nFor the numerical solutions presented below we are\nguided by Ref. 1, which uses parameters for bulk cobalt,\nand take ω1= 100 s−1,ω3= 7000 s−1, anisotropy fre-\nquencyω0= 108s−1, Gilbert constant α= 0.01, tem-\nperature T= 300 K, and N=S//planckover2pi1= 1012. First, we elu-\ncidate the effects of Gilbert dissipation, and consider the\nshorttimebehaviorin aweakmagneticfield, ωB= 1s−1.\nThe initial spin direction is intentionally chosen notto be\nalong the easy magnetic axis; n(0) = (1 /2,1/√\n2,1/2),\nm(0) = (1 /√\n2,1/√\n2,0),ℓ(0) = (0 ,0,0). Figure 1(a)\nshows the fast spin dissipation as it aligns with the easy\naxis of the needle, i.e., m(t)→n(t) after a short time,\nand Fig. 1(b) shows relaxation of the oscillations in ℓ(t),\nwhileℓx(t) andℓy(t) approach finite values. Figure 1(c)\nshowsthe innerproduct m·n, which clearlytendstounity\nonthe timescaleofthe figure. Increasing αleadsto faster\ndissipation of m(t), but the short-time saturation values\nof bothm(t) andℓ(t) are almost independent of α.\nWe consider now the long time dynamics (still in\nthe weak field regime) and take the initial value of the\nspin to coincide with the easy magnetization axis, e.g.,\nm(0) =n(0) = (1 /√\n2,1/√\n2,0), with all other param-\neters unchanged. The spin versus time is plotted in\nFig. 2(a). The unit vectors m(t) andn(t) are almost\nidentical, andsincetheir z-componentisnearlyzero,they\nmove together in the x-yplane. In this weak field case,\nthe nutation is small, and the fast small-oscillations due\nto nutation are barely visible. The orbital angular mo-\nmentum dynamics is plotted in Fig. 2(b) [note the differ-\nenttimescalein (a)and(b)] andshowsthat ℓ(t) oscillates\nwith a frequency equal to that of the fast tiny-oscillation\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0002\n-\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0002\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003\u0007\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0001\u0001\u0002\u0007\u0001\u0001\u0002\u0007\u0003\u0004\u0002\u0001\u0001{\u0002\u0002\u0003}\nFIG. 1: (color online) (a) The normalized spin vector mver-\nsus time for the low-field case at short times (5 orders of\nmagnitude shorter than in Fig. 2) when the initial spin is\nnot along the fast axis. (b) The reduced orbital angular mo-\nmentum vector ℓ(t). (c) The inner product m(t)·n(t) (the\nprojection of the spin on the fast magnetic axis of the needle .\nofm(t) [the oscillation amplitude is 0 .02|m(t)|]. Fig-\nure 2(c) shows a parametric plot of m(t) versus time.\nThe nutation is clearly very small; the dynamics of m(t)\nconsists almost entirely of precession at frequency ωB.\nFigure 3 shows the dynamics at high magnetic field\n(ωB= 105s−1) with all the other parametersunchanged.\nFigure 3(a) shows mversus time, and now the nutation\nis clearly significant. For the high magnetic field case,\nm(t) is also almost numerically equal to n(t).ℓ(t) is\nplotted in Fig. 3(b). Its amplitude is very large, ℓ(t)≈\n40m(t). However, its oscillation frequency is comparable\nwith that of m(t). In contrast with the results in Fig. 2,\nhere, in addition to precession of the needle, significant\nnutation is present, as shown clearly in the parametric\nplot of the needle spin vector m(t) in Fig. 3(c).\nWe now determine the uncertainty of the MNM due to\ninternal magnetic field fluctuations related to the Gilbert\ndamping. A stochastic force ξ(t), whose strength is de-\ntermined by the fluctuation–dissipation theorem [6], is3\n\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0003\u0004\u0005\u0006\u0002\n-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\b\u0005\u0007\u0006{\u0001\u0001\t\u0001\u0002\t\u0001\u0003\t\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\nFIG. 2: (color online) Dynamics for thelow-field case ( ωB= 1\ns−1), over relatively long timescales relative to those in Fig. 1.\n(a)mversus time in units of seconds (note that nis indistin-\nguishable from mon the scale of the figure). (b) ℓ(t) (note\nthat it stays small compared to S). (c) Parametric plot of the\nneedle spin vector m(t) showing that nutation is almost im-\nperceptible for small fields [contrast this with the large fie ld\nresult in Fig. 3(c)]; only precession is important.\nadded to Eq. (6), in direct analogy with the treatment of\nBrownianmotionwherebothdissipationandastochastic\nforce are included [12]:\n˙m=m×(ωBb+ξ)+ω0(m×n)(m·n)\n−αm×(˙m−Ω×m). (11)\nξ(t) is internal to the needle and therefore it does not\naffect the total angular momentum jdirectly, i.e., ξ(t)\ndoes not appear in Eq. (9) [since the term −m×ξis also\nadded to the RHS of (7)]. However, as shown below, ξ(t)\naffectsℓas well as m, causing them to wobble stochas-\ntically. This, in turn, makes jstochastic as well via the\nZeeman torque [see Eq. (9)].\nThe fluctuation-dissipation theorem [6] implies\n∝angbracketleftξαξβ∝angbracketrightω≡/integraldisplay\ndt∝angbracketleftξα(t)ξβ(0)∝angbracketrighteiωt\n=δαβαωcoth(/planckover2pi1ω/2kBT)\nN≈δαβ2αkBT\n/planckover2pi1N,(12)\u0001\u0001\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0007\u0002\u0001-\u0001\u0002\b\u0001\u0002\b\u0007\u0002\u0001{\u0001\u0001\t\u0001\u0002\t\u0001\u0003}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0004\u0001-\u0003\u0001\u0003\u0001\u0004\u0001{\u0001\u0001\n\b\u0001\u0002\n\u0000\u0001\u0003}\nFIG. 3: (color online) High-field case ( ωB= 105s−1). (a)\nm(t) [which is almost numerically equal to n(t)]. (b)ℓ(t)\n(note the ordinate axis scale is [ −40,40]). (c) Parametric plot\nof the needle spin vector m(t) showing that strong nutation\noccurs for large fields in addition to precession.\nwhereN=S//planckover2pi1, and the last approximation is ob-\ntained under the assumption that /planckover2pi1ω≪kBT. Note that\nEq. (11) should be solved together with Eqs. (8) and (9).\nThe presence of the anisotropy term in Eq. (11) makes\nnumerical solution difficult for large ω0. Hence, we con-\nsider a perturbative expansion in powers of λ≡ω1/ω0:\nm(t) =n0(t)+λδm(t)+...,n(t) =n0(t)+λδn(t)+...,\nj(t) =j0(t) +λδj(t) +.... Sinceω0is the largest fre-\nquency in the problem, the inequalities αω0≫ωB,ω1,ω3\nhold. Moreover, the Gilbert constant αis large enough\nto effectively pin m(t) ton(t) [hencej(t) =ℓ(t)+m(t)≈\nℓ(t)+n(t)]. Therefore, anadiabaticapproximationtothe\nset of dynamical stochastic equations can be obtained.\nThe zero order term in λreads:\n˙j0=ωBn0×b,˙n0=ω1j0×n0,(13)4\nwhereΩwas approximated by Ω0= (ω3−ω1)(j0·n0−\n1)n0+ω1(j0−n0) in Eqs. (8) and (10) in obtaining (13)\n[7]. The solution to Eqs. (13) [for times beyond which\nGilbert dissipation is significant so m(t)≈n(t)] is very\nclose to that obtained from Eqs. (6)-(8).\nExpanding Eq. (11) in powers of λand keeping only\nthe first order terms (the zeroth order term on the LHS\nvanishes since m0=n0), we get: ω1(δm−δn)×n0=\n˙n0−ωBn0×b+αn0×(˙n0−Ω0×n0)−n0×ξ. Taking\nEq. (13) into account and introducing the notation δη≡\nδm−δn, we obtain\nδη×n0=j0×n0−(ωB/ω1)n0×b−(1/ω1)n0×ξ,(14)\nand from Eqs. (8) and (9) we find\nd\ndtδj=ωB(δn+δη)×b, (15)\nd\ndtδn=ω1(j0−n0)×δn+ω1(δj−δn−δη)×n0\n=ω1j0×δn+ω1(δj−δη)×n0. (16)\nTo first order in λ,δn⊥n0(sincenmust be a unit\nvector), and δm⊥n0, henceδη⊥n0. Therefore, δη×\nb= [j0−(j0·n0)n0]×b+(ωB/ω1)[b−(b·n0)n0]×b+\nω−1\n1[ξ−(ξ·n0)n0]×bon the RHS of Eq. (15) and\nd\ndtδj=ωBδn×b+ωB[j0−(j0·n0)n0]×b\n−ω2\nB\nω1(b·n0)n0×b+ωB\nω1[ξ−(ξ·n0)n0]×b.(17)\nEquations (13), (16) and (17) form a closed system of\nstochastic differential equations [upon using Eq. (14) to\nsubstitute for δη×n0on the RHS of Eq. (16)]. With\nthe largest frequency ω0eliminated, a stable numerical\nsolution is obtained. Moreover, for small magnetic field\n(whereωBis the smallest frequency in the system), an\nanalytic solution of these equations is achievable. To ob-\ntain an analytic solution to Eqs. (13), let us transform\nto the frame rotating around Bwith frequency ωBto\nget equations of the formd\ndτv=d\ndtv+ωBb×v(which\ndefinesτ):\nd\ndτn0=−ω1n0×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n,(18)\nd\ndτj0=ωBb×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n.(19)\nIf the initial condition is n0(0)−j0(0)+(ωB/ω1)b= 0,\nthen, in the rotating frame j0(τ) andn0(τ) are constant\nvectors. Note that this initial condition is only slightly\ndifferent from the “ordinary” initial condition n0(0) =\nj0(0)since( ωB/ω1)≪1forsmallmagneticfields. Hence,\nin the rotating frame,\nd\ndτδn=ω1n0×(δn−δj+δη),(20)d\ndτδj=−ωBb×(δn−δj+δη).(21)\nWith the special initial conditionbeing satisfied, Eq. (14)\nbecomes δη×n0=−(1/ω1)n0×ξ, and Eqs. (20)-(21)\nbecome a set of first order differential equations with\ntime-independent coefficients. Their solution for initial\nconditions, δn(t= 0) = 0, δj(t= 0) = 0 is,\n/parenleftbiggδn(t)\nδj(t)/parenrightbigg\n=t/integraldisplay\n0dt1exp[C(t−t1)]C/parenleftbiggδη(t1)\n0/parenrightbigg\n,(22)\nwhere the constant matrix C=/parenleftbiggA−A\n−B B/parenrightbigg\nhas di-\nmension 6 ×6 and the 3 ×3 matrices AandBare given by\nAij=−ω1ǫijknk\n0,Bij=−ωBǫijkbk. Without loss of gen-\neralitywecanchoose n0=ˆzandb=ωB(cosθˆz+sinθˆx),\nwhereθis the angle between the easy magnetization\naxis and the magnetic field. In this basis, ∝angbracketleftδηxδηx∝angbracketrightω=\n∝angbracketleftδηyδηy∝angbracketrightω≈ω−2\n0∝angbracketleftξxξx∝angbracketrightω=ω−2\n0∝angbracketleftξyξy∝angbracketrightω=Sa(ω), and\n∝angbracketleftδηzδηz∝angbracketrightω= 0. Here ∝angbracketleftxx∝angbracketrightω≡/integraltext\ndteiωt∝angbracketleftx(t)x(0)∝angbracketrightand [see\nEq. (12)] Sa(ω) =αωcoth(/planckover2pi1ω/2kBT)\nω2\n0N≈2αkBT\nN/planckover2pi1ω2\n0.\nWe are particularly interested in the quantities\n∝angbracketleftδn2\ny(t)∝angbracketright ≡ ∝angbracketleftδny(t)δny(t)∝angbracketrightand∝angbracketleftδj2\ny(t)∝angbracketright ≡ ∝angbracketleftδjy(t)δjy(t)∝angbracketright\nbecause, in the basis chosen above, the y-axis is the di-\nrection of precession of n0aroundb. Using Eq. (22) we\nobtain∝angbracketleftδn2\ny(t)∝angbracketright ≈tω2\n1Sa(ω∼ω1). Assuming the pre-\ncession of nis measured, [or equivalently, the precession\nofm, since they differ only for short timescales of or-\nder (αω0)−1], the uncertainty in the precession angle is\n∝angbracketleft(∆ϕ)2∝angbracketright ≈tω2\n1Sa(ω∼ω1). We thus arrive at our central\nresult: the precision with which the precession frequency\ncan be measured is, ∆ ωB=√\n/angbracketleft(∆ϕ)2/angbracketright\nt≈ω1\nω0/radicalBig\n2αkBT\n/planckover2pi1N1√\nt.\nEquivalently, the magnetic field precision is,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω1\nω0/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt.(23)\nFor the parameters used in this paper we find ∆ B≈\n5×10−18√\nt[s]Tesla (independent of ωB). This result should\nbe compared with the scaling ∆ B∝t−3/2obtained in\nRef. 1. Therein, the initial uncertainty of the spin di-\nrection relative to the needle axis was estimated from\nthe fluctuation-dissipation relation and the deterministic\nprecession resulted in the t−3/2scaling of the precession\nangle uncertainty (in addition this angle was assumed to\nbe small). In contrast, we consider the uncertainty ac-\nquired due to Gilbert dissipation duringthe precession,\nallowing the precession angle to be large. Thus, the stan-\ndard1/√\ntdiffusion scalingis obtained and dominates for\ntimes that are even much longer than those considered\nin Ref. 1.\nIntheSupplementalMaterial[13]wediscussthreerele-\nvant related issues. (a) The time at which diffusion stops\nbecause equipartition is reached (we estimate the time5\nwhen the energy stored in stochastic orbital motion be-\ncomes of order kBT). (b) The uncertainty of the mag-\nnetic field for experiments in which the fast precession of\nnaroundjis averaged out in the measurement, and the\ndiffusion of jdetermines ∆ B. (c) We consider the related\nproblem of the dynamics and stability of a rotating MN\nin an inhomogeneous field (e.g., levitron dynamics in a\nIoffe-Pritchard trap [14, 15]).\nIn conclusion, we show that ∆ Bdue to Gilbert damp-\ning is very small; external noise sources, as discussed in\nRef. [1], will dominate over the Gilbert noise for weak\nmagnetic fields. A closed system of stochastic differen-\ntial equations, (13), (16) and (17), can be used to model\nthe dynamics and estimate ∆ Bfor large magnetic fields.\nA rotating MN in a magnetic trap can experience levi-\ntation, although the motion does not converge to a fixed\npoint or a limit cycle; an adiabatic–invariant stability\nanalysis confirms stability [13].\nThis work was supported in part by grants from the\nDFG through the DIP program (FO703/2-1). Useful\ndiscussions with Professor Dmitry Budker are gratefully\nacknowledged. A. S. was supported by DFG Research\nGrant No. SH 81/3-1.\n[1] D. F. J. Kimball, A. O. Sushkov, and D. Budker, Phys.\nRev. Lett. 116, 190801 (2016).\n[2] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004)\n[3] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8153\n(1935). In L. D. Landau, Collected Papers. Ed. by D. ter\nHaar, (Gordon and Breach, New York, 1967), p. 101.\n[4] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).\n[5] H. Keshtgar, et al., Phys. Rev. B 95, 134447 (2017).\n[6] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).\n[7] We note in passing that Eqs. (13) are equivalent to the\nequations of motion of a symmetric top in a gravita-\ntional field when the top is anchored at a point a=an\non its axis a distance afrom the center of mass. The\nequations of motion are: dL/dt=T, where Land\nT=an×(−mgz) are taken with respect to the fixedpoint, and dn/dt=Ω×n. The angular velocity is given\nbyΩ=I−1\n1[L−(L·n)n] +I−1\n3(L·n)n, where the mo-\nments of inertia ( I1,I1,I3) are calculated relative to the\nfixed point. Introducing a characteristic scale L0so that\nL=L0j(jis not a unit vector and its length is not\nconserved) we obtain Eqs. (13) with ωB=mga/L 0and\nω1=L0/I1. Here, the analog of the magnetic field is the\ngravitational field and the analog of bisz.\n[8] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein.\n[9] O. Zobay and B. M. Garraway, Phys. Rev. A 61, 033603\n(2000); J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi,\nB. Wu, and Q. Niu, Phys. Rev. A 66, 023404 (2002).\n[10] Y. B. Band, I. Tikhonenkov, E. Pazyy, M. Fleischhauer,\nand A. Vardi, J. of Modern Optics 54, 697-706 (2007).\n[11] Y. B. Band, Phys. Rev. E 88, 022127 (2013); Y. B. Band\nand Y. Ben-Shimol, Phys. Rev. E 88, 042149 (2013).\n[12] H. P. Breuer and F. Petruccione, The Theory of\nOpen Quantum Systems (Oxford University, Cambridge,\n2002); M. Schlosshauer, Decoherence and the Quantum-\nto-Classical Transition (Springer, Berlin, 2007).\n[13] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.121.160801\nwhich contains a discussion of the three issues enumer-\nated in the text, and which includes Refs. 16-20.\n[14] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[15] A movie showing the dynamics of a Levitron can be seen\nathttps://www.youtube.com/watch?v=wyTAPW_dMfo .\n[16] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a\nMagnetic Needle Magnetometer: Sensitivity to Landau–\nLifshitz–Gilbert Damping”, Phys. Rev. Lett. (to be pub-\nlished).\n[17] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[18] C. C.Rusconi, V.P¨ ochhacker, K.Kustura, J.I.Ciracan d\nO. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017);\nC. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac\nand O. Romero-Isart, Phys. Rev B 96, 134419 (2017); C.\nC. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427\n(2016).\n[19] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[20] D. R. Merkin, Introduction to the Theory of Stability ,\n(Springer–Verlag, New York, 1997); F. Verhulst, Non-\nlinear Differential Equations and Dynamical Systems ,\n(Springer–Verlag, Berlin, 1990).arXiv:1803.10064v2  [physics.gen-ph]  19 Oct 2018Supplemental Material for “Dynamics of a Magnetic Needle Ma gnetometer:\nSensitivity to Landau–Lifshitz–Gilbert Damping”\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nIn this supplemental material we expand the discussion of the main t ext [1] and address the following three issues.\n(a) The time τeat which the diffusion of the magnetic needle axis direction nand the magnetic needle orbital\nangular momentum ℓstops because equipartition is reached, i.e., we estimate the time req uired for the energy stored\nin stochastic orbital motion to become of order kBT. (b) The uncertainty ∆ Bof the magnetic field for experiments\nin which the fast precession of naroundjis averaged out in the measurement process and the uncertainty ∆ Bis\ndetermined by the diffusion of j. (c) The dynamics of a magnetic needle in an inhomogeneous field, e.g., levitron\ndynamics of a rotating magnetic needle in a Ioffe-Pritchard trap [2], s ee Refs. [3–5].\n(a):τecan be estimated by noting that the diffusion determined in [1] stops once equipartition is reached. The\nenergy ∆ Estored in stochastic orbital motion is given by\n∆E∼/planckover2pi1ω1N/angbracketleftδℓ2/angbracketright, (1)\nwhere where N=S//planckover2pi1(note that δj−δn=δℓ). By requiring ∆ E∼kBTwe can estimate that the diffusion given\nby Eqs. (20-21) of [1] stops when τe∼ω2\n0/(αω3\n1) (this result can also be obtained by expanding Eq. (11) further in\npowers of λ≡ω1/ω0). For the parameters used in [1] this is an extremely long time ( τe∼1012s∼5 years). Hence,\nwe conclude that the diffusion of Eqs. (20-21) and the error estima tes given for ∆ Bin Ref. [1] are relevant for all\nreasonable times.\n(b): In [1] we calculate ∆ Bassuming the experimental measurement follows the temporal dyn amics of nandj.\nAn alternative assumption is that the precession of naroundjis averaged out by the measurement process and one\nmeasures the diffusion of j. For the latter we obtain the leading term\n/angbracketleftδj2\ny(t)/angbracketright ≈tω2\nBcos2θSa(ω∼ω1), (2)\nwhereSa(ω) is given in Eq. (23) of [1]. At θ=π/2 the leading contribution obtained in Eq. (2) vanishes and the\nremaining sub-leading term is\n/angbracketleftδn2\ny(t)/angbracketright ≈t2ω4\nB\nω2\n1Sa(ω∼ω1), (3)\nhence for θ/negationslash=π/2 we obtain\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBωB\nω0cosθ/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt, (4)\nwhereas at θ=π/2,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω2\nB\nω0ω1/radicalbigg\n4αkBT\n/planckover2pi1N1√\nt. (5)\nTakingωB= 1s−1we obtain ∆ B≈cosθ×5×10−23√\nt[s]Tesla for θ/negationslash=π/2, and ∆ B≈7×10−25√\nt[s]Tesla for θ=π/2.2\n(c): A rotating magnet can be levitated in an inhomogeneous magnet ic field [3–5]. This is possible despite Earn-\nshaw’s theorem [6] from which one can conclude that levitation of a non-rotating ferromagnetin a static magnetic field\nis not possible. Two important factors regarding magnetic levitation are the forces on the magnet and its stability\n(ensuring that it does not spontaneously slide or flip into a configura tion without lift). The dynamics of a magnetic\nneedle in an inhomogeneous magnetic field can be modelled using Eqs. (6 ), (7) and (8) of [1] augmented by the\nequations of motion for the center of mass (CM) degrees of freed om of the needle,\n˙p=∇(µ·B(r)), (6)\n˙r=p/m , (7)\nwhererandpare the needle CM position and momentum vectors. Our numerical re sults show levitation of the\nmagnetic needle when the initial rotational angular momentum vecto r of the needle is sufficiently large and points\nin the direction of magnetic field at the center of the trap. We shall s ee that the dynamical variables do not evolve\nto a fixed point or a simple cyclic orbit. Moreover, a linear stability analy sis yields a 15 ×15 Jacobian matrix with\neigenvalues having a positive real part, so the system is unstable. However, a stability analysis of the system using\nthe adiabatic invariant |µ||B|[3] does yield a stable fixed point (contrary to the full numerical re sults which show a\nmore complicated levitation dynamics).\nFigure 1 shows the dynamics of the system over time in the trap. We u se the same magnetic needle parameters\nused in Fig. 2 of [1] and a Ioffe-Prichard magnetic field [2]\nB(r) =ex/parenleftbigg\nB′x−B′′\n2xz/parenrightbigg\n+ey/parenleftbigg\nB′y−B′′\n2zy/parenrightbigg\n+ez/parenleftbigg\nB0+B′′\n2(z2−x2+y2\n2)/parenrightbigg\n, (8)\nwith field bias B0, gradient B′, and curvature B′′parameters chosen so that the Zeeman energy and its variation ov er\nthe trajectory of the needle in the trap are substantial (as is clea r from the results shown in the figure). We start\nthe dynamics with initial conditions: r(0) = (0,0,0),p(0) = (0,0,0),m(0) = (0,0.0011/2,−(1−0.001)1/2) (almost\nalong the −zdirection), n(0) =m(0),ℓ(0) = (0 ,0,0.001) [this is large orbital angular momentum since ℓis the\norbital angular momentum divided by S]. Figure 1(a) shows the needle CM position r(t) versus time. Fast and slow\noscillations are seen in the xandymotion, whereas z(t) remains very close to zero. Figure 1(b) shows oscillations of\nthe CM momentum p(t) with time. px(t) andpy(t) oscillate with time, and pz(t) remains zero. Figure 1(c) plots the\nspinm(t) versus time. Initially, m(0) points almost in the −zdirection, and the tip of the needle n(t) =m(t) carries\nout nearly circular motion in the nx-nyplane. Figure 1(d) plots the orbital angular momentum ℓ(t). The components\nℓx(t) andℓy(t) undergo a complicated oscillatory motion in the ℓx(t)-ℓy(t) plane but ℓz(t)≈ℓz(0). Figure 1(e) is a\nparametric plot of m(t); the motion consists of almost concentric rings that are slightly dis placed one from the other.\nThe full dynamics show levitation but they do not converge to a fixed point or a limit cycle.\nQuite generally, for a system of dynamical equations, ˙ yi(t) =fi(y1,...,y n),i= 1,...n, a linear stability analysis\nrequires calculating the eigenvalues of the Jacobian matrix evaluate d at the equilibrium point y∗wheref(y∗) =0,\nJij=/parenleftBig\n∂fi\n∂yj/parenrightBig\ny∗[7]. The system is unstable against fluctuations if any of the eigenvalu es ofJijhave a positive real\npart. Equations (6), (7) and (8) of [1] together with Eqs. (6) and (7) above have a Jacobian matrix with eigenvalues\nwhose real part are positive, so the linear stability test fails. Howev er, if the Zeeman force −∇HZin Eq. (6) is\nreplaced by the gradient of the adiabatic invariant, µ·∇|B(r)|, none of the eigenvalues of the Jacobian matrix have\na positive real part and the system is linearly stable, i.e., the stability a nalysis using the adiabatic-invariant predicts\nstability. Note that substituting the adiabatic invariant for the Zee man energy in the full equations of motion yields\nr(t) andp(t) vectors that are constant with time and n(t),m(t) andℓ(t) are similar to the results obtained with\nthe full equations of motion (but the parametric plot of m(t) is a perfectly circular orbit). Thus, adiabatic–invariant\nstability analysis of a rotating magnetic needle in a magnetic trap confi rms stability of its levitation as obtained in\nthe numerical solution of the dynamical equations.3\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0004-\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0004\n\u0001{\u0001\u0001\u0002\u0001\u0003}\u0002\u0003\u0004\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0003\u0001-\u0001\u0005\u0001\u0002\u0001\u0005\u0001\u0001\u0001\u0005\u0001\u0002\u0001\u0005\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0003\u0005\u0001-\u0001\u0005\u0006-\u0001\u0005\u0007-\u0001\u0005\b-\u0001\u0005\u0004\u0001\u0005\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0001\u0006-\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0006\u0001\u0005\u0001\u0001\u0001\u0007\u0001\u0005\u0001\u0001\u0001\b\u0001\u0005\u0001\u0001\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\nFIG. 1: (color online) Dynamics of a needle in a Ioffe-Pritcha rd magnetic field. (a) rversus time, (b) pversus time, (c) m\nversus time (note that n(t) is indistinguishable from m(t) on the scale of the figure). (d) ℓversus time (note that |ℓ(t)|is small\ncompared to Sbut rotational angular momentum L(t) =Sℓ(t) is large since S= 1012). (e) Parametric plot of the needle spin\nvectorm(t) (nutation is very small for this case of small magnetic field ).4\n[1] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnet ic Needle Magnetometer: Sensitivity to Landau–Lifshitz–\nGilbert Damping”, Phys. Rev. Lett. (to be published).\n[2] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[3] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[4] A movie a a Levitron can be seen at https://www.youtube.com/watch?v=wyTAPW_dMfo .\n[5] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C.\nRusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romer o-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and\nO. Romero-Isart, Phys. Rev B 93, 054427 (2016).\n[6] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[7] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Nonlinear Differential\nEquations and Dynamical Systems , (Springer–Verlag, Berlin, 1990)."    },    {        "title": "1602.07317v2.Relaxation_of_a_classical_spin_coupled_to_a_strongly_correlated_electron_system.pdf",        "content": "Relaxation of a classical spin coupled to a strongly correlated electron system\nMohammad Sayad, Roman Rausch and Michael Pottho\u000b\nI. Institute for Theoretical Physics, University of Hamburg, Jungiusstra\u0019e 9, D-20355 Hamburg, Germany\nA classical spin which is antiferromagnetically coupled to a system of strongly correlated con-\nduction electrons is shown to exhibit unconventional real-time dynamics which cannot be described\nby Gilbert damping. Depending on the strength of the local Coulomb interaction U, the two main\nelectronic dissipation channels, transport of excitations via correlated hopping and via excitations of\ncorrelation-induced magnetic moments, become active on largely di\u000berent time scales. We demon-\nstrate that correlations can lead to a strongly suppressed relaxation which so far has been observed\nin purely electronic systems only and which is governed here by proximity to the divergent magnetic\ntime scale in the in\fnite- Ulimit.\nPACS numbers: 75.78.Jp, 71.10.Fd, 75.10.Hk, 05.70.Ln\nMotivation. A classical spin in an external magnetic\n\feld shows a precessional motion but when exchange-\ncoupled to a conduction-electron system the spin addi-\ntionally relaxes and \fnally aligns to the \feld direction.\nThis is successfully described on a phenomenological level\nby the Landau-Lifschitz-Gilbert (LLG) equation [1] and\nextensions of this concept [2, 3]. The Gilbert damping\nconstant\u000bis often taken as a phenomenological parame-\nter but can also be computed ab initio for real materials\n[4{6] within a framework of e\u000bectively independent elec-\ntrons using band theory [7] and then serves as an impor-\ntant input for atomistic spin-dynamics calculations [8].\nElectron correlations are expected to have an impor-\ntant e\u000bect on the spin dynamics. This has been demon-\nstrated in a few pioneering studies [9{11] { within di\u000ber-\nent models and using various approximations { but only\nindirectly by computing the e\u000bect of the Coulomb inter-\naction on the Gilbert damping. One hallmark of strong\ncorrelations, however, is the emergence and the separa-\ntion of energy (and time) scales { with the correlation-\ninduced Mott insulator [12] as a paradigmatic example.\nWith the present study we address correlation e\u000bects\nbeyond an LLG-type approach and keep the full tempo-\nral memory e\u000bect. It is demonstrated that correlation-\ninduced time-scale separation has profound and qualita-\ntively new consequences for the spin dynamics. These\nare important, e.g., for the microscopic understanding\nof the emerging relaxation time scales in modern nano-\nspintronics devices involving various transition metals\nand compounds [13{15].\nConcretely, we consider a generic model with a clas-\nsical spinSthat is antiferromagnetically exchange cou-\npled (J > 0) to a Hubbard system and study the spin\ndynamics as a function of the Hubbard- U. To tackle\nthis quantum-classical hybrid problem, we develop a\nnovel combination of linear-response theory [6, 16, 17] for\nthe spin dynamics with time-dependent density-matrix\nrenormalization group (t-DMRG) [18{20] for the corre-\nlated electron system. For technical reasons we consider\na Hubbard chain but concentrate on generic e\u000bects which\nare not bound to the one-dimensionality of the model.In the metallic phase at quarter \flling, a complex phe-\nnomenology is found where two di\u000berent channels for en-\nergy and spin dissipation, namely dissipation via corre-\nlated hopping and via excitations of local magnetic mo-\nments, become active on characteristic time scales, de-\npending on U. While magnetic excitations give the by\nfar dominating contribution to the Gilbert damping in\nthe strong-coupling limit, they contribute to the spin dy-\nnamics to a much lesser extent and on later and later\ntime scales when Uis increased.\nIt is demonstrated that electron correlations can have\nextreme consequences: At half-\flling and strong U, the\nspin relaxation is incomplete on intermediate time scales.\nThis represents a novel e\u000bect in a quantum-classical hy-\nbrid model which is reminiscent of prethermalization [21{\n24] or metastability of excitations due to lack of phase\nspace for decay [25{28], i.e., physics which so far has been\nobserved in purely electronic quantum systems only.\nGilbert damping. We consider the Hubbard model for\nNelectrons on an open chain of length Las a prototypical\nmodel of correlated conduction electrons:\nHe=\u0000Tn:n:X\ni<jX\n\u001b(cy\ni\u001bcj\u001b+ H.c.) +ULX\ni=1ni\"ni#:(1)\nThe nearest-neighbor (n.n.) hopping T= 1 sets the en-\nergy and time scale ( ~= 1). Using standard arguments\n[16, 17], the Gilbert damping parameter \u000bcan be com-\nputed as\n\u000b=\u0000J2Z1\n0dtt\u001f loc(t) (2)\nand depends on the Hubbard interaction Uvia the local\n(diagonal and isotropic) retarded spin susceptibility\n\u001floc(t) =\u0000i\u0002(t)h0j[si0z(t);si0z(0)]j0i (3)\nat the site i0where the classical spin is coupled to. J\nis the strength of the exchange interaction [see Eq. (4)\nbelow]. Furthermore, j0iis the ground state of He,\nsiz(t) =eiHetsize\u0000iHet, andsizis thez-component of thearXiv:1602.07317v2  [cond-mat.str-el]  16 Sep 20162\nlocal conduction-electron spin si=P\n\u001b\u001b0cy\ni\u001b\u001c\u001b\u001b0ci\u001b0=2,\nwith the vector of Pauli matrices \u001cand with\u001b;\u001b0=\";#.\nWe choose i0= 1 for two reasons: (i) As compared to\nthe symmetric choice i0=L=2, this allows us to double\nthe accessible time scale (before \fnite-size e\u000bects set in).\n(ii) The time-integral in (2) is sensitive to the long-time\nbehavior of \u001floc(t) which, at least for U= 0, is related\nto the strength of the van Hove singularities in the local\ndensity of states [17]. At the edge of the open chain,\nthose are weak and characteristic for a three-dimensional\nsystem.\nLocal spin correlations at quarter \flling. To compute\n\u001floc(t), we apply t-DMRG and the framework of matrix-\nproduct states [18] for systems with L= 80{120 sites.\nConcretely, we use the two-site version of the algorithm\nsuggested in Ref. [19, 20] which is based on the time-\ndependent variational principle (TDVP).\nElectron correlations are expected to speed up the re-\nlaxation of the classical spin since electron scattering fa-\ncilitates the transport of energy and spin density from\ni0to the bulk of the system. An increasingly e\u000ecient\ndissipation implies an increase of \u000bwithU. This can be\nnicely seen in \u001floc(t), which determines \u000bvia Eq. (2) and\nwhich is shown in Fig. 1 (upper panel) for quarter \flling\nn\u0011N=L = 0:5 where we have a (correlated) metal in the\nentireUrange. In fact, the absolute value of the integral\nweightR\ndt\u001floc(t) grows with increasing U.\nNote that via the \ructuation-dissipation theorem, the\ntotal weightR\ndt\u001floc(t) is given by the negative local\nstatic spin susceptibility. This explains that \u001floc(t) is\nmainly negative.\nSeparation of time scales. A central observation is\nthat\u001floc(t) develops a pronounced two-peak structure for\nstrongU. ForU= 0 and in the weak-coupling regime,\nthere is essentially a single (negative) peak around t\u00181\nonly. This corresponds to fast correlated-hopping pro-\ncesses on a scale set by the inverse hopping 1 =T. As is\nseen in the \fgure, the contribution of these processes to\nthe Gilbert damping grows with increasing U.\nThe second (negative) peak is clearly present for U&\n8. The almost linear shift of its position with Uhints\ntowards a time scale set by an e\u000bective magnetic interac-\ntionJH\u00181=Ubetween local magnetic moments formed\nby strong correlations in the conduction-electron system.\nEven forU!1 , however, local-moment formation is\nnot perfect at quarter \flling: We have hs2\nii=3\n4n=3\n8<\ns(s+ 1) withs= 1=2 for the size of the correlated lo-\ncal moment [12]. This explains the residual contributions\nfrom correlated hopping processes (\frst peak).\nFor strong Uthe Gilbert damping is dominated by\nmagnetic processes: Because of the extra factor tunder\nthe integral in Eq. (2), the contribution of the second\npeak in\u001floc(t) by far exceeds the hopping contribution\n(note the logarithmic scale in Fig. 1). Clearly, \u000bstrongly\nincreases with Uthough a precise value cannot be given\ndue to limitations of the t-DMRG in accessing the long-\n-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=011Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842\n1\nn=0.584100101102\n6080100U!0U!1⌧(U)UFIG. 1: (Color online) Upper panel: Local spin correlation\n\u001floc(t) ati0= 1 for an open Hubbard chain with L= 80 sites\nas obtained by t-DMRG for quarter \flling and di\u000berent Uas\nindicated. ( U\u001532:t\u0000Jmodel with three-site terms [29],\nL= 100 sites; U=1:L= 120). Energy and time scales are\n\fxed by the n.n. hopping T= 1. Lower panel: Resulting real-\ntime dynamics of a classical spin S(t) (withjS(t)j=1\n2, only\nSzis shown) coupled at i0to the local conduction-electron\nspin as obtained from Eq. (5) for J= 1 and di\u000berent U. The\nspin dynamics is initiated by switching the local magnetic\n\feld in Eq. (5) at time t= 0 fromx- toz-direction ( B\fn=\n1).Inset:U-dependence of the relaxation time \u001c, de\fned as\nSz(\u001c) = 0:98jSj.\ntime limit.\nSpin-dynamics model. The simple LLG equation for a\nclassical spin, i.e., _S=S\u0002B\u0000\u000bS\u0002_S, can only provide\nan overall picture of the spin dynamics and in fact ig-\nnores the electronic time-scale separation. We therefore\napply a re\fned approach which explicitly accounts for\nthe conduction-electron degrees of freedom in a model\nHwhich, besides He, Eq. (1), includes the local and\nisotropic coupling between si0and the classical spin S\n(jSj= 1=2):\nH=He+He\u0000spin=He+Jsi0S\u0000BS: (4)\nWe have also added a local magnetic \feld Bwhich, at\ntimet= 0, is suddenly switched from B=Bini^x, forcing\nthe spin to point in xdirection, toB=B\fn^zwithB\fn=\n1 to initiate the spin dynamics. Complete relaxation is\nachieved ifS(t)!1\n2^zfort!1 .\nThe Hamiltonian (4) represents a semiclassical Kondo-\nimpurity model with \fnite Hubbard- U. For a strong\nlocal \feld B\fn, the dynamical Kondo e\u000bect [30] that\nwould show up in case of a quantum-spin S=1\n2is sup-\npressed { this justi\fes the classical-spin approximation.\nThe quantum-classical hybrid (4) also results in the limit3\nof large spin quantum numbers S[31] of a correlated\nquantum-spin Kondo impurity model [32].\nS(t) satis\fes the classical equation of motion _S(t) =\nS(t)\u0002B\u0000JS(t)\u0002hsi0it[33]. Applying lowest-order per-\nturbation theory in J, the Kubo formula yields hsi0it=\nJRt\n0dt0\u001floc(t\u0000t0)S(t0) where the retarded local spin cor-\nrelation\u001floc(t) now plays the role of the linear-response\nfunction. As has been demonstrated in Ref. [17] for\nU= 0, this approach is perfectly reliable even for fairly\nstrong couplings Jand up to the time scale necessary\nfor complete spin relaxation. Here, we choose J= 1\nto generate relaxation times accessible to the t-DMRG\napproach.\nThe resulting e\u000bective integro-di\u000berential equation of\nmotion [6, 16] (see also Refs. [34, 35]),\n_S(t) =S(t)\u0002B\u0000J2S(t)\u0002Zt\n0dt0\u001floc(t\u0000t0)S(t0);(5)\nis numerically solved using a high-order Runge-Kutta [36]\nand quadrature techniques.\nCorrelation e\u000bects in the spin dynamics. The result-\ning spin dynamics (Fig. 1, lower panel) is characterized\nby precessional motion ( Sx(t);Sy(t) not shown) with Lar-\nmor frequency !L/Baround the ^ zaxis, and by relax-\nation driven by dissipation of energy and spin into the\nbulk of the electronic system. In the \fnal state there is\ncomplete alignment, S(t)\"\"B.\nComparing the results for the di\u000berent U, we \fnd that\n(i) signi\fcant relaxation starts at times t\u00181=T, i.e., on\nthe time scale for dissipation through correlated-hopping\nprocesses. (ii) Correlation e\u000bects lead to a considerably\nshorter relaxation time, e.g., by about a factor two when\ncomparing the results for U= 0 andU= 16 (see inset).\n(iii) To some extent this is due to an additional damping\nmechanism, namely via excitations of correlation-induced\nmagnetic moments { at least for moderate U. (iv) For\nstrongU, however, the relaxation time increases again.\nThis is counterintuitive but easily explained: Since the\nsecond, \\magnetic\" peak in \u001floc(t) shifts with increasing\nUto later and later times, relaxation is already com-\npleted before dissipation through spin-\rip processes can\nbecome active. This is most obvious for U!1 where\nmagnetic damping is never activated, and where a renor-\nmalized band picture may apply. (v) At intermediate U,\nhowever, the picture is di\u000berent. Here, spin-\rip processes\ndo contribute to the relaxation but more than an order\nof magnitude later ( t&U=T2) than the hopping time\nscale. Despite their dominating contribution to \u000b, their\ne\u000bect is weaker as compared to the correlated-hopping\nprocesses. Still, spin-\rips leave a clear characteristics in\nSz(t): their additional torque produces oscillations (with\na period largely independent of U) which superimpose\nthe monotonic relaxation dynamics. The onset of these\n\\magnetic\" oscillations linearly grows with U. Note that\nthe interpretation of these (and the following) \fndings\ndoes not rely on the one-dimensionality of the model.Gilbert damping of a Mott insulator. Dissipation\nthrough correlated hopping is impeded or even sup-\npressed at half-\flling where the system is a Mott insula-\ntor for allU > 0. Fig. 2 (upper panel) shows the t-DMRG\ndata for\u001floc(t) atn= 1 and di\u000berent U. Its time depen-\ndence is dominated by a single (negative) structure which\ngrows with increasing Uup to, say,U\u00198. In the weak-\ncoupling regime, U.4, the local magnetic moments are\nnot yet well-formed since the charge gap \u0001 \u0018e\u00001=U(as\nobtained from the Bethe ansatz [37] for U!0) is small\nas compared to T. Hence, residual hopping processes still\ncontribute signi\fcantly.\nIn the strong-coupling limit, on the other hand, spin-\n\rip processes dominate. Here, we observe scaling behav-\nior,\u001floc(t) =F(4tT2=U) with a universal function F(x).\nIndeed, due to the suppression of charge \ructuations, the\nlong-time, low-energy dynamics is captured by a Heisen-\nberg chainHHeis:=JHP\nisisi+1with antiferromagnetic\ninteraction JH= 4T2=Ubetween rigid s= 1=2-spins. As\nJHis the only energy scale remaining, F(tJH) is the re-\ntarded local susceptibility of the Heisenberg chain. With\nF(x) obtained numerically by means of t-DMRG applied\ntoHHeis:atJH= 1, the t-DMRG data for strong Uare\n\ftted perfectly (see Fig. 2). Signi\fcant deviations from\nthe scaling behavior can be seen in Fig. 2 for U= 8 and\nt\u00193, for instance.\nScaling can be exploited to determine the U-\ndependence of the Gilbert damping for a Mott insulator.\nFrom Eq. (2) we get\n\u000b=J2\nJ2\nHZ1\n0dxxF (x) =J2\nJ2\nH\u000b0=J2U2\n16T4\u000b0;(6)\nand thus, for \fxed J;T, we have\u000b/U2. For the univer-\nsal dimensionless Gilbert damping constant \u000b0we \fnd\n\u000b0\u00194:8: (7)\nFor a correlated Mott insulator, Eqs. (6) and (7) com-\npletely describe the U-dependence of the classical-spin\ndynamics in the weak- J, weak-Blimit where the t-\ndependence of S(t) is so slow, as compared to the typical\nmemory time \u001cmemcharacterizing \u001floc(t), that the Taylor\nexpansionS(t0)\u0019S(t) +_S(t)(t0\u0000t) can be cut at the\nlinear order under the t0-integral in Eq. (5), such that the\nLLG equation is obtained as a Red\feld equation [38].\nIncomplete spin relaxation. As demonstated with\nFig. 2 (lower panel), there is an anomalous U-dependence\nof the spin dynamics at n= 1. Only in the weak-\ncoupling regime, U.2, do damping e\u000bects increase and\nlead to a decrease of the relaxation time with increasing\nU. ForU= 4, however, the relaxation time increases\nagain. This behavior is clearly beyond the LLG the-\nory and is attributed to the fact that the memory time,\n\u001cmem/1=JH/Ufor strongU, becomes comparable to\nand \fnally exceeds the precession time scale \u001cB= 2\u0019=B\n(see the Supplementary Material [40]).4\n-0.4-0.3-0.2-0.10.0\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)56Sx(t)Sz(t)U=84U=01201U=3216876542-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842\nU=27U=81632HeisenbergmodelHeisenbergmodel\nn=1.0\nFIG. 2: (Color online) The same as Fig. 1 but for n= 1\n(L= 60). Thin black lines: Heisenberg model with JH=4T2\nU\n(L= 400) and, for improved accuracy at U= 8, with n.n.\nand n.n.n. couplings JH=4T2\nU\u000016T4\nU3andJ0\nH=4T4\nU3[39]\n(L= 300).\nIn addition, as for n= 0:5, we note a non-monotonic\nbehavior of Sz(t) with superimposed oscillations (see U=\n6, for example). With increasing Uthese oscillations die\nout, and at a \\critical\" interaction Uc\u00188 and for all\nU > U cthe relaxation time seems to diverge. Namely,\nthez-component of S(t) approaches a nearly constant\nvalue which decreases with increasing UwhileSx(and\nSy) still precess around B(see inset). Hence, on the\naccessible time scale, Ucmarks a transition or crossover\nto an incompletely relaxed but \\stationary\" state.\nThe same type of dynamical transition is also seen for\na classical spin coupled to a Heisenberg chain for which\nmuch larger system sizes ( L= 400) and thus about an\norder of magnitude longer time scales are accessible to t-\nDMRG. Here, the crossover coupling is JH;c\u00180:5. How-\never, these calculations as well as analytical arguments\nclearly indicate that a state with Sz= const.6= 1=2 is\nunstable and that \fnally, for t!1 , the fully relaxed\nstate withS(t)\"\"Bis reached (see [40] for details).\nThe \\stationary state\" on an intermediate time scale\noriginates when the bandwidth of magnetic excitations\ngets smaller than the \feld { as can be studied in detail\nalready for U= 0 (and very strong B). On the time\naxis, the missing relaxation results from a strong memory\ne\u000bect which, in the strong- Ulimit, shows up for JH.B.\nHere,\u001cmem&\u001cBwhich implies that the z-component of\nthe spin torque on S(t) averages to zero [40].\nThe incomplete spin relaxation can also be understood\nas a transient \\phase\" similar to the concept of a prether-\nmalized state. The latter is known for purely electronicsystems [21{24] which, in close parametric distance to\nintegrability, do not thermalize directly but are trapped\nfor some time in a prethermalized state. Here, for the\nquantum-classical hybrid, the analogue of an \\integrable\"\npoint is given by the U!1 limit where, for every \fnite\nt, the integral kernel \u001floc(t)\u00110, and Eq. (5) reduces to\nthe simple (linear) Landau-Lifschitz equation [1].\nThe situation is also reminiscent of quantum excita-\ntions which are metastable on an exponentially long time\nscale due to a small phase space for decay. An example\nis given by doublons in the Hubbard model which, for U\nmuch larger than the bandwidth and due to energy con-\nservation, can only decay in a high-order scattering pro-\ncess [25{28]. The relaxation time diverges in the U!1\nlimit where the doublon number is conserved. Here, for\naclassical spin, one would expect that relaxation via dis-\nsipation of (arbitrarily) small amounts of energy is still\npossible. Our results show, however, that this would hap-\npen on a longer time scale not accessible to the linear-\nresponse approach while the \\stationary state\" on the\nintermediate time scale is well captured [40].\nOutlook. Correlation-induced time-scale separation\nand incomplete relaxation represent phenomena with fur-\nther general implications. While slow correlation-induced\nmagnetic scales dominate the Gilbert damping \u000b, their\nactivation has been found to depend on microscopic\ndetails. This calls for novel correlated spin-dynamics\napproaches. The combination of t-DMRG with non-\nMarkovian classical spin dynamics is an example how to\nlink the \felds of strongly correlated electron systems and\nspin dynamics, but further work is necessary. Combi-\nnation with dynamical mean-\feld theory [42] is another\npromising option. Also spin dynamics based on LLG-\ntype approaches combined with ab initio band theory\ncould be successful in the case of very strong Uwhere\ndue to the absence of magnetic damping a renormalized\nband picture may be adequate. Further progress is even\nneeded for the very theory of a consistent hybrid-system\ndynamics [33, 43]. Generally, hybrid systems are not\nwell understood and call for a merger of known quantum\nand classical concepts, such as eigenstate thermalization,\nprethermalization, (non-)integrability etc [44]. However,\nalso concrete practical studies with classical spins cou-\npled to conduction electrons [17] are needed, as those\nhold the key for the microscopic understanding of nano-\nspintronics devices [15] or skyrmion dynamics [45, 46].\nAcknowledgements. We thank F. Hofmann for in-\nstructive discussions. Support of this work by the DFG\nwithin the SFB 668 (project B3) and within the SFB 925\n(project B5) is gratefully acknowledged.\n[1] L. D. Landau and E. M. 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Nan-\notechnol. 8, 742 (2013).6\nSupplementary material\nValidity of linear-response theory.\nThe reliability of the linear-response approach (see Eq.\n(5) of the main text) can be tested by comparing with the\nresults of the full (non-perturbative) quantum-classical\nhybrid dynamics for the model given by Eq. (4) of the\nmain text. This is easily accessible for the case U= 0 (see\nRef. [17] for details). Fig. 3 displays the time dependence\nof thez-component of the classical spin for J= 1, for a\nhalf-\flled system ( n= 1) of non-interacting conduction\nelectrons (U= 0) and for di\u000berent strengths of the \feld\nBafter switching from x- toz-direction.\nWhile there are some discrepancies visible, as ex-\npected, the \fgure demonstrates that the agreement on\na qualitative level is in fact excellent for weak as well\nas for strong \felds. Both approaches also predict a\ncrossover from complete to incomplete spin relaxation\natB=Bc\u00194. We conclude that the linear-response\napproach provides reliable results for the classical spin\ndynamics.\nThis can be explained by the observation that jhsi0itjis\nsmall and that the classical spin S(t) and the conduction-\nelectron momenthsi0itare nearly collinear at any instant\nof time (see Ref. [17] for a detailed and systematic discus-\nsion). Hence, even for moderately strong couplings J, the\nlinear-response contribution J2S(t)\u0002hsi0itto the equa-\ntion of motion for S(t) is small (and the quadratic and\nhigher-order corrections are expected to be even smaller).\n10\u00001100101102timet0.00.10.20.30.40.5SzJ=1B=1B=2B=3B=3.5B=4B=5\nFIG. 3: Time dependence of the z-component of the classical\nspin forJ= 1,n= 1,U= 0 and di\u000berent values of the \feld\nBas indicated. Calculations based on the linear-response\napproach (fat solid lines) are compared to the results of the\nfull quantum-classical hybrid theory (thin solid lines) for L=\n500.Mechanism for incomplete relaxation.\nFig. 3 shows that the relaxation of the classical spin\nbecomes incomplete for strong B. On the basis of the\nlinear-response theory this can be explained as follows:\nThexandycomponents of the linear response\nhsi0it=Zt\n0d\u001c\u001f loc(\u001c)S(t\u0000\u001c); (8)\ntend to zero if the characteristic memory time \u001cmem of\nthe kernel\u001floc(\u001c) is much larger than the precession time\nscale\u001cB= 2\u0019=B since the integral produces a vanishing\naverage in this case. This means that the corresponding\ntorque,\u0000J2S(t)\u0002hsi0it, is perpendicular to the \feld\ndirection and hence there is no relaxation of the spin.\nThe same argument can also be formulated after trans-\nformation to frequency space: After some transient ef-\nfect, we havehsi0i!=\u001floc(!)S(!), and thus the x;y-\ncomponents of the linear response will vanish if \u001floc(!=\nB) = 0, i.e., if Bis stronger than the bandwidth of the\nmagnetic excitations (here: Bc\u00194). Note that this\nrequires an unrealistically strong \feld in case of non-\ninteracting conduction electrons.\nIn the case of correlated conduction electrons, jS(t)\u0002\nhsi0itjremains small (of the order of 0.1 or smaller), for\nweak and for strong B, as has been checked numerically.\nWe therefore expect the linear-response approach to pro-\nvide qualitatively correct results for U > 0 as well.\nAt half-\flling and for strong U, the memory time\n\u001cmem/J\u00001\nH/U, i.e.,\u001cmem can easily become large as\ncompared to \u001cB, and thus incomplete spin relaxation can\noccur at comparatively weak and physically meaningful\n\feld strengths. For example, from the Bethe ansatz [41]\nwe have\nWspinon = 2Z1\n0dx\nxJ1(x)\ncosh(Ux=4)!\u0019\n2JHforU!1\n(9)\nfor the spinon bandwidth Wspinon whereJ1(x) is the \frst\nBessel function. Hence, for strong Hubbard interaction,\nBc\u00192Wspinon =\u0019JH.\nClassical spin coupled to a Heisenberg model.\nAt half-\flling and in the limit U!1 the low-energy\nphysics of the Hubbard model is captured by an antifer-\nromagnetic Heisenberg model,\nHs=X\ni(JHsisi+1+J0\nHsisi+2); (10)\nwhere up to order O(T2=U) the nearest-neighbor and\nthe next-nearest-neighbor couplings [39] are JH= 4T2=U\nandJ0\nH= 0, and up to order O(T4=U3),\nJH=4T2\nU\u000016T4\nU3; J0\nH=4T4\nU3: (11)7\n10\u00001100101102timet-0.4-0.3-0.2-0.10.0\u0000loc(t)\nJH=1\nFIG. 4: Local susceptibility at the edge of an open Heisen-\nberg chain ( JH= 1).\nAnalytically, by perturbation theory in x= 4tT2=U=\ntJH, one veri\fes the linear short-time behavior\n\u001floc(t) = \u0002(t)t2\n3(JHhsi0si0+1i+J0\nHhsi0si0+2i) +O(x2);\n(12)\nvalid to leading order for both, the Hubbard and the\ne\u000bective Heisenberg model. However, as can be seen in\nFig. 2 of the main text, the Heisenberg dynamics also\napplies to intermediate times; the e\u000bective model with\ncoupling constants (11) almost perfectly reproduces the\nresults of the Hubbard model for U\u00158.\nHere, we treat the Heisenberg model with n.n. cou-\npling as an independent system. Fig. 4 shows the cor-\nresponding local spin susceptibility for JH= 1 as ob-\ntained by t-DMRG calculations with L= 400 Heisen-\nberg spins. Since JHis the only energy scale, we have\n\u001floc(t) =F(tJH) for arbitrary JHwhereF(x) is a func-\ntion independent of JH. This implies that the dominant\n(negative) peak of \u001floc(t) shifts to later and later times\nasJHdecreases.\nFig. 5 displays the spin dynamics resulting from the\nfull model\nH=Hs+Hs\u0000spin=Hs+Jsi0S\u0000BS; (13)\nas obtained by the linear-response approach. One clearly\nnotes that for J.Jc\u00180:5 (corresponding to Uc\u00188)\nthe time dependence of Szdevelops a prethermalization-\nlike plateau on an intermediate time scale t\u0018100 (in\nunits of 1=JH).\nFor the Heisenberg model, using the scaling property\nof\u001floc(t), it is easily possible to perform calculations up\ntot= 1000. On this longer time scale, it is clearly vis-\nible (see Fig. 5) that Szdoes notapproach a constant\nvalue asymptotically. For JH= 0:4 andJH= 0:2 thez-\ncomponent of S(t) is even found to decrease and appears\nto approach the trivial solution Sz(t)\u00110.\nHowever, it is straightforwardly seen that a \\stationary\nstate\" of the form\nS(t) =Sz^z+S?cos(!t+')^x+S?sin(!t+')^y(14)\n10\u00001100101102timet-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5S(t)Sx(t)Sz(t)JH=1.0JH=0.80.60.50.40.2FIG. 5: Classical spin dynamics for J= 1 and di\u000berent JH.\nwith arbitrary parameters S?;!;' and with constant\n(time-independent) Szdoes not solve the integro-\ndi\u000berential equation (5) of the main text for t! 1 .\nThere is one exception only, namely the trivial case where\n\u001floc(t)\u00110 which can be realized, up to arbitrarily long\ntimes, in the limit JH!0.\nFor small but \fnite JH>0, we therefore expect that\nthe classical spin develops a dynamics on an extremely\nlong time scale t\u001d103, the onset of which is already seen\nin Fig. 5, which \fnally terminates in the fully relaxed\nstate withS(t)!S0\"\"B.\nIt is in fact easy to see from the integro-di\u000berential\nequation that, if there is spin relaxation to a time-\nindependent constant, S(t)!S0fort!1 , the relaxed\nstate hasS0= 0:5^z. This implies that if there is com-\nplete relaxation at all, the spin relaxes to the equilibrium\ndirection.\nOscillations at short times.\nAs can be seen in Fig. 5 for weak JH, thez-component\nof the spin develops oscillations at short times, which can\nalso be seen for the case of the Hubbard model (cf. Fig. 2\nof the main text). These oscillations can be understood\nin the following way: Inserting the expression (12) with\nJ0\nH= 0 for the behavior of \u001floc(t) at short times into Eq.\n(5) of the main text,\n_S(t) =S(t)\u0002B\u00002\n3J2JHhsi0si0+1iS(t)\n\u0002Zt\n0dt0(t\u0000t0)S(t0) +O(t3J4); (15)\nand approximating S(t) by theJ= 0 resultS0(t) =\nS(cos!t;sin!t;0) (withB=B^z,!=B,S= 1=2) in\nthe second term on the right-hand side, a straightforward\ncalculations yields:\nSz(t) =2\n3J2JHhsi0si0+1iS2!tsin!t+ 2 cos!t\u00002\n!3\n+O(t4J2J2\nH): (16)8\nThis is found to perfectly describe the short-time oscil-\nlations for weak JHin Fig. 5 and for strong Uin Fig. 2in the main text. For longer times the oscillations are\ndamped and eventually die out."    },    {        "title": "0911.4628v1.Origin_of_adiabatic_and_non_adiabatic_spin_transfer_torques_in_current_driven_magnetic_domain_wall_motion.pdf",        "content": "arXiv:0911.4628v1  [cond-mat.mes-hall]  24 Nov 2009Origin of adiabatic and non-adiabatic spin transfer torque s in current-driven magnetic\ndomain wall motion\nJun-ichiro Kishine\nDepartment of Basic Sciences, Kyushu Institute of Technolo gy, Kitakyushu, 804-8550, Japan\nA. S. Ovchinnikov\nDepartment of Physics, Ural State University, Ekaterinbur g, 620083 Russia\n(Dated: 24 November 2009)\nAconsistent theorytodescribe thecorrelated dynamics ofq uantummechanical itinerantspins and\nsemiclassical local magnetization is given. We consider th e itinerant spins as quantum mechanical\noperators, whereas local moments are considered within cla ssical Lagrangian formalism. By appro-\npriately treating fluctuation space spanned by basis functi ons, including a zero-mode wave function,\nwe construct coupled equations of motion for the collective coordinate of the center-of-mass motion\nand the localized zero-mode coordinate perpendicular to th e domain wall plane. By solving them,\nwe demonstrate that the correlated dynamics is understood t hrough a hierarchy of two time scales:\nBoltzmann relaxation time τel, when a non-adiabatic part of the spin-transfer torque appe ars, and\nGilbert damping time τDW, when adiabatic part comes up.\nSpin torque transfer (STT) process is expected to rev-\nolutionize the performance of memory device due to non-\nvolatility and low-power consumption. To promote this\ntechnology, it is essential to make clear the nature of the\ncurrent-driven domain wall (DW) motion[1, 2]. Recent\ntheoretical [3, 4, 5, 6, 7, 8, 9, 10] and experimental[11]\nstudies have disclosed that the STT consists of two vec-\ntors perpendicular to the local magnetization m(x) and\ncan be written in general as N=c1∂xm+c2m×\n∂xm[3]. The c1andc2-terms respectively come from\nadiabatic[1, 4] and non-adiabatic [5] processes between\nconduction electrons and local magnetization, and the\nterminal velocity of a DW is controlled by not c1but\nsmallc2term. The origin of the c2term is ascribed to\nthe spatial mistracking of spins between conduction elec-\ntrons and local magnetization[5]. Behind appearance of\nthec2term is the so called transverse spin accumula-\ntion (TSA) of itinerant electrons generated by the elec-\ntric current[6, 7]. Now, any consistent theory should ex-\nplain how the adiabatic and non-adiabatic STT come up\nstarting with microscopic model. In particular, it should\nbe made clear how the TSA caused by the non-adiabatic\nSTTeventuallyleadstotranslationalmotionofthewhole\nDW. In this letter, to solve this highly debatable prob-\nlem, we propose a consistent theory to describe the cor-\nrelated dynamics of quantum mechanical itinerant spins\nand semiclassical local magnetization.\nWe consider a single head-to-head N´ eel DW through\na magnetic nanowire with an easy xaxis and a hard\nzaxis. Fee electrons travel along the DW axis ( x-\naxis). We describe a local spin by a semiclassical vec-\ntorS=Sn=S(sinθcosϕ,sinθsinϕ,cosθ) whereS=|S|\nand the polar coordinates θandϕare assumed to be\nslowly varying functions of one-dimensional coordinate x\n[Fig.1(a)]. The DW formation is described by the Hamil-\ntonian (energy per unit area) in the continuum limit,\nHDW=JS2\n2a/integraldisplay∞\n−∞dx/bracketleftBig\n(∂xn)2−λ−2ˆn2\nx+κ−2ˆn2\nz/bracketrightBig\n,(1)\nwhereais the cubic lattice constant, Jis the ferromag-\nnetic exchange strength, λ=/radicalbig\nJ/Kandκ=/radicalbig\nJ/K⊥respectively represent the single-ion easy and hard axis\nanisotropies measured in the length dimension. The\nstationary N´ eel wall ( θ0=π/2) is described by n0=\n(cosϕ0,sinϕ0,0) withϕ0(z) = 2arctan( ex/λ). In the in-\nfinite continuum system, the DW configuration has con-\ntinuous degeneracy labeled by the center of mass posi-\ntion,X, of the DW. This degeneracy apparently leads to\nrigidtranslationoftheDW,i.e., n0(x)→n0(x−X)[12].\nAs explicitly shown below, however, the translation in\noff-equilibrium accompanies internal deformation of the\nDW.\nThe creation operator of a conduction electron is writ-\nten in a spinor form as c†(x) = (c†\n↑(x),c†\n↓(x)). By per-\nforming the local gauge transformation c(x) =ˆU(x)¯c(x)\nwith the unitary operator ˆU(x) =eiˆσzϕ0(x)/2(ˆσzis a\n(a) \n(b)\nzx−y−\n0n\nxy\nTSAT1\nT2\nOPZAy\nxz0nx−y−\nX\n0\n00\nT2TT\nns'x\n'0\nȟx\nFIG. 1: (a) Stationary configuration of local spins ( n0) asso-\nciated with a single N´ eel wall. Labotatory frame x, y, zand\nlocal frame ¯ x,¯y, zare indicated. (b) Schematic view of the\ntransverse spin accumulation (TSA) of itinerant spin sand\nthe out-of-plane ( θ) zero-mode accumulation (OPZA) of local\nspinn. These magnetic accumulations respectively cause the\nnon-adiabatic torque T2and adiabatic torque T1.2\nPauli matrix) the quantization axis becomes parallel to\nthe local spin located at x. Assuming |a∂xϕ0(x)| ≃\na/λ≪1, i.e. wall thickness is much larger than atomic\nlattice constant, this procedure leads to the single-\nparticle Hamiltonian,\nHel=/planckover2pi12\n2m∗a/integraldisplay∞\n−∞dx/bracketleftbigg1\n2|∂x¯c|2+i(∂x¯c†)ˆAz¯c/bracketrightbigg\n+c.c,(2)\nwhere the effective mass of the conduction electron is\nm∗. The SU(2) gauge field[8, 13] is introduced as ˆAz≡\ni−1ˆU−1∂xˆU=−(∂xϕ0)ˆσz/2. The conduction electrons\nare assumed to interact with the local spins by a s-d\ncoupling represented in the form,\nHsd=−Jsd\na3/integraldisplay∞\n−∞dxˆs(x)·S(x−X),(3)\nwhereˆsandS=Snare respectively the spins of itin-\nerant and localized electrons .We treat ˆs(x) =1\n2c†ˆσcas\nfully quantum mechanical operator, while nis a semi-\nclassical vector.\nBoltzmann relaxation : let switch on the electric field\nEatt= 0. We introduce the Boltzmann relaxation\ntimeτeland the number density of the conduction elec-\ntronsfkσin the state k,σ. We assume that the devia-\ntionfromequilibriumFermi-Diracdistribution f0(εkσ) =\n[exp[(εkσ−µ)/kBT]+1]−1issmall, where εkσ(σ=↑,↓)\nis the single-particle energy, µis the chemical potential.\nUsing standard Boltzmann kinetic equation with relax-\nation time approximation[6], the distribution function is\nwritten as\nfkσ≃f0(εkσ)+eEτelvkσ∂f0(εkσ)\n∂εkσ,(4)\nwhere the electron charge is −eand the spin-dependent\nvelocity is vkσ≡/planckover2pi1−1∂εkσ/∂k. The spin-dependence of\nεkσoriginates from the SU(2) gauge fields ( ˆAz)↑↑and\n(ˆAz)↓↓. In the process of approaching to stationary cur-\nrentflowingstate aroundthe time t∼τel, aswewillshow\nexplicitly, the statistical average of the conduction elec-\ntron’s spin component perpendicular to the local quan-\ntization axis accumulates and acquires finite value. As\nschematically depicted in Fig.1(b), this process is ex-\nactly the TSA. The TSA causes an additional magnetic\nfield actingon the localspins and exertthe non-adiabatic\ntorque on the local spins.\nLocal spin dynamics : next we formulate dynamics of\nthe local spins coupled with the conduction electrons.\nWe introduce the δθ(x,t) (out-of-plane) and δϕ(x,t)\n(in-plane) fluctuations of the local spins around the sta-\ntionary DW configuration n0(x).We say “out-of-plane”\nand “in-plane” with respect to the DW plane. The\nfluctuations are spanned by the orthogonal basis func-\ntionsvqanduqasϕ(x) =ϕ0(x−X)+δϕ(x−X) and\nθ(x) =π/2+δθ(x−X),where\nδϕ(x) =/integraldisplay∞\n−∞dq ηq(t)vq(x), δθ(x)=/integraldisplay∞\n−∞dq ξq(t)uq(x).\n(5)\nAt this stage, Xis not a dynamical variable, but just\na parameter. The basis functions obey the Schr¨ odingerequations,/parenleftbig\nJS2/2/parenrightbig\n(−∂2\nx−2λ−2sin2ϕ0+λ−2)vq(x) =\nεϕ\nquq(x) and/parenleftbig\nJS2/2/parenrightbig\n(−∂2\nx−2λ−2sin2ϕ0+λ−2+κ−2)\nuq(x) =εθ\nquq(x).Bothθandϕmodes consist of a sin-\ngle bound state ( zero mode ) and continuum states ( spin-\nwave modes ). The dimensionless zero mode wave func-\ntions are given by u0(x) =v0(x) = Φ0(x), where\nΦ0(x)≡/radicalbigg\naλ\n2∂xϕ0(x) =/radicalbigga\n2λ1\ncosh(x/λ),(6)\nwith the corresponding energies respectively given by\nεθ\n0=JS2/(2κ2) andεϕ\n0= 0. The normalization is\ngiven by a−1/integraltext∞\n−∞dx[Φ0(x)]2= 1.Although to excite\nthe out-of-plane ( θ) zero mode costs finite energy gap\nεθ\n0coming from the hard-axis anisotropy, we still call\nthis “zero mode.” The spin-wave states have energy\ndispersions given by εθ\nq=1\n2JS2/parenleftbig\nq2+λ−2+κ−2/parenrightbig\nand\nεϕ\nq=1\n2JS2/parenleftbig\nq2+λ−2/parenrightbig\n. Because the zero mode and the\nspin-wave states are orthogonal to each other and sepa-\nrated by the anisotropy gaps, the spin-wave modes are\ntotally irrelevant to a low energy effective theory. There-\nfore, we ignore the spin-wave modes from now on.\nOut-of-plane zero-mode(OPZ) coordinate ξ0: in order\nto obtain the correct form of the dynamical Hamiltonian,\none has to regard the variable Xas a dynamical variable\nX(t) and replace the zero mode coordinate η0withX(t).\nFollowing this idea, the zero-mode fluctuations should be\ngiven by,\nϕ(x,t) =ϕ0[x−X(t)], (7)\nθ(x,t) =π/2+ξ0(t)Φ0[x−X(t)].(8)\nEq. (8) is a key ingredient of this letter, which has\nneverbeen explicitly treated sofar[14]. That is to say, we\nnaturally include the out-of-plane(OPZ) zero-mode, in\naddition to the in-plane ( ϕ) zero-mode replaced by X(t).\nThe zero-mode wave function Φ 0[x−X(t)] serves as the\nbasis function of the θ-fluctuations localized around the\ncenter of the DW and ξ0(t) is the OPZ coordinate . Now,\nour effective theory is fully described by two dynamical\nvariables X(t) andξ0(t) which naturally give physical\ncoordinates along the Hilbert space of orthogonal θand\nϕfluctuations. As we will see, we have ξ0(t)/ne}ationslash= 0 only\nfor inequilibrium current flowing state under E/ne}ationslash= 0 [Fig.\n2(a)].\nIt is here important to note an essential difference be-\ntween Tatara and Khono’s approach[8] and ours. Tatara\nand Khono used X(t) and the weighted average, θ0(t)\n=/integraltext∞\n−∞dx θ(x,t)sin2ϕ[x−X(t)], as dynamical vari-\nables. Later, they systematically used complex coordi-\nnateξ=eiϕtan(θ/2) and described the fluctuations in\nthe form ξ=e−u(x,t)+iϕ0+η[x−X(t)][9](their notation is\nreproduced by putting θ→π/2−θ,ϕ→ϕin our nota-\ntion). Inourunderstanding, thesedescriptionsinevitably\ncause redundant coupling between uandvmodes in Eq.\n(5). Actually, our natural choice of the dynamical vari-\nables is essential to appropriately derive relaxational dy-\nnamics described by the following equations of motion\ngiven by (12a) and (12b).\nEquations of motion of the DW : now, we construct\nan effective Lagrangian L=LDW+Lsdto describe the3\nxπ\n0\nXπ/2(a)\nj (b)\n(c)0\nȟ'0\nj OPZAȥX.\n0,\nˢk\nˢkJsd ȥ0ȍ0\nFIG. 2: (a) Spatial profile of the polar angles ϕ(x,t) =\nϕ0[x−X(t)] andθ(x,t) =π/2 +ξ0(t)Φ0[x−X(t)] in the\ncurrent flowing state. (b) Linear dependence of ξ0and˙X(t)\non thecurrentdensity j. (c) Single- particle propagation (rep-\nresented by solid line) with spin flip process by the s-d inter -\naction (represented by wavy line) which leads to the STT.\nDW motion and resultant equations of motion (EOM).\nUsing (7) and (8), the local spin counterpart is given by\nLDW=/planckover2pi1S\na3/integraltext∞\n−∞dx(cosθ−1) ˙ϕ−HDWexplicitly written\nas\nLDW=/planckover2pi1S\na3/parenleftBigg/radicalbigg\n2a\nλξ0+π/parenrightBigg\n˙X−JS2\n2κ2ξ2\n0.(9)\nTo understand the effect of the s-d coupling, it is use-\nful to note n[θ0+δθ,ϕ0+δϕ]≃n0−ezδθ−n0δθ2/2,\nwhere we dropped δϕbecause this degree of freedom is\neliminated by the global gauge fixing[14]. We have thus\ns-d Lagrangian,\nLsd=a−3JsdS/parenleftbig\nF0−S/bardblξ02/2/parenrightbig\n, (10)\nwhere,F0[X(t)]≡/integraltext∞\n−∞dxˆn0[x−X(t)]· /an}bracketle{ts(x,t)/an}bracketri}ht\nandS/bardbl[X(t)]≡/integraltext∞\n−∞dx{Φ0[x−X(t)]}2n0[x−X(t)]·\n/an}bracketle{ts(x,t)/an}bracketri}ht. Finally, to take account of dissipative\ndynamics, we use the Rayleigh dissipation function\nWRayleigh=α\n2/planckover2pi1S\na3/integraltext∞\n−∞dx˙n2explicitly written as\nWRayleigh=α\n2/planckover2pi1S\na3/parenleftbigg\na˙ξ2\n0+2\nλ˙X2/parenrightbigg\n,(11)\nwhereαis the Gilbert damping parameter. It is simple\nto write down the Euler-Lagrange-Rayleigh equations,\nd(∂L/∂˙qi)/dt−∂L/∂q i=−∂W/∂˙qi,for the dynami-\ncal variables q1=Xandq2=ξ0. We obtain the EOMs\nwhich contain the dynamical variables in linear order,\n/planckover2pi1/radicalbigg\n2a\nλ˙ξ0+JsdT⊥=−2α/planckover2pi1\nλ˙X, (12a)\n−/planckover2pi1/radicalbigg\n2a\nλ˙X+/parenleftbigga3JS\nκ2+JsdS/bardbl/parenrightbigg\nξ0=−α/planckover2pi1a˙ξ0,(12b)\nwhere the quantities\nT⊥≡ −∂F0\n∂X=/integraldisplay∞\n−∞dx ∂xϕ0[x−X(t)]/an}bracketle{t¯sy(x)/an}bracketri}ht,(13a)\nS/bardbl≡/integraldisplay∞\n−∞dxΦ2\n0[x−X(t)]/an}bracketle{t¯sx(x)/an}bracketri}ht, (13b)respectively give the non-adiabatic STT and longitudinal\nspin accumulation[7]. The statistical average of the con-\nduction electron’s spin component is denoted by /an}bracketle{t···/an}bracketri}ht.\nThe gauge-transformed spin variables are introduced by\n¯s(x) =ˆU−1[x−X(t)]ˆs(x)ˆU[x−X(t)] which has lo-\ncal quantization axis tied to the local spin at the po-\nsition of x−X(t). To obtain Eq.(13a), we used rela-\ntions∂xn0[x−X(t)] =−∂xϕ0(x)ez×n0[x−X(t)] and\n/an}bracketle{t¯sy/an}bracketri}ht=−/an}bracketle{tˆsx/an}bracketri}htsinϕ0+/an}bracketle{tˆsy/an}bracketri}htcosϕ0.The relation (13a) im-\nplies that the translation of the DW ( x→x−X) natu-\nrally gives rise to the TSA, /an}bracketle{t¯sy/an}bracketri}ht, along the local ¯ yaxis.\nThe appearance of /an}bracketle{t¯sy/an}bracketri}htcauses local magnetic moment\nwhich triggers the local spins to precess around the local\n¯yaxis and consequently produce finite deviation of the\npolar angle δθ=θ−θ0.It is seen that upon switching the\nexternal electric field, the deviation δθrelaxes to finite\nmagnitude in the stationary current-flowing state, i.e.,\nthe OPZ coordinate ξ0(t) accumulates and reaches finite\nterminal value ξ∗\n0. We call this process out-of-plane zero-\nmode accumulation(OPZA)as schematically depicted in\nFig.1(b). This effect is physically interpreted as appear-\nance of demagnetization field phenomenologically intro-\nduced by D¨ oring, Kittel, Becker[15], and Slonczewski[1].\nIt is also to be noted that we ignored the term ∂S/bardbl/∂X.\nThis simplification is legitimate for the case of of small\nsd-coupling.\nGilbert relaxation : coupled equations of motion (12a)\nand(12b)arereadilysolvedtogiverelaxationalsolutions,\nξ0=ξ∗\n0(1−e−t/τDW), V≡˙X=V∗(1−e−t/τDW),(14)\nwhere the OPZA reaches the terminal value,\nξ∗\n0=−1\nα/radicalbig\naλ/2JsdT⊥/parenleftbig\na3JSκ−2+JsdS/bardbl/parenrightbig≃ −α−1/radicalbigg\nλ\n2a/parenleftBigκ\na/parenrightBig2Jsd\nJST⊥,\n(15)\nand correspondingly the terminal velocity of the DW\nreachesV∗=−λ\n2α/planckover2pi1JsdT⊥.The relaxation time of the\nDW magnetization, τDW, is given by\nτDW=/planckover2pi1aα−1+α\nκ−2a3JS+JsdS/bardbl≃α−1/parenleftBigκ\na/parenrightBig2/planckover2pi1\nJS.(16)\nThis result clearly shows that the DW magnetization try\nto relax through the Gilbert damping toward the direc-\ntion of the newly established precession axis. We stress\nthat without the OPZ coordinate ξ0in Eqs. (12a) and\n(12b), onlytheterminalvelocityisavailableandthetran-\nsient relaxational dynamics is totally lost.\nAs depicted in Fig.2(a), the OPZA[Eq. (15)] gives rise\nto finite out-of-plane ( z) component of the local spin,\nnz(x,t) = cosθ≃1\n2α/parenleftBigκ\na/parenrightBig2Jsd\nJS1\ncosh[(x−X(t))/λ]T⊥.\n(17)\nThe resultant local spin S⊥=Seznz(x,t) gives the de-\nmagnetization field phenomenologically treated by Slon-\nczewski and gives rise to the adiabatic torque T1=\nc1∂xn(x) =c1(∂xϕ0)(−sinϕ0,cosϕ0,0). At the in-\nterface of the DW boundary, ϕ0=π/2 andT1=\nc1(∂xϕ0)(−1,0,0), i.e., the adiabatic torque rotate the\nlocal spin to counterclockwise direction when the electric4\ncurrent flows in the (1 ,0,0)-direction. As is clear from\nthe above discussion, this adiabatic torque is established\nafterthe stationary current-flowing [ j= (ne2τel/m∗)E]\nstate establishes the non-adiabatic torque, T⊥. Around\nthe time scale of t≃τel+τDW, the whole system (in-\ncluding conduction electrons and DW) reaches non-\nequilibrium but stationary state. In this state, the DW\nmagnetizations continuously feel the OPZA and macro-\nscopically rotate around it. This process exactly corre-\nsponds to stationary translation of the DW.\nComputation of T⊥: the final step is to compute an\nexplicit form of T⊥. By taking Fourier transform ¯ ckσ(t)\n=1√\nL/summationtext\nkeikx¯cσ(x,t), and retaining only the momentum\nconserving process, we have\nT⊥=1\n2/integraldisplayπ/a\n−π/adkReG<\nk↑,k↓(t,t), (18a)\nS/bardbl=a\n2π/integraldisplayπ/a\n−π/adkImG<\nk↑,k↓(t,t).(18b)\nHere, the expectation values are computed by using the\nlesser component of the path-oriented Green function\nG<\nkσ,k′σ′(t,t′) =i/an}bracketle{t¯c†\nk′σ′(t′)¯ckσ(t)/an}bracketri}ht, wheret(t′) is defined\non the upper (lower) branch of Keldysh contour. Since\nS/bardbldoes not play an essential role, we pay attention to an\nessential quantity T⊥. To evaluate the Green functions,\nwe perturbatively treat the s-d coupling and write down\nthe Dyson equation. Then, we truncate the Dyson equa-\ntion by using the Born approximation including the s-d\ncoupling in linear order which causes a single spin flip\nprocess [Fig.2(c)] and gives rise to off-diagonal compo-\nnent in spin space,\nG<\nk↑,k↓(t,t) =−iJsd\n2fk↑−fk↓\nεk↑−εk↓−i0.(19)\nTo obtain the explicit form of εkσ, we write the single-\nparticle Hamiltonian (2) in Fourier space and obtain\nHel=H0+Hgauge, whereH0represents free conduction\nandHgaugecomes form the second term in Eq. (2). By\nretaining only momentum conserving process, we have\nHel=/summationtext\nk,σεkσ¯c†\nkσ¯ckσ,whereεk↑,↓=/planckover2pi12(k∓δk)2/2m∗,\nwhere the shift of the Fermi wave numbers due to the\nbackground DW is given by δk=π/(2a).\nUsing Eqs. (4), (19), and (18a), we finally obtain the\nSTT which points in the z-direction, T1=T⊥ez, where\nits magnitude is given in a form,\nT⊥=1\n4Jsd\nkBT1\ncosh2[(ε0−µ)/2kBT]j\nj0,(20)\nwherej0= 4ne/planckover2pi1/(πam∗) andε0=/planckover2pi12π2/(8m∗a2) corre-\nspondstothechemicalpotentialathalf-filling. Wehavea\nmaster formula which gives relation between the current\ndensity and the terminal velocity of the DW,\nV∗=−1\n8αλJsd\n/planckover2pi1Jsd\nkBT1\ncosh2[(ε0−µ)/2kBT]j\nj0.(21)\nAs shown in Fig.2(b), we see there is no threshold for the\nvelocity, which is consistent with the result obtained byThiaville et al.[10]. Standard choice of parameters, j0≃\n1016[A·m−2],λ= 10−8[m],α= 10−2,j≃1011[A·m−2]\ngive a rough estimate V∗≃ −100(Jsd/kBT)2[m/s].Of\ncourse, to pursuit more quantitative result needs numer-\nical estimation of T⊥taking account of real band struc-\nture.\nIt is essential that the Gilbert damping coefficient, α,\nentersEq.(21). Therelaxationprocessofthe DWdynam-\nics is governed by the Boltzmann relaxation followed by\nthe Gilbert damping in hierarchical manner. As summa-\nrized in Figs.2(a) and (b), in our treatment, it is crucial\nto recognize that the OPZ coordinate ξ0acquires finite\nvalue (i.e., accumulation) only for the current flowing\nstate which is non-equilibrium but stationary. This is the\ncase where dynamical relaxation leads to finite accumula-\ntion of physical quantities which are zero in equilibrium.\nAlthough essential role of the sliding mode to describe\nlocalized spin dynamics was pointed out before[8, 12]\nand importance of out-of-plane canting of the local spins\nwas stressed[1, 8], the OPZA presented in this letter has\nnot been discussed before. For example, the sliding mo-\ntion in Ref.[12] does not contain internal deformation\nof the DW. The OPZA is an outcome of time-reversal-\nsymmetry breaking by electric current. This interpre-\ntation seems natural because current-flowing state is off\nequilibrium.\nJ. K. acknowledges Grant-in-Aid for Scientific Re-\nsearch (C) (No. 19540371) from the Ministry of Educa-\ntion, Culture, Sports, Science and Technology, Japan.\n[1] J.C. Slonczewski, J. Magn. Magn. Mat. 159L1 (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] M. D. Stiles and A. Zangwill, Phys. Rev. B66,\n014407(2002).\n[4] Ya.B. Bazaliy, B.A. Jones, and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[5] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[6] J. Xiao, A. Zangwill and M. D. Stiles, Phys. Rev. B73,\n054428 (2006).\n[7] S. Zhang, P.M. Levy and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[8] G. Tatara, H.Kohno, Phys.Rev.Lett. 92, 086601 (2004).\n[9] G. Tatara, H. Kohno and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[10] A. Thiaville, et al.,Europhys. Lett. 69, 990 (2005).\n[11] S. Petit, et al.,Phys. Rev. Lett. 98, 077203 (2007); Z.\nLi,et al.,Phys. Rev. Lett. 100, 246602 (2008).\n[12] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95,\n107204 (2005).\n[13] G. E. Volovik, J. Phys. Condens. Matter 20, 83(1987).\n[14] The out-of-plane zero-mode was discussed in the con-\ntext of chiral helimagnet by the present authors:\nI. G. Bostrem, J. Kishine, and A. S. Ovchinnikov, Phys.\nRev.B77, 132405 (2008); Phys. Rev. B78, 064425\n(2008); I.G. Bostrem, J. Kishine, R. V. Lavrov, A.S.\nOvchinnikov, Phys. Lett. A 373, 558(2009).\n[15] W. D¨ oring, Zeits. f. Naturforschung 3a, 374 (1948); R.\nBecker, Proceedings of the Grenoble Conference, July\n(1950); C. Kittel, Phys. Rev. 80, 918 (1950)."    },    {        "title": "2109.05901v1.Control_of_magnetization_dynamics_by_substrate_orientation_in_YIG_thin_films.pdf",        "content": "1 \n Control of magnetization dynamics by substrate orientation in  YIG thin films \nGanesh Gurjar1, Vinay Sharma3, S. Patnaik1*, Bijoy K. Kuanr2,* \n1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n2Special Centre for Nanosciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n3Morgan State University, Department of Physics, Baltimore, MD, USA 21251  \n \n \nAbstract  \nYttrium Iron Garnet (YIG) and b ismuth (Bi)  substituted YIG (Bi 0.1Y2.9Fe5O12, BYG) films are \ngrown  in-situ on single crystalline Gadolinium Gallium Garnet (GGG) substrates  [with (100) and \n(111) orientation s] using  pulsed laser deposition  (PLD ) technique .  As the orientation of the Bi-\nYIG film changes from (100) to (111) , the lattice constant  is enhanced  from 12.384  Å to 12.401 Å \ndue to  orientation dependent distribution of Bi3+ ions at dodecahedral  sites in the lattice cell. \nAtomic force microscopy (AFM) images show smooth  film surfaces  with roughness 0.308 nm in \nBi-YIG (111) . The change in substrate orientation leads to the modification of Gilbert damping \nwhich , in turn, gives rise to the enhancement of  ferromagnetic resonance (FMR) line width . The \nbest value s of Gilbert damping are found to be  (0.54±0.06 )×10-4, for YIG (100)  and \n(6.27±0.33) ×10-4, for Bi-YIG (111)  oriented  films . Angle variation ( ) measurements of the H r are \nalso performed, that shows  a four -fold symmetry for the resonance field  in the  (100) g rown film. \nIn addition, the value of  effective magnetization  (4πM eff) and extrinsic linewidth  (ΔH 0) are \nobserved to be  dependent on  substrate orientation . Hence PLD growth can assist single -crystalline \nYIG and BY G films with  a perfect interface that can be used for  spintronics  and related device \napplications.  \n \n \n \nKeyword s: Pulse Laser Deposition, Epitaxial YIG thin films, lattice strain, ferromagnetic \nresonance, Gilbert damping, inhomogeneous broa dening  \n \nCorresponding authors:  bijoykuanr@mail.jnu.ac.in , spatnaik@mail.jnu.ac.in  2 \n 1. Introduction  \nOne of the most widely studied material s for the realization of spintronic devices appears to be the \niron garnets , particularly the  yttrium iron garnet (YIG , Y3Fe5O12) [1,2] .  In thin film form  of YIG  \nseveral potential applications have been envisaged that include spin-caloritronics  [3,4] , magneto -\noptical (MO) devices, and microwave resonators, circulators, and  filters [5–8]. The attraction of \nYIG over other ferroic materials is primarily due to their strong magnet o-crystalline  anisotropy \nand low magnetization damping  [2]. Furthermore, towards  high frequency  applications, YIG’s \nmain advantage s are its electrically insulating behavior  along with low ferromagnetic resonance \nline-width (H) and low Gilbert damping  parameter [9–11]. These are important parameters for \npotential  use in high fr equency filters and actuators [12–14]. In this paper, we report optimal \ngrowth parameters for pure and Bi -doped YIG on oriented subs trates and identify the conditions \nsuitable for their  prospective  applications.  \n \nIn literature, YIG is known to be a room temperature  ferrimagnetic insulator with a  Tc near 560 K \n[15]. It has a cubic structure (space group Ia3̅d). The y ttrium  (Y) ions occupy the  dodecahedral  \n24c sites ( in the Wyckoff notation), two Fe ions  at octahedral 16a and three at tetrahedral 24d sites, \nand oxygen the 96h sites  [16,17] . The d site is resp onsible for the ferri magnetic  nature of YIG. It \nis already reported that substitution of Bi in place of Y in YIG leads to substantial improvement in \nthe magneto -optical response [7,18 –25]. It was also observed that MO performance increa ses \nlinearly with Bi/Ce doping concentration [22]. Furthermore, substitution of  Bi in YIG  (BYG)  is \ndocumented to provide growth -induced anisotropy  that is useful in applications such as magnetic \nmemory and logic devices [26–30]. The study of basic properties of  Bi -substituted YIG materials \nis of great current interest  due to their applications  in magneto -optical devices , magnon -3 \n spintronics , and related fields such as caloritronics  due to its high uniaxial anisotropy and faraday \nrotation  [21,31 –35]. The structural and magnetic pr operties can be changed via change in Bi3+ \nconcentration in YIG or via choosing a proper  substrate orientation. Therefore, t he choice of \nperfect substrate orientation is crucial for the identification of the growth of Bi  substituted YIG \nthin films.  \nIn this  work, we have studied the structural and magnetic properties  of Bi-substituted YIG \n[Bi0.1Y2.9Fe5O12 (BYG)] and YIG thin films with two different single crystalline Gadolinium \nGallium Garnet (GGG)  substrate orientation s: (100) and (111) . The YIG and BYG films of  \nthickness  ~150 nm  were grown by pulsed laser deposit ion (PLD) method  [23,36,37]  on top of  \nsingle -crystalline GGG  substrates . The structural and magnetic properties of all grown films were \ncarried out using x -ray diffraction (XRD), surface morphology by atomic force microscopy \n(AFM) , and magnetic properties via vibrating sample magnetometer (VSM) and ferromagnetic \nresonance (FMR) techniques.  The FMR is the most useful  technique to study the magnetization \ndynamics by measuring the  properties of magnetic materials through evaluation of their damping \nparameter and linewidth .  Furthermore, it provides insightful information on the static magnetic \nproperties such as the saturation magnetization and the anisotropy field. FMR is also extremely \nhelpful to study fundamentals of spin wave dynami cs and towards characte rizing the relaxation \ntime and L ande g factor of  magnetic material s [11]. \n \n2. Experiment  \n \nYIG a nd BY G target s were synthesized  via the solid -state reaction method . Briefly, y ttrium oxide \n(Y2O3) and iron oxide (Fe 2O3) powder s were ground for ~14 hours  before calcination at 1100 oC. 4 \n The calcined  powders were pressed into pellets and sintered at 1300 oC.  Using thes e YIG and \nBYG targets,  thin films of thickness ~150 nm  were grown  in-situ on (100) - and (111) -oriented \nGGG  substrate s by the PLD technique . The prepared samples have been labeled as YIG (100) , \nYIG (111) , BYG (100), and BYG (111).  GGG substrates were cleaned using acetone and \nisopropanol.  Before deposition, the deposition chamber  was thoroughly cleaned and evacuated to \na base vacuum of 2 ×10-6 mbar. We have used KrF excimer laser (248  nm), with pulse frequency \n10 Hz to ablate the target s at 300mJ  energy . During deposition , target to substrate distance, \nsubstrate temperature , and oxygen pressure w ere kept at  ~4.8 cm, 825 oC, and 0.15 mbar , \nrespectively.  Best films were grown at a rate  of 6 nm/min . The as -grown thin film s were annealed \nin-situ for 2 hours at 825  oC and cooled down to 300 oC in the presence of oxygen  (0.15 mbar)  \nthroughout the process . The structural properties of the thin film were determined  by XRD  using \nCu-Kα radiation  (1.5406  Å) and surface morphology as well as  the thickness of the film were \ncalculated with atomic force microscopy  by WITec  Gmb H, Germany . Magnetic properties were \nstudied using a 14 tesla PPMS (Cryogenic) . FMR  measurements were carried out  by the  Vector \nNetwork Analyzer ( VNA ) (Keysight , USA)  using a coplanar waveguide ( CPW ) in a flip -chip \ngeometry with dc magnetic field applied parallel  to the film plan e. \n \n3. Results and Discussion  \n3.1 Structural properties  \nThe room temperature XRD  data for the polycrystalline targets of YIG and BYG are plotted in \nfigure  1 (a) and 1  (b) respectively.  Rietveld refinement patterns after fitting XRD data are also \nincluded in the panel s. XRD peaks are indexed according to the JCPDS card no. ( # 43-0507) . Inset 5 \n of figure  1 (a) shows crystallographic sub -lattices  of YIG that elucidates  Fe13+ tetrahedral  site, \nFe23+ octahedral site , and Y3+ dodecahedral site. Inset (i) of figure  1 (b)  shows evidence for \nsuccessful incorporation of Bi into YIG ; the lattice constant  increases when Bi is substituted into \nYIG due to larger ionic radii of Bi (1.170 Å) as compared with Y (1.019 Å) [19]. From Rietveld \nrefinement  we estimate the lattice constant of YIG and BYG to be  12.377 Å [38] and 12.401 Å  \nrespectively .  \n \nFigure  2 (a) and 2  (b) show the XRD pattern of  bare (100) and (111) oriented GGG substrates . \nThis is followed by figure  2 (c)  & 2(d) for YIG and figure  2 (e)  & 2 (f) for BYG as grown thin \nfilms. XRD patterns confirm  the single -crystalline grow th of YIG and BYG thin film s over GGG \nsubstrates . The l attice  constant, lattice mismatch  (with respect to substrate) , and lattice volume \nobtain ed from XRD  data are listed in Table  1. Lattice cons tant a for the cubic structure is evaluated \nusing the [39]. \n𝒂=𝜆√ℎ2+𝑘2+𝑙2\n2sin𝜃                                                        (1) \n Where  𝜆 is the wavelength of Cu -Kα radiation , 𝜃 is the diffraction angle , and [h , k, l ] are the \nmiller indices of the corresponding XRD peak. Lattice  misfit  (𝛥𝑎\n𝑎) is evaluated using equation 2  \n[24,38] . \n𝛥𝑎\n𝑎=(𝑎𝑓𝑖𝑙𝑚− 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )\n𝑎𝑓𝑖𝑙𝑚 100               (2) \nWhere  𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒  are the lattice constant of film and substrate respectively.  Lattice \nconstant of pure YIG bulk  is 12.377  Å, whereas we have observed  a larger value of lattice constants \nof YIG and BY G films  than th at of bulk YIG as shown in T able 1 . Sim ilarly, to these  obtained \nresults, a larger value of lattice constants than that of bulk YIG has been reported as well [40–44]. 6 \n The obtained values of the lattice constant are in agreement with the previous reports \n[18,21,25,34,45] . In the case of  BYG (111), the value of lattice constant slightly increases \ncompared to BYG (100) because the distribution of Bi3+ in the dodecahedral site depends on the \norientation of the substrate [28,46] . Inset (ii) of figure  1 (b) shows  plane (111) has more \ncontribution of Bi3+ ions [(ionic radius of bismuth (1.170 Å) is larger as compared with YIG (1.019 \nÅ) [19]]. This slight increase in the lattice constant  (in 111 direction)  implies a more lattice \nmismatch (or strain ) in BYG films . Positive  value of lattice mismatch indicates the slightly larger \nlattice constant of films (YIG and BYG) were observed as compared to substrates (GGG). We \nwould like to emphasize  that lattice plane dependence growth  is important to signify  the changes \nin the struc tural and magnetic properties.  \n3.2 Surface morphology study  \nRoom temperature AFM images  with roughness are  shown in figure 3 (a)-(d). Roughness plays \nan important role from the application prospective as it is related to  Gilbert  damping factor α. \nLower roughness (root mean square height) is observed for the (111) oriented films of YIG and \nBYG compared to those grown on (100) oriented substrates. Available literature [61] indicate that \nroughness would depend more on variation on growth parameters ra ther than on substrate \norientation.  In this sense further study is needed to clarify substrate dependence of roughness. No \nsignificant change in the roughness is observed between YIG and BYG films  [38,47] . Table 1  \ndepicts a comparison between the roughne ss measured in YIG and the BY G thin films.  7 \n 3.3 Static magnetization p roperties  \nVSM  magnetization  measurements were performed  at 300 K with magnetic field appl ied parallel \nto the film plane  (in-plane) .  Figure 3 (e) and 3  (f) show the magnetization plot s for YIG and BYG \nrespectively after careful subtraction of paramagnetic contr ibution  that is assigned to the substrate.  \nThe m easured saturation magnetization ( 4πM S) data are given in Table 1 which are in general \nagreement with the reported values [11,40,48] . Not much change in the measured  4πM S value of \nYIG and Bi -YIG films  are observed . The ferrimagnetism nature of YIG arises from super -\nexchange interaction between the non -equivalent Fe3+ ions at octahedral  and tetrahedral  sites [49]. \nBismuth located at dodecahedral site does not affect the tetrahedral and octahedral Fe3+ ions. So, \nBismuth does not show a significant change in saturation magnetization at  room temperature. I t is \nreported in literature th at Bi addition leads to increase  in Curie temperature, so in t hat sense there \nis an decreasing  trend in saturation magnetization in BYG films in contrast to YIG films [50,51] . \nError bars in saturation magnetization relate to uncertainty in sample volume.  \n3.4 Ferromagnetic r esonance properties  \nThe FMR absorption spectroscopy  is shown in figure 4. These measurements were  performed  at \nroom temperature .  The external dc magnetic field  was appli ed parallel to the plane of the film . \nLorentzian  fit of the calibrated experimental data are used to calculate t he FMR linewidth (∆H) \nand resonance  magnetic  field (H r). From the e nsemble of all the FMR data at different resonance \nfrequencies  (f = 1 GHz -12 GHz ), we have calculated the gyromagnetic ratio (γ) , effective \nmagnetization field ( 4𝜋𝑀𝑒𝑓𝑓) from the  fitting  of Kittel’s  in-plane equation  [52].  \n 8 \n In general, t he uniform precession of magnetization can be described by the Landau -Lifshitz -\nGilbert (LLG) equation of motion;  \n𝜕𝑀⃗⃗ \n𝜕𝑡=−𝛾(𝑀⃗⃗ ×𝐻⃗⃗ 𝑒𝑓𝑓)+𝐺\n𝛾𝑀𝑠2[𝑀⃗⃗ ×𝜕𝑀⃗⃗ \n𝜕𝑡]   (3) \nHere, t he first term corresponds to the precessional torque in the effective magnetic field and the \nsecond term is the Gilbert damping torque.  The gyromagnetic ratio  is given by 𝛾=𝑔𝜇𝐵/ℏ , where  \n𝑔 is the Lande’s factor, 𝜇𝐵 is Bohr magnetron and ℏ is the Planck’s constant. Similarly, 𝐺=𝛾𝛼𝑀𝑠 \nis related to the intrinsic relaxation rate in the nanocomposites and 𝛼 represents the Gilbert \ndamping constant. Ms (or 4πMs) is the saturation magnetization. It can be shown that t he solution \nfor in -plane resonance frequency can be written as;  \n𝑓𝑟=𝛾′√(𝐻𝑟)(𝐻𝑟+4𝜋𝑀𝑒𝑓𝑓)     (4), \nWhere  𝛾′=𝛾/2𝜋, 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 is the effective field and 𝐻𝑎𝑛𝑖=2𝐾1\n𝑀𝑠 is the anisotropy \nfield.  Following through, we have obtained Gilbert damping parameter (α) and inhomogeneous \nbroadening (∆H 0) linewidth from the fitting of Landau –Lifshitz –Gilbert equation (LLG) [53] \n𝛥𝐻(𝑓)=𝛥𝐻0+4𝜋𝛼\n√3𝛾𝑓                                      (5) \n \n \n \n \nDerived parameters from the FMR study are listed in T able 2 . The obtai ned Gilbert damping (α) \nis in agreement with the  reported thin films used for the study  of spin-wave  propagation  \n[2,27,54,55] .  In the case of YIG no t much change  in the value of α is seen . However , a substantial \nincrease is observed in case of BYG with (111) orientation. Qualitatively this could be assigned to  9 \n the presence of  Bi3+ ions which induce s spin-orbit coupling (SOC)  [56–58] and also  due to  electron \nscattering inside the lattice as lattice  mismatch (or  strain ) increases  [59]. We have seen  more \ndistribution of Bi3+ ions along (111) planes ( see inset (ii) of figure  1 (b) ) and also slightly larger \nlattice mismatch  in BYG (111) from our XRD  results. These results also explain higher  value of \nGilbert damping  and ΔH 0 in case of BYG (111).  The change in 4𝜋𝑀𝑒𝑓𝑓 could be attributed to \nuniaxial in -plane magnetic anisotropy . This is because no change in 4πM S is observed from \nmagnetization measurements and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 [38,40,60] . The uniaxial inplane magnetic \nanisotropy  is induced due to lattice mismatch between films and GGG substrates  [38,40] . The \ncalculated gyromagnetic ratio (γ)  and ΔH 0 are also included in Table 2 . The magnitude of  ΔH 0 is \nclose to reported values for same  substrate orientation  [38]. In summary we find that YIG with \n(100) orientation yields lowest damping fact or and extrinsic contribution to  linewidth.  These are \nthe r equired optimal parameters for spintronic s application with high spin diffusion length.  \nHowever,  MOKE signal is usual ly very low in bare YIG thin films because of its lower magnetic \nanisotropy and strain [61]. But previous reports suggest that magnetic anisotropy and magnetic \ndomains formation can be achieved in YIG system by doping rare earth materials like Bi and Ce \n[18,61]. We have shown that anisotropic characteristic  with Bi doping in YIG is more pron ounced \nalong <111> direction which can lead to the enhanced MOKE signal in Bi -YIG films on <111> \nsubstrate.  \nWe have also recorded polar angle  () data of resonance field ( Hr) versus magnetic field \n(H) at frequency 12  GHz  for the BYG (100) and BYG (111)  films (figure  5 (c) & 5(d) respectively  \nwhere inset shows the azimuthal angle ( ) variation of Hr measured at frequency of 3 GHz ). The \ndata are fitted with modified Kittel equation . From figure 5 (c) & (d),  we can see that Hr increases \nup to 2.5 kOe in BYG (100) and 3.0 kOe in BYG (111)  by varying the  direction  of H from 0 to 90 10 \n degree with respect to sample surface  (inset of Fig 5 (a)) . Obtained parameters from angular \nvariation of FMR magnetic field H r (θH) are listed in the inset of figure 5 (c) & (d) . From  variation \ndata (by varying the  direction  of H from 0 to 18 0 degree with respect to sample  edge (Fig 5 (a)  \nInset ), we see clear four -fold and two -fold in-plane anisotropy in BYG (100) and BYG (111)  films  \n[61,62] . This further consolidates single -crystalline characteristics of our films. The change \nobserved in Hr with respect to  variation  is 79.52 Oe in BYG (100)  (H=0 to 45)  and 19.25 Oe in \nBYG (111) ( H=0 to 45). Thus,  during  in-plane rotation, higher change in FMR field is observed \nalong  (100) orientation . \n \n4. Conclusion  \nIn conclusion , we have  grown high quality YIG and B i-YIG thin film s on GGG substrates with \n(100)  and (111)  orientation . The films were gr own by pulsed laser deposition. The optimal \nparameters  i.e. target to substrate distance, substrate temperature, and oxygen pressure are \ndetermined to be  ~ 4.8 cm, 825 oC, and 0.15 mbar, respectively. The as  grown thin films have \nsmooth surfaces and  are found to be  phase pure  from AFM and XRD characterizations.  From FMR \nmeasurements , we have found  lower value of damping parameter in (100) YIG that indicates \nhigher spin diffusion length for potential spintronics application.  On the other -hand bismuth \nincorporation to YIG leads to dominance of anisotropic characteristics  that augers well for \napplication in magnetic bubble memory and  magneto -optic  devices . The enhanced value of α in \nBi-YIG films is ascribed to the spin orbit coupled Bi3+ ions. We also ta bulate the values of \nmagnetic parameters such as linewidth ( ∆H0), gyromagnetic ratio ( γ), and effective magnetization  \n4𝜋𝑀𝑒𝑓𝑓 with respect to substrate orientation.  Unambiguous four-fold in -plane anisotropy is \nobserved in (100) oriented films. We  find high-quality magnetization dynamics and lower Gilbert 11 \n damping parameter  is possible in  Bi-YIG grown on (111) GGG  in conjunction with enhanced \nmagnetic anisotropy.  The choice of perfect substrate orientation is therefore found to be crucial \nfor the growth of YIG and Bi-YIG thin films  for high frequency applications.  \n \nAcknowledgments  \nThis work is supported by the MHRD -IMPRINT grant, DST (SERB, AMT , and PURSE -II) grant \nof Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support . We \nacknowledge AIRF, JNU for access of PPMS facility.   \n \nReferences  \n[1] S.A. Manuilov, C.H. Du, R. Adur, H.L. Wang, V.P. Bhallamudi, F.Y. Yang, P.C. \nHammel, Spin pumping from spinwaves in thin film YIG, Appl. Phys. Lett. 107 (2015) \n42405.  \n[2] C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. \nSawicki, S.G. Ebbinghaus, G. 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YIG (111)  (1.05±0.13)  26.51±0.21  2331 .38±65.78  2.86±0.02  \n3. BYG (100)  (1.66±0.10)  26.52 ±0.17  1701.67±31.87  2.89±0.11  \n4. BYG (111)  (6.27±0.33)  29.28 ±0.62  2366 .85±62.60  2.86±0.02  \n \n  21 \n Figure Captions  \nFigure 1: XRD with Rietveld refinement pattern of (a) YIG target ( inset shows  crystallographic \nsub-lattices, Fe 13+ tetrahedral  site, Fe 23+ octahedral site and Y3+ dodecahedral site ) (b) BYG target  \n(inset (i) shows effect of Bi doping into YIG , inset  (ii) shows contribution of the Bi3+ along the \n(100) and (111) planes ). \n \nFigure 2: XRD pattern of  (a) GGG (100) , (b) GGG (111), (c)  YIG (100) , (d) YIG (111) , (e) BYG \n(100) , and (f) BYG (111) . \n \nFigure 3: AFM images of  (a) YIG  (100), (b ) YIG (111) , (c) BYG (100), (d) BYG (111) and  static \nmagnetization graph of (e ) YIG (100), YIG (111);  and (f) BY G (100), BYG (111) . \n \nFigure 4: FMR absorption spectra of (a) YIG (100), (b) YIG (111), (c) BYG (100), and (d) BYG \n(111).  \n \nFigure 5: (a) FMR magnetic field Hr is plotted as a function of frequency f.  Experiment data fitted \nwith Kittel equation for YIG and BYG oriented films. Inset shows how the applied field  angle is \nmeasured from sample surface  (b) Frequency -dependent FMR linewidth data fitted with LLG \nequation for YIG and BYG oriented films.  Inset shows the magnified version to illustrate  the effect \nof Bi doping in YIG . (c) and (d) show angular variation   of FMR magnetic field (Hr (θH)) fitted \nwith modified Kittel  equation at 12  GHz frequency for BYG (100) and  BYG (111)  films .  Insets \nshow the FMR magnetic  field (H r) as a function of azimuthal angle ( ). \n 22 \n  \nFigure 1  \n \n23 \n  \nFigure 2 \n \n \n \n \n \n24 \n  \nFigure 3 \n \n \n \n \n25 \n  \n \n \n \n \nFigure 4 \n \n \n \n \n \n \n26 \n  \n \n \n \nFigure 5 \n \n \n"    },    {        "title": "2201.06060v2.Ferromagnetic_resonance_modulation_in__d__wave_superconductor_ferromagnetic_insulator_bilayer_systems.pdf",        "content": "Ferromagnetic resonance modulation in d-wave superconductor/ferromagnetic\ninsulator bilayer systems\nYuya Ominato,1Ai Yamakage,2Takeo Kato,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China.\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: May 6, 2022)\nWe investigate ferromagnetic resonance (FMR) modulation in d-wave superconductor\n(SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert\ndamping in these systems re\rects the existence of nodes in the d-wave SC and shows power-law\ndecay characteristics within the low-temperature and low-frequency limit. Our results indicate the\ne\u000bectiveness of use of spin pumping as a probe technique to determine the symmetry of unconven-\ntional SCs with high sensitivity for nanoscale thin \flms.\nI. INTRODUCTION\nSpin pumping (SP)1,2is a versatile method that can\nbe used to generate spin currents at magnetic junctions.\nWhile SP has been used for spin accumulation in vari-\nous materials in the \feld of spintronics3,4, it has recently\nbeen recognized that SP can also be used to detect spin\nexcitation in nanostructured materials5, including mag-\nnetic thin \flms6, two-dimensional electron systems7{9,\nand magnetic impurities on metal surfaces10. Notably,\nspin excitation detection using SP is sensitive even for\nsuch nanoscale thin \flms for which detection by con-\nventional bulk measurement techniques such as nuclear\nmagnetic resonance and neutron scattering experiment is\ndi\u000ecult.\nRecently, spin injection into s-wave superconductors\n(SCs) has been a subject of intensive study both theoret-\nically11{20and experimentally21{34. While the research\ninto spin transport in s-wave SC/magnet junctions is ex-\npected to see rapid development, expansion of the devel-\nopment targets toward unconventional SCs represents a\nfascinating research direction. Nevertheless, SP into un-\nconventional SCs has only been considered in a few recent\nworks35,36. In particular, SP into a d-wave SC, which is\none of the simplest unconventional SCs that can be real-\nized in cuprate SCs37, has not been studied theoretically\nto the best of our knowledge, although experimental SP\nin ad-wave SC has been reported recently38.\nIn this work, we investigate SP theoretically in a bi-\nlayer magnetic junction composed of a d-wave SC and\na ferromagnetic insulator (FI), as shown in Fig. 1. We\napply a static magnetic \feld along the xdirection and\nconsider the ferromagnetic resonance (FMR) experiment\nof the FI induced by microwave irradiation. In this setup,\nthe FMR linewidth is determined by the sum of the in-\ntrinsic contribution made by the Gilbert damping of the\nbulk FI and the interface contribution, which originates\nfrom the spin transfer caused by exchange coupling be-\nMicrowavex yz\nSpin current\nFerromagnetic resonanceInteractionFIG. 1. Schematic of the d-wave SC/FI bilayer system. The\ntwo-dimensional d-wave SC is placed on the FI. Precessional\nmotion of the magnetization is induced by microwave irradia-\ntion. The spins are injected and the magnetization dynamics\nare modulated because of the interface magnetic interaction.\ntween thed-wave SC and the FI. We then calculate the\ninterface contribution to the FMR linewidth, which is\ncalled the modulation of the Gilbert damping hereafter,\nusing microscopic theory based on the second-order per-\nturbation39{41. We show that the temperature depen-\ndence of the modulation of the Gilbert damping exhibits\na coherent peak below the transition temperature that\nis weaker than that of s-wave SCs11,13{15. We also show\nthat because of the existence of nodes in the d-wave SCs,\nthe FMR linewidth enhancement due to SP remains even\nat zero temperature.\nThe paper is organized as follows. In Sec. II, we in-\ntroduce the model Hamiltonian of the SC/FI bilayer sys-\ntem. In Sec. III, we present the formalism to calculate\nthe modulation of the Gilbert damping. In Sec. IV, we\npresent the numerical results and explain the detailed\nbehavior of the modulation of the Gilbert damping. In\nSec. V, we brie\ry discuss the relation to other SC sym-\nmetries, the proximity e\u000bect, and the di\u000berence between\nd-wave SC/FI junctions and d-wave SC/ferromagnetic\nmetal junctions. We also discuss the e\u000bect of an e\u000bectivearXiv:2201.06060v2  [cond-mat.mes-hall]  5 May 20222\nZeeman \feld due to the exchange coupling. In Sec. VI,\nwe present our conclusion and future perspectives.\nII. MODEL\nThe model Hamiltonian of the SC/FI bilayer system\nHis given by\nH=HFI+HdSC+HT: (1)\nThe \frst term HFIis the ferromagnetic Heisenberg\nmodel, which is given by\nHFI=\u0000JX\nhi;jiSi\u0001Sj\u0000~\rhdcX\njSx\nj; (2)\nwhereJ>0 is the exchange coupling constant, hi;ji\nrepresents summation over all the nearest-neighbor sites,\nSjis the localized spin at site jin the FI,\ris the gy-\nromagnetic ratio, and hdcis the static magnetic \feld.\nThe localized spin Sjis described as shown using the\nbosonic operators bjandby\njof the Holstein-Primako\u000b\ntransformation42\nS+\nj=Sy\nj+iSz\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (3)\nS\u0000\nj=Sy\nj\u0000iSz\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (4)\nSx\nj=S\u0000by\njbj; (5)\nwhere we require [ bi;by\nj] =\u000ei;jto ensure that S+\nj,S\u0000\nj,\nandSx\njsatisfy the commutation relation of angular mo-\nmentum. The deviation of Sx\njfrom its maximum value S\nis quanti\fed using the boson particle number. It is conve-\nnient to represent the bosonic operators in the reciprocal\nspace as follows\nbk=1p\nNX\nje\u0000ik\u0001rjbj; by\nk=1p\nNX\njeik\u0001rjby\nj;(6)\nwhereNis the number of sites. The magnon opera-\ntors with wave vector k= (kx;ky;kz) satisfy [bk;by\nk0] =\n\u000ek;k0. Assuming that the deviation is small, i.e., that\nhby\njbji=S\u001c1, the ladder operators S\u0006\njcan be approx-\nimated asS+\nj\u0019(2S)1=2bjandS\u0000\nj\u0019(2S)1=2by\nj, which\nis called the spin-wave approximation. The Hamiltonian\nHFIis then written as\nHFI\u0019X\nk~!kby\nkbk; (7)\nwhere we assume a parabolic dispersion ~!k=Dk2+\n~\rhdcwith a spin sti\u000bness constant Dand the constant\nterms are omitted.\nThe second term HdSCis the mean-\feld Hamiltonian\nfor the two-dimensional d-wave SC, and is given by\nHdSC=X\nk(cy\nk\";c\u0000k#)\u0012\n\u0018k \u0001k\n\u0001k\u0000\u0018k\u0013\u0012ck\"\ncy\n\u0000k#\u0013\n;(8)wherecy\nk\u001bandck\u001bdenote the creation and annihilation\noperators, respectively, of the electrons with the wave\nvectork= (kx;ky) and thexcomponent of the spin\n\u001b=\";#, and\u0018k=~2k2=2m\u0000\u0016is the energy of conduc-\ntion electrons measured from the chemical potential \u0016.\nWe assume that the d-wave pair potential has the form\n\u0001k= \u0001 cos 2\u001ekwith the phenomenological temperature\ndependence\n\u0001 = 1:76kBTctanh \n1:74r\nTc\nT\u00001!\n; (9)\nwhere\u001ek= arctan(ky=kx) denotes the azimuth angle of\nk. Using the Bogoliubov transformation given by\n\u0012ck\"\ncy\n\u0000k#\u0013\n=\u0012\nuk\u0000vk\nvkuk\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n; (10)\nwhere\ry\nk\u001band\rk\u001bdenote the creation and annihilation\noperators of the Bogoliubov quasiparticles, respectively,\nandukandvkare given by\nuk=r\nEk+\u0018k\n2Ek; vk=r\nEk\u0000\u0018k\n2Ek; (11)\nwith the quasiparticle energy Ek=p\n\u00182\nk+ \u00012\nk, the mean-\n\feld Hamiltonian can be diagonalized as\nHdSC=X\nk(\ry\nk\";\r\u0000k#)\u0012\nEk 0\n0\u0000Ek\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n:(12)\nThe density of states of the d-wave SC is given by43\nD(E)=Dn= Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\n; (13)\nwhereDn=Am= 2\u0019~2is the density of states per spin of\nthe normal state, Ais the system area, and K(x) is the\ncomplete elliptic integral of the \frst kind in terms of the\nparameterx, where\nK(x) =Z\u0019=2\n0d\u001ep\n1\u0000xcos2\u001e: (14)\nD(E) diverges at E=\u0001 = 1 and decreases linearly when\nE=\u0001\u001c1 because of the nodal structure of \u0001 k. The\ndensity of states for an s-wave SC, in contrast, has a\ngap forjEj<\u0001. This di\u000berence leads to distinct FMR\nmodulation behaviors, as shown below.\nThe third term HTdescribes the spin transfer between\nthe SC and the FI at the interface\nHT=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (15)\nwhereJq;kis the matrix element of the spin transfer pro-\ncesses, and \u001b\u0006\nq= (\u001by\nq\u0006i\u001bz\nq)=2 andS\u0006\nk=Sy\nk\u0006iSz\nkare3\n(a) Spin transfer process (b) Self-energy\nJq,kJ*q,k\np/uni2191p+q/uni2193\np/uni2191p+q/uni2193\n−k −k\n/uni03A3R\nk(/uni03C9)=\nFIG. 2. (a) Diagrams of the bare vertices of the spin transfer\nprocesses at the interface. (b) Self-energy within the second-\norder perturbation.\nthe Fourier components of the ladder operators and are\ngiven by\n\u001b+\nq=X\npcy\np\"cp+q#; \u001b\u0000\n\u0000q=X\npcy\np+q#cp\"; (16)\nS\u0000\n\u0000k\u0019(2S)1=2by\nk; S+\nk\u0019(2S)1=2bk: (17)\nUsing the expressions above, HTcan be written as\nHT\u0019p\n2SX\np;q;k\u0010\nJq;kcy\np\"cp+q#by\n\u0000k+J\u0003\nq;kcy\np+q#cp\"b\u0000k\u0011\n:\n(18)\nThe \frst (second) term describes a magnon emission\n(absorption) process accompanying an electron spin-\rip\nfrom down to up (from up to down). A diagrammatic\nrepresentation of the interface interactions is shown in\nFig. 2 (a).\nIn this work, we drop a diagonal exchange coupling at\nthe interface, whose Hamiltonian is given as\nHZ=X\nq;kJq;k\u001bx\nqSx\nk: (19)\nThis term does not change the number of magnons in\nthe FI and induces an e\u000bective Zeeman \feld on electrons\nin the two-dimensional d-wave SC. We expect that this\nterm does not a\u000bect our main result because the coupling\nstrength is expected to be much smaller than the super-\nconducting gap and the microwave photon energy. We\nwill discuss this e\u000bect in Sec. V brie\ry.\nIII. FORMULATION\nThe coupling between the localized spin and the mi-\ncrowave is given by\nV(t) =\u0000~\rhacX\ni(Sy\nicos!t\u0000Sz\nisin!t); (20)wherehacis the amplitude of the transverse oscillating\nmagnetic \feld with frequency !. The microwave irra-\ndiation induces the precessional motion of the localized\nspin. The Gilbert damping constant can be read from\nthe retarded magnon propagator de\fned by\nGR\nk(t) =1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (21)\nwhere\u0012(t) is a step function. Second-order perturbation\ncalculation of the magnon propagator with respect to the\ninterface interaction was performed and the expression of\nthe self-energy was derived in the study of SP39{41. Fol-\nlowing calculation of the second-order perturbation with\nrespect to Jq;k, the Fourier transform of the retarded\nmagnon propagator is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (22)\nwhere\u000bis the intrinsic Gilbert damping constant that\nwas introduced phenomenologically44{46. The diagram\nof the self-energy \u0006R\nk(!) is shown in Fig. 2 (b). From the\nexpressions given above, the modulation of the Gilbert\ndamping constant is given by\n\u000e\u000b=\u00002SIm \u0006R\nk=0(!)\n~!: (23)\nWithin the second-order perturbation, the self-energy\nis given by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (24)\nwhere\u001fR\nq(!) represents the dynamic spin susceptibility\nof thed-wave SC de\fned by\n\u001fR\nq(!) =\u00001\ni~Z\ndtei(!+i0)t\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(25)\nSubstituting the ladder operators in terms of the Bogoli-\nubov quasiparticle operators into the above expression\nand performing a straightforward calculation, we then\nobtain434\n\u001fR\nq(!) =\u0000X\npX\n\u0015=\u00061X\n\u00150=\u00061\u0012(\u0018p+\u0015Ep)(\u0018p+q+\u00150Ep+q) + \u0001 p\u0001p+q\n4\u0015Ep\u00150Ep+q\u0013f(\u0015Ep)\u0000f(\u00150Ep+q)\n\u0015Ep\u0000\u00150Ep+q+~!+i0; (26)\nwheref(E) = 1=(eE=kBT+ 1) is the Fermi distribution\nfunction.\nIn this paper, we focus on a rough interface modeled in\nterms of the mean J1and variance J22of the distribution\nofJq;k(see Appendix A for detail). The con\fgurationally\naveraged coupling constant is given by\njJq;k=0j2=J12\u000eq;0+J22: (27)\nIn this case, \u000e\u000bis written as\n\u000e\u000b=2SJ12\n~!Im\u001fR\nq=0(!) +2SJ22\n~!X\nqIm\u001fR\nq(!):(28)\nThe \frst term represents the momentum-conserved spin-\ntransfer processes, which vanish as directly veri\fed from\nEq. (26). This vanishment always occurs in spin-singlet\nSCs, including sandd-wave SCs, since the spin is\nconserved43. Consequently, the enhanced Gilbert damp-\ning is contributed from spin-transfer processes induced\nby the roughness proportional to the variance J22\n\u000e\u000b=2SJ22\n~!X\nqIm\u001fR\nq(!): (29)\nThe wave number summation can be replaced as\nX\nq(\u0001\u0001\u0001)!Dn\n2\u0019Z1\n\u00001d\u0018Z2\u0019\n0d\u001e(\u0001\u0001\u0001): (30)\nChanging the integral variable from \u0018toEand substi-\ntuting Eq. (26) into Eq. (29), we \fnally obtain\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\nRe\u00142\n\u0019K\u0012\u00012\n(E+~!)2\u0013\u0015\n:\n(31)\nNote that the coherence factor vanishes in the above ex-\npression by performing the angular integral. The en-\nhanced Gilbert damping in the normal state is given by\n\u000e\u000bn= 2\u0019SJ 22D2\nn; (32)\nfor the lowest order of !. This expression means that \u000e\u000b\nis proportional to the product of the spin-up and spin-\ndown densities of states at the Fermi level7.IV. GILBERT DAMPING MODULATION\nFigure 3 shows the enhanced Gilbert damping constant\n\u000e\u000bas a function of temperature for several FMR frequen-\ncies, where \u000e\u000bis normalized with respect to its value in\nthe normal state. We compare \u000e\u000bin thed-wave SC shown\nin Figs. 3 (a) and (c) to that in the s-wave SC shown in\nFigs. 3 (b) and (d). The enhanced Gilbert damping for\nthes-wave SC is given by13\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002\u0012\n1 +\u00012\nE(E+~!)\u0013\n\u0002Re\u0014jEjp\nE2\u0000\u00012\u0015\nRe\"\njE+~!jp\n(E+~!)2\u0000\u00012#\n;\n(33)\nwhere the temperature dependence of \u0001 is the same as\nthat for the d-wave SC, given by Eq. (9). Note that\nthe BCS theory we are based on, which is valid when\nthe Fermi energy is much larger than \u0001, is described by\nonly some universal parameters, including Tc, and inde-\npendent of the detail of the system in the normal state.\nWhen ~!=k BTc= 0:1,\u000e\u000bshows a coherence peak just\nbelow the transition temperature Tc. However, the co-\nherence peak of the d-wave SC is smaller than that of\nthes-wave SC. Within the low temperature limit, \u000e\u000bin\nthed-wave SC shows power-law decay behavior described\nby\u000e\u000b/T2. This is in contrast to \u000e\u000bin thes-wave SC,\nwhich shows exponential decay. The di\u000berence in the low\ntemperature region originates from the densities of states\nin thed-wave ands-wave SCs, which have gapless and full\ngap structures, respectively. When the FMR frequency\nincreases, the coherence peak is suppressed, and \u000e\u000bde-\ncays monotonically with decreasing temperature. \u000e\u000bhas\na kink structure at ~!= 2\u0001, where the FMR frequency\ncorresponds to the superconducting gap.\nFigure 4 shows \u000e\u000batT= 0 as a function of !. In\nthed-wave SC,\u000e\u000bgrows from zero with increasing !as\n\u000e\u000b/!2. When the value of \u000e\u000bbecomes comparable to\nthe normal state value, the increase in \u000e\u000bis suppressed,\nand\u000e\u000bthen approaches the value in the normal state.\nIn contrast, \u000e\u000bin thes-wave SC vanishes as long as the\ncondition that ~! < 2\u0001 is satis\fed. When ~!exceeds\n2\u0001,\u000e\u000bthen increases with increasing !and approaches\nthe normal state value. This di\u000berence also originates\nfrom the distinct spectral functions of the d-wave and\ns-wave SCs. Under the low temperature condition that\nT= 0:1Tc, the frequency dependence of \u000e\u000bdoes not5\n0.1 5.0\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0/uni210F/uni03C9/kBTc\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n(a)  d-wave (b)  s-wave\n(c)  d-wave (d)  s-wave\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0\nFIG. 3. Enhanced Gilbert damping \u000e\u000bas a function of tem-\nperatureT. The left panels (a) and (c) show \u000e\u000bin thed-\nwave SC in the low and high frequency cases, respectively.\nThe right panels (b) and (d) show \u000e\u000bin thes-wave SC in the\nlow and high frequency cases, respectively. \u000e\u000bnis the normal\nstate value.\nchange for the s-wave SC, and it only changes in the\nlow-frequency region where ~!.kBTfor thed-wave SC\n(see the inset in Fig. 4).\nV. DISCUSSION\nWe discuss the modulation of the Gilbert damping\nin SCs with nodes other than the d-wave SC consid-\nered in this work. Other SCs with nodes are expected\nto exhibit the power-law decay behavior within the low-\ntemperature and low-frequency limit as the d-wave SCs.\nHowever, the exponent of the power can di\u000ber due to\nthe di\u000berence of the quasiparticle density of states. Fur-\nthermore, in the p-wave states, two signi\fcant di\u000berences\narise due to spin-triplet Cooper pairs. First, the uni-\nform spin susceptibility \u001fR\nq=0(!) can be \fnite in the spin-\ntriplet SCs because the spin is not conserved. Second,\nthe enhanced Gilbert damping exhibits anisotropy and\nthe value changes by changing the relative angle between\nthe Cooper pair spin and localized spin35.\nIn our work, proximity e\u000bect between FIs and SCs\nwas not taken into account because the FMR modula-\ntion was calculated by second-order perturbation based\non the tunnel Hamiltonian. Reduction of superconduct-\n/uni210F/uni03C9=2/uni0394(T =0)\ns-waved-wave\n00.2\n2 4 6 81.2\n0.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n\n/uni210F/uni03C9/kBTc0.4\n0.0\n100 10.05\n0.000.1T/Tc=0.0FIG. 4. Enhanced Gilbert damping \u000e\u000bas a function of fre-\nquency!. The vertical dotted line indicates the resonance\nfrequency ~!= 2\u0001(T= 0). The inset shows an enlarged\nview in the low-frequency region.\ning gap due to the proximity e\u000bect15and e\u000bect of the\nsubgap Andreev bound states that appear in the ab-axis\njunction47would also be an important problem left for\nfuture works.\nPhysics of the FMR modulation for d-wave\nSC/ferromagnetic metal junctions is rather di\u000ber-\nent from that for d-wave SC/FI junctions. For d-wave\nSC/ferromagnetic metal junctions, spin transport is\ndescribed by electron hopping across a junction and\nthe FMR modulation is determined by the product\nof the density of states of electrons for a d-wave SC\nand a ferromagnetic metal. (We note that the FMR\nmodulation is determined by a spin susceptibility of\nd-wave SC, which in general includes di\u000berent informa-\ntion from the density of states of electrons.) While the\nFMR modulation is expected to be reduced below a SC\ntransition temperature due to opening an energy gap, its\ntemperature dependence would be di\u000berent from results\nobtained in our work.\nFinally, let us discuss e\u000bect of the diagonal exchange\ncoupling given in Eq. (19) (see also the last part of\nSec. II). This term causes an exchange bias, i.e., an e\u000bec-\ntive Zeeman \feld on conduction electrons in the d-wave\nSC, which is derived as follows. First, the x-component\nof the localized spin is approximated as hSx\nji \u0019S,\nwhich gives Sx\nk\u0019Sp\nN\u000ek;0. Next, the matrix element\nJq;k=0is replaced by the con\fgurationally averaged value\nJq;k=0=J1\u000eq;0. Consequently, the e\u000bective Zeeman\n\feld term is given by\nHZ\u0019EZX\np(cy\np\"cp\"\u0000cy\np#cp#); (34)\nwhere we introduced a Zeeman energy as EZ=J1Sp\nN.\nThis term induces spin splitting of conduction electrons6\nin thed-wave SC and changes the spin susceptibility of\nthe SC. The spin-splitting e\u000bect causes a spin excitation\ngap and modi\fes the frequency dependence in Fig. 4, that\nwill provide additional information on the exchange cou-\npling at the interface. In actual experimental setup for\nthed-wave SC, however, the Zeeman energy, that is less\nthan the exchange bias between a magnetic insulator and\na metal, is estimated to be of the order of 0 :1 erg=cm2.\nThis leads to the exchange coupling that is much less\nthanJ\u00180:1 meV for YIG48. Therefore, we expect that\nthe interfacial exchange coupling is much smaller than\nthe superconducting gap and the microwave photon en-\nergy though it has not been measured so far. A detailed\nanalysis for this spin-splitting e\u000bect is left for a future\nproblem.\nVI. CONCLUSION\nIn this work, we have investigated Gilbert damping\nmodulation in the d-wave SC/FI bilayer system. The\nenhanced Gilbert damping constant in this case is pro-\nportional to the imaginary part of the dynamic spin sus-\nceptibility of the d-wave SC. We found that the Gilbert\ndamping modulation re\rects the gapless excitation that\nis inherent in d-wave SCs. The coherence peak is sup-\npressed in the d-wave SC when compared with that in\nthes-wave SC. In addition, the di\u000berences in the spec-\ntral functions for the d-wave ands-wave SCs with gap-\nless and full-gap structures lead to power-law and ex-\nponential decays within the low-temperature limit, re-\nspectively. Within the low-temperature limit, \u000e\u000bin the\nd-wave SC increases with increasing !, while\u000e\u000bin the\ns-wave SC remains almost zero as long as the excitation\nenergy ~!remains smaller than the superconducting gap\n2\u0001.\nOur results illustrate the usefulness of measurement of\nthe FMR modulation of unconventional SCs for determi-\nnation of their symmetry through spin excitation. We\nhope that this fascinating feature will be veri\fed exper-\nimentally in d-wave SC/FI junctions in the near future.\nTo date, one interesting result of FMR modulation in\nd-wave SC/ferromagnetic metal structures has been re-\nported38. This modulation can be dependent on metallic\nstates, which are outside the scope of the theory pre-\nsented here. The FMR modulation caused by ferromag-\nnetic metals is another subject that will have to be clar-\ni\fed theoretically in future work.\nFurthermore, our work provides the most fundamental\nbasis for application to analysis of junctions with vari-\nous anisotropic SCs. For example, some anisotropic SCs\nare topological and have an intrinsic gapless surface state.\nSP can be accessible and can control the spin excitation of\nthe surface states because of its interface sensitivity. The\nextension of SP to anisotropic and topological supercon-\nductivity represents one of the most attractive directions\nfor further development of superconducting spintronics.\nAcknowledgments.| This work is partially supportedby the Priority Program of Chinese Academy of Sciences,\nGrant No. XDB28000000. We acknowledge JSPS KAK-\nENHI for Grants (No. JP20H01863, No. JP20K03835,\nNo. JP20K03831, No. JP20H04635, and No.21H04565).\nAppendix A: Magnon self-energy induced by a\nrough interface\nThe roughness of the interface is taken into account\nas an uncorrelated (white noise) distribution of the ex-\nchange couplings35, as shown below. We start with an\nexchange model in the real space\nHex=X\njZ\nd2rJ(r;rj)\u001b(r)\u0001Sj\n=X\nq;kJq;k\u001bq\u0001Sk: (A1)\nThe spin density \u001b(r) in the SC and the spin Sjin the\nFI are represented in the momentum space as\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (A2)\nSj=1p\nNX\nkeik\u0001rjSk; (A3)\nwhereAdenotes the area of the system and Nis the\nnumber of sites. The exchange coupling constant is also\nobtained to be\nJq;k=1\nAp\nNX\njZ\nd2rei(q\u0001r+k\u0001rj)J(r;rj): (A4)\nThe exchange model Hexis decomposed into the spin\ntransfer term HTand the e\u000bective Zeeman \feld term HZ\nasHex=HT+HZ.\nNow we consider the roughness e\u000bect of the interface.\nUncorrelated roughness is expressed by the mean J1and\nvarianceJ22as\n1p\nNX\njJ(r;rj) =J1; (A5)\n1\nNX\njj0J(r;rj)J(r0;rj0)\u0000J12=J22A\u000e2(r\u0000r0);(A6)\nwhereOis the con\fgurational average of Oover the\nroughness. 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These topological solitons can be tailored in multilayered\nthin films, but the experimental observation of their spin wave dynamics remains elusive, in particular due to\nlarge damping. Here, we study Pt /FeCoB /AlO xmultilayers hosting dense and robust skyrmion lattices at room\ntemperature with Gilbert damping of ∼0.02. We use magnetic force microscopy to characterise their static mag-\nnetic phases and broadband ferromagnetic resonance to probe their high frequency response. Micromagnetic\nsimulations reproduce the experiments with accuracy and allow us to identify distinct resonant modes detected\nin the skyrmion lattice phase. Low ( <2 GHz) and intermediate frequency (2 −8 GHz) modes involve excitations\nlocalised to skyrmion edges in conjunction with precession of the uniform background magnetisation, while a\nhigh frequency ( >12 GHz) mode corresponds to in-phase skyrmion core precession emitting spin waves into\nuniform background with wavelengths in the 50–80 nm range commensurate with the lattice structure. These\nfindings could be instrumental in the investigation of room temperature wave scattering and the implementation\nof novel microwave processing schemes in reconfigurable arrays of solitons.\nThe dynamic response of magnetic materials at microwave\nfrequencies represents a rich field of research for its funda-\nmental interest and applications in information processing.\nLinear excitations in the form of spin waves (SWs) underpin\nthe field of magnonics, which describes the paradigm of trans-\nmitting and processing information with such waves1–3. SW-\nbased devices may allow for fast and energy-e fficient logic\napplications4, and a growing number of proposals have shown\npotential uses for computing5and spectral analysis6by SW\ninterference. Magnonic circuit elements such as transistors7,\ndiodes8and filters9have also been demonstrated in experi-\nments based on the manipulation of dipole-dominated SWs\nwith micrometer wavelengths. The generation and detection\nof shorter wavelength SWs, a prerequisite for miniaturiza-\ntion, is challenging, owing to the limitations of nanoscale fab-\nrication. Recent studies have shown that short wavelength\nemission can be achieved by broadband antennae10, grating\neffects11–13and spin torques14.\nIn this light, nonuniform magnetic textures have been ex-\nplored for generating and manipulating spin waves, and gen-\nerally o ffer an alternative route to expand the range of useful\nphenomena for applications15. Such textures can appear spon-\ntaneously at the micro- and nano-scale in magnetic materials\nas a result of the competing interactions, namely, exchange,\ndipolar, and anisotropy. Magnetic textures like bubbles16,17,\nstripes18–20, and vortices21,22have been shown to exhibit a\nrich diversity of magnetisation dynamics. For example, it has\nbeen demonstrated that magnetic domain walls can serve as\nnanoscale waveguides23,24, while the cores of magnetic vor-\ntices can be used to generate omnidirectional dipole-exchange\nSWs with sub-100 nm wavelengths25,26.\nRecently, topologically non-trivial chiral magnetic con-\nfigurations called skyrmions have generated much interest\nowing to their robust and particle-like nature27. They can\nbe stabilised at room temperature in thin films28,29with\nperpendicular magnetic anisotropy (PMA) and interfacialDzyaloshinskii-Moriya interaction (iDMI), which provides\ndifferent handles for tuning the desired magnetic parameters,\nboth statically and dynamically30. On one hand, their dc\ncurrent-driven dynamics31,32could be exploited for racetrack\nmemory and logic devices27, and on the other, their unique\nmicrowave response opens up the possibilities of skyrmion-\nbased spin-torque oscillators33, rf detectors34, and recon-\nfigurable magnonic crystals for microwave processing35–38.\nSkyrmions exhibit a rich variety of eigenmodes39–43, among\nwhich azimuthal bound states and breathing modes have been\nexperimentally observed at low temperatures in bulk crys-\ntals44–46, which have low damping parameters. Only very\nrecently, resonant dynamics with specific spectroscopic sig-\nnatures of thin-film multilayers hosting skyrmions has been\nevidenced47,48. However, much remains to be explored and\nunderstood concerning their individual and collective resonant\nresponse to microwave excitations.\nIn this work, we report a study of the resonant dynam-\nics of ultrathin film multilayers with perpendicular magnetic\nanisotropy, which host stable skyrmion lattices under ambi-\nent conditions with typical periods of 250 nm and skyrmion\ndiameters of 100 nm, while exhibiting Gilbert damping in\nthe range of α/similarequal0.02. By combining magnetic force mi-\ncroscopy (MFM) and ferromagnetic resonance (FMR) exper-\niments with micromagnetic simulations, we can identify dis-\ntinct SW modes associated with the skyrmion lattice phase.\nAt low frequency ( /lessorsimilar2 GHz), we observe a number of modes\nrelated to the precession of the uniform background state of\nindividual layers close to or at the surfaces of the stack, along\nwith eigenmodes localised to the skyrmion edges. At inter-\nmediate frequencies (2 −8 GHz), the precession of the uni-\nform background near the centre of the stack dominates the\nresponse. Similar modes were previously described and re-\nported in bulk crystals45,46, and lately in thin films47,48. In-\ntriguingly, we also observe a well-defined mode at high fre-\nquency (>12 GHz), which corresponds to the in-phase pre-arXiv:2111.11797v2  [cond-mat.mes-hall]  28 Jun 20222\ncession of the magnetisation within the skyrmion cores. The\ncores possess a distinct three-dimensional structure due to\nthe competition between all the existing magnetic interactions\nin these multilayers, notably the interlayer dipolar e ffects49.\nStrikingly, this precession is accompanied by the emission of\nspin waves, with wavelengths in the range of 50 to 80 nm,\ninto the uniformly magnetised background. These SWs in-\nterfere with those generated at neighbouring skyrmion cores,\nyielding a collective dynamical state governed by the subtle\ninterplay between the skyrmion diameter, the wavelength of\nthe emitted SWs, and the skyrmion lattice periodicity.\nRESULTS\nMultilayer composition\nThe basic element of the multilayer film studied is the\nPt(1.6) /Fe0.7Co0.1B0.2(1.2)/AlO x(1.0) trilayer (hereafter re-\nferred to as Pt /FeCoB /AlO x), where the figures in parenthe-\nses indicate the nominal film thickness in nm. This trilayer\nlacks inversion symmetry along the film thickness direction\nwhereby the Dzyaloshinskii-Moriya interaction is promoted\nat the interfaces of the ferromagnetic FeCoB film with Pt (and\npossibly AlO x50). The trilayer is repeated 20 times to form our\nmultilayer sample (see Methods), as shown in Fig. 1(a). The\nchoice of Fe-rich FeCoB and the aforementioned optimised\nthicknesses of Pt and FeCoB allow having su fficient DMI and\nPMA51–53to stabilise skyrmions, while limiting spin pumping\neffects which would otherwise lead to an increased damping\ncoefficient54. We estimate by FMR a Gilbert damping con-\nstant ofα=0.022 and an inhomogeneous broadening of ∼\n18 mT (see Supplementary Figure 1) for our samples, which\nare relatively low for such multilayer systems47,54. The over-\nall magnetic volume is enhanced with the 20 repetitions of the\ntrilayer, which not only increases the thermal stability of the\nskyrmions28but also provides a larger signal-to-noise ratio for\ninductive measurements.\nStatic characterization\nFigure 1(b) shows the out-of-plane hysteresis curve of the\nsample measured by alternating gradient field magnetome-\nter (AGFM), which is characteristic of thin films hosting\nskyrmions. The quasi-static magnetisation configuration of\nthe sample is probed by MFM using a low moment tip by\nsweeping the out-of-plane (OP) field from negative ( −z) to\npositive ( +z) saturation as illustrated in Fig. 1(d). On decreas-\ning the field from negative saturation, we observe nucleation\nof skyrmions with an average diameter around 100 nm (zone\n(i)), which develops into a dense lattice structure upon further\nreduction of the field (zone (ii)). In zone (iii) some of the\nskyrmions elongate and /or coalesce to form stripes leading to\na mixture of skyrmions and stripe domains. At small positive\nfields, labyrinthine domains (zone (iv)) become energetically\nstable and, upon increasing the field further, lead to the forma-\ntion of a dense skyrmion lattice once again (zone (vi)). From\nFIG. 1. Static characterization of the magnetic phases. (a)\nSchematics of the sample stack. (b) The out-of-plane (OP) mag-\nnetisation curve on sweeping the field from negative to positive val-\nues. (c) The corresponding skyrmion density and apparent skyrmion\ndiameter as a function of the field. The dashed red line indicates\nthe possible extrapolation of the observed skyrmion density at higher\npositive field values in accordance with the OP hysteresis curve. The\nskyrmion diameter is extracted only for the field ranges where the\nskyrmion density is close to maximum (see Methods). (d) Mag-\nnetic domain configurations of the sample: (i) randomly distributed\nskyrmions, (ii and vi) skyrmion lattice, (iii and v) skyrmion-stripe\nmix, and (iv) labyrinthine domains, measured by MFM as a function\nof OP magnetic field (corresponding to the values marked in (b))\nswept from negative to positive values.\nthe MFM images we extract the density of skyrmions as a\nfunction of the applied field µ0Hzand apparent skyrmion di-\nameter for the ranges of magnetic field where the skyrmion\ndensity is close to maximum, as shown in Fig. 1(c).\nIt is interesting to note that the observed skyrmion lattice is\nalmost quasi periodic hexagonal (see Supplementary Figure\n2) whereas more amorphous states were observed in previ-\nous studies47,48in such kind of multilayered films, indicating\nonly minor inhomogeneities and defects induced during the\nfilm growth in our samples. The skyrmion lattice phase is\nalso remarkably stable at positive field, as shown by the con-\nstant skyrmion density of about 21 per square micron between\n200 mT and 350 mT.\nBy using the period of the labyrinthine domain pattern in\nFig. 1(d)(iv) and the measured values of saturation magneti-3\nsation ( Ms=1.2 MA /m) and uniaxial anisotropy ( Ku=\n0.7 MJ/m3), we estimate the DMI to be D=1.2 mJ/m2with\nan exchange constant of A=15 pJ /m for our sample (see\nMethods), which are in good agreement with direct experi-\nmental determination of these parameters in Pt /FeCoB /MgO\nstacks55.\nDynamic characterization\nThe resonant dynamics of the sample was probed by broad-\nband FMR using a coplanar waveguide [Fig. 2(d)] and a vector\nnetwork analyser (VNA). The dc out-of-plane field is swept\nfrom negative to positive values while the frequency of the in-\nplane ac field is scanned over a range of 0.1–20 GHz for each\ndc field step. The real and the imaginary parts of the transmis-\nsion signal S21are recorded and processed to remove a back-\nground signal that is independent of the dc field, which im-\nproves the contrast (see Methods and Supplementary Figure\n3). The corresponding frequency versus field map is shown\nin Fig. 2(a). In the saturated state, we observe the high in-\ntensity Kittel mode (KM) along with two additional low in-\ntensity secondary modes at higher field (see also Supplemen-\ntary Figure 1). The latter are attributed to localised modes\nin the multilayer thickness that result from inhomogeneous\ninterfacial couplings of our multilayer system56. In addition\nto the KM observed above saturation fields, three groups of\nmodes with lower intensities appear when the magnetisation\nenters a non-uniform state. On ramping the field from nega-\ntive saturation towards zero, the KM softens close to µ0Hz=\n−335 mT where skyrmions start to nucleate (see MFM im-\nages in Fig. 1(d)). It then evolves into a mode with negative\nfield dispersion ( i.e.,∂f/∂|Hz|<0) as the skyrmion lattice\ngrows denser, which we call the intermediate frequency mode\n(IFM). At this point, a weak amplitude mode also emerges\nat high frequency ( >12 GHz) which has a positive field dis-\npersion ( i.e.,∂f/∂|Hz|>0) denoted as the high frequency\nmode (HFM). The IFM and HFM fade away close to zero\nfield, where the static magnetisation configuration consists of\nlabyrinthine domains. While the IFM reappears as the field\nis increased towards positive values corresponding to a mix-\nture of skyrmions and stripes, the HFM is visible only above\n200 mT when the magnetisation profile consists of a dense\nskyrmion lattice network (see Fig. 1(c)), and the IFM disper-\nsion nearly flattens. We also notice that at the very same field\na mode appears at low frequency ( <2 GHz) followed by an-\nother at even lower frequencies, which we refer to as low fre-\nquency modes (LFM). The HFM mode fades away at fields\nwhen the skyrmion lattice transforms into isolated skyrmions.\nThe IFM continues beyond this point however with an abrupt\nchange of slope until the magnetisation becomes uniform be-\nyond 390 mT. The line cuts along fixed fields ( −295 mT and\n295 mT) and fixed frequencies (14 GHz and 4 GHz) are shown\nin Figs. 2(b) and 2(c) respectively. The modes at negative and\npositive fields are not symmetric but instead depend on the\nmagnetic field history (see Supplementary Figure 4), as does\nthe static magnetisation profile.Micromagnetic simulations\nWe performed simulations with the finite-di fference mi-\ncromagnetic code M umax 357in order to gain better insight\ninto the static and dynamic properties of our sample (see\nMethods). We modelled the full 20-layer repetition of the\nPt/FeCoB /AlO xtrilayer (with periodic boundary conditions\nin the film plane) in order to account for dipolar e ffects as\naccurately as possible, since it is known that the skyrmion\ncore deviates from the usual tubular structure to complex\nthree-dimensional configurations due to inhomogeneous dipo-\nlar fields along the multilayer thickness49. Figures 3(a) and\n3(b) show the simulated hysteresis loop and the skyrmion den-\nsity and apparent diameter variation as a function of the OP\nfield swept from negative to positive values. The MFM im-\nages calculated from the simulated magnetisation profiles (see\nMethods) are shown in Fig. 3(c). The field evolution of the\nsimulated static characteristics presented in Figs. 3(a)–(c) is\nfound to be in good agreement with the experiments.\nA striking feature of the labyrinthine domain and skyrmion\nstructures found is that their micromagnetic configuration ex-\nhibits strong variations along the multilayer thickness direc-\ntion. An example of such complex structures is shown in\nFig. 3(d), where the mzcomponent is shown for a single\nskyrmion core in the lattice phase at µ0Hz=285 mT. The\nfigure shows the contours for mz=0.9 (red), mz=0 (yel-\nlow), and mz=−0.9 (blue) as surfaces where cubic interpo-\nlation has been used across the nonmagnetic layers. We note\nthat the skyrmion core, in particular the region of reversed\nmagnetisation mz≤−0.9, does not extend across the entire\nthickness of the multilayer. Recall that the FeCoB layers are\nonly coupled together through dipolar interactions, which are\nsufficiently large to maintain an alignment of the core cen-\ntre but too weak to promote a coherent magnetisation profile\nacross the di fferent layers. We can also observe that the mag-\nnetisation in the uppermost layers at the core centre is not re-\nversed at all, but slightly tilted away from the film normal as\nindicated by the presence of the inverted cone. The in-plane\ncomponents of the magnetic texture shown in Fig. 3(d) are\npresented in Fig. 3(e). Here, each cube represents the mag-\nnetic state of a finite-di fference cell, where the colour repre-\nsents the orientation of the magnetisation in the cell. In order\nto highlight the role of the in-plane components to comple-\nment the data shown in Fig. 3(d), the relative size and opacity\nof each cube is scaled with the function 1 −m2\nz; this renders\nthe regions of the uniform background and the reversed mag-\nnetisation transparent. A clear skyrmion profile can be seen\nfor the nine bottom layers, where the same left-handed ( i.e.\ncounterclockwise) N ´eel chirality is found in each layer. In\nlayers 10 to 14, on the other hand, the in-plane component\nof the magnetisation at the skyrmion boundary becomes more\nuniform and does not exhibit the same winding as in the bot-\ntom half of the stack. This means that the reversed domain\nstructure is non-topological. A skyrmion profile reappears in\nlayers 15, 16 and 17 but with a reversed chirality, where the\nin-plane magnetisation components of the right-handed ( i.e.\nclockwise) N ´eel structure are rotated by 180 degrees in the\nplane with respect to their left-handed counterparts. Finally in4\nFIG. 2. Broadband ferromagnetic resonance. (a) Frequency-field dispersion map measured by VNA-FMR over a frequency range of\n0.1 GHz to 20 GHz with the applied OP field swept from -500 mT to 500 mT. Apart from the Kittel mode (KM) above saturation, several\ndistinct modes appear in the non-saturated state which are labelled as HFM: high frequency mode ( f>12 GHz), IFM: intermediate frequency\nmode ( f<8 GHz) and LFM: low frequency modes ( f<2 GHz). (b) Line cuts for fixed OP field values: -295 mT and 295 mT. (c) Line cuts\nfor fixed frequency values: 4 GHz and 14 GHz. (d) Schematic of the VNA-FMR setup.\nlayers 18 to 20, we observe another type of non-topological\ntexture where the core magnetisation is closely aligned with\nthe background magnetisation, which corresponds to the in-\nverted cone at the top of the stack in Fig. 3(d). Figure 3(f)\nshows the variation of the topological charge density per layer\nas a function of the layer number, which shows that a similar\nthickness dependence is observed across the skyrmion lattice.\nThe skyrmions in the bottom half of the stack remain topo-\nlogical, while the top half comprises largely non-topological\nbubbles. Finally, Fig. 3(g) illustrates the magnetic configura-\ntion of layer 12 across the entire region simulated, where we\ncan observe that the mainly uniformly-magnetised regions of\nthe magnetic bubble walls can vary greatly from one bubble\nto the next, with no discernible spatial order. We have ver-\nified that these features persist for finite di fference cell sizes\ndown to∼2 nm, which indicates that the complex magneti-\nsation structure does not arise from discretization e ffects (see\nSupplementary Figure 5).\nWe next discuss the dynamical response of the system,\nwhere the frequency-dependent susceptibility is computed un-\nder di fferent applied fields as in Fig. 2(a) (see Methods). The\nsimulated susceptibility map is shown in Fig. 4(a), which is\ndetermined from response of the static configurations com-\nputed in Fig. 3 to sinusoidal in-plane fields in the frequencyrange of 0.1 to 20 GHz. The Kittel mode (KM) is eas-\nily identified for the uniform state for fields above the sat-\nuration field. In the regime in which the magnetisation is\nnonuniform,−330< µ 0Hz<365 mT, three distinct types\nof modes can be identified, as illustrated by the line cut at\n285 mT shown in Fig. 4(b). As in experiments, the KM trans-\nforms into a negative field dispersion mode, an intermediate\nfrequency mode (IFM), at fields where skyrmion nucleation\nbegins. The IFM dispersion is rather rugged on the negative\nfield side, where the skyrmion density rapidly evolves with\nfield, and a faint splitting of IFM is seen around −260 mT.\nThe IFM is asymmetric with respect to zero field and exhibits\nsmoother variations on the positive field side. Similar to the\nnegative field side, another branch of the IFM is seen for pos-\nitive fields appearing at around 200 mT, which is relatively\nflat around 4 GHz in the range of positive magnetic fields\n240≤µ0Hz≤345 mT, where a dense skyrmion lattice is\nstable. It then merges to a single IFM and then varies sharply\nuntil saturation. Several low frequency modes (LFMs), com-\nprising several closely spaced branches in the frequency range\nof 0.5<f<2.5 GHz, extend over the entire magnetic field\nrange in which the magnetisation is non-uniform. Finally, a\nhigh frequency mode (HFM) is also observed above 10 GHz\nwith a positive field dispersion. It exhibits a sawtooth-like5\nFIG. 3. Micromagnetic simulations of equilibrium magnetic configuration. (a) Simulated magnetisation curve, (b) skyrmion density and\ndiameter evolution and (c) the corresponding simulated MFM images for the OP field swept from negative to positive values. The grey shaded\narea in (b) results from uncertainties in determining the skyrmion diameter, as discussed in the Methods section. (d) mzcontours and (e)\nin-plane components of the magnetisation texture of a single skyrmion in the lattice phase at 285 mT. (f) Topological charge density Qper\nlayer 285 mT. (g) Magnetic configuration of layer 12 at 285 mT.\nvariation at negative fields and a smoother variation for posi-\ntive fields. It is attenuated in the field range where magneti-\nsation profile consists of randomly oriented stripes (region\n(iv) in Fig. 3). The simulated susceptibility map reproduces\nwell the three families of modes observed experimentally in\nthe non-saturated state and gives good quantitative agreement\nfor the mode frequencies. Note that in experiments, the LFM\nbranches could not be resolved for the negative field side in the\nS21transmission data of Fig. 2(a) (as opposed to the S11and\nS22reflection data, presented in Supplementary Figure 6). It\nis because these modes have a nearly flat frequency–field dis-\npersion, and hence get diluted in the signal processing which\nis necessary to remove the field independent o ffset. Only two\nof the LFM branches become resolvable when they acquire\na certain slope on the positive field side, where the skyrmion\ndensity is constant and the skyrmion size strongly varies with\nthe field.\nThe spatial profiles of the excited modes are found by driv-\ning the system at the mode frequencies identified in Fig. 4(a)\nand by recording the resonant response. In Fig. 4(b), we\npresent the frequency response for µ0Hz=285 mT at which\nthe equilibrium ground state is a skyrmion lattice, where wecan identify well-defined peaks corresponding to the LFM,\nIFM, and HFM. Fig. 4(c) gives layer-resolved susceptibility\nwhose sum over all layers results in the curve in Fig. 4(b).\nFor the LFM, we observe that the peaks in Fig. 4(b) corre-\nspond to two kinds of resonances. For LFM 1 at f=0.94\nGHz, the primary response arises from the central part of the\nstack in layers 13 and 14, and also from the top of the stack in\nlayer 20. A similar behaviour is seen for LFM 2 at 1.50 GHz,\nwhere layers 1 and 19, close to the stack surfaces, resonate\nalong with the central part of the multilayer, as with LFM 3\nat 1.92 GHz, where a strong response is seen in layers 2 and\n18. This trend continues as the frequency increases, where\nsuccessive layers farther away from the surfaces exhibit local\nresonances in conjunction with the central part. These mode\nprofiles are visualised in Fig. 4(d), where we show the res-\nonant fluctuations in the xcomponent of the magnetisation,\n/angbracketleftδmx/angbracketright(see Methods). Blue regions give the main response to\nthe susceptibility, while the sum of equally intense blue and\nred regions cancel each other out. Similarly to Fig. 3(e), the\nopacity of the cells increase with the strength of the response\nin order to provide a clearer picture of their spatial profiles.\nLFM 1 mainly corresponds to the uniform precession of the6\nFIG. 4. Micromagnetic simulations of the dynamic response. (a) FMR spectra simulated by calculating the frequency-dependent suscepti-\nbility of multilayer to uniform excitation fields. (b) Imaginary part of the magnetic susceptibility at 285 mT (indicated by a dashed line in (a)),\nshowing three LFM (0.94, 1.50, and 1.92 GHz), two IFM (3.90 and 4.24 GHz) and one HFM (15.10 GHz). (c) Thickness-resolved susceptibil-\nity at 285 mT. (d, e, f) Three-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightfor (d) LFM, (e) IFM, and (f) HFM. (g, h, i) Two-dimensional\nview of the magnetic response /angbracketleftδmx/angbracketrightin layer 10 of the modes in (d,e,f) for (g) LFM, (h) IFM, and (i) IFM.\nbackground magnetisation at the top of the multilayer stack,\nwhile the skyrmion edges are also excited in the central part\nof the stack with a small overall contribution to the suscepti-\nbility. This can be seen more clearly in Fig. 3(g), where the\nfluctuations are shown for layer 10. For LFM 2 and LFM 3, a\nsimilar behaviour is seen where individual layers precess uni-\nformly close to the surfaces, while excitations in the central\npart are localised to skyrmion edges.\nFor the IFM, two broad resonances are identified at µ0Hz=\n285 mT and correspond to excitations that are mainly lo-\ncalised to the central part of the multilayer [Fig. 4(c,e)]. The\nmain contribution comes from the uniform precession of thebackground magnetisation between the skyrmion cores, i.e.,\nthe inter-skyrmion region which is aligned parallel to the ap-\nplied field, which can been seen from the predominantly blue\nregions in Fig. 4(h) that appear outside the cores. In addition,\nsome SW channelling is also observed at the skyrmion edges,\nas shown by alternating red and blue patches encircling the\nskyrmions.\nThe HFM involves the in-phase precession of the reversed\nmagnetisation within the skyrmion cores and extends across\nmost of the multilayer [Fig. 4(c,f)], with much of the power\nconcentrated in the central part of the stack. The coherence\nof the precession can be seen in Fig. 4(i) by the uniform blue7\nzones of/angbracketleftδmx/angbracketrightwithin the mz=0 contours denoted by the con-\ntinuous black circles. Moreover, this precession results in the\ngeneration of short wavelength SWs in the inter-skyrmion re-\ngion, which are almost as intense as the core precession. How-\never, only the volume occupied by the skyrmion cores pro-\nvides a coherent response to the susceptibility for the HFM,\nwhich explains its much weaker intensity compared to the\nIFM and KM.\nThese features are also present at other applied field val-\nues at which the equilibrium state comprises nonuniform\nmagnetisation. Supplementary Figure 7 shows an analo-\ngous behaviour at µ0Hz=−285 mT, where the lattice pos-\nsesses a lower skyrmion density with deformed core profiles.\nAtµ0Hz=140 mT where the equilibrium state comprises\nlabyrinthine domains, only the LFM and IFM are present,\nwhich further highlights the link between the HFM and the\nprecessing skyrmion cores (Supplementary Figure 8). At zero\nfield, the equilibrium state comprises a mixture of bubbles\nand elongated stripes in which no clear background orienta-\ntion can be defined. Here, LFM persist while the IFM and\nHFM are absent. Instead, we can observe the existence of cav-\nity modes that represent confined spin waves (Supplementary\nFigure 9).\nSpin wave emission from the high frequency mode\nThe short wavelength SWs associated with the HFM can-\nnot be excited directly with the uniform driving fields used\nin FMR, so their presence indicates the skyrmion cores act as\nnanoscale transducers that convert a spatially-uniform drive\ninto a spatially-nonuniform excitation. To obtain deeper in-\nsight into this process, we focus on the HFM in the simulation\ndata at two values of the applied fields, 300 and 364 mT, as\npresented in Fig. 5. The equilibrium state at 300 mT com-\nprises a skyrmion lattice, with a skyrmion density of 20 µm−2\nand an average core-to-core distance of L=262 nm, while\nonly one metastable skyrmion is present at 364 mT within the\nsame area. The skyrmion diameters are roughly equivalent at\nthese field values, which are 95 and 94 nm, respectively. The\nsusceptibility at these two fields, shown in Fig. 5(a) for the\nrange of 12–17 GHz, display a clear resonance peak around\n15.5 GHz, which is over an order of magnitude larger at\n300 mT than at 364 mT due to the di fference in the skyrmion\ndensity.\nAt both field values, the coherent precession of the reversed\nmagnetisation within the skyrmion cores emit spin waves into\nthe ferromagnetic background. Their wavelengths are simi-\nlar at both fields and are estimated from the /angbracketleftδmx/angbracketrightprofiles in\nlayer 10 shown in Figs. 5(b) and 5(c), yielding λ/similarequal60 nm and\nλ/similarequal54 nm atµ0Hz=300 mT and µ0Hz=364 mT, respec-\ntively. Despite the strong variations in the static core profile\nacross the multilayer thickness, the emitted spin waves are re-\nmarkably coherent across the thickness. This can be seen in\nFig. 5(d) for 300 mT and Fig. 5(e) for 364 mT where coher-\nent wavefronts, driven in the central part of the multilayer at\nwhich the core diameter is greatest, extend across much of the\nmultilayer thickness. The left inset in Figs. 5(d,e) indicate thenumber of topological charges per layer, which shows that the\nspin wave emission is just as e ffective for the topological bub-\nbles residing primarily in the bottom of the stack as for the\nnon-topological bubbles residing primarily in the top.\nThe overall coherence of these spin wave excitations\ncan also be seen from the layer-averaged spatial discrete\nFourier transform of the dynamical magnetisation shown in\nFigs. 5(b,c). These Fourier maps, shown in Figs. 5(f,g), dis-\nplay a ring pattern with wave vectors 100 <kSW<150µm−1\ncorresponding to wavelengths 40 < λ < 60 nm. At 300 mT,\nthe pattern in the Fourier transformed data also manifests the\nperiodic structure of the skyrmion lattice, from which the lat-\ntice periodicity can be estimated as shown in Fig. 5(f), using\n2π/k=262 nm, where k=24µm−1.\nThe spin wave dynamics can be understood from the lin-\nearised equations of motion for the dynamic magnetisation\nδm,∂tδm=−|γ0|(δm×Heff,0+m0×δHeff), where m0is\nthe static magnetisation configuration and Heff,0is the static\neffective (internal) field associated with m0, whileδHeffis the\ndynamic e ffective (internal) field associated with δm. The full\n3D structure of the core and the surrounding configuration de-\ntermines the spatial profile of the static internal field and the\nlocalisation of the mode. The average value of this static in-\nternal field Heff,0(r) inside the skyrmion core can be extracted\nfrom micromagnetic simulations and is about −200 mT, which\nleads to resonances at 5.8 GHz without any confinement ef-\nfects. To this, one needs to add the dynamic dipolar and ex-\nchange contributions δHeff(r,t) due to the localisation of mag-\nnetisation dynamics in a volume of ∼100×100×50 nm3,\nwhich increases the resonant frequency to about 15 GHz (see\nSupplementary Note 2). Similarly, we can estimate the wave-\nlength of the emitted SWs by using the dispersion relation\nof forward-volume SWs in the inter-skyrmion region. At\n300 mT, the average internal field in this region is around\n110 mT, which results in a wavelength of 60 nm at 15.5 GHz, a\ngroup velocity of 960 m /s and an attenuation length of 490 nm\n(see Supplementary Figure 10). These estimations are in good\nagreement with the SW characteristics observed in Fig. 5.\nThe di fference between the mode profiles shown in\nFigs. 5(b) and 5(c) is due to interference e ffects. At low\nskyrmion densities, the SW emitted at one core attenuates be-\nfore reaching another, resulting in the familiar concentric rip-\nples around each excitation source. At higher skyrmion densi-\nties, on the other hand, the inter-skyrmion distance is smaller\nthan the attenuation length and interference patterns can arise.\nThe cores can emit SWs in a broad spectral range, which de-\npends on the excitation frequency and the skyrmion size (see\nSupplementary Figure 11). As such, the overall collective be-\nhavior is governed by a subtle interplay between the charac-\nteristic length scales of the system. The skyrmion diameter,\nwhich varies with the applied magnetic field, determines the\nprecession frequency of the core thereby the wavelength of\nthe emitted SWs, whereas constructive or destructive SW in-\nterference patterns are determined by the commensurability of\nSW wavelength and lattice parameters.8\nFIG. 5. Emission and interference of spin waves in skyrmion lattice. (a) Simulated susceptibility curves at 300 mT and 364 mT with the\ndynamic magnetisation profiles /angbracketleftδmx/angbracketrightin (b) and (c) of layer 10, respectively, corresponding to the resonance at 15.5 GHz. Cross-sections\nof/angbracketleftδmx/angbracketrightmarked by dotted lines in (b) and (c) are also shown respectively in panels (d) and (e), highlighting the in-phase skyrmions core\nprecession (in blue) and spin wave emission in the surrounding medium (alternate blue and red zones) (f,g) Power spectral density (PSD) of\nthe layer averaged spatial discrete Fourier transform of the dynamic magnetisation profiles indicating the average k S W-vector of the emitted\nSWs at 300 mT and 364 mT respectively. At 300 mT, the skyrmion lattice parameter can also be deduced from the k=24µm−1periodic\npattern seen in the spatial Fourier transform.\nDISCUSSION\nSW emission related to the HFM can have advantages over\nother reported methods of sub-100 nm wavelength genera-\ntion. For example, magnetic vortex cores, which can act as\nSW emitters, require confined geometries such as dots for\ntheir existence and do not spontaneously form lattice struc-\ntures in continuous films, which precludes the appearance of\nSW interference phenomena seen here. The precession fre-\nquency of the skyrmions in our samples is also above the FMR\ngap of the surrounding domain background and hence SWs\ncan be excited at this frequency in the same medium, unlike\nin other hybrid structures such as Pt /Co/YIG58. Moreover,\nthe SWs generated in PMA materials are forward-volume\nSWs, which propagate isotropically59compared to backward-\nvolume and Damon-Eshbach modes in in-plane magnetised\nsystems, while maintaining high group velocities.\nAn interesting analogy of our skyrmion lattice as SW emit-\nters can be made with arrays of spin-torque nano-oscillators,\nwhich can generate propagating SWs and get synchronised via\nthem60. These structures have been proposed for implement-\ning unconventional computing schemes5. We have shown\nhere that SWs can be emitted in the uniform background by\nthe skyrmion dynamics, but one also expects that in return,\nSWs can influence the skyrmion dynamics39, which could en-\nlarge the processing capabilities of this type of devices.\nOur study also paves the way toward creating reconfig-\nurable magnonic crystals, since the skyrmion lattices we ob-serve are quasi-periodic and dynamically tunable37. We can\nget a glimpse of the role of this lattice in the observation\nof SW interference patterns in the simulated HFM. How-\never, the properties of the skyrmion lattice as a magnonic\ncrystal also needs to be explored using broadband propagat-\ning SW spectroscopy10or advanced SW imaging methods9,25.\nLastly, our system could serve as a platform to study magnon-\nskyrmion scattering39, emergent magnon motifs61, or chiral\nmagnonic edge states62.\nMETHODS\nExperimental details\nTa(10) /Pt(8)/[FeCoB(1.2) /AlO x(1.0)/Pt(1.6)] 19/FeCoB(1.2)\n/AlO x(1.0)/Pt(3) multilayer stacks (where FeCoB ≡\nFe70Co10B20, and thicknesses given in nm) were deposited by\ndc magnetron sputtering at room temperature on thermally-\noxidised silicon substrates, under Ar gas flow at a pressure\nof 0.25 Pa. The base pressure of the sputtering equipment\nwas 5×10−6Pa. magnetisation cycles were measured by\nalternating gradient force magnetometry on 3 ×3 mm2films.\nMFM images were obtained at room temperature and ambient\npressure using a double-pass lift mode, detecting the phase\nshift (MFM signal) of the second pass after a topographic\nmeasurement. The employed low moment tips were home\nmade49. Small permanent magnets with tunable gap and9\ncalibrated values of field were used to bias the samples while\nimaging them. VNA-FMR was conducted on 2 ×4 mm2films\nfacing the 50 µm wide central conductor of a 50 Ωmatched\ncoplanar waveguide. The OP magnetic field was ramped up\nand down between −550 mT and +550 mT. For each of the\n400 field steps, the VNA frequency was swept from 0.1 GHz\nto 20 GHz with a number of points set to 1491 and a sweep\ntime of 45 s. To improve the signal to noise ratio, the data\nwere averaged over 30 magnetic field cycles. To remove the\nfield independent background and normalise the transmission\nsignal data S21, we calculate d DS21using the derivative\ndivide method63. To increase the contrast further, we apply an\nadditional frequency di fferentiation, and plot the imaginary\npart of the obtained quantity in Fig. 2(a).\nExtraction of magnetic parameters\nThe magnetisation of the studied multilayers was deter-\nmined by SQUID magnetometry. The 1.2 nm thick FeCoB\nis found to have a saturation magnetisation Ms=1.20±\n0.1 MA /m. Using angle dependent cavity-FMR, the gyro-\nmagnetic ratio γ0/(2π)=28.75±0.10 GHz /T and the total\nuniaxial anisotropy field including demagnetising and PMA\ncontributions µ0(Ms−Hk)=354.7±0.5 mT are determined\n(see Supplementary Note 1). In the case of uniaxial anisotropy\nµ0Hk=2Ku/Ms, hence Ku=0.69±0.11 MJ /m3. By as-\nsuming Ms=1.2 MA /m and Ku=0.7 MJ/m3, the DMI and\nexchange parameters were iteratively determined by adjusting\nthe period of the simulated labyrinthine domains to the one\n(/similarequal170 nm) observed by MFM at remanence49. We find the\nDMI constant to be D=1.2 mJ/m2for an exchange constant\nA=15 pJ /m.\nDetermination of skyrmion diameter\nFrom the MFM measurements of the quasi static mag-\nnetic configuration as a function of the applied field, we ex-\ntract skyrmion density by using a MATLAB program which\nuses binarisation. However, as the image resolution is not\ngood enough to estimate the skyrmion diameters directly from\nthe MFM images (which would anyway require to know the\ntransfer function of the tip64, and might be complicated by\nthe influence of the tip’s stray field), we extract the appar-\nent skyrmion diameters (2 Rsk) indirectly, for the field values\nwhere the skyrmion density ( dsk) is close to maximum, us-\ning the out-of-plane experimental magnetisation curve: Rsk=/radicalbig\n(1−mz)/(2πdsk). In Fig. 1(c) the error bars constitute the er-\nrors on the skyrmion density (proportional to the density and\ninversely proportional to the image size: ∼8%) and magnetic\nfield values to account for the slight misalignment of the mag-\nnetic pole and the magnetic tip in the MFM setup (10%). In\nthe case of simulated MFM images, we used the same MAT-\nLAB program that binarises the simulated MFM images, de-\ntects and finds the diameters of all the skyrmions, and calcu-\nlates the mean and the standard deviation. For the simula-\ntions, we also compared the skyrmion diameter i /calculatedby the program and ii /estimated from skyrmion density and\nmzcurve as a function of field when the skyrmion density is\nclose to maximum. The small di fference in the skyrmion di-\nameters for the two methods lies within the standard deviation\nof the former method, indicated by the shaded area in grey in\nFig. 3(b).\nMicromagnetic simulations\nSimulations were carried out with the open-source finite-\ndifference micromagnetics code M umax 357. We used values\nofMs=1.3375 MA /m and Ku=0.9 MJ /m3, which are\nslightly larger than the experimentally-determined values but\nwere chosen to reproduce quantitatively the mode frequencies\n(see Supplementary Figure 12). The geometry consists of a\nrectangular slab of volume 1 µm×1µm×72 nm discretised\nusing 128×128×60 rectangular cells for the susceptibility\ncalculations and a volume of 2 µm×2µm×72 nm with\n256×256×60 cells for the quasi-static hysteresis loop calcu-\nlations. The simulated stack consists of 20 ferromagnetic lay-\ners with thickness 1.2 nm separated by 2.4 nm empty spacers,\nwhich accurately mimics the experimental stack, in which the\ntotal thickness of non-magnetic Pt and AlO xlayers separating\nadjacent FeCoB layers is 2.6 nm. Periodic boundary condi-\ntions are used in xandydirections in the film plane. The code\nperforms a numerical time integration of the Landau-Lifshitz-\nGilbert equation for the magnetisation dynamics,\ndm\ndt=−|γ0|m×Heff+αm×dm\ndt, (1)\nwhere/bardblm(r,t)/bardbl=1 is a unit vector representing the ori-\nentation of the magnetisation field, γ0=µ0gµB//planckover2pi1is the\ngyromagnetic constant, and αis the Gilbert damping con-\nstant. The e ffective field, Heff=−(1/µ0Ms)δU/δm, repre-\nsents a variational derivative of the total magnetic energy U\nwith respect to the magnetisation, where Ucontains contri-\nbutions from the Zeeman, nearest-neighbour Heisenberg ex-\nchange, interfacial Dzyaloshinskii-Moriya, and dipole-dipole\ninteractions along with a uniaxial anisotropy with an easy axis\nalong ˆz, perpendicular to the film plane. Note that while the\nlateral cell size of 7.8125 nm used for the simulations pre-\nsented here is larger than the characteristic exchange length,\nlex=/radicalbig\n2A/µ0M2s/similarequal3.65 nm, we have verified that using\nsmaller discretisation cells do not a ffect the salient features\npresented here (see Supplementary Figure 13).\nTo simulate quasi-static processes such as hysteresis loops,\nwe begin with a uniform magnetic state oriented along −z\nunder an applied field of µ0Hz=−500 mT. A Langevin-\ndynamics simulation with T=300 K is then performed over 1\nns to mimic experimental conditions in which thermal fluctu-\nations help to sample di fferent metastable states in the energy\nlandscape. The system is then relaxed without the preces-\nsional term in Eq. (1) toward an energy minimum. This proce-\ndure of running Langevin dynamics, followed by relaxation, is\nthen repeated two more times. After this, the micromagnetic\nconfiguration is recorded and assigned as the equilibrium state\nfor the applied field value. This equilibrium state then serves10\nas the initial state for the next value of the applied field, which\nis increased to µ0Hz=500 mT in increments of 1 mT. The\nsimulated MFM images are computed from the equilibrium\nstate by assuming a sample-to-tip distance of 50 nm.\nTo simulate the microwave susceptibility of the FeCoB\nmultilayer, we compute the response of the magnetisation to\nharmonic excitation fields. While it is common practice (and\nusually more e fficient computationally) to calculate the eigen-\nmode spectra from the Fourier transform of the transient re-\nsponse to pulsed fields, such an approach for the present sys-\ntem results in strong artifacts arising from non-resonant drift\nof the background magnetic state, such as the displacement\nof the skyrmion lattice or labyrinthine domain structure. In-\nstead, we adopt an approach that more closely resembles the\nexperimental protocol. For each value of the external field\nHz, we compute the response of the precomputed static con-\nfiguration, m0(r), to sinusoidal in-plane fields with frequency\nf,hx=hrfsin(2πf t), over 20 periods τ=1/f, where the\nspatial average of the resulting component mx(t) is recorded\nat intervals of 0.1 τ. In this procedure, the Gilbert damping\nparameter is set to α=0.02 and the excitation amplitude to\nµ0hrf=0.1 mT (1 mT in Fig. 5 and Supplementary Figure 12).\nWe estimate the imaginary part of the susceptibility χat this\nfrequency, Im[ χ(f)], by projecting/angbracketleftmx(t)/angbracketrightonto−cos(2πf t),\nIm/bracketleftbigχ(f)/bracketrightbig/similarequal−1\n20τ/integraldisplay20τ\n0/angbracketleftmx(t)/angbracketrightcos(2πf t)dt, (2)\nwhich is the quantity shown in Figs. 4(a) and 4(b). Im[ χ(f)]\nexhibits peaks at resonance when mxis in perfect quadrature\nwith respect to hx. This procedure is repeated as a function of\nfranging from 0.1 GHz to 20 GHz in steps of 0.1 GHz for\nthe map in Fig. 4(a) and in steps of 0.02 GHz for the data in\nFigs. 4(b,c).\nThe spatial mode profile at a given mode resonance fn\nis extracted by driving the system harmonically at fnwith\nhx=hrfsin(2πfnt)and by recording, cell-by-cell, the compo-\nnents of magnetisation at intervals of 0.1 τn, as for the calcula-\ntion of the susceptibility. The mode profiles are characterised\nby the fluctuation amplitude /angbracketleftδmx(r)/angbracketright, which is computed in\nan analogous way to the susceptibility, where we drive the\nsystem over 100 periods and then compute, after transients\nare washed out,\n/angbracketleftδmx(r)/angbracketright=−1\n2τn/integraldisplay2τn\n0[mx(r,t)−mx,0(r)] cos (2πfnt)dt,(3)\nover two additional periods. Visually, dark blue regions con-\ntribute most to the overall Im[ χ(f)], while equally dark redand blue regions cancel each other out because they represent\nfluctuations with opposite sign.\nDATA A V AILABILITY\nData are available from the corresponding authors upon rea-\nsonable request.\nACKNOWLEDGEMENTS\nThis work was partially supported by the French National\nResearch Agency (ANR) under grant no. ANR-17-CE24-\n0025 (TOPSKY) and the Horizon2020 Research Framework\nProgramme of the European Commission under grant nos.\n824123 (SKYTOP) and 899646 (k-NET). It is also supported\nby a public grant overseen by the ANR as part of the “In-\nvestissements d’Avenir” program (Labex NanoSaclay, refer-\nence: ANR-10-LABX-0035).\nAUTHOR CONTRIBUTIONS\nN.R., V .C., T.D., J.-V .K and G.d.L. conceived the project.\nThe samples were designed, grown and characterised by T.S.,\nY .S., F.A., N.R. and V .C. T.S. performed the MFM experi-\nments with the help of A.V . and K.B. H.H. conducted the\ncavity-FMR measurements. T.S. and I.N. performed initial\nFMR characterizations. T.D. conducted the VNA-FMR ex-\nperiments and analyzed the results with T.S. and G.d.L. J.-\nV .K. performed the micromagnetic simulations and analyzed\nthem with T.S. and G.d.L. T.S., J.-V .K. and G.d.L. wrote the\nmanuscript. All the authors discussed the data and commented\non the manuscript.\nCOMPETING INTERESTS\nThe authors declare no competing interests.\nADDITIONAL INFORMATION\nSupplementary information The online version contains\nsupplementary material available at https: //doi.org /xxxxx.\nCorrespondence and requests for materials should be ad-\ndressed to T.S., J.-V .K. or G.d.L.\n∗titiksha.srivastava@cea.fr\n†joo-von.kim@c2n.upsaclay.fr\n‡gregoire.deloubens@cea.fr\n1V . V . Kruglyak, S. O. Demokritov, and D. Grundler, “Magnonics,”\nJ. Phys. D Appl. Phys. 43, 264001 (2010).\n2M. Krawczyk and D. 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Mater. 551, 169073 (2022).Supplementary Information —\n“Resonant dynamics of skyrmion lattices in thin film multilayers: Localised modes and spin\nwave emission”\nTitiksha Srivastava,1, 2,∗Yanis Sassi,1Fernando Ajejas,1Aymeric Vecchiola,1\nIgor Ngouagnia,2Herv ´e Hurdequint,2Karim Bouzehouane,1Nicolas Reyren,1\nVincent Cros,1Thibaut Devolder,3Joo-V on Kim,3,†and Gr ´egoire de Loubens2,‡\n1Unit´ e Mixte de Physique, CNRS, Thales, Universit´ e Paris-Saclay, 91767, Palaiseau, France\n2SPEC, CEA, CNRS, Universit´ e Paris-Saclay, 91191, Gif-sur-Yvette, France\n3Centre de Nanosciences et de Nanotechnologies, CNRS, Universit´ e Paris-Saclay, 91120, Palaiseau, France\n∗titiksha.srivastava@cea.fr\n†joo-von.kim@c2n.upsaclay.fr\n‡gregoire.deloubens@cea.frarXiv:2111.11797v2  [cond-mat.mes-hall]  28 Jun 20222\nCONTENTS\nSupplementary Note 1. Additional FMR measurements in the saturated state 3\nAngle dependent cavity-FMR 3\nDamping parameter 3\nSupplementary Note 2. Analysis of the high frequency mode 4\nInternal field in skyrmion lattice phase 4\nContributions to the eigenfrequency 4\nCharacteristics of the spin waves emitted by the skyrmion cores 5\nSupplementary Figures 6\nSupplementary References 193\nSUPPLEMENTARY NOTE 1. ADDITIONAL FMR MEASUREMENTS IN THE SATURATED STATE\nAngle dependent cavity-FMR\nCavity-FMR measurements are performed at a fixed frequency of 9.725 GHz to probe the FMR of\nthe multilayer sample as a function of the angle θHbetween the applied field and the film plane. Out-\nof-plane (OP) and in-plane (IP) configurations correspond to θH=0◦andθH=90◦, respectively. The\nresonant field measured at θH=0◦is higher than at θH=90◦, meaning that the overall anisotropy is in-\nplane. From the values of the resonant field in these two configurations, one can extract the gyromagnetic\nratioγ0/(2π)=28.75 GHz /T and the total uniaxial anisotropy field including demagnetising and PMA\ncontributions µ0(Ms−Hk)=354.7 mT. Using these values, the full angular variation of the resonance field is\naccurately reproduced by the theoretical expectation [1], as shown by the continuous line in Supplementary\nFigure 1(a). The angular dependence of the full width ∆Hat half maximum of the resonance peak is\ndisplayed in Supplementary Figure 1(b). The minimal value of 33.2 mT is observed in the IP configuration,\nwhile the linewidth is only slightly larger in the OP configuration, ∆H=35.5 mT; it is rather low for this\ntype of multilayer film, which evidences its good dynamical quality.\nDamping parameter\nTo separate the intrinsic and extrinsic contributions to the FMR linewidth, broadband-FMR measure-\nments are performed in the IP configuration. At 10 GHz, the extracted linewidth is 32.5 mT, in good\nagreement with the cavity-FMR results. The frequency dependence of ∆His plotted in Supplementary Fig-\nure 1(c) together with its linear fit. The latter yields the frequency independent contribution to the linewidth,\nfound to be about 18 mT, which indicates a fair amount of inhomogeneities in the film. The frequency linear\npart of the linewidth yields the Gilbert damping parameter of the film, α=0.022±0.003.4\nSUPPLEMENTARY NOTE 2. ANALYSIS OF THE HIGH FREQUENCY MODE\nInternal field in skyrmion lattice phase\nThe spatial profile of the total internal field ( i.e., the sum of the applied, anisotropy, and dipolar fields)\ncan be obtained from micromagnetic simulations. The map obtained at 300 mT is shown for layer 12 in\nSupplementary Figure 10(a). It shows that su fficiently far away from the skyrmions, the static internal field\nis rather homogeneous in the background domains, around 110 to 120 mT. Due to the strong dipolar fields\nin the vicinity of the oppositely magnetised skyrmion cores, the internal field substantially increases while\napproaching the skyrmion boundaries, as shown by the closely spaced isocontours. Inside the skyrmion\ncores it becomes negative, about −200 mT.\nContributions to the eigenfrequency\nAs shown by Fig. 5 of the main text the high frequency mode (HFM) corresponds to a precession of\nthe magnetisation localized inside the skyrmion cores. Its frequency can be calculated as the convolution\nof its spatial profile with a frequency operator taking into account all the contributions to the magnetic\nenergy (Eq. 8 in ref. [2]): Zeeman, anisotropy, dipolar and exchange energies. Even for homogeneously\nmagnetised non-ellipsoidal bodies, it is di fficult to evaluate the magnetic self-interactions (magneto-dipolar\nand inhomogeneous exchange) contributing to the eigenfrequency. As shown in Fig. 3 of the main text,\nthe static configuration in the 20 dipolarly coupled magnetic layers is quite complex in the skyrmion phase.\nHowever, to roughly estimate the HFM frequency, one can consider the averaged value of the static internal\nfield inside the skyrmion core where the precession is localized, to which the dynamic magneto-dipolar\nself-interaction ( i.e.the depolarization energy of the mode profile on itself) and the exchange contributions\nshould be added. The averaged value µ0Heff,0≈−200 mT of the internal field in the core translates into a\nfrequency of 5 .8 GHz, which is less than the HFM frequency of 15.5 GHz determined by micromagnetic\nsimulations. The exchange contribution to the frequency is given by ( γ0Ak2\nHFM)/(πMs) where kHFM is the\neffective wavevector of the mode. Assuming an e ffective wavelength of the order of the skyrmion diameter\n≈100 nm, one obtains an exchange contribution of 2.6 GHz. The evaluation of the contribution due to the\ndynamic magneto-dipolar self-interaction is more di fficult, because it requires the knowledge of static and\ndynamic magnetisation profiles to perform the required integration. However, the missing 7.1 GHz fall in\nthe frequency range of this contribution in submicron samples [2].5\nCharacteristics of the spin waves emitted by the skyrmion cores\nTo find the characteristics of the spin waves (SWs) emitted by the precession of the skyrmion cores in\nthe surrounding background domains, one should consider that the latter are homogeneously magnetised\nOP along the applied field and use the dispersion relation of forward-volume waves [3]:\nω(k)=/radicalBigg/parenleftBigg\nωHeff,0+2Aγ0\nMsk2/parenrightBigg/parenleftBigg\nωHeff,0+2Aγ0\nMsk2+ωM/parenleftBigg\n1−1−e−kt\nkt/parenrightBigg/parenrightBigg\n, (S1)\nwhereωHeff=γ0Heff,0andωM=γ0M. To take into account the multilayer nature of the film, the dynamic\ndipolar term is evaluated using an e ffective film medium of total thickness t=20×(tFeCoB +tPt+tAlOx)=\n76 nm and magnetisation M=tFeCoB/(tFeCoB +tPt+tAlOx)Ms=0.42 MA /m. This dispersion relation is\nplotted in Supplementary Figure 10(b) using µ0Heff,0=110 mT, which corresponds to the mean value of\nthe static internal field in the interskyrmion region at the bias field of 300 mT shown in Supplementary\nFigure 10(a). The skyrmion core precession frequency at this field is equal to 15.5 GHz, as indicated by the\nhorizontal dashed line that crosses the dispersion curve at k=105µm−1, corresponding to a wavelength of\nλ=59.8 nm. This value is in excellent agreement with the one directly determined from the SW radiation\npattern of Fig. 5(c) in the main text. Moreover, it is also possible to compute the group velocity Vg=∂ω/∂ k\nof the forward-volume wave in the e ffective film, which is plotted in Supplementary Figure 10(c). The\ngroup velocity of the SWs propagating at 15.5 GHz is found to be Vg=960 m /s. Based on Eq. (S1),\nforward volume SWs exhibit almost circular precession in our material, therefore one can use τ=1/(αω)\nto evaluate their lifetime. At ω/(2π)=15.5 GHz,τ=1/(αω)=0.513 ns, which allows to estimate the\nattenuation length of this forward-volume SW, Latt=Vgτ=493 nm.6\nSUPPLEMENTARY FIGURES\nSupplementary Figure 1 |Ferromagnetic resonance in the saturated state . (a) Evolution of the resonance field as\na function of the angle θHbetween the applied field and the film plane, determined by cavity-FMR at 9.725 GHz. The\ncontinuous line is the theoretical expectation using the values of gyromagnetic ratio and total uniaxial anisotropy field\nextracted from out-of-plane (OP) and in-plane (IP) configurations given in the text. (b) Corresponding dependence of\nthe FMR linewidth on θH. The dashed line is a guide to the eye. Inset: spectrum measured in the OP configuration.\nThe resonance field of the uniform Kittel mode is 691.2 mT, additional modes are observed at 718.3 mT and 737.2 mT.\n(c) Dependence of the in-plane FMR linewidth on frequency measured by broadband FMR. The red line is the linear\nfit yielding the damping parameter and the inhomogeneous contribution.7\nSupplementary Figure 2 |Skyrmion lattice. (a) MFM image of skyrmions in disordered state at -280 mT, and\n(b) nearest neighbour (NN) probability distribution. (c) Skyrmion lattice at -190 mT, and (d) its nearest neighbour\nprobability distribution, demonstrating its almost quasi periodic hexagonal structure.8\nSupplementary Figure 3 |VNA-FMR data processing . (a) Intensity map of the 30-cycle averaged transmission\nsignal S21in the field–frequency coordinates, the field being swept upwards. (b) Map of the S21data processed\nusing the derivative method. (c) Map of the S21data processed using the derivative method followed by a frequency\ndifferentiation. (d–f) same as (a–c) for the magnetic field being swept downwards. In addition to the narrow signals\nascribed to ferromagnetic resonances in the multilayer sample, there is a very broad background signal coming from\nthe measurement cell which is symmetric between negative and positive field values, as can be seen from the distorted\n”V” shape in panels (b) and (e), and independent of the sweep direction – in strong contrast with the resonances\noriginating from the multilayer sample (see also Suppl. Fig. 4). The frequency di fferentiation attenuates the broad\nbackground signal and improves the signal-to-noise ratio of the narrow resonances from the sample, as seen from\npanels (c) and (f).9\nSupplementary Figure 4 |Hysteretic character of the FMR modes . (a) OP magnetisation curves determined using\nalternating gradient force magnetometry by sweeping the field up and down. (b) Corresponding dependence of FMR\nmodes determined by broadband VNA-FMR.10\nSupplementary Figure 5 |Mesh size dependence of static states . The simulated static properties, as discussed in\nFig. 3 of the main text, are shown for three di fferent discretisation schemes for 1 µm of the simulated geometry: 128 (as\ndiscussed in the main text), 256, and 512 cells. (a–c), Hysteresis loops. (d–f), Density plots of the topological charge\ndensity per layer as a function of perpendicular applied field Bz. (g–i), In-plane components of the magnetisation\nconfiguration for a skyrmion at Bz=285 mT. (j–l), Topological charge density as a function of layer number at\nBz=285 mT. (m–o), Magnetisation configuration of layer 12 at Bz=285 mT, where the dashed boxes indicate the\nstate shown in (g–i), respectively.11\nSupplementary Figure 6 |VNA-FMR reflection parameters . Real parts of the (a) S11and (b) S22parameters\nprocessed using the derivative divide method. A broad signal between 0.5 GHz and 1.5 GHz is observed on the full\nfield range between −330 mT and 390 mT, which corresponds to the simulated low frequency modes (LFM).12\nSupplementary Figure 7 |Micromagnetic simulations of the dynamic response at µ0Hz=−285mT. (a) Field-\ndependent susceptibility map, reproduced from Fig. 4(a). (b) Imaginary part of the magnetic susceptibility at -285 mT,\nshowing six LFM (0.6, 1.06, 1.16, 1.4, 1.6, and 2.08 GHz), two IFM (2.92 and 3.1 GHz) and one HFM (13.52 GHz).\n(c) Thickness-resolved susceptibility at -285 mT. (d, f, g) Three-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightfor\n(d) LFM, (f) IFM, and (g) HFM. (e, h, i) Two-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightin layer 10 of the\nmodes in (d,f,g) for (e) LFM, (h) IFM, and (i) IFM, respectively.13\nSupplementary Figure 8 |Micromagnetic simulations of the dynamic response at µ0Hz=140mT. (a) Field-\ndependent susceptibility map, reproduced from Fig. 4(a). (b) Imaginary part of the magnetic susceptibility at 140 mT,\nshowing five LFM (1.02, 1.2, 1.36, 1.96, and 3 GHz) and one IFM (6.54 GHz). No HFM is observed because\nno skyrmions are present. (c) Thickness-resolved susceptibility at 140 mT. (d, e) Three-dimensional view of the\nmagnetic response /angbracketleftδmx/angbracketrightfor (d) LFM and (e) IFM. (f, g) Two-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightin\nlayer 10 of the modes in (d,e) for (f) LFM and (g) IFM, respectively.14\nSupplementary Figure 9 |Micromagnetic simulations of the dynamic response at µ0Hz=0mT. (a) Field-\ndependent susceptibility map, reproduced from Fig. 4(a). (b) Imaginary part of the magnetic susceptibility at 0 mT,\nshowing four LFM (0.7, 1.06, 1.56, and 2.1 GHz) and three cavity modes (CM) (8.02, 10.62, and 13.14 GHz), where\nthe latter represent spin wave excitations confined within the bubble or elongated stripe domains. (c) Thickness-\nresolved susceptibility at 0 mT. (d, e) Three-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightfor (d) LFM and (e)\nCM. (f, g) Two-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightin layer 10 of the modes in (d,e) for (f) LFM and\n(g) CM, respectively.15\nSupplementary Figure 10 |Internal field and spin wave dispersion . (a) Map of the internal field extracted from\nmicromagnetic simulations at 300 mT in layer 12. Values are given in units of mT. (b) Dispersion relation of forward-\nvolume SWs calculated using Eq. S1 for a uniform internal field of 110 mT. (c) Corresponding group velocity. The\ndashed lines in panels (b) and (c) highlight the SWs excited at 15.5 GHz, whose wavelength and group velocity are\nrespectively found to be λ=59.8 nm and Vg=960 m /s.16\nSupplementary Figure 11 |Frequency dependence of the SW interference pattern . Dynamic magnetisation\nexcited in layer 10 at 300 mT (top) and 364 mT (bottom) by an in-plane microwave field of frequency ranging\nfrom 10 GHz to 18 GHz, and associated layer averaged spatial FFT. The most e fficient frequency to drive the core\nprecession corresponds to the resonance frequency of 15.5 GHz. When using a lower or higher frequency to drive the\nsystem, its overall response has a lower amplitude. In addition, following the trend of the dispersion relation shown\nin Supplementary Figure 10(b), the wave vector of the emitted SWs increases with the excitation frequency of the\nskyrmion core, as seen from the FFT maps. As a result, the resulting interference patterns also change, in a way\nthat depends on the skyrmion lattice, as shown by the comparison between 300 mT and 364 mT. More generally, the\ninterference patterns depend on the characteristic lengths of the system, which can be tuned by the applied OP field.\nBoth the density and the diameter of the skyrmions indeed depend on the field (see Fig. 3(b) of the main text), which\ndetermines the travel length of the SWs between the boundaries of neighbouring skyrmions.17\nSupplementary Figure 12 |Optimisation of magnetic parameters used in micromagnetic simulations . Simulated\nimaginary part of the dynamic susceptibility, Im[ χ(f)], as a function of applied perpendicular field, µ0Hz, for devia-\ntions from experimentally-determined values of the micromagnetic parameters, (a) Ms,0=1.2 MA /m,A0=15 pJ /m,\nKu,0=0.70 MJ /m3, and D0=1.2 mJ /m2. (b) Ku=Ku,0+0.04 MJ /m3, (c) Ku=Ku,0−0.04 MJ /m3, (d) A=A0+5\npJ/m, (e) A=A0−5 pJ/m, (f) D=D0+0.03 mJ /m2, (g) Experimental values with applied field tilted 2◦from z-axis\nalong the xdirection, (h) Ms=Ms,0+0.1 MA /m and Ku=Ku,0+0.14 MJ /m3, (i)Ms=Ms,0+0.175 MA /m and\nKu=Ku,0+0.26 MJ /m3, and (j) Ms=Ms,0+0.1375 MA /m and Ku=Ku,0+0.20 MJ /m3, which represents the best\nfit. The blue dots represent the experimental data.18\n-500 -400 -300 -200 -100 0 100 200 300 400 50002468101214161820\n-500 -400 -300 -200 -100 0 100 200 300 400 50002468101214161820\n-500 -400 -300 -200 -100 0 100 200 300 400 50002468101214161820\nSupplementary Figure 13 |Mesh size dependence of dynamic susceptibility . Simulated imaginary part of the\ndynamic susceptibility, Im[ χ(f)], as a function of applied perpendicular field, µ0Hz, for three finite-di fference dis-\ncretization schemes of the 1000 ×1000×72 nm simulation box: (a) 128 ×128×60, (b) 256×256×60, and (c) 512\n×512×60 finite di fference cells. The discretisation scheme and corresponding cell size is indicated on each figure.\nThe micromagnetic parameters used correspond to the experimental values [see Supplementary Figure 12(a).].19\nSUPPLEMENTARY REFERENCES\n[1] Hurdequint, H. FMR studies of ultrathin permalloy layers sandwiched by Al 2O3.J. Magn. Magn. Mater. 242-245 ,\n521–524 (2002).\n[2] Naletov, V . V . et al. Identification and selection rules of the spin-wave eigenmodes in a normally magnetized\nnanopillar. Phys. Rev. B 84, 224423 (2011).\n[3] Kalinikos, B. A. & Slavin, A. N. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with\nmixed exchange boundary conditions. J. Phys. C 19, 7013 (1986)."    },    {        "title": "1706.01185v1.Consistent_microscopic_analysis_of_spin_pumping_effects.pdf",        "content": "Consistent microscopic analysis of spin pumping e\u000bects\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS),\n2-1 Hirosawa, Wako, Saitama, 351-0198 Japan\nShigemi Mizukami\nWPI - Advanced Insitute for Materials Research,\nTohoku University Katahira 2-1-1, Sendai, Japan\n(Dated: October 20, 2018)\nAbstract\nWe present a consistent microscopic study of spin pumping e\u000bects for both metallic and insulating\nferromagnets. As for metallic case, we present a simple quantum mechanical picture of the e\u000bect as\ndue to the electron spin \rip as a result of a nonadiabatic (o\u000b-diagonal) spin gauge \feld. The e\u000bect of\ninterface spin-orbit interaction is brie\ry discussed. We also carry out \feld-theoretic calculation to\ndiscuss on the equal footing the spin current generation and torque e\u000bects such as enhanced Gilbert\ndamping constant and shift of precession frequency both in metallic and insulating cases. For thick\nferromagnetic metal, our study reproduces results of previous theories such as the correspondence\nbetween the dc component of the spin current and enhancement of the damping. For thin metal\nand insulator, the relation turns out to be modi\fed. For the insulating case, driven locally by\ninterfacesdexchange interaction due to magnetic proximity e\u000bect, physical mechanism is distinct\nfrom the metallic case. Further study of proximity e\u000bect and interface spin-orbit interaction would\nbe crucial to interpret experimental results in particular for insulators.\n1arXiv:1706.01185v1  [cond-mat.mes-hall]  5 Jun 2017I. INTRODUCTION\nSpin current generation is of a fundamental importance in spintronics. A dynamic method\nusing magnetization precession induced by an applied magnetic \feld, called the spin pumping\ne\u000bect, turns out to be particularly useful1and is widely used in a junction of a ferromagnet\n(F) and a normal metal (N)(Fig. 1). The generated spin current density (in unit of A/m2)\nhas two independent components, proportional to _nandn\u0002_n, wherenis a unit vector\ndescribing the direction of localized spin, and thus is represented phenomenologically as\njs=e\n4\u0019(Arn\u0002_n+Ai_n); (1)\nwhereeis the elementally electric charge and ArandAiare phenomenological constants\nhaving unit of 1 =m2. Spin pumping e\u000bect was theoretically formulated by Tserkovnyak et al.2\nby use of scattering matrix approach3. This approach, widely applied in mesoscopic physics,\ndescribes transport phenomena in terms of transmission and re\rection amplitudes (scattering\nmatrix), and provides quantum mechanical pictures of the phenomena without calculating\nexplicitly the amplitudes. Tserkovnyak et al. applied the scattering matrix formulation\nof general adiabatic pumping4,5to the spin-polarized case. The spin pumping e\u000bect was\ndescribed in Ref.2in terms of spin-dependent transmission and re\rection coe\u000ecients at the\nFN interface, and it was demonstrated that the two parameters, ArandAi, are the real\nand the imaginary part of a complex parameter called the spin mixing conductance. The\nspin mixing conductance, which is represented by transmission and re\rection coe\u000ecients,\nturned out to be a convenient parameter for discussing spin current generation and other\ne\u000bects like the inverse spin-Hall e\u000bect. Nevertheless, scattering approach hides microscopic\nphysical pictures of what is going on, as the scattering coe\u000ecients are not fundamental\nmaterial parameters but are composite quantities of Fermi wave vector, electron e\u000bective\nmass and the interface properties.\nE\u000bects of slowly-varying potential is described in a physically straightforward and clear\nmanner by use of a unitary transformation that represents the time-dependence. (See Sec.\nII A for details.) The laboratory frame wave function under time-dependent potential, j (t)i,\nis written in terms of a static ground state ('rotated frame' wave function) j\u001eiand a unitary\nmatrixU(t) asj (t)i=U(t)j\u001ei. The time-derivative @tis then replaced by a covariant\nderivative,@t+ (U\u00001@tU), and the e\u000bects of time-dependence are represented by (the time-\ncomponent of) an e\u000bective gauge \feld, A \u0011 \u0000i(U\u00001@tU) (See Eq. (12)). In the same\n2FIG. 1. Spin pumping e\u000bect in a junction of ferromagnet (F) and normal metal (N). Dynamic\nmagnetization n(t) generates a spin current jsthrough the interface.\nmanner as the electromagnetic gauge \feld, the e\u000bective gauge \feld generates a current if\nspatial homogeneity is present (like in junctions) and this is a physical origin of adiabatic\npumping e\u000bect in metals.\nIn the perturbative regime or in insulators, a simple picture instead of e\u000bective gauge \feld\ncan be presented. Let us focus on the case driven by an sdexchange interaction, Jsdn(t)\u0001\u001b,\nwhereJsdis a coupling constant and \u001bis the electron spin. Considering the second-order\ne\u000bect of the sdexchange interaction, the electron wave function has a contribution of a\ntime-dependent amplitude\nU(t1;t2) = (Jsd)2(n(t1)\u0001\u001b)(n(t2)\u0001\u001b) = (Jsd)2[(n(t1)\u0001n(t2)) +i[n(t1)\u0002n(t2)]\u0001\u001b];(2)\nwheret1andt2are the time of the interactions. The \frst term on the right-hand side,\nrepresenting the amplitude for charge degrees of freedom, is neglected. The spin contribution\nvanishes for static spin con\fguration, as is natural, while for slowly varying case, it reads\nU(t1;t2)'\u0000i(t1\u0000t2)(Jsd)2(n\u0002_n)(t1)\u0001\u001b: (3)\nAs a result of this amplitude, spin accumulation and spin current is induced proportional to\nn\u0002_n. The fact indicates that n\u0002_nplays a role of an e\u000bective scalar potential or voltage in\nelectromagnetism, as we shall demonstrate in Sec. VII B for insulators. (The factor of time\ndi\u000berence is written in terms of derivative with respect to energy or angular frequency in a\nrigorous derivation. See for example, Eqs. (129)(132).) The essence of spin pumping e\u000bect\nis therefore the non-commutativity of spin operators. The above picture in the perturbative\nregime naturally leads to an e\u000bective gauge \feld in the strong coupling limit6.\n3The same scenario applies for cases of spatial variation of spin, and an equilibrium spin\ncurrent proportional to n\u0002rinemerges, where idenotes the direction of spatial variation7.\nThe spin pumping e\u000bect is therefore the time analog of the equilibrium spin current induced\nby vector spin chirality. Moreover, charge current emerges from the third-order process from\nthe identity6\ntr[(n1\u0001\u001b)(n2\u0001\u001b)(n3\u0001\u001b)] = 2in1\u0001(n2\u0002n3); (4)\nand this factor, a scalar spin chirality, is the analog of the spin Berry phase in the pertur-\nbative regime. The spin pumping e\u000bect and spin Berry's phase and spin motive force have\nthe same physical root, namely the non-commutative spin algebra.\nFrom the scattering matrix theory view point the cases of metallic and insulating fer-\nromagnet make no di\u000berence as what conduction electrons in the normal metal see is the\ninterface. From physical viewpoints, such treatment appears too crude. Unlike the metallic\ncase discussed above, in the case of insulator ferromagnet, the coupling between the mag-\nnetization and the conduction electron in normal metal occurs due to a magnetic proximity\ne\u000bect at the interface. Thus the spin pumping by an insulator ferromagnet seems to be a\nlocally-induced perturbative e\u000bect rather than a transport induced by a driving force due to\na generalized gauge \feld. We therefore need to apply di\u000berent approaches for the two cases\nas brie\ry argued above. In the insulating case, one may think that magnon spin current is\ngenerated inside the ferromagnet because magnon itself couples to an e\u000bective gauge \feld8\nsimilarly to the electrons in metallic case. This is not, however, true, because the gauge \feld\nfor magnon is abelian (U(1)). Although scattering matrix approach apparently seems to\napply to both metallic and insulating cases, it would be instructive to present in this paper\na consistent microscopic description of the e\u000bects to see di\u000berent physics governing the two\ncases.\nA. Brief overview of theories and scope of the paper\nBefore carrying out calculation, let us overview history of theoretical studies of spin\npumping e\u000bect. Spin current generation in a metallic junction was originally discussed by\nSilsbee9before Tserkovnyak et al. It was shown there that dynamic magnetization induces\nspin accumulation at the interface, resulting in a di\u000busive \row of spin in the normal metal.\n4Although of experimental curiosity at that time was the interface spin accumulation, which\nenhances the signal of conduction electron spin resonance, it would be fair to say that Silsbee\npointed out the `spin pumping e\u000bect'.\nIn Ref.2, spin pumping e\u000bect was originally argued in the context of enhancement of\nGilbert damping in FN junction, which had been a hot issue after the study by Berger10,\nwho studied the case of FNF junction based on a quantum mechanical argument. Berger\ndiscussed that when a normal metal is attached to a ferromagnet, the damping of ferromagnet\nis enhanced as a result of spin polarization formed in the normal metal, and the e\u000bect was\nexperimentally con\frmed by Mizukami11. Tserkovnyak et al. pointed out that the e\u000bect has\na di\u000berent interpretation of the counter action of spin current generation, because the spin\ncurrent injected into the normal metal indicates a change of spin angular momentum or a\ntorque on ferromagnet. In fact, the equation of motion for the magnetization of ferromagnet\nreads\n_n=\u0000\rB\u0002n\u0000\u000bn\u0002_n\u0000a3\neSdjs; (5)\nwhere\ris the gyromagnetic ratio, \u000bis the Gilbert damping coe\u000ecient, dis the thickness\nof the ferromagnet, Sis the magnetude of localized spin, and ais the lattice constant. Spin\ncurrent of Eq. (1) thus indicates that the gyromagnetic ratio and the the Gilbert damping\ncoe\u000ecient are modi\fed by the spin pumping e\u000bect to be2\n~\u000b=\u000b+a3\n4\u0019SdAr\n~\r=\r\u0014\n1 +a3\n4\u0019SdAi\u0015\u00001\n: (6)\nThe spin pumping e\u000bect is therefore detected by measuring the e\u000bective damping constant\nand gyromagnetic ratio. The formula (6) is, however, based on a naive picture neglecting\nthe position-dependence of the damping torque and the relation between the pumped spin\ncurrent amplitude and damping or \rwould not be so simple in reality. (See Sec. V.)\nThe issue of damping in FN junction was formulated based on linear-response theory by\nSimanek and Heinirch12,13. They showed that the damping coe\u000ecient is given by the \frst-\norder derivative with respect to the angular frequency !of the imaginary part of the spin\ncorrelation function and argued that the damping e\u000bect is consistent with the Tserkovnyak's\nspin pumping e\u000bect. Recently, a microscopic formulation of spin pumping e\u000bect in metallic\njunction was provided by Chen and Zhang14and one of the author15by use of the Green's\n5functions, and a transparent microscopic picture of pumping e\u000bect was provided. Scattering\nrepresentation and Green's function one are related14because the asymptotic behaviors of\nthe Green's functions at long distance are governed by transmission coe\u000ecient16. In the\nstudy of Ref.15, the uniform ferromagnet was treated as a dot having only two degrees\nof freedom of spin. Such simpli\fcation neglects the dependence on electron wave vectors\nin ferromagnets and thus cannot discuss the the case of inhomogeneous magnetization or\nposition-dependence of spin damping.\nThe aim of this paper is to provide a microscopic and consistent theoretical formula-\ntion of spin pumping e\u000bect for metallic and insulating ferromagnets. We do not rely on\nthe scattering approach. Instead we provide elementary quantum mechanical argument to\ndemonstrated that spin current generation is a natural consequence of magnetization dy-\nnamics (Sec. II). Based on the formulation, the e\u000bect of interface spin-orbit interaction is\ndiscussed in Sec. III. We also provide a rigorous formulation based on \feld-theoretic ap-\nproach emploied in Ref.15in Sec. IV. We also reproduce within the same framework Berger's\nresult10that the spin pumping e\u000bect is equivalent to the enhancement of the spin damping\n(Sec. V). E\u000bect of inhomogeneous magnetization is brie\ry discussed in Sec. VI.\nCase of insulating ferromagnet is studied in Sec. VII assuming that the pumping is\ninduced by an interface exchange interaction between the magnetization and conduction\nelectron in normal metal, namely, by magnetic proximity e\u000bect. The interaction is treated\nperturbatively similarly to Refs.17,18. The dominant contribution to the spin current, the\none linear in the interface exchange interaction, turns out to be proportional to _n, while the\none proportional to n\u0002_nis weaker if the proximity e\u000bect is weak.\nThe contribution from the magnon, magnetization \ructuation, is also studied. As has\nbeen argued8, a gauge \feld for magnon emerges from magnetization dynamics. It is, however,\nan adiabatic one diagonal in spin, which acts as chemical potential for magnon giving rise\nonly to adiabatic spin polarization proportional to n. This is in sharp contrast to the\nmetallic case, where electrons are directly driven by spin-\rip component of spin gauge \feld,\nresulting in perpendicular spin accumulation, i.e., along _nandn\u0002_n. The excitation\nin ferromagnet when magnetization is time-dependent is therefore di\u000berent for metallic and\ninsulating cases. We show that magnon excitation nevertheless generates perpendicular spin\ncurrent,n\u0002_n, in the normal metal as a result of annihilation and creation at the interface,\nwhich in turn \rips electron spin. The result of magnon-driven contribution agrees with the\n6one in previous study19carried out in the context of thermally-driven spin pumping ('spin\nSeebeck' e\u000bect). It is demonstrated that the magnon-induced spin current depends linearly\non the temperature at high temperature compared to magnon energy. The amplitude of\nmagnon-driven spin current provides the magnitude of magnetic proximity e\u000bect.\nIn our analysis, we calculate consistently the pumped spin current and change of the\nGilbert damping and resonant frequency and obtain the relations among them. It is shown\nthat the spin mixing conductance scenario saying that the magnitude of spin current pro-\nportional ton\u0002_nis given by the enhancement factor of the Gilbert damping constant2,\napplies only the case of thick ferromagnetic metal. For thin metallic case and insulator case,\ndi\u000berent relations hold (See Sec. VIII.).\nII. QUANTUM MECHANICAL DESCRIPTION OF METALLIC CASE\nIn this section, we derive the spin current generated by the magnetization dynamics\nof metallic ferromagnet by a quantum mechanical argument. It is sometimes useful for\nintuitive understanding, although the description may lack clearness as it cannot handle\nmany-particle nature like particle distributions. In Sec. IV we formulate the problem in the\n\feld-theoretic language.\nA. Electrons in ferromagnet with dynamic magnetization\nThe model we consider is a junction of metallic ferromagnet (F) and a normal metal (N).\nThe magnetization (or localized spins) in the ferromagnet is treated as spatially uniform but\nchanging with time slowly. As a result of strong sdexchange interaction, the conduction\nelectron's spin follows instantaneous directions of localized spins, i.e., the system is in the\nadiabatic limit. The quantum mechanical Hamiltonian for the ferromagnet is\nHF=\u0000r2\n2m\u0000\u000fF\u0000Mn(t)\u0001\u001b; (7)\nwheremis the electron's mass, \u001bis a vector of Pauli matrices, Mrepresents the energy split-\nting due to the sdexchange interaction and n(t) is a time-dependent unit vector denoting\nthe localized spin direction. The energy is measured from the Fermi energy \u000fF.\n7As a result of the sdexchange interaction, the electron's spin wave function is given by20\njni\u0011cos\u0012\n2j\"i+ sin\u0012\n2ei\u001ej#i (8)\nwherej\"iandj#irepresent the spin up and down states, respectively, and ( \u0012;\u001e) are polar\ncoordinates for n. To treat slowly varying localized spin, we switch to a rotating frame where\nthe spin direction is de\fned with respect to instantaneous direction n7. This corresponds\nto diagonalizing the Hamiltonian at each time by introducing a unitary matrix U(t) as\njn(t)i\u0011U(t)j\"i; (9)\nwhere\nU(r) =0\n@cos\u0012\n2sin\u0012\n2e\u0000i\u001e\nsin\u0012\n2ei\u001e\u0000cos\u0012\n21\nA; (10)\nwhere states are in vector representation, i.e., j\"i=0\n@1\n01\nAandj#i=0\n@0\n11\nA. The rotated\nHamiltonian is diagonalized as (in the momentum representation)\neHF\u0011U\u00001HFU=\u000fk\u0000M\u001bz; (11)\nwhere\u000fk\u0011k2\n2m\u0000\u000fFis the kinetic energy in the momentum representation (Fig. 2). In general,\nFIG. 2. Unitary transformation Ufor conduction electron in ferromagnet converts the original\nHamiltonian HFinto a diagonalized uniformly spin-polarized Hamiltonian eHFand an interaction\nwith spin gauge \feld, As;t\u0001\u001b.\nwhen a statej ifor a time-dependent Hamiltonian H(t), satisfying the Schr odinger equation\ni@\n@tj i=H(t)j i, is written in terms of a state j iconnected by a unitary transformation\nj\u001ei\u0011U\u00001j i, the new state satis\fes a modi\fed Schr odinger equation\n\u0012\ni@\n@t+iU\u00001@\n@tU\u0013\nj\u001ei=~Hj\u001ei; (12)\n8where ~H\u0011U\u00001HU. Namely, there arises a gauge \feld \u0000iU\u00001@\n@tUin the new frame j\u001ei. In\nthe present case of dynamic localized spin, the gauge \feld has three components (su\u000ex t\ndenotes the time-component);\nAs;t\u0011\u0000iU\u00001@\n@tU\u0011As;t\u0001\u001b; (13)\nexplicitly given as7\nAs;t=1\n20\nBBB@\u0000@t\u0012sin\u001e\u0000sin\u0012cos\u001e@t\u001e\n@t\u0012cos\u001e\u0000sin\u0012sin\u001e@t\u001e\n(1\u0000cos\u0012)@t\u001e1\nCCCA: (14)\nIncluding the gauge \feld in the Hamiltonian, the e\u000bective Hamiltonian in the rotated frame\nreads\neHe\u000b\nF\u0011eHF+As;t\u0001\u001b=0\n@\u000fk\u0000M\u0000Az\ns;tA\u0000\ns;t\nA+\ns;t\u000fk+M+Az\ns;t1\nA (15)\nwhereA\u0006\ns;t\u0011Ax\ns;t\u0006iAy\ns;t. We see that the adiabatic ( z) component of the gauge \feld, Az\ns;t,\nacts as a spin-dependent chemical potential (spin chemical potential) generated by dynamic\nmagnetization, while non-adiabatic ( xandy) components causes spin mixing. In the case\nof uniform magnetization we consider, the mixing is between the electrons with di\u000berent\nspin\"and#but having the same wave vector k, because the gauge \feld A\u0006\ns;tcarries no\nmomentum. This leads to a mixing of states having an excitation energy of Mas shown in\nFig. 3. In low energy transport e\u000bects, what concern are the electrons at the Fermi energy;\nThe wave vector kshould be chosen as kF+andkF\u0000, the Fermi wave vectors for \"and#\nelectrons, respectively. (E\u000bects of \fnite momentum transfer is discussed in Sec. VI. )\nThe Hamiltonian Eq. (15) is diagonalized to obtain energy eigenvalues of ~ \u000fk\u001b=\u000fk\u0000\n\u001bq\n(M+Az\ns;t)2+jA?\ns;tj2, wherejA?\ns;tj2\u0011A+\ns;tA\u0000\ns;tand\u001b=\u0006represents spin (\"and#cor-\nrespond to + and \u0000, respectively). We are interested in the adiabatic limit, and so the\ncontribution lowest-order, namely, the \frst order, in the perpendicular component, A?\ns;t, is\nsu\u000ecient. In the present rotating-frame approach, the gauge \feld is treated as a static po-\ntential, since it already include time-derivative to the linear order (Eq. (14)). Moreover, the\nadiabatic component of the gauge \feld, Az\ns;t, is neglected, as it modi\fes the spin pumping\nonly at the second-order of time-derivative. The energy eigenvalues, \u000fk\u001b'\u000fk\u0000\u001bM, are\n9thus una\u000bected by the gauge \feld, while the eigenstates to the linear order read\njk\"iF\u0011jk\"i\u0000A+\ns;t\nMjk#i\njk#iF\u0011jk#i+A\u0000\ns;t\nMjk\"i; (16)\ncorresponding to energy of \u000fk+and\u000fk\u0000, respectively. For low energy transport, states we\nneed to consider are the following two having spin-dependent Fermi wave vectors, kF\u001bfor\n\u001b=\";#, namely\njkF\"\"iF=jkF\"\"i\u0000A+\ns;t\nMjkF\"#i\njkF##iF=jkF##i+A\u0000\ns;t\nMjkF#\"i: (17)\nFIG. 3. For uniform magnetization, the non-adiabatic components of the gauge \feld, A\u0006\ns;t, induces\na spin \rip conserving the momentum.\nB. Spin current induced in the normal metal\nSpin pumping e\u000bect is now studied by taking account of the interface hopping e\u000bects on\nstates in Eq. (17). The interface hopping amplitude of electron in F to N with spin \u001bis\ndenoted by ~t\u001band the amplitude from N to F is ~t\u0003\n\u001b. We assume that the spin-dependence\nof electron state in F is governed by the relative angle to the magnetization vector, and\nhence the spin \u001bis the one in the rotated frame. Assuming moreover that there is no spin\n\rip scattering at the interface, the amplitude ~t\u001bis diagonal in spin. (Interface spin-orbit\ninteraction is considered in Sec. III.) The spin wave function formed in the N region at the\n10interface as a result of the state in F (Eq. (17)) is then\njkF\"iN\u0011~tjkF\"i=~t\"jkF\"i\u0000 ~t#A+\ns;t\nMjkF#i\njkF#iN\u0011~tjkF#i=~t#jkF#i+~t\"A\u0000\ns;t\nMjkF\"i; (18)\nwherekFis the Fermi wave vector of N electron. The spin density induced in N region at\nthe interface is therefore\nes(N)=1\n2(NhkF\"j\u001bjkF\"iN\u0017\"+NhkF#j\u001bjkF#iN\u0017#) (19)\nwhere\u0017\u001bis the spin-dependent density of states of F electron at the Fermi energy. It reads\nes(N)=1\n2X\n\u001b\u0017\u001bT\u001b\u001b^z\u0000\u0017\"\u0000\u0017#\nM\u0000\nRe[T\"#]A?\ns;t+ Im[T\"#](^z\u0002A?\ns;t)\u0001\n(20)\nwhereA?\ns;t= (Ax\ns;t;Ay\ns;t;0) =As;t\u0000^zAz\ns;tis the transverse (non-adiabatic) components of\nspin gauge \feld and\nT\u001b\u001b0\u0011~t\u0003\n\u001b~t\u001b0: (21)\nSpin density of Eq. (20) is in the rotated frame. The spin polarization in the laboratory\nframe is obtained by a rotation matrix Rij, de\fned by\nU\u00001\u001biU\u0011Rij\u001bj; (22)\nas\ns(N)\ni=Rijes(N)\nj: (23)\nExplicitly,Rij= 2mimj\u0000\u000eij, wherem\u0011\u0000\nsin\u0012\n2cos\u001e;sin\u0012\n2sin\u001e;cos\u0012\n2\u00017. Using\nRij(A?\ns;t)j=\u00001\n2(n\u0002_n)i\nRij(^z\u0002A?\ns;t)j=\u00001\n2_ni; (24)\nandRiz=ni, the induced interface spin density is \fnally obtained as\ns(N)=\u0010s\n0n+ Re[\u0010s](n\u0002_n) + Im[\u0010s]_n (25)\n11where\n\u0010s\n0\u00111\n2X\n\u001b\u0017\u001bT\u001b\u001b\n\u0010s\u0011\u0017\"\u0000\u0017#\n2MT\"#: (26)\nSince the N electrons contributing to induced spin density is those at the Fermi energy,\nthe spin current is simply proportional to the induced spin density as jsN=kF\nms(N), resulting\nin\nj(N)\ns=kF\nm\u0010s\n0n+kF\nmRe[\u0010s](n\u0002_n) +kF\nmIm[\u0010s]_n: (27)\nThis is the result of spin current at the interface. The pumping e\u000eciency is determined\nby the product of hopping amplitudes t\"andt\u0003\n#. The spin mixing conductance de\fned in\nRef.2corresponds to iT\"#. If spin mixing e\u000bects due to spin-orbit interaction is neglected at\nthe interface, the hopping amplitudes t\u001bare chosen as real, and Im[ \u0010s] = 0. If spin current\nproportional to _nis measured, it would be useful tool to estimate the strength of interface\nspin-orbit interaction, as discussed in Sec. III.\nIt should be noted that the spin pumping e\u000bect at the linear order in time-derivative is\nmapped to a static problem of spin polarization formed by a static spin-mixing potential in\nthe rotated frame as was mentioned in Ref.15. The rotate frame approach employed here\nprovides clear physical picture, as it grasps the low energy dynamics in a mathematically\nproper manner. In this approach, as we have seen, it is clearly seen that pumping of spin\ncurrent arises as a result of o\u000b-diagonal components of the spin gauge \feld that cause\nelectron spin \rip. Important role of nonadiabaticity is also indicated in a recent analysis\nbased on the full counting statistics21. In the strict sense, spin pumping e\u000bect is a result of a\nnon-adiabatic process including state change. The same goes for general adiabatic pumping;\nSome sort of state change is necessary for current generation, although the nonadiabaticity\nis obscured in the conventional \\adiabatic\\ argument focusing on the wave function in\nthe laboratory frame. In the case of slowly-varying external potential with frequency \nacting on electrons, the state change is represented by the Fermi distribution di\u000berence,\nf(!+ \n)\u0000f(!)'\nf0(!), where!is the electron frequency3,4. The existence of a factor of\nf0clearly indicates that a state change or nonadiabaticity is necessary for current pumping.\n12III. EFFECTS OF INTERFACE SPIN-ORBIT INTERACTION\nIn this section, we discuss the e\u000bect of spin-orbit interaction at the interface, which\nmodi\fes hopping amplitude ~t\u001b. We particularly focus on that linear in the wave vector,\nnamely the interaction represented in the continuum representation by a Hamiltonian\nHso=a2\u000e(x)X\nij\rijki\u001bj; (28)\nwhere\rijis a coe\u000ecient having the unit of energy representing the spin-orbit interaction,\nais the lattice constant, and the interface is chosen as at x= 0. Assuming that spin-\norbit interaction is weaker than the sdexchange interaction in F, we carry out a unitary\ntransformation to which diagonalize the sdinteraction to obtain\nHso=a2\u000e(x)X\nije\rijki\u001bj; (29)\nwheree\rij\u0011P\nl\rilRlj, withRijbeing a rotation matrix de\fned by Eq. (22). This spin-\norbit interaction modi\fes diagonal hopping amplitude ~tiin the direction iat the interface\nto become a complex as\neti=~t0\ni\u0000iX\nje\rij\u001bj: (30)\n(In this section, we denote the total hopping amplitude including the interface spin-orbit\ninteraction by etand the one without by et0.) We consider the hopping amplitude perpendic-\nular to the interface, i.e., along the xdirection, and suppress the su\u000ex irepresenting the\ndirection. In the matrix representation for spin the hopping amplitude is\net(\u0011etx) =0\n@et\"et\"#\net#\"et#1\nA; (31)\nwhere\net\"=~t0\n\"\u0000ie\rxzet#=~t0\n#+ie\rxz\net\"#=i(e\rxx+ie\rxy) et#\"=i(e\rxx\u0000ie\rxy): (32)\nLet us discuss how the spin pumping e\u000bect discussed in Sec. II B is modi\fed when the\nhopping amplitude is a matrix of Eq. (31). The spin pumping e\u000eciency is written as in Eqs.\n13(21)(26). In the absence of spin-orbit interaction hopping amplitude ~tis chosen as real, and\nthus the contribution proportional to n\u0002_nin Eq. (27) is dominant. Spin-orbit interaction\nenhances the other contribution proportional to _nbecause it gives rise to an imaginary part.\nMoreover, it leads to spin mixing at the interface, modifying the spin accumulation formed\nin the N region at the interface.\nThe electron states in the N region at the interface are now given instead of Eq. (18) by\nthe following two states (choosing basis as0\n@jkF\"i\njkF#i1\nA)\njkF\"iN\u0011etjkF\"\"iF=0\n@et\"\u0000et\"#A+\ns;t\nM\net#\"\u0000et#A+\ns;t\nM1\nA\njkF#iN\u0011etjkF##iF=0\n@et\"#+et\"A\u0000\ns;t\nM\net#+et#\"A\u0000\ns;t\nM1\nA: (33)\nThe pumped (i.e., linear in the gauge \feld) spin density for these two states are\nNhkF\"j\u001bjkF\"iN=\u00002\nM(A?\ns;tRe[Ttot\n\"#] + (^z\u0002A?\ns;t)Im[Ttot\n\"#]\n+Re[(et\"#)\u0003et#\"]0\nBBB@Ax\ns;t\n\u0000Ay\ns;t\n01\nCCCA+ Im[(et\"#)\u0003et#\"]0\nBBB@Ay\ns;t\nAx\ns;t\n01\nCCCA1\nCCCA\n\u0000^z(Ax\ns;tRe[(et\")\u0003et\"#\u0000et#(et#\")\u0003]\u0000Ay\ns;tIm[(et\")\u0003et\"#\u0000et#(et#\")\u0003]) (34)\nNhkF#j\u001bjkF#iN=2\nM(A?\ns;tRe[Ttot\n\"#] + (^z\u0002A?\ns;t)Im[Ttot\n\"#]\n+Re[(et\"#)\u0003et#\"]0\nBBB@Ax\ns;t\n\u0000Ay\ns;t\n01\nCCCA+ Im[(et\"#)\u0003et#\"]0\nBBB@Ay\ns;t\nAx\ns;t\n01\nCCCA1\nCCCA\n+^z(Ax\ns;tRe[(et\")\u0003et\"#\u0000et#(et#\")\u0003]\u0000Ay\ns;tIm[(et\")\u0003et\"#\u0000et#(et#\")\u0003]) (35)\nWe here focus on the linear e\u000bect of interface spin-orbit interaction and neglect the\nspin polarization along the magnetization direction, n. The expression for the pumped\nspin current then agrees with Eq. (27) with the amplitude \u0010swritten in terms of hopping\nincluding the interface spin-orbit,\nT\"#= ((~t0\n\")\u0003+i(e\rxz)\u0003)(~t0\n#+ie\rxz): (36)\n14If bulk spin-orbit interaction is neglected, bare hopping amplitude ~t0\n\u001bis real and we may\nreasonably assume that e\rijis real. The interface spin-orbit then leads to an imaginary part\nas (usinge\rxz=ni\rxi)\nIm[\u0010s] =\u0017\"\u0000\u0017#\n2M(~t0\n\"+~t0\n#)\rxini: (37)\nThe amplitude of spin current proportional to _nthus works as a probe for interface spin-orbit\ninteraction strength, \rxi.\nLet us discuss some examples. Of recent particular interest is the interface Rashba\ninteraction, represented by antisymmetric coe\u000ecient\n\r(R)\nij=\u000fijk\u000bR\nk; (38)\nwhere\u000bRis a vector representing the Rashba \feld. In the case of interface, \u000bRis perpen-\ndicular to the interface, i.e., \u000bRk^x. Therefore the interface Rashba interaction leads to\n\r(R)\nxj= 0 and does not modify spin pumping e\u000bect at the linear order. (It contributes at\nthe second order as discussed in Ref.14.) In other words, vector coupling between the wave\nvector and spin in the form of k\u0002\u001bexists only along the x-direction, and does not a\u000bect\nthe interface hopping (i.e., does not include kx).\nIn contrast, a scalar coupling \u0011(D)(k\u0001\u001b) (\u0011(D)is a coe\u000ecient), called the Dirac type\nspin-orbit interaction, leads to \r(D)\nij=\u0011(D)\u000eij. The spin current along _nthen reads\nj_n\ns=\u0011(D)kF(\u0017\"\u0000\u0017#)\n2mM(~t0\n\"+~t0\n#)nx_n: (39)\nFor the case of in-plane easy axis along the zdirection and magnetization precession given\nbyn(t) = (sin\u0012cos!t;sin\u0012sin!t;cos\u0012), where\u0012is the precession angle and !is the angular\nfrequency, we expect to have a dc spin current along the ydirection, as nx_n=\u0000!\n2sin2\u0012^y\n(nx_ndenotes time average).\nIV. FIELD THEORETIC DESCRIPTION OF METALLIC CASE\nHere we present a \feld-theoretic description of spin pumping e\u000bect of metallic ferromag-\nnet. The many-body approach has an advantage of taking account of particle distributions\nautomatically. Moreover, it describes propagation of particle density in terms of the Green's\n15functions, and thus is suitable for studying spatial propagation as well as for intuitive under-\nstanding of transport phenomena. All the transport coe\u000ecients are determined by material\nconstants.\nThe formalism presented here is essentially the same as in Ref.15, but treating the fer-\nromagnet of a \fnite size and taking account of electron states with di\u000berent wave vectors.\nInterface spin-orbit interaction is not considered here.\nConduction electron in ferromagnetic and normal metals are denoted by \feld operators\nd,dyandc,cy, respectively. These operators are vectors with two spin components, i.e.,\nd\u0011(d\";d#). The Hamiltonian describing the F and N electrons is HF+HN, where\nHF\u0011Z\nFd3rdy\u0012\n\u0000r2\n2m\u0000\u000fF\u0000Mn(t)\u0001\u001b\u0013\nd\nHN\u0011Z\nNd3rcy\u0012\n\u0000r2\n2m\u0000\u000fF\u0013\nc: (40)\nWe set the Fermi energies for ferromagnet and normal metal equal. The hopping through\nthe interface is described by the Hamiltonian\nHI\u0011Z\nIFd3rZ\nINd3r0\u0000\ncy(r0)t(r0;r;t)d(r) +dy(r)t\u0003(r0;r;t)c(r0)\u0001\n; (41)\nwheret(r0;r;t) represents the hopping amplitude of electron from rin ferromagnetic regime\nto a siter0in the normal region and the integrals are over the interface (denoted by I F\nand I Nfor F and N regions, respectively). The hopping amplitude is generally a matrix\ndepending on magnetization direction n(t), and thus depends on time t. Hopping is treated\nas energy-conserving. Assuming sharp interface at x= 0, the momentum perpendicular to\nthe interface is not conserved on hopping.\nWe are interested in the spin current in the normal region, given by\nj\u000b\ns;i(r;t) =\u00001\n4m(r(r)\u0000r(r0))itr[\u001b\u000bG<\nN(r;t;r0;t)jr0=r; (42)\nwhereG<\nN(r;t;r0;t0)\u0011i\nc(r;t)cy(r0;t0)\u000b\ndenotes the lesser Green's function for the normal\nregion. It is calculated from the Dyson's equation for the path-ordered Green's function\nde\fned for a complex time along a complex contour C\nGN(r;t;r0;t0) =gN(r\u0000r0;t\u0000t0)\n+Z\ncdt1Z\ncdt2Z\nd3r1Z\nd3r2gN(r\u0000r1;t\u0000t1)\u0006N(r1;t1;r2;t2)GN(r2;t2;r0;t0);\n(43)\n16whereg<\nNdenotes the Green's function without interface hopping and \u0006 N(r1;t1;r2;t2) is the\nself-energy for N electron, given by the contour-ordered Green's function in the ferromagnet\nas\n\u0006N(r1;t1;r2;t2)\u0011Z\nIFd3r3Z\nIFd3r4t(r1;r3;t1)G(r3;t1;r4;t2)t\u0003(r2;r4;t2): (44)\nHerer1andr2are coordinates at the interface I Nin N region and r3andr4are those in\nIFfor F.Gis the contour-ordered Green's function for F electron in the laboratory frame\nincluding the e\u000bect of spin gauge \feld. We denote Green's functions of F electron by G\nandgwithout su\u000ex and those of N electron with su\u000ex N. The lesser component of the\nnormal metal Green's function is obtained from Eq. (43) as (suppressing the time and space\ncoordinates)\nG<\nN= (1 +Gr\nN\u0006r\nN)g<\nN(1 + \u0006a\nNGa\nN) +Gr\nN\u0006<\nNGa\nN: (45)\nFor pumping e\u000bects, the last term on the right-hand side is essential, as it contains the\ninformation of excitation in F region. We thus consider the second term only;\nG<\nN'Gr\nN\u0006<\nNGa\nN; (46)\nand neglect spin-dependence of the normal region Green's functions, Gr\nNandGa\nN. The\ncontribution is diagramatically shown in Fig. 4.\nA. Rotated frame\nTo solve for the Green's function in the ferromagnet, rotated frame we used in Sec. II A\nis convenient. In the \feld representation, the unitary transformation is represented as (Fig.\n5(c))\nd=U~d; c =U~c; (47)\nwhereUis the same 2\u00022 matrix de\fned in Eq. (10). We rotate N electrons as well as F\nelectrons, to simplify the following expressions. The hopping interaction Hamiltonian reads\nHI=Z\nIFd3rZ\nINd3r0\u0010\n~cy(r0)~t(r0;r)~d(r) +~dy(r)~t\u0003(r0;r)~c(r0)\u0011\n; (48)\n17FIG. 4. (a) Schematic diagramatic representations of the lessor Green's function for N electron\nconnecting the same position r,G<\nN(r;r)'Gr\nN\u0006<\nNGa\nNrepresenting propagation of electron density.\nIt is decomposed into a propagation of N electron from rto the interface at r2, then hopping to r4\nin the F side, a propagation inside F, followed by a hopping to N side (to r1) and propagation back\ntor. (Position labels are as in Eqs. (43)(44).) (b): The self energy \u0006<\nNrepresents all the e\u000bects\nof the ferromagnet. (c) Standard Feynman diagram representation of lessor Green's function for\nN atr, Eqs. (46) and (44).\nFIG. 5. Unitary transformation Uof F electron converts the original system with \feld operator\nd(shown as (a)) to the rotated one with \feld operator ~d\u0011U\u00001d(b). The hopping amplitude\nfor representation in (b) is modi\fed by U. If N electrons are also rotated as ~ c\u0011U\u00001c, hopping\nbecomes ~t\u0011U\u00001tU, while the N electron spin rotates with time, as shown as (c).\nwhere\n~t(r0;r)\u0011Uy(t)t(r0;r;t)U(t); (49)\nis the hopping amplitude in the rotated frame. The rotated amplitude (neglecting interface\nspin-orbit interaction) is diagonal in spin;\n~t=0\n@~t\"0\n0~t#1\nA: (50)\nIncluding the interaction with spin gauge \feld, the Hamiltonian for F and N electrons in\n18the momentum representation is\nHF+HN=X\nk~dy\nk0\n@\u000fk\u0000M\u0000Az\ns;tA\u0000\ns;t\nA+\ns;t\u000fk+M+Az\ns;t1\nA~dk+X\nk\u000f(N)\nk~cy\nk~ck (51)\nAs for the hopping, we consider the case the interface is atomically sharp. The hopping\nHamiltonian is then written in the momentum space as\nHI=X\nkk0\u0010\n~cy(k)~t(k;k0)~d(k0) +~dy(k0)~t\u0003(k;k0)~c(k)\u0011\n; (52)\nwherek= (kx;ky;kz),k0= (k0\nx;ky;kz), choosing the interface as the plane of x= 0. Namely,\nthe wave vectors parallel to the interface are conserved while kxandk0\nxare uncorrelated.\nB. Spin density induced by magnetization dynamics in F\nPumped spin current in N is calculated by evaluating \u0006<\nNand using Eqs. (42)(45)(46).\nBefore discussing the spin current, let us calculate spin density in ferromagnet induced by\nmagnetization dynamics neglecting the e\u000bect of interface, HI. (E\u000bects of HIare discussed\nin Sec. V.) The spin accumulation in N is discussed by extending the calculation here as\nshown in Sec. IV C.\nThe lessor Green's function in F in the rotated frame including the spin gauge \feld to\nthe linear order is calculated from the Dyson's equation\nG<=g<+gr(As;t\u0001\u001b)g<+g<(As;t\u0001\u001b)ga; (53)\nwhereg\u000b(\u000b=<;r,a) represents Green's functions without spin gauge \feld. The lessor\nGreen's function satis\fes for static case g<=F(ga\u0000gr), whereF\u00110\n@f\"0\n0f#1\nAis spin-\ndependent Fermi distribution function. We thus obtain the Green's function at the linear\norder as15\n\u000eG<=gr[As;t\u0001\u001b;F]ga+gaF(As;t\u0001\u001b)ga\u0000gr(As;t\u0001\u001b)Fgr: (54)\nThe last two terms of the right-hand side are rapidly oscillating as function of position and\nare neglected. The commutator is calculated as (sign \u0006denotes spin\"and#)\n[As;t\u0001\u001b;F] = (f+\u0000f\u0000)X\n\u0006(\u0006)A\u0006\ns;t\u001b\u0007: (55)\n19FIG. 6. Feynman diagram for electron spin density of ferromagnet induced by magnetization\ndynamics (represented by spin gauge \feld As) neglecting the e\u000bect of normal metal. The amplitude\nis essentially given by the spin \rip correlation function \u001f\u0006(Eq. (58)).\nIn the rotated frame, the spin density in F pumped by the spin gauge \feld is therefore\n(diagrams shown in Fig. 6)\n~s(F)\n\u000b(k;k0)\u0011\u0000iZd!\n2\u0019tr[\u001b\u000b\u000eG<(k;k0;!)]\n=\u0000iZd!\n2\u0019X\nk00(fk00+\u0000fk00\u0000)X\n\u0006(\u0006)A\u0006\ns;ttr[\u001b\u000bgr(k;k00;!)\u001b\u0007ga(k00;k0;!)]\n=8\n<\n:\u0007iRd!\n2\u0019P\nk00(fk00+\u0000fk00\u0000)A\u0006\ns;tgr\n\u0007(k;k00;!)ga\n\u0006(k00;k0;!) (\u000b=\u0006)\n0 ( \u000b=z): (56)\nLet us here neglect the e\u000bects of interface in dicussing spin polarization of F electrons; Then\nthe Green's functions are translationally invariant, i.e., ga(k;k0) =\u000ek;k0ga(k) (a= r;a).\nUsing the explicit form of the free Green's function, ga\n\u001b(k;!) =1\n!\u0000\u000fk;\u001b\u0000i0, and\nZd!\n2\u0019gr\n\u0007(k;k00;!)ga\n\u0006(k00;k0;!) =i\n\u000fk;\u0006\u0000\u000fk;\u0007+i0; (57)\nthe spin density in the rotated frame then reduces to\n~s(F)\n\u0006(k) =\u0000A\u0006\ns;t\u001f\u0006; (58)\nwhere\n\u001f\u0006\u0011\u0000X\nkfk;\u0006\u0000fk;\u0007\n\u000fk;\u0006\u0000\u000fk;\u0007+i0; (59)\nis the spin correlation function with spin \rip, + i0 meaning an in\fnitesimal positive imaginary\npart. Since we focus on adiabatic limit and spatially uniform magnetization, the correlation\nfunction is at zero momentum- and frequency-transfer. We thus easily see that\n\u001f\u0006=n+\u0000n\u0000\n2M; (60)\n20wheren\u0006=P\nkfk\u0006is spin-resolved electron density.\nThe spin polarization of Eq. (58) in the rotated frame is proportional to A?\ns;t, and\nrepresents a renormalization of total spin in F. In fact, it corresponds in the laboratory\nframe tos(F)/n\u0002_n, and exerts a torque proportional to _nonn.\nIt may appear from Eq. (60) that a damping of spin, i.e., a torque proportional to\nn\u0002_n, arises when the imaginary part for the Green's function becomes \fnite, because\n1\nMis replaced by1\nM\u0007i\u0011i, where\u0011iis the imaginary part. This is not always the case. For\nexample, nonmagnetic impurities introduce a \fnite imaginary part inversely proportional to\nthe elastic lifetime ( \u001c),i\n2\u001c. They should not, however, cause damping of spin. The solution\nto this apparent controversy is that Eq. (56) is not enough to discuss damping even including\nlifetime. In fact, there is an additional process called vertex correction contributing to the\nlesser Green's function, and it gives rise to the same order of small correction as the lifetime\ndoes, and the sum of the two contributions vanishes. Similarly, we expect damping does not\narise from spin-conserving component of spin gauge \feld, Az\ns;t. This is indeed true as we\nexplicitly demonstrate in Appendix A. We shall show in Sec. V that damping arises from\nthe spin-\rip components of the self energy.\nC. Spin polarization and current in N\nFIG. 7. Feynman diagram for electron spin density of normal metal driven by the spin gauge \feld\nof ferromagnetic metal, As. The spin current is represented by the same diagram but with spin\ncurrent vertex.\nThe spin polarization of N electron lesser Green's function including the self-energy to\n21the linear order is calculated from Eqs. (46) (54)(55) as (diagram shown in Fig. 7)\n\u0000itr[\u001b\u0006G<\nN(r;t;r0;t)] =\u0000iX\nkk0k00eik\u0001re\u0000ik0\u0001r0gr\nN(k;!)ga\nN(k0;!)\n\u0002X\n\u0006(fk00\u0006\u0000fk00\u0007)A\u0006\ns;t~t\u0007(k;k00)~t\u0003\n\u0006(k00;k0)gr\n\u0007(k00;!)ga\n\u0006(k00;!): (61)\nWe assume that dependence of N Green's functions on !is weak and useP\nkeik\u0001rgr\nN(k;!) =\n\u0000i\u0019\u0017NeikFxe\u0000jxj=`\u0011gr\nN(r), where`is elastic mean free path, \u0017NandkFare the density of\nstates at the Fermi energy and Fermi wave vector, respectively, whose !-dependences are ne-\nglected. (For in\fnitely wide interface, the Green's function becomes one-dimensional.) As a\nresult of summation over wave vectors, the product of hopping amplitudes ~t\u0007(k;k00)~t\u0003\n\u0006(k00;k0)\nis replaced by the average over the Fermi surface, ~t\u0007~t\u0003\n\u0006\u0011T\u0006\u0007, i.e.,\n~t\u0007(k;k00)~t\u0003\n\u0006(k00;k0)!T\u0006\u0007: (62)\nThe spin polarization of N electron induced by magnetization dynamics (the spin gauge\n\feld) is therefore obtained in the rotated frame as\n~s(N)\n\u0006(r;t) =\u0000jgr\nN(r)j2X\n\u0006A\u0006\ns;t\u001f\u0006T\u0006\u0007; (63)\nor using\u001f\u0003\n+=\u001f\u0000\n~s(N)(r;t) =\u00002jgr\nN(r)j2\u0002\nA?\ns;tRe[\u001f+T+\u0000] + (^z\u0002A?\ns;t)Im[\u001f+T+\u0000]\u0003\n: (64)\nIn the laboratory frame, we have (using s(N)\ni=Rij~s(N)\nj)\ns(N)(r;t) =jgr\nN(r)j2\u0002\nRe[\u001f+T+\u0000](n\u0002_n) + Im[\u001f+T+\u0000]_n\u0003\n: (65)\nThe spin current induced in N region is similarly given by (neglecting the contribution\nproportional to n)\njs(r;t) =kF\nmjgr\nN(r)j2\u0002\nRe[\u001f+T+\u0000](n\u0002_n) + Im[\u001f+T+\u0000]_n\u0003\n=e\u0000jxj=`(Re[\u0010s](n\u0002_n) + Im[\u0010s]_n); (66)\nwhere\n\u0010s\u0011\u00192kF\u00172\nN\n2mM(n+\u0000n\u0000)T+\u0000: (67)\n22The coe\u000ecient \u0010sis essentially the same as the one in Eq. (27) derived by quantum mechan-\nical argument, as quantum mechanical dimensionless hopping amplitude corresponds to \u0017N~t\nof \feld representation.\nFor 3d ferromagnet, we may estimate the spin current by approximating roughly M\u0018\n1=\u0017N\u0018\u000fF\u00181eV andn\u001b\u0018kF3. The hopping amplitude jT+\u0000jin metallic case would be\norder of\u000fF. The spin current density then is of the order of (including electric charge eand\nrecovering ~),js\u0018e~kF\nmh~!\n\u000fF\u00185\u00021011A/m2if precession frequency is 10 GHz.\nV. SPIN ACCUMULATION IN FERROMAGNET\nThe spin current pumping is equivalent to the increase of spin damping due to magne-\ntization precession, as was discussed in Refs.2,10. In this section, we con\frm this fact by\ncalculating the torque by evaluating the spin polarization of the conduction electron spin in\nF region.\nThere are several ways to evaluate damping of magnetization. One way is to calculate the\nspin-\rip probability of the electron as in Ref.10, which leads to damping of localized spin in\nthe presence of strong sdexchange interaction. The second is to estimate the torque on the\nelectron by use of equation motion22. The relation between the damping and spin current\ngeneration is clearly seen in this approach. In fact, the total torque acting on conduction\nelectron is ( ~times) the time-derivative of the electron spin density,\nds\ndt=i\u0000\n[H;dy]\u001bd\u000b\n+\ndy\u001b[H;d]\u000b\u0001\n: (68)\nAt the interface, the right-hand side arises from the interface hopping. Using the hopping\nHamiltonian of Eq. (41), we have\nds\ndt\f\f\f\f\ninterface=i\u0000\ncyt\u001bd\u000b\n\u0000\ndy\u001btyc\u000b\u0001\n; (69)\nas the interface contribution. As is natural, the the right -hand side agrees with the de\fnition\nof the spin current passing through the interface. Evaluating the right-hand side, we obtain in\ngeneral a term proportional to n\u0002_n, which gives the Gilbert damping, and term proportional\nto_n, which gives a renormalization of magnetization. In contrast, away from the interface,\nthe commutator [ H;d] arises from the kinetic term H0\u0011R\nd3rjrdj2\n2mdescribing electron\n23propagation, resulting in\nds\u000b\ndt=i\u0000\n[H0;dy]\u001bd\u000b\n+\ndy\u001b[H;d]\u000b\u0001\n=r\u0001j\u000b\ns (70)\nwherej\u000b\ns(r)\u0011\u0000i\n2m(rr\u0000r r0)\ndy(r0)\u001b\u000bd(r)\u000b\njr0=ris the spin current. Away from the inter-\nface, the damping therefore occurs if the spin current has a source or a sink at the site of\ninterest.\nHere we use the third approach and estimate the torque on the localized spin by calculat-\ning the spin polarization of electrons as was done in Refs.7,23. The electron spin polarization\nat positionrin the ferromagnet at time tiss(F)(r;t)\u0011\ndy\u001bd\u000b\n, which reads in the rotated\nframes(F)\n\u000b=R\u000b\f~s(F)\n\f, where\n~s(F)\n\f(r;t) =\u0000itr[\u001b\fG<(r;r;t;t)]; (71)\nwhereG<\n\u001b\u001b0(r;r0;t;t0)\u0011iD\n~dy\n\u001b0~d\u001bE\nis the lesser Green's function in F region, which is a\nmatrix in spin space ( \u001b;\u001b0=\u0006). We are interested in the e\u000bect of the N region arising from\nthe hopping. We must note that the hopping interaction of Eq. (48) is not convenient for\nintegrating out N electrons, since the ~ celectrons' spins are time-dependent as a result of a\nunitary transformation, U(t). We thus use the following form (Fig. 5(b)),\nHI=Z\nIFd3rZ\nINd3r0\u0010\ncy(r0)U~t(r0;r)~d(r) +~dy(r)~t\u0003(r0;r)Uyc(r0)\u0011\n; (72)\nnamely, the hopping amplitude between ~dandcelectrons includes unitary matrix U.\nLet us argue in the rotated frame why the e\u000bect of damping arising from the interface.\nIn the totally rotated frame of Fig. 5(c), the spin of F electron is static, while that of N\nelectron varies with time. When F electron hops to N region and comes back, therefore,\nelectron spin gets rotated with the amount depending on the time it stayed in N region. This\ne\u000bect is in fact represented by a retardation e\u000bect of the matrices UandU\u00001in Eq. (72). If\no\u000b-diagonal nature of UandU\u00001are neglected, the interface e\u000bects are all spin-conserving\nand do not induce damping for F electron (See Sec. A).\nWe now proceed calculation of induced spin density in the ferromagnetic metal. Diagra-\nmatic representation of the contribution is in Fig. 8. Writing spatial and temporal positions\nexplicitly, the self-energy of F electron arising from the hopping to N region reads ( r1and\n24FIG. 8. Diagramatic representation of the spin accumulation in ferromagnetic metal induced as\na result of coupling to the normal metal (Eqs. (71)(73)). Conduction electron Green's functions\nin ferromagnet and normal metal are denoted by gandgN, respectively. Time-dependent matrix\nU(t), de\fned by Eq. (10), represents the e\u000bect of dynamic magnetization. Expanding UandU\u00001\nwith respect to slow time-dependence of magnetization, we obtain gauge \feld representation, Eq.\n(75).\nr2are in F)\n\u0006a(r1;r2;t1;t2) =Z\nINd3r0\n1Z\nINd3r0\n2~t(r1;r0\n1)U\u00001(t1)ga\nN(r0\n1;r0\n2;t1\u0000t2)U(t2)~ty(r2;r0\n2) (73)\nwherea= r;a;<. We assume the Green's function in N region is spin-independent; i.e., we\nneglect higher order contribution of hopping. Moreover, we treat the hopping to occur only\nat the interface, i.e., at x= 0. The self-energy is then represented as\n\u0006a(r1;r2;t1;t2) =a2\u000e(x1)\u000e(x2)~tU\u00001(t1)U(t2)~tyX\nkga\nN(k;t1\u0000t2); (74)\nwhereais the interface thickness, which we assume to be the order of the lattice constant.\nDiagramatic representation of Eqs. (71)(73) are in Fig. 8. Expanding the matrix using\nspin gauge \feld as U\u00001(t1)U(t2) = 1\u0000i(t1\u0000t2)As;t+O((As;t)2), we obtain the gauge \feld\ncontribution of the self-energy as\n\u0006a(r1;r2;t1;t2) =a2\u000e(x1)\u000e(x2)Zd!\n2\u0019de\u0000i!(t1\u0000t2)\nd!~tAs;t~tyX\nkga\nN(k;!)\n=\u0000a2\u000e(x1)\u000e(x2)Zd!\n2\u0019e\u0000i!(t1\u0000t2)~tAs;t~tyX\nkd\nd!ga\nN(k;!) (75)\n25The linear contribution of the lessor component of the o\u000b-diagonal self-energy is\nG<(r;t;r0;t) =gr\u0006rga+gr\u0006<ga+g<\u0006aga\n=a2Zd!\n2\u0019X\nk\u0014\ngr(r;!)dgr\nN(k;!)\nd!~tAs;t~tyg<(\u0000r;!)\n+gr(r;!)dg<\nN(k;!)\nd!~tAs;t~tyga(\u0000r;!) +g<(r;!)dga\nN(k;!)\nd!~tAs;t~tyga(\u0000r;!)\u0015\n(76)\nFor \fnite distance from the interface, r, dominant contribution arises from the terms contain-\ning bothgr(r;!) andga(\u0000r;!), as they do not contain a rapid oscillation like ei(kF++kF\u0000)r\nande2ikF\u001br. Using an approximationP\nkgr\nN(k;!)\u0018\u0000i\u0019\u0017Nand partial integration with\nrespect to!, Eq. (76) reduces to\nG<(r;t;r0;t) = 2\u0019i\u0017Na2Zd!\n2\u0019f0\nN(!)gr(r;!)~tAs;t~tyga(\u0000r;!) (77)\nFor damping, o\u000b-diagonal contributions, A\u0006\ns;t, are obviously essential. The result of the spin\ndensity in F in the rotated frame, Eq. (71), is\n~s(F)\n\u000b(r;t) = 2\u0019i\u0017Na2Zd!\n2\u0019f0\nN(!)A\f\ns;ttr[\u001b\u000bgr(r;!)~t\u001b\f~tyga(\u0000r;!)]\n= 2\u0019i\u0017Na2Zd!\n2\u0019f0\nN(!)A\f\ns;tX\nkk0ei(k\u0000k0)\u0001rtr[\u001b\u000bgr(k;!)~t\u001b\f~tyga(k0;!)] (78)\nEvaluating the trace in spin space, we obtain\n~s(F)(r;t) =\u0000\u0017N\u0014\nA?\ns;t\r1(r) + ( ^z\u0002A?\ns;t)\r2(r)\u0015\n(79)\nwhere\n\r1(r)\u0011X\n\u001b~t\u0000\u001b~ty\n\u001bgr\n\u0000\u001b(r)ga\n\u001b(\u0000r)\n\r2(r)\u0011X\n\u001b(\u0000i\u001b)~t\u0000\u001b~ty\n\u001bgr\n\u0000\u001b(r)ga\n\u001b(\u0000r): (80)\nWe consider an interface with in\fnite area and consider spin accumulation averaged over\nthe plane parallel to the interface. The wave vector contributing is then those with \fnite kx\nbut withky=kz= 0 and Green's function become one-dimensional like\nX\nkeik\u0001rgr\n\u001b(k) =im\nkF\u001beikF\u001bjxje\u0000jxj=(2`\u001b); (81)\n26where`\u001b\u0011vF\u001b\u001c\u001b(vF\u001b\u0011kF\u001b=m) is electron mean free path for spin \u001b. The induced spin\ndensity in the ferromagnet is \fnally obtained from Eq. (79) as\ns(F)(r;t) =m2\u0017Na2\n2kF+kF\u0000X\n\u001b\u0014\n(n\u0002_n)T\u001b;\u0000\u001be\u0000i\u001b(kF+\u0000kF\u0000)x+_n(\u0000i\u001b)T\u001b;\u0000\u001be\u0000i\u001b(kF+\u0000kF\u0000)x\u0015\n=m2\u0017Na2\n2kF+kF\u0000X\n\u001b\u0014\n(n\u0002_n)\u0002\nRe[T\";#] cos((kF+\u0000kF\u0000)x) + Im[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n+_n\u0002\nIm[T\";#] cos((kF+\u0000kF\u0000)x)\u0000Re[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n(82)\nand the torque on localized spin, \u0000Mn\u0002s(F), is\n\u001c(r;t) =\u0000m2\u0017Na2M\n2kF+kF\u0000X\n\u001b\u0014\n\u0000_n\u0002\nRe[T\";#] cos((kF+\u0000kF\u0000)x) + Im[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n+ (n\u0002_n)\u0002\nIm[T\";#] cos((kF+\u0000kF\u0000)x)\u0000Re[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n: (83)\nA. Enhanced damping and spin renormalization of ferromagnetic metal\nThe total induced spin accumulation density in ferromagnet is\ns(F)\u00111\ndZ0\n\u0000ddxs(F)(x)\n=1\nM\u0010\n(n\u0002_n)h\n\u0000Im[\u000e](1\u0000cos~d) + Re[\u000e] sin~di\n+_nh\nRe[\u000e](1\u0000cos~d) + Im[\u000e] sin~di\u0011\n;\n(84)\nwhere ~d\u0011(kF+\u0000kF\u0000)d,dis the thickness of ferromagnet and\n\u000e\u0011m2\u0017Na2M\nkF+kF\u0000(kF+\u0000kF\u0000)dT\";#: (85)\nAs a result of this induced electron spin density, s(F), the equation of motion for the averaged\nmagnetization is modi\fed to be10\n_n=\u0000\u000bn\u0002_n\u0000\rB\u0002n\u0000Mn\u0002s(F); (86)\nwhereBis the external magnetic \feld.\nLet us \frst discuss thin ferromagnet case, d\u001djkF+\u0000kF\u0000j\u00001, where oscillating part with\nrespect to ~dis neglected. The spin density then reads s(F)'1\nM(\u0000Im[\u000e](n\u0002_n) + Re[\u000e]_n)\nand the equation of motion becomes\n(1 + Im\u000e)_n=\u0000~\u000bn\u0002_n\u0000\rB\u0002n; (87)\n27where\n~\u000b\u0011\u000b+ Re\u000e; (88)\nis the Gilbert damping including the enhancement due to the spin pumping e\u000bect. The\nprecession angular frequency !Bis modi\fed by the imaginary part of T\";#, i.e., by the spin\ncurrent proportional to _n, as\n!B=\rB\n1 + Im\u000e: (89)\nThis is equivalent to the modi\fcation of the gyromagnetic ratio ( \r) or theg-factor.\nFor most 3d ferromagnets, we may approximatem2\u0017NaM\u000fF2\n2kF+kF\u0000(kF+\u0000kF\u0000)'O(1) (askF+\u0000\nkF\u0000/M), resulting in \u000e\u0018a\ndT\";#. As discussed in Sec. III, when interface spin-orbit\ninteraction is taken into account, we have T\";#=~t0\n\"~t0\n#+ie\rxz(~t0\n\"+~t0\n#) +O((e\r)2), where ~t0\n\u001b\nande\rxzare assumed to be real. Moreover, ~t0\n\u001bcan be chosen as positive and thus T\";#>0.\n(~t0\n\u001bhere is \feld-representaion, and has unit of energy.) Equations (88) and (89) indicates\nthat the strength of the hopping amplitude ~t0\n\u001band interface spin-orbit interaction e\rxzare\nexperimentally accessible by measuring Gilbert damping and shift of resonance frequency\nas has been known2. A signi\fcant consequence of Eq. (88) is that the enhancement of the\nGilbert damping,\n\u000e\u000b\u0018a\nd1\n\u000fF2~t0\n\"~t0\n#; (90)\ncan exceed in thin ferromagnets the intrinsic damping parameter \u000b, as the two contributions\nare governed by di\u000berent material parameters. In contrast to the positive enhancement of\ndamping, the shift of the resonant frequency or g-factor can be positive or negative, as it is\nlinear in the interface spin-orbit parameter e\rxz.\nExperimentally, enhancement of the Gilbert damping and frequency shift has been mea-\nsured in many systems11. In the case of Py/Pt junction, enhancement of damping is observed\nto be proportional to 1 =din the range of 2nm <d< 10nm, and the enhancement was large,\n\u000e\u000b=\u000b'4 atd= 2 nm11. These results appears to be consistent with our analysis. Same 1 =d\ndependence was observed in the shift of g-factor. The shift was positive and magnitude is\nabout 2% for Py/Pt and Py/Pd with d= 2nm, while it was negative for Ta/Pt11. The exis-\ntence of both signs suggests that the shift is due to the linear e\u000bect of spin-orbit interaction,\nand the interface spin-orbit interaction we discuss is one of possible mechanisms.\n28For thin ferromagnet, ~d.1, the spin accumulation of Eq. (84) reads\ns(F)=1\nM((n\u0002_n)Re[\u000ethin] +_nIm[\u000ethin]); (91)\nwhere\n\u000ethin\u0011\u000e~d=m2\u0017Na2M\n2kF+kF\u0000T\";#: (92)\nEquation (91) indicates that the roles of imaginary and real part of T\";#are interchanged\nfor thick and thin ferromagnet, resulting in\n~\u000b=\u000b+ Im\u000ethin\n!B=\rB\n1\u0000Re\u000ethin; (93)\nfor thin ferromagnet. Thus, for weak interface spin-orbit interaction, positive shift of reso-\nnance frequency is expected (as Re \u000ethin>0). Signi\fcant feature is that the damping can be\nsmallened or even be negative if strong interface spin-orbit interaction exists with negative\nsign of Im\u000ethin. Our result indicates that 'spin mixing conductance' description of Ref.2\nbreaks down in thin metallic ferromagnet (and insulator case as we shall see in Sec. VII D).\nIn this section, we have discussed spin accumulation and enhanced Gilbert damping in\nferromagnet attached to a normal metal. In the \feld-theoretic description, the damping\nenhancement arises from the imaginary part of the self-energy due to the interface. Thus\na randomness like the interface scattering changing the electron momentum is essential for\nthe damping e\u000bect, which sounds physically reasonable. The same is true for the reaction,\nnamely, spin current pumping e\u000bect into N region, and thus spin current pumping requires\nrandomness, too. (In the quantum mechanical treatment of Sec. II, change of electron\nwave vector at the interface is essential.) The spin current pumping e\u000bect therefore ap-\npears di\u000berent from general pumping e\u000bects, where randomness does not play essential roles\napparently3.\nSpin accumulation and enhanced Gilbert damping was discussed by Berger10based on a\nquantum mechanical argument. There 1 =ddependence was pointed out and the damping\ne\u000bect was calculated by evaluating the decay rate of magnons. Comparison of enhanced\nGilbert damping with experiments was carried out in Ref.2but in a phenomenological man-\nner.\n29VI. CASE WITH MAGNETIZATION STRUCTURE\nField theoretic approach has an advantage that generalization of the results is straightfor-\nward. Here we discuss brie\ry the case of ferromagnet with spatially-varying magnetization.\nThe excitations in metallic ferromagnet consist of spin waves (magnons) and Stoner excita-\ntion. While spin waves usually have gap as a result of magnetic anisotropy, Stoner excitation\nis gapless for \fnite wave vector, ( kF+\u0000kF\u0000)<jqj<(kF++kF\u0000), and it may be expected\nto have signi\fcant contribution for magnetization structures having wavelength larger than\nkF+\u0000kF\u0000. Let us look into this possibility.\nOur result of spin accumulation in ferromagnet, represented in the rotated frame, Eq.\n(63), indicates that when the magnetization has a spatial pro\fle, the accumulation is deter-\nmined by the spin gauge \feld and spin correlation function depending on the wave vector q\nas\nX\nqA\u0006\ns;t(q)\u001f\u0006(q;0); (94)\nwhere\n\u001f\u0006(q;\n)\u0011\u0000X\nkfk+q;\u0006\u0000fk;\u0007\n\u000fk+q;\u0006\u0000\u000fk;\u0007+ \n +i0; (95)\nis the correlation function with \fnite momentum transfer qand \fnite angular frequency \n.\nFor the case of free electron with quadratic dispersion, the correlation function is24\n\u001f\u0006(q;\n) =Aq+i\nBq\u0012st(q) +O(\n2); (96)\nwhere\nAq=ma3\n8\u00192\u0014\n(kF++kF\u0000)\u0012\n1 +(kF+\u0000kF\u0000)2\nq2\u0013\n+1\n2q3((kF++kF\u0000)2\u0000q2)(q2\u0000(kF+\u0000kF\u0000)2) ln\f\f\f\fq+ (kF++kF\u0000)\nq\u0000(kF++kF\u0000)\f\f\f\f\u0015\nBq=m2a3\n4\u0019jqj; (97)\nand\n\u0012st(q)\u00118\n<\n:1 (kF+\u0000kF\u0000)<jqj<(kF++kF\u0000)\n0 otherwise: (98)\n30describes the wave vectors where Stoner excitation exists. As we see from Eq. (96), the\nStoner excitation contribution vanishes to the lowest order in \n, and thus the spin pumping\ne\u000bect in the adiabatic limit (\n !0) is not a\u000bected. Moreover, the real part of the correlation\nfunction,Aq, is a decreasing function of qand thus the spin pumping e\u000eciency would\ndecrease when ferromagnet has a structure. However, for rigorous argument, we need to\ninclude the spatial component of the spin gauge \feld arising form the spatial derivative of\nthe magnetization pro\fle.\nAs for the e\u000bect of the Stoner excitation on spin damping (Gilbert damping), it was\ndemonstrated for the case of a domain wall that the e\u000bect is negligibly small for a wide wall\nwith thickness \u0015\u001d(kF+\u0000kF\u0000)\u00001(Refs.24,25). Simanek and Heinrich presented a result of\nthe Gilbert damping as the linear term in the frequency of the imaginary part of the spin\ncorrelation function integrated over the wave vector12. The result is, however, obtained for a\nmodel where ferromagnet is atomically thin layer (a sheet), and would not be applicable for\nmost experimental situations. Discussion of Gilbert damping including \fnite wave vector\nand the impurity scattering was given in Ref.26. Inhomogenuity e\u000bects of damping of a\ndomain wall was studied recently in detail27.\nVII. INSULATOR FERROMAGNET\nIn this section, we discuss the case of ferromagnetic insulator. It turns out that the\ngeneration mechanisms for spin current in the insulating and metallic cases are distinct.\nA. Magnon and adiabatic gauge \feld\nThe Lagrangian for the insulating ferromagnet is\nLIF=Z\nd3r\u0014\nS_\u001e(cos\u0012\u00001)\u0000J\n2(rS)2\u0015\n\u0000HK; (99)\nwhereJis the exchange interaction between the localized spin, S, andHKdenotes the\nmagnetic anisotropy energy.\nWe \frst study low energy magnon dynamics induced by slow magnetization dynamics.\nFor separating the classical variable and \ructuation (magnon), rotating coordinate descrip-\ntion used in the metallic case is convenient. For magnons described by the Holstein-Primakov\n31boson, the unitary transformation is a 3 \u00023 matrix de\fned as follows28.\nS=UeS; (100)\nwhere\nU=0\nBBB@cos\u0012cos\u001e\u0000sin\u001esin\u0012cos\u001e\ncos\u0012sin\u001ecos\u001esin\u0012sin\u001e\n\u0000sin\u0012 0 cos\u00121\nCCCA=\u0010\ne\u0012e\u001en\u0011\n: (101)\nThe diagonalized spin eSis represented in terms of annihilation and creation operators for\nthe Holstein-Primakov boson, bandby, as29\neS=0\nBBB@q\nS\n2(by+b)\niq\nS\n2(by\u0000b)\nS\u0000byb1\nCCCA: (102)\nWe neglect the terms that are third- and higher-order in boson operators. Derivatives of the\nlocalized spin then read\n@\u0016S=U(@\u0016+iAU;\u0016)eS; (103)\nwhere\nAU;\u0016\u0011\u0000iU\u00001r\u0016U; (104)\nis the spin gauge \feld represented as a 3 \u00023 matrix. The spin Berry's phase of the Lagrangian\n(99) is written in terms of magnon as (derivation is in Sec. B)\nLm= 2S\r2Z\nd3ri[by(@t+iAz\ns;t)b\u0000by( \n@t\u0000iAz\ns;t))b]; (105)\nnamely, magnons interacts with the adiabatic component of the same spin gauge \feld for\nelectrons,Az\ns;t, de\fned in Eq. (14). As magnon is a single-component \feld, the gauge \feld\nis also single-component, i.e., a U(1) gauge \feld. This is a signi\fcant di\u000berence between\ninsulating and metallic ferromagnet; In the metallic case, conduction electron couples to\nan SU(2) gauge \feld with spin-\rip components, which turned out to be essential for spin\ncurrent generation. In contrast, in the insulating case, the magnon has diagonal gauge \feld,\ni.e., a spin chemical potential, which simply induces diagonal spin polarization. Pumping of\nmagnon was discussed in a di\u000berent approach by evaluating magnon source term in Ref.30.\n32The exchange interaction at the interface is represented by a Hamiltonian\nHI=JIZ\nd3rIS(r)\u0001cy\u001bc; (106)\nwhereJIis the strength of the interface sdexchange interaction and the integral is over\nthe interface. We consider a sharp interface at x= 0. Using Eq. (100), the interaction is\nrepresented in terms of magnon operators up to the second order as\nHI=JIZ\nd3rI\"\n(S\u0000byb)cy(n\u0001\u001b)c+r\nS\n2\u0002\nbycy\b\u0001\u001bc+bcy\b\u0003\u0001\u001bc\u0003#\n; (107)\nwhere\n\b\u0011e\u0012+ie\u001e=0\nBBB@cos\u0012cos\u001e\u0000isin\u001e\ncos\u0012sin\u001e+icos\u001e\n\u0000sin\u00121\nCCCA: (108)\nEquation (107) indicates that there are two mechanisms for spin current generation; namely,\nthe one due to the magnetization at the interface (the term proportional to n) and the one\ndue to the magnon spin scattering at the interface (described by the term linear in magnon\noperators).\nLet us brie\ry demonstrate based on the expression of Eq. (107) that spin-\rip processes\ndue to magnon creation or annihilation lead to generation of spin current in the normal\nmetal. At the second order, the interaction induces a factor on the electron wave function\n(\b\u0003(t)\u0001\u001b)(\b(t0)\u0001\u001b) for magnon creation and ( \b(t)\u0001\u001b)(\b\u0003(t0)\u0001\u001b) for annihilation (we\nallow an in\fnitesimal di\u000berence in time tandt0). The factor for the creation has charge\nand spin contributions, ( \b\u0003(t)\u0001\u001b)(\b(t0)\u0001\u001b) =\b\u0003(t)\u0001\b(t0) +i\u001b\u0001(\b\u0003(t)\u0002\b(t0)). For\nmagnon annihilation, we have ( \b\u0003(t)\u0002\b(t0))\u0003, and thus the sum of the magnon creation\nand annihilation processes give arise to a factor\nX\nq[(nq+ 1)(\b\u0003(t)\u0002\b(t0)) +nq(\b\u0003(t)\u0002\b(t0))\u0003] =X\nq[(2nq+ 1)Re[ \b\u0003(t)\u0002\b(t0)] +iIm[\b\u0003(t)\u0002\b(t0)]:\n(109)\nFor adiabatic change the amplitude is expanded as\n(\b\u0003(t)\u0002\b(t0)) = 2i(1 +i(t\u0000t0) cos\u0012_\u001e)n\u0000(t\u0000t0)(n\u0002_n\u0000i_n) +O((@t)2); (110)\n33where we see that an retardation e\u000bect from the adiabatic change of magnetization (rep-\nresented by the second term on the right-hand side) gives rise to a magnon state change\nproportional to n\u0002_nand _n. The retardation contribution for the spin part (Eq. (109)) is\n(t\u0000t0)X\nq[\u0000(2nq+ 1)(n\u0002_n) +i_n]: (111)\nWe therefore expect that a spin current proportional to n\u0002_nemerges proportional to the\nmagnon creation and annihilation number,P\nq(2nq+ 1). (As we shall see below, the factor\nt\u0000t0reduces to a derivative with respect to angular frequency of the Green's function.) A\nrigorous estimation using Green's function method is presented in Sec. VII C.\nIn Eq. (111), the last term proportional to _nis an imaginary part arising from the\ndi\u000berence of magnon creation and annihilation probabilities of vacuum, nq+ 1 andnq.\nThe term is, however, unphysical one corresponding to a real energy shift due to magnon\ninteraction, and is removed by rede\fnition of the Fermi energy.\nB. Spin current pumped by the interface exchange interaction\nHere we study the spin current pumped by the classical magnetization at the interface,\nnamely, the one driven by the term proportional to Snin Eq. (107). We treat the ex-\nchange interaction perturbatively to the second order as the exchange interaction between\nconduction electron and insulator ferromagnet is localized at the interface and is expected\nto be weak. The weak coupling scheme employed here is in the opposite limit as the strong\ncoupling (adiabatic) approach used in the metallic ferromagnet (Sec. IV).\nIn the perturbative regime, the issue of adiabaticity needs to be argued carefully. In\nthe strong sdcoupling limit, the adiabaticity is trivially satis\fed, as the time needed for\nthe electron spin to follow the localized spin is the fastest timescale. In the weak coupling\nlimit, this timescale is long. Nevertheless, the adiabatic condition is satis\fed if the electron\nspin relaxation is strong so that the electron spin relaxes quickly to the local equilibrium\nstate determined by the localized spin. Thus the adiabatic condition is expected to be\nMI\u001csf=~\u001c1, whereMIand\u001csfare the interface spin splitting energy, and conduction\nelectron spin relaxation time, respectively. In the following calculation, we consider the case\nof\u000fF\u001csf=~\u001d1, i.e., ~(\u001csf)\u00001\u001c\u000fF, as the spin \rip lifetime is by de\fnition longer than the\nelastic electron lifetime \u001c, which satis\fes \u000fF\u001c=~\u001d1 in metal. Our results therefore cover\n34both adiabatic and nonadiabatic limits.\nThe calculation is carried out by evaluating Feynmann diagrams of Fig. 9, similar to the\nstudy of Refs.17,18. A di\u000berence is that while Refs.17,18assumed a smooth magnetization\nstructure and used a gradient expansion, the exchange interaction we consider is localized.\nFIG. 9. T\nhe Feynmann diagrams for spin current pumped by interface sdexchange interaction.\nThe lesser Green's function for normal metal including the interface exchange interaction\nto the linear order is\nG(1)<\nN(r;t;r;t) =MIZd!\n2\u0019Zd\n2\u0019X\nkk0e\u0000i\ntei(k0\u0000k)\u0001r\u0002\n(f(!+ \n)\u0000f(!))gr\nk0;!+\nga\nk!\u0000f(!)gr\nk0;!+\ngr\nk!+f(!+ \n)ga\nk0;!+\nga\nk!\u0003\n(n\n\u0001\u001b);\n(112)\nwhereMI\u0011JISis the local spin polarization at the interface. Expanding the expression\nwith respect to \n and keeping the dominant contribution at long distance, i.e., the terms\ncontaining both gaandgr. UsingP\nkga\nk!eik\u0001r'im\nkFeikre\u0000jxj\n`(\u0011ga(r)), the result of spin\ncurrent is\nj(1)\ns(r;t) =\u0000MIm\nkF_ne\u0000jxj=`: (113)\nThe second-order contribution is similarly calculated to obtain\nG(2)<\nN(r;t;r;t)'(MI)2Zd!\n2\u0019Zd\n1\n2\u0019Zd\n2\n2\u0019\n\u0002X\nkk0k00e\u0000i(\n1+\n2)tei(k0\u0000k)\u0001rf0(!)gr\nk0;!ga\nk!(\n1ga\nk00!+ \n 2gr\nk00!)(n\n1\u0001\u001b)(n\n2\u0001\u001b)\n=\u00002\u0019i\u0017(MI)2jgr(r)j2(n\u0002_n)\u0001\u001b: (114)\nThe corresponding spin current at the interface ( x= 0) is thus\nj(2)\ns(x= 0;t) =\u0017(MI)2m\nkF(n\u0002_n); (115)\n35and the total spin current reads\njs(x= 0;t) =\u0000MIm\nkF_n\u00002\u0017(MI)2m\nkF(n\u0002_n): (116)\nIn the perturbation regime, the spin current proportional to _nis dominant (larger by a\nfactor of (\u0017M I)\u00001) compared to the one proportional to n\u0002_n.\nExpression of spin current induced by the interface exchange interaction was presented\nin Ref.31in the limit of strong spin relaxation, MI\u001csf\u001c1, where\u001csfis the spin relaxation\ntime of electron. By solving the Landau-Lifshitz-Gilbert equation for the electron spin, they\nobtained an result of Eq. (116) with \u0017M Ireplaced by MI\u001csf.\nC. Calculation of magnon-induced spin current\nHere magnon-induced spin current due to the magnon interaction in Eq. (107) is cal-\nculated. As magnon is a small \ructuation of magnetization, the contribution here is a\nsmall correction to the contribution of Eq. (116). Nevertheless, the magnon contribution\nhas a typical linear dependence on the temperature, and is expected to be experimentally\nidenti\fed easily.\nSpin current induced in normal metal is evaluated by calculating the self-energy arising\nfrom the interface magnon scattering of Eq. (107). The contribution to the path-ordered\nGreen's function of N electron from the magnon scattering to the second order is\nGN(r;t;r0:t0) =Z\nCdt1Z\nCdt2X\nr1r2gN(r;t;r1;t1)\u0006I(r1;t1;r2;t2)gN(r2;t2;r0;t0); (117)\nwhere\n\u0006I(r1;t1;r2;t2)\u0011iSJ2\nI\n2D\u000b\f(r1;t1;r2;t2)\u001b\u000bgN(r1;t1;r2;t2)\u001b\f; (118)\nrepresents the self energy. Here\nD\u000b\f(r1;t1;r2;t2)\u0011\u0000ihTCB\u000b(r1;t1)B\f(r2;t2)i; (119)\nis the Green's function for magnon dressed by the magnetization structure ( \bis de\fned in\nEq. (108)),\nB\u000b(r;t)\u0011\b\u000b(t)by(r;t) + \by\n\u000b(t)b(r;t): (120)\n36FIG. 10. T\nhe Feynmann diagrams for spin current pumped by magnons at the interface. Green's functions\nfor magnons and electrons in the normal metal are denoted by DandgN, respectively. \b\nrepresents the e\u000bects of magnetization dynamics (Eq. (108)).\nDiagramatic representation is in Fig. 10. In the present approximation including the inter-\nface scattering to the second order, the electron Green's function in Eq. (118) is treated as\nspin-independent, resulting in a self energy (de\fned on complex time contour)\n\u0006I(r1;t1;r2;t2) =iSJ2\nI\n2(\u000e\u000b\f+i\u000f\u000b\f\r\u001b\r)D\u000b\f(r1;t1;r2;t2)gN(r1;t1;r2;t2): (121)\nWe focus on the spin-polarized contribution,\n\u0006I;\r(r1;t1;r2;t2)\u0011\u0000SJ2\nI\n2eD\r(r1;t1;r2;t2)gN(r1;t1;r2;t2); (122)\nwhereeD\r\u0011\u000f\u000b\f\rD\u000b\f. The spin-dependent contribution of lessor Green's function, Eq. (117),\nreads (time and spatial coordinates partially suppressed)\nG<\nN(r;t;r0;t0) =\u001b\rZ1\n\u00001dt1Z1\n\u00001dt2\u0014\ngr\nN(t\u0000t1)\u0006r\nI;\r(t1;t2)g<\nN(t2\u0000t0) +gr\nN\u0006<\nI;\rga\nN+g<\nN\u0006a\nI;\rga\nN\u0015\n\u0011\u001b\rG<\nN;\r(r;t;r0:t0):\n(123)\nFor the self energy type of the Green's functions, depending on two time as g(t1\u0000t2)D(t1\u0000t2)\n(Eq. (122)), real-time components are written as (suppressing time and su\u000ex of N) (See\nSec. C)\n[g(t1\u0000t2)D(t1\u0000t2)]r=grD<+g>Dr=g<Dr+grD>\n[g(t1\u0000t2)D(t1\u0000t2)]a=gaD>+g<Da=gaD<+g>Da\n[g(t1\u0000t2)D(t1\u0000t2)]<=g<D<: (124)\nThe Green's function eDis that of a composite \feld B\u000bde\fned in Eq. (120), and is decom-\n37posed to elementary magnon Green's function, D, as\neD\r(r1;t1;r2;t2) = [\by(t1)\u0002\b(t2)]\rD(r1;t1;r2;t2)\u0000[\by(t2)\u0002\b(t1)]\rD(r2;t2;r1;t1);\n(125)\nwhere\nD(r1;t1;r2;t2)\u0011\u0000i\nTCb(r1;t1)by(r2;t2)\u000b\n: (126)\nThe spin-dependent factor in Eq. (125) is calculated for adiabatic dynamics as\n\by(t1)\u0002\b(t2) = 2in(t1) + (t2\u0000t1)[\t+i_n] +O((@t)2); (127)\nwhere\n\t\u00112 cos\u0012_\u001en+n\u0002_n: (128)\nThe real-time Green's functions are therefore ( D(1;2)\u0011D(r1;t1;r2;t2))\neD<\n\r(r1;t1;r2;t2) = 2in(t1)[D<(r1;t1;r2;t2)\u0000D>(r2;t2;r1;t1)]\n+ (t2\u0000t1)\u0014\n\t[D<(r1;t1;r2;t2) +D>(r2;t2;r1;t1)] +i_n[D<(r1;t1;r2;t2)\u0000D>(r2;t2;r1;t1)]\u0015\neDr\n\r(1;2) =\u0012(t1\u0000t2)(eD<\n\r(1;2)\u0000eD>\n\r(1;2))\neDa\n\r(1;2) =\u0000\u0012(t2\u0000t1)\u000f\u000b\f\r(D<\n\u000b\f(1;2)\u0000D>\n\u000b\f(1;2)); (129)\nandeD<\n\ris obtained by exchanging <and>ineD<\n\r. Elementary Green's functions are\ncalculated as\nD<(r1;t1;r2;t2) =\u0000iX\nqeiq\u0001(r1\u0000r2)nqe\u0000i!q(t1\u0000t2)\nD>(r1;t1;r2;t2) =\u0000iX\nqeiq\u0001(r1\u0000r2)(nq+ 1)e\u0000i!q(t1\u0000t2); (130)\nwhere!qis magnon energy and nq\u00111\ne\f!q\u00001. In our model, the interface is atomically \rat\nand has an in\fnite area, and thus ri(i= 1;2) are atx= 0. Fourier components de\fned as\n(a= r;a;<;> )\neDa\n\r(x1= 0;t1;x2= 0;t2)\u0011X\nqZd\n2\u0019e\u0000i\n(t1\u0000t2)eDa\n\r(q;\n); (131)\n38are calculated from Eq. (129) as\neD<\n\r(q;\n) =\u0000i\u0014\n2n(D<\n\u0000\u0000D>\n+) +d\nd\n\u0002\n\t(D<\n\u0000+D>\n+) +i_n(D<\n\u0000\u0000D>\n+)\u0003\u0015\neDr\n\r(q;\n) =\u0000i\u0014\n2n(Dr\n\u0000+Dr\n+) +d\nd\n\u0002\n\t(Dr\n\u0000\u0000Dr\n+) +i_n(Dr\n\u0000+Dr\n+)\u0003\u0015\neDa\n\r(q;\n) =\u0000i\u0014\n2n(Da\n\u0000+Da\n+) +d\nd\n\u0002\n\t(Da\n\u0000\u0000Da\n+) +i_n(Da\n\u0000+Da\n+)\u0003\u0015\n; (132)\nwhere\nDa\n\u0006\u00111\n\n\u0006!q\u0000i0; Dr\n\u0006\u00111\n\n\u0006!q+i0\nD<\n\u0000\u0011nq(Da\n\u0000\u0000Dr\n\u0000); D>\n+\u0011(1 +nq)(Da\n+\u0000Dr\n+): (133)\nThe spin part of the Green's function, Eq. (123), is\nG<\nN;\r(r;t;r0;t) =\u0000SJ2\nI\n2Zd!\n2\u0019Zd\n2\u0019X\nkk0X\nk00q\u0014\ngr\nN;k!\u0010\neDr\n\r(q;\n)g>\nN;k00;!\u0000\n+eD<\n\r(q;\n)gr\nN;k00;!\u0000\n\u0011\ng<\nN;k0!\n+gr\nN;k!eDr\n\r(q;\n)g>\nN;k00;!\u0000\nga\nN;k0!+g<\nN;k!\u0010\neDa\n\r(q;\n)g>\nN;k00;!\u0000\n+eD<\n\r(q;\n)ga\nN;k00;!\u0000\n\u0011\nga\nN;k0!\u0015\n:\n(134)\nThe contribution survives at long distance is the one containing gr\nN;!(r) andga\nN;!(\u0000r), i.e.,\nG<\nN;\r(r;t;r0;t)'Zd!\n2\u0019X\nkk0gr\nN;k!ga\nN;k0!eik\u0001re\u0000ik0\u0001r0e\u0006I;\r; (135)\nwhere\ne\u0006I;\r\u0011\u0000SJ2\nI\n2Zd\n2\u0019X\nk00q\u0014\u0010\nfk0eDr\n\r(q;\n)\u0000fkeDa\n\r(q;\n)\u0011\n(fk00\u00001)(ga\nN;k00;!\u0000\n\u0000gr\nN;k00;!\u0000\n)\n+eD<\n\r(q;\n)(fk0gr\nN;k00;!\u0000\n\u0000fkga\nN;k00;!\u0000\n+fk00(ga\nN;k00;!\u0000\n\u0000gr\nN;k00;!\u0000\n))\u0015\n: (136)\nWe focus on the pumped contribution, containing derivative with respect to \n in Eq. (132).\nThe result is, using partial integration with respect to \n ( e\u0006Iis a vector representation of\ne\u0006I;\r),\ne\u0006I'\u0000iSJ2\nI\n2Zd\n2\u0019X\nk00q\u0014\n\u0000\nfk0[\t(Dr\n\u0000\u0000Dr\n+) +i_n(Dr\n\u0000+Dr\n+)]\u0000fk[\t(Da\n\u0000\u0000Da\n+) +i_n(Da\n\u0000+Da\n+)]\u0001\n(fk00\u00001)d\nd\n(ga\nN;k00;!\u0000\n\u0000gr\nN;k00;!\u0000\n)\n+ [\t(D<\n\u0000+D>\n+) +i_n(D<\n\u0000\u0000D>\n+)]d\nd\n[(fk00\u0000fk)ga\nN;k00;!\u0000\n\u0000(fk00\u0000fk0)gr\nN;k00;!\u0000\n]\u0015\n:\n(137)\n39Usingd\nd\nga\nk00;!\u0000\n= (ga\nk00;!)2+O(\n) and an approximation, we obtainP\nk00(ga\nk00;!)2'\u0000\u0019i\u0017\n2\u000fF,\ne\u0006I'\u0019\u0017\n\u000fFSJ2\nI\n2Zd\n2\u0019X\nqk00\u0014\n\t\u0012\n(fk00\u00001)[fk0(Dr\n\u0000\u0000Dr\n+)\u0000fk(Da\n\u0000\u0000Da\n+)] +1\n2(2fk00\u0000fk\u0000fk0)(D<\n\u0000+D>\n+)\u0013\n+i_n\u0012\n(fk00\u00001)[fk0(Dr\n\u0000+Dr\n+)\u0000fk(Da\n\u0000+Da\n+)] +1\n2(2fk00\u0000fk\u0000fk0)(D<\n\u0000\u0000D>\n+)\u0013\u0015\n:\n(138)\nAs argued for Eq. (111), only the imaginary part of self energy contributes to the induced\nspin current, as the real part, the shift of the chemical potential, is compensated by redis-\ntribution of electrons. The result is thus\ne\u0006I'i\t\u0019\u0017\n\u000fFSJ2\nI\n2X\nqk00(1 + 2nq)(2fk00\u0000fk\u0000fk0): (139)\nWe further note that the component of \tproportional to n(Eq. (128)) does not contribute\nto the current generation, as a result of gauge invariance. (In other words, the contribution\ncancels with the one arising from the e\u000bective gauge \feld for magnon.)\nThe \fnal result of the spin current pumped by the magnon scattering is therefore\njm\ns(r;t) =\u0019\u0017\n\u000fFSJ2\nI\n2jgr(r)j2X\nq(1 + 2nq)(n\u0002_n): (140)\nAt high temperature compared to magnon energy, \f!q\u001c1, 1 + 2nq'2kBT\n!q, and the\nmagnon-induced spin current depends linearly on temperature. The result (140) agrees\nwith previous study carried out in the context of thermally-induced spin current19.\nD. Correction to Gilbert damping in the insulating case\nIn this subsection, we calculate the correction to the Gilbert damping and g-factor of\ninsulating ferromagnet as a result of spin pumping e\u000bect. We study the torque on the\nferromagnetic magnetization arising from the e\u000bect of conduction electron of normal metal,\ngiven by\n\u001cI=BI\u0002n=MI(n\u0002sI); (141)\n40where\nBI\u0011\u0000\u000eHI\n\u000en=\u0000MIsI; (142)\nis the e\u000bective magnetic \feld arising from the interface electron spin polarization, sI(t)\u0011\n\u0000itr[\u001bG<\nN(0;t)]. The contribution to the electron spin density linear in the interface ex-\nchange interaction, Eq. (106), is\ns(1);\u000b\nI(t) =\u0000iZ\ndt1MIn\f(t1)tr[\u001b\u000bgN(t;t1)\u001b\fgN(t1;t)]<; (143)\nwhere the Green's functions connect positions at the interface, i.e., from x= 0 tox= 0,\nand are spin unpolarized. (The Feynman diagrams for the spin density are the same as the\none for the spin current, Fig. 9 with the vertex jsreplaced by the Pauli matrix.) Pumped\ncontribution proportional to the time variation of magnetization is obtained as\ns(1)\nI(t) =\u0000MI_nZd!\n2\u0019X\nkk0f0(!)(ga\nN;k0\u0000gr\nN;k0)(ga\nN;k\u0000gr\nN;k)\n=\u0000MI(\u0019\u0017)2_n: (144)\nThe second order contribution similarly reads\ns(2);\u000b\nI(t) =\u0000i\n2Z\ndt1Z\ndt2(MI)2n\f(t1)n\r(t2)tr[\u001b\u000bgN(t;t1)\u001b\fgN(t1;t2)\u001b\rgN(t2;t)]<\n'\u00002(MI)2(\u0019\u0017)3(n\u0002_n): (145)\nThe interface torque is therefore\n\u001cI=\u0000(MI\u0019\u0017)2(n\u0002_n) + 2(MI\u0019\u0017)3_n: (146)\nIncluding this torque in the LLG equation, _n=\u0000\u000bn\u0002_n\u0000\rB\u0002n+\u001c, we have\n(1\u0000\u000eI)_n=\u0000\u000bI(n\u0002_n)\u0000\rB\u0002n; (147)\nwhere\n\u000eI= 2\u0016d(\u0019M I\u0017)3\n\u000bI=\u000b+\u0016d(\u0019M I\u0017)2; (148)\nwhere\u0016d\u0018dmp=dis the ratio of the length of magnetic proximity ( dmp) and thickness of\nthe ferromagnet, d. The Gilbert damping constant therefore increases as far as the interface\n41spin-orbit interaction is neglected. The resonance frequency is !B=\rB\n1\u0000\u000eI, and the shift can\nhave both signs depending on the sign of interface exchange interaction, MI.\nThere may be a possibility that magnon excitation induce torque that corresponds to\ne\u000bective damping. In fact, such torque arises of hbior\nby\u000b\nare \fnite, i.e., if magnon Bose\ncondensation glows, as seen from Eq. (102). Such condensation can in principle develop from\nthe interface interaction of magnon creation or annihilation induced by electron spin \rip,\nEq. (107). However, conventional spin relaxation processes arising from the second order\nof random spin scattering do not contribute to such magnon condensation and additional\ndamping.\nComparing the result of pumped spin current, Eq. (116), and that of damping coe\u000ecient,\nEq. (148), we notice that the 'spin mixing conductance' argument2, where the coe\u000ecients\nfor the spin current component proportional to n\u0002_nand the enhancement of the Gilbert\ndamping constant are governed by the same quantity (the imaginary part of 'spin mixing\nconductance') does not hold for the insulator case. In fact, our result indicates that the\nspin current component proportional to n\u0002_narises from the second order correction to\nthe interaction (the second diagram of Fig. 9), while the damping correction arises from the\n\frst order process (the \frst diagram of Fig. 9). Although the magnitudes of the two e\u000bects\nhappen to be both second order of interface spin splitting, MI, physical origins appear to\nbe distinct. From our analysis, we see that the 'spin mixing conductance' description is not\ngeneral and applies only to the case of thick metallic ferromagnet (see Sec. V A for metallic\ncase).\nVIII. DISCUSSION\nOur results are summarized in table I. Let us discuss experimental results in the light of\nour results. In the early ferromagnetic resonance (FMR) experiments, consistent studies of\ng-factor and the Gilbert damping were carried out on metallic ferromagnets11. The results\nappear to be consistent with theories (Refs.2,10and the present paper). Both the damping\nconstant and the gfactor have 1 =d-dependence on the thickness of ferromagnet in the range\nof 2nm<d< 10nm11. The maximum additional damping reaches \u000e\u000b\u00180:1 atd=2nm, which\nexceeds the original value of \u000b\u00180:01. Theg-factor modulation is about 1% at d= 2nm, and\nits sign depends on the material; the g-factor increases for Pd/Py/Pd and Pt/Py/Pt while\n42Ferromagnet(F) Ai Ar \u000e\u000b \u000e! B Assumption Equations\nMetal ImT+\u0000 ReT+\u0000ReT+\u0000\nImT+\u0000ImT+\u0000\nReT+\u0000Thick F\nThin F(27)(66) (88)(89)\n(93)\nInsulator MI\u0017 (MI\u0017)2(MI\u0017)2(MI\u0017)3Weak spin relaxation\u0003(116) (148)\n- (MI\u0017)2P\nq(1 + 2nq) - - Magnon (140)\nTABLE I. Summary of essential parameters determining spin current js, corrections to the Gilbert\ndamping\u000e\u000band resonance frequency shift \u000e!Bfor metallic and insulating ferromagnets. Coe\u000e-\ncientsAiandArare for the spin current, de\fned by Eq. (1). Label \u0000indicates that it is not\ndiscussed in the present paper.\u0003: For strong spin relaxation case, the density of states \u0017is\nreplaced by inverse of electron spin-\rip time, \u001csf.31\ndecreases for Ta/Py/Ta. These results appear consistent with ours, because \u000e!Bis governed\nby ImT+\u0000, whose sign depends on the sign of interface spin-orbit interaction. In contrast,\ndamping enhancement proportional to Re T+\u0000is positive for thick metals. However, other\npossibilities like the e\u000bect of a large interface orbital moment playing a role in the gfactor,\ncannot be ruled out at present.\nRecently, inverse spin Hall measurement has become common for detecting the spin cur-\nrent. In this method, however, only the dc component proportional to n\u0002_nis accessible so\nfar and there remains an ambiguity for qualitative estimates because another phenomeno-\nlogical parameter, the conversion e\u000eciency from spin to charge, enters. Qualitatively, the\nvalues ofArobtained by the inverse spin Hall measurements32and FMR measurements are\nconsistent with each other.\nThe cases of insulating ferromagnets have been studied recently. In the early experiments,\norders of magnitude smaller value of Arcompared to metallic cases were reported31, while\nthose small values are now understood as due to poor interface quality. In fact, FMR\nmeasurements on epitaxially grown samples like YIG/Au/Fe turned out to show Arof 1\u0018\n5\u00021018m\u00002(Refs.33,34), which is the same order as in the metallic cases. Inverse spin Hall\nmeasurements on YIG/Pt reports similar values35, and the value is consistent with the \frst\nprinciples calculation36. Systematic studies of YIG/NM with NM=Pt, Ta, W, Au, Ag, Cu,\nTi, V, Cr, Mn etc. were carried out with the result of Ar\u00181017\u00001018m\u00002(Refs.37{40). If we\nuse naive phenomenological relation, Eq. (6), Ar= 1018m\u00002corresponds to \u000e\u000b= 3\u000210\u00004if\n43a= 2\u0017A,S= 1 andd= 20\u0017A. Assuming interface sdexchange interaction, the value indicates\nMI\u0017\u00180:01, which appears reasonable at least by the order of magnitude from the result of\nx-ray magnetic circular dichroism (XMCD) suggesting spin polarization of interface Pt of\n0:05\u0016B41.\nOn the other hand, FMR frequency shift of insulators cannot be explained by our theory.\nIn fact, the shift for YIG/Pt is \u000e!B=!B\u00181:6\u000210\u00002, which is larger than \u000e\u000b\u00182\u000210\u00003,\nwhile our perturbation theory assuming weak interface sdinteraction predicts \u000e!B=!B<\u000e\u000b .\nWe expect that the discrepancy arises from the interface spin-orbit interaction that would\nbe present at insulator-metal interface, which modi\fes the magnetic proximity e\u000bect and\ndamping torque signi\fcantly. It would be necessary to introduce anomalous sdcoupling at\nthe interface like the one discussed in Ref.42. Experimentally, in\ruence of interface spin-\norbit interaction43and proximity e\u000bect needs to be carefully characterized by using the\nmicroscopic technique, such as MCD, to compare with theories.\nIX. SUMMARY\nWe have presented a microscopic study of spin pumping e\u000bects, generation of spin cur-\nrent in ferromagnet-normal metal junction by magnetization dynamics, for both metallic and\ninsulating ferromagnets. As for the case of metallic ferromagnet, a simple quantum mechan-\nical picture was developed using a unitary transformation to diagonal the time-dependent\nsdexchange interaction. The problem of dynamic magnetization is thereby mapped to the\none with static magnetization and o\u000b-diagonal spin gauge \feld, which mixes the electron\nspin. In the slowly-varying limit, spin gauge \feld becomes static, and the conventional spin\npumping formula is derived simply by evaluating the spin accumulation formed in the nor-\nmal metal as a result of interface hopping. The e\u000bect of interface spin-orbit interaction was\ndiscussed. Rigorous \feld-theoretical derivation was also presented, and the enhancement of\nspin damping (Gilbert damping) in the ferromagnet as a result of spin pumping e\u000bect was\ndiscussed. The case of insulating ferromagnet was studied based on a model where spin\ncurrent is driven locally by the interface exchange interaction as a result of magnetic prox-\nimity e\u000bect. The dominant contribution turns out to be the one proportional to _n, while\nmagnon contribution leads to n\u0002_n, whose amplitude depends linearly on the temperature.\nOur analysis clearly demonstrate the di\u000berence in the spin current generation mechanism\n44for metallic and insulating ferromagnet.\nThe in\ruence of atomic-scale interface structure on the spin pumping e\u000bect are open and\nurgent issues, in particular for the case of ferrimagnetic insulators which have two sub-lattice\nmagnetic moments.\nACKNOWLEDGMENTS\nGT thanks H. Kohno, C. Uchiyama, K. Hashimoto and A. Shitade for valuable discus-\nsions. This work was supported by a Grant-in-Aid for Exploratory Research (No.16K13853)\nand a Grant-in-Aid for Scienti\fc Research (B) (No. 17H02929) from the Japan Society for\nthe Promotion of Science and a Grant-in-Aid for Scienti\fc Research on Innovative Areas\n(No.26103006) from The Ministry of Education, Culture, Sports, Science and Technology\n(MEXT), Japan.\nAppendix A: E\u000bect of spin-conserving spin gauge \feld on spin density\nHere we calculate contribution of spin-conserving spin gauge \feld, Az\ns;t, on the interface\ne\u000bects of spin density in F. It turns out that spin-conserving spin gauge \feld combined with\ninterface e\u000bects does not induce damping. This result is consistent with a naive expectation\nthat only the nonadiabatic components of spin current should contribute to damping.\nFIG. 11. Diagramatic representation of the contribution to the lessor Green's function for F\nelectron arising from the interface hopping (represented by tandt\u0003) and spin gauge \feld ( As;t).\nThe diagram (c) includes the spin gauge \feld implicitly in unitary matrices UandU\u00001.\nThe contribution to the lesser Green's function in F from the interface hopping (lowest,\nthe second-order in the hopping) at the linear order in the spin gauge \feld reads (diagra-\n45matically shown in Fig. 11)\n\u000eG<=\u000eG<\n(a)+\u000eG<\n(b)+\u000eG<\n(c)\n\u000eG<\n(a)=gr(As;t\u0001\u001b)gr\u0006r\n0g<+gr(As;t\u0001\u001b)gr\u0006<\n0ga+gr(As;t\u0001\u001b)g<\u0006a\n0ga+g<(As;t\u0001\u001b)ga\u0006a\n0ga\n\u000eG<\n(b)=gr\u0006r\n0gr(As;t\u0001\u001b)g<+gr\u0006<\n0g<(As;t\u0001\u001b)ga+gr\u0006a\n0ga(As;t\u0001\u001b)ga+g<\u0006a\n0ga(As;t\u0001\u001b)ga\n\u000eG<\n(c)=gr\u0006rg<+gr\u0006<ga+g<\u0006aga: (A1)\nHere\n\u0006a\u0011~tU\u00001ga\nNU~ty;(a= a;r;<)\n\u0006a\n0\u0011~tga\nN; (A2)\nare self energy due to the interface hopping, where \u0006ais the full self energy including the\ntime-dependent unitary matrix U, which includes spin gauge \feld. \u0006a\n0is the contribution of\n\u0006awith the spin gauge \feld neglected. We here focus on the contribution of the adiabatic ( z)\ncomponent,Az\ns;t. Usingg<=F(ga\u0000gr) for F (Fis a 2\u00022 matrix of the spin-polarized Fermi\ndistribution function) and g<\nN=fN(ga\nN\u0000gr\nN) and noting that all the angular frequencies of\nthe Green's function are equal, we obtain\n\u000eG<\n(a)+\u000eG<\n(b)'Az\ns;t\u001bz\u0002\n\u00002F[(gr)3\u0006r\n0\u0000(ga)3\u0006a\n0]\u0000(F\u0000fN)[(gr)2ga+gr(ga)2](\u0006a\n0\u0000\u0006r\n0)\u0015\n:\n(A3)\nThe contribution \u000eG<\n(c)is calculated noting that\n~tU\u00001ga\nNU~ty=ga\nN~t~ty\u0000dga\nN\nd!~t(As;t\u0001\u001b)~ty+O((As;t)2): (A4)\nThe linear contribution with respect to the zcomponent of the gauge \feld turns out to be\n\u000eG<\n(c)'Az\ns;t\u001bz\u0014\nF\u0014\n(gr)2@\n@!\u0006r\n0\u0000(ga)2@\n@!\u0006a\n0\u0015\n+ (F\u0000fN)grga@\n@!(\u0006a\n0\u0000\u0006r\n0)\u0015\n: (A5)\nWe therefore obtain the e\u000bect of spin-conserving gauge \feld as\n\u000eG<=Az\ns;t\u001bz@\n@!\u0014\nF\u0002\n(gr)2\u0006r\n0\u0000(ga)2\u0006a\n0\u0003\n+ (F\u0000fN)grga(\u0006a\n0\u0000\u0006r\n0)\u0015\n; (A6)\nwhich vanishes after integration over !. Therefore, contribution from spin-conserving gauge\n\feld and interface hopping vanishes in the spin density, leaving the damping una\u000bected.\n46Appendix B: Magnon representation of spin Berry's phase term\nHere we derive the expression for the spin Berry's phase term of the Lagrangian (99)\nin terms of magnon operator. The time-integral of the term is written by introducing an\narti\fcial variable uas44\nZ\ndtL B=SZ\ndt_\u001e(cos\u0012\u00001) =S\u00002Z\ndtZ1\n0duS\u0001(@tS\u0002@uS); (B1)\nwhereS(t;u) is extended to a function of tandu, but onlyS(t;u= 1) is physical. Noting\nthat the unitary transformation matrix element of Eq. (101) is written as\nUij= (ej)i; (B2)\nwherer1\u0011e\u0012,e2\u0011e\u001eande3\u0011n, we obtain\nS\u0001(@tS\u0002@uS) =eS\u0001[(@t+iAU;t)eS\u0002(@u+iAU;u)eS)]: (B3)\nEvaluating to the second order in the magnon operators, we have\n@teS\u0002@ueS= 2i\r^z[(@uby)(@tb)\u0000(@tby)(@ub)]: (B4)\nUsing the explicit form of AU;\u0016, the gauge \feld contribution is\n@ueS\u0001[eS\u0002iAU;teS)] =S2\r[(@uby)(\u0000sin\u0012_\u001e+i_\u0012) + (@ub)(\u0000sin\u0012_\u001e\u0000i_\u0012)]\u00002S\r2cos\u0012(@t\u001e)@u(byb):\n(B5)\nThe terms linear in the boson operators vanish by the equation of motion, and the second-\norder contribution is\nS\u0001(@tS\u0002@uS) = 2S\r2\u0014\ni@u[by(@tb)\u0000(@tby)b]\n\u0000@u[cos\u0012(@t\u001e)byb] +@t[cos\u0012(@u\u001e)byb] + sin\u0012((@t\u0012)(@u\u001e)\u0000(@u\u0012)(@t\u001e))byb\u0015\n:\n(B6)\nIntegrating over tandu, the term total derivative with respect to tof Eq. (B6) vanishes,\nresulting in\nZ\ndtZ1\n0duS\u0001(@tS\u0002@uS) = 2S\r2Z\ndt\u0014\ni[by(@tb)\u0000(@tby)b]\u0000cos\u0012(@t\u001e)byb\n+ sin\u0012((@t\u0012)(@u\u001e)\u0000(@u\u0012)(@t\u001e))byb\u0015\n: (B7)\n47The last term of Eq. (B7) represents the renormalization of spin Berry's phase term, i.e.,\nthe e\u000bectS!S\u0000byb, which we neglect below. The Lagrangian for magnon thus reads\nLm= 2S\r2Z\nd3ri[by(@t+iAz\ns;t)b\u0000by( \n@t\u0000iAz\ns;t))b]; (B8)\nnamely, magnons interacts with the adiabatic component of spin gauge \feld, Az\ns;t.\nAppendix C: Decomposition of contour-ordered self energy\nHere we summarize decomposition formula of self energy. 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Tak-\ning axial symmetry into account, we study the existence of domain w all\ntype coherent structure solutions, with focus on one space dimen sion and\nspin-polarization, but our results also apply to vanishing spin-torqu e term.\nUsing methods from bifurcation theory for arbitrary constant ap plied fields,\nwe prove the existence of domain walls with non-trivial azimuthal pro file,\nreferred to as inhomogeneous . We present an apparently new type of do-\nmain wall, referred to as non-flat, whose approach of the axial magnetiza-\ntion has a certain oscillatory character. Additionally, we present th e leading\norder mechanism for the parameter selection of flatandnon-flat inhomoge-\nneous domain walls for an applied field below a threshold, which depends on\nanisotropy, damping, and spin-transfer. Moreover, numerical c ontinuation\nresults of all these domain wall solutions are presented.\n1 Introduction\nMagnetic domain walls (DWs) are of great interest both from a theor etical perspective\nand for applications, especially in the context of innovative magnetic storages [1]. Re-\ncent developments in controlled movement of DWs via spin-polarized c urrent pulses in\nnanomagnetic structures, in particular in nanowires, are thought to lead to a new class\nof potential non-volatile storage memories, e.g. racetrack memor y [1, 2, 3, 4]. These\ndevices make use of the fact that spin-transfer driven effects ca n change the dynamics\nin sufficiently small ferromagnetic structures (e.g. nanowires), wh ere regions of uniform\n∗Universit¨ at Bremen, lars.siemer@uni-bremen.de ; Corresponding author\n†Universit¨ at Hamburg, Lobachevsky State University of Nizhny No vgorod\n‡Universit¨ at Bremen\n1magnetization, separated by DWs, can appear [5, 6, 7]. This motivat es further studies of\nthe existence of magnetic domains and their interaction with spin-po larized currents as\na building block for the theory in this context. In this paper we take a mathematical per-\nspective and, in a model for nanomagnetic wires, rigorously study t he existence of DWs.\nThis led us to discover an apparently new kind of DWs with a certain inho mogeneous\nand oscillatory structure as explained in more detail below.\nThe description of magnetization dynamics in nanomagnetic structu res, governed by the\nLandau-Lifschitz-Gilbert (LLG) equation, is based on works by Berger and Slonczewski\nassuming a spin-polarized current [8, 9]. In the presence of a const ant applied field and\na spin-polarized current, the dynamics driven by the joint action of magnetic field and\nspin torque can be studied by adding a spin-transfer term in the dire ction of the current\n(current-perpendicular-to-plane (CPP) configuration). In cas e of a spatially uniform\nmagnetization, the resulting Landau-Lifschitz-Gilbert-Slonczewski (LLGS) equation for\nunit vector fields ( m1,m2,m3) =m=m(x,t)∈S2(cf. Figure 1) reads\n∂tm−αm×∂tm=−m×heff+m×(m×J). (LLGS)\nwith effective field heff, Gilbert damping factor α>0, and the last term is the so-called\npolarized spin transfer pseudotorque.\nNote that the above equation reduces to the LLG equation for J≡0, see§2 for more\ndetails.\nIn this paper we consider the axially symmetric case and set\nheff:=∂2\nxm+h−µm3e3,J:=β\n1+ccpm3e3, (1)\nwhereh=he3with a uniform and time-independent field strength h∈R, and m 3=\n/an}bracketle{tm,e3/an}bracketri}ht,e3∈S2. This effective field heffalso includes the diffusive exchange term ∂2\nxm,\nthe uniaxial anisotropy and demagnetization field. The specific here with parameter\nµ∈Rderives from a first order approximation in the thin film/wire limit for a u niformly\nmagnetized body [6, 10]. In the axially symmetric structure, β≥0 andccp∈(−1,1)\ndescribe the strength of the spin-transfer and the ratio of the p olarization [7, 11]. The\nspin-transfer torque term may provide energy to the system und er certain conditions\nand counterbalance dissipation associated to the Gilbert damping te rm, which gives rise\nto coherent non-variational dynamics, see e.g. [12].\nNotably, for β= 0 one obtains the LLG-equation that does not account for spin tr ansfer\neffects. Moreover, as shown in [12], this also holds up to parameter change in case\nccp= 0. Hence, solutions to the LLGS equation for β= 0 orccp= 0 are also solutions\nto the LLG equation, so that all the analytical as well as numerical r esults forccp= 0\nin this paper directly transfer to the LLG equation.\nA key ingredient for the separation of uniformly magnetized states in space are interfaces\nbetweentwomagneticdomains. Themostcoherentformofsuchint erfacesintheuniaxial\nsetting are relative equilibria with respect to translation and rotatio n symmetry of the\nform\nm(ξ,t) =m0(ξ)eiϕ(ξ,t),whereξ=x−standϕ(ξ,t):=φ(ξ)+Ωt,\n2(a)\n (b)-5 5-101\nxm1,m2\n(c)\nFigure 1: Homogeneous DW profile ( q≡0) withα= 0.5,β= 0.1,µ=−1,h= 50,ccp= 0.\n(a) (m2,m3)-profile. (b) Projection onto S2. (c) Zoom-in on m1(blue solid) and m2\n(red dashed).\nwithspeedsandfrequency Ω. Here the complex exponential acts on m0∈S2by\nrotation about the e3-axis, i.e., the azimuth, and in spherical coordinates we can choose\nm0(ξ) = (sin(θ(ξ)),0,cos(θ(ξ))) with altitude angle θ.\nWe refer to such solutions with m0(ξ)→ ±e3asξ→ ±∞orξ→ ∓∞asdomain walls .\nA first classification of DWs is based on the local wavenumber q:=φ′, which determines\nφuniquely due to the axial rotation symmetry and satisfies\nq(ξ) =/an}bracketle{t(m′\n1,m′\n2),(−m2,m1)/an}bracketri}ht\n1−m2\n3(ξ). (2)\nDefinition 1. We call a DW with constant φhomogeneous (hom) , i.e.,q≡0, and\ninhomogeneous otherwise.\nInhomogeneous DWs have a spatially inhomogeneous varying azimuth al angle, compare\nFigures 1 and 2.\nIn the case of uniaxial symmetry and the LLG case β= 0, an explicit family of homo-\ngeneous DWs was discovered in [13] for applied fields with arbitrary st rength and time\ndependence, cf. Figure 1. Furthermore, for constant applied fie lds and in case of ccp/ne}ationslash= 0\nit was shown in [12] that DWs cannot be homogeneous, and the existe nce of inhomoge-\nneous DWs was proven, whose spatial profile slowly converges to ±e3and where |s| ≫1.\nThis latter type of DWs is ‘weakly localized’ and has large ‘width’ in the se nse that the\ninverse slope of m3atm3(0) = 0 tends to infinity as |s| → ∞.\nAn apparently thus far unrecognized distinction of DWs is based on t he convergence\nbehavior of qasξ→ ±∞.\nDefinition 2. We call a DW flatif|q(ξ)|has a limit on R∪ {∞}as|ξ| → ∞and\nnon-flatotherwise.\nNote that homogeneous DWs are flat ones by definition (recall φ′=q). Moreover, for\nall DWsm0(ξ) converges to e3or−e3as|ξ| → ∞.\n3small applied field\n(a)large applied field\n(b)\n-5 5-101\nξm1\n(c)-5 5-101\nξm1,m2\n(d)\nFigure 2: Shown are profiles of inhomogeneous DWs m(ξ) computed by parameter continua-\ntion, cf.§4, inccptoccp= 0.5 with fixed α= 0.5,β= 0.1,µ=−1. (a,c) codim2case\nh= 0.5,s= 0.112027,Ω = 0.447173, (b,d) codim 0 case h= 50,s= 19.92,Ω = 40.4.\n(c) magnification of the m1-component; note the change of frequency for small vs.\nlargeξ. (d) Magnification of m1(blue solid) as well as m2(red dashed) component.\nThe main result of this paper is an essentially complete understanding of the existence\nand the type of DWs near the aforementioned explicit solution family f or a nanowire\ngeometry, i.e., µ <0. This includes the LLG case β·ccp= 0, but our focus is on the\nspintronic case β·ccp/ne}ationslash= 0 for which these results pertain 0 <|ccp| ≪1 and any value of\nthe (constant) applied field h.\nThe different types of DWs occur in parameter regimes close to ccp= 0 in the (spatial)\nODE which results from the coherent structure ansatz. Since the parameters αandµ\nare material-dependent we take the applied field strength has the primary parameter.\nIn brief, organized by stability properties of the steady states ±e3in the ODE, this leads\nto the following cases and existence results for localized DWs in nanow ires (µ<0):\n•‘codim-2’ (h∗<h<h∗) : existence of flat inhomogeneous DWs with s,Ω selected\nby the other parameters,\n•‘center’ (h=h∗orh=h∗) : existence of flat and non-flat inhomogeneous DWs,\n•‘codim-0’ (h<h∗orh>h∗) : existence of flat inhomogeneous DWs,\nwhereh∗:=β/α+2µ\nα2(1+α2) as well as h∗:=β/α−2µ\nα2(1+α2). Note that h∗<h∗always\n(recallα>0 andµ<0). Due to symmetry reasons, we mainly discuss the existence of\n4Figure 3: Stability diagram of homogeneous states ±e3inhandccpforα= 1,β= 0.5, and\nµ=−1. State + e3unstable to the left and stable to the right of Γ+,−e3stable\nto the left and unstable to the right of Γ−. Homogeneous DWs (hom) exist only on\ntheh-axis, i.e., ccp≡0. See text for further explanations. Note that also negativ e\napplied fields are shown.\nright-moving DWs close to the explicit solution family and thus focus on an applied field\nβ/α≤h(cf.§3). The main results can be directly transferred to the case of left -moving\nDWs (h≤β/α). Notably, the codim-0 case occurs for ‘large’ magnetic field habove a\nmaterial dependent threshold. In the center and codim-2 cases t here is a selection of s\nand Ω by the existence problem.\nThebasicrelationbetween thePDEandtheODEstabilitypropertiesw .r.t.handccpare\nillustrated in Figure 3 for α= 1,β= 0.5,µ=−1 fixed. Due to the fact that sand Ω are\nODE parameters only, the diagram illustrates a slice in the four dimens ional parameter\nspace with axes h,ccp,s, and Ω. Note that homogeneous DWs (hom) can occur only\non the line ccp≡0 (see [12, Theorem 5] for details). The stability regions are defined\nas follows. monostable−(blue): + e3unstable and −e3stable,bistable(shaded blue):\nboth +e3and−e3stable,monostable+(red): +e3stable and −e3unstable, unstable\n(shaded red): both + e3and−e3unstable. For a more detailed stability discussion, see\nRemark 5. Note that the transition from bistable to monostable in th e PDE does not\ncoincide with the transition, of the homogeneous family, from codim- 2 to codim-0 in\nthe ODE. In contrast, the analogous transitions occur simultaneo usly for example in the\nwell-known Allen-Cahn orNagumo equation.\nIn Figure 4 below, we present numerical evidence that inhomogeneo us DWs are indeed\nalso dynamically selected states, especially for large applied fields, als o in the LLG case\n(β,ccp= 0).\nThe understanding of DW selection by stability properties generally d epends on the exis-\ntence problem discussed in this paper, which is therefore a prerequ isite for the dynamical\nselection problem, cf. Remark 5.\n5(a)\n (b)\n(c)\nFigure 4: Direct simulation of full PDE (LLGS) for α= 0.5,β= 0.1,µ=−1,h= 50, and\nccp= 0 with dynamical selection of an inhomogeneous DW. Initial condition near\nhomogeneous DW (9) in codim-0 regime ( h∗= 10.2,s0= 19.92, and Ω 0= 40.04).\n(a) Profile at t= 20 projected onto the sphere. (b) Speed and frequency of DW\nover time with asymptotic (selected) values s= 12.5 and Ω = 78 .28. (c) Space-time\nplots of DW components (without co-moving frame), range as i n black box in (b).\nFinal profile is heteroclinic connection in (7), cf. Proposi tion 2.\nTo our knowledge, existence results of DWs for ccp/ne}ationslash= 0 are new. In more detail, the\nexistence of localised inhomogeneous, i.e. flat as well as non-flat, DW s forccp/ne}ationslash= 0 and\nespecially for ccp= 0 are new results. Indeed, the existence proof of non-flat DWs is\nthe most technical result and entails an existence proof of hetero clinic orbits in an ODE\nbetween an equilibrium and a periodic orbit. These solutions indicate th e presence of\nDWs in other regimes of spin driven phenomena and may be of interest for spin-torque\ntransfer MRAM (Magnetoresistive random-access memory) syst ems [14].\nThis paper is organized as follows. In §2, the LLGS equation and coherent structures as\nwell as first properties are discussed. Section 3 more precisely intr oduces homogeneous\nand inhomogeneous as well as flat and non-flat DWs and it also includes the main\nresults of this paper (Theorem 1, 2, and 3). The technical proofs of Theorem 2 as\nwell as Theorem 3 are deferred to Appendix 6.1 and 6.2. Section 4 pre sents results of\n6numerical continuation in parameter ccpfor the three regimes of the applied field (codim-\n2, center, and codim-0), where the center case is studied in more d etail. We conclude\nwith discussion and outlook in §5.\nAcknowledgements\nL.S. and J.R. acknowledge support by the Deutsche Forschungsge meinschaft (DFG, Ger-\nmanResearchFoundation)-Projektnummer 281474342/GRK222 4/1. J.R.alsoacknowl-\nedges support by DFG grant Ra 2788/1-1. I.O. acknowledges fund ing of a previous\nposition by Uni Bremen, where most of this paper was written, as we ll as support by\nthe recent Russian Scientific Foundation grant 19-11-00280.\n2 Model equations and coherent structure form\nThe classical model for magnetization dynamics was proposed by La ndau and Lifschitz\nbased on gyromagnetic precession, and later modified by Gilbert [15, 16]. See [17] for\nan overview. The Landau-Lifschitz-Gilbert equation for unit vector fields m(x,t)∈S2\nin one space dimension x∈Rand in terms of normalized time in dimensionless form is\n∂tm−αm×∂tm=−m×heff. (LLG)\nHerem=M/MSrepresents thenormalizedmagnetization, heff=Heff/MStheeffective\nfield, i.e.thenegativevariationalderivativeofthetotalmagneticf reeenergywithrespect\ntom, both normalized by the spontaneous magnetization MS. For gyromagnetic ratio\nγand saturation magnetization MSthe time is measured in units of ( γMS)−1, and it\nis assumed that the temperature of the magnetic body is constant and below the Curie\ntemperature [5]. Finally, Gilbertdampingα>0 turnsmtowardsheffand both vectors\nare parallel in the static solution.\nIn modern spin-tronic applications, e.g. Spin-Transfer Torque Mag netoresistive Random\nAccess Memories (MRAM), the spin of electrons is flipped using a spin- polarized current.\nTo take these effects into account, the LLG equation is supplement ed by an additional\nspin transfer torque term. Using a semiclassical approach, Sloncz ewski derived an ex-\ntended effective field\nHeff=heff−m×J,\nwhereJ=J(m) depends on the magnetization and the second term is usually called\nSlonczewski term [9]. In contrast to the LLG equation, which can be written as the\ngradient of free ferromagnetic energy, this generalized form is no longer variational and\nthe energy is no longer a Lyapunov functional.\nAs to the specific form of Heff, including a leading order form of exchange interaction,\nuniaxial crystal anisotropy in direction e3, andZeemanas well as stray-field interactions\nwith an external magnetic field, see e.g. [6], gives the well known form (1).\nIn this paper we consider a constant applied magnetic field h∈Ralonge3and uniaxial\nanisotropy with parameter µ∈R, for which the anisotropy energy density is rotationally\n7symmetric w.r.t. e3. According to the energetically preferred direction in the uniaxial\ncase, minima of the anisotropy energy density correspond to easydirections, whereas\nsaddles or maxima correspond to medium-hard orharddirections, respectively. There-\nfore, one refers to µ <0 aseasy-axis anisotropy and µ >0 aseasy-plane , both with\nregard to e3.\nAs mentioned before, the LLG equation with its variational structu re appears as a\nspecial case of (LLGS) for β= 0 orccp= 0. While our main focus is the non-variational\nspintronic case β·ccp/ne}ationslash= 0, all results contain the case β·ccp= 0 and thus carry over\nto (LLG).\nIt iswell-known that(LLGS) admitsanequivalent formasanexplicit ev olution equation\nof quasilinear parabolic type in the form, see e.g. [12],\n∂tm=∂x(A(m)∂xm)+B(m,∂xm).\nAs a starting point, we briefly note the existence of spatially homoge neous equilibrium\nsolutions of (LLGS) for which m(x,t) is constant in xandt.\nRemark 1. The only (spatially)homogeneous equilibria of (LLGS)forβ >0are the\nconstant up- and down magnetization states ±e3. Indeed, setting ∂tm=∂2\nxm= 0in\n(LLGS), forβ/ne}ationslash= 0the last equation implies that m1=m2= 0and thus the only solutions\nm∗\n±∈S2arem∗\n±= (0,0,±1)T.\nRemark 2. In caseβ= 0as well as |h/µ|<1there exist a family of additional\nhomogeneous solutions of (LLGS) given by m∗= (m1,m2,h/µ)T,withm2\n1+m2\n2=\n1−(h/µ)2. Note that similar cases occur for symmetry axis being e1ande2, respectively\n(cf. Brown’s equations ).\n2.1 Coherent structure ODE\nDuetotherotationsymmetryaroundthe e3-axisof (LLGS), itisnaturaltousespherical\ncoordinates\nm=\ncos(ϕ)sin(θ)\nsin(ϕ)sin(θ)\ncos(θ)\n,\nwhereϕ=ϕ(x,t) andθ=θ(x,t). This changes (LLGS) to\n/parenleftbigg\nα−1\n1α/parenrightbigg/parenleftbigg\n∂tϕsin(θ)\n−∂tθ/parenrightbigg\n=/parenleftbigg\n2∂xϕ∂xθcos(θ)\n−∂2\nxθ/parenrightbigg\n+sin(θ)/parenleftbigg\n∂2\nxϕ+β/(1+ccpcos(θ))\n(∂xϕ)2cos(θ)+h−µcos(θ)/parenrightbigg (3)\nNote that the rotation symmetry has turned into the shift symmet ry in the azimutal\nangleϕ, as (3) depends on derivatives of ϕonly.\n8Recall that DW solutions spatially connect the up and down magnetiza tion states ±e3\nin a coherent way as relative equilibria with respect to the translation symmetry in x\nandφ, which yields the ansatz\nξ:=x−st, θ=θ(ξ), ϕ=φ(ξ)+Ωt. (4)\nSuch solutions are generalized travelling waves that move with const ant speeds∈R\nin space and rotate pointwise with a constant frequency Ω ∈Raround the e3-axis;\nsolutions with Ω = 0 are classical travelling waves.\nAs in [12], applying ansatz (4) to (3) leads to the so-called coherent structure ODE\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbigg(Ω−sφ′)sin(θ)\nsθ′/parenrightbigg\n=/parenleftbigg2φ′θ′cos(θ)\n−θ′′/parenrightbigg\n+sin(θ)/parenleftbiggφ′′+β/(1+ccpcos(θ))\n(φ′)2cos(θ)+h−µcos(θ)/parenrightbigg\n,(5)\nwhere′=d/dξ. This system of two second-order ODEs does not depend on φand thus\nreduces to three dynamical variables ( θ,ψ=θ′,q=φ′). Following standard terminology\nfor coherent structures, we refer to qas thelocal wavenumber .\nWriting (5) as a first-order three-dimensional system gives\nθ′=ψ\nψ′= sin(θ)[h−Ω+sq+(q2−µ)cos(θ)]−αsψ\nq′=αΩ−β/(1+ccpcos(θ))−αsq−s+2qcos(θ)\nsin(θ)ψ, (6)\nand DWs in the original PDE are in 1-to-1-correspondence with the O DE solutions\nconnecting θ= 0 andθ=π.\n2.1.1 Blow-up charts and asymptotic states\nAs in [12], the singularities at zeros of sin( θ) in (6) can be removed by the singular\ncoordinate change ψ:=psin(θ), which is a blow-up transformation mapping the poles\nof the sphere ±e3to circles thus creating a cylinder. The resulting desingularized system\nreads\nθ′= sin(θ)p\np′=h−Ω−αsp+sq−(p2−q2+µ)cos(θ)\nq′=αΩ−β/(1+ccpcos(θ))−sp−αsq−2pqcos(θ).(7)\nThe coherent structure system (6) is equivalent to the desingular ized system (7) for\nθ/ne}ationslash=nπ,n∈Zand therefore also for maway from ±e3. Furthermore, the planar blow-\nup chartsθ= 0 andθ=πare invariant sets of (7), which are mapped to the single\npointse3and−e3, respectively by the blow-down transformation. System (7) has a\nspecial structure (cf. Figure 6) that will be relevant for the subs equent DW analysis. In\nthe remainder of this section we analyze this in some detail.\n9Lemma 1. Consider the equations for pandqin(7)for an artificially fixed value of θ.\nIn terms of z:=p+iqthis subsystem can be written as the complex (scalar) ODE\nz′=Az2+Bz+C, (8)\nwhereA:=−cos(θ),B:=−(α+i)s,andCθ:=h−Ω+Aµ+i/parenleftig\nαΩ−β\n1−Accp/parenrightig\n.\nForA/ne}ationslash= 0the solution with z0=z(ξ0)away from the equilibria zθ\n+=−B\n2A+ iγθ\n2Aand\nzθ\n−=−B\n2A−iγθ\n2A, withγθ=γ(θ):=√\n4ACθ−B2, reads\nz(ξ) =γθ\n2Atan/parenleftbiggγθ\n2ξ+δ0/parenrightbigg\n−B\n2A, (9)\nwhere\nδ0= arctan/parenleftbigg2Az0+B\nγθ/parenrightbigg\n−γθ\n2ξ0.\nForA= 0, the solution away from the equilibrium zπ/2=−Cπ/2/Bis given by\nz(ξ) =/parenleftbigg\nz0+Cπ/2\nB/parenrightbigg\neB(ξ−ξ0)−Cπ/2\nB.\nClearly, the solution of (8) relates only to those solutions of (7) for whichθis constant,\ni.e.,θ= 0,π. Although we are mostly interested in the dynamics on the blow-up ch arts,\nwe consider θas a parameter in order to demonstrate the special behaviour of ( 7) forθ\nartificially fixed. Notably, the equilibria zθ\n±of (8) forθ/ne}ationslash= 0,πare not equilibria in the\nfull dynamics, due to the fact that (7) is only invariant for θat the blow-up charts.\nProof.We readily verify the claimed form of the ODE and directly check the cla imed\nsolutions. /squaresolid\nRemark 3. Lemma 1 states in particular that the desingularized ODE sys tem(7)can\nbe solved explicitly on the invariant blow-up charts, where θ= 0,πand thusA=−1,1,\nrespectively. System (7)possesses two real equilibria on each blow-up chart, Z0\n±:=\n(0,p0\n±,q0\n±)TandZπ\n±:= (π,pπ\n±,qπ\n±)T. Herepθ\nσ:= Re(zθ\nσ),qθ\nσ:= Im(zθ\nσ)forθ= 0,π,\nσ=±and\nz0\n+:= 1/2(B−iγ0), z0\n−:= 1/2(B+iγ0)\nand analogously\nzπ\n+:= 1/2(−B+iγπ), zπ\n−:= 1/2(−B−iγπ),\nwhere we set\nγ0:=γ/vextendsingle/vextendsingle\nA=−1=√\n−4C0−B2andγπ:=γ/vextendsingle/vextendsingle\nA=1=√\n4Cπ−B2\nwithC0:=C/vextendsingle/vextendsingle\nA=−1andCπ:=C/vextendsingle/vextendsingle\nA=1.\nDue to the analytic solution (9), we obtain the following more detailed r esult in case\nθ/ne}ationslash=π/2 (cf. Figure 6).\n10Lemma 2. For each given 0≤θ≤πwithθ/ne}ationslash=π/2as a parameter, the fibers of (7)with\nconstantθconsists entirely of heteroclinic orbits between zθ\n−andzθ\n+in caseIm(γθ)/ne}ationslash= 0,\norγθ/ne}ationslash= 0, except for the equilibrium states. In case Im(γθ) = 0andRe(γθ)/ne}ationslash= 0, the\nfiber at fixed θis filled with periodic orbits away from the invariant line {q=s\n2A}, for\nwhich the period of solutions close to it tends to infinity.\nProof.Forθfixed in (8), consider the case Re( γθ) = 0 and also Im( γθ)/ne}ationslash= 0 forA/ne}ationslash= 0\nwhich leads to\nz(ξ) = iIm(γθ)\n2A·tan/parenleftbigg\ni/parenleftigIm(γθ)\n2ξ+Im(δ0)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:ˇξ/parenrightig\n+Re(δ0)/parenrightbigg\n−B\n2A\n=Im(γθ)\n2A·isin(2Re(δ0))−sinh/parenleftbig\n2ˇξ/parenrightbig\ncos(2Re(δ0))+cosh/parenleftbig\n2ˇξ/parenrightbig−B\n2A.\nFor Re(γθ)/ne}ationslash= 0 as well as Im( γθ)/ne}ationslash= 0, we obtain\nz(ξ) =γθ\n2Atan/parenleftbiggRe(γθ)\n2ξ+Re(δ0)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:˜ξ+iIm(γθ)\n2ξ+iIm(δ0)/parenrightbigg\n−B\n2A\n=γθ\n2A·sin(2˜ξ)+isinh/parenleftig\n2/parenleftig\nIm(γθ)\nRe(γθ)˜ξ−Im(γθ)\nRe(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig\ncos(2˜ξ)+cosh/parenleftig\n2/parenleftig\nIm(γθ)\nRe(γθ)˜ξ−Im(γθ)\nRe(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig−B\n2A,\nThe asymptotic states are\nIm/parenleftbig\nγθ/parenrightbig\n>0 : lim\nξ→−∞z(ξ) =−iγθ\n2A−B\n2A,lim\nξ→+∞z(ξ) = iγθ\n2A−B\n2A,\nas well as\nIm/parenleftbig\nγθ/parenrightbig\n<0 : lim\nξ→−∞z(ξ) = iγθ\n2A−B\n2A,lim\nξ→+∞z(ξ) =−iγθ\n2A−B\n2A,\nwhich simplify in case Re( γθ) = 0 to\nIm/parenleftbig\nγθ/parenrightbig\n>0 : lim\nξ→−∞z(ξ) =Im(γθ)−B\n2A,lim\nξ→+∞z(ξ) =−Im(γθ)+B\n2A,\nas well as\nIm/parenleftbig\nγθ/parenrightbig\n<0 : lim\nξ→−∞z(ξ) =−Im(γθ)+B\n2A,lim\nξ→+∞z(ξ) =Im(γθ)−B\n2A.\nNote that the asymptotic states coincide if γθ= 0.\n11The last case to consider is Re/parenleftbig\nγθ/parenrightbig\n/ne}ationslash= 0 and Im/parenleftbig\nγθ/parenrightbig\n= 0, where the solutions are\nz(ξ) =Re/parenleftbig\nγθ/parenrightbig\n2A·sin(2ˆξ)+isinh(2Im( δ0))\ncos(2ˆξ)+cosh(2Im( δ0))−B\n2A,\nwithˆξ:=Re(γθ)\n2ξ+Re(δ0) and which leads to periodic solutions of (8) iff\nIm(δ0)/ne}ationslash= 0⇔2AIm(z0)+Im(B)/ne}ationslash= 0⇔q0/ne}ationslash=s\n2A,\nwherez0=p0+iq0and recall that B=−(α+i)s. /squaresolid\nBased on Lemma 2, explicitly onthe blow-up chart θ= 0 the heteroclinic orbits are from\nz0\n−toz0\n+in case Im( −4C0−B2)>0, or Im( −4C0−B2) = 0 and Re( −4C0−B2)≤0,\nand fromz0\n+toz0\n−if Im(−4C0−B2)<0 . Forθ=π, if Im(4Cπ−B2)>0, or\nIm(4Cπ−B2) = 0 and Re(4 Cπ−B2)≤0 they are connections from zπ\n−tozπ\n+, and if\nIm(4Cπ−B2)<0 fromzπ\n+tozπ\n−.\nForA/ne}ationslash= 0, the case s= 0 is a special situation, which will be also discussed in the\ncontext of DWs in §3 later. It turns out that on the blow-up charts θ= 0 (orθ=π),\nthe solution with appropriate initial conditions has a limit as |ξ| → ∞if and only if\nIm(√\n−C0)/ne}ationslash= 0 (Im(√\nCπ)/ne}ationslash= 0). In terms of the parameters in (7) and with\nβ−:=β\n1−ccpandβ+:=β\n1+ccp,\nthis leads to the conditions for θ= 0 given by:\nΩ/ne}ationslash=β+\nαor Ω =β+\nαand Ω≤h−µ, (10)\nand forθ=πgiven by:\nΩ/ne}ationslash=β−\nαor Ω =β−\nαand Ω≥h+µ, (11)\nIn caseccp= 0, i.e. for the LLG equation, the conditions in (10) and (11) reduce to\nΩ/ne}ationslash=β\nαor Ω =β\nαand 2µ≤β\nα,\nwhere the latter inequality always holds in case of a nanowire geometr y (µ<0). Hence\nstanding domain walls in nanowires in case ccp= 0 can only connect equilibria, if they\nexist.\nLemma 2 also states that the equilibria on the blow-up charts θ∈ {0,π}are surrounded\nby periodic orbits in case Im( γ0) = 0 and Re( γ0)/ne}ationslash= 0 (Im(γπ) = 0 and Re( γπ)/ne}ationslash= 0).\nIn fact, system (7) is Hamiltonian (up to rescaling) on the blow-up ch arts for certain\nfrequencies Ω, as follows\n122p\n-11q\n(a)-5 5ξ\n-1\n2\np \u0000q\n(b)\nFigure 5: (a) Phase plane streamplot with Mathematica of (14) around the equilibrium zπ\n−,\ni.e., (7) at θ=π, forα= 0.5,β= 0.1,µ=−1,h= 10.2,s= 4,Ω = 8.2 andccp= 0,\nwhich leads to/parenleftbig\npπ\n−,qπ\n−/parenrightbigT= (1,0)T. The red solid line marks the trajectory with\ninitial condition ( p0,q0) = (7/4,0) (cf. plot of solutions in b). (b) Plot of the profile\nfor the solution highlighted in (a), where p(solid blue line) and q(dashed red line)\nare given by (9) for the parameter set as in (a).\nProposition 1. The dynamics of (7)on the invariant blow-up chart θ= 0in case\nΩ =β+\nα−s2\n2possesses the invariant line {q=−s\n2}and, after time-rescaling, for q/ne}ationslash=−s\n2\nthe Hamiltonian\nH0(p,q) =−p2+q2+αsp+sq−h+β+/α+µ\nq+s\n2\nalong solutions of (8). Analogously on the chart θ=π, in case\nΩ =β−\nα+s2\n2(12)\npossesses the invariant line {q=s\n2}and forq/ne}ationslash=s/2the Hamiltonian\nHπ(p,q) =p2+q2−αsp−sq+h−β−/α+µ\nq−s\n2. (13)\nMoreover, each half plane {θ= 0,q≤ −s\n2},{θ= 0,q≥ −s\n2}/parenleftbig\n{θ=π,q≤s\n2}and\n{θ=π,q≥s\n2}/parenrightbig\nis filled with periodic orbits encircling the equilibria at z0\n±/parenleftbig\nzπ\n±/parenrightbig\nif addi-\ntionallyΩ>h−µ+s2\n4(α2−1)/parenleftig\nΩ<h+µ+s2\n4(1+α2)/parenrightig\n.\nProof.For the sake of clarity, we will only present the computation for the blow-up chart\nθ=π; the computation on θ= 0 can be done in the same manner. With respect to the\nparameters of equation (7), the condition Im(4 C−B2) = 0 is equivalent to Ω =β−\nα+s2\n2\nand in this case the system (7) on {θ=π}is given by\np′=p2−q2−αsp+sq+h+µ−β−\nα−s2\n2\nq′= 2pq−sp−αsq+αs2\n2(14)\n13We readily compute that for solutions of this\ndHπ\ndξ=∂Hπ\n∂pp′+∂Hπ\n∂qq′=q′·p′−p′·q′\n(q−s\n2)2= 0,\nwhich shows the canonical Hamiltonian structure of (14) up to time r escaling. If addi-\ntionally Ω< h+µ+s2\n4(1+α2), it follows from Lemma 2 that each half plane is filled\nwith periodic orbits. /squaresolid\nProposition 1 concerns the special case that ccp∈(−1,1) andβ,Ω are such that (12)\nholds, whichishenceforthreferredtoasthe center-case . Inparticular, eachorbitexcept\nthe lineq≡s/2 on the blow-up chart θ=πcan by identified via the quantity (13),\nand each equilibrium zπ\n±has a neighborhood filled with periodic orbits if additionally\nh >β−\nα−µ+s2\n4(1−α2) (cf. Figure 5). Note the relation between the conditions (10)\nand (11) and the conditions in Proposition 1 in case s= 0.\nBased on Lemma 2, we also state the following uniqueness result.\nProposition 2. ForΩ<β+\nα−s2\n2/bracketleftig\nΩ>β+\nα−s2\n2/bracketrightig\n, orΩ =β+\nα−s2\n2andΩ≤h−µ+\ns2\n4(α2−1)there is a unique orbit with (θ,p,q)T(ξ)withθ(ξ)→0asξ→ −∞, and it\nholds that (p+iq)(ξ)→z0\n−/bracketleftbig\n(p+iq)(ξ)→z0\n+/bracketrightbig\nasξ→ −∞.\nProof.The conditions on Ω are equivalent to those in Lemma 2. If the statem ent were\nfalse, it nevertheless follows from Lemma 2 that ( p+iq)(ξ)→z0\n−asξ→ −∞. However,\ntransverse to the blow-up chart, the equilibrium state Z0\n−is stable for increasing ξand\nthus repelling for decreasing ξ. This contradicts the requirement θ(ξ)→0 asξ→ −∞.\nTogether with the fact that Z0\n−has a one-dimensional unstable manifold uniqueness\nfollows. Analogously in case Ω >β+\nα−s2\n2. /squaresolid\nDomain walls are heteroclinic orbits between the blow-up charts and d ecisive for their\nbifurcation structure are the dimensions (and directions) of un/s table manifolds of the\nequilibria on these charts. Hence, we next discuss the equilibria Z0\n±andZπ\n±and their\nstability.\nTransverse to the blow-up charts in θ-direction we readily compute the linearization\n∂θ(sin(θ)p) = cos(θ)p, i.e., the transverse eigenvalue is −Aθ·Re(zθ\n±) atθ= 0 andπ,\nrespectively. The eigenvalues within the blow-up charts are determ ined by±iγ. With\nσ=±, respectively, the eigenvalues for Z0\nσare\nν0\n1,σ=−σiγ0, ν0\n2,σ=ν0\n1,σ, ν0\n3,σ= Re(z0\nσ) (15)\nand forZπ\nσ\nνπ\n1,σ=σiγπ, νπ\n2,σ=νπ\n1,σ, νπ\n3,σ=−Re(zπ\nσ). (16)\nTherefore, the signs of the real parts within each blow-up chart a re opposite at Zπ\n+\ncompared to Zπ\n−and determined by the sign of Re/parenleftbig\nν0,π\n1,+/parenrightbig\n. Hence, within the blow-up\ncharts each equilibrium is either two-dimensionally stable, unstable or a linearly neutral\ncenter point.\n142p\n-11q\n(a)2p\n-11q\n(b)2p\n-11q\n(c)\n2p\n-11q\n(d)2p\n-11q\n(e)2p\n-11q\n(f)\nFigure 6: Phase plane streamplots (with Mathematica ) in blow-up charts near the equilib-\nriumzπ\n−= (1,0) forα= 0.5,β= 0.1,µ=−1,ccp= 0, i.e., the second and third\nequation of (7). (a-c) θ= 0 and (d-f) θ=π. (a,d) codim-2 regime, (b,e) center\ncase, where Ω = β/α+s2/2 holds on the chart θ=π, and (c,f) codim-0 regime.\nThe remaining parameters and equilibria in (a,d): h= 0.5,s0= 0.12, Ω0= 0.44,\nandz0\n+=−1.06−0.12i,zπ\n+=−0.94+0.12i. In (b,e): h= 10.2,s0= 4, Ω 0= 8.2,\nandz0\n+=−3−4i,zπ\n+= 1 + 4i. In (c,f): h= 50.0,s0= 19.92, Ω0= 40.04, and\nz0\n+=−10.96−19.92i,zπ\n+= 8.96+19.92i.\nFor completeness, we next notethat the equilibria onbothblow-up c harts can beneutral\ncenters simultaneously (cf. Figure 3). However, this requires a ne gative spin polarization\nand a small Gilbert damping factor, and is not further studied in this p aper.\nRemark 4. The equilibria of both blow-up charts are centers simultane ously, if and only\nifIm(±γ0,π) = 0andγ0,π/ne}ationslash= 0(compare Lemma 2). For example if α= 0.5,β= 0.1,µ=\n−1,ccp=−0.99,h= 10\ns2=3960\n199,Ω =β/α\n1−ccp+s2\n2=2000\n199,\nwe obtain\nγ0= 3.33551, γπ= 3.27469.\n3 Domain Walls\nAll domain walls between ±e3that we are aware of are of coherent structure type,\nand thus in one-to-one correspondence to heteroclinic connectio ns between the blow-up\n15charts{θ= 0}and{θ=π}in(7). Typically we expect these to beheteroclinics between\nequilibria within the charts, but this is not necessary. Based on the p revious analysis,\nthere are three options for heteroclinics between the charts: po int-to-point, point-to-\ncycle, and cycle-to-cycle. We study the first two in this section, fo r which Proposition 2\nimplies uniqueness oftheDW(uptotranslations/rotations) foragiv enset ofparameters.\nThe third case can occur at most in a relatively small set of paramete rs (see Remark 4).\nIts analysis is beyond the scope of this paper.\nNote that in case of an existing connection between an equilibrium and a periodic orbit\n(see Proposition 1), the domain wall is automatically an inhomogeneou s non-flat one.\nMoreover, via the singular coordinate change any such heteroclinic solution is hetero-\nclinic between θ= 0,πin (6) and through the spherical coordinates it is a heteroclinic\nconnection between ±e3in the sphere, possibly with unbounded ϕ.\n3.1 Homogeneous Domain Walls\nIt is known from [13] in case β= 0 and from [12] in case ccp= 0 (and arbitrary β)\nthat (7) admits for µ<0 a family of explicit homogeneous DWs m0given by\n\nθ0\np0\nq0\n=\n2arctan/parenleftbig\neσ√−µξ/parenrightbig\nσ√−µ\n0\n (17)\nand parameterized by Ω =h+αβ\n1+α2,s2=−(β−αh)2\nµ(1+α2)2>0, andσ= 1 for positive speed s\nandσ=−1 for negative s; the family extends to s= 0 in the limit h→β\nαwith scaling\nof the frequency by Ω =β\nα+√−µ\nαs. Fors= 0 (standing) fronts with both orientations\nexist simultaneously ccp= 0 and are given by\n\nθ0\np0\nq0\n=\n2arctan/parenleftbig\ne±√−µξ/parenrightbig\n±√−µ\n0\n.\nHence, the branches of left and right moving walls as parametrized b yseach have\ntermination point at s= 0 (cf. Figure 13).\nThe family of explicit DWs (17) have domain wall width√−µ, a profile independent\nof the applied field hand propagate along a nanowire ( µ <0) with velocity swhile\nprecessing with azimuthal velocity Ω. Since these are unique up to sp atial reflection\nsymmetry, the direction of motion is related to the spatial direction of connecting ±e3\nthroughσ,\nθ(−∞) = 0θ(+∞) =π⇔s>0 (wall moves to the right)\nθ(−∞) =π θ(+∞) = 0⇔s<0 (wall moves to the left) .(18)\nTo simplify some notation we will focus on the case of right-moving walls including\nstanding walls ( s≥0) and thus make the standing assumptions that h≥β/αas well as\n16µ <0. We therefore have a 1-to-1 relation of parameters ( α,β,h,µ) and right-moving\nDWs from\nm(ξ,t) =m0(ξ,t;α,β,h,µ)\nwith speed and frequency given by\ns0=s0(α,β,h,µ) :=αh−β√−µ(1+α2),Ω0= Ω0(α,β,h,µ) :=h+αβ\n1+α2(19)\nwhere the subindex 0 emphasizes that ccp= 0. Sinces0is surjective on R≥0any velocity\ncan be realised. Spatial reflection covers the case h≤β/α.\nBased on Lemma 1 as well as Remark 3 for ccp= 0 and (homogeneous) speed and\nfrequency (19), one readily finds the asymptotic states of (7) giv en by\nE0:=Z0\n−/vextendsingle/vextendsingle\n(s0,Ω0)=/parenleftbig\n0,√−µ,0/parenrightbigTandEπ:=Zπ\n−/vextendsingle/vextendsingle\n(s0,Ω0)=/parenleftbig\nπ,√−µ,0/parenrightbigT,\nwith (spatial) eigenvalues (15), (16) given by\nν0\nk,−:=−αs0−2√−µ−(−1)kis0, ν0\n3,−=√−µ,\nνπ\nk,−:=−αs0+2√−µ−(−1)kis0, νπ\n3,−=−√−µ,(20)\nwherek= 1,2. Note that the above equilibria cannot be centers simultaneously ( recall\nµ <0), hence a cycle-to-cycle connection can not exist close to it (see Remark 4 for\ndetails). For this reason, we focus on point-to-point as well as poin t-to-cycle connections.\n3.2 Inhomogeneous Domain Walls\nHomogeneous DWs exist only in case ccp= 0 [12, Theorem 5], are explicitly given\nby (17) and completely characterized by (19). By [12, Theorem 6], f ast inhomogeneous\nDW solutions with |s| ≫1 exist for any ccp∈(−1,1), but in contrast to (17), the\ngradient of these profiles is of order 1 /|s|and thus have a large ‘width’. The natural\nquestion arises what happens for any sin caseccp/ne}ationslash= 0.\nThis section contains the main results of this paper: the existence, parameter selection\nand structure of inhomogeneous DW solutions in case of small |ccp|for any value of the\napplies field h, and thus any speed s. This will be achieved by perturbing away from the\nexplicit solution m0given by (17), where the bifurcation structure is largely determine d\nby comparing the dimensions of the un/stable eigenspaces at the as ymptotic equilibrium\nstates, which are determined by (20).\nLetW0\ns/uandWπ\ns/udenote the stable and unstable manifolds associated to E0and\nEπ, respectively, and w0\ns/uas well aswπ\ns/ube the dimension of these manifolds so that\nw0\ns+w0\nu=wπ\ns+wπ\nu= 3. Notably w0\ns= 2 andw0\nu= 1 for all values of the parameters,\nandwπ\nsis either 1 or 3. Recall the standing assumption s0≥0.\nIfwπ\ns= 1, the heteroclinic connection of E0andEπgenerically has codimension-2 ,\nwhile forwπ\ns= 3 it has codimension-0 , and we refer to the transition point between\n17these cases, following the discussion in §2.1.1, as the center case . From (20) we have\nwπ\ns= 1⇔0≤s0<2√−µ\nα,wπ\ns= 3⇔s0>2√−µ\nαand the center case at s0=2√−µ\nα.\nHence, within the family of homogeneous DWs given by (17) and satisf ying (19), the\ndifferent bifurcation cases have speed and frequency relations\ncodim-0:s0>2√−µ\nαand Ω 0>β\nα−2µ\nα2,\ncenter:s0=2√−µ\nαand Ω 0=β\nα−2µ\nα2,\ncodim-2: 0 ≤s0<2√−µ\nαandβ\nα≤Ω0<β\nα−2µ\nα2.\nUsing (19) these can be written in terms of the parameters of (LLG S), which gives the\ncharacterization mentioned in the introduction §1.\nRemark 5. The case distinction is also related to the spectral stabili ty of the asymptotic\nstatesm=±e3in the dynamics of the full PDE (LLGS)which is beyond the scope\nof this paper, but see Figure 3 for an illustration. In short, it follows from, e.g., [12,\nLemma 1] that e3isL2-stable forh>β/α, while−e3isL2-stable forh<β/α −µand\nunstable for h>β/α −µ. Based on this, the stability curves in Figure 3 are defined as\nfollows\nΓ+:=β/α\nh−µ−1,Γ−:= 1−β/α\nh+µ,\nwhich intersect at\nh=β\n2α+/radicalbigg\nβ2\n4α2+µ2.\nSince the destabilisation of −e3ifβ >0corresponds to a Hopf-instability of the (purely\nessential)spectrum, it is effectively invisible in the coherent struct ure ODE, which detects\nchanges in the linearization at zero temporal eigenvalue on ly. Visible from the PDE\nstability viewpoint is a transition of absolute spectrum th rough the origin in the complex\nplane of temporal eigenmodes, cf. [18]. Now in the center cas e, the state −e3is already\nL2-unstable since α>0as well asµ<0andh>β/α implies\nh=h∗=β\nα−2µ\nα2(1+α2)>β\nα−µ\nand therefore that Γ−never intersects the line ccp≡0ath=h∗.\nMoreover, it was shown in [19] that the family of explicit hom ogeneous DWs (9)is\n(linearly) stable for sufficiently small applied fields, actually for h <−µ/2, in case\nβ= 0, hence in the bi-stable case where ±e3areL2-stable. As mentioned before, β= 0\nis equivalent to ccp= 0in the LLGS equation with an additional shift in handβ, which\nleads to the (LLG)case. We expect these DWs are also stable for small perturbat ions in\nccp, due to the properties of the operator established in [19], b ut further analysis also on\nthe transition from convective/transient to absolute inst ability will be done elsewhere.\n18With these preparations, we next state the mainresults, which con cern existence of DWs\nin the three regimes.\nTheorem 1. For any parameter set (α0,β0,h0,µ0)in the codim-0 case, i.e., µ0<0and\nh0> β0/α0−2µ0−2µ0/α2\n0, the following holds. The explicit homogeneous DWs m0\nin(9)lies in a smooth family mccpof DWs parameterized by (ccp,α,β,h,µ,s, Ω)near\n(0,α0,β0,h0,µ0,s0,Ω0)withs0,Ω0from(19)evaluated at (α0,β0,h0,µ0). Moreover,\nin caseccp= 0and(s,Ω)/ne}ationslash= (s0,Ω0)evaluated at (α,β,h,µ), orccp/ne}ationslash= 0, these are\ninhomogeneous flat DWs.\nProof.As mentioned, in the codim-0 case we have wπ\ns= 3 and for all parameters w0\nu=\n1. Due to the existence of the heteroclinic orbit (17), this means W0\nuintersectsWπ\ns\ntransversely and non-trivially for ccp= 0 in a unique trajectory. Therefore, this DW\nperturbs to a locally unique family by the implicit function theorem for p erturbations\nof the parameters in (7).\nForccp/ne}ationslash= 0 sufficiently small these are inhomogeneous DWs since the derivativ e of the\nthird equation, the q-equation, in (7) with respect to ccpis nonzero in this case; hence\nalready the equilibrium states move into the inhomogeneous regime.\nForccp= 0 but (s,Ω)/ne}ationslash= (s0,Ω0) at (α,β,h,µ), it follows from [12, Theorem 5] that\nthese DWs cannot be homogeneous. /squaresolid\nNext we consider the center case, where h=h∗=β/α−2µ−2µ/α2. We start with\na result that follows from the same approach used in the codim-2 cas e and give refined\nresults below.\nCorollary 1. The statement of Theorem 1 also holds for a parameter set in th e center\ncase if the perturbed parameters (ccp,α,β,h,µ,s, Ω)satisfyΩ> s2/2 +β−/α. IfΩ =\ns2/2 +β−/αandΩ< h+µ+s2\n4(1 +α2)the same holds except the DW is possibly\nnon-flat.\nProof.It follows from Proposition 1 and the discussion before that Ω >s2/2+β−/αfor\nthe parameter perturbation implies that the eigenvalues of the per turbed equilibrium\nZπ\n−≈Eπsatisfy Re(νπ\nk,−)<0,k= 1,2. Hence, the stable manifold at the target equi-\nlibrium is two-dimensional and lies in a smooth family with the center-sta ble manifold\nat the transition point. Then the proof is the same as in the codim-0 c ase. If the pertur-\nbation has Ω = s2/2+β−/αthen we consider as target manifold the three-dimensional\nstable manifold of a neighborhood of Zπ\n−within the blow-up chart θ= 0. This neigh-\nborhood consists of periodic orbits by Proposition 1 if Ω < h+µ+s2\n4(1 +α2). By\ndimensionality the intersection with the unstable manifold of Z0\n−persists and yields a\nheteroclinic orbit from the perturbed equilibrium at θ= 0 to the blow-up chart at θ=π.\nPerturbing ccpaway from zero moves the left-asymptotic state into the inhomoge neous\nregime and thus generates an inhomogeneous DWs. Note from Prop osition 1 that the\nright-asymptotic state is either an equilibrium with q/ne}ationslash= 0 or a periodic orbit along which\nq= 0 happens at most at two points. /squaresolid\n19Next we present a refined result in which we show that typical pertu rbations indeed give\nnon-flat DWs, i.e., heteroclinic connections with right-asymptotic st ate being a periodic\norbit. The existence of flat DWs for ccp/ne}ationslash= 0 is severely constrained, but not ruled out\nby this result. Our numerical results, such as those presented in §4, always lead to a\nselected solution with a periodic asymptotic state.\nIn addition, attempts to perform numerical continuation (see §4) of flat DWs to ccp/ne}ationslash= 0\nfailed. Here we added the constraint /tildewideH= 0 and allowed adjustment the parameters h\nands, but the continuation process did not converge, which confirms nu merically the\ngeneric selection of a periodic orbit.\nAs mentioned before, the right asymptotic state is e3in either case in the PDE coordi-\nnates; the difference between flat and non-flat lies in the finer deta ils of how the profile\napproaches e3in term ofpand alsoq, which relates to mthrough (2).\nTheorem 2. Consider the smooth family of DWs from Corollary 1 with ccp= 0for\nparameters satisfying (12)with fixedα >0,β≥0, andµ <0. Then there is a\nneighborhood (ccp,s,h)of(0,s0,h∗)suchthat the followingholds. FlatDWs occurat most\non a surface in the (ccp,s,h)-parameter space and, for β/ne}ationslash= 0, satisfy|h−h0|2+|s−s0|2=\nO(|ccp|3), more precisely (31)holds, where h0=h∗ands0= 2√−µ/α. Otherwise DWs\nare non-flat, in particular all DWs not equal to m0forccp= 0orβ= 0are non-flat.\nDue to its more technical nature, the proof of this theorem is defe rred to Appendix 6.1.\nIt remains to consider the codim-2 case.\nTheorem 3. For any parameter set (α0,β0,h0,µ0)in the codim-2 case, i.e., µ0<0and\nβ0/α0≤h0< β0/α0−2µ0−2µ0/α2\n0, the following holds. The explicit homogeneous\nDWsm0in(9)lies in a smooth family of DWs parameterized by (ccp,α,β,h,µ )near\n(0,α0,β0,h0,µ0). Here the values of (s,Ω)are functions of the parameters (ccp,α,β,h,µ )\nand lie in a neighbourhood of (s0,Ω0)from(19). This family is locally unique near m0\nand forccp/ne}ationslash= 0consists of inhomogeneous flat DWs.\nThe proof of Theorem 3 is presented in Appendix 6.2 and is based on th e Melnikov\nmethod for perturbing from m0. As the unperturbed heteroclinic orbit has codimension\ntwo, thebifurcationisstudied inathree-parametricfamilywithpert urbationparameters\nη:= (ccp,s,Ω)T, which yields a two-component splitting function M(η) that measures\nthe mutual displacement of the manifolds W0\nuandWπ\ns.\nDue to the fact that Re( νπ\nj,−)<0 also forβ= 0 in the codim-0 regime and the case\nβ= 0 is included in Theorem 2 as well as Theorem 3, we immediately get the f ollowing\nresult.\nCorollary 2. Inhomogeneous flat DWs also exist in the LLG equation (β= 0), which\ncan be flat or non-flat, respectively.\nTheorem3 completes theexistence study ofDWs. Therefore, for any valueoftheapplied\nfieldhthere exists a heteroclinic connection between the blow-up charts withccp/ne}ationslash= 0\nandq/ne}ationslash≡0, thus an inhomogeneous (typically flat) DW. Recall that we have fo cused on\n20right moving DWs, but all results are also valid for left moving walls due t o symmetry.\nTherefore inhomogeneous DWs exist with ccp/ne}ationslash= 0 for any value of the applied field\nh∈R.\n4 Numerical Results\nNumerical continuation for ordinary differential equations is an est ablished tool for bi-\nfurcation analysis in dynamical system. In this section we present c ontinuation results\nto illustrate the analytical results discussed in §3. In particular, we will focus on con-\ntinuation in the parameter ccpin the range of ( −0.5,0.5) as this perturbs away from the\nknown family m0from (17) (cf. Figure 1b) with speed and frequency determined by (19)\nfor a given applied field. Note that we also focus only on right-moving f ronts in this\nsection for reasons of clarity. All results were produced by contin uation in AUTO-07P\nand graphics were created with Mathematica as well as MATLAB .\nHeteroclinic orbits were detected as solutions to the boundary valu e problem given by\nthe desingularized system (7) plus a phase condition andboundary c onditions at ξ=−L\nandξ=Ltaken from the analytic equilibrium states in pandqon the blow-up charts\n(Remark 3). In the codim-2 case, the four required conditions are thep,qvalues at the\ncharts. In the center case, the three required conditions are: ( 1,2) the two p,qvalues\nat the left chart and (3) the energy difference determined by the f unction (13). In the\ncodim-0 case, the two required conditions are the pvalues at both charts. Moreover, we\nfoundL= 50 was sufficiently large.\nIn order to relate to (LLGS), we plot most of the profiles after blow ing down to the\nsphere rather than using the ODE phase space.\n(a)-5 5-0.04-0.02\nξq\n- 5 \u0001-101\nξ\nm1\n(b)\nFigure 7: DWs obtained from continuation of m0in system (7) in the codim-2 regime h= 0.5\n(h∗= 10.2) with initial speed and frequency s0= 0.12 as well as Ω 0= 0.44, and\n(ccp,s,Ω) = (−0.5,0.11221,0.44077). (a) Projection onto the sphere. (b) Zoom-in\nof corresponding q-profile (red) and m1component (blue).\nFollowing the standing assumption on positive speeds and using ccpas well ashas the\nmain parameters, we keep the other parameters fixed with values\nα= 0.5,β= 0.1,µ=−1.\n21(a)\n (b)\nFigure 8: DWs obtained from continuation of m0in system (7) projected onto the sphere in\nthe codim-2 regime h= 10.1 (h∗= 10.2) with initial speed and frequency s0=\n3.96 and Ω 0= 8.12. (a) ( ccp,s,Ω) = (−0.5,3.99541,8.05973). (b) ( ccp,s,Ω) =\n(0.5,4.08089,8.22402).\nThe value of the applied field for the center case, given the fixed par ameters, is h∗= 10.2\n(cf.§3.2), which leads to s0= 4.0 as well as Ω 0= 8.2 (cf. (19)).\n4.1 Codim-2 case\nThe lower boundary for values of the applied field hlies in the codim-2 regime and\nis given by h=β/α= 0.2. As a first numerical example we consider the slightly\nlarger value h= 0.5. The results upon continuation in the negative as well as positive\ndirection of ccpare presented in Figures 2a, 2c, and 7. The inhomogeneous nature of\nthese solutions ( ccp/ne}ationslash= 0) is reflected in the significantly varying azimuthal angles, also\nvisible in the oscillatory nature of the m1component in Figures 2c as well as 7b.\nThe linear part of the splitting function (33) (see Theorem 3), which predicts the direc-\ntion of parameter variation for the existence of inhomogeneous DW s (ccp/ne}ationslash= 0) to leading\norder, reads in this example\nM(ccp,s,Ω) =/parenleftbigg\n−0.00147567 −0.499245 0.245945\n−0.000577908 −0.245945 −0.499245/parenrightbigg\n·\nccp\ns\nΩ\n,\nsothatM= (0,0)Tfor(s,Ω) = (−0.00283744 ·ccp,0.000240252 ·ccp). Fortheparameter\nvalues in Figure 7 and 2a we obtain, respectively,\nM(−0.5,−0.007788,0.000771) = (0 .00481558,0.00181945)T,\nM(0.5,−0.007973,0.007173) = (0 .00648248,−0.00133122)T.\nNote that here the splitting of the (1-dimensional) unstable manifold of the left equilib-\nrium and the (1-dimensional) stable manifold of the right equilibrium diffe r, i.e., are in\nopposite directions (signs) in frequency and speed for variations in ccp.\n22In addition note the decrease in frequency in the m1component, and thus also in the\nm2as a result of the increase of the qcomponent towards zero, cf. Figure 7b. Here, the\nazimuthal angle decreases since φ=/integraltext\nqandq <0.\nAs a further example in the codim-2 regime, we consider h= 10.1<10.2 =h∗near the\nupper boundary of the codim-2 regime in terms of the applied field h. The results of the\ncontinuation in ccpare presented in Figure 8. The linear approximation of the splitting\nin this case is given by M(−0.5,0.03541,−0.06027) = ( −0.000175537,−0.00104378)Tas\nwellasM(0.5,0.12089,0.10402) = ( −0.00149519,0.0014975)Tin(a)and(b),respectively.\nNote that the direction of splitting of the two components in this cas e is also dependent\non the polarity sign, as in the previous example. In both cases ( h= 0.5 andh= 10.1),\nthecontinuationresultslookbasicallythesameinthecodim-2regime, wherethesolution\nis, roughly speaking, constantly spiraling down from the north to th e south pole.\n4.2 Center case\n(a)-20 200.0450.055\nξq\n(b)\nFigure 9: DWs obtained from continuation of m0in system (7) in the center regime h=h∗=\n10.2 withs=s0= 4,Ω = Ω 0= 8.2, andccp= 0.5. (a) Projection onto the sphere.\n(b) Profile of corresponding q-component.\nWe perform computations in the center case with applied field h=h∗= 10.2 and fixed\nfrequency Ω = β−/α+s2/2 (see Proposition 1 and its discussion details). Theorem 2\nshows that the right asymptotic state is generically a periodic orbit a nd more precisely\nthat in case ccp= 0, no constellation of handsexists, both not equal to zero, for which\nthe right asymptotic state is the (shifted) equilibrium. The results o f continuation in ccp\nprojected on the sphere look quite the same, which is why only the re sult forccp= 0.5\nis presented in Figure 9. The fact that the right asymptotic state is a periodic orbit on\nthe blow-up chart θ=πis reflected by the nearly constant oscillations in the q-profile\nforξclose to the right boundary (cf. Figure 9b).\nThat the right state is not the equilibrium in the blow-up chart is furth er corroborated\nby computing the difference in energy /tildewideHbetween this equilibrium state (see Remark 3)\nand the approximate right asymptotic state obtained from continu ation. The analytic\nprediction of this difference up to second order is given by (31), whic h reads, for the\n23-0.5 0 0.5\nccp-2-1010-6\n(a)9.2 10.2 11.2\nh-1010-4\n(b)3.5 4 4.5\ns-2010-3\n(c)\nFigure 10: Continuation of m0insystem(7)inthecenter casewithappliedfield h=h∗= 10.2\nand fixed frequency Ω = β−/α+s2/2; heres0= 4. Shown is the energy difference\n(solid blue line) between the equilibrium and asymptotic st ate from continuation\non right boundary against the continuation parameter ccpin (a),hin (b), and sin\n(c). The red dashed curve in (b) and (c) is the quadratic appro ximation (21).\nchosen parameters,\n−0.006612+0.00673s−0.00183s2−0.00134h−0.000086h2+0.00077hs.(21)\nAs this analytic prediction is independent of ccpthe dependence of /tildewideHonccp≈0 is of\ncubic or higher power. Indeed, the results plotted in Figure 10a sug gest an at least\nquartic dependence since a maximum lies at ccp= 0. The asymmetric nature of the\ngraph suggests that odd powers appear in the expansion beyond o ur analysis, but also\nnote the order of 10−6in/tildewideH. In addition to the dependence on ccp, continuations for\nccp= 0 of/tildewideHinhwith fixeds=s0= 4 and in swith fixedh=h∗= 10.2 are plotted in\nFigure 10b and 10c, respectively. Here we also plot the quadratic pr ediction (21).\n4.3 Codim-0 case\nNext, we consider an applied field h= 10.3 in the codim-0 regime, just above the applied\nfield value for the center case h∗= 10.2. The results of continuation in ccpare plotted\non the sphere in Figure 11. The azimuthal profile in φand hence in qare non-trivial as\npredicted for inhomogeneous DWs.\nIn the ODE, qpossesses an oscillating profile and has a monotonically decreasing am -\nplitude in both cases. This is a consequence of the proximity to the ce nter case and the\nconvergence to equilibria (see §3.2 for details). Recall that the speed and frequency are\nnot selected by the existence problem during continuation in ccp, but are taken as the\nfixed parameters ( s0,Ω0) defined in (19).\nThe final example is for a relatively large applied field h= 50 in the codim-0 regime,\nfar away from the center case, and the results of continuation in ccpprojected on the\nsphere are presented in Figure 2b as well as Figure 12a. Moreover, the corresponding\nm1andm2profiles for ccp= 0.5 are presented in Figure 2d, and for ccp=−0.5, in\nFigure 12b. As in the previous example, the inhomogeneous nature is visible in the\nnon-trivial azimuthal profile.\nIn summary, switching on the parameter ccpleads to a variety of inhomogeneous flat as\nwell as non-flat DW solutions, but also in case ccp= 0 there exist inhomogeneous DWs\n24(a)\n (b)\nFigure 11: DWs obtained from continuation of m0in system (7) projected onto the sphere in\nthe codim-0 regime h= 10.3 (h∗= 10.2) and initial speed and frequency s0= 4.04\nand Ω0= 8.28. (a)ccp=−0.5. (b)ccp= 0.5.\n(a)-5 5-101\nξm1,m2\n(b)\nFigure 12: DWs obtained from continuation of m0in system (7) in the codim-0 regime h= 50\n(h∗= 10.2) withinitial speedandfrequency s0= 19.92,Ω0= 40.04, andccp=−0.5.\n(a) Projection onto the sphere. (b) Zoom-in of the correspon dingm1(solid blue)\nandm2(dashed red) component.\n(cf. Figure 4) which are much more complex than the homogeneous o ne given by (9) (cf.\nFigure 1b).\nFinally, recall from §3.1 that in the explicit family (9), the right moving DWs terminate\nats= 0. The question arises what happens for ccp/ne}ationslash= 0 along the parameter s. To\nstudy this, we performed a continuation in the parameter sfor different values of ccp.\nFor decreasing swe found that numerical continuation failed at some s>0 forccp/ne}ationslash= 0\n(cf. Figure 13). The details of this apparent existence boundary a re beyond the scope of\nthis paper. Note that the special role of sis reflected in the splitting function (34). In\ncases0=s= 0, the first column of (32) is zero (see (34)) and thus the parame terization\nofccpcan not be written as a function in s.\nIn detail, we continued the analytic solution (9) in ccpaway from zero for different initial\nvalues ofβ/α < h < h∗. This led to inhomogeneous ( ccp/ne}ationslash= 0) DWs, which we in turn\ncontinued in the parameter stowards zero for different fixed values of ccpuntil the con-\ntinuation process fails to converge. Based on this, there is numeric al evidence that DWs\nwith opposite speed sign (counter-propagating fronts) can only e xist simultaneously for\n25s= 0 (standing fronts). We took a polynomial fit on these points as an approximation of\nthe existence boundary (cf. blue curve in Figure 13a). The continu ation process towards\nthe boundary is indicated by red arrows in Figure 13a for positive ccp. Additionally, the\ncorresponding results in the parameter space Ω and ccpis presented in 13b.\n0 0.10.20.40.6\nsc\nc\u0002\n(a)0.2 0.50.20.40.6\nΩc\n\u0003\u0004\n(b)\nFigure 13: Continuation results in sand Ω for fixed ccpin the codim-2 parameter regime.\nFurther parameters are α= 0.5,β= 0.1,µ=−1, andhfree. Blue solid line\nrepresents interpolated termination boundaryfor sin (a) and Ω in (b), respectively.\nRed arrows indicate continuation approach towards boundar y.\nAs a last point, we briefly describe the numerical method for time-int egration near DWs,\nincluding freezing of speed and frequency (cf. Figure 4). All calcula tions were done\nwith the (free) software package pde2path which is based on a Finite Element Method\n(FEM), cf. [20] and the references therein. Time-integration in pde2path with the so-\ncalled ‘freezing’ method is discussed in [21]. In addition to the phase co ndition for the\nspeedweaddedaphaseconditionfortherotationandtime-integra tedvia asemi-implicit\nEuler scheme.\n5 Discussion and Outlook\nWe have presented results pertaining the existence of different ty pes of domain walls for\nthe LLG as well as LLGS equation. Our main focus has been on a nonze ro polarisation\nparameterccp/ne}ationslash= 0 for any value of the applied field, including the high-field case, and\nthus for any domain wall speed. These results extend what is known in particular for\ninhomogeneously structured DWs, and we have discovered an appa rently new type of\nDWs with certain oscillatory tails, referred to as non-flat here.\nIn detail, we have provided a classification of DWs based on co-dimens ion properties\nin a reduced (spatial) coherent structure ODE, which relates to st ability and selection\nproperties that we review next. First, we have proven the existen ce of inhomogeneous\nflat DWs in case ccp= 0 as well as ccp/ne}ationslash= 0 for an applied field above a certain threshold,\nwhichismainlymaterialdepending. Toourknowledge, theonlypreviou sexistence result\nforccp/ne}ationslash= 0 with ’large’ applied fields concerns less relevant non-localized DWs [ 12]. Here\nthe existence problem does not select speed and frequency.\nSecond, we have discussed the so-called center case, which is char acterized by non-\nhyperbolic equilibria in the underlying coherent structure ODE. In th is case, we have\n26shown the existence of inhomogeneous DWs including the leading orde r selection mech-\nanism. These solutions are non-flat in case ccp= 0 and generically also non-flat for ccp\naway from zero, which was substantiated by numerical results. Th e fundamental obser-\nvation has been the existence of a Hamiltonian function in a certain pa rameter regime\nin the corresponding coherent structure ODE.\nThird, we have proven the existence of inhomogeneous DWs in the so -called codim-2\nregime, which is a range of values for the applied field in which the speed sis between\nzero and the center case speed. In this regime, each solution in cas eccp/ne}ationslash= 0 is uniquely\ndetermined by its speed as well as frequency. Here we have also pre sented the leading\norder selection function in the coherent structure ODE variables pandq, which depends\non the speed s, the frequency Ω, and is independent of ccpfor standing fronts.\nWe believe that these results are not only interesting and relevant f rom a theoretical and\nmathematical viewpoint, but also from an application viewpoint. They could help to\nbetter understand the interfaces between different magnetic do mains in nanostructures,\ne.g. in the development of racetrack memories, which are a promising prospective high\ndensity storage unit that utilize a series of DWs by shifting at high spe ed along magnetic\nnanowires through nanosecond current pulses.\nIn order to illustrate and corroborate these theoretical results , we have presented numer-\nical computations for a variety of values for the applied field in §4. On the one hand,\nthe examples in essence show that large applied fields lead to more com plex profiles of\nthe DWs in case ccp/ne}ationslash= 0. On the other hand, while in the center case the DWs projected\non the sphere appear similar to those for small applied fields, these s olutions approach\nthe poles in a qualitatively different ‘non-flat’ manner – as predicted b y our analysis.\nMoreover, we compared the numerical and analytical results of th e selection mechanism\nin the center case, showing that the analytical leading order appro ximation predicts the\neffect of small perturbations in the parameters. Notably, for app lied fields above a cer-\ntain threshold, where the existence analysis does not provide a sele ction of speed and\nfrequency, numerically the DWs selected in the PDE dynamics are in th e center case,\nboth forccp= 0 as well as ccp/ne}ationslash= 0. Hence, it might be possible to detect these solutions\nin a high-field regime in real materials.\nOne question concerning existence beyond our analysis is whether in homogeneous (flat\nor non-flat) solutions exist for any value of ccp∈(−1,1), and whether this class could\nbe utilized in applications.\nA natural step towards the understanding of domain wall motion in n anowires beyond\nthe question of existence concerns the dynamic stability. For inhom ogeneous solutions\nthere appears to be no rigorous result in this direction. In particula r for larger applied\nfields, stability results would be an essential step towards underst anding the selection\nmechanismofsolutionsintermsofspeedandfrequency; ourfirstn umericalinvestigations\nshow that solutions inthe center parameter regime areselected, i.e ., inhomogeneous non-\nflat DWs.\nMoreover, preliminary analytic results, for ccp= 0 as well as ccp/ne}ationslash= 0, show that selection\nmechanism is mainly determined by the value of the applied field, where in the bi-stable\n27case (±e3linearly stable) homogeneous DWs are selected, and in the mono-sta ble case\ninhomogeneous non-flatDWsareselected, which will bestudied indet ail inanupcoming\nwork.\nReferences\n[1] Stuart SP Parkin, Masamitsu Hayashi, and Luc Thomas. Magnetic domain-wall\nracetrack memory. Science, 320(5873):190–194, 2008.\n[2] William Reohr, Heinz Honigschmid, Raphael Robertazzi, Dietmar Gog l, Frank Pe-\nsavento, Stefan Lammers, Kelvin Lewis, Christian Arndt, Yu Lu, Ha ns Viehmann,\net al. 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Physical Review B , 84(10):104445,\n2011.\n[20] T Dohnal, J Rademacher, H Uecker, and D Wetzel. pde2path 2.0. ENOC, 2014.\n[21] Jens DM Rademacher and Hannes Uecker. Symmetries, freezin g, and hopf bifurca-\ntions of traveling waves in pde2path, 2017.\n[22] Yuri A Kuznetsov. Elements of applied bifurcation theory , volume 112. Springer\nScience & Business Media, 2013.\n6 Appendix\n6.1 Proof of Theorem 2\nWe use the notation\nu=u(ξ;η,α,β,µ)= (θ(ξ;η,α,β,µ),p(ξ;η,α,β,µ),q(ξ;η,α,β,µ))T\nand bifurcation parameters η= (ccp,s,h)T, wheres0andh0=h∗are defined below (see\n§3.2 for details). The starting point for our perturbation analysis ar e the unperturbed\nparameters and explicit heteroclinic solution in the center case (12) , where the frequency\nis Ω0=s2\n0/2+β/α. These are given by\nη0:=\nccp0\ns0\nh0\n:=\n0\n2√−µ\nαβ\nα−2µ−2µ\nα2\n\n29as well as\nu0=u0(ξ;η0,α,β,µ) :=\nθ0(ξ;η0,α,β,µ)\np0(ξ;η0,α,β,µ)\nq0(ξ;η0,α,β,µ)\n:=\n2arctan(exp(√−µξ))√−µ\n0\n.\nUnless stated otherwise, we suppress the explicit dependence of uonα,β, andµin the\nfollowing discussion. Let us write Zπ:=Zπ\n−with the notation from Remark 3 so that\nthe unperturbed right asymptotic state is given by\nZπ(η0) =/parenleftbigg\nπ,αs0\n2,s0\n2−/radicalbigg\n−µ\nα2/parenrightbiggT\n=/parenleftbig\nπ,√−µ,0/parenrightbigT\nand its derivative with respect to ηis given by\nZπ\nη(η0) =\n0 0 0\n0α\n20\nβ\n2√−µ2+α2\n2−α\n2√−µ\n.\nWe write system (7) for brevity as\nu′=f(u;η), (22)\nsof(u;η) denotes the right side of (7). The linearization w.r.t. ηin the unperturbed\nheteroclinic connection u0, given by (17), is the non-autonomous linear equation\nu′\nη=fu(u0;η0)uη+fη(u0;η0)η, (23)\nwhereuη= (θη,pη,qη)T. Its homogeneous part is\nθ′\nη=√−µcos(θ0)θη+pηsin(θ0)\np′\nη=−(αs0+2√−µcos(θ0))pη+s0qη\nq′\nη=−s0pη−(αs0+2√−µcos(θ0))qη, (24)\nwithθ0(ξ) = 2arctan(exp(√−µξ)) due to (17). We next solve (24) and determine its\nfundamental solution matrix.\nThe first obvious vector-solution of it is U1=u′\n0= (θ′\n0,0,0) since the second and the\nthird equation of (24) do not depend on θη. The other solutions can be obtained from\nU1and the result of Lemma 1. Changing to polar coordinates\npη=rcosϕ, q η=rsinϕ,\nthe equations for pηandqηbecome\nr′=−(αs0+2√−µcos(θ0))r\nϕ′=−s0,\n30whose general solution can be written as\npη=r0r(ξ)cos(−s0ξ+ϕ0)\nqη=r0r(ξ)sin(−s0ξ+ϕ0)\nwhere\nr(ξ) = exp\n−αs0ξ−2√−µ/integraldisplay\nξcos(θ0(τ))dτ\n=/parenleftig\n1+e2√−µξ/parenrightig2\ne(−2√−µ−αs0)ξ,\nandr0,ϕ0arearbitrary integrationconstants corresponding to suitable init ial conditions.\nNote that lim\nξ→±∞r(ξ) =∞for 0≤s0<2√−µ/α.\nNext, the values of the integration constants have to be selected in order for the second\nand the third vector-solutions\nU2=\nθ1\n1\nr1r(ξ)cos(−s0ξ+ϕ1)\nr1r(ξ)sin(−s0ξ+ϕ1)\n, U3=\nθ2\n1\nr2r(ξ)cos(−s0ξ+ϕ2)\nr2r(ξ)sin(−s0ξ+ϕ2)\n(25)\ntobelinearlyindependent. Here θ1\n1,θ2\n1arenotrelevantforwhatfollows. Thedeterminant\nof the fundamental matrix reads\ndetΦ(ξ) = det(U1(ξ),U2(ξ),U3(ξ)) =r1r2r2(ξ)θ′\n0(ξ)sin(ϕ2−ϕ1),\nwhich is non-zero for r1=r2= 1,ϕ1= 0 andϕ2=π/2, i.e. detΦ( ξ) =r2(ξ)θ′\n0(ξ).\nTogether, we get the fundamental solution matrix of the homogen eous part as\nΦ(ξ) =\nθ′\n0(ξ)θ1\n1(ξ) θ2\n1(ξ)\n0r(ξ)cos(−s0ξ)−r(ξ)sin(−s0ξ)\n0r(ξ)sin(−s0ξ)r(ξ)cos(−s0ξ)\n. (26)\nThe derivative of (22) with respect to ηis given by (23) and from the variation of\nconstants formula we get for some ξ0that\nuη(ξ) = Φξ,ξ0uη(ξ0)+ξ/integraldisplay\nξ0Φξ,τfη(u0(τ);η0)dτ,\nwhere Φ ξ,τ= Φ(ξ)·Φ−1(τ) is the evolution operator. Using (26) we find\nΦξ,τ(ξ,τ;η) =\nΘ1 Θ2 Θ3\n0r(ξ)\nr(τ)cos(−s0(ξ−τ))−r(ξ)\nr(τ)sin(−s0(ξ−τ))\n0r(ξ)\nr(τ)sin(−s0(ξ−τ))r(ξ)\nr(τ)cos(−s0(ξ−τ))\n,(27)\nwhere the explicit forms of the functions Θ 1,2,3(ξ) are not relevant for the remainder\nof this proof. Since uη(ξ) tends to∂ηZ0\n−forξ→ −∞the hyperbolicity of Z0\n−(more\n31precisely the resulting exponential dichotomy) implies Φ ξ,ξ0uη(ξ0)→0 asξ0→ −∞and\nso\nuη(ξ) =ξ/integraldisplay\n−∞Φξ,τfη(u(τ;η0);η0)dτ. (28)\nRegarding the limiting behavior as ξ→ ∞, recall that Corollary 1 states that the right\nasymptotic limit of the perturbed heteroclinic orbit is either the pert urbed equilibrium\nZπ(η) or a periodic orbit around it in the blow-up chart at θ=π. The integral (28)\ndistinguishes these case in the sense that either it has a limit as ξ→+∞so the\nheteroclinic orbit connects the two equilibria, or it does not and the h eteroclinic orbit\nconnects to a periodic solution.\nWe next determine uη(ξ) componentwise\nuη(ξ) =v:=\nv11v12v13\nv21v22v23\nv31v32v33\n,\nwherevijare the components of (28) and index i= 1,2,3 relates to θ,p,qas well as\nj= 1,2,3 toccp,s,h.\nTowards this, we compute\nfη(u0(τ),η0) =\n0 0 0\n−β/α−√−µ\nα(2+α2) 1\n2β\n1+e2√−µ τ√−µ0\n,\nand together with (28) and (27) we obtain\nv21=−β\nαIC−2βJS, v22=−√−µ\nα(2+α2)IC−√−µIS, v23=IC,\nv31=−β\nαIS+2βJC, v32=−√−µ\nα(2+α2)IS+√−µICv33=IS,\nwhere\nIC=IC(ξ):=ξ/integraldisplay\n−∞(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))2cos(−s0(ξ−τ))dτ,\nIS=IS(ξ):=ξ/integraldisplay\n−∞(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))2sin(−s0(ξ−τ))dτ,\nJC=JC(ξ):=ξ/integraldisplay\n−∞exp(−2√−µτ)(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))3cos(−s0(ξ−τ))dτ,\n32ξ ξ0iπ\n2√−µ\nI1I2I3\nI4\nFigure 14: Contour CinCfor the integrals IandJ.\nJS=JS(ξ):=ξ/integraldisplay\n−∞exp(−2√−µτ)(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))3sin(−s0(ξ−τ))dτ.\nNote that we do not provide explicit formulas for v11,v12andv13, because they are\nnot needed for further computations. This is the reason why we ne glected the explicit\nexpressions of Θ 1,Θ2, and Θ 3before. We now introduce the following complex-valued\nintegrals for further computations:\nI(ξ):=IC(ξ)+iIS(ξ) =ξ/integraldisplay\n−∞/parenleftbig\n1+e−2√−µξ/parenrightbig2\n/parenleftbig\n1+e−2√−µτ/parenrightbig2exp/parenleftbigg\n−i2√−µ\nα(ξ−τ)/parenrightbigg\ndτ,\nforICandISas well as\nJ(ξ):=JC(ξ)+iJS(ξ) =ξ/integraldisplay\n−∞e−2√−µτ/parenleftbig\n1+e−2√−µξ/parenrightbig2\n/parenleftbig\n1+e−2√−µτ/parenrightbig2exp/parenleftbigg\n−i2√−µ\nα(ξ−τ)/parenrightbigg\ndτ\nforJCandJS. We extend the above integrals to the complex plane and integrate a long\nthe counter-clockwise oriented rectangular contour Cas illustrated in Figure 14, and\nletξ0→ −∞. We will provide the details of the computation of Ionly, asJcan be\ncalculated in a fully analogous way.\nThe complex integrand of Iis\ng(z;ξ):=/parenleftbig\n1+e−2√−µξ/parenrightbig2\n/parenleftbig\n1+e−2√−µz/parenrightbig2exp/parenleftbigg\n−i2√−µ\nα(ξ−z)/parenrightbigg\n,\nwith singularities in Cat the pointsi(π+2kπ)\n2√−µ,k∈Z, one of which lies in the interior of C,\nnamelyz0:=iπ\n2√−µ. The contour integral Ican now be written via the residue theorem\nas\nI1(ξ)+I2(ξ)+I3(ξ)+I4(ξ) = 2πi/summationdisplay\nintCResg(z;ξ),\nwhereI1,...,I 4are given by\n33I1:z=x,\nI1(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbiggξ/integraldisplay\nξ0exp/parenleftig\ni2√−µ\nαx/parenrightig\n/parenleftbig\n1+e−2√−µx/parenrightbig2dx\nI(ξ) = lim\nξ0→−∞I1(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbiggξ/integraldisplay\n−∞exp/parenleftig\ni2√−µ\nαx/parenrightig\n/parenleftbig\n1+e−2√−µx/parenrightbig2dx,\nI2:z=ξ+iy,\nI2(ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2π√−µ/integraldisplay\n0iexp/parenleftig\n−2√−µ\nαy/parenrightig\n/parenleftbig\n1+e−2√−µ(ξ+iy)/parenrightbig2dy,\nI3:z=x+π√−µi,\nI3(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbigg\ne−2π\nαξ0/integraldisplay\nξexp/parenleftig\ni2√−µ\nαx/parenrightig\n/parenleftbig\n1+e−2√−µξ/parenrightbig2dx\n=−e−2π\nαI1(ξ0,ξ),\nI4:z=ξ0+iy,\nI4(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\ni2√−µ\nα(ξ0−ξ)/parenrightbigg0/integraldisplay\nπ√−µiexp/parenleftig\n−2√−µ\nαy/parenrightig\n/parenleftbig\n1+e−2√−µ(ξ0+iy)/parenrightbig2dy,\nlim\nξ0→−∞I4(ξ0,ξ) = 0.\nUtilizing the Laurent series of gwe obtain\nResg(z;ξ)|z=z0=α+i\n2α√−µe−π\nα/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbigg\n,\nwhich leads to\nI(ξ) =/parenleftig\n1−e−2π\nα/parenrightig−1/parenleftbiggπi(α+i)\nα√−µe−π\nα/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbigg\n−I2(ξ)/parenrightbigg\n.\nNow we can write\nIC(ξ) = ReI(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\nα√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\n−cos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n−αsin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1\n1−e−2π\nαIr\n2(ξ)\n34as well as\nIS(ξ) = ImI(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\nα√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\nαcos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n−sin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1\n1−e−2π\nαIi\n2(ξ),\nwhereIr\n2(ξ) andIi\n2(ξ) are the real and imaginary part of I2(ξ), respectively.\nStudying the integral Jin a similar fashion, we obtain\nJC(ξ) = ReJ(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\n2α2√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\nαcos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n−sin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1/parenleftig\n1−e−2π\nα/parenrightigJr\n2(ξ)\nas well as\nJS(ξ) = ImJ(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\n2α2√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\ncos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n+αsin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1/parenleftig\n1−e−2π\nα/parenrightigJi\n2(ξ),\nwhere also here Jr\n2(ξ) andJi\n2(ξ) are the real and imaginary part of J2(ξ). Direct compu-\ntations show also that\nlim\nξ→+∞Ii\n2(ξ) =α\n2√−µ/parenleftbigg\n1−exp/parenleftbigg\n−2π\nα/parenrightbigg/parenrightbigg\nand\nlim\nξ→+∞Ir\n2(ξ) = lim\nξ→+∞Jr\n2(ξ) = lim\nξ→+∞Ji\n2(ξ) = 0.\nSumming up, the second and third component of uη(ξ)·ηfor sufficiently large ξare\n/parenleftigg\nα\n2s\nβ\n2√−µccp+2+α2\n2s−α\n2√−µh/parenrightigg\n+\nπ\nρ\n/parenleftig\n−1\nα√−µh+2\nα2s/parenrightig\ncos/parenleftig\n−2√−µ\nαξ/parenrightig\n+/parenleftig\n−1√−µh+(3+α2)\nαs/parenrightig\nsin/parenleftig\n−2√−µ\nαξ/parenrightig\n/parenleftig\n1√−µh−(3+α2)\nαs/parenrightig\ncos/parenleftig\n−2√−µ\nαξ/parenrightig\n+/parenleftig\n−1\nα√−µh+2\nα2s/parenrightig\nsin/parenleftig\n−2√−µ\nαξ/parenrightig\n+\n+O(e−2√−µξ),\nwhereρ:= exp(π/α)−exp(−π/α). One readily verifies that the oscillatory part in the\nexpression above vanishes if and only if sandhare zero and thus we infer that the\nheteroclinic connection cannot be between equilibria to first order in the parameters.\nIn order to detect cancellations of these oscillatory parts for high er orders of sandh, we\nnext consider the behavior of the quantity (13) with respect to pa rameter perturbations.\nWith slight abuse of notation, for u= (θ,p,q)Twe writeH(u;η) :=H(p,q) evaluated\n35at parameters η, and other parameters at some fixed value, and we always consider the\nheteroclinic solutions from Corollary 1.\nOur strategy in the following steps is as follows: we utilize the quantity Hbecause\nlimξ→+∞Halways exists along these solutions. In order to distinguish whether this\nlimit is an equilibrium or a periodic orbit, we consider\n/tildewideH(u;η):=H(u;η)−H(Zπ;η),\ni.e., the difference of the H-values of the (parameter dependent) equilibrium Zπand the\nlimit ofuasξtends to infinity. Expanding /tildewideHin the limit ξ→ ∞with respect to the\nparameterηyields conditions for periodic asymptotics. In the following, subindice s of\nHdenote partial derivatives, e.g. Hu=∂uH.\nClearly,H(u0;η0) =H(Zπ(η0);η0), thus/tildewideH0= 0 and, since equilibria are critical points\nofH, we have/tildewideHu(u0;η0) =/tildewideHη(u0;η0) = (0,0,0)T. The second derivative is given by\nd2\ndη2/tildewideH=/an}bracketle{tuη,/tildewideHuuuη/an}bracketri}ht+/an}bracketle{tuη,/tildewideHuη/an}bracketri}ht+/an}bracketle{t/tildewideHηu,uη/an}bracketri}ht, (29)\nsince/tildewideHηηis the zero matrix, /tildewideHuthe zero vector, and /tildewideHuη=/tildewideHT\nηu. Thus\n/tildewideH(u0+uηη;η) =1\n2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η\n−1\n2/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig\nZπ\nη(η0)η/parenrightbig\n−/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuη(Zπ(η0);η0)η+O/parenleftbig\n/bardblη/bardbl3/parenrightbig\n.(30)\nWith the derivatives Huu,Huηin (30) given by\nHuu(u;η) =\n0 0 0\n02\nq−s/2−2p−αs\n(q−s/2)2\n0−2p−αs\n(q−s/2)22p2−αsp+h−β−/α+µ−s2/4\n(q−s/2)3\n,\nHuη(u;η) =\n0 0 0\n0p−αq\n(q−s/2)2 0\nβ/α\n(1−ccp)2(q−s/2)2−p2−αpq−αs\n2p−s\n2q+h−β−/α+µ\n(q−s/2)3 −1\n(q−s/2)2\n,\nfor the right hand side of (30) in the limit ξ→+∞we obtain\n1\n2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η=−αβ2\n4µ√−µc2\ncp\n+(4+5α2+α4)(α2ρ2−4(1+α2)π2)\n4α3ρ2√−µ(s−s0)2−α4ρ2−4(1+α2)π2\n4αρ2µ√−µ(h−h0)2\n−αβ(2+α2)\n2µccp(s−s0)+α2β\n2µ√−µccp(h−h0)\n+(2+α2)(α4ρ2−4(1+α2)π2)\n2α2ρ2µ(s−s0)(h−h0),\n36as well as\n1\n2/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig\nZπ\nη(η0)η/parenrightbig\n+/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuη(Zπ(η0);η0)η=\n−αβ2\n4µ√−µc2\ncp+α(4+5α2+α4)\n4√−µ(s−s0)2−α3\n4µ√−µ(h−h0)2\n−αβ(2+α2)\n2µccp(s−s0)+α2β\n2µ√−µccp(h−h0)+α2(2+α2)\n2µ(s−s0)(h−h0).\nTherefore, the expansion in the limit ξ→+∞is independent of ccpand reads\nlim\nξ→∞/tildewideH(u0+uηη;η) =−(1+α2)2(4+α2)π2\nα3ρ2√−µ(s−s0)2\n−2(1+α2)(2+α2)π2\nα2ρ2µ(s−s0)(h−h0)\n+(1+α2)π2\nαρ2µ√−µ(h−h0)2+O/parenleftbig\n/bardblη−η0/bardbl3/parenrightbig\n.(31)\nRecallρ= exp(π/α)−exp(−π/α). One readily verifies that the resulting (binary)\nquadratic form of (31) is negative definite for all α >0 so the only solution to the\nleading order problem\nd2\ndη2/tildewideH(u0;η0) = 0\nis thetrivial one( s,h) = (0,0), and anynon-trivial solution satisfies |s−s0|2+|h−h0|2=\nO(|ccp|3).\nIn particular, for ccp= 0 there is a neighborhood of ( s0,h0) such that the only solution\nis the trivial one, which is therefore also the case in the LLG equation . In caseccp/ne}ationslash= 0,\nhigher orders may lead to a solution with non-zero sand/orh, but there is numerical\nevidence that such solutions do not exist (see §4 for details).\n6.2 Proof of Theorem 3\nThe idea of the proof is to apply Lyapunov-Schmidt reduction, i.e., to determine a bi-\nfurcation equation whose solutions are in one-to-one correspond ence with heteroclinic\nconnections between equilibria (7) near one of the explicit solutions u0from (17) con-\nnecting the equilibria Z0\n−andZπ\n−. In the present context this is known as Melnikov’s\nmethod, see for example [22].\nRecallu0corresponds to a homogeneous DW for ccp= 0 with speed s0and rotation\nfrequency Ω 0given by (19). In the present codim-2 parameter regime we will show\nthat the bifurcation equation defines a codimension two bifurcation curve in the three-\ndimensional parameter space ( ccp,s,Ω), which passes through the point (0 ,s0,Ω0). The\nmain part of the proof is to show the existence of certain integrals f or the considered\nparameter set. These integrals are almost identical to the ones st udied within the proof\nof Theorem 2 and we use the same approach.\n37In this section we denote the parameter vector by η:= (ccp,s,Ω)T∈R3, with initial\nvalueη0= (0,s0,Ω0)Tcorresponding to the unperturbed values. The solutions of the\nperturbedsystemcloseto u0hastheform u(ξ;η) =u0(ξ)+uη(ξ;η0)(η−η0)+O(/bardblη−η0/bardbl2),\nwhereuη= (θη,pη,qη)T=O(/bardblη−η0/bardbl).\nAs discussed in Appendix 6.1, the linearization (23) of system (7) aro und the unper-\nturbed heteroclinic connection u0has the fundamental solution matrix Φ( ξ) as defined\nin(26). Inthepresent codim-2casewithdim( W0\nu) = dim(Wπ\ns) = 1inR3, thebifurcation\nequationM(η) = 0 entails two equations. Here M(η) measures the displacement of the\nmanifoldsW0\nuandWπ\ns, andwe willchoose thistobenearthepoint u0(0) =/parenleftbigπ\n2,√−µ,0/parenrightbigT\nin the directions given by vectors v1(0) andv2(0) from adjoint solutions as detailed be-\nlow. From the Taylor expansion M(η) =Mη(η0)(η−η0)+O(/bardblη−η0/bardbl2) we infer by the\nimplicit function theorem that a full rank of Mη(η0) implies a one-to-one correspondence\nof solutions to the bifurcation equation with elements in the kernel o fM(η0).\nIn order to compute Mη(η0) and its rank, we project onto the transverse directions to\nu0, which means to project the inhomogeneous part of equation (23) onto two linearly\nindependent bounded solutions v1,v2of the adjoint variational equation v′=−AT·v,\nwhere\nAT=\n√−µcos(θ0) 0 0\nsin(θ0)−αs0−2√−µcos(θ0) −s0\n0 s0 −αs0−2√−µcos(θ0)\n.\nThe solutions are given in terms of (25) by\nv1=U1×U2\ndetΦ=\n0\n−1\nr(ξ)sin(−s0ξ)\n1\nr(ξ)cos(−s0ξ)\nandv2=U3×U1\ndetΦ=\n0\n1\nr(ξ)cos(−s0ξ)\n1\nr(ξ)sin(−s0ξ)\n.\nImplementing the projection onto these, we obtain the so-called Melnikov integral\nMη(η0):=+∞/integraldisplay\n−∞(v1,v2)T·fη(u0;η0)dξ\n=/parenleftbigg\nβICCα√−µIS−√−µICIS+αIC\nβICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg\n,(32)\nwhere\nICC:=+∞/integraldisplay\n−∞/parenleftbig\n1−e2√−µξ/parenrightbig\neαs0ξ+2√−µξcos(−s0ξ)\n/parenleftbig\n1+e2√−µξ/parenrightbig3dξ,\nICS:=+∞/integraldisplay\n−∞/parenleftbig\n1−e2√−µξ/parenrightbig\neαs0ξ+2√−µξsin(−s0ξ)\n/parenleftbig\n1+e2√−µξ/parenrightbig3dξ,\n38IC:=+∞/integraldisplay\n−∞eαs0ξcos(−s0ξ)/parenleftbig\n1+e2√−µξ/parenrightbig\n·/parenleftbig\n1+e−2√−µξ/parenrightbigdξ,\nIS:=+∞/integraldisplay\n−∞eαs0ξsin(−s0ξ)/parenleftbig\n1+e2√−µξ/parenrightbig\n·/parenleftbig\n1+e−2√−µξ/parenrightbigdξ.\nWe next show that the second and thirdcolumns in (32) have non-va nishing determinant\nso that the rank is always 2, in particular also for β= 0.\nFor brevity, we present the calculations of ICCandICSonly, which are based on the\nsame idea as the computations in Appendix 6.1. The solutions for ICandIScan be\ncomputed in an analogous way.\nFrom Appendix 6.1 we know that the following integral would not exist in cases0=\n2√−µ/αand one readily verifies the existence for s0= 0. Therefore, we first assume\n0<s0<2√−µ/αfor the moment and discuss the case s0= 0 later. We set\nI:=+∞/integraldisplay\n−∞g(ξ)dξ, g(ξ):=/parenleftbig\n1−e2√−µξ/parenrightbig\ne(αs0+2√−µ)ξe−is0ξ\n/parenleftbig\n1+e2√−µξ/parenrightbig3,\nso thatICC= Re(I) andICS= Im(I). Utilizing the same idea for the contour integral\nas in Appendix 6.1 (cf. Figure 14) and the residue theorem, we obtain\nI−eαs0iπ√−µ·es0π√−µ·I= 2πi/summationdisplay\nRes(g).\nThe function ghas a pole of order three atiπ\n2√−µand theLaurentseries gives\nRes(g) =−(α−i)2s2\n0\n8µ√−µexp/parenleftbigg\n(α−i)iπs0\n2√−µ/parenrightbigg\n,\nand therefore\nI=2πs2\n0eπs0\n2√−µ/bracketleftig\n−2αcos/parenleftig\nπαs0\n2√−µ/parenrightig\n+(α2−1)sin/parenleftig\nπαs0\n2√−µ/parenrightig/bracketrightig\n8µ√−µ/parenleftig\n1−eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\n−ieπs0√−µsin/parenleftig\nπαs0√−µ/parenrightig/parenrightig\n+i2πs2\n0eπs0\n2√−µ/bracketleftig\n(1−α2)cos/parenleftig\nπαs0\n2√−µ/parenrightig\n−2αsin/parenleftig\nπαs0\n2√−µ/parenrightig/bracketrightig\n8µ√−µ/parenleftig\n1−eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\n−ieπs0√−µsin/parenleftig\nπαs0√−µ/parenrightig/parenrightig.\nSeparating the real and imaginary parts of Iwe get\nICC=πs2\n0√−µeπs0\n2√−µ\n4µ2·2α/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n+(1−α2)/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\n39ICS=πs2\n0√−µeπs0\n2√−µ\n4µ2·(1−α2)/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n+2α/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig .\nAs mentioned before, one analogously gets\nIC=πs0eπs0\n2√−µ\n2µ·/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n−α/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\nIS=πs0eπs0\n2√−µ\n2µ·α/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n+/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig .\nHaving the explicit expressions for IC,IS,ICC, as well as ICS, we can study the rank\nof (32) in case 0 < s0<2√−µ/α. The determinant of the second and third column\nof (32) simplifies to\n(αIS−IC)2+(IS+αIC)2=/parenleftbig\n1+α2/parenrightbig2π2s2\n0eπs0√−µ/ne}ationslash= 0,∀s0/ne}ationslash= 0.\nTherefore,Mη(η0) has full rank and we obtain\nM(η) =/parenleftbiggβICCα√−µIS−√−µICIS+αIC\nβICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg\n·(η−η0)+O/parenleftbig\n/bardblη−η0/bardbl2/parenrightbig\n.(33)\nIn the remaining case s0= 0, which means h=β/α, we haveICC=ICS=IS= 0,\nIC=1\n2√−µ, and thus\nMη(η0) =/parenleftigg\n0−1\n2α\n2√−µ\n0−α\n2−1\n2√−µ/parenrightigg\n, (34)\nwhose rank is always 2. Thus the splitting directions are independent ofccpin first order\nin caseh=β/α. Moreover, note that the splitting in sis independent of the anisotropy\nµto first order in case h=β/α. This completes the proof of Theorem 2.\n40"    },    {        "title": "2002.12255v1.Ultrafast_magnetization_dynamics_in_half_metallic_Co__2_FeAl_Heusler_alloy.pdf",        "content": "Ultrafast magnetization dynamics in half-metallic Co 2FeAl\nHeusler alloy\nR. S. Malik,1E. K. Delczeg-Czirjak,1D. Thonig,2R. Knut,1I. Vaskivskyi,1\nR. Gupta,3S. Jana,1R. Stefanuik,1Y. O. Kvashnin,1S. Husain,3A.\nKumar,3P. Svedlindh,3J. S oderstr om,1O. Eriksson,1, 2and O. Karis1\n1Department of Physics and Astronomy,\nUppsala University, Box 516, SE- 75120 , Uppsala, Sweden\n2School of Science and Technology,\nOrebro University, SE- 70182 Orebro, Sweden\n3Department of Materials Science and Engineering,\nUppsala University, Box 534, SE- 75121 , Uppsala, Sweden\nAbstract\nWe report on optically induced, ultrafast magnetization dynamics in the Heusler alloy Co 2FeAl,\nprobed by time-resolved magneto-optical Kerr e\u000bect. Experimental results are compared to results\nfrom electronic structure theory and atomistic spin-dynamics simulations. Experimentally, we \fnd\nthat the demagnetization time ( \u001cM) in \flms of Co 2FeAl is almost independent of varying structural\norder, and that it is similar to that in elemental 3d ferromagnets. In contrast, the slower process\nof magnetization recovery, speci\fed by \u001cR, is found to occur on picosecond time scales, and is\ndemonstrated to correlate strongly with the Gilbert damping parameter ( \u000b). Our results show\nthat Co 2FeAl is unique, in that it is the \frst material that clearly demonstrates the importance of\nthe damping parameter in the remagnetization process. Based on these results we argue that for\nCo2FeAl the remagnetization process is dominated by magnon dynamics, something which might\nhave general applicability.\n1arXiv:2002.12255v1  [cond-mat.mtrl-sci]  27 Feb 2020Studies of ultrafast demagnetization was pioneered by Beaurepaire et al. [1], who demon-\nstrated that the optical excitation of a ferromagnetic material - using a short pulsed laser\ncould quench the magnetic moment on sub-picosecond timescales. The exact underlying mi-\ncroscopic mechanisms responsible for the transfer of angular momentum have been strongly\ndebated for more than 20 years [2{4]. Ultrafast laser-induced demagnetization has now be-\ncome an intense \feld of research not only from fundamental point of view but also from\na technological aspect, due to an appealing possibility to further push the limits of oper-\nation of information storage and data processing devices [5]. Both experiment [4{15] and\ntheory [16{23] report that all of the 3d ferromagnets (Fe, Ni and Co) and their alloys, show\ncharacteristic demagnetization times in the sub-picosecond range, while 4f metals exhibit a\ncomplicated two-step demagnetization up to several picoseconds after the excitation pulse\n[4, 24].\nIn this work, we have made element speci\fc investigations of the ultrafast magnetiza-\ntion dynamics of a half-metallic Heusler alloy. This class of alloys has been investigated\nintensively, especially concerning the magnetic properties, ever since the discovery in 1903,\nwhen Heusler et al. reported that alloys like Cu 2MnAl exhibit ferromagnetic properties,\neven though none of its constituent elements was in itself ferromagnetic [25]. The ferro-\nmagnetic properties were found to be related to the chemical ordering [26]. One of the key\nfeatures of several Heusler alloys is their unique electronic structure, where the majority\nspin band-structure has a metallic character while the minority spin band is semiconducting\nwith a band gap. Such materials are also referred to as half-metallic ferromagnets (HMFs)\nand were initially predicted by de Groot et al. [27], based on electronic structure theory.\nHalf-metals ideally exhibit 100% spin-polarization at the Fermi level. This exclusive prop-\nerty makes them candidates to be incorporated in spintronic devices, e.g. spin \flters, tunnel\njunctions and giant magneto-resistance (GMR) devices [28{31]. One of the advantages of\nHeusler alloys with respect to other half-metallic system, like CrO 2and Fe 3O4, are their\nrelatively high Curie temperature ( Tc) and low coercivity ( Hc) [32, 33]. Heusler alloys are\nalso appealing for spintronic applications due to the low Gilbert damping, which allows for\na long magnon di\u000busion length [34{39]. It has been shown that the low value of the Gilbert\ndamping constant is related with the half-metallicity [37, 38]. The origin of the band gap\nand the mechanism of half-metallicity in these materials have been studied by using \frst\nprinciple electronic structure calculations [40{43]. The half-metallic property is furthermore\n2known to be very sensitive to structural disorder [41{45]. From a fundamental point of view,\nit is intriguing to ask, how the band gap in the minority spin channel e\u000bects the ultrafast\nmagnetization dynamics of Heusler alloys [46, 47]. It has already been reported that some\nof the half-metals like CrO 2and Fe 3O4exhibit very slow dynamics, involving time-scales of\nhundreds of picoseconds, [46, 47] while several Co-based Heusler alloys show a much faster\ndemagnetization, similar to the time-scales of the elemental 3d-ferromagnets [48{50]. The\nfaster dynamics of these Heuslers has been discussed in Ref.[46] to be due to the fact that\nthe band gap in the minority spin channel is typically around 0 :3\u00000:5 eV, which is smaller\nthan the photon energy (1 :5 eV) of the exciting laser. It is also smaller than the band gap\nof CrO 2and Fe 3O4. Importantly, the Heusler alloys o\u000ber the possibility to study magneti-\nzation dynamics, as a function of structural order, since they normally can be prepared to\nhave a fully ordered L21phase, a partially ordered B2 phase, and a completely disordered\nA2 phase. The structural relationships of these phases are described in the Supplemental\nMaterial (SM) [51].\nWe have here studied the optically induced, ultrafast magnetization dynamics of Co 2FeAl\n(CFA) \flms, using time-resolved magneto-optical Kerr e\u000bect (TR-MOKE) as described in\nRef. [52]. By control of the growth temperature, CFA alloy forms with varying degree of\nstructural order, in a continuous way between the A2 and B2 phases, as well as between\ntheB2 andL21phases [53, 54]. We present data from four CFA samples, grown at 300 K,\n573 K, 673 K, and 773 K respectively. We henceforth denote each sample by its growth\ntemperature as a subscript, e.g. CFA 300K. As evidenced by X-ray di\u000braction, the sample\ngrown at 300 K is found to exhibit the A2 phase, while the samples grown at 573 K and\n673 K predominantly exhibit the B2 phase. The sample grown at 773 K is found to exhibit a\npure B2 phase [54]. The value of the Gilbert damping \u000bis found to monotonously decrease\nwith annealing temperature and is thus lowest for the sample grown at 773 K [55].\nCalculations based on density functional theory (DFT) of the magnetic moment, Heisen-\nberg exchange interaction and the Gilbert damping parameter are described in detail in\n(SM) [51]. These parameters were used in a multiscale approach to perform atomistic mag-\nnetization dynamics simulations, described in Sec.S1 of (SM) [51]. Here we employed the\ntwo temperature model (2TM) for the temperature pro\fle of the spin-system. In the 2TM,\nthe spin temperature increases due to the coupling to the hot-electron bath, that is excited\nby the external laser pulse. In the simulations we used a peak temperature in the 2TM of\n30.70.750.80.850.90.951.01.051.1M/M(0)05 1 0 1 5A2CoFe05 1 0 1 5t( p s )B2\n05 1 0 1 5 2 0L21FIG. 1. (Color online) Simulations of ultrafast dynamics of Co 2FeAl in the di\u000berent structural\nphasesA2 (left panel), B2 (central panel), and L21(right panel). The demagnetization is shown\nelement resolved (blue line - Co, red line - Fe). The peak temperature is 1200 K. The dotted line\nindicates the equilibrium magnetization at T= 300 K.\n1200 K. A full description of the 2TM and the details of all spin-dynamics simulations are\ndescribed in Sec.S2 of SM [51].\nThe results of the simulations are shown in Fig. 1, for the A2,B2 andL21phases. It can\nbe seen that the di\u000berent phases react di\u000berently to the external stimulus. In general, this\nmodel provides a dynamics that is controlled by i)the temperature of the spin-subsystem,\nii)the strength of the magnetic exchange interaction and iii)the dissipation of angular\nmomentum and energy during the relaxation of the atomic magnetic moments (Gilbert\ndamping) [56]. Before continuing the discussion, we note that the average magnetization,\nM, of element Xis calculated as MX=P\nicX\niMX\ni=4P\nicX\ni, wherecX\niis the concentration of the\nparticular element Xin the particular phase and iruns over the four nonequivalent sites of\nthe unit cell. After the material demagnetizes, the spin temperature eventually drops and\nthe average magnetization returns to its initial value after 10 \u000020 ps (cf. Fig. 1).\nTo estimate the time constants of the demagnetization \u001cMand remagnetization ( \u001cR)\n4processes, in an element-speci\fc way, we \ft both the theoretical and experimental transient\nmagnetizations by a double exponential function [57]. We show results of \u001cMand\u001cRin\nFig. 2 for the A2 and B2 phase, as well as for alloys with intermediate degree of disorder\n(described in Sec.S2 of SM [51]).\n4.5\n4.0\n3.5\n3.0\n2.5\n2.0R (ps)\n0 20 50 80 100\nAmount of B2 order [%] Fe\n Co\n Remagnetization (b)2.0\n1.5\n1.0\n0.5\n0.0M (ps)\n Fe\n CoDemagnetization (a)\nFIG. 2. (Color online) Element resolved relaxation times of Co 2FeAl, from simulations of alloys\nwith varying amounts of A2!B2 phase. 0 corresponds to pure A2 phase while 100 corresponds to\npureB2 phase. Panel (a) shows the demagnetization time and panel (b) shows the remagnetization\ntime. Both time constants are obtained from \ftting the time trajectory of MX(t) by a double-\nexponential function (see text).\nThe theoretical demagnetization time is seen from Fig. 2 to typically be around 1 ps,\nwhereas the remagnetization time is 2 \u00005 ps. Going from the A2 to the B2 alloy, both\ntimes increase, albeit the simulations show a stronger increase of the remagnetization time\n5as function of alloy composition. We also note that the relevant time scale is somewhat\nlarger for Fe than for Co, and the ratio between them,\u001cFe=\u001cCo, grows when going from A2 to\nB2 phase.\nFigures 3 (a-d) shows the measured magnetization dynamics of CFA \flms that were\ngrown at di\u000berent temperatures (see SM, Sec.S3 for thin \flms synthesis, and Sec.S4 for\ndetails on the experimental measurements [51]). The inset shows the observed magnetization\ndynamics up to\u00181 ps. For all samples, the data for Fe (red) and Co (blue) show similar\ndemagnetization dynamics in the \frst few hundred femtoseconds, whereupon di\u000berences in\nthe magnetization dynamics become visible, especially on the picosecond timescale.\n1.000.950.900.850.800.75 Normalized Asymmetry (arb.units)1086420 Time Delay (ps) A2(a)\n1.000.950.900.850.800.75A/A00.80.40.0FeCo1086420 Time Delay (ps) 90% B2(b)\n1.000.950.900.850.800.75A/A00.80.40.0FeCo  1086420 Time Delay (ps) 90% B2(c)\n1.00.90.80.7A/A00.80.40.0 Fe Co1086420 Time Delay (ps) 100% B2(d)\n1.00.90.80.7A/A00.80.40.0FeCo\nFIG. 3. Measured element-speci\fc Fe (red) and Co (blue) magnetization dynamics of Co 2FeAl.\nSamples are denoted by the growth temperature in each case. The red and blue lines correspond\nto \ftted data (see text). (a) 300 K (100% A2phase), (b) 573 K (90% B2 phase), (c) 673 K (90%\nB2 phase), and (d) 773 K (100% B2 phase). The insets show the demagnetization dynamics up to\n\u00181 ps. All of the measurements were performed with similar pump-\ruence (for details, see Sec.S4\nof SM).\nFigure 4 (a-b) shows the measured values of the demagnetization and remagnetization\ntime constants, for the four di\u000berent growth temperatures, representing di\u000berent degree of\ndisorder in Co 2FeAl, along the alloy path A2!B2. It may be seen that the \u001cMfor Fe\nand Co is the same within the error bars for all four samples, regardless of the degree of\nstructural ordering (Fig. 4a). It may also be noted that the measured \u001cMfor CFA is similar\nto that of 3d transition metals [48, 49] and very much shorter than that of CrO 2or Fe 3O4.\nDemagnetization times that are independent on degree of structural ordering is interest-\n6ing, since it can be expected that the presence of structural disorder in Heusler alloys ought\nto result in a lower degree of spin polarization of the electronic states (i.e. an increased\ndensity of states (DOS) at the Fermi level in the minority band). This is expected to en-\nhance spin-\rip scattering, with an accompanying speed-up of the demagnetization dynamics\n[46, 47]. The electronic structure calculation of CFA also shows that the DOS at the Fermi\nlevel varies with di\u000berent structural phases (analyzed in the Sec.S1 of SM [51]). The A2\nphase has a large number of states at the Fermi level, while the L21phase, and to some\nextent the B2 phase, has a low amount [54]. Despite these di\u000berences in the electronic struc-\nture, the measured demagnetization dynamics shown in Fig. 4(a) is essentially independent\non degree of structural ordering.\nOn longer time-scales, there is a signi\fcant e\u000bect of structural ordering on the observed\nmagnetization dynamics, which becomes particularly relevant for the remagnetization pro-\ncess. As seen in Fig. 4b, there is a monotonous increase of remagnetization time, \u001cR, with\nincreasing growth temperature and hence the degree of ordering along the A2!B2 path.\nThe sample grown at 300 K with A2 phase, exhibits the fastest remagnetization dynam-\nics (\u001cR). With increasing growth temperature and corresponding increase in the structural\nordering along the A2!B2 path, a distinct trend of increasingly slower remagnetization\ndynamics is observed.\nThe most conspicuous behaviour of the measured magnetization dynamics, and its depen-\ndence on the degree of ordering, concerns the remagnetization time (Fig. 4b). The time-scale\nof the remagnetization process is su\u000eciently long to allow for an interpretation based on\natomistic spin-dynamics. Two materials speci\fc parameters should be the most relevant to\ncontrol this dynamics; the exchange interaction, as revealed by the local Weiss \feld, and\nthe damping parameter. In the Sec.S2 of SM [51], we report on the calculated Weiss \felds\nand damping parameters. It is clear from these results that the trend in the experimental\ndata shown in Fig. 4b, can not be understood from the Weiss \feld alone, whereas an expla-\nnation based on the damping is more likely. In order to illustrate this, we show in Fig. 5\nthe inverse of the measured remagnetizatiom time compared to the theoretically calculated\ndamping and experimental measured damping through ferromagnetic resonance (FMR) (de-\nscribed in Sec.S6 of SM)[51]. The \fgure shows that the damping is large in the completely\ndisordered A2 phase and for a large range of structural orderings, which comes out from\nboth theory and experiment. The \fgure also demonstrates that the inverse of the measured\n7300250200150100500 τM (fs)  Fe Co(a)\n3.02.52.01.5 τR (ps)CFA300 K CFA573 K CFA673 K CFA773 K (b)FeCo FIG. 4. Measured magnetization times for the investigated Co 2FeAl alloys. In (a) the demagneti-\nzation time, \u001cM, is shown and in (b) the remagnetization time, \u001cR, is plotted.\nremagnetization time scales very well with both the calculated damping and experimentally\nmeasured damping . According to the \fgure, a large damping parameter corresponds to\nfaster remagnetization dynamics in the measurements.\nCo2FeAl is, to the best of our knowledge, the \frst system where experimental observations\nand theory point to the importance of damping in the process of ultrafast magnetization\ndynamics. We note that this primarily is relevant for the remagnetization process; the\ninitial part of the magnetization dynamics (\frst few hundred fs) is distinctly di\u000berent. In\nthe demagnetization we observe a similar behaviour for Fe and Co in all samples, and\nan insensitivity of the demagnetization times in relation to structural ordering. Also, the\n80.550.500.450.400.350.300.25 (τr )-1 (1/ps) 43210Damping ( x 10-3)\n100806040200Amount of B2 ordering (%)Remagnetization timeTheoretical dampingExperimental dampingFIG. 5. The relationship of inverse of the measured remagnetization time (right y-axis) and the-\noretically calculated and experimentally measured Gilbert damping (left y-axis) in Co 2FeAl for\nvarying amount of B2 order along the A2!B2 path, i.e. 0 corresponds to pure A2 phase while\n100 corresponds to pure B2 phase.\nmeasured and theoretical demagnetization times evaluated from atomistic spin-dynamics\nsimulations, do not agree. Other mechanisms, of electronic origin, most likely play role in\nthis temporal regime.\nThe remagnetization process of Co 2FeAl alloys with varying degree of structural order,\nhighlights clearly the importance of the Gilbert damping and that magnon dynamics domi-\n9nates the magnetization at ps time-scales. The relevance of the Gilbert damping parameter\nfor ps dynamics is natural, since this controls angular momentum (and energy) transfer to\nthe surrounding. What is surprising with Co 2FeAl is the fact that other interactions (e.g.\nthe Weiss \feld) show such a weak dependence on the amount of structural diorder. This\nis fortuitous, since it allows to identify the importance of the Gilbert damping. A picture\nemerges from the results presented here, that the magnetization dynamics in general have\ntwo regimes; one which is primarily governed by electronic processes, and is mainly active\nin the \frst few hundered fs ( \u001cM), and a second regime where it is primarily magnons that\ngovern the remagnetiztion dynamics ( \u001cR).\nWe acknowledge support from the Swedish Research Council (VR, contracts 2019-\n03666,201703799, 2016-04524 and 2013-08316), the Swedish Foundation for Strategic Re-\nsearch, project SSF Magnetic materials for green energy technology under Grant No. EM16-\n0039, the Knut and Alice Wallenberg foundation, STandUP and eSSENCE, for \fnancial\nsupport. 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B\n81, 174401 (2010).\n13"    },    {        "title": "1803.00017v2.Roles_of_chiral_renormalization_on_magnetization_dynamics_in_chiral_magnets.pdf",        "content": "Roles of chiral renormalization on magnetization dynamics in chiral magnets\nKyoung-Whan Kim,1,\u0003Hyun-Woo Lee,2,yKyung-Jin Lee,3,4Karin Everschor-Sitte,5Olena Gomonay,5,6and Jairo Sinova5,7\n1Institut für Physik, Johannes Gutenberg-Universität Mainz, Mainz 55128, Germany\n2Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea\n3Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea\n4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea\n5Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 55128, Germany\n6National Technical University of Ukraine “KPI,\" Kyiv 03056, Ukraine\n7Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Praha 6, Czech Republic\n(Dated: July 28, 2021)\nInmetallicferromagnets,theinteractionbetweenlocalmagneticmomentsandconductionelectronsrenormal-\nizesparametersoftheLandau-Lifshitz-GilbertequationsuchasthegyromagneticratioandtheGilbertdamping,\nand makes them dependent on the magnetic configurations. Although the effects of the renormalization for\nnonchiral ferromagnets are usually minor and hardly detectable, we show that the renormalization does play a\ncrucial role for chiral magnets. Here the renormalization is chiral and as such we predict experimentally identi-\nfiableeffectsonthephenomenologyofmagnetizationdynamics. Inparticular,ourtheoryfortheself-consistent\nmagnetization dynamics of chiral magnets allows for a concise interpretation of domain wall creep motion. We\nalso argue that the conventional creep theory of the domain wall motion, which assumes Markovian dynamics,\nneedscriticalreexaminationsincethegyromagneticratiomakesthemotionnon-Markovian. Thenon-Markovian\nnature of the domain wall dynamics is experimentally checkable by the chirality of the renormalization.\nRenormalization is a useful concept to understand interac-\ntion effects between a physical system and its environment.\nIn metallic ferromagnets, magnetic moments experience such\nrenormalization due to their coupling to conduction electrons\nthrough exchange interactions. Spin magnetohydrodynamic\ntheory [1–3] examines the renormalization of dynamical pa-\nrameters in the Landau-Lifshitz-Gilbert (LLG) equation as\nfollows. Magnetization dynamics exerts a spin motive force\n(SMF) [4, 5] on conduction electrons, and the resulting spin\ncurrentgeneratesspin-transfertorque(STT)[6–8]thataffects\nthemagnetizationdynamics itself. Thisself-feedbackofmag-\nnetizationdynamics[9]renormalizestheGilbertdampingand\nthe gyromagnetic ratio. However, its consequences rarely go\nbeyond quantitative corrections in nonchiral systems [10–14]\nand are commonly ignored.\nChiralmagnetsareferromagnetsthatpreferaparticularchi-\nrality of magnetic texture due to spin-orbit coupling (SOC)\nand broken inversion symmetry. Examples include ferro-\nmagnets in contact with heavy metals, such as Pt [15] and\nthose with noncentrosymmetric crystal structures [16]. Mag-\nnetization dynamics in chiral magnets are usually described\nby generalizing the conventional LLG equation to include\nthe chiral counterpart of the exchange interaction called the\nDzyaloshinskii-Moriya interaction (DMI) [17–19] and that of\nSTTcalledspin-orbittorque(SOT)[20–23]. Thisdescription\nis incomplete, however, since it ignores the renormalization\nby the self-feedback of magnetization dynamics. Although\nthe renormalization in chiral magnets has been demonstrated\ntheoretically for a few specific models [24–27], most experi-\nmentalanalysesofchiralmagnetsdonottakeintoaccountthe\nrenormalization effect.\nInthiswork,wedemonstratethattherenormalizationinchi-\nralmagnetsshouldbechiralregardlessofmicroscopicdetails\nand these effects should be nonnegligible in chiral magnets\nwith large SOT observed in many experiments [21–23, 28–30]. Unlike in nonchiral systems, the chiral renormalization\ngenerates experimentally identifiable effects by altering the\nphenomenology of magnetization dynamics. This provides a\nuseful tool to experimentally access underlying physics. We\nillustratethiswiththefield-drivenmagneticdomainwall(DW)\nmotion with a controllable chirality by an external magnetic\nfield [31, 32]. We find that not only is the steady state DW\nvelocity chiral due to the chiral damping [25], but also the\neffective mass of the DW [33] is chiral due to the chiral gy-\nromagnetic ratio. The chiral gyromagnetic ratio also signifi-\ncantly affects the DW creep motion, which is one of the tech-\nniques to measure the strength of the DMI [32]. We argue\nthat the chiral gyromagnetic ratio is the main mechanism for\nthe non-energetic chiral DW creep velocity [34], contrary to\nthe previous attribution to the chiral damping [25, 34]. We\nalso highlight the importance of the tilting angle excitation\nand its delayed feedback to the DW motion. This has been\nignoredinthetraditionalcreeptheory[35,36]foralongtime,\nsince its effects merely alter the velocity prefactor which is\nindistinguishable from other contributions, such as the impu-\nrity correlation length [37]. However, in chiral magnets, it is\ndistinguishablebymeasuringtheDWvelocityasa functionof\nchirality (not a single value).\nTo get deep insight into the chiral renormalization, we\nadopt the self-feedback mechanism of magnetization dynam-\nics through conduction electrons and develop a general, con-\ncise, and unified theory for chiral magnets. There are several\nprevious reports on the anisotropic or chiral renormalization\nof the magnetic damping [24–26, 38] and the gyromagnetic\nratio[27,38,39]intheRashbamodel[40]. Tounifyandgen-\neralize the previous works, we start from the general Onsager\nreciprocityrelationandpredictallthecoreresultsoftheprevi-\nous reports. Our theory can be generalized to situations with\nanyphenomenologicalspintorqueexpression,whichcaneven\nbedeterminedbysymmetryanalysisandexperimentswithoutarXiv:1803.00017v2  [cond-mat.mes-hall]  14 Mar 20182\nMagnetization under\nchiral self-feedback Effective  equation of \nmotion for magnetization SOT \nchiral \nSMF\nLLG ( γ, α)\nchiral LLG ( ζγ , G )(a) (b) \nFIG.1. (a)Magnetizationdynamicsdescribedbytheunrenormalized\nLLG equation. The dynamics of magnetization and that of electrons\narecoupledtoeachotherbytheexchangeinteraction. (b)Aftertracing\nout the electron degree of freedom, the gyromagnetic ratio ( \u0010\r) and\nthe magnetic damping ( G) are chirally renormalized [Eq. (1)].\nknowing its microscopic mechanism. We provide a tabular\npicture(SeeTableIbelow)forphysicalunderstandingofeach\ncontribution to the chiral renormalization. Furthermore, one\ncan utilize the generality of the Onsager relation to include\nmagnon excitations [26], thermal spin torques [41], and even\nmechanical vibrations [42] in our theory.\nToexaminetheconsequencesofthechiralrenormalization,\nwestartfromthefollowingrenormalizedLLGequation,which\nwe derive in the later part of this paper,\n(\u0010\r)\u00001\u0001@tm=\u0000m\u0002He\u000b+\r\u00001m\u0002G\u0001@tm+\r\u00001Text;(1)\nwhere mis the unit vector along magnetization, \ris the un-\nrenormalized gyromagnetic ratio, He\u000bis the effective mag-\nneticfield,and Textreferstospintorqueinducedbyanexternal\ncurrent.\u0010andG,whicharegenerallytensorsandfunctionsof\nmand its gradients, address respectively the renormalization\nofthegyromagneticratioandthemagneticdamping,depicted\nin Fig. 1. If the renormalization is neglected, Eq. (1) reduces\nto the conventional LLG equation with \u0010= 1andG=\u000b,\nwhere\u000bistheunrenormalizedGilbertdamping. Otherwise \u0010\nandGare dependent on the chirality of magnetic texture. At\nthe end of this paper, we show that the chiral renormalization\nis completely fixed once the expressions of STT and SOT are\ngiven.\nWe first examine implications of the chiral renormaliza-\ntion on a few exemplary types of field-driven DW dynamics\n(Fig.2). Westartfrom He\u000b=H0+Hext+Hth,where H0is\nthe energetic contribution (without an external field), Hext=\n(Hx;0;Hz)is the external field, and Hthis a thermal fluctu-\nation field. We use the DW profile m(x) = (sin\u001esech[(x\u0000\nX)=\u0015];cos\u001esech[(x\u0000X)=\u0015];tanh[(x\u0000X)=\u0015])whereX,\n\u001eand\u0015aretheposition,thetiltingangle,andthewidthofthe\nDW, respectively. Taking Xand\u001eas the collective coordi-\nxyz\nφφ\nHxHz v(   )FIG. 2. Chiral dynamics of a DW between domains with m=\u0007^z\n(red and blue respectively). The DW chirality is characterized by\nthe DW tilting angle \u001e[the positivity (negativity) of \u001ecorresponds\nto the left-handed (right-handed) chirality], and can be controlled\nby an in-plane field ( Hx). The DW motion is driven by an applied\nfield (Hz). Measuring the DW velocity as a function of \u001e(orHx),\nthe difference between v(\u001e)andv(\u0000\u001e)gives the information of the\nchiral renormalization.\nnates, Eq. (1) gives\n\u000bX\ne\u000b\n\u0015dX\ndt+1\n\u0010e\u000bd\u001e\ndt=FX+\u0018X; (2a)\n\u00001\n\u0010e\u000bdX\ndt+\u000b\u001e\ne\u000b\u0015d\u001e\ndt=F\u001e+\u0018\u001e; (2b)\nwhereFX=\u001e= (\r=2)R\n(H0+Hext)\u0001(@X=\u001em)dxrefertothe\nforce onXand\u001e.\u0018X=\u001e= (\r=2)R\nHth\u0001(@X=\u001em)dxis the\nthermal force on Xand\u001e.\nTheeffectivedamping \u000bX=\u001e\ne\u000bandthegyromagneticratio \u0010e\u000b\nare given by\n\u000bX\ne\u000b=\u0015\n2Z\n(@Xm\u0001G\u0001@Xm)dx; (3a)\n\u000b\u001e\ne\u000b=1\n2\u0015Z\n(@\u001em\u0001G\u0001@\u001em)dx; (3b)\n\u0010\u00001\ne\u000b=1\n2Z\u0002\n(m\u0002@\u001em)\u0001\u0010\u00001\u0001@Xm\u0003\ndx:(3c)\nNote that without the chiral renormalization, Eq. (2) reduces\ntotheThieleequations[43]with \u000bX=\u001e\ne\u000b=\u000band\u0010e\u000b= 1. We\nemphasizethat \u000bX=\u001e\ne\u000band\u0010e\u000bdependonthetiltingangle \u001eand\nthus on the chirality of the DW. Figure 3 shows the \u001edepen-\ndenciesoftheseparameters. Theasymmetricdependenceson\n\u001econfirmtheirchiraldependences. Notethat,evenforpurely\nfield-drivenDWmotion,thechiraldependencesoftheparam-\netersaredeterminedbytheexpressionof current-induced spin\ntorque.\nWe first consider the steady-state dynamics of DW in the\nflow regime, where the effects of the pinning and the thermal\nforces are negligible. Then, translational symmetry along\nXguarantees the absence of contribution from H0toFX,\nthus only the external field contribution survives in the right-\nhand side of Eq. (2a), FX+\u0018X\u0019\u0000\rHz. In a steady state\n(d\u001e=dt = 0), Eq. (2a) gives the DW velocity as\nv\row=\u0000\r\u0015\n\u000bX\ne\u000bHz; (4)3\nα\u0001\u0002\u0002\u0001ϕα\u0001\u0002\u0002\u0001\u0001\nα\u0001\u0002\u0002ϕϕα\u0001\u0002\u0002ϕ\u0001\nζ\u0001\u0002\u0002\u0003ϕζ\u0001\u0002\u0002\u0003\u0001\n\u0001\u0002\u0003 \u0001\u0002\u0004 \u0001\u0002\u0001 \u0001\u0002\u0004 \u0001\u0002\u0003\u0001\u0002\u0005\u0001\u0002\u0004\u0001\u0002\u0006\u0001\u0002\u0001\u0001\u0002\u0006\u0001\u0002\u0004\u0001\u0002\u0005\nϕ π\u0007\b\tRighthandedchirality Left handedchirality\nFIG. 3. The effective dynamical parameters, \u000bX\ne\u000b(the red, solid\ncurve),\u000b\u001e\ne\u000b(the red, dashed curve), and \u0010\u00001\ne\u000b(the blue curve), as a\nfunction of the DW tilting angle \u001e. We take the phenomenological\nexpression of spin torque in magnetic bilayers [21–23, 30], which is\na typical example with large SOT: T= (\r~=2eMs)f(js\u0001r)m\u0000\n\f1m\u0002(js\u0001r)m+kSO(^z\u0002js)\u0002m\u0000\f2kSOm\u0002[(^z\u0002js)\u0002\nm]g, where each term refers to the adiabatic STT [44], nonadiabatic\nSTT[45,46],fieldlikeSOT[47,48],anddampinglikeSOT[30,49–\n51], induced by the spin current js. Here,Ms= 1000 emu =cm3is\nthesaturationmagnetization, e>0isthe(negative)electroncharge,\n^zistheinterfacenormaldirection, kSO= 1:3 (nm)\u00001characterizes\nthestrengthoftheSOT.Wetake \f1= 0:05,\f2= 5,\u0015= 8 nm,and\nthe electrical conductivity \u001b\u00001\n0= 6\u0016\ncm. The parameters are on\nthe order of the typical values for Pt/Co systems [28, 45, 52].\nwhichisinverselyproportionaltothechiraldamping \u000bX\ne\u000beval-\nuatedatthesteady-statetiltingangle \u001eeqforwhichd\u001e=dt = 0.\nAs\u001eeqcanbemodulatedby Hx,themeasurementof v\rowasa\nfunction ofHxprovides a direct test of the chiral dependence\non\u000bX\ne\u000b.\nAs an experimental method to probe the chiral dependence\nof\u0010e\u000b, we propose the measurement of the DW mass, called\nthe Döring mass [33]. It can be performed by examining\nthe response of DW under a potential trap to an oscillating\nfieldHz[53]. Unlike v\row,\u001eis not stationary for this case,\nand dynamics of it is coupled to that of X. Such coupled\ndynamicsof \u001eandXmakes\u0010e\u000brelevant. IntheSupplemental\nMaterial[54],weintegrateoutthecoupledequations[Eq.(2)]\nto obtain the effective Döring mass,\nmDW=1\n\u00102\ne\u000b2MsS\n\rjF0\n\u001e(\u001eeq)j; (5)\nwhereSis the cross-sectional area of the DW. Here, \u0010e\u000brep-\nresents a measurement of its value for \u001e=\u001eeq, which can be\nvariedbyHx.mDWprovidesanexperimentalwaytomeasure\nthe chiral dependence of \u0010e\u000b.\nIn the creep regime of the DW where the driving field is\nmuch weaker than the DW pinning effects, the implication of\nthechiralrenormalizationgobeyondmerelychiralcorrections\ntotheDWvelocity. TherecentcontroversiesonthechiralDW\ncreep speed vcreepmeasured from various experiments [32,\n34, 55, 56] require more theoretical examinations. Typically,vcreepis believed to follow the Arrhenius-like law vcreep =\nv0exp(\u0000\u0014H\u0000\u0016\nz=kBT)[35, 36], where v0is a prefactor, \u0016is\nthe creep exponent typically chosen to be 1/4 [57], and \u0014is\na parameter proportional to the DW energy density. Based\non the observation that the DMI affects \u0014, an experiment [32]\nattributedthechiraldependenceof vcreeptotheDMI.However\nlater experiments [34, 55, 56] found features that cannot be\nexplained by the DMI. In particular, Ref. [34] claimed that\nthe chiral dependence of vcreepis an indication of the chiral\ndamping [25], based on the observation v0/(\u000bX\ne\u000b)\u00001. On\nthe other hand, our analysis shows that the explanation of the\nchirality dependence may demand more fundamental change\nto the creep law, which assumes the dynamics of \u001eto be\nessentially decoupled from that of Xand thus irrelevant for\nvcreep. As a previous experiment on the DW creep motion in\na diluted semiconductor [58] argued the coupled dynamics of\n\u001eandXto be important, it is not a prioriclear whether the\nassumptionofdecoupling Xand\u001eholdsinthecreepregime.\nWe consider the coupling between the dynamics of Xand\n\u001easfollows. Afterthedynamicsof Xexcites\u001e,thedynamics\nof\u001eresults in a feedback to Xwith a delay time \u001c. Since the\ndynamics at a time tis affected by its velocity at past t\u0000\u001c,\nit is non-Markovian. The traditional creep theory takes the\nMarkovian limit ( \u001c!0), thus\u001e=\u001eeqat any instantaneous\ntime,decoupledfromthedynamicsof X. Toshowthecrucial\nroleofafinitefeedbacktime \u001c,wecalculatetheescaperateof\nthe DW over a barrier, which is known to be proportional to\nv0[37] and apply the Kramer’s theory [59] for barrier escape\nand its variations for non-Markovian systems [60, 61]. After\nsomealgebraintheSupplementalMaterial[54],Eq.(2)gives\nv0/\u001a(\u000bX\ne\u000b)\u00001\u001c\u00170\u001c\u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b(Markovian );\n\u0010e\u000b\u001c\u00170&\u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b(non-Markovian );\n(6)\nwhere\u00170is called the reactive frequency [61] and is on the\norder of 2\u0019times the attempt frequency ( \u00191 GHz[37]). We\nemphasizethatthetworegimesshowverydifferentbehaviorin\nthesenseofunderlyingphysicsaswellasphenomenology. The\nvalidityoftheMarkovianassumptiondependsonthetimescale\nof\u001ccomparedto \u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b. Sincethedampingissmall,itis\nnotguaranteedforoursituationtobeintheMarkovianregime.\nIndeed,wedemonstrateintheSupplementalMaterial[54]that\nthesecondregime(non-Markovian)inEq.(6)ismorerelevant\nwithrealisticvalues,thusthechiralityof v0mainlyoriginates\nfrom the gyromagnetic ratio, not the damping [34]. One can\nmeasure the chiral dependence of \u000bX\ne\u000band\u0010e\u000bfrom the flow\nregime[Eqs.(4)and(5)]andcomparetheirchiraldependences\ntothecreepregimetoobservethenon-Markoviannatureofthe\nDWdynamics. Thisadvantageoriginatesfromthepossibility\nthatonecanmeasuretheDWvelocityasa functionofchirality,\nin contrast to nonchiral magnets where one measures the DW\nvelocity as a single value.\nSo far, we present the role of the chiral renormalization for\ngiven renormalized tensors Gand\u0010. To provide underlying\nphysical insight into it, we present a analytic derivation of\nEq. (1) in general situations. We start from the LLG equation4\n\r\u00001@tm=\u0000m\u0002He\u000b+\r\u00001\u000bm\u0002@tm+\r\u00001Tandreferto\nthe scenario illustrated in Fig. 1. Note that There includes a\ncontribution from an internally generated SMF ( Tint) as well\nas that from an external current [ Textin Eq. (1)]. We write\ndown the spin torque in a general current-linear form T=\n\u0000(\r~=2eMs)m\u0002P\nuAu(m)js;u,whereurunsoverx;y;z.\nHere the spin current jsis split into an internally generated\nSMF [4, 5] js;intand the external current js;ext. The former\nisproportionalto @tm,thusitrenormalizesthegyromagnetic\nratio and the damping. The latter generates Textin Eq. (1).\nTheexpressionof js;intisgivenbytheOnsagerreciprocityof\nSTT and SMF [62]: js;int;u=\u0000(\u001b0~=2e)Au(\u0000m)\u0001@tm,\nwhere\u001b0is the charge conductivity [63]. Substituting this to\nTint= (\r~=2eMs)m\u0002P\nuAu(m)js;int;ugivestheeffective\nLLGequation \r\u00001@tm=\u0000m\u0002He\u000b+\r\u00001m\u0002A\u0001@tm+\n\r\u00001Text, whereA=\u000b+\u0011P\nuAu(m)\nAu(\u0000m),\u0011=\n\r~2\u001b0=4e2Msand\nisthedirecttensorproduct. Asaresult,\nTintis taken care of by renormalizing \u000bintoAin the LLG\nequation.\nTherenormalizeddampingandgyromagneticratioaregiven\nby separating different contributions of Awith different time\nreversal properties. A damping contribution is required to\nbe dissipative (odd in time reversal), whereas a gyromagnetic\nterm should be reactive (even in time reversal). Separating\nthese gives Eq. (1) where G= (A+AT)=2and\u0010\u00001=\n1\u0000m\u0002(A\u0000AT)=2. Theparticularchoicefortheadiabatic\nSTTandthenonadiabaticSTT Au(m) =m\u0002@um+\f@um\nreproduces the renormalized LLG equation for nonchiral sys-\ntems [1–3]. When one uses Au(m)for a particular chiral\nsystem, Eq. (1) produces the effective LLG equation for it, as\nreported by a numerical study for a one-dimensional Rashba\nmodel [27].\nIn chiral magnets, it is known that spin torque includes two\nmore contributions called fieldlike SOT [47, 48] and damp-\ninglikeSOT[30,49–51]. Thecharacterizationoffieldlikeand\ndampinglikeSOTisregardlessofthechoiceofSOC,sinceitis\ndetermined by the time reversal characteristic. Since Au(m)\nconsistsoffourcontributions,thereare16contributionsinthe\nfeedback tensor \u0001A=\u0011P\nuAu(m)\nAu(\u0000m)for each\nu. We tabulate all terms of \u0001Ain Table I. The contributions\nSTT:Ax(m)\nSMF:\nAx(\u0000m)Adiabatic\nm\u0002@xmNonadiabatic\n\f1@xmFLT\nkSOm\u0002(^y\u0002m)DLT\n\f2kSO^y\u0002m\nm\u0002@xmG\u0010\u00001G\u0010\u00001\n\u0000\f1@xm\u0010\u00001G\u0010\u00001G\nkSOm\u0002(^y\u0002m)G\u0010\u00001G\u0010\u00001\n\u0000\f2kSO^y\u0002m\u0010\u00001G\u0010\u00001G\nTABLE I. Example characterization of contributions in Ax(m)\nAx(\u0000m). Counting orders of gradients and mgives the conven-\ntional (white), chiral (lighter gray), or anisotropic (darker gray) con-\ntributions to the gyromagnetic ratio ( \u0010\u00001) or the damping (G). The\nform of the fieldlike SOT (FLT) and dampinglike SOT (DLT) are\ntaken from magnetic bilayers [30, 47–51] for illustration, but the\ncharacterization procedure works generally.withthewhitebackgroundarezerothorderinSOCbutsecond\norder in gradient and are the conventional nonlocal contribu-\ntions [3, 65]. Those with the lighter gray background are first\norder in gradient and chiral [27]. Those with the darker gray\ncolor are zeroth order in gradient and anisotropic [66]. In\nthisway,ourtheoryprovidesaunifiedpictureontheprevious\nworks. Whether a term contributes to \u0010\u00001orGis deter-\nmined by the order in m. After a direct product of STT and\nSMF, a term even (odd) in mgivesG(\u0010\u00001), since it gives\na time irreversible (reversible) contribution appearing in the\nLLGequationas m\u0002A\u0001@tm. Thesameanalysiswithsimple\norder countings works for any Au(m). It holds even if our\ntheory is generalized to other physics, such as magnons [26],\nthermal effects [41], and mechanical effects [42].\nAsanexampleofapplicationsofTableI,weadoptthespin-\nHall-effectdrivenSOT[21,67,68],wherealargedampinglike\nSOTarises. FromTableI,onecanimmediatelyfigureoutthat\nthecombinationofthedampinglikeSOTandtheconventional\nSMF(themosttoprightcell)givesachiralgyromagneticratio\ncontribution. As another example, one notices that the com-\nbination of the dampinglike SOT and its Onsager counterpart\n(the fourth term in the SMF) gives an anisotropic damping\ncontribution. Note that the Onsager counter part of the spin-\nHall-effect driven SOT is the inverse spin Hall effect driven\nby spin pumping current generated by the magnetization dy-\nnamics. In this way, Table I provides useful insight for each\ncontribution.\nTable I also allows for making the general conclusion that\nthe magnitude of the chiral gyromagnetic ratio is determined\nbythesizeofthedampinglikeSOT( \f2)andthatofthenona-\ndiabatic STT ( \f1). This is an important observation since\nmany experiments on magnetic bilayers and topological in-\nsulators [21–23, 30] shows a large dampinglike SOT. This\nconclusionisregardlessofthemicroscopicdetailsoftheSOT,\nbecauseadampinglikecontributionissolelydeterminedbyits\ntime-reversal property.\nTo summarize, we demonstrate that the chiralities of the\ngyromagnetic ratio and Gilbert damping have significant im-\nplicationswhichgofurtherbeyondmerelythechangeinmag-\nnetization dynamics. The chirality plays an important role in\ninvestigating underlying physics because physical quantities,\nwhich were formerly treated as constants, can now be con-\ntrolled through their chiral dependence. An example is the\nnon-Markovian character of the DW creep motion, which is\ndifficult to be verified in nonchiral systems. From the non-\nMarkovian nature of the DW creep motion, we conclude that\nthe non-energetic origin of chiral DW creep originates from\nthe chiral gyromagnetic ratio rather than the chiral damping.\nWealsoprovideageneral,concise,andunifiedtheoryoftheir\nchiralities, which provide useful insight on the self-feedback\nof magnetization.\nWe acknowledge M. D. Stiles, Y. Tserkovnyak, A. Thiav-\nille, S.-W. Lee, V. Amin, and D.-S. Han for fruitful discus-\nsion. This work is supported by the Alexander von Humboldt\nFoundation, the ERC Synergy Grant SC2 (No. 610115), the\nTransregionalCollaborativeResearchCenter(SFB/TRR)1735\nSPIN+X, and the German Research Foundation (DFG) (No.\nEV 196/2-1 and No. SI 1720/2-1). 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Phys. 87, 1213 (2015).Supplementary Materials for\n“Roles of chiral renormalization of magnetization dynamic s in chiral magnets\"\nKyoung-Whan Kim,1Hyun-Woo Lee,2Kyung-Jin Lee,3, 4Karin Everschor-Sitte,1Olena Gomonay,1, 5and Jairo Sinova1, 6\n1Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 5512 8, Germany\n2PCTP and Department of Physics, Pohang University of Science and Te chnology, Pohang 37673, Korea\n3Department of Materials Science and Engineering, Korea University, S eoul 02841, Korea\n4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea\n5National Technical University of Ukraine “KPI\", Kyiv 03056, Ukraine\n6Institute of Physics, Academy of Sciences of the Czech Republic, Cuk rovarnická 10, 162 53 Praha 6 Czech Republic\nI. THE NON-MARKOVIAN NATURE OF THE DW DYNAMICS\nA. Integrating out φ\nIn the linear response regime, we may take Fφ≈ −|F′\nφ(φeq)|(φ−φeq)and the dynamical coefficients ζeffandαX/φ\neffto be\nevaluated at φ=φeq. Without loss of generality, we assume the initial conditio nX(0) = 0 andφ(0) = φeq. We then define the\nLaplace transforms L[f(t)](s) =/integraltext∞\n0e−stf(t)dt. We denote L[X] =QandL[φ−φeq] =P. Then the Laplace transform of\nEq. (2) in the main text is\nsαX\neff\nλQ+s\nζeffP=L[FX] +L[ξX], (S1a)\n−s\nζeffQ+sαφ\neffλP=−|F′\nφ(φeq)|P+L[ξφ], (S1b)\nEliminating Pin Eq. (S1) gives\n1\nζ2\neffs2\n|F′\nφ(φeq)|+sαφ\neffλQ+sαX\neff\nλQ=−γHz\ns+L[Fpin] +/parenleftBigg\nL[ξX]−s\nζeffL[ξφ]\nb+sαφ\neffλ/parenrightBigg\n, (S2)\nwhich is an equation of Xonly. Taking the inverse Laplace transform, we obtain the fo llowing non-Markovian equation:\n1\nλ/integraldisplayt\n0f(t−u)X′(u)du=FX+˜ξX(t). (S3)\nHere f(t)is a feedback function from φ, whose explicit form is\nf(t) =L−1/bracketleftBigg\nαX\neff+1\nζ2\neffαφ\neffsτ\n1 +sτ/bracketrightBigg\n=/parenleftBigg\nαX\neff+1\nζ2\neffαφ\neff/parenrightBigg\nδ(t)−1\nζ2\neffαφ\neffτe−t/τΘ(t), (S4)\nandτ=αφ\neffλ/|F′\nφ(φeq)|is the relaxation time of φdegree of freedom. The correlation relation for the effectiv e thermal\nfluctuation field ˜ξX(t)is given by the fluctuation-dissipation theorem ∝angbracketleft˜ξX(t)˜ξX(t′)∝angbracketright ∝Tf(|t−t′|)where Tis the temperature.\nThe noise is ‘colored’ in the sense that it is no longer a white random noise.\nB. Order-of-magnitude estimation of τ\nTo estimate the order of magnitude of τ, we use the fact that the magntude of |Fφ|is determined by the DMI or the hard axis\nanisotropy: |F′\nφ(φeq)| ≈γλ(π/2)×(2H⊥orDλ−1). We take the DMI field Dλ−1being 30 mT [1] for a rough estimation.\nThen, |F′\nφ(φeq)|/λ≈γ×30 mT ≈5 GHz , so that τ=αφ\neffλ/|F′\nφ(φeq)| ≈αφ\neff×0.2 ns, which is small compared to the time\nscale of the dynamics of X.2\nC. First order approximation - chiral mass correction\nSince τis small, compared to the times scale of the dynamics of X, we may expand f(t)byτ, in the sense of the gradient\nexpansion in time space. Then, f(t)≈ L[αX\neff+ (1/ζ2\neffαφ\neff)sτ] =αX\neffδ(t) + (τ/ζ2\neffαφ\neff)δ′(t). Putting this into Eq. (S3) gives\nτ\nζ2\neffαφ\neff1\nλd2X\ndt+αX\neff\nλdX\ndt=FX+˜ξX(t), (S5)\nwhere the first term represents a massive term. To obtain the D W mass, we need to find the factor which makes FXhave the\ndimension of force. Note that the force generated by pushing the DW is calculated by Ms/integraltext\nHeff·∂Xmd3x= (2MsS/γ)FX.\nTherefore, the mass is defined by multiplying the factor 2MsS/γ,\nmDW=1\nζ2\neff2MsSτ\nγαφ\neffλ, (S6)\nwhich is equivalent to Eq. (5) in the main text.\nD. Higher order contributions - chiral creep\nTo calculate v0, one needs to solve a barrier escape problem. For an energy ba rrierEb, Kramer [2] derived the thermal escapes\nrate\nΓ =ν\n2π/radicalBigg\n|F′(Xm)|\n|F′(XM)|e−Eb/kBT, (S7)\nwhere F′(Xm)andF′(XM)are the derivatives of the force (second derivatives of the p inning energy landscape) evaluated at\nthe potential well and the saddle point respectively. νis called the reactive frequency [3] which we calculate belo w. Then, v0\nis proportional to Γ. According to the Kramer’s theory, ν∝1/αX\nefffor a high damping and Markovian limit, which was also\nconfirmed by the functional renormalization group techniqu e [4].\nHowever, we generalize this result to a non-Markovian situa tion [Eq. (S3)]. To do this, we apply the theory of escape rate for\na non-Markovian equation of motion [3, 5], based on which, th e reactive frequency νcorresponding to Eq. (S3) is given by the\npositive root of the following algebraic equation:\n1\nλνL[f(t)](ν) =|F′(XM)|, (S8)\nwhose exact solution can be calculated from Eq. (S4). As a res ult,\nν=2ν0\n(1−τν0) +/radicalBig\n(1 +τν0)2+ 4τν0/ζ2\neffαX\neffαφ\neff≈\n\nν0∝1\nαX\neffν0τ≪ζ2\neffαX\neffαφ\neff,\nζeff/radicalBigg\nν0αX\neffαφ\neff\nτ∝ζeffν0τ/greaterorsimilarζ2\neffαX\neffαφ\neff,(S9)\nwhere ν0=λ|F′(XM)|/αX\neffis the reactive frequency for τ= 0. In the second limit, we assume that the damping parameters\nare small, thus the last term in the denominator in Eq. (S9) do minates the other terms in the denominator. The two limits sh ows\ncompletely different dependences of νon the dynamical parameters. Therefore, it is important to d etermine the relevant regime.\nAssuming Fpinis random, |F′(XM)| ≈ |F′(Xm)|in Eq. (S7) gives ν0/2πto be the typical attempt frequency ≈1 GHz [6].\nFrom τ≈αφ\neff×0.2 ns estimated above, we obtain τν0≈αφ\neffwhich is an order of magnitude larger than ζ2\neffαX\neffαφ\neff, thus the\nsecond regime in Eq. (S9) is more relevant, contrary to the tr aditional creep theory just taking τ= 0.\n[1] S.-G. Je, D.-H. Kim, S.-C. You, B.-C. Min, K.-J. Lee, and S.- B. Choe, Phys. Rev. B 88, 214401 (2013).\n[2] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940).\n[3] E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989).\n[4] P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62, 6241 (2000)\n[5] R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).\n[6] K. Gorchon, S. Bustingorry, J. Ferré, V. Jeudy, A. B. Kolton, a nd T. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014)."    },    {        "title": "1703.01879v5.Damping_dependence_of_spin_torque_effects_in_thermally_assisted_magnetization_reversal.pdf",        "content": "IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  1 \nDamping dependence of spin-torque effects in thermally assisted \nmagnetization reversal   \nY.P. Kalmykov,1 D. Byrne,2 W.T. Coffey,3 W. J. Dowling,3 S.V.Titov,4 and J.E. Wegrowe5 \n1Univ. Perpignan Via Domitia, Laboratoire de Mathématiques et Physique, F -66860,  Perpignan, France  \n2School of Physics, University College Dublin, Belfield, Dublin 4, Ireland  \n3Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland  \n4Kotel’nikov Institute of Radio Engineering and Electronics of the Russia n Academy of Sciences, Vvedenskii Square 1, \nFryazino, Moscow Region, 141120, Russia  \n5Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France  \nThermal fluctuations of nanomagnets driven by spin -polarized currents are treated via the Landau -Lifshitz -Gilbert equation as \ngeneralized to include both the random thermal noise field and Slonczewski spin -transfer torque (STT) term s. The magnetization \nreversal time of such a nanomagnet is then  evaluated for wide ranges of damping by using a method which  generalizes the solution of \nthe so -called Kramers turnover problem for mechanical Brownian particles  thereby  bridging the very low damping (VLD) and \nintermediate damping (ID) Kramers escape rates , to the analogous magnetic turnover problem. The reversal time is then evaluated for \na nanomagnet with the free energy density given in the standard form of superimposed easy -plane and in -plane easy -axis anisotropies \nwith the dc bias field along the easy axis.  \n \nIndex Terms — Escape rate, Nanomagnets, Reversal time of the magnetization, Spin -transfer t orque .  \n \nI. INTRODUCTION  \nue to the  spin-transfer torque (STT) effect [1 -6], the \nmagnetization of a nanoscale ferromagnet may be altered \nby spin -polarized currents . This phenomenon occurs because \nan electri c current with spin polarization in a ferromagnet has \nan associated flow of angular momentum [3,7] ther eby \nexerting a macroscopic spin torque.  The phenomenon  is the \norigin of the novel  subject of spintronics  [7,8], i.e., current -\ninduced control over magnet ic nanostructures . Common \napplications are very high-speed  current -induced \nmagnetization switching by  (a) reversing the orien tation of \nmagnetic bits [3,9 ] and (b) using spin polarized currents  to \ncontrol steady state microwave oscillations [9 ]. This is \naccomplished via the steady state magnetization precession \ndue to STT representing the conversion of DC input into an \nAC output voltage [3]. Unfortunately , thermal fluctuations \ncannot now be ignored due to the nanometric  size of STT \ndevices, e.g., leading to mainly noise -induced switching at \ncurrents far less than the critical switching current without \nnoise [10] as corroborated by experiments (e.g., [11]) \ndemonstrating that STT near room temperature significantly \nalters thermally activated switching processes . These now \nexhibit a pronounced dependence on both material and \ngeometrical parameters. Consequently, an accurate account of \nSTT switching effects at finite temperatures is necessary in \norder to achieve further improvements in the design and \ninterpretatio n of experiments, in view of the manifold practical applications in spintronics, random access memory \ntechnology, and so on.  \nDuring the last decade, various analytical and numerical \napproaches to the study of STT effects in the thermally \nassisted  magnetiza tion reversal (or switching) time in \nnanoscale ferromagnets  have been developed  [6,7,12 -26]. \nTheir objective being to generalize  methods originally \ndeveloped for zero STT  [12,27 -32] such as  stochastic \ndynamics simulations (e. g., Refs. [21 -25]) and  extensio ns to \nspin Hamiltonians  of the mean first passage time (MFPT) \nmethod (e.g., Refs. [16] and [17] ) in the Kramers escape rate \ntheory  [33,34]. However, unlike zero STT substantial progress \nin escape rate theory including STT effects has so far been \nachieved o nly in  the limit  of very low damping (VLD), \ncorresponding to vanishingly small values of the damping \nparameter \n  in the Landau -Lifshitz -Gilbert -Slonczewski \nequation  (see Eq. (5) below). Here  the pronounced time \nseparation between fast precessional and slow energy changes \nin lightly  damped  closed phase space trajectories (ca lled \nStoner -Wohlfarth orbits) has been exploited in Refs. \n[7,14, 16,17]  to formulate a one -dimensional Fokker -Planck \nequation for the energy distribution function  which may be \nsolved by quadratures. This equation is  essentially similar to \nthat derived by Kramers [ 33] in treating the VLD noise -\nactivated escape rate of a point Brownian particle from a \npotential well although the Hamiltonian of the magnetic \nproblem is no longer separable and additive and the barrier \nheight is now STT depend ent. The Stoner -Wohlfarth orbits \nand steady precession along such an orbit of constan t energy \noccur if the spin -torque is strong enough to cancel out the \ndissipative torque. The origin of the orbits arises from the \nbistable (or, indeed, in general multistable) structure of the \nanisotropy potential. This structure allows one to define a \nnonconservative “effective” potential with damping - and D \nManuscript received April 6, 2017; revised June 27, 2017; accepted July \n24, 2017. Date of publication July 27, 2017; date of current ver -sion \nSeptember 18, 2017. Correspondin g author: Y. P. Kalmykov (e -mail: \nkalmykov@univ -perp.fr).  \nColor versions of one or more of the figures in this paper are available \nonline at http://ieeexplore.ieee.org . \nDigital Object Identifier: 10.1109/TMAG.2017. 2732944  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 2 \ncurrent -dependent potential barrier s between stationary self -\noscillatory states of the magnetization, thereby permitting one  \nto estimate the reversal (switching) time between these states . \nThe magnetizat ion reversal time in the VLD  limit  is then \nevaluated  [16,17,35 ] both for zero and nonzero STT. In \nparticular, for nonzero STT , the VLD reversal time has been \nevaluated analytically in Refs. [16,17 ]. Here it has been shown \nthat in the high barrier limit, an asymptotic equation for the \nVLD magnetization reversal time from a single  well in the \npresence of the STT is given by  \n \nVLD\nTST1\nCES . (1) \nIn Eq. (1), \n is the damping parameter  arising from the \nsurroundings , \nTST\nAE\nEfe  is the escape rate render ed by \ntransition state theory (TST) which ignores effects due to the \nloss of spins at the barrier [34], \nAEf  is the well precession \nfrequency, \nE  is the damping and spin -polarized -current \ndependent effective en ergy barrier, and \nCES  is the \ndimensionless action  at the saddle point C (the action is given \nby Eq. (13) below).  \nThe most essential fea ture of the results obtained in Refs. \n[16,17,35 ] and how they pertain to this  paper is that they apply \nat VLD only where  the inequality \n1\nCES  holds meaning \nthat the energy loss per cycle of the almost periodic motion at \nthe critical en ergy is much less than the thermal energy . \nUnfortunately  for typical values of the material parameters \nCES\n may be very high (\n310 ), meaning that this inequality  \ncan be fulfilled only for \n0.001 . In addition, both \nexperimental and theoretical estimates suggest  higher  values \nof  of the order of 0.001 -0.1 ( see, e.g., Refs. [6,36 -38]), \nimplying that the VLD asymptotic results are no longer valid \nas they will now differ substantially from the true  value of the \nreversal time . These considerations suggest that the \nasymptotic calculations for STT should be extended to include \nboth the VLD and intermediate damping (ID) regions. This is \nour primar y objective here . Now like point Brownian particles  \nwhich  are governed by a separable and additive Hamiltonian , \nin the escape rate problem as it pertains to magnetic moments \nof nanoparticles, three regimes of damping appear [ 12,33,34]. \nThese are  (i) very low damping \n( 1)\nCES , (ii) intermediate -\nto-high damping (IHD)  \n( 1)\nCES , and  (iii) a more or less \ncritically damped turnover regime \n( ~ 1)\nCES . Also , Kramers \n[33] obtained his now -famous VLD and IHD escape rate \nformulas for point Brownian particles by assuming in  both \ncases that the energy barrier is much greater than the thermal \nenergy so that the concept of an escape rate applies. He \nmentioned, however, that he could not find a general method \nof attack in order to obtain an escape rate formula valid for \nany damp ing regime. This problem, namely the Kramers \nturnover, was initially solved by Mel’nikov and Meshkov \n[39]. They obtained an escape rate that is valid for all values \nof the damping by a semi heuristic argument, thus constituting a solution of the Kramers tu rnover problem for point particles. \nLater, Grabert [40] and Pollak et al . [41] have presented  by \nusing a coupled oscillator model of the thermal bath , a \ncomplete solution of the Kramers turnover problem and have \nshown that the turnover escape rate formula can be obtained \nwithout  the ad hoc  interp olation between the VLD  and IHD \nregimes  as used by Mel’nikov and Meshkov . Finally, Coffey \net al. [42,43 ] have shown  for classical spins  that at zero STT , \nthe magnetization reversal  time for values of damping up to \nintermediate values, \n1,  can also be evaluated via the \nturnover formula for the escape rate bridging the  VLD  and ID  \nescape rates, namely,  \n \nTST1\n()\nCE AS , (2) \nwhere \n()Az  is the so-called depopulation factor, namely  [39-\n42] \n \n 2\n2\n0ln 1 exp[ ( 1/4)]1\n1/4()z\nd\nA z e\n   \n\n . (3) \nNow the ID reversal time (or the lower bound of the reversal \ntime) may always be evaluated via TST as [32,34]  \n \nID\nTST1 . (4) \nTherefore b ecause \n()\nCCEE A S S  is the energy loss per \ncycle at the critical energy  \n0\nCES  [39] (i.e. , in the VLD \nlimit) , Eq. (2) transparently reduces to the VLD Kramers \nresult, Eq.  (1). Moreover  in the ID range, where \n( ) 1\nCE AS , \nEq. (2) reduces to the TST Eq.  (4). Nevertheless in the high \nbarrier limit \n1,\nCES  \n given by Eq. (2) can substantially  \ndeviate in the damping range \n0.001 1  both from \nID , \nEq. (4), and \nVLD , Eq. (1). Now, the approach of Coffey et al. \n[42,43 ] generalizing the Kramers turnover results to classical \nspins (nanomagnets)  was developed for zero STT, \nnevertheless, it can also be used to account for STT effects. \nHere  we shall  extend th e zero STT results of Refs. \n[14,16,17,39 -42] treating the damping dependence of STT \neffects  in the magnetization reversal of  nano scaled \nferro magnets  via escape rate theory in the most important \nrange of damping comprising  the VLD and ID ranges , \n1.   \nII. MODEL  \nThe object of our study is the role played by STT effects in the  \nthermally assist ed magnetization reversal  using an adaptation \nof the theory of thermal fluctuations in nanomagnets \ndeveloped in the seminal work s of Néel [27] and Brown  \n[28,29]. The Néel -Brown theory i s effect ively  an adaptation of \nthe Kramers theory [ 33,34 ] originally given for point \nBrownian particles to magnetization  relaxa tion governed by a \ngyromagnetic -like equation  which is taken as the Langevin \nequation of the pro cess. Hence, the verification of that theory \nin the pure (i.e., without STT) nanomagnet context nicely \nillustrates the Kramers conception of a thermal relaxation \nprocess as escape over a potential barrier arising from the IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 3 \nshuttling action of the Brownian m otion. However, it should \nbe recalled throughout that unlike nanomagnets at zero STT \n(where the giant spin escape rate theory may be effectively \nregarded as fully developed),  devices  based on STT , due to the \ninjection of the spin -polarized current, invaria bly represent an \nopen system in an out-of-equilibrium steady state.  This is in \nmarked contrast to the conventional steady state of \nnanostructures characterized by the Boltzmann equilibrium \ndistribution that arises when STT is  omitted .  Hence both the \ngover ning Fokker -Planck and Langevin equations and the \nescape rate theory based on these must be modified .  \nTo facilitate our di scussion, we first describe a schematic \nmodel of the STT effect.  The archetypal model (Fig. 1  (a)) of \na STT device is a nanostructure  compr ising two magnetic \nstrata label ed the free and fixed layers and a nonmagnetic \nconducting spacer. The fixed layer is much more strongly \npinned along its orientation than the free one. If an electric \ncurrent  is passsed  through the fixed layer it become s spin -\npolarized . Thus , the current , as it encounters the free layer, \ninduces a STT . Hence, the magnetization \nM  of the free  layer  \nis altered . Both ferromagnetic layers are assumed to be \nuniformly  magnetized [3,6]. Although th is gia nt coherent spin  \napproximation cannot explain all observations of the \nmagnetization dynamics in spin -torque systems, nevertheless \nmany qualitative features needed to interpret experimental \ndata are satisfactorily reproduced. Indeed,  the current -induced \nmagnetization dynamics in the free layer may be described by \nthe Landau -Lifshitz -Gilbert -Slonczewski equation  including thermal fluctuations , i.e., the usual Landau -Lifshitz -Gilbert \nequation [ 44] incl uding STT, however  augmented by a \nrandom magnetic field  \n()tη  which is regarded as white noise. \nHence it now becomes a magnetic Langevin equation \n[3,6,7,12 ], viz.,  \n \nS             u u H η u u u u I\n . (5) \nHere  \n/SMuM  is the unit vector directed along \nM , \nSM  is \nthe saturation magnetization, and  is the gyromagnetic -type \nconstant . The effective  magnetic field \nH  comprising the \nanisotropy and external applied fields  is defined as  \n \n0SkT E\nvMHu . (6) \nHere  E is the  normalized free energy density of the free layer  \nconstituting a conservative potential, \nv  is the free layer  \nvolume , \n7 2 1\n04 10 JA m    in SI units,  and \nkT  is the \nthermal energy.  For purposes of illustration , we sh all take  \n,)(E\n in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies  plus the Zeeman term due to \nthe applied  magnetic  field \n0H  [45] (in our notation):  \n \n22 2, ) sin cos sin cos )( ( 2 cos h E         . (7) \nIn Eq. (7)  and  are the polar and azimuthal angles  in the \nusual spherical polar coordinate system , \n0S/ (2 ) h H M D\n  \nand \n2\n0S / ( ) v M D kT\n  are the external field and anisotropy \nparameters, \n/1DD \n  is the biaxiality parameter \ncharacterized by  \nD\n  and \nD  thereby encompassing both \ndemagnetizing and magnet ocrystalline anisotropy effects \n(since \n  and \n  are determined by both  the volume an d the \nthickness of the free layer, th eir numerical values may vary \nthrough  a very large range, in particular, they can be very \nlarge , > 100  [45]). The form of Eq. (7) implies that  both the \napplied field \n0H  and the unit vector \nPe  identifying the \nmagnetization direction in the fixed layer are directed along \nthe easy X-axis (see Fig. 1(a)) . In general,  \n,()E  as \nrendered by Eq. (7) has two equivalent saddle points  C and \ntwo nonequivalent  wells  at \nA  and \nA  (see Fig.1(b) ). Finally , \nthe STT induced field \nSI  is given by   \n \n0S\nSkT\nvMIu , (8) \nwhere  \n is the normalized non conservative potential due to \nthe spin -polarized current, which in its simplest form i s \n \n ( , )PJ   eu . (9) \nIn Eq.  (9), \n()P J b I e kT\n  is the dimensionless STT \nparameter , I is the spin -polarized current regarded  as positive \nif electrons flow  from the free into the fixed layer, e is the \nelectronic charge, \n  is Planck’s reduced constant , and \nPb is a \nparameter determined  by the spin polarization factor \nP  [1]. \nAccompanying  the magnetic Langevin equation  (5) (i.e., the \nstochastic differential equation of the random magnetization \nprocess) , one has the Fokker -Planck equation  for th e evolution \nof the  associated probability density function \n( , , )Wt  of \norientations of \nM  on the unit sphere, viz., [ 6,12,16 ] \n \nX e   u Z \nY M  \n \neasy axis  H0  \nfixed layer  free layer   I eP \n(a)   \n \n \n       (b)  \nFig. 1. (a)  Geometry of the problem: A STT device consists of two \nferromagnetic strata labelled the free and fixed layers, respectively, and a \nnormal conducting spacer all sandwiched on a pillar between two ohmic \ncontact s [3,6]. Here I is the spin -polarized current, M is the magnetization of \nthe free layer, H0 is the dc bias magnetic field. The magnetization of the \nfixed layer is directed along  the unit vector  eP. (b) Free energy potential of \nthe free layer presented in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies, Eq. (7), at  = 20 and h = 0.2 .  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 4 \n \nFPLWWt , (10) \nwhere \nFPL  is the Fokker -Planck operator in phase space \n( , )\n defined via [6,12,26]  \n \n1\nN\n1\n11FP1 ( )L sin2 sin\n1 ( ) 1\nsin sin\n( ) ( )sinWEWW\nEW\nEEW   \n    \n\n\n\n\n\n         \n           \n \n\n \n    \n   (11) \nand \n0 N1\nS( ) / (2 ) v M kT     is the free diffus ion time \nof the magnetic moment. If \n0=  (zero STT), Eq. (10) \nbecomes the original Fokker -Planck equation derived by \nBrown  [33] for magnetic nanoparticles . \nIII. ESCAPE RATES AND REVE RSAL TIME IN THE DAM PING \nRANGE \n1  \nThe magnetization  reversal tim e can be calculated  exactly by \nevaluating  the smallest nonvanishing eigenvalue \n1  of the \nFokker -Planck operator L FP in Eq. (10) [32,34 ,42]. Thus \n1  is \nthe inverse of the longest relaxation time of the magnetization \n11/\n, which is usually associated with the reversal time . \nIn the manner of zero STT [42,43], the calculation of \n1  can \nbe approximately accomplished using the Mel’nikov -Meshkov \nformalism  [39]. This relies on the fact that  in the high barrier  \nand underda mped limit s, one may rewrite the Fokker -Planck \nequation, Eq. (10), as an energy -action diffusion equation. \nThis in turn is very similar to that for translating point \nBrownian particles moving along the x-axis in an external \npotential V(x) [7,17,42] . In the under damp ed case, which is the \nrange of interest, for the escape of spins from a single \npotential well with a minimum at a point A of the \nmagnetocristalline anisotropy over a single saddle point C, the \nenergy distribution function  \n()WE  for magnetic moments \nprecessing in the  potential well can  then be found via an \nintegral equation [42], which  can be  solved for \n()WE  by the \nWiener –Hopf method. Then, the flux -over-population method \n[33,34]  yields the decay ( escape ) rate as \n1/CAJN . Here \nconstCJ\n is the probability current density  over the sadd le \npoint and \n()C\nAE\nAEN W E dE  is the well population while the \nescape rate is rendered as the product of the depopulation \nfactor  \n( ),\nCE AS  Eq. (3), and the TST  escape rate \nTST\nAE\nEfe\n. In the preceding equation \nE  is the effective \nspin-polari zed current dependent energy barrier  given by  \n \n1\nAC\nCE\nE EAEVdE E E ES    , (12) \nwhere \nAE  is the energy at the bottom of the potential well, \nCE\n is the energy at the saddle  point, and the dimensionless \naction \nES  and the dimensionless work \nEV  done by the STT are defined as [7,17]  \n \nEEd SE  \nuuu\n , (13) \n \nEE d Vuuu\n , (14) \nrespectively. T he contour integrals in Eqs. (13) and (14) are \ntaken along the energy trajectory  \nconstE  and are to be \nevaluated in the vanishing damping sense.  \nFor the bistable potential, Eq. (7), having  two nonequivalent \nwells  \nA and \nA  with minima  \n( 1 2 ) Eh\n  at \n0A  \nand \nA , respectively,  and two equivalent saddle points C \nwith \n2\nCEh  at \ncosC h  (see Fig. 1(b)) we see that two \nwells and two escape routes over two saddle points  are \ninvolved in the relaxation process . Thus, a finite probability \nfor the magnetic dipole to return to the initi al well having \nalready visited the second one exists. This possibility cannot \nbe ignored in the underdamped regime  because then the \nmagnetic dipole having entered the second well loses its \nenergy so slowly that even after several precessions, thermal \nfluctuations may still reverse it back over the potential barrier. \nIn such a situation, on applying the Mel’nikov -Meshkov \nformalism  [39] to the free energy potential, Eq. (7), and the \nnonconservative potential, Eq. (9), the energy distribution \nfunction s \n()WE  and \n()WE  for magne tic moments \nprecessing in the  two potential well s can then be found by \nsolving two coupled integral equations for \n()WE  and \n()WE\n. These then yield the depopulation factor \n, ()\nCCEE A S S\n via the Mel’nik ov-Meshkov formula for two \nwells, viz., [39]  \n \n( ) ( )\n((), )CC\nCC\nCCEE\nEE\nEEA S A S\nA S SA S S\n\n\n .  \nHere \n()Az  is the depopulation factor for a single well \nintroduced in accordance with Eq. (3) above while \nCES  are the \ndimensionless action s at the energy saddle point s for two \nwells. These are to be calculated via Eq. (13) by integrating \nalong the energy trajectories \nC EE  between two saddle \npoints and are explicitly given by  \n     \n2\n3/2\n12\n21\n12(1 )\n(1(1 2 a4\n(1\nrct)\n)1an )(1 )\n)1 (1CCEEh\nhhhES\nhd\nh\nh \n\n\n\n\n\n      \n   \n \n\nuuu\n  (15) \n(at zero dc bias field, h = 0, these simplify to \nCCEESS  \n4\n). Furthermore, the overall TST escape rate \nTST  for \na bistable potential, Eq. (7), is estimated via the individual  \nescape rates \nTST\n  from each of the two wells as  \n \n TST TSTTST2.EEffee   \n      (16) \nIn Eq. (16), the factor 2 occurs because two magnetization \nescape r outes from each well over the two saddle points exist, \nwhile \nE  are the effective spin -polarized current dependent IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 5 \nbarrier heights  for two wells (explicit equations for \nE  are \nderived in Appendix A). In addit ion  \n \n01(1 )(1 )2f h h      (17) \nare the corresponding well precession frequencies, where \n1\n0S 2MD\n is a precession time  constant . Thus, the \ndecay rate \n1  becomes  \n2\n2(1 ) ( , )1\n0\n(1 ) ( , )(1 )(1 )\n(( ) ( )\n()\n, 1 )(1 )CC\nCCJh F hEE\nEE\nJh F hA S A S\nAh h e\nhhS\neS\n\n\n\n\n\n\n\n\n  \n\n \n\n \n\n(18) \nwhere both the functions \n( , )Fh  occurring  in each \nexponential are given by the analytical formula:  \n  \n22\n2\n1\n12\n2 1 1 2(1 ) (1( , ) 12 2 2 (1\n(1 21 arctan\n(1)(1 )\n)\n)1\n)(1 ) 1 (1 )hhFhhh h\nhh h\nh h h  \n\n\n\n\n       \n    \n\n\n\n  (19) \nand  0.38 is a numerical par ameter  (see Eq.  (A.6), etc. in \nAppendix A ). For zero STT, J = 0, Eq. (18) reduces to the \nknown results  of the Néel -Brown theory [32,43] for classical \nmagnetic moments with superimposed easy -plane and in -plane \neasy-axis anisotropies  plus the Zeeman term due to the applied  \nmagnetic  field. In contrast to zero STT, for  normalized spin \ncurrents J  0,  depends on \n  not only through the \ndepopulation factors \n()\nCE AS  but also through the spin-\npolarized current dependent  effective barrier heights \nE . \nThis i s so because part s of the arguments of the exponentials \nin Eq.  (18) , namely Eq. (19), are markedly dependent on the \nratio \n/J  and the dc bias field parameter.  The turnover Eq. \n(18) also yields a n asymptotic estimate for the inverse of the \nsmallest nonvanishing eigenvalue of the Fokker -Planck \noperator \nFPL  in Eq. (10). In additio n, one may  estimate two \nindividual  reversal times, namely, \n  from the deeper well \naround the energy minimum at \n0A  and \n  from the \nshallow well around the energy minimum at \nA  (see Fig. \n1(b))  as \n \n2(1 ) ( , )\n02\n( ) (1 )(1 )\nCJh F h\nEe\nA S h h\n\n\n  \n . (20) \nThe individual  times are in general unequal, i.e., \n . In \nderiving Eqs. (18) and (20), all terms of order \n22, , ,JJ  etc. \nare neglected. This hypothesis is true only for the \nunderdamped regime , α < 1, and weak  spin-polarized currents, \nJ<<1. ( Despite these restrictions as we will see below Eqs. \n(18) and (20) still yield accurate estimates for \n  for much \nhigher values of J). Now,  \n can also be calculated  \nnumerically  via the method of statistical moments developed  \nin Ref. [26] whereby t he solution of the Fokker -Planck \nequation (10) in configuration space  is reduced to the task of solving  an infinite hierarchy of differential -recurrence \nequations for the averaged spherical harmonics  \n( , ) ( )lmYt  \ngoverning the magnetization relaxation . (The \n( , )lmY  are the \nspherical harmonics [46 ], and the angular brackets denote the \nstatistical aver aging ). Thus one can evaluate  \n numerically \nvia \n1  of the Fokker -Planck operator L FP in Eq. (10) by using \nmatrix continued fr actions as described in Ref. [47 ]. We \nremark that the r anges of applicability of the escape rate \ntheory and the matrix continued -fraction method are in a sense  \ncomplementary because  escape rate theory cannot be used for  \nlow potential barrie rs, \n3E , while  the matrix continued -\nfraction method encounters substantial computation al \ndifficulties for  very high  potential barriers \n25E  in the \nVLD range, \n410 . Thus , in the foregoing se nse, numerical \nmethods and escape rate theory are very useful for the \ndetermination of τ for low and very high potential barriers, \nrespectively. Nevertheless , in certain  (wide) ranges of model \nparameters both methods yield accurate results for the reversal  \ntime ( here these ranges are \n5 30, 3,     and \n410 ). \nThen the numerically exact  benchmark solution provided by \nthe matrix continued fraction method  allows one  to test the \naccuracy of the analytical es cape rate equations given above.  \nIV. RESULTS AND DISCUSSIO N \nThroughout the calculations, the anisotropy and spin -\npolarization parameters will be taken as \n0.034 D\n , \n20 , \nand \n0.3P  (\n0.3 0.4P  are typical  of ferromagnetic \nmetals)  just as in Ref. 6. Thus for \n5 1 1mA s . 10 , 22  \n300T\nK\n, \n24~10v\n3m , and a current density of the order \nof \n7~ 10\n2A cm  in a 3 nm thick layer of cobalt with \n61\nS 1 1. Am 04 M\n, we have  the following estimates for the \nanisotropy (or inverse temperature ) parameter \n20.2 , \ncharacteristic time \n1\n0S2()MD\n0.48 ps, and spin -\npolarized current parameter \n( ) ~1P J b I e kT\n . In Figs. 2 \nand 3,  we compare  from the asymptotic escape rate Eq . (18) \nwith \n1\n1  of the Fokker –Planck operator as calculated \nnumerically via matrix continued fraction s [26]. Apparently,  \nas rendered by  the turnover equation (18) and \n1\n1  both lie \nvery close to each other in the high barrier limit, where the \nasymptotic Eq. (18) provides an  accurate approximation \nto\n1\n1.  In Fig. 2,  is plotte d as a function of \n  for various J. \nAs far as STT effects are concerned they are governed by the \nratio \n/J  so that by altering \n/J  the ensuing variation of  \nmay exceed  several or ders of magnitude (Fig. 2) . Invariably \nfor J << 1, which is a condition of applicability of the escape \nrate equations (1) and (18), STT effects on the magnetization \nrelaxation are pronounced only at very low damping,  << 1 . \nFor \n1 , i.e. high damping, STT influences  the reversal \nprocess very weak ly because the STT term in Eq. (5) is then \nsmall compared to the damping and random field terms . \nFurthermore,  may greatly exceed or, on the other hand, be  \nvery much less than the value for zero STT , i.e., J = 0 (see Fig. \n2). For example, as J decreases from positive values, \n  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 6 \nexponentially increases  attaining a  maximum at a critical \nvalue of the spin -polarized current and then smoothly switches \nover to exponential decrease  as \nJ  is further increased \nthrough negative values of J [26]. Now, t he temperature , \nexternal d.c. bias field,  and dam ping dependence of  can \nreadily be understood in terms of  the effective potential \nbarriers  \nE  in Eq.  (18). For example, for  \n5,  the \ntemperature dependence  of  has the customary Arrhenius \nbehavior \n~,Ee  where \nE , Eq. (19), is markedly \ndependent on \n/J  (see Fig. 3 a). Furthermore, the slope of \n1()T\n significantly decreases as the dc bias field parameter h increases due to lowering  of the barrier height \nE  owing to \nthe action of  the external field  (see Fig. 3b) . Now, although \nthe range of applicability of  Eqs. (18) and (20) is ostensibly \nconfined to weak spin -polarized currents, J << 1, they can still \nyield accurate estimates for the reversal time for much higher \nvalues of J far exceeding this condition (see Fig. 3 a).  \nThus , the turnover formula for , Eqs. (18) and (20), \nbridgi ng the Kramers VLD  and ID escape rates as a function \nof the damping  parameter for point  particles [35,39 -41] as \nextended by Coffey et al.  [42,43] to the magnetization \nrelaxation in nanoscale ferromagnets allows us (via the further \nextension to include STT embodied in Eq. (18)) to accurately  \nevaluate STT effects in the magnetization reversal time of  a \nnanomagnet driven by spin -polarized current in the highly \nrelevant ID to VLD damping range. This (underdamped) range  \nis characterized by  \n1  and the asymptotic escape rates are \nin complete agreement with independent  numerical results  \n[17]. Two particular merits of the escape rate equations  for the \nreversal time are that (i)  they are relatively simple ( i.e., \nexpressed via elementary functions) and (ii) that they can be \nused in those parameter ranges, where numerical methods \n(such as matrix continued fractions [17]) may be no longer \napplicable , e.g., for very high barriers , \n25E . Hence , one \nmay conclude that the damping dependence of the \nmagnetization reversal time is very marked in the \nunderdamped regime \n1 , a fact which may be very \nsignificant  in int erpreting  many  STT experiments.  \nV. APPENDIX A: CALCULATION OF \n( , )Fh  IN EQ. (19) \nFor the bistable potential  given by  Eq. (7), and the \nnonconservative potential, Eq. (9), the spin -polarized current \ndependent effective barrier heights  \nE  for each of the two \nwells are given by (cf. Eq. (12)) \n \n21(1 ) ( , )h J F E h     \n , (A.1) \nwhere  \n \n( , )C\nAVFhSd\n\n \n\n\n , (A.2) \nwith \n/E , \n/ 1 2AAEh \n , \n2/CCEh . The \ndimensionless action  \nS and the dimensionless work  done by \nthe STT \nV for the deeper well can be calculated analytically \nvia elliptic integrals as described in detail in Ref. [17] yielding  \n   \n2 2\n0\n2\n22(1 )\n1\n2 ( )11 ( ) ( )\n)(2\n1\n(1 )( )\n() (142),(1 )( ( ( ) ) 1)p Ehd hpf\nEm hqq q m K m\nq h q mhpq q mS\nm\nKm\n\n\n\n\n   \n \n\n\n\n \n\n\n\n \n         \n   \n    \n\n  \n\n \n uuu\n  (A.3) \n\n54321: J = 0.2\n2: J = 0.1\n3: J = 0\n4: J = 0.1\n5: J = 0.2/ \nh =0.15\n =20\n = 201 \nFig. 2. Reversal time \n0/  vs the damping parameter \n  for various values \nof the  spin-polarized  current  parameter J. Solid lines : numerical calculations \nof the inverse of t he smallest nonvanishing eigenvalue \n1\n01()  of the \nFokker –Planck operator , Eq. (11). Asterisks: the turnover  formula, Eq. (18). \n \n54\n/ 3211: J = 1\n2: J = \n3: J = \n4: J = \n5: J = \nh = 0.1\n = 0.01\n  = 20\n(a)\n \n  (b)\n4\n/ 321 1: h = 0.0\n2: h = 0.1\n3: h = 0.2\n4: h = 0.3\n = 0.01\n  = 20\nJ  = \n\n \nFig. 3 . Reversal time \n0/  vs. the anisotropy  (inverse temperature) \nparameter  for various spin-polarized currents J (a) and dc bias field \nparameters h (b). Solid lines: numerical solution for the inverse of the \nsmallest nonvanishing eigenvalue \n1\n01() of the Fokker –Planck operator , \nEq. (11). Asterisks: Eq. (18). IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 7 \n    \n\n0\n2 23\n2\n2( 1)\n1 ( 1)\n( | )1\n2\n21 2 1()\n( ) (1\n(|)121 ( ) (,))phV\nm\nm\nqhdhf\nq hpK\nhp q E m m qqq q m K mm\n\n\n\n\n     \n\n\n\n \n\n    \n\n \n    \n\n \n    \n\nueu\n  (A.4) \nwhere  \n \n2\n2\n21( 1)ph\n\n , \n1\n1eqe\n ,  \n \n\n1 (1 )\n(1 ) 1eemee\n , \n2\n( 1)hepph\n\n  , \n()Km\n, \n()Em , and \n( | )am  are the complete  elliptic integrals \nof the fir st, seco nd, and third kinds, respectively [48], and  \nf \nis the precession frequency in the deeper well at a given \nenergy, namely,  \n \n0( 1)(1\n())(1 )\n8p e efKm\n\n     . (A.5) \nThe quantities \nS , \nV , and \nf  for the shallower well are \nobtained simply by replacing  the dc bias field parameter  \nh by \nh\n in all the equations for \nS , \nV , and \nf . We remark that  \nS\n and \nV in Eqs. (A.3) and (A.4) differ by a factor 2 from \nthose given in Ref. [17]. This is so because \nS  and \nV are \nnow calculated between the saddle points and not over the \nprecession period . When \n( , )C     , \nS  in Eqs. (A.3) \nreduces to \nCES , Eq. (15). \nIn the parameter ranges \n01h  and \n1 , the integral in \nEq. (A.2) can be accurately evaluated analytically using an \ninterpolation function for \n/VS  between t he two limiting \nvalues \n/\nAAVS  and \n/\nCCVS  at \n1A h\n  and \n2\nCh , \nnamely  \n \n11\nC AA\nA C AA\nCAV VV V\nS S S S\n \n   \n  \n          , (A.6) \nwhere  0.38 is an interpolation parameter yielding the best \nfit of \n/VS  in the interval \n.AC    These limiting \nvalues can be calculated from Eqs. (A.3) and (A.4) yielding \nafter tedious algebra:  \n \n1\n22A\nAV\nh S\n\n  (A.7) \nand  \n2\n2\n1\n12\n2 1 1 2)(1 ) 1 (112 (1\n(1 21 arct)\n)1an\n(1 (1 )(1 ) ) 1C\nCV h\nh Sh\nhh h\nh h h\n\n\n\n\n\n\n\n\n\n\n \n    \n. (A.8) \nHence with Eqs. 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Khersonskii, Quantum \nTheory of Angular Momentum . World Scientific, Singapore, 1988 . \n[47] Y.P. Kalmykov, “Evaluation of the smallest nonvanishing eigenvalue of \nthe Fokker -Planck equation for the Brownian motion in a potential. II. \nThe matrix continued fraction approach ”, Phys. Rev. E , vol.  62, p. 227, \n2000. \n[48] M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical \nFunctions . Dover, New York, 1964 . IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 9 \n "    },    {        "title": "2008.13061v3.Exploring_a_quantum_information_relevant_magnonic_material__Ultralow_damping_at_low_temperature_in_the_organic_ferrimagnet_V_TCNE_x.pdf",        "content": " \n 1  \nExploring a quantum-information-relevant magnonic material: ultralow \ndamping at low temperature in the organic ferrimagnet V[TCNE] x \n \nH. Yusuf*1, M. Chilcote*1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E.  \nFlatté3, E. Johnston-Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210 \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853 \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242 \n \n* These authors contributed equally to this work. \n \nAbstract:  Quantum information science and engineering requires novel low-loss magnetic \nmaterials for magnon-based quantum-coherent operations. The search  for low-loss \nmagnetic materials, traditionally driven by applications in microwave electronics near \nroom-temperature, has gained additional constraints from the need to operate at cryogenic \ntemperatures for many applications in quantum information science and technology. \nWhereas yttrium iron garnet (YIG) has been the material of choice for decades, the \nemergence of molecule-based materials with robust magnetism and ultra-low damping has \nopened new avenues for exploration. Specifically, thin-films of vanadium \ntetracyanoethylene (V[TCNE] x) can be patterned into the multiple, connected structures \nneeded for hybrid quantum elements and have shown room-temperature Gilbert damping \n(α = 4 × 10-5) that rivals the intrinsic (bulk) damping otherwise seen only in highly-polished \nYIG spheres (far more challenging to integrate into arrays). Here, we present a \ncomprehensive and systematic study of the low-temperature magnetization dynamics for \nV[TCNE] x thin films, with implications for their application in quantum systems. These \nstudies reveal a temperature-driven, strain-dependent magnetic anisotropy that \ncompensates the thin-film shape anisotropy, and the recovery of a magnetic resonance \nlinewidth at 5 K that is comparable to room-temperature values (roughly 2 G at 9.4 GHz). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 2 We can account for the se variations of the V[TCNE] x linewidth within the context of \nscattering from very dilute paramagnetic impurities, and anticipate additional linewidth \nnarrowing as the temperature is further reduced.  \n \nThe search for low-loss magnetic materials dates to the early days of radio and \nmicrowave electronics [1–3], and the study of elementary excitations, or magnons, in these \nmagnetically-ordered materials has proven to be a rich area of research for both \nfundamental physics and their potential technological applications. More recently, interest \nin these low-loss systems has expanded to include applications in the field of quantum \ninformation technology such as quantum sensing and quantum transduction [4 –7], wherein \nlow-temperature operation allows for the freeze-out of thermal excitations and access to \nthe single-quantum regime . In this regime the field of quantum magnonics utilizes hybrid \narchitectures for coupling magnons to other quantum degrees of freedom, such as \nmicrowave photons, with the aim of extending their functionality in the quantum limit \n[8,9]. It has been demonstrated that magnons can be resonantly excited over a wide range \nof microwave frequencies, allowing for precise control of qubit states mediated by coherent \nexchange via cavity-mode photon excitations [4 ,7]. Magnons also exhibit the potential to \ncoherently couple localized spin-qubits with high cooperativity [10] . However, while \nmagnons exist in a wide range of materials, the same delocalized electrons that are most \noften responsible for stabilizing ferromagnetic order also contribute to electron-magnon \nscattering [11], leading to substantial losses in most metallic ferromagnets. As a result, the \nstudy of low-dissipati on magnon dynamics for quantum applications has focused on \ninsulating ferromagnets and ferrimagnets, with yttrium iron garnet (YIG) and its close \nrelatives holding pride of place as the benchmark low-loss materials for more than 50 \nyears  [ 4,12–14]. As a result, despite these longstanding and emerging needs, applications \nare still constrained by the materials limitations of YIG; namely the need for growth or \nannealing at high temperatures (typically 800° C)  [15–17] and the resulting difficulty in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 3 integrating and patterning YIG thin-films with other microwave electronic structures and \ndevices.   \nIn this context, the emergence of the molecule-based ferrimagnet vanadium \ntetracyanoethylene (V[TCNE] x) has dramatically expanded the playing field for low-loss \nmagnets. Despite what one might expect from its molecular building blocks, V[TCNE] x \nhas a magnetic ordering temperature of over 600 K and shows sharp hysteresis at room-\ntemperature  [ 18–20]. Moreover, its dynamic properties are exceptional, showing ultra-\nnarrow ferromagnetic resonance (FMR) linewidth (typically ~ 1 – 1.5 G at 9 .4 GHz) with \na Gilbert damping parameter,  of 4 × 10-5 for thin-films [ 18,21]. As a comparison, the \nbest YIG thin-films typically show  = 6.5 × 10-5  [22] and a value of 4 × 10-5 is competitive \nwith the intrinsic  damping of bulk YIG  = 3 × 10-5  [15,23]. From an applications \nperspective, V[TCNE] x has been shown to deposit on a wide variety of substrates without \ncompromising material quality  [24–26], facile encapsulation allows for direct integration \nwith pre-patterned microwave structures for operation under ambient conditions [ 27], and \nrecent work has demonstrated patterning at length scales down to 10 m without increased \ndamping  [21]. However, while these properties clearly establish the potential of \nV[TCNE] x for new applications in traditional microwave electronics, very little is known \nabout its low-temperature magnetization dynamics and therefore its potential for \napplications in quantum information science and engineering (QISE ). \nHere we present a detailed study of the low-temperature magnetic resonance of \nV[TCNE] x films. We identify two regimes. In the high-temperature regime, extending from \n300 K down to 9 K, we observe a monotonic shift in the resonance frequency consistent \nwith a temperature-dependent strain. This strain results in a crystal-field anisotropy that \nincreases with decreasing temperature with a magnitude of at least 140 Oe and the same \nsymmetry, but opposite sign, to the shape anisotropy of the thin-film.  In addition, we \nobserve an increase in linewidth consistent with magnon scattering from paramagnetic \nimpurities similar to what has been observed in YIG  [23,28,29], but with an amplitude 3 \ntimes smaller ( i.e. an increase in linewidth by 9 times in V[TCNE] x as compared to 28 \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 4 times in YIG  [23, 30]). In the low-temperature regime, starting at 9 K and extending to 5 \nK, we observe a discontinuous change in both anisotropy and linewidth: the anisotropy \nabruptly reverts to the room-temperature symmetry (in-plane easy-axis) and the linewidth \napproaches room-temperature values (2.58 G) at 5 K. This linewidth variation can be \nexplained using a model for scattering between magnons and paramagnetic impurities that \ntakes into account the finite spin-lifetime of the impurity spins  [23,31]. At high \ntemperatures (above 100 K) the spin-lifetime is sufficiently short that changes in \ntemperature do not lead to significant changes in scattering rate, and at low-temperatures \n(below 9 K) the spin-lifetime becomes long with respect to the spin-magnon scattering \ntime, resulting in a saturation of the excited state. At intermediate temperatures (from 9 K \nto 100 K) this spin-magnon scattering dominates relaxation due to the increase of the \nground state impurity population, which results in a local maximum in the linewidth that \nis 9 times larger than the room-temperature value. These results are extremely promising \nfor low-temperature applications of V[TCNE] x magnonics, promising low-temperature \nmagnon resonators with unprecedented low-loss that can be integrated on-chip into \nmicrowave electronic circuits and devices [20,21]. \nFor this study, thin-films of V[TCNE] x are deposited on sapphire (Al 2O3 (0001)) \nsubstrates using chemical vapor deposition (CVD) growth process consistent with prior \nreports [18,19].  Briefly, argon gas transfers the two precursors tetracyanoethylene (TCNE) \nand vanadium hexacarbonyl (V(CO) 6) into the reaction zone of a custom-built CVD reactor \n(Fig. 1(a)) where V[TCNE] x is deposited onto polished sapphire substrates. The system is \ntemperature controlled to maintain the TCNE, V(CO) 6 and the reaction zones at 65° C, 10° \nC and 50° C respectively . After growth the sample is mounted on a custom, microwave-\ncompatible sample holder and sealed using a septa cap in an electron paramagnetic \nresonance (EPR) grade quartz tube in an argon environment. When the sample is not being \nmeasured, it is stored in a - 35° C freezer housed in an argon glovebox and is stable for over \none month [ 27].  \nFerromagnetic resonance (FMR) measurements are performed using a Bruker EMX \nPlus X-band EPR spectrometer at temperatures ranging from 300 K down to 5 K. The \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 5 microwave frequency of the spectrometer is tuned between 9 and 10 GHz for optimal \nmicrowave cavity performance before the measurement, and then the frequency is fixed \nwhile the DC field is swept during data collection.  Figure 1(b) shows a representative \nroom-temperature FMR measurement of a typical V[TCNE] x thin-film with the external \nmagnetic field applied in the plane of the sample. The resonance feature is consistent with \npreviously reported high-quality V[TCNE] x thin-film growth, showing a peak- to-peak \nlinewidth of 1.5 G at 9.4 GHz  [18,19].  \nComparing this data to FMR measurements at temperatures of 80 K and 40 K \n(Figure 1(c)) shows an increase in the resonance field of over 40 G (roughly half of the \nsaturation magnetization, 4𝜋𝑀 𝑠) as the temperature decreases. Since the applied \nmicrowave frequency is held constant at 9.4 GHz , this shift must arise from fields internal \nto the V[TCNE] x film, i.e. magnetic anisotropy fields. Note that since the value of the DC \napplied field varies between 3350 G and 3450 G, well above 4𝜋𝑀 𝑠, changes in the \nmagnetization of the film are not expected to contribute to this field shift.  In a similar \nfashion, changes in the shape-dependent anisotropy fields can be ruled out, leaving only \nchanges to the crystal-field anisotropy as a potential source of this phenomenon. Crystal-\nfield anisotropy originates from the interaction of a material’s mean exchange field and the \nangular momenta of neighboring atoms (ions) in the material , indicating that there is a \ntemperature dependence to the local atomic environment within the V[TCNE] x films , e.g. \ndue to a temperature-dependent strain within the film.  \nIn order to more comprehensively map out this phenomenon angle dependent FMR \nmeasurements are performed to quantitatively track changes in the magnetic anisotropy at \ntemperatures of 300 K, 80 K, and 40 K (Fig. 2). Variation of the magnetic resonance field \nas a function of the angle between the applied field and the princip al axes of the film can \nbe modeled by considering the free energy of the magnetic system with anisotropic \ncontributions. If we consider the case of a uniaxial anisotropy with the hard-axis \nperpendicular to the easy-axis, and where the magnetization is parallel to the external field \n(i.e. external field is much larger than the saturation magnetization, as is the case here) the \ntotal magnetostatic energy is as follows  [ 32]: \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 6   \n                                              𝐸 = −𝑴 · 𝑯 + 2𝜋 (𝑴 · 𝒏 )𝟐− 𝐾 (𝑴 · 𝒖 𝑀 ⁄)2                          (1) \n \nwhere M is the magnetization, H is the applied magnetic field, n is the unit vector parallel \nto the normal of the magnetic sample , u is the unit vector parallel to the easy-axis and K is \nan anisotropy constant. For the case of in-plane uniaxial anisotropy, this simplifies to \n \n           𝐸 = − 𝑀𝐻 (sin𝜙sin2𝜃 + cos2𝜃)+ 2𝜋𝑀2cos2𝜃 − 𝐾 sin2𝜃sin𝜙2               (2) \n    \nwhere 𝜃 is the angle between M and the sample normal and 𝜙  \nis the azimuthal angle. Minimizing the magnetostatic energy with respect to 𝜃 , one will \nfind that the easy-axis orientation occurs when 𝜃 = 2 𝑛𝜋±  𝜋\n2, where n is an integer . Using \nthis simple symmetry analysis, we can see that the data in Fig. 2 indicates that the easy-\naxis lies in-plane at a temperature of 300 K (i.e. the resonance field is smallest when the \napplied magnetic field lies in-plane) and out- of-plane at a temperature of 40 K (i.e. the \nresonance field is smallest when the applied magnetic field is out- of-plane). In this context, \nthe lack of variation in resonance field at 80 K indicates a nearly isotropic magnetic \nresponse. This switch in magnetic easy-axis from in-plane to out- of-plane further supports \nthe proposition that there is an additional temperature-dependent crystal-field contribution \nto the magnetic anisotropy.  \nIn previous studies, templated growth of V[TCNE] x resulting in nanowire \nmorphologies induced an additional in-plane magnetic anisotropy with easy-axis \nperpendicular to the long-axis of the nanowires, strongly suggesting the presence of a \nstrain -dependent contribution to the crystal-field anisotropy  [ 33]. In the thin-films studied \nhere, such a strain-dependent crystal-field effect would be expected to generate anisotropy \nparallel to the surface normal, i.e. in the out- of-plane direction. The anisotropy field would \nthen be parallel to the expected shape anisotropy from a thin-film, though not necessarily \nwith the same sign. As a result, if there is a difference in the coefficient of thermal \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 7 expansion between the V[TCNE] x film and the sapphire substrate then the temperature \ndependence of magnetic anisotropy can potentially be understood as a proxy for a \ntemperature dependence of strain in the thin- film; such variations in strain leads to changes \nin the local atomic structure, leading to the observed changes in magnetic anisotropy. We \nnote that while the coefficient of thermal expansion for V[TCNE] x has not yet been \nmeasured, the value for sapphire is 5.4 ppm/K and typical values for molecular-based solids \ncan range somewhere between 28 –500 ppm/K [ 34]. Assuming no strain at room-\ntemperature, this would then imply a compressive strain between 0.67% to 15% at the \nsapphire –V[TCNE] x interface at 5 K , leading to an out- of-plane distortion whose symmetry \nis consistent with the observed anisotropy.    \nA schematic describing how these two anisotropy fields would be expected to \ninteract as a function of temperature can be found in Fig. 3(a). At a temperature of 300 K \n(Fig. 3(a), upper panels), the orientation of the easy-axis is determined by the shape \nanisotropy, resulting in an in-plane easy-axis for thin-films. But at a temperature of 40 K \n(Fig. 3(a) lower panels), there is an additional crystal-field anisotropy, 𝐻⊥, proposed that \ndominates the shape anisotropy, reorienting the easy-axis to be out-of-plane. This \nsymmetry analysis also explains the lack of orientation dependence at a temperature of 80 \nK, which is apparently the temperature at which the strain-driven crystal-field anisotropy \nperfectly cancels out the shape anisotropy. We note that similar phenomenology is also \nobserved in vanadium methyl tricyanoethylenecarboxylate (V[MeTCEC] x) thin-films (see \nsupplementary materials), indicating that this temperature- and strain-dependent \nanisotropy is a general property of this class of metal-ligand ferrimagnets. \nThe fact that the shape and proposed crystal-field anisotropies have the same \nsymmetry make it challenging to distinguish between the two; therefore, an effective field \nis defined as 𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓= 4𝜋𝑀 𝑠− 𝐻 ⊥, where 𝑀𝑠 is the saturation magnetization and \n𝐻⊥is the crystal-field anisotropy. Figure 2 shows the effects of this net anisotropy field in \nthe form of resonance field shifts and a change in the easy-axis orientation . Quantitatively \nextracting the magnitude and direction of this anisotropy field provides detailed insight \ninto the role of crystal-field anisotropy in tuning the magnetic response of V[TCNE] x thin-\nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 8 films. To this end, each scan is fit to the sum of the derivatives of absorption and dispersion \nfrom a Lorentzian function to extract the resonance frequency and linewidth (experimental \ndata are obtained using a modulated-field technique that yields the derivative of the \nexpected Lorentzian resonance lineshape). For scans showing an out- of-plane easy-axis a \nsingle derivative sum provides good agreement with the data, while for scans showing in-\nplane easy-axis more complex structure is observed requiring the addition of up to three \nderivative sums. In the results discussed below we focus on the behavior of the primary , \ni.e. largest amplitude, peak (a full description of the fitting and resulting phenomenology \ncan be found in the supplemental material).  \nFigure 3(b) shows the extracted resonance field plotted against sample rotation \nangle for the high-and low-temperature data shown in Fig. 2, 300 K and 40 K, respectively . \nTaking into account a uniaxial out- of-plane anisotropy defined by 𝐻𝑒𝑓𝑓, as described \nabove, the angular dependence for in-plane to out- of-plane rotation of a thin-film sample \nis given by  [19,35,36]: \n \n𝜔\n𝛾 =  √(𝐻 − 𝐻 𝑒𝑓𝑓cos2𝜃)(𝐻 − 𝐻 𝑒𝑓𝑓cos 2𝜃)\n=  √(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥) cos2𝜃)(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥)cos 2𝜃)  \n                                        \nwhere  is the resonance frequency and  is the gyromagnetic ratio. As a result, the \nphenomenology of the data presented in Fig. 2 can be understood as an 𝐻𝑒𝑓𝑓 that is positive \nat 300 K and negative at 40 K, as 𝐻⊥ increases with decreasing temperature, consistent \nwith the mechanism for anisotropy switching described in Fig. 3(a). This qualitative \nunderstanding can be made quantitative by fitting the data in Fig. 2 using Eq. (3) to extract \n𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓 of 91.2 G  1.6 G and -22.8 G  0.4 G, respectively. \nFigure 3 (c) shows this 𝐻𝑒𝑓𝑓 plotted against temperature over the temperature range \nfrom 300 K to 5 K, extracted from angular dependencies such as the measurements \npresented in Fig. 2 . It should be noted that each anisotropy point in Figure 3(c) represents (3) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 9 a fit to a complete angular dependence such as the data shown in Figure 3(b).The effective \nfield mak es a smooth transition through zero from positive (in-plane) to negative (out- of-\nplane) at a temperature of roughly 80 K . This behavior is qualitatively consistent with the \nphenomenological model presented above and reveals a magnitude of the variation in 𝐻𝑒𝑓𝑓, \nfrom +91 .2 G ± 1.6 G at 300 K to - 45.2 G ± 1.1 G at 10 K, that is roughly 150% of the \nroom-temperature value.  \nNotably, this more comprehensive study also reveals new phenomenology at the \nlowest temperature of 5 K, where the anisotropy abruptly shifts back to in-plane with a \nvalue of +26.2 G ± 0.6 G (roughly 25% of the room-temperature value). This behavior \nreproduces across all samples measured and is quantitatively reproduced upon temperature \ncycling of individual films. The abruptness of this change is distinct from the broad and \nmonotonic behavior observed for temperatures greater than 9 K . The origin of this abrupt \nchange is unclear, but there are two potential explanations consistent with this \nphenomenology. First, it is possible that the increase in strain results in an abrupt relaxation \nthrough the creation of structural defects. This explanation would require some level of \nself-healing upon warming in order to explain the reproducibility of the transition. Given \nthe lack of long-range structural order in V[TCNE] x films as-grown [ 37] it is possible that \nany residual structural defects do not contribute to additional magnetic loss (damping). \nSecond, it is possible that there exist paramagnetic spins in the system that magnetically \norder at temperatures below 9 K. If such spins were preferentially located in an interface \nlayer, for example, their ordering could create an exchange bias that would then pull the \neasy-axis back to an in-plane orientation. \nThe temperature dependence of the linewidth of the magnetic resonance provides \nan additional avenue for evaluating these potential explanations. Figure 4 shows the \nlinewidth for the in-plane magnetic resonance from 300 K to 5 K, with additional data to \nmore clearly resolve the sharp change between 5 K and 9 K. The linewidth data presented \nin Figure 4 is extracted from a single (in-plane applied magnetic field orientation) scan. As \na result, the initial dataset underlying Figure 3 was supplemented by a second temperature \ndependent scan at fixed angle in Figure 4. This data reveals a monotonic increase in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 10 linewidth with decreasing temperature from 300 K down to 9 K followed by a dramatic \ndecrease in linewidth between 9 K and 5 K, coincident with the abrupt change in magnetic \nanisotropy. We note that in studies of YIG thin-films broadly similar phenomenology is \nobserved, though with a maximum in linewidth that is both higher amplitude (roughly 28 \ntimes the room-temperature value) and at higher temperature (typically 25 K) than is \nobserved here [23,30]. Prior work [23,28 ] has explained this behavior using a model of \nmagnon scattering from paramagnetic defect spins (also referred to as two-level \nfluctuators, TLF) wherein the scattering cross-section at high temperature increases with \ndecreasing temperature as the thermal polarization of the spins increases. This \nphenomenology competes with magnon-pumping of the paramagnetic spins into their \nexcited state, a process that saturates as the spin-lifetime of the defects becomes long \nrelative to the spin-magnon scattering time. The competition between these two processes \nyields a local maximum in the damping (linewidth) that depends on the temperature \ndependent spin lifetime, ts, the energy separation between majority and minority spin states, \nℏωeg, and the difference between that energy splitting and the uniform magnon energy,  \n(ℏω - ℏωeg ). \nIn this model, the linewidth expression is proportional to the square of the exchange \ninteraction energy between V[TCNE] x atoms and the impurity level ( ℏωint)2 ~ (ℏωeg)2, a \nline-shape factor accounting for the finite spin lifetime, 1/𝑡 𝑠/(ℏ2/𝑡𝑠2 + (ℏω - ℏωeg )2ts2), and \nthe ratio between the ground and excited impurity states for fast impurity relaxation , \ntanh(ℏω/2k BT)  [23, 28], \n \nΔ𝐻 =  𝑆\n𝛾 𝑁𝑖𝑚𝑝\n𝑁 (ℏ𝜔 𝑖𝑛𝑡)2 1/𝑡 𝑠 \nℏ2/𝑡𝑠2+ (ℏ𝜔 − ℏ𝜔 𝑒𝑔)2 𝑡𝑠2tanh (1\n2 ℏ𝜔\n𝑘𝐵𝑇) + 𝐻 𝑂 \n \nwhere Nimp/N is the ratio between number of impurit ies and number of V[TCNE] x atoms, \nand S is the averaged V[TCNE] x spin per site , 𝛾 is the gyromagnetic ratio and 𝐻𝑂 is a \nconstant offset due to other relaxation mechanisms. In addition, we assume spin lifetime ts (4) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 11 = t∞ 𝑒𝐸𝑏𝑘𝐵𝑇⁄ [31, 38, 39] where t∞ is the spin lifetime limit at very high temperatures, and \nEb is a phenomenological activation energy. Figure 4 includes a fit of Eq. (4) to the \nexperimental linewidth (orange line) that yields for S ~ 1 and ωint ~ ωeg the parameters: \nωegt∞ = 0.98, Eb = 1meV, ω egNimp/N = 36.5GHz and 𝐻𝑜= 1 G . Interestingly, if we assume \na reasonable value for ℏωeg of 1.3 meV, a value of Nimp/N = 0.1 follows, thus indicating \nthat V[TCNE] x is an exceptional low-loss magnetic material even if we assume an impurity \nconcentration as high as 10%. This observation is consistent with the hypothesis of \ninsensitivity to structural defects discussed above. \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out- of-plane easy-axis to an in-plane easy-axis. This \nchange in magnetic anisotropy has the potential to have a substantial impact on spin-\nmagnon scattering efficiency. For example, this change will result in a shift of the energy \nof the magnon bands (see Eq. 1), and if this change involves a commensurate change in the \nstrain there will also be a modification to the spin-orbit coupling and exchange parameters \nat the paramagnetic defects. It should be noted that although this reentrant anisotropy is an \nintriguing feature, the fits to our model for TLFs in Figure 4 are able to reproduce our \nlinewidth data without reference to this effect. As a result, we interpret this fit as an upper \nbound on Eb. This is represented by the additional fits shown in Figure S7 within the \nSupplemental Material wherein we assume a lower temperature for the nominal peak in \nlinewidth occurring due to spin-magnon scattering that is experimentally preempted by the \nchange in magnetic anisotropy. These alternate fits agree with experimental observations \nat temperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>> ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins would also \nenhance the suppression of spin-magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K.  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 12 When considering the expected behavior as the temperature is further reduced below \n5 K, as would be the case for many applications in QISE, it is useful to consider recent \nmilliKelvin-range measurements of YIG films [40]. That work confirms the expected \ncontinued narrowing down to 500 mK followed by a modest increase from 500 mK down \nto 20 mK, for an overall line narrowing of roughly a factor of 2. The model of scattering \nfrom TLFs described above is consistent with this result in YIG if one supposes a second \npopulation of TLFs that are dipole coupled to the magnons rather than exchange coupled , \nfor example dilute magnetic impurities in the substrate or environment. We note that \nextending this model into V[TCNE] x requires taking into account: i) the substantial \ndifference in structure and chemistry between V[TCNE] x and YIG, and ii) the fact that Ms. \nin V[TCNE] x is roughly 20 times smaller than in YIG. The former consideration indicates \nthat the presence of these dipole coupled TLFs need not correlate between the two systems, \nwhile the latter predicts that any relaxation associated with their presence should be \nreduced by a factor of 20 from Ref. [40]. As a result, the overall factor of 2 decrease in \nlinewidth observed in YIG between temperatures of 5 K and 20 mK should be taken as an \nextremely conservative lower bound on the performance of V[TCNE] x. Given that the \nlinewidth in V[TCNE] x at 5 K is already on par with its room temperature value, these \nresults firmly establish the suitability for this material for applications in quantum \nmagnonics and related aspects of QISE. \nIn conclusion, this work presents the first systematic study of the magnetization \ndynamics of V[TCNE] x at low temperatures. A strong variation in resonance frequency and \nanisotropy with temperature is observed , and attributed to a temperature-dependent strain \narising from the mismatch in thermal expansion coefficients between V[TCNE] x films and \ntheir sapphire substrates. The resonance linewidth of these films is found to increase with \ndecreasing temperature up to a maximum value of 15 G (roughly 9 times the room-\ntemperature value) and is well fit by a model based on magnon scattering from \nparamagnetic defect spins. At 5 K the magnetic anisotropy reverts to in-plane, coinciding \nwith a nearly complete recovery of the resonance linewidth to room-temperature values; \nquantitative modeling suggests the linewidth behavior arises from scattering from \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 13 paramagnetic defect spins that is suppressed at very low-temperature. This suppression of \nspin-magnon scattering is expected to strengthen as temperature is further decreased into \nthe milli-Kelvin range due to freeze-out of thermal magnons and phonons, providing a \ncompelling case for the utility of V[TCNE] x for low-temperature microwave applications, \nsuch as those emerging in the field of quantum information science and technology. \n \nAcknowledgements:  The authors would like to thank A. Franson for providing a \nsoftware suite for fitting FMR spectra as well as general fitting assistance, and G. Fuchs \nfor fruitful discussions . The work presented in the main text, both experiment and theory, \nwas primarily supported by the U.S. Department of Energy, Office of Basic Energy \nSciences, under Award Number DE-SC0019250. S. Kurfman was supported by NSF \nEFMA-1741666 and grew V[TCNE] x calibration samples used for preliminary \nmeasurements not explicitly included in this paper . Work on V[MeTCEC] x presented in \nthe supplementary material was performed by M. Chilcote and Y. Lu with the support of \nNSF Grant No. DMR- 1741666. \n \nData availability statement:  See supplementary material at URL will be inserted by AIP \nPublishing for datasets pertaining to temperature-dependent anisotropy of V[MeTCEC] x, \nmethod for extracting linewidth of V[TCNE] x from FMR scans and additional fits to \nexperimental data highlighting temperature dependence of V[TCNE] x linewidth. \n \nReferences:  \n \n[1] A. Raveendran, M. T. Sebastian, and S. Raman, “Applications of Microwave \nMaterials: A Review” J. Electron. Mater. 48, 2601 (2019). \n[2] Ü. Özgür, Y. Alivov, and H. Morkoç, “Microwave ferrites, part 1: Fundamental \nproperties” J. Mater. Sci. Mater. Electron. 20, 789 (2009). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 14 [3] J. M. Silveyra, E. Ferrara, D. L. Huber, and T. C. 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Hammel, F. Y. Yang, and D. C. Ralph, “Increased low -temperature \ndamping in yttrium iron garnet thin films” Phys. Rev. B 95, 174411 (2017). \n[31] W. A. Yager, J. K . Galt, and F. R. Merritt, “Ferromagnetic resonance in two nickel -\niron ferrites” Phys. Rev. 99, 1203 (1955). \n[32] H. Puszkarski and M. Kasperski, On the Interpretation of the Angular Dependence \nof the Main FMR/SWR Line in Ferromagnetic Thin Films  (2012). \n[33] M. Chilcote, M. Harberts, B. Fuhrmann, K. Lehmann, Y. Lu, A. Franson, H. Yu, \nN. Zhu, H. Tang, G. Schmidt, and E. Johnston- Halperin, “Spin -wave confinement \nand coupling in organic- based magnetic nanostructures” APL Mater. 7, (2019).    \n[34]   Y. Mei, P. J. Diemer, M. R. Niazi, R. K. Hallani. K. Jarolimek, C. S. Day, C. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 18 Risko, J. E. Anthony, A. Amassian and O. D. Jurchescu, “Crossover from band -\nlike to thermally activated charge transport in organic transistors due to strain-\ninduced traps” P NAS 114, 33 (2017). \n[35] H. Suhl, “Ferromagnetic Resonance in Nickel Ferrite Between One and Two \nKilomegacycles” Phys. Rev. 97, 555 (1955). \n[36] J. Smit and H. G. Beljers., “Ferromagnetic resonance absorption in BaFe 12O19” \nPhilips Res. Rep. 10, 113 (1955). \n[37] M. Chilcote, Y. Lu, and E. Johnston-Halperin, Organic-Based Magnetically \nOrdered Films  (World Scientific, 2018). \n[38]   J. K. Galt and E. G. Spencer, “Loss Mechanism in Spinel Ferrites ” Phys. Rev. 127, \n1572, 1962. \n[39]   H. Maier-Flaig , S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. \n          Huebl, and S. T. B. Goennenwein , “Temperature dependent damping of yttrium \n          iron garnet spheres ” Phys. Rev. B 95, 214423 (2017). \n[40]   S. Kosen, A. F. van Loo, D. A Bozhko, L. Mihalceanu, R. Gross, and A. D. \n          Karenowska , “Microwave magnon damping in YIG films at millikelvin \n          temperatures ” APL Mater. 7, 101120  (2019 ). \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 19 Figure Legends:  \n \nFigure 1  \n \n(a) Schematic (planar view) of the CVD growth system; (b) FMR scan of V[TCNE] x \nthin film at 300 K with the applied magnetic field applied in the plane (IP) of the \nsample with 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. ΔH pp denotes the peak-\nto-peak linewidth measured as the difference between the positive and negative peak \npositions; (c) FMR line scans for in-plane field orientation at 300 K, 80 K and 40 K \nwith 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. \n \nFigure 2  \n \nAngle-dependent FMR spectra at temperatures of 300 K, 80 K and 40 K at different \nfield orientations with respect to the sample normal. Nominally the sample is rotated \nfrom  𝜃 = − 10𝑜 to  𝜃 =  100𝑜 in increments of 10𝑜, where 𝜃 =  90𝑜 and 𝜃 =  0𝑜 \nare in-plane and out- of-plane field orientations respectively. Angle corrections have \nbeen taken into account (through fitting with Eq. ( 3)) to reflect the actual rotation \nangles, denoted by the black arrows to the right of each of the temperature-labeled \npanels. \n \nFigure 3  \n \n(a) Schematic of the changes in anisotropy at 300 K and 40 K. 𝑯𝒂𝒑𝒑 denotes the \nexternal magnetic field, 𝑯𝒅𝒆𝒎𝒂𝒈  represents the demagnetizing field of the \nV[TCNE] x film and 𝑯𝒄𝒓𝒚𝒔𝒕𝒂𝒍  is the crystal-field anisotropy. It should be noted that \na finite thin-film has a (negligibly) small demagnetization field when the external \nfiled is applied in the plane since this is not a truly infinite film; (b) Resonance field \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 20 at different field orientations plotted against sample rotation angles for 300 K (open \ncircles)  and 40 K (filled circles) and fits to Eq. (3) (dashed and solid line, \nrespectively) to extract the effective field 𝑯𝒆𝒇𝒇; (c) 𝑯𝒆𝒇𝒇 plotted against temperature \nranging from 300K – 5K. The inset shows the FMR lineshapes at 300 K and 5 K; \nfitting the data to extract the linewidth at FWHM gives 1.63 G and 2. 58 G \nrespectively , this shows that the two linewidths are indeed comparable with the \nlinewidth at 5 K only about 1.66 times larger than the room-temperature value . For \nboth (b) and (c), experimental errors are smaller than the point size.   \n \nFigure 4  \n \nV[TCNE] x linewidth as a function of temperature (black points) and \ncorresponding curve fit (orange line) using Eq. (4). \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 21  \n \n \n \n \n \n \n \n \n \n  \n  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n     Figure 1 H. Yusuf  et al. \n  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 22  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2 H. Yusuf  et al. \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 23  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3 H. Yusuf  et al. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 24  \n \n \n \n \n \n \n \nFigure 4 H. Yusuf  et al. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 25  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 1 Supplementary Materials for “ Exploring  a quantum -information -relevant  \nmagnonic  material:  ultralow  damping  at low temperature  in the organic  \nferrimagnet  V[TCNE] x” \nH. Yusuf *1, M. Chilcote *1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E.  \nFlatté3, E. Johnston -Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210  \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853  \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242  \n \n* These authors contributed equally to this work.  \n \n1. Temperature -dependent anisotropy of V [MeTCEC ]x \nHere, we investigate the magnetic properties of  vanadium methyl tricyanoethylene  \ncarboxylate  V[MeTCEC ]x thin-films using temperature -dependent cavity ferromagnetic \nresonance (FMR). The MeTCEC ligand is similar to the TCNE described in the main text, \nand these results demonstrate that strain -dependent anisotropy is a general feature of this \nclass of metal -ligand materials. Figure  S1a shows the molecular structures of both the \nTCNE molecule and the MeTCEC molecule discussed below.  Figure  S1b shows \ntemperature -dependent magnetization  data for zero field -cooled (ZFC; open black squares) \nand field -cooled (FC; open red circles) measurements , and electron transport data (filled \nblack squares) collected for V[MeTCEC] x thin-films on the same temperature axis. Notice \nthat the maximum in the ZFC magnetization curve – sometimes referred to as the blocking \ntemperature[13,14] – corresponds to the rapid rise observed in the resistance data. This \nchange in electronic and magnetization properties has been associated with carrier freeze  \nout and a magnetic phase transition in related materials such as magnatites, but in light of  2 the results presented in the main text we note \nthat a structural transition associated with \nincreased strain in the films may also play a role \nin these measurements.  \nV[MeTCEC] x samples are deposited on \nAl2O3(0001) substrates using a previously \nreported synthesis and chemical vapor \ndeposition (CVD) growth process.[4,16] During \nthe deposition, argon gas carries the two \nprecursors, MeTCEC and V(CO) 6, into the \nreaction zone where V [MeTCEC ]x is deposited \nonto one or more substrates. The system \nemploys three independently temperature -\ncontrolled regions for the MeTCEC , V(CO) 6, \nand reaction zone with typical setpoints of \n55 °C, 10 °C, and 50  °C, respectively and with \ntypical flow rates for each precursor of 50 \nsccm. Sample growth, manipulation, and \nhandling is p erformed in an argon glovebox \n(O2 < 1.0 ppm; H 2O < 1.0 ppm).   \nAfter growth, samples are mounted onto custom microwave compatible sample \nholders in the appropriate orientation, protected from undesired rotation, and flame -sealed \nin evacuated electron paramagnetic resonance (EPR) grade quartz tubes without exposure \nto air. When not being measured, the sealed samples are stored in a -55 °C freezer and are \nfound to be stable for weeks.  \nFigure  S2 shows four ferromagnetic resonance (FMR) spectra of V[MeTCEC] x \noriented both in  plane (90 °; see inset to Fig.  3c) and out of plane (0 °) at 140  K and 80  K. \nThe FMR response of magnetic materials is sensitive to the local field environment of the \nFigure S1 (a) The molecular structures of \ntetracyanoethylene (TCNE) and \ntricyanoethylenecarboxylate (MeTCEC). (b) \nMagnetization vs. temperature curves for zero field -\ncooled (ZFC; open black squares) and zero field -\ncooled (FC; open red circles) measurements. On the \nsame  temperature axis, resistance vs. temperature \ndata is shown for a V[MeTCEC]x thin -film (filled \nblack squares). The corresponding dependent -axis is \nshown on the right axis. Note the maximum in the \nmagnetization data corresponds to the rapid rise \nobserved in  the resistance data.   3 sample and therefore allows for sensitive characterization of the anisotropy fields in \nV[MeTCEC] x. FMR measurements are performed using a Bruker electron paramagnetic \nresonance spectrometer setup for X -band measurements with 200 µW of applied \nmicrowave power and fitted with an Oxford Instruments ESR900 cryostat insert. The \ncryostat is cooled by flowing liquid nitrogen and operates at tempe ratures ranging from \n80 K to 300  K with better than 50  mK stability during FMR measurements. In standard \noperation, the microwave frequency of the spectrometer is tuned between 9 and 10  GHz \nfor optimal microwave cavity performance before the measurement, a nd then the frequency \nis fixed while the DC field is swe pt during the measurement.   \nFigure  S2a shows FMR spectra collected at 140  K for the magnetic field applied in  \nplane (𝜃 = 90°) and out  of plane ( 𝜃 = 0°). Consistent with prior FMR measurements of \norganic -based magnetic  materials,[6,16,17] the center field associated with th e resonant \nfeature in the in -plane spectrum is at a lower field than that of the out -of-plane spectrum, \nand therefore the easy magnetization axis is oriented in the plane of the film. This easy -\naxis orientation is the expected outcome resulting from the sh ape anisotropy present in \nthin-film samples. Figure  S2b also shows FMR spectra collected with the magnetic field \napplied in  plane ( 𝜃 = 90° ) and out of plane (𝜃 = 0°).  However, this data is collected at 80  K, \nfurther below the maximum in the V [MeTCEC ]x ZFC magnetization curve than the data \nshown in Fig . S2a. Surprisingly, the center field of the dominant resonance feature in the \nin-plane spectrum is at a higher field than that of the out -of-plane spectrum. This behavior \nseems to indicate that th e sample has an easy  axis oriented out of the plane of the sample; \nthe spectra show signs of a switch in the magnetic easy  axis from in  plane to out  of plane \nas it is cooled from 140  K to 80  K.  4 To investigate this behavior in greater \ndetail, angular -dependent data is collected in 10 ° \nincrements as the ap plied field is rotated from in  \nplane ( 𝜃 = 90° ) to out of plane (𝜃 = 0°)  of the \nsample. The data is shown in Figs . 3a and b for \n140 K and 80  K respectively. A gray dashed line \nis overlaid on the data to serve as a guide to the \neye. The field shifts shown in Figs . S3a and S3b \nare consistent with those shown in Fig . 2 above. \nFigure  S3c shows the center fields extracted from \nthe two -angle series, emphasizing the magnitude \nof the change in the anisotropy.  \nThis switch in the magnetic easy  axis from \nin plane to out  of plane present in the data \nsuggests the presence of an additional \ncontribution to the anisotropy beyond simply \nshape anisotropy. Previously, given the isotropic \nin-plane response of thin  films at room  \ntemperature, additional contributions to  the anisotropy had been excluded. However, the \nresults here warrant the inclusion of an additional term 𝐻#, which is responsible for \ninducing perpendicular anisotropy in thin  films. This phenomenology is consistent with the \nmeasurements of V[TCNE] x thin films presented in the main text. Following that \ndevelopment,  the angular  dependence of the FMR response for in  plane to out of plane \nrotation of a thin -film sample can therefore be described by,[17–19] \n𝜔\n𝛾='(𝐻−4𝜋𝑀-..cos2𝜃)\t(𝐻−4𝜋𝑀-..\tcos2𝜃)\t\n='(𝐻−(4𝜋𝑀6−𝐻7)cos2𝜃)\t(𝐻−(4𝜋𝑀6−𝐻7)\tcos2𝜃), (1) \nFigure S2 (a) Single FMR line scans at 140  K for \na sample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield. (b) Single FMR line scans at 80  K for a \nsample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield.   5 where ω is the resonance fequency, γ is the gyromagnetic ratio, 𝐻 is the applied field, and \n𝜃 is the polar angle of the magnetization.  The FMR resonance fields are more than an order \nof magnitude larger than the typical saturation field for V[TCNE] x, and therefore we have \nassumed that the magnetization is effectively p arallel to the applied magnetic field (i.e. \n𝜙≈𝜙> and 𝜃≈𝜃? where 𝜃, 𝜙 and 𝜃?, 𝜙? are the polar and azimuthal angles of the \nmagnetization 𝑀 and the applied bias field 𝐻, respectively). Also, note that the in -plane \nFMR response remains isotropi c, with the 𝜙 dependence dropping out:  \n𝜔\n𝛾='𝐻\t(𝐻+4𝜋𝑀-..)\t\n='𝐻\t(𝐻+(4𝜋𝑀6−𝐻7)) \nThe data in Fig.  S3c are then fit according to the dispersion relation in Eq.  1. The fits \nare shown as solid and dashed lines in Fig.  3c. The 140  K data yields an 𝐻eff=4π𝑀eff= \n15.4 Oe ± 0.1 Oe while fi tting to the 80  K data result in an 𝐻eff value of  -28.3 Oe ± 1.0 Oe. \nThe negative value of 𝐻eff for the 80  K data means that 𝐻#>4π𝑀S and that the film has \nperpendicular magnetic anisotropy. Note that the magnetic energy landscape, and therefore \nthe angular dependence contained in Eq.  1, does not allow for the easy  magnetization axis \nto take on an intermediate vector between in  plane or  out of plane for this set of anisotropy \nfields. This result also implies that prior measurements of the anisotropy of thin  films are \nin fact measuring 4π𝑀eff rather than the bare 4π𝑀S as previously assumed.[17,20] However,  (2) \n  (𝜃=90°).   6 as with previous studies of uniform thin  films, \nit is challenging to disentangle this form of \nanisotropy from 4π𝑀S, leading us to use the \nmore general 𝐻eff=4π𝑀eff. Temperature -\ndependent FMR studies combined with careful \nDC magnetization measurements provide a \npromising avenue to decoupling the two \nanisotropy fields.  \nIn comparing the data shown in Fig.  S3a \nand b, also note that at lower temperatures, the \nresonance response becomes markedly multi -\nmodal and appears to broaden. To investigate \nthis behavior in greater detail, FMR data is \ncollected over a range of temperatures with the \napplied field oriented in the plane of the \nsample. The dat a is shown in Fig.  S4a. Note \nthe clear shift of the resonant features towards \nhigher field at lower temperatures as the in -\nplane orientation, which is the geometry being \nmeasured in this data set, changes from the \neasy magnetization axis to the hard \nmagnet ization axis.  \nThe effective magnetization, 4π𝑀eff, \nextracted from the data shown in Fig.  S3 \ncontains contributions from a perpendicular \nmagnetic  anisotropy energy. This 𝐻# does not \narise from shape anisotropy in thin  films and  \nFigure S3 (a) Shows FMR spectra as the sample is \nrotated from in -plane ( 𝜃 = 90°) to out of plane \n(𝜃 = 0°) with respect to the externally applied \nmagnetic field at 140  K. (b)  Shows the FMR \nspectra as the sample is rotated from in -plane \n(𝜃 = 90°) to out of plane ( 𝜃 = 0°) with respect to \nthe externally applied magnetic field at 80  K. (c) \nShows the extracted center fields from the angular \nseries shown in (a) and (b) with fits shown as solid \nand dashed lines. The inset shows the coordinate \nsystem with respect to the sam ple geometry.  0.1 Oe \n1.1 Oe  7   \n  \n  \n  \nmust instead come  from a crystal -field anisotropy wherein the local exchange vector \nacquires some anisotropy due to some combination of lattice symmetry and strain. Given \nthe large differences in the coefficients of thermal expansion for organic and in organic \nmaterials (often varying by an order of magnitude or more) , stain due to differential thermal \nexpansion at the interface between the substrate and organic -based materials is likely \ncreating an anisotropic strain field in the magnetic material. As the sample temperature is \nlowered, this strain field increases until 𝐻# becomes larger in magnitude than 4π𝑀S and \n4π𝑀eff takes on a negative value. The result is a magnet with an easy -axis out of plane as \nshown in Fig.  S3b.  \nWe note that qualitatively similar results were obtained for vanadium ethyl \ntricyanoethylene carb oxylate ( V[ETCEC ]x). V[ETCEC ]x is a third member of this class of \nmetal ligand ferrimagnets[23,24], supporting the thesis that strain -dependent anisotropy is a \ncommon feature of this class of materials.  \n2. Method for extracting linewidth from FMR scans  \nThe FMR scans are obtained through phase -sensitive detection, where in addition to the \nstatic DC magnet ic field the sample sees a sinusoidally modulated field component that is  Figure S4 (a) Shows FMR spectra of V[MeTCEC ]x sample mounted  in-plane \n(𝜃 = 90°) with respect to the externally applied magnetic field as a function of \ntemperature . (b)  Shows the extracted peak -to-peak linewidths from the \ntemperature -dependent spectra shown in (a)  \n 8 varied  at the same frequency as the amplitude modulation  of the microwaves reflected from \nthe cavity. If there is an EPR signal , that signal is converted into a sine wave whose \namplitude is proportional to the  derivative of  the signal  (change in microwave power \nrelative to field modulation)  and appears as the first derivative of a Lorentzian function . In \naddition, it should be noted that some FMR scans show multi  peaks (for examp le, the 300 \nK scans shown in Figure 2 of the main  text) and a possible reason for that could be \ninhomogeneous strain. As discussed in our main text, strain in our films is induced by \ndifference in thermal expansion coefficients between V[TCNE] x and the substrate. Given \nthat we have taken no special precautions to prevent it, we believe it is likely that this strain \nwill be inhomogeneous, resulting in regions of our sample with differing magnetic \nanisotropy, and therefore the potential for additi onal peaks in FMR spectra. It has been \nreported that strain -induced distortions can alter the local electronic and crystal -field \nenvironment by changing the orbital occupancy, tilt angle between neighboring spins[25] or \nmagnetocrystalline anisotropy[26,27], for instance, leading to local changes in magnetic \nanisotropy which result in the appearance of additional resonance peaks . \n Since the asymmetry of the FMR lineshape and the multi -peaks need to be \naccounted for , scans are  not simply fit by the derivative of a symmetric Lorentzian . In \nphase -sensitive measurements the microwave electric field generates oscillati ng electric \ncurrents in the sample ; the oscillating magnetization due to the microwave magnetic field \nresults in oscilla ting angles between the current flow and magnetization, leading to local \nlattice distortions  which may c ause the observed asymmetry in signal lineshape  due to \ninhomogeneous broadening[28,29]. Another possible source of this asymmetry could be the \nresult of high cavity loading[30] and the resulting phase error introduced by the automatic \nfrequency controller of the EPR spectrometer when the sample is resonantly excited. This \nwarrants the inclusion of a dispersion or antisymmetric term that takes int o account this \nasymmetry, therefore the FMR scans are fitted to the sum of the derivative of an absorption \n(symmetric  term) and dispersion (antisymmetric term ) from a Lorentzian . The derivative s \nhave the following form : (3) (4)  9 absorption derivative =\t−32\t√3\t𝐴\t𝐹𝑊𝐻 𝑀K(𝐵−𝐵M)\n9\t[FWHM \t2+4(𝐵−𝐵M)\t2]\t2 \n \ndispersion derivative =\t−4\t𝐷\t𝐹𝑊𝐻𝑀 \t(𝐵−𝐵M)\nFWHM \t2+4(𝐵−𝐵M)\t2 \nwhere FWHM  is the full -width at half -max, A is the height of the absorption derivative, D \nis the height of the dispersion derivative, 𝐵M is the location of the resonance (center) field  \nand B is the amplitude of the magnetic field that is being swept at each data point. \nTherefore, the resulting li ne shape depends on the relative contributions of these two terms .  \nFor scans with an out -of-plane easy  \naxis fitting with a single  derivative sum \nprovides good agreement with the data  \n(Figure S5). But for scans with in -plane \neasy axis, due to the appearance of a \nmodest satellite peak , obtaining a good fit \nto the data requires addition of up to three \nderivative sum s. For FMR scans in the \nrange 9 K  –  80 K (out -of-plane easy  axis \nbetween 9 K – 100 K and negligible \nanisotropy at 80 K) the date is well fit with a single derivate sum . On the other hand,  fits \nfor scans in the high temperatures between 120 K – 300 K (in -plane easy  axis) give good \nagreement with data when  two derivative sums  are used , a few requiring up to three \nderivative sums (Figure  S6b).  However, FMR scans at 5 K (in-plane easy  axis) and 6 K \n(out-of-plane easy  axis) mimic the high temperature fits by requiring two derivative sums.  \nFor the purposes of this study, which explores the fundamental FMR mode, in scans \nshowing multiple peaks we focus on the contribution fr om the peak that persists to low Figure S5 Shows a single derivative fit to the \nFMR data collected at 2 2 K \n 10 temperatures.  If we plot the individual Lorentzian \ncomponents of the FMR fit, we find that the first \ncomponent YL1 (component with the highest \noverall peak -to-peak magnitude) is present in all \nthe temperatures being consid ered in the range 300 \n– 5 K. Therefore, the linewidth date plotted against \ntemperatures in Figure 4 of the main text is the \nlinewidth at f ull width  half max (FWHM) of YL1.  \nIn Figure S6a it can be seen that fitting the \nFMR scan at 300 K with a single derivative sum \ndoes not provide a great fit to the data. However , \nfrom  Figure S6b it becomes clear that fitting the \nsame data with the superposition of three \nderivative sums or components (each with their \ndistinct A, D and FWHM ) gives a decent  fit. In \nFigure S6c the amplitude of each individual \ncomponent is plotted against magnetic field sweep \nrange to provide a visual understanding of how \neach component contributes to the overall FMR \nline shape.  \n3. Temperature dependent linewidth  \nThe V [TCNE ]x linewidth dependence on \ntemperature can be well explained from the \ninteraction between magnons and defects or Figure S6 (a) Shows a single derivative fit to the \nFMR data collected at 300 K. (b) Shows FMR \nscan at 300 K fitted to superposition of three \nLorentzian derivative  sums. (c) Amplitude of \neach sum or component plotted against magnetic \nfield sweep range.  YL1, YL2 and Y L3 are the \nfirst, second and third components respectively.  \n 11 impurities in V [TCNE ]x.  The \ndefects or impurities are \nconsidered to be a two -level spin \nsystem s. These experience spin -\nflip transitions excited by the \nannihilation of a uniform -magnon \nmode  [31,32]. This process \nintroduces a finite magnon \nlifetime, which in turn leads to the \nlinewidth expression Eq. (4)  in the \nmain  text. In Fig. S7, we use four \ndiffere nt parameter sets to fit the \nhigh temperature experimental \ndata using Eq. (4). All the different \nsets yield a good fitting for T> 9 K, \nalthough the smaller the E b, the \nsmaller the nominal peak in \nlinewidth. As discussed in the main text, this imposes an upp er bound on E b ~ 1meV.   \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out -of-plane easy  axis to an in -plane easy  axis. This change \nin magnetic anisotropy has the potential to have a substantial impact on spin -magnon \nscattering efficiency. For example, this change will result in a shift of the energy of the \nmagnon bands (see Eq. 1  in main text ), and if this change involves a commensurate change \nin the strain there will also be a modifi cation to the spin -orbit coupling and exchange \nparameters at the paramagnetic defects. This is represented by the fits shown in Fig. S 7 \nwherein we assume a lower temperature for the nominal peak in linewidth occurring due \nto spin -magnon scattering that is experimentally preempted  by the change in magnetic \nanisotropy.  As a result, we interpret th e fit in Fig. 4 of the main text as an upper bound on \nFigure S7 (a) (b), (c) and (d) show V [TCNE ]x linewidth as a \nfunction of temperature and the corresponding fit curves using \nfitting parameters of Eq. (4)  \n 12 Eb. The alternate fits presented in Fig. S7 agree with experimental observations at \ntemperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>>  ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins  would also \nenhance the suppression of spin -magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  13 References  \n[1] C. J. Brabec, Sol. Energy Mater. Sol. Cells  2004 , 83, 273.  \n[2] H. Shirakawa, E. J. Louis, A. G. MacDiarmid, C. K. Chiang, A. J. Heeger, J. Chem. \nSoc. Chem. Commun.  1977 , 578.  \n[3] C. W. Tang, S. A. Vanslyke, Appl. Phys. Lett.  1987 , 51, 913.  \n[4] Y. Lu, M. Harberts, C. -Y. Y. Kao, H. Yu, E. Johnston -Halperin, A. J. Epstein, Adv. \nMater.  2014 , 26, 7632.  \n[5] Y. Lu, H. Yu, M. Harberts, A. J. Epstein, E. Johnston -Halperin, J. Mater. Chem. C  \n2015 , 3, 7363.  \n[6] Y. Lu, H. Yu, M. Harberts, A. J. Epstein, E. Johnston -Halperin, RSC Adv.  2015 , 5, \n82271.  \n[7] J. L. Arthur, S. H. Lapidus, C. E. Moore, A. L. Rheingold, P. W. Stephens, J. S. \nMiller, Adv. Funct. Mater.  2012 , 22, 1802.  \n[8] J. P. Fitzgeral d, B. B. 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Celinski and B. Heinrich , Journal of Applied Physics  1991 , 70, 5935 .  \n[29]    Y. Li, F. Zeng, S. -L. Zhang, H. Shin, H. Saglam, V. Karakas, O. Ozatay, J. E.  \n           Pearson, O. G. Heinonen, Y. Wu, A. Hoffman and W. Zhang , Phys. Rev. Lett.  \n           2019 , 122, 117203 .  \n [30]    I. B. Goldberg and H. R. Crowe, Anal. Chem. 1977 , 49, 9, 1353 -1357.  \n [31]    M. Sparks, Ferromagnetic -Relaxation Theory (McGraw Hill, New York, 1964 ). \n [32]    P. E. Seiden, Phys. Rev.  1964 , 133, A728.  \n \n \n "    },    {        "title": "1911.12786v1.Transport_properties_of_spin_superfluids__comparing_easy_plane_ferro__and_antiferromagnets.pdf",        "content": "Transport properties of spin superfluids—comparing easy-plane ferro- and\nantiferromagnets\nMartin Evers and Ulrich Nowak\nFachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany\n(Dated: December 2, 2019)\nWe present a study on spin-superfluid transport based on an atomistic, classical spin model. Easy-\nplane ferro- as well as antiferromagnets are considered, which allows for a direct comparison of these\ntwo material classes based on the same model assumptions. We find a spin-superfluid transport\nwhich is robust against variations of the boundary conditions, thermal fluctuations, and dissipation\nmodeled via Gilbert damping. Though the spin accumulations is smaller for antiferromagnets the\nrange of the spin-superfluid transport turns out to be identical for ferro- and antiferromagnets. Fi-\nnally, we calculate and explore the role of the driving frequency and especially the critical frequency,\nwhere phase slips occur and the spin accumulation breaks down.\nI. INTRODUCTION\nSpin transport in magnetic insulators [1, 2] has been\nintensively studied beacause of the fundamental interest\nin the various physical phenomena that occur in these\nmaterials and because of their potential for future appli-\ncations. Magnetic insulators do not exhibit Joule heat-\ning [3] as no electron transport is involved and many of\nthese are oxides with exceptionally low magnetic damp-\ning [4], which hopefully allows for energy efficient trans-\nport properties. It has even been shown that the realiza-\ntion of logic elements is possible [5], such that devices are\ncompatible and integratable with CMOS technology [6].\nStudies on transport in this material class focuses mostly\non transport of magnons [7], i.e. quanta of spin waves—\nthe elementary excitations of the magnetic ground state.\nAs magnons are quasi particles, their number is not con-\nserved and each magnon mode shows an exponential de-\ncay upon transport through the system on a length scale\nξcalled magnon propagation length [8–13]. This is even\ntrue at zero temperature and in a clean system without\nany disorder due to the coupling of the magnons to elec-\ntronic and phononic degrees of freedom, a fact which is\ndescribed phenomenologically via Gilbert damping in the\nequation of motion as will be explained below.\nIn contrast to this damped magnonic transport, a pro-\nposal for spin transport was made that carries the name\nspin superfluidity. The original idea is in fact quite old\n[14, 15] and rests on a similarity of the magnetic or-\nder parameter—either the magnetization of a ferromag-\nnet or the Néel vector of an antiferromagnet—compared\nto the order parameter of superfluidity—the macroscopic\nwave function—as it occurs for He-4 below the lambda\ntransition. For instance, in easy-plane ferromagnets the\nmagnetizationfeaturesaspontaneouslybrokenrotational\nsymmetry in the easy plane ( SO(2)symmetry) that is\nequivalent to the spontaneously broken gauge invariance\nof the macroscopic wave function ( U(1)symmetry). This\nsymmetry leads in both cases to currents that are sta-\nble against small deviations—the supercurrents. [16] One\nstriking difference of spin-superfluid transport to spin-\nwave transport is its distance dependence: for spin su-perfluidity it is expected to be non-exponential, pushing\nthe limit of the range of magnonic transport.\nThe first experimental realizations of a spin superfluid\nwas achieved in a system of nuclear spins of He-3 atoms\n[17]—a model system which is not in a solid state. Only\nrecently the physics of spin superfluidity has drawn again\nattention for the case of solid magnets [18–23], including\na proposed dissipationless transport in metallic magnets\n[18]. However, König et al. neglected spin-orbit inter-\naction in their model for the electrons, which is one of\nthe reasons for Gilbert damping in magnets [24]. But ev-\nery known material exhibits spin-orbit interaction—since\nspinandangularmomentumofanatomareneverexactly\nzero—and therefore also magnetic damping, even if it is\nsmall. Consequently, spinsuperfluidsdoalwaysshowdis-\nsipation in contrast to their conventional counterparts.\nRecent theoretical work has focused on insulators\nrather than metals, usually based of phenomenological\nmodelsincluding theLandau-Lifshitz-Gilbertequationof\nmotion for both ferro- and antiferromagnets. [16, 19, 20]\nThe experimental detection of spin superfluidity in solid-\nstate magnets has been reported for magnon condensates\n[25], where the origin of the spin-superfluid order param-\neter is different to the cases described above, and also\nin antiferromagnetic solids [23]. However, the interpre-\ntation of the experimental findings is still controversially\ndiscussed [16, 26–28].\nIn the following, we will investigate and compare spin\nsuperfluidityinferro-andantiferromagneticmodels. The\ngeometry of our model resembles that of an experimen-\ntal non-local spin-transport investigation as sketched in\nfig. 1. In the corresponding experiments [29] at one side\n(here on the left) a spin current is injected into the mag-\nnet viathe spin-Halleffect causedby an electricalcurrent\nthrough an attached heavy-metal stripe. The resulting\nspin current is detected using the inverse spin-Hall ef-\nfect at another position (here the right-hand side). In\nour model we avoid the details of the excitation mech-\nanism and model the effect of the injected spin current\nby an appropriate boundary condition that triggers the\ndynamics of the spin systems that we investigate. This\nis done from the perspective of an atomistic, classicalarXiv:1911.12786v1  [cond-mat.mes-hall]  28 Nov 20192\nFigure 1. Basic concept of non-local spin transport as in an\nexperimental setup: heavy metal stripes are attached to the\nmagnet to inject a spin current via the spin-Hall effect (here\non the left hand side). The spin current in a certain distance\n(here at the right end) is detected via inverse spin-Hall effect.\nspin model, which has some advantages: the approach is\nnot restricted to small deviations from the ground state,\nfinite temperatures can be investigated and our calcu-\nlations are not limited to the steady state only. Fur-\nthermore, we are able to compare ferro- and antiferro-\nmagnetic systems. Their behavior turns out to be very\nsimilar, except for the resulting spin accumulation that is\nmuchlowerforthelatter. However,fromanexperimental\npoint of view antiferromagnets are much more promising,\nsince these are not prone to a breakdown of spin super-\nfluidity as a consequence of dipolar interactions, which is\nhard to avoid in ferromagnets. [22]\nII. ATOMISTIC SPIN MODEL\nWe consider the following classical, atomistic spin\nmodel of Heisenberg type [30], comprising Nnormal-\nized magnetic moments Sl=µl/µSon regular lattice\nsitesrl. We assume a simple cubic lattice with lattice\nconstanta. The Hamiltonian for these moments, in the\nfollowing called “spins”, is given by\nH=−J\n2/summationdisplay\n/angbracketleftn,m/angbracketrightSn·Sm−dz/summationdisplay\nn(Sn\nz)2,(1)\ntaking into account Heisenberg exchange interaction of\nnearest neighbors quantified by the exchange constant\nJ, where each spin has Nnbnearest neighbors. Further-\nmore, a uniaxial anisotropy with respect to the zdirec-\ntion with anisotropy constant dzis included. In this work\nwe consider the easy-plane case dz<0, where the mag-\nnets ground state readsgSl=±(cos(gϕ),sin(gϕ),0)with\nsome arbitrary, but uniform anglegϕ∈[0,2π](SO(2)\nsymmetry) and an alternating sign ±in case of antifer-\nromagnetic order ( J <0).\nThe time evolution of the spins Slis governed by\nthe stochastic Landau-Lifshitz-Gilbert (LLG) equationof motion [31–33]\ndSl\ndt=−γ\nµS(1 +α2)/bracketleftbig\nSl×/parenleftbig\nHl+αSl×Hl/parenrightbig/bracketrightbig\n(2)\nHl=−∂H\n∂Sl+ξl\n/angbracketleftbig\nξl\nβ(t)/angbracketrightbig\n= 0,/angbracketleftBig\nξl\nβ(t)ξl/prime\nη(t/prime)/angbracketrightBig\n=δll/primeδβηδ(t−t/prime)2µSαkBT\nγ\ndescribing the motion of a spin in its effective field Hl,\nwhereγisthegyromagneticratio, αtheGilbertdamping\nconstant,kBthe Boltzmann constant and Tthe absolute\ntemperature. The properties of the thermal noise ξlare\nchosen such that the dissipation-fluctuation theorem is\nsatisfied [34]. The material parameters define our sys-\ntem of units,|J|for the energy, tJ:=µS/γ|J|for the\ntime,afor the distance. Numerically the LLG equation\nis solved either by the classical Runge-Kutta method in\ncase of zero temperature, or at finite temperature using\nstochastic Heun’s method. At zero temperature the dis-\nsipated power per spin due to Gilbert damping follows\ndirectly from the time evolution of the spins Sl(t)[35]:\nPdiss=1\nNdH\ndt=1\nN/summationdisplay\nn∂H\n∂Sn/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\neff.field·∂Sn\n∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nLLG.(3)\nWe study a magnetic wire extended along xdirection.\nThe system size for our numerical simulations is given\nbyN=Nx×Ny×Nzspins along x-,y- andzdirec-\ntion, where Nx/greatermuchNy,Nz. For transverse directions we\nuse periodic boundary conditions if not noted otherwise.\nBoundary spins at x=Nxa(the right-hand side) are\ndenotedSl/vextendsingle/vextendsingle\nrightand at this side an open boundary con-\ndition is applied, Sl/vextendsingle/vextendsingle\nright= 0. At the opposite side, at\nx= 0, we use a time-dependent boundary condition,\nSl/vextendsingle/vextendsingle\nleft=±(cos(ω0t),sin(ω0t),0), (4)\nin form of an in-plane precession with frequency ω0that\ninjects a spin current from this side. The alternating sign\n(±) is used only for antiferromagnetic systems, according\nto the sublattices with antiparallel spin orientation.\nThe use of this boundary condition creates an ex-\ncitation with well-defined frequency ω0. Alternatively,\nwe also assumed an externally given spin accumulation\nµ=µezat the left-hand side that causes additional\ntorques on the spins and drives them out of equilibrium,\nwhich directly maps an experimental implementation us-\ning a spin-Hall-generated spin accumulation to the model\nutilized here. This method has been used for instance in\n[22]. In appendix B we calculate how this spin accumula-\ntion maps to the excitation frequency ω0and we further-\nmore confirmed numerically that both mechanisms lead\nto the same response for ferro- and antiferromagnets.\nAlthough an atomistic picture—comprising discrete\ndegrees of freedom—is studied numerically, the micro-\nmagnetic approximation is of particular value for analyt-\nical considerations of ferromagnets. This approximation3\nassumes that spatial variations of magnetic structures\nare small compared to the atomic distance a. In this\ncase differences can be approximated as derivatives and\nthe spins form a continuous field S(r,t). It is handy to\nuse cylindrical coordinates\nS=/parenleftBig/radicalbig\n1−S2zcosϕ,/radicalbig\n1−S2zsinϕ, Sz/parenrightBig\n,where definitions Sz(rl) :=Sl\nzandϕ(rl) :=ϕllink the\natomistic picture to the micromagnetics. Note that for\na spin superfluid Szis considered as the spin-superfluid\ndensity and ϕits phase. The use of the micromagnetic\napproximationforferromagnetsallowstoreformulatethe\nLLGequationintermsofdifferentialequationsfor Szand\nϕthat read\nµS\nγ˙ϕ=Ja2/bracketleftBigg\n1\n1−S2z∆Sz+Sz|∇Sz|2\n(1−S2z)2+Sz|∇ϕ|2/bracketrightBigg\n+ 2dzSz−αµS\nγ˙Sz\n1−S2z(5)\nµS\nγ˙Sz=−Ja2/bracketleftbig/parenleftbig\n1−S2\nz/parenrightbig\n∆ϕ−2Sz∇Sz·∇ϕ/bracketrightbig\n+α/parenleftbig\n1−S2\nz/parenrightbigµS\nγ˙ϕ. (6)\nThese two equations are strictly equivalent to the LLG\nequation eq. (2) for zero temperature with the only as-\nsumption of the micromagnetic approximation. If one\nexpands these equations in lowest order in ∇ϕ,∆ϕ,∇Sz,\nand∆Szfor an easy-plane magnet, which implies espe-\ncially assuming|Sz|/lessmuch1, but keeping|∇ϕ|2, one ends up\nwith\nµS\nγ˙ϕ=Ja2∆Sz+Ja2Sz|∇ϕ|2+ 2dzSz−αµS\nγ˙Sz(7)\nµS\nγ˙Sz=−Ja2∆ϕ+αµS\nγ˙ϕ. (8)\nImportantly, keeping the |∇ϕ|2term is actually required\nif the damping takes relatively high values, a fact which\nwe checked numerically. Furthermore, these equations\nare very similar to others already reported in [19, 21], but\nnot exactly equivalent. Ref. [19] uses more approxima-\ntions, especially neglecting the |∇ϕ|2-term, and ref. [21]\nconsiders a different starting point, namely a quantum\ntheory at low temperatures, where this term has a dif-\nferentSz-dependence. Because of this difference, the re-\nsult from [21] does not exactly match our numerical re-\nsults of the atomistic spin model, nor does it match the\nclassical micromagnetic theory. Hence, we use eqs. (7)\nand (8) that do describe the atomistic spin simulations\nwell. However, eqs. (7) and (8) can be solved in steady\nstate for a special case: a ferromagnet that is of length L\nalongxdirection and exhibits translational invariance in\ny- andzdirection as carried out in appendix A. Steady\nstate means a coherent precession of all spins with a fre-\nquency ˙ϕ=ω0and a stationary profile Sz(x). This so-\nlution of eqs. (7) and (8) reads:\nsϕ(x,t) =α\n2µSω0\nγJ(x−L)2\na2+ω0t+ϕ0(9)\nsSz(x) =sSz(L)\n1 +µ2\nSω2\n0\n2γ2Jdzα2/parenleftbigx−L\na/parenrightbig2, (10)\nwith a spin accumulation at the right end of the sys-\ntem (atx=Nxa=:L) ofsSz(L) =µSω0/2γdz, avalue which is independent of L—one of the striking fea-\ntures of spin superfluidity. Another feature is the mono-\ntone increase of ϕwhich implies the formation of an in-\nplane spin spiral with winding number Nw, which reads\n2πNw=/integraltext\ndϕ=ϕ(L)−ϕ(0). Note furthermore, that an\nopen boundary condition at the right end is an assump-\ntion that leads to solutions eqs. (9) and (10), correspond-\ning to a Neumann condition ∇ϕ/vextendsingle/vextendsingle\nright= 0, which must be\njustified as a realistic choice.\nFor the numerical study of eq. (2) we assume an open\nboundary at the right end. Equation (10) assumes the\nsame and results in a finite Szatx= 0, which contra-\ndicts the numerical driving boundary at this side, eq. (4),\nthat forces Sz(x=0) = 0. Furthermore, in an experiment\nan open boundary at the right end might not be feasi-\nble because of outflowing spin currents, for example into\nan attached heavy metal. Thus, the real behavior at\nthe boundaries for sure deviates from the ideal solution\neq. (10) and raises the question how strong that devia-\ntion is and in how far the boundary conditions influence\nthe overall bulk behavior of the spin transport. This\nis examined numerically from the full model eq. (2) by\nvarying the boundary conditions on the left and right.\nOne example of the variations we tested is an absorbing\nboundary condition on the right, modeling an outflow-\ning spin current by an enhanced damping. As result we\nobserve the profile Sl\nzto show only little change in that\ncase compared to an open boundary and also that in all\ncases the numerical profiles well follow eq. (10) (see in\nthe following fig. 2 a) as example). Other variations of\nthe boundary condition which we tested have also hardly\nany impact on the magnets overall response.\nIII. EASY-PLANE FERROMAGNET\nIn a first step of the numerical investigation, we con-\nsider a collinear ferromagnet as most simple case, with\nparameters J > 0for the ferromagnetic state and dz=\n−0.01Jas in-plane anisotropy. Let us describe the phe-4\n0 1000 2000 3000 4000 5000012345610-3\n010203040506070\n0 0.5 1 1.5 2 2.5\n10-300.010.020.03\n5 6\n10-40.0240.026\nFigure 2. Spin superfluidity in a 1D ferromagnet at T= 0in the steady state: a)depicts the spin accumulation Szand the\nin-plane angle ϕforω0tJ=−2×10−4; numerical data (blue and red symbols) follow perfectly the theoretical curve eqs. (9)\nand (10) (black, dashed lines), except for the vicinity of the left boundary. This is an artifact of the boundary condition, eq. (4),\nused for the numerics. b)shows the spin accumulation at the right end of the system SN\nzversus driving frequency ω0; for small\ndriving frequencies up to a critical value ωcritthe numerical data follow the analytical curvesSz(L); for larger frequencies the\nspin accumulation breaks down, deviating form the theoretical curve, due to phase slips and spin wave excitations.\nnomenology of the spin superfluid in a 1D system of size\nNx×Ny×Nz= 5000×1×1at temperature T= 0.\nThis model is equivalent to a 3D system with transla-\ntional invariance in y- andzdirection. Furthermore, we\nsetα= 0.05andω0tJ=−2×10−4.\nStarting from a uniform ferromagnet as initial condi-\ntion, the boundary spin starts to rotate and due to ex-\nchange the next spin will follow this rotation and ac-\ncordingly drive its neighbor and so on. But since a spin\ncannot immediately follow the dynamics of its neighbor,\nthere is a certain phase difference Dϕbetween the spins,\ni.e., the neighbor to the right is lagging behind. In the\nmicromagnetic approximation this effect is described by\na phase gradient ∇ϕ≈Dϕ/a. The rotation of the spins\nspeeds up, until it reaches the final precession frequency,\ngiven by the driving frequency ω0. At the same time\ntheout-of-planecomponent Sz—thespinaccumulation—\nincreases until it reaches a steady state profile. The time\nscale of this transient phase for reaching a steady state\ncan be quantified: ˙ϕ(t)andSz(t)follow a limited expo-\nnential growth on a characteristic time τt≈5×105tJ\nfor the parameters used here. τtscales positively with\nsystem size Nxand damping α.\nEventually, the numerical time evolution reaches a\nsteady state as shown in fig. 2 a). This steady state\nverifies the analytical solution eqs. (9) and (10) in bulk\nwith a deviation only at the left boundary as anticipated\nand described above. Note that the finite spin accumu-\nlationSzas a consequence of this type of dynamics has\nimportantfeatures: itisalong-rangespintransportsince\nit decays non-exponential and it would allow to measure\nspin transport by means of the inverse spin-Hall effect.\nFurthermore, it could also be addressed, for instance, by\nmagneto-optical measurements—if sensitive to the out-\nof-plane magnetization for a geometry as studied here.\nFor a further investigation, we vary the frequency ω0\nand find two different regimes, one for sufficiently smallω0, where the system is able to follow the excitation\nwithout disturbance, and one for large ω0where the sys-\ntems response deviates from the theoretical expectation.\nThesetworegimes, whichwewillcalllinearandnonlinear\nregime in the following, are sharply separated by a crit-\nical frequency ωcrit. The existence of these two regimes\ncan be seen from the data depicted in fig. 2 b). Here, as\na measure, we consider the spin accumulation of the last\nspinSN\nzat the right end of the system. Below ωcritwe\nfind just the analytical valuesSz(L), see eq. (10), which\nscales linearly with ω0. Atωcritthis behavior breaks\ndown and the spin accumulation SN\nzdecreases with in-\ncreasing pumping frequency. This breakdown can be un-\nderstood in terms of the phase gradient ∇ϕwhich scales\nlinearly with the driving frequency ω0, see eq. (9). How-\never, one can expect a maximum phase gradient ∇ϕfor a\nspin-superfluid state given by the Landau criterion [36]:\nif the phase gradient exceeds locally a critical value, it\nis energetically favorable for the spins at this position to\nrotate out of the x-yplane and return to the plane by\nunwinding the spiral. Hence, the winding number Nw\ndecreases by one—an effect which is called a phase slip.\nThe Landau criterion for the stability of a spin superfluid\nwith respect to phase slips reads [36]\n|∇ϕ|</radicalbigg\n−2dz\nJa2. (11)\nNote that this relation is not exact as a uniform Szis\nassumed for its derivation. Nevertheless do we observe\nthese phase slips numerically. In the linear regime the\nwinding number is constant in the steady state, whereas\nin the nonlinear regime it relaxes by one at a regular\nrate Γpsas shown in fig. 3 at the example of ω0tJ=\n−6.5×10−4.\nTheω0dependence of the phase-slip rate Γpsis de-\npicted in fig. 4. Each phase slip is accompanied by the\nexcitation of a broad spin-wave spectrum on top of the5\n4 4.005 4.01 4.015 4.02\n10748.54949.550\n00.511.510-7\nFigure 3. Winding number Nwand dissipated power Pdiss\nin the steady state of a driven ferromagnet for ω0tJ=\n−6.5×10−4, well in the nonlinear regime. At a rate of Γps\nphase slips relax the winding number of the in-plane spiral.\nFor each such event the dissipated power spikes.\nspin superfluid. These spin waves are visible as oscil-\nlations ofSzaround the spin-superfluid magnitude and,\nhence, there is strictly speaking no steady state any more\nas the phase slips and the spin-wave excitation are def-\ninitely time dependent. In particular, for systems with\nlow Gilbert damping this dynamics leads to determinis-\nticchaos, thoughthe spin-superfluidbackground remains\nvisible. These findings have some severe implications as\nthere is a maximum spin accumulation, which is achieved\nright at the edge between the linear and the nonlinear\nregime. Furthermore, driving the system in the nonlin-\near regime means also to waste energy to the phase slips\nand the excitation of incoherent spin waves.\nFor the parameters here the critical frequency takes\nthe valueωcrittJ≈−5.15×10−4, which is determined\nfrom fig. 2 b). We also tested different parameters, vary-\ningαandL(data not shown), and find that ωcritscales\nnegatively with αandL. Our numerical result can be\ncompared to the analytical prediction above, eq. (11).\nFrom eq. (9) follows that the maximum phase gradient\nis given by∇sϕ(0) =αµSω0L/γJa2, which, inserted into\neq. (11), implies\n|ωcrit|=γJa\nαµSL/radicalbigg\n−2dz\nJ. (12)\nFor our parameter set this takes value 4×10−41/tJ,\nwhich is slightly lower compared to the numerical value\nabove. This discrepancy is probably due to the fact that\neq. (11) ignores the spatial dependence of Sz. Further-\nmore, a test for very low damping α= 10−4showed that\neq. (11) is even more inaccurate in that case.\nAnother important quantity is the dissipated power\ngiven by eq. (3), which takes negative sign as it lowers\nthe total energy. Figure 4 depicts its dependence on ω0.\nBelowthecriticalfrequency, inthesteadystate, itistime\nindependent as the dynamics is completely stationary.\nIn this regime it scales quadratically with the excitation\nfrequency,Pdiss∝ω2\n0, a result which has already been\n10-410-310-210-1010-910-810-7\n10-610-410-2Figure 4. Time-averaged dissipated power /angbracketleftPdiss/angbracketrightand phase-\nslip rate Γpsin the steady state versus driving frequency\nω0comparing ferro- (FM) and antiferromagnets (AFM). The\nperpendicular dash-dotted lines mark ωcritfor the FM and\nthe AFM, where the latter takes on the higher value. For\nω0< ω critthe dissipated power scales as /angbracketleftPdiss/angbracketright∝ω2\n0and is\nidentical for FMs and AFMs. Above ωcritthe increase slows\ndown and the curve flattens for very high ω0. In this regime,\n/angbracketleftPdiss/angbracketrightis higher for AFMs as compared to FMs. For the\nphase-slip rate we find Γps= 0for|ω0|<|ωcrit|and similar,\nincreasing values for the FM and the AFM above ωcrit. The\ndeviation between FM and AFM near ωcritis due to the dif-\nferent critical frequency, i.e. the data almost coincide when\nplotted versus ω0−ωcrit.\nreported before [37]. This behavior changes above ωcrit.\nThetimeevolutionofthedissipationinthisregimeshows\nthat the phase slips notably contribute to the dissipated\npower, i.e. for each phase slip Pdissspikes as shown in\nfig. 3. Because of this time-dependence of Pdiss, we have\nto consider an average over time for the evaluation of the\ndissipated power. Still, the dissipated power increases\nfurtherwith ω0butlessthanlinearandthecurvenotably\nflattens.\nSo far, our results were obtained from zero-\ntemperature simulations. In the following we address\nthe robustness of spin-superfluid transport at finite tem-\nperature. For this, we consider a finite cross section\nNy×Nz>1andNx= 2000 and vary the tempera-\nture. An average over Navrealizations of thermal noise\nis carried out and, furthermore, data are averaged over\nthecrosssectioninordertoreducethenoise. Thespecific\nchoice of parameters in provided is table I.\nFigure 5 presents the numerical results for the exam-\nple ofkBT/J = 10−2forSzandϕ. The spin-superfluid\ntransport remains in tact but, in particular, the spin ac-\ncumulation Szshows strong thermal fluctuations despite\nthe averages taken over the cross section and the Navre-\nalizations. However, on average the spin accumulation\nclearly deviates from its equilibrium value, which is zero.\nTo quantify the influence of the temperature we calculate\nthe spatial average over the xdirection/angbracketleftSz/angbracketrightxand com-\nparethistothezero-temperaturevalue, givenbyeq.(10).\nThe results are included in Table I. Furthermore, the in-\nplane angle/angbracketleftϕ/angbracketrightNavshows only little fluctuations and its6\n00.020.04\n0 500 1000 1500 200001020\nFigure 5. Spin superfluidity in a ferromagnet at finite tem-\nperaturekBT/J = 10−2and forω0tJ=−2×10−4: shown is\nthe spin accumulation Szand the in-plane angle ϕ. Blue lines\nrepresent the numerical data, black dash-dotted lines the an-\nalytical solution at zero-temperature. The spin accumulation\nis subjected to strong thermal fluctuations but still has a fi-\nniteaveragevalue /angbracketleftSz/angbracketrightx=/summationtext\nnSn\nz/Nx, depictedasreddashed\nline. Its value is only slightly lower than the zero-temperature\nvalue. Thermal fluctuations are much less pronounced for the\nin-plane angle.\nTable I. Averaged spin accumulation of a ferromagnet driven\nwithω0tJ=−2×10−4for different temperatures. The cor-\nresponding zero-temperature value is /angbracketleftSz/angbracketright= 0.01, from which\nno significant deviation is observed.\nkBT/JNx×NyNav/angbracketleftSz/angbracketrightx\n10−44×4 38 0.010\n10−24×4 15 0.009\n0.05 8×8 5 0.010\n0.10 8×8 4 0.011\n0.20 14×145 0.012\nspatial profile shows hardly any deviation from the zero-\ntemperature behavior, given by eq. (9). Overall, we find\nno significant difference to the zero temperature case.\nWe also checked whether phase slips due to thermal ac-\ntivation can be observed, but from the available data\nwe could not observe a single one with the conclusion\nthat ΓpstJ<4×10−5. Hence, spin superfluidity is very\nrobust against thermal fluctuations, even though these\nfluctuations are a problem in our simulations in terms of\nthe signal-to-noise ratio.\nIV. EASY-PLANE ANTIFERROMAGNETS\nFor antiferromagnets, the magnetic unit cells comprise\ntwoatoms—denotedAandBinthefollowing—thatform\ntwo sublattices. We write all properties using this label-\ning so thatASlandBSlare spins of the corresponding\nsublattices. In the ground state both sublattices have\nopposite orientation,ASl=−BSl. The field equations,\neqs. (5) and (6), do not hold as these require a small\nin-plane angle difference between two neighboring spins\nDϕ, which is obviously not true in this case. However,it is reasonable to define phase differences and gradients\nwithin each sublattice, i.e.ADϕas phase difference be-\ntween a spin of sublattice A and its next-nearest neigh-\nbor, which is the nearest neighbor within sublattice A.\nAccordingly,BDϕdefines the phase difference of sublat-\nticeB. Assuming sufficiently weak excitation, spatial\nvariationswithineachsublatticearesmallsuchthatami-\ncromagnetic approximation inside the sublattices reads\n∇A,Bϕ≈A,BDϕ/2a. Interestingly, numerical results re-\nveal that the antiferromagnetic system in bulk fulfills\nfield equation (8), applied separately to each sublattice.\nThe other eq. (7) is not valid, as has been reported before\n[20] for a phenomenological model for antiferromagnets.\nConsequently, the antiferromagnet is expected to exhibit\nthe same in-plane angleA,Bϕ(up to phase difference of π\nbetweensublattices)asaferromagnetwithcorresponding\nparameters, but not the same spin accumulationA,BSz.\nBefore we discuss the numerical results in detail, let\nus first introduce two differences to the ferromagnet that\nare essential for understanding the following results: the\nroleofexchangeand(interlinkedwiththis)thetransverse\ngeometry. Just as in a ferromagnet, a spin-superfluid dy-\nnamics imposes a finite spin accumulationA,BSzwhich,\nremarkably, carries the same sign for both sublattices\nleading to a small out-of-plane magnetization. But this is\nof course antagonized by the antiferromagnetic exchange\nthat favors antiparallel orientation of all components be-\ntween sublattices. Consequently, the exchange interac-\ntions must lower the spin accumulation Sztremendously\nas compared to the ferromagnet (compare fig. 6 a) and\nfig. 2 a)). This also implies that the behavior of Szis\ndetermined by the number of nearest neighbors Nnbof\na spin as more neighbors imply stronger exchange cou-\npling. Consequently, a 1D spin chain is less prone to this\nexchange reduction than a 3D system. We checked this\nnumerically by comparing 1D, 2D and 3D models and,\nindeed, the spin accumulation of the spin superfluid Sz\nscales linearly with Nnb.\nThere is another implication: at a boundary the num-\nberofneighborsislocallyreduced—andthereforetheim-\nportance of the exchange—, resulting in deviations of the\nsublattice componentsA,BSz, see fig. 6 a) for a 1D setup\n(the effect is less pronounced in 3D). This 1D setup owns\nonly boundaries along the xdirection and the question\nwhether for finite cross section Ny×Nz>1these devi-\nations aty- andzboundaries significantly influence the\nbulk behavior has also been tested numerically. Fortu-\nnately, deviations at transversal boundaries quickly fall\noff with distance to the boundary over a few lattice con-\nstants. The bulk then behaves qualitatively and quan-\ntitatively just as a 1D system, except for the reduced\nspin accumulation due to the number of neighbors as\ndiscussed above. The study of 1D systems is preferable\nto keep computational costs feasible.\nWe turn now to the presentation of the numerical data\nfor a 1D system. The model parameters are the same as\ngiven above for the ferromagnet, except for the exchange\nconstant which is now negative. Similarly to the ferro-7\n050100024610-5\n10002000300040004900 5000\n0 0.5 1 1.5 2 2.5\n10-300.511.510-4\n5.566.5\n10-41.31.41.510-4\nFigure 6. Spin superfluidity in antiferromagnetic spin chains: a)the spin accumulation in the stead state resolved for the two\nsublattices A and B. In the bulk both take the same value, leading to a finite total spin accumulation, which is two orders of\nmagnitude lower as compared to the ferromagnet. At the boundaries the profiles show deviations from bulk behavior because\nof the broken exchange right at the boundary. b) the spin accumulation at the right end of the system as function of driving\nfrequencyω0; as for the ferromagnet there are two regimes separated by a critical frequency ωcrit.\nmagnet, the system reaches a steady state after a tran-\nsientphasecharacterizedbyalimitedexponentialgrowth\non a time scale τt, which is roughly the same as for the\nferromagnet. In the steady state the sublattice-resolved\nin-plane anglesA,Bϕboth follow exactly the same profile\nas the ferromagnet, i.e. eq. (9), but with a phase differ-\nence ofπbetween the two sublattices because of the an-\ntiferromagnetic order (data for the antiferromagnet not\nshown).\nThe spin accumulation deviates from the behavior of\na ferromagnet as depicted in fig. 6 a). The bulk profiles\n(away from boundaries at x= 0andx=Nxa) are iden-\ntical in the two sublattices,ASz=BSz. Hence, a measur-\nable spin accumulation is present, but it is two orders\nof magnitude lower than in comparable ferromagnetic\ncases. This is the aforementioned exchange reduction.\nIf we consider the spin accumulation Szin bulk, in the\ndata in fig. 6 a) hardly a space dependence is observed in\ncontrast to the ferromagnet, where Sl\nzhas a finite slope.\nThe antiferromagnet exhibits this in the same way, but\nit is also much smaller and the profile becomes roughly\nconstant. Contrary to the ferromagnet, there are distur-\nbances at the boundaries in the profile of Szwhich we\nalready discussed before.\nDriving the antiferromagnet with the time-dependent\nboundary condition eq. (4) at frequency ω0leads to the\nvery same two different regimes as for ferromagnets, a\nlinear regime up to a critical frequency ωcritand above—\nin the nonlinear regime—phase slips occur. These phase\nslips reduce the winding number, lead to the excitation\nof spin waves, and a further increase of the spin accu-\nmulation is not possible. We quantify this behavior in a\nsimilar way as for the ferromagnet. It is, however, not\npossible to use the spin accumulation of the last spin\nSN\nzas a measure because of the deviating profile at the\nboundary. Instead, we take the spin accumulation at the\nend of the bulk in form of a spatial average over the spins\nin the range xl/a∈[4900,4920],Send\nz:=/angbracketleftbig\nSl\nz/angbracketrightbig\n[4900,4920].This range is chosen such that it is sufficiently separated\nfrom the boundary. The data for the ω0dependence of\nthespinaccumulationareshowninfig.6, panelb): These\nshow that critical frequencies takes roughly same values\nfor ferro- and antiferromagnets, a result which has been\ntested and confirmed for another parameter set with dif-\nferentNx,α, anddz. For the data set shown here the\nvalue isωcrittJ≈−5.75×10−4. However, the decrease\nof the spin accumulation Send\nzwith increasing driving fre-\nquencyω0in the nonlinear regime is less pronounced for\nantiferromagnets. We also calculated the ω0dependence\nof the time-averaged dissipated power /angbracketleftPdiss/angbracketrightand of the\nphase-slip rate Γps, both shown in fig. 4. Similar to other\nfeatures these properties behave for the antiferromagnets\nvery much as for ferromagnets: below ωcritthe dissipated\npower shows exactly the same dependence and above it\nis dominated by phase slips. However, a difference is that\naboveωcritthe dissipated power increases faster with ω0.\nOne reason for this might be the dynamics of spin waves\nthat very much differ between ferro- and antiferromag-\nnets. The phase-slip rate differs slightly, however, this\nseems to be solely due to the fact that ωcritdiffers for\nferro- and antiferromagnets. When Γpsis plotted versus\nω0−ωcrit, both curves match almost.\nThe next step is to consider finite temperature. Again\nthis requires a finite cross section for which we use\nNx×Ny×Nz= 2000×4×4and we test two temper-\natures,kBT/J = 10−2andkBT/J = 10−4. As before,\nthe magnetic response is very similar to that of a ferro-\nmagnet: the in-plane angles follow the zero-temperature\nprofiles, as well as does the average spin accumulation\nfor the lower of the two temperatures. The only major\ndifference is the ratio of the spin-superfluid spin accumu-\nlation to the thermal fluctuations, which is two orders of\nmagnitude smaller as a result of the lower spin-superfluid\nsignal and an equal strength of the fluctuations. For the\nhigher temperature, this even leads to an average Szthat\nis essentially zero. This does not mean that there is no8\nspin-superfluid spin accumulation, but rather that the\navailable numerical data are not sufficient to resolve it\nand more averaging is needed. Note that the in-plane\nangle is not affected by this—it is as robust against the\nfluctuations just as for the ferromagnet.\nV. DISCUSSION AND CONCLUSION\nOur comparative study addresses spin superfluidity in\nferro- and antiferromagnets, where one should bear in\nmind that the former are less promising for spin super-\nfluidity as the latter because of the negative influence of\nthe stray field [22]. Nevertheless, the former can help to\nunderstand the behavior of the latter, which we utilize\nin this work. One of the striking features of spin super-\nfluidity is the transport range that leads to a spin ac-\ncumulation at the end of the system Sz(L)(see eqs. (9)\nand (10)) that does depend on the system length L—\na completely different situation compared to spin-wave\ntransport where the intensity decays exponentially with\nthe distance. However, this non-exponential decay does\nnot imply the possibility of an infinite transport range\nsince with increasing system size the critical frequency\nlowers until no undisturbed spin superfluid is possible\nanymore.\nWe present a full analytical solution for the steady\nstate of the ferromagnet, which slightly deviates from the\nanalytical theory reported before [19, 21]. This theory\nis tested numerically by the full atomistic model, which\nallows to test the robustness of the spin-superfluid trans-\nport against varying boundary conditions, against high\nexcitation frequencies and finite temperature. We show\nthat this kind of transport is remarkably robust: bound-\nary conditions and also elevated temperature hardly\nhamper the magnets spin-superfluid response.\nFurthermore, we identify the critical frequency ωcrit—\na manifestation of the Landau criterion—as the limiting\nfactor for the range of this transport. Above this critical\nfrequency phase slips occur, which also sets a limit to\nthe spin accumulation that can be achieved. In ref. [38]\nanother limitation on the spin current of such a spin su-\nperfluid is discussed, which rests on the fact that |Sz|\nis bounded above. But the estimated values would re-\nquire an out-of-plane component that takes quite large\nvalues|Sz|>0.1, which our simulations reveal to be\nhardly possible even for low damping. This is in particu-lar true for the case for antiferromagnets and, therefore,\nwe conclude that the critical frequency—and therefore\nthe phase slips—is a more relevant limitation on spin su-\nperfluid transport.\nThe direct comparison of antiferromagnets to ferro-\nmagnets shows that both exhibit the very same behavior:\nDriven by an in-plane rotation, both form an in-plane\nspin spiral that exhibits exactly the same behavior, in-\ncluding a spin accumulation in form of an out-of-plane\nmagnetization. Antiferromagnets show in principle the\nsame transport range as ferromagnets with a spin accu-\nmulation at the end of the system independent of the\nsystem length, provided the excitation frequency ω0is\nkept constant ( ω0itself depends on the magnets geome-\ntry in experimental setups, see eq. (B12)). Furthermore,\nthe critical frequency takes very similar value for the two\ntypes of magnets. This general accordance of spin super-\nfluidity for both types of magnets is in contrast to spin-\nwavetransportthatisknowntobedifferentforferro-and\nantiferromagnets[39]. Yetthereisamajordeviation: the\nantiferromagnetic exchange lowers tremendously the spin\naccumulation.\nOurstudyalsocoversanexaminationofthedissipation\nofaspinsuperfluidandoftheeffectoffinitetemperature.\nWe proof the principle robustness of spin superfluidity\nagainst thermal fluctuations, i.e. that quite high temper-\natures are required before thermal phase slips start to\nhamper the transport. 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Sonin, Advances in Physics 59, 181 (2010),\nhttps://doi.org/10.1080/00018731003739943.\n[37] H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev.\nB98, 054424 (2018).\n[38] Y. Tserkovnyak and M. Kläui, Phys. Rev. Lett. 119,\n187705 (2017).\n[39] F. Keffer, H. Kaplan, and Y. Yafet, Amer-\nican Journal of Physics 21, 250 (1953),\nhttps://doi.org/10.1119/1.1933416.\n[40] A. Manchon, J. Železný, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nRev. Mod. Phys. 91, 035004 (2019).\nAppendix A: Analytical theory for a 1D ferromagnet\nThe ferromagnet in the micromagnetic approximation\nunder the assumption of small out-of-plane component,\n|Sz|/lessmuch1, is described by the LLG equation in cylindrical\ncoordinates, eqs. (7) and (8). Assuming translational\ninvariancealong y-andzdirectionleadstoa1Dproblem:\nµS\nγ˙ϕ=Ja2∂2\nxSz+Ja2Sz(∂xϕ)2+ 2dzSz−αµS\nγ˙Sz\n(A1)\nµS\nγ˙Sz=−Ja2∂2\nxϕ+αµS\nγ˙ϕ. (A2)\nSteady state means ˙ϕ=ω0and ˙Sz= 0. This allows to\nintegrate the latter equation,\nsϕ(x,t) =α\n2µSω0\nγJ/parenleftbiggx−L\na/parenrightbigg2\n+ω0t+ϕ0,(A3)\nwhere the first integration constant follows from the Neu-\nmann boundary condition at the right end, ∂xϕ(L) = 0\n(no outflow of spin current), and the second one satisfies\nthe condition ˙ϕ=ω0and allows for an arbitrary phase\nϕ0. This is inserted in the first equation, which then\nreads\n−Ja2∂2\nxSz=−µSω0\nγ+µ2\nSω2\n0\nγ2J/parenleftbigg\nαx−L\na/parenrightbigg2\nSz+ 2dzSz.\n(A4)\nWe argue that the second-derivative term can be ne-\nglected−Ja2∂2\nxSz≈0. This is justified in a twofold\nmanner: first we compared the relevance of all terms\nin that equation numerically by calculating those three\nterms from simulations of the full atomistic spin model,\neq. (2). Indeed the result is that in steady state the\nsecond-derivative term is several orders of magnitude\nsmaller compared to the other two. The second reason\nfollows a-posteriori from the calculated solution and is10\nspin injector (using SHE)\nspins not subjected\nto SHEspins driven\nby SHEspins not subjected\nto SHE\nFigure 7. 1D setup for calculation of the excitation frequency\nω0of a magnet driven by a spin injector utilizing the spin-Hall\neffecttoexertexternaltorquesonthespins. Thesetorquesare\napplied in the region [l1,l2]and vanish outside. Furthermore,\nthe Gilbert damping in [l1,l2]is enhanced by αd. The ground\nstateSis in-plane, the spin accumulation µperpendicular.\nexplainedbelow. From −Ja2∂2\nxSz≈0followsthesteady-\nstate solution for Sz:\nsSz=µSω0\n2γdz\n1 +µ2\nSω2\n0\n2γ2Jdz/parenleftbig\nαx−L\na/parenrightbig2. (A5)\nThis solution does not fulfill eq. (A4), however, we can\ninsert it and calculate the deviation by calculating\n∂2\nxsSz=−2µSω0\nγJα2\na2sS2\nz+ 4/parenleftbiggµSω0\nγJ/parenrightbigg2α4(x−L)2\na4sS3\nz\n=O/parenleftbig\nS2\nz/parenrightbig\n.\nThis allows the conclusion that the correction by taking\nthe second derivative into account is of higher order in\nSzand neglecting this is consistent with the original as-\nsumption|Sz|/lessmuch1. Hence, eqs. (A3) and (A5) form the\nanalytical solution for a 1D setup.\nAppendix B: Frequency of a spin superfluid\nThe usual excitation of a spin current in a magnet\nrests on a spin accumulation µat an interface between\nthe magnet and a heavy metal, which is created by an\nelectrical current. Normally for that the spin-Hall effect\nis utilized. The aim of this appendix is to calculate the\nresulting excitation frequency ω0of a spin superfluid.\nWe assume here that the spin accumulation is per-\npendicular to the magnets ground state, i.e. µ∝ez.\nConsequently, there is an additional damping-like torque\n[22, 40] in the LLG equation (here written as viscousdamping):\n˙Sl=−γ\nµSSl×Hl+αlSl×˙Sl+α/prime\nlSl×/parenleftbigg\nSl×µl\n~/parenrightbigg\n.\n(B1)\nA subsetVdof the total volume of the magnet is driven,\ni.e. subjected to the additional torques and the driving\nalso creates an enhanced damping α/prime\nlwithinVd:\nµl=/braceleftbigg\nµdezforrl∈Vd\n0else(B2)\nαl=α0+α/prime\nlwithα/prime\nl=/braceleftbigg\nαdforrl∈Vd\n0else.(B3)\nα0is the intrinsic Gilbert damping of the magnet.\nTo proceed we consider the LLG equation in the fol-\nlowing form, resolved for the time derivative:\n˙Sl=−γ\nµS(1 +α2\nl)Sl×/parenleftbig\nHl+αlSl×Hl/parenrightbig\n+Tl\n1Sl×Al+Tl\n2Sl×/parenleftbig\nSl×Al/parenrightbig\n.(B4)\nTl\n1andTl\n2parameterize arbitrary additional torques with\nrespecttoanaxis Alandforthespecificchoice Al=µl/~,\nTl\n1=αlα/prime\nl/(1+α2\nl)andTl\n2=−α/prime\nl/(1+α2\nl)eq. (B4) is equiva-\nlent to eq. (B1). However, for the sake of generality we\nconsider for the calculation eq. (B4). Assuming Al∝ez\nand using cylindrical coordinates and again the micro-\nmagnetic approximation, this form of the LLG reads\nµS\nγ˙ϕ=Ja2Sz|∇ϕ|2+ 2dzSz−αµS\nγ˙Sz\n−µS\nγAz(T1+αT2) (B5)\nµS\nγ˙Sz=−Ja2∆ϕ+αµS\nγ˙ϕ+µS\nγAz(αT1−T2),(B6)\nan extension of eqs. (7) and (8). In the same spirit as\nin appendix A we can solve these equations in one di-\nmension in steady-state (assuming ˙Sz= 0and ˙ϕ=ω0),\nwhere the geometry depicted in fig. 7 is assumed. We ap-\nply the external spin accumulation in the interval [l1,l2],\nwhereas the total magnet expands over [0,L]. Therefore,\nT1,2(x) =/braceleftbigg\nTd\n1,2forx∈[l1,l2]\n0else\nA(x) =/braceleftbigg\nAd\nzezforx∈[l1,l2]\n0else.11\nIn the 1D setup eq. (B6) reads\n∂2\nxϕ=α(x)µS\nγJa2ω0+µS\nγJa2Az(x) [α(x)T1(x)−T2(x)]\n=\n\n=:¯ω0/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nα0µS\nγJa2ω0 forx∈[0,l1]\n(α0+αd)µS\nγJa2ω0\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:¯ω/prime\n0+µS\nγJa2Ad\nz/bracketleftbig\n(α0+αd)Td\n1−Td\n2/bracketrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:tforx∈[l1,l2]\nα0µS\nγJa2ω0 forx∈[l2,L], (B7)\nwhich can be integrated. There are six boundary conditions to consider, each one at the left and right end of the\nmagnet, where we assume a Neumann condition ∂xϕ(0) =∂xϕ(L) = 0, i.e. no outflow of spin currents. Furthermore, ϕ\nand∂xϕmust be continuous at l1andl2, delivering four internal boundary conditions. But there is another condition,\na gauge condition for ϕ, which allows to add an arbitrary constant phase to ϕ(x)without altering the physics. (In\npractice this gauge phase depends on the prehistory of the magnet, i.e. on how it had reached its steady state, and also\nwhich exact instant in time is considered.) As gauge we use ϕ(0) = 0. Altogether there are 6 integration constants\nand the unknown frequency ω0in combination with 6 boundary conditions and a gauge, such that the problem has a\nunique solution.\nAs result we obtain\nϕ=\n\n1\n2¯ω0x2forx∈[0,l1]\n1\n2(¯ω/prime\n0+t)x2+ (¯ω0−¯ω/prime\n0−t)l1x+1\n2(¯ω/prime\n0−¯ω0+t)l2\n1forx∈[l1,l2]\n1\n2¯ω0x2+ (¯ω/prime\n0−¯ω0+t)/bracketleftbig\n(l2−l1)x+1\n2(l2\n1−l2\n2)/bracketrightbig\nforx∈[l2,L](B8)\nSz=µS\nγω0+Az(x) [T1(x) +α(x)T2(x)]\nJa2(∂xϕ)2+ 2dz(B9)\nand, importantly, we also gain\nω0=−Ad\nz/bracketleftbig\n(α0+αd)Td\n1−Td\n2/bracketrightbig\n(l2−l1)\nα0L+αd(l2−l1). (B10)\nThis holds true for arbitrary torques taking form\neq. (B4). If the specific case of the spin injector utiliz-\ning the spin-Hall effect is considered, then inserting the\nparameters T1,T2andAreads\nω0=−µd\n~αd\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:τ·l2−l1\nα0L+αd(l2−l1).(B11)\nThe former factor τis the strength of the spin-Hall effect\non the magnet [40]:\nτ=γ\nMs~\n2eηϑSHjel1\nd,\nwith spin transparency of the interface η, spin-Hall an-\ngleϑSH, saturation magnetization Msand thickness d\nof the magnet. jelis the electric current density. The\nlatter factor in eq. (B11) is a geometric factor that is ba-\nsically the ratio between the driven volume l2−l1and the\ntotal volume L, weighted with the total damping of the\nmagnet, where the Gilbert-damping enhancement can beexpressed as [22]\nαd=g⊥~2\n2e2γ\nMsd,\nwith transverse spin mixing conductance g⊥of the in-\nterface. This rigorous derivation holds only true for 1D\nferromagnets, however, the natural extension to 2D and\n3D is given by\nω0=−τ·Vd\nα0V+αdVd, (B12)\nwhereVis the magnets total and Vdthe driven volume.\nThe validity of this expression has been checked numeri-\ncally for 1D and 2D systems using various geometries by\ninvestigating the full atomistic LLG eq. (B4). As a result\nwe obtain very good agreement with the analytical calcu-\nlation except for two cases. First, when the assumption\n|Sz|/lessmuch 1is violated and second if the setup is not ef-\nfectively one dimensional, i.e. if the system is not driven\nover the entire transverse width. However, such a mis-\nmatch in usually small for realistic experimental setups.\nWe furthermore did not only simulate ferromagnets, but12\nalso antiferromagnets with same parameters except for\nthe sign of J. These simulations result in exactly the\nsame frequencies ω0as the corresponding ferromagnetsand thus eqs. (B10) to (B12) are also valid for antiferro-\nmagnets, even though note that the resulting spin accu-\nmulation deviates."    },    {        "title": "2103.07008v1.Magnetoelastic_Gilbert_damping_in_magnetostrictive_Fe___0_7__Ga___0_3___thin_films.pdf",        "content": "Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films\nW. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1,∗\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA\n2Department of Materials Science and Engineering,\nUniversity of Maryland, College Park, Maryland 20742, USA\n3Department of Physics, Pukyong National University, Busan 48513, South Korea\nWe report an enhanced magnetoelastic contribution to the Gilbert damping in highly magne-\ntostrictive Fe 0.7Ga0.3thin films. This effect is mitigated for perpendicular-to-plane fields, leading\nto a large anisotropy of the Gilbert damping in all of the films (up to a factor of 10 at room tem-\nperature). These claims are supported by broadband measurements of the ferromagnetic resonance\nlinewidths over a range of temperatures (5 to 400 K), which serve to elucidate the effect of both the\nmagnetostriction andphonon relaxation on the magnetoelastic Gilbert damping.\nAmong the primary considerations in the design of\nspintronics devices is Gilbert damping. However, a full\nunderstanding of the mechanisms which cause damping\nof magnetization dynamics in ferromagnets remains elu-\nsive. Reports of anisotropy in the Gilbert damping have\nproven to be useful tools in the understanding of the un-\nderlying mechanisms involved [1–3], but there is much\nthat is yet unclear. Studies of the temperature depen-\ndence also promise to be a uniquely powerful tool for a\ncomplete physical understanding [4, 5], however, there\nare few such reports in existence.\nRecently, it has been shown that spins can be co-\nherently coupled over large distances ( ∼1 mm) using\nmagnon-phonon coupling [6–8]. It is also well known\nthat magnetization dynamics can be excited elastically\nthrough this phenomenon [9], but its effect on Gilbert\ndamping has been largely confined to theoretical calcu-\nlations [10–13] and lacks clear experimental validation.\nFurthermore, most studies have focused on yttrium iron\ngarnet (YIG), which is weakly magnetostrictive.\nIn this Letter, we observe a large and anisotropic mag-\nnetoelastic contribution to the Gilbert damping in highly\nmagnetostrictive Fe 0.7Ga0.3films through broadband\nmeasurements of the ferromagnetic resonance (FMR)\nlinewidths over a wide range of temperatures. The\nperpendicular-to-plane linewidths exhibit a relatively low\nminimum in the Gilbert damping of approximately 0.004,\nsimilar to that of bcc Fe [14]. At room temperature, the\nGilbert damping is as large as a factor of 10 greater with\nfield applied in plane relative to out of plane. In fact, for\nany given sample and temperature, the anisotropy is, at\nminimum, about a factor of 2. We argue this is due to\na mitigation of the magnetoelastic contribution for per-\npendicular magnetization, arising from finite-thickness\nboundary conditions and weak elastic coupling to the\nsubstrate. The nonmonotonic temperature dependence\nof the Gilbert damping also shows the competing effects\nof the magnetostriction, which increases at low tempera-\nture, and the phonon viscosity, which generally decreases\nat low temperature.\nThe Fe 0.7Ga0.3films studied in this letter were de-\nposited on SiO 2/Si wafers at room temperature by dcmagnetron sputtering of an Fe 0.7Ga0.3target. The base\npressure of the deposition chamber was 5 ×10−8torr,\nand the working pressure was kept at 5 ×10−3torr\nwith Ar gas. The composition of the Fe 0.7Ga0.3films\nwas quantitatively analyzed by energy dispersive spec-\ntroscopy (EDS). Films were grown with thicknesses of\n21 nm, 33 nm, 57 nm, and 70 nm (the 21 nm, 57 nm,\nand 70 nm belong to the same growth series). An addi-\ntional 33 nm film was grown at 200◦C. The 33 nm room\ntemperature deposition was etched using an ion mill to\nobtain films with thicknesses of 17 nm and 26 nm. The\nthicknesses of the films were measured using x-ray reflec-\ntometry (see Supplemental Material).\nThe FMR linewidths were measured using a setup in-\nvolving a coplanar waveguide and modulation of the ap-\nplied magnetic field for lock-in detection as described in\nRef. [15]. Measurements were done with the field applied\nin the plane (IP) and perpendicular to the plane (PP) of\nthe film. The sample temperature was varied from 5 K\nto 400 K for both IP and PP configurations [16] with\nmicrowave excitation frequencies up to 52 GHz. The res-\nonance fields and linewidths were isotropic in the plane,\nand the absence of in-plane magnetic anisotropy was ver-\nified with vibrating sample magnetometry (see Supple-\nmental Material). This is also consistent with the abun-\ndance of grain boundaries observed with atomic force mi-\ncroscopy (AFM). In analyzing the FMR linewidths, we\nconsider three contributions: Gilbert damping 4 παf/γ\n(αis the Gilbert damping coefficient, fis the microwave\nfrequency, and γis the gyromagnetic ratio), inhomo-\ngeneous broadening ∆ H0, and two-magnon scattering\n∆HTMS (for IP fields). Eddy current damping and ra-\ndiative damping contributions [17] are neglected because\nwe expect them to be small ( <10−4) for these films.\nLinewidths of the 70 nm film at 300 K for both con-\nfigurations of the applied field are shown in Fig. 1(a),\nand the IP linewidths with individual contributions to\nthe linewidth plotted separately in Fig. 1(b). We fit the\nIP linewidths using a model of two-magnon scattering\nbased on granular defects [15, 18, 19]. The fit for the\n70 nm film is shown in Fig. 1(b), along with the two-\nmagnon contribution alone given by the magenta curve.arXiv:2103.07008v1  [cond-mat.mtrl-sci]  11 Mar 20212\n01 02030405005001000150020000500100015002000/s61508\nH0/s61508HTMS/s61508H (Oe)F\nrequency (GHz)H in-plane/s61508\nHGilbertT = 300 K/s61537\n = 0.0390 /s61617 0.0005/s61560\n = 17 nm(a)(\nb)/s61508 H (Oe)T  = 300 K/s61537\nIP = 0.0390 /s61617 0.0005/s61537\nPP = 0.0035 /s61617 0.0001/s61508HIP/s61508\nHPP70 nm film\nFIG. 1. (a) FMR linewidths for IP (black squares) and\nPP (red circles) configurations for the 70 nm film. The IP\nlinewidths are fit to a model of two-magnon scattering and\nthe PP linewidths are fit using the standard Gilbert damping\nmodel. (b) Total linewidth (solid black), Gilbert linewidth\n(dotted blue), two-magnon scattering linewidth (dashed ma-\ngenta), and inhomogeneous broadening (dashed/dotted red)\nfor the 70 nm film with IP field.\nThe fit parameters are the Gilbert damping α(indicated\non the figure) and the RMS inhomogeneity field H/prime. The\ndefect correlation length ξis fixed to 17 nm based on the\nstructural coherence length obtained with x-ray diffrac-\ntion (XRD), which agrees well with the average grain di-\nameter observed with AFM (see Supplemental Material).\nFurthermore, the high-frequency slope of the linewidths\napproaches that of the Gilbert damping since the two-\nmagnon linewidth becomes constant at high frequencies\n[see Fig. 1(b)].\nWe now compare the IP and PP linewidths of the\n70 nm film shown in Fig. 1(a). The two-magnon scat-\ntering mechanism is inactive with the magnetization per-\npendicular to the plane [20], and so the PP linewidths are\nfit linearly to extract the Gilbert damping. We obtain a\nvalue of 0.0035±0.0001 for PP fields and 0 .039±0.0005\nfor IP fields, corresponding to an anisotropy larger than\na factor of 10. Li et al. [3] recently reported a large\nanisotropy (∼factor of 4) in epitaxial Co 50Fe50thin films.\nFirst we discuss the dependence of the PP Gilbert\ndampingαPPon temperature for all of the films, shown\nin Fig. 2. We observe a significant temperature depen-\n01 002 003 004 00024685\n7 nm 17 nm2\n6 nm3\n3 nm (RT dep)/s61537 (×10-3)T\nemperature (K)21 nm 3\n3 nm (200 °C dep)H\n perpendicular-to-plane70 nmFIG. 2. Gilbert damping αfor PP field shown as a func-\ntion of temperature for the 17 nm (orange), 21 nm (blue),\n26 nm (green), 33 nm room temperature deposition (ma-\ngenta), 33 nm 200◦C deposition (gold), 57 nm (red), and\n70 nm (black) Fe 0.7Ga0.3films.\ndence in all cases (with the exception of the 33 nm room\ntemperature deposition), characterized by a maximum\nat around 50 K. Then, at the lowest temperatures (5 to\n10 K),αPPapproaches the same value for all of the films\n(/similarequal0.004).\nNow we turn to the temperature dependence of the\nIP Gilbert damping αIPshown in Fig. 3. The values\nobtained here were obtained by fitting the linewidths lin-\nearly, but excluding the low-frequency points ( <∼20 GHz)\nsince the two-magnon scattering becomes constant at\nhigh frequencies [21]. Here we note, upon comparison\nwith Fig. 2, that a large anisotropy of the Gilbert damp-\ning exists for all of the samples. In the 70 nm film, for\ninstance,αIPis more than a factor of 10 larger than αPP\nat 300 K. In the temperature dependence of αIP, we ob-\nserve behavior which is similar to that seen in αPP(Fig.\n2), namely, a maximum at around 50 K (with the excep-\ntion of the 21 nm film). Here, however, αIPdoes not\napproach a common value at the lowest temperatures in\nall of the samples as it does in the PP case.\nThe IP Gilbert damping is larger than the PP Gilbert\ndamping for all of the samples over the entire range of\ntemperatures measured. This anisotropy of the Gilbert\ndamping—along with the nonmonotonic temperature\ndependence—in all seven samples implies a contribution\nto the Gilbert damping in addition to Kambersk´ y damp-\ning. We have verified that the orientation of FeGa(110)3\nplanes is completely random with XRD for the 33 nm\n(both depositions) and 70 nm films (see Supplemen-\ntal Material), and it is therefore not possible that the\nanisotropy is due to Kambersk´ y damping. Interface\nanisotropy has reportedly led to anisotropic Kambersk´ y\ndamping in ultrathin ( ∼1 nm) films of Fe [2], but this\nis highly unlikely in our case due to the relatively large\nthicknesses of the films. In addition, the fact that the\ndamping anisotropy shows no clear correlation with film\nthickness furthers the case that intrinsic effects, which\ntend to show a larger anisotropy in thinner films [2],\ncannot be the cause. The longitudinal resistivity ρxxof\nthe 33 nm (both depositions) and 70 nm films (see Sup-\nplemental Material) shows very weak temperature de-\npendence. In the Kambersk´ y model, the temperature\ndependence of the damping is primarily determined by\nthe electron momentum relaxation time τ, and we would\ntherefore not expect the Kambersk´ y damping to show\na significant temperature dependence for samples where\nthe residual resistivity ratio is approximately unity. It is\nplausible that the Kambersk´ y damping would still show\na temperature dependence in situations where the spin\npolarization is a strong function of temperature, due to\nchanges in the amount of interband spin-flip scattering.\nThis kind of damping, however, would be expected to\ndecrease at low temperature [22, 23]. The temperature\ndependence we observe for both αPPandαIPis therefore\ninconsistent with Kambersk´ y’s model, and the similarity\nbetween the two cases in this regard suggests that the\nenhanced Gilbert damping has a common cause that is\nmitigated in the PP configuration.\nIt has been proposed that magnetoelastic coupling\ncan lead to Gilbertlike magnetization damping through\nphonon relaxation processes [10, 12, 24]. Similar treat-\nments calculate the magnetoelastic energy loss through\ninteraction with the thermal population of phonons\n[11, 25]. The Kambersk´ y mechanism is often assumed to\nbe the dominant Gilbert damping mechanism in metal-\nlic samples, so magnetoelastic Gilbert damping is usually\nstudied in magnetic insulators, particularly yttrium iron\ngarnet (YIG). There is the possibility, however, for the\nmagnetoelastic damping to dominate in metallic samples\nwhere the magnetostriction is large, such as in Fe-Ga al-\nloys. Later we will discuss how magnetoelastic damping\ncan be mitigated in thin films by orienting the magneti-\nzation perpendicular to the plane, and how the degree to\nwhich it is mitigated depends on the boundary conditions\nof the film.\nHere we outline a theory of magnetoelastic damping,\nwhich relies on the damping of magnetoelastic modes\nthrough phonon relaxation mechanisms. Figure 4 illus-\ntrates the flow of energy through such a process. Analyt-\nically, the procedure is to equate the steady-state heating\nrate due to Gilbert damping to the heating rate due to\ncrystal viscosity, and solve for the Gilbert damping α\nin terms of the crystal shear viscosity ηand the mag-\n01 002 003 004 000123453\n3 nm (200 °C dep)17 nm2\n6 nm/s61537 (×10-2)T\nemperature (K)33 nm (RT dep)57 nm2\n1 nmH in-plane7\n0 nmFIG. 3. Gilbert damping αfor IP field shown as a func-\ntion of temperature for the 17 nm (orange), 21 nm (blue),\n26 nm (green), 33 nm room temperature deposition (ma-\ngenta), 33 nm 200◦C deposition (gold), 57 nm (red), and\n70 nm (black) Fe 0.7Ga0.3films.\nnetostrictive coefficients λhkl. Shear strain uijresult-\ning from the magnetoelastic interaction can be expressed\nasuij=λ111mimj[26], where mi≡Mi/Msare the\nreduced magnetizations. The leading-order shears thus\nhave equations of motion given by ˙ uiz=λ111˙mi, where\ni=xory, andzis the direction of the static magnetiza-\ntion so that mz≈1. Longitudinal modes are quadratic\nin the dynamical component of the magnetization [24]\nand so will be neglected in this analysis.\nThe heating rate due to Gilbert damping can be writ-\nten as ˙Qα=Ms\nγα( ˙m2\nx+ ˙m2\ny), and the heating rate due to\nthe damping of phonon modes as ˙Qη= 4η( ˙u2\nxz+ ˙u2\nyz) =\n4ηλ2\n111( ˙m2\nx+ ˙m2\ny) [12], with the factor of 4 accounting\nfor the symmetry of the strain tensor. Equating the two,\nand solving for α(henceforward referred to as αme), we\nobtain\nαme=4γ\nMsηλ2\n111. (1)\nWe will restrict our attention to the case of isotropic mag-\nnetostriction, and set λ111=λ.\nIn order to use Eq. 1 to estimate αmein our films,\nwe first estimate the shear viscosity, given for transverse\nphonons with frequency ωand relaxation time τas [27]\nη=2ρc2\nt\nω2τ, (2)4\n(b)\nu(t)\nphonon pumpingM(t) H0\nM(t)\nu(t)(a)\nkph\ndH0\nFIG. 4. (a) Depiction of magnetoelastic damping process\nfor magnetization in plane and (b) perpendicular to plane,\nwhere M(t) is the magnetization vector and u(t) is the lattice\ndisplacement. In panel (b), the magnon-phonon conversion\nprocess is suppressed when d < π/k ph, wheredis the film\nthickness and kphis the transverse phonon wavenumber at\nthe FMR frequency.\nwhereρis the mass density and ctis the transverse\nspeed of sound. Using ω/2π= 10 GHz, τ= 10−11s,\nandct= 2.5 km/s, we obtain η≈2.3 Pa s. (The\nestimate of the phonon relaxation time is based on a\nphonon mean free path of the order of the grain size:\n∼10 nm.) Furthermore, the magnetostriction of an equiv-\nalent sample has been measured to be ∼100 ppm at room\ntemperature [28]. Then, with γ/2π= 29 GHz/T and\nMs= 1123 emu/cc (extracted from FMR data taken at\n300 K), we estimate αme≈0.016. This estimate gives us\nimmediate cause to suspect that magnetoelastic Gilbert\ndamping is significant (or even dominant) in these films.\nWe now discuss why the magnetoelastic damping can\nbe much weaker for PP magnetization in sufficiently thin\nfilms. We will start by assuming that there is no coupling\nbetween the film and substrate, and later we will relax\nthis assumption. In this case the only phonons excited\nby the magnetization, to leading-order in the magneti-\nzations and strains, are transverse modes propagating in\nthe direction of the static magnetization [24]. One may\nassume that the minimum allowable phonon wavenum-\nber is given by π/d, wheredis the film thickness, since\nthis corresponds to the minimum wavenumber for a sub-\nstrate having much lower acoustic impedance than the\nfilm (requiring the phonons to have antinodes at the in-\nterfaces) [13]. (We also assume an easy-axis magnetic\nanisotropy at the interfaces, so that the dynamical mag-\nnetizations have antinodes at the interfaces.) We expect\nthen that the magnetoelastic damping will be suppressed\nfor cases where the phonon wavelength, at the frequencyof the precessing magnetization, is greater than twice the\nfilm thickness [see Fig. 4(b)]. Thus, in sufficiently thin\nfilms (with weakly-coupled substrates), the magnetoelas-\ntic damping process can be suppressed when the mag-\nnetization is perpendicular to the plane. However, the\nmagnetoelastic damping can be active (albeit mitigated)\nwhen there is nonnegligible or “intermediate” coupling\nto the substrate.\nBefore moving on, we briefly note the implications of\nEq. (1) for the temperature dependence of the Gilbert\ndamping. On the basis of the magnetostriction alone,\nαmewould be expected to increase monotonically as tem-\nperature is decreased ( λhas been shown to increase by\nnearly a factor of 2 from room temperature to 4 K in\nbulk samples with similar compositions [29]). However,\nthe viscosity ηwould be expected to decrease at low tem-\nperature, leading to the possibility of a local maximum\ninαme. In polycrystalline samples where the grain size\nis smaller than the phonon wavelength, viscous damping\nof phonons due to thermal conduction caused by stress\ninhomogeneities can be significant [27, 30]. (In our case\nthe phonon wavelengths are ∼100 nm and the grain\nsizes are∼10 nm.) This effect scales with temperature\nasη∼Tα2\nT/Cχ [30], where αTis the thermal expansion\ncoefficient, Cis the specific heat at constant volume, and\nχis the compressibility. At higher temperatures, αTand\nCwill approach constant values, and χwill always de-\npend weakly on temperature. We therefore expect that\nthe viscosity is approximately linear in T. In this case,\nαmeis maximized where λ2(T) has an inflection point.\nWe proceed to explain our data in terms of the mecha-\nnism described above, turning our attention again to the\nPP Gilbert damping for all of the films shown in Fig.\n2. We previously argued that the magnetoelastic damp-\ning mechanism will be suppressed for the case where the\nacoustic impedances of the film and substrate are mis-\nmatched. However, the clear dependence on tempera-\nture, which we have already shown is inconsistent with\nKambersk´ y damping, appears to be consistent with the\nmagnetoelastic damping mechanism. We estimate that\nthe acoustic impedance of the film (defined as the product\nof mass density ρand transverse speed of sound ct[13])\nis about a factor of 2 larger than the substrate. This sug-\ngests that the elastic coupling between the film and sub-\nstrate, albeit weak, may be nonnegligible. Furthermore,\nexperiments with YIG/GGG heterostructures (where the\nacoustic match is good) have demonstrated magnetic ex-\ncitation of phononic standing waves that have boundary\nconditions dictated by the combined thickness of the film\nand substrate, rather than the film thickness alone (i.e.,\nthe wavelengths are much larger than the film thickness)\n[6, 31]. In this case, the Gilbert damping may contain\nsome contribution from the magnetoelastic mechanism.\nA final point is that αPPapproaches/similarequal0.004 at 5 to\n10 K for all of the films. Both the magnetostriction and\nthe viscosity are quantities which could have significant5\n01 0020030040002468/s61537me (×10-2)T\nemperature (K)Ref. [29]2\n1 nm70 nm5\n7 nm0\n.000.250.500.751.00/s61548\n2(T)//s615482(0)010020030001/s61544 (arb. units)T\n (K)\nFIG. 5. Magnetoelastic Gilbert damping αmefor the 21 nm\n(blue), 57 nm (red), and 70 nm (black) films (left ordinate)\nandλ2(T)/λ2(0) from Clark et al. [29] (magenta; right or-\ndinate) shown as a function of temperature. Inset shows the\nratio ofαmeandλ2(T)/λ2(0), labeled as η(T), along with lin-\near fits for the 21 nm (blue), 57 nm (red), and 70 nm (black)\nfilms.\nvariation between samples, leading to variations in αme.\nHowever, the viscosity becomes small at low temperature,\nwhich means that the Gilbert damping will approach the\nKambersk´ y “limit,” a property that is determined by the\nelectronic structure, implying that the Kambersk´ y damp-\ning is/similarequal0.004 in these films and that it is the primary\ncontribution to the Gilbert damping near T= 0.\nNow we revisit the IP Gilbert damping shown in Fig.\n3. In this configuration, there is a strong temperature\ndependence of the Gilbert damping similar to that of\nthe PP case, again implying the presence of magnetoe-\nlastic damping. However, the overall magnitude is much\nhigher. That is because in this case arbitrarily long wave-\nlength phonons can be excited regardless of the thick-\nness of the film. Although we cannot directly measure\nthe magnetostriction as a function of temperature, we\nestimate the scaling behavior of λby interpolating the\ndata in Ref. [29] taken for bulk samples of similar com-\nposition. In order to demonstrate that αIPscales with\ntemperature as expected from the model, we have plot-\nted the quantities αmeandλ2(T)/λ2(0) as functions of\ntemperature in Fig. 5—where we define the quantity\nαme≡αIP−0.004—for the 21 nm, 57 nm, and 70 nm\nfilms (which are part of the same growth). The corre-\nlation between the two quantities is not completely con-\nvincing. There is, however, an additional temperature\ndependence in αmebesidesλ2(T), namely, the viscosity\nη(T). The inset of Fig. 5 shows the ratio of αmeand\nλ2(T), which [from Eq. (1)] is proportional to η(T). The\nlinear fits provide strong evidence that the mechanism\nbehind the viscosity is indeed the thermal conduction\nprocess that we have argued is approximately linear inT. It is noteworthy that the maximum in αme(∼50 to\n75 K for all of the samples) coincides approximately with\nthe inflection point in λ2(T). This was a consquence of\nour assumption that η(T) should be roughly linear. We\nalso obtain a significant value for the zero-temperature\nviscosity, which is around 25 % of the value at 300 K. This\nis likely due to boundary-scattering processes which will\npreventαmefrom going to zero at low temperatures, par-\nticularly for in-plane magnetization where αmeis much\nlarger than 0.004 (our estimate for the Kambersk´ y damp-\ning). For the PP case, αmeis much smaller due to limita-\ntions on the wavelengths of phonons that can be excited,\nso the Gilbert damping of all the samples approaches the\nKambersk´ y limit of 0.004 near zero temperature. We also\nfound that η(T) was linear for the 33 nm (200◦C depo-\nsition) film, but had a more complicated dependence on\nTfor the 17 nm, 26 nm, and 33 nm (room temperature\ndeposition) films (the latter three being notably of the\nsame growth). The viscosity near zero temperature is\nwithin roughly a factor of 2 for all seven of the samples,\nhowever.\nFinally, we propose that this mechanism may be re-\nsponsible for a Gilbert damping anisotropy of similar\nmagnitude reported in Ref. [3], observed in an epitax-\nial Co 0.5Fe0.5thin film. The authors attributed the\nanisotropy to the Kambersk´ y mechanism [22, 23, 32, 33],\narising from tetragonal distortions of the lattice. The\nmagnetostriction is known to be highly anisotropic in\nbulk Co 0.5Fe0.5,viz.,λ100= 150 ppm and λ111= 30 ppm\n[34]. We therefore expect that the Gilbert damping aris-\ning from the mechanism we have described may be much\nlarger for M/bardbl(110) than M/bardbl(100), which is precisely\nwhat the authors observed.\nIn summary, we observe large and anisotropic magne-\ntoelastic Gilbert damping in Fe 0.7Ga0.3polycrystalline\nthin films (thicknesses ranging from 17 to 70 nm). At\n300 K, the damping coefficient is more than a factor of\n10 larger for field in plane than it is for field perpendicu-\nlar to the plane in the 70 nm film. The large anisotropy\nis caused by a mitigation of the magnetoelastic effect for\nperpendicular-to-plane fields due to a dependence on the\nelastic coupling of the film to the substrate, which in our\ncase is weak. Finally, there is a nonmonotonic tempera-\nture dependence of the Gilbert damping, which we show\nis consistent with our model.\nWe acknowledge Rohit Pant and Dyland Kirsch for as-\nsistance with thin film deposition and characterization.\nThis work was supported by SMART, a center funded\nby nCORE, a Semiconductor Research Corporation pro-\ngram sponsored by NIST. Parts of this work were carried\nout in the Characterization Facility, University of Min-\nnesota, which receives partial support from NSF through\nthe MRSEC program, and the Minnesota Nano Cen-\nter, which is supported by NSF through the National\nNano Coordinated Infrastructure Network, Award Num-\nber NNCI - 1542202.6\n∗Author to whom correspondence should be addressed:\ncrowell@umn.edu\n[1] K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and\nM. F¨ ahnle, Anisotropic damping of the magnetization\ndynamics in Ni, Co, and Fe, Phys. Rev. B 81, 174414\n(2010).\n[2] L. Chen, S. Mankovsky, S. Wimmer, M. A. W. Schoen,\nH. S. 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Brockmeyer, P. E. Wigen, and H. D¨ otsch,\nMagnetoelastic resonances in epitaxial garnet films, Le\nJ. Phys. Colloq. 49, C8 (1988).\n[32] V. Kambersk´ y, On the Landau-Lifshitz relaxation in fer-7\nromagnetic metals, Can. J. Phys. 48, 2906 (1970).\n[33] V. Kambersk´ y, On ferromagnetic resonance damping in\nmetals, Czechoslov. J. Phys. 26, 1366 (1976).\n[34] R. C. Hall, Magnetic Anisotropy and Magnetostrictionof Ordered and Disordered Cobalt-Iron Alloys, J. Appl.\nPhys. 31, S157 (1960).Supplemental Material for\n“Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films”\nW. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA\n2Department of Materials Science and Engineering,\nUniversity of Maryland, College Park, Maryland 20742, USA\n3Department of Physics, Pukyong National University, Busan 48513, South Korea\nCONTENTS\nS1. Magnetization dynamics 1\nS2. Ferromagnetic resonance linewidths of 70 nm film 2\nS3. X-ray reflectivity 2\nS4. X-ray diffraction 3\nS5. Atomic force microscopy 3\nS6. Vibrating sample magnetometry 3\nS7. Longitudinal resistivity 3\nReferences 4\nS1. MAGNETIZATION DYNAMICS\nThe treatment of magnetization dynamics begins with the Landau-Lifshitz-Gilbert equation of motion\ndM\ndt=−γM×Heff+α\nMsM×dM\ndt(S1)\nwhere the relaxation is characterized by the Gilbert damping parameter α. Upon linearizing this equation in the\ndynamic component of the magnetization, one obtains for the ac magnetic susceptibility of the uniform q= 0 mode\nχac(q= 0,ω)∝αω/γ\n(H−HFMR )2+ (αω/γ )2(S2)\nso that the field-swept full-width-at-half-maximum linewidth is given by ∆ HFWHM = 2αω/γ . Therefore, the Gilbert\ndamping parameter αis obtained by measuring ∆ HFWHM as a function of ω.\nRelaxation of the uniform mode can include mechanisms which are not described by Gilbert damping. The most\ncommon of these is inhomogeneous broadening, which results from inhomogeneities in the system and is constant\nas a function of frequency. Another mechanism is two-magnon scattering, which is also extrinsic in nature. Two-\nmagnon scattering originates from the negative group velocity at low qof the backward volume mode magnons for\nin-plane magnetization. The negative group velocity is due to a lowering of the magnetostatic surface charge energy\nfor increasing q. The existence of negative group velocity at low qleads to the appearance of a mode at nonzero q\nthat is degenerate with the uniform mode. Two-magnon scattering refers to the scattering of the uniform mode to\nthe nonuniform degenerate mode.\nMuch work has been done on the treatment of two-magnon scattering [S1–S3], and here we will simply give an\nexpression for the contribution of two-magnon scattering to the field-swept linewidth\n∆HTMS =γ2ξ2H/prime2\ndω/dH/integraldisplay\nd2qΛ0q1\n(1 + (qξ)2)3/21\nπωα\n(ωα)2+ (ω−ωFMR )2(S3)\nwithξthe defect correlation length, H/primethe RMS inhomogeneity field, and Λ 0qthe magnon-magnon coupling. In\ngeneral, this leads to a nonlinear dependence of the linewidth on frequency. Eq. S3 is used to fit the IP linewidths.arXiv:2103.07008v1  [cond-mat.mtrl-sci]  11 Mar 20212\n0102030405005001000150020002500(b) \n5 K \n50 K \n150 K \n225 K \n300 K \n400 K/s61508H (Oe)F\nrequency (GHz)H in-plane\n010203040500100200300400500 5 K \n50 K \n150 K \n225 K \n300 K \n400 K/s61508H (Oe)F\nrequency (GHz)H perpendicular-to-plane(a)\nFIG. S1. FMR linewidths of the 70 nm film with field PP (a) and IP (b) for sample temperatures of 5 K (blue), 50 K (gold),\n150 K (black), 225 K (magenta), 300 K (red), and 400 K (orange). The solid lines are linear fits in both panels. In (b), the\nvertical dashed line indicates the lower bound of the points included in the fit.\n12345610-1100101102103104105106(c)Intensity (arb. units)/s61553\n/2/s61553 (degrees)d = 56.6 nm\n12345610-1100101102103104105106(b)Intensity (arb. units)/s61553\n/2/s61553 (degrees)200 °C depositiond\n = 33.1 nm \n123410-1100101102103104105106(a)R\nT depositiond\n = 33.2 nm Intensity (arb. units)/s61553\n/2/s61553 (degrees)\nFIG. S2. X-ray reflectivity data (black) overlaid with fits (red) for the (a) 33 nm (room temperature deposition), (b) 33 nm\n(200◦C deposition), and (c) 57 nm films. Thicknesses dobtained from the fits are indicated on the figure.\nS2. FERROMAGNETIC RESONANCE LINEWIDTHS OF 70 nm FILM\nThe field-swept FMR linewidths of the 70 nm film are shown in Fig. S1 for field PP and IP. For the case of field\nIP, the data above 23 GHz were fit linearly to obtain the Gilbert damping. (This value varied between different\nsamples since the characteristic roll-off frequency depends on both defect lengthscale and film thickness, but remained\nin the range 20 to 25 GHz.) It is safe to do this provided there are no inhomogeneities at lengthscales smaller than a\nfew nm, which could cause the two-magnon scattering contribution to the linewidth to roll off at higher frequencies.\nWe believe that defects at such small lengthscales are highly unlikely given the characterization performed on these\nsamples.\nS3. X-RAY REFLECTIVITY\nIn Fig. S2 we show x-ray reflectivity measurements at grazing incidence for 33 nm (room temperature and 200◦C\ndepositions) and 57 nm films. The measurements were taken using a Rigaku SmartLab diffractometer. The thicknesses\ndyielded by the fits of the data are indicated on the figure.3\n4243444546036912F\ne0.7Ga0.3(110)(a)I\nntensity (arb. units)/s61553\n/2/s61553 (degrees)33 nm (RT deposition)2\n/s61553c= 44.14 /s61617 0.03 °F\nWHM = 0.78 /s61617 0.09 °\n4243444546036912F\ne0.7Ga0.3(110)(b)I\nntensity (arb. units)/s61553\n/2/s61553 (degrees)33 nm (200 °C  deposition)2\n/s61553c = 44.07 /s61617 0.05 °F\nWHM = 1.18 /s61617 0.17 °\n4243444546036912(c)F\ne0.7Ga0.3(110)Intensity (arb. units)/s61553\n/2/s61553 (degrees)2/s61553c = 44.36 /s61617 0.02 °F\nWHM = 0.50 /s61617 0.07 °70 nm\nFIG. S3. X-ray diffraction symmetric θ/2θscans for (a) 33 nm room temperature deposition, (b) 33 nm 200◦C deposition,\nand (c) 70 nm films. Full width at half maxima (FWHM) and 2 θcenter positions are indicated on the figure.\nS4. X-RAY DIFFRACTION\nX-ray diffraction (XRD) measurements were performed in order to determine both the degree of orientation and\nthe structural coherence length of the films.\nSymmetric θ/2θscans were taken with a Rigaku Smartlab diffractometer using Cu Kα 1(λ= 1.54˚A) radiation.\nThe data for both samples are shown in Fig. S3. The grain size was estimated using the Scherrer formula for spherical\ngrains [S4] as 13 nm, 9 nm, and 17 nm for the 33 nm (room temperature deposition), 33 nm (200◦C deposition), and\n70 nm films respectively.\nTwo-dimensional images were collected with a Bruker D8 Discover diffractometer using Co Kα 1(λ= 1.79˚A)\nradiation. Detector images showing the “ring” corresponding to the Fe 0.7Ga0.3(110) peak in four different samples\nare shown in Fig. S4. The ring indicates that the Fe 0.7Ga0.3(110) planes are randomly oriented over the range of the\ndetector, which we take to be evidence that there is no texture over a macroscopic scale in these samples. Furthermore,\nthe films were grown directly on top of amorphous SiO 2layers, so we do not expect an epitaxial relationship between\nthe film and substrate. The Fe 0.7Ga0.3(110) peaks were the only measurable Bragg peaks since the structure factor\nis highest for this case.\nS5. ATOMIC FORCE MICROSCOPY\nAtomic force microscopy data are shown in Fig. S5 for the 33 nm (room temperature and 200◦C depositions),\n57 nm, and 70 nm films. The field-of-view is 250 nm for the 33 nm films and 500 nm for the 57 nm and 70 nm films.\nThe root-mean-square (RMS) roughness of the sample surfaces is 0.7 nm, 0.4 nm, 1.5 nm, and 1.3 nm for the 33 nm\n(room temperature deposition), 33 nm (200◦C deposition), 57 nm, and 70 nm films, respectively .\nS6. VIBRATING SAMPLE MAGNETOMETRY\nVibrating sample magnetometry (VSM) data for the 33 nm (room temperature and 200◦C depositions) and 70 nm\nfilms are shown in Fig. S6. The magnetic field was applied in 3 different directions, with no discernible difference in the\nhysteresis loops. We conclude that there is no in-plane magnetocrystalline anisotropy over macroscopic lengthscales,\nwhich is consistent with the FMR measurements.\nS7. LONGITUDINAL RESISTIVITY\nLongitudinal resistivity ρxxwas measured as a function of temperature for the 33 nm (room temperature and 200◦C\ndepositions) and 70 nm films (Fig. S7) by patterning Hall bars and performing 4-wire resistance measurements.4\n65 60 55 50 45 40 35\n2/s61553(°)\n0100200300400Intensity\n(arb.units)\n65 60 55 50 45 40 35\n2/s61553(°)\n0100200300400Intensity\n(arb.units)\n65 60 55 50 45 40 35\n2/s61553(°)\n020406080Intensity\n(arb.units)\n65 60 55 50 45 40 35\n2/s61553(°)\n020406080Intensity\n(arb.units)(b) (a)\n(c) (d)q\nxyz\nq\nxyz33 nm (RT) 33 nm (200 °C) \n57 nm 70 nm\nFIG. S4. Two-dimensional detector images of the Fe 0.7Ga0.3(110) peak for (a) 33 nm (room temperature deposition), (b)\n33 nm (200◦C deposition), (c) 57 nm, and (d) 70 nm films. The total scattering angle is 2 θand is shown on the abscissa. The\nmeasurement is conducted such that the symmetric configuration corresponds to the center of the detector, which is to say\nthat the incident radiation is at an angle ω/similarequal26◦relative to the sample surface. In panel (a), the effect of moving vertically\nfrom the center of the detector on the scattering vector qis shown ( qis canted into the y-zplane).\n[S1] R. Arias and D. L. Mills, Extrinsic contributions to the ferromagnetic resonance response of ultrathin films, Phys. Rev. B\n60, 7395 (1999).\n[S2] R. D. McMichael and P. Krivosik, Classical Model of Extrinsic Ferromagnetic Resonance Linewidth in Ultrathin Films,\nIEEE Trans. Magn. 40, 2 (2004).\n[S3] P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two magnon scattering microwave relaxation:\nTheory and applications, J. Appl. Phys. 101, 083901 (2007).5\n-2-1012Height(nm)(a)\n-2-1012Height(nm)(b)\n(c) (d)\n-2-1012Height(nm)\n-2-1012Height(nm)100 nm\n100 nm100 nm\n100 nm33 nm (RT)\n57 nm33 nm (200 °C) \n70 nm\nFIG. S5. Atomic force microscopy for (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), (c) 57 nm,\nand (d) 70 nm films. RMS roughnesses are (a) 0.7 nm, (b) 0.4 nm, (c) 1.5 nm, and (d) 1.3 nm.\n[S4] M. Birkholz, Thin Film Analysis by X-Ray Scattering (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006).6\n-2000-1000010002000-101M/MsF\nield (Oe) H || Si[100] \nH || Si[110] \nH || Si[010]70 nm(c)\n-200-1000100200-101 \nH || Si[100] \nH || Si[110] \nH || Si[010]M/MsF\nield (Oe)33 nm (\n200 °C)(b)\n-50-2502550-101 \nH || Si[100] \nH || Si[110] \nH || Si[010]M/Ms F\nield (Oe)33 nm (RT)(a)\nFIG. S6. Vibrating sample magnetometry of (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), and\n(c) 70 nm films for H/bardblSi[100] (black), H/bardblSi[110] (red), and H/bardblSi[010] (blue).\n0100200300400020040060080010007\n0 nm(c)/s61554 xx (µΩ cm)T\nemperature (K)\n0100200300400050100150200250/s61554xx (µΩ cm)T\nemperature (K)33 nm (200 °C deposition)(b)\n0100200300400050100150200250(a)/s61554xx (µΩ cm)T\nemperature (K)33 nm (RT deposition)\nFIG. S7. Longitudinal resistivity ρxxas a function of temperature for the (a) 33 nm (room temperature deposition), (b) 33 nm\n(200◦C deposition), and (c) 70 nm films."    },    {        "title": "1807.07897v2.Another_view_on_Gilbert_damping_in_two_dimensional_ferromagnets.pdf",        "content": "Another view on Gilbert damping in two-dimensional\nferromagnets\nAnastasiia A. Pervishko1, Mikhail I. Baglai1,2, Olle Eriksson2,3, and Dmitry Yudin1\n1ITMO University, Saint Petersburg 197101, Russia\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75 121 Uppsala, Sweden\n3School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden\nABSTRACT\nA keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical\ndescription, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far\napproached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological\ngrounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to\nlattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which describes relaxation of the\nmagnetization to equilibrium. In this study we work out a microscopic Kubo-St ˇreda formula for the components of the Gilbert\ndamping tensor and apply the elaborated formalism to a two-dimensional Rashba ferromagnet in the weak disorder limit. We\nshow that an exact analytical expression corresponding to the Gilbert damping parameter manifests linear dependence on the\nscattering rate and retains the constant value up to room temperature when no vibrational degrees of freedom are present\nin the system. We argue that the methodology developed in this paper can be safely applied to bilayers made of non- and\nferromagnetic metals, e.g., CoPt.\nIntroduction\nIn spite of being a mature field of research, studying magnetism and spin-dependent phenomena in solids still remains one of\nthe most exciting area in modern condensed matter physics. In fact, enormous progress in technological development over the\nlast few decades is mainly held by the achievements in spintronics and related fields1–11. However the theoretical description of\nmagnetization dynamics is at best accomplished on the level of Landau-Lifshitz-Gilbert (LLG) equation that characterizes\ntorques on the magnetization. In essence, this equation describes the precession of the magnetization, mmm(rrr;t), about the effective\nmagnetic field, HHHeff(rrr;t), created by the localized moments in magnetic materials, and its relaxation to equilibrium. The latter,\nknown as the Gilbert damping torque12, was originally captured in the form ammm\u0002¶tmmm, where the parameter adetermines the\nrelaxation strength, and it was recently shown to originate from a systematic non-relativistic expansion of the Dirac equation13.\nThus, a proper microscopic determination of the damping parameter a(or, the damping tensor in a broad sense) is pivotal to\ncorrectly simulate dynamics of magnetic structures for the use in magnetic storage devices14.\nFrom an experimental viewpoint, the Gilbert damping parameter can be extracted from ferromagnetic resonance linewidth\nmeasurements15–17or established via time-resolved magneto-optical Kerr effect18, 19. In addition, it was clearly demonstrated\nthat in bilayer systems made of a nonmagnetic metal (NM) and a ferromagnet material (FM) the Gilbert damping is drastically\nenhanced as compared to bulk FMs20–24. A strong magnetocrystalline anisotropy, present in CoNi, CoPd, or CoPt, hints\nunambiguously for spin-orbit origin of the intrinsic damping. A first theoretical attempt to explain the Gilbert damping\nenhancement was made in terms of sdexchange model in Ref.25. Within this simple model, magnetic moments associated with\nFM layer transfer angular momentum via interface and finally dissipate. Linear response theory has been further developed\nwithin free electrons model26, 27, while the approach based on scattering matrix analysis has been presented in Refs.28, 29. In\nthe latter scenario spin pumping from FM to NM results in either backscattering of magnetic moments to the FM layer or\ntheir further relaxation in the NM. Furthermore, the alternative method to the evaluation of the damping torque, especially\nin regard of first-principles calculations, employs torque-correlation technique within the breathing Fermi surface model30.\nWhile a direct estimation of spin-relaxation torque from microscopic theory31, or from spin-wave spectrum, obtained on the\nbasis of transverse magnetic field susceptibility32, 33, are also possible. It is worth mentioning that the results of first-principles\ncalculations within torque-correlation model34–38and linear response formalism39, 40reveal good agreement with experimental\ndata for itinerant FMs such as Fe, Co, or Ni and binary alloys.\nLast but not least, an intensified interest towards microscopic foundations of the Gilbert parameter ais mainly attributed\nto the role the damping torque is known to play in magnetization reversal41. In particular, according to the breathing Fermi\nsurface model the damping stems from variations of single-particle energies and consequently a change of the Fermi surfacearXiv:1807.07897v2  [cond-mat.mes-hall]  21 Nov 2018z\nyx\nFM\nNMFigure 1. Schematic representation of the model system: the electrons at the interface of a bilayer, composed of a\nferromagnetic (FM) and a nonmagnetic metal (NM) material, are well described by the Hamiltonian (1). We assume the\nmagnetization of FM layer depicted by the red arrow is aligned along the zaxis.\nshape depending on spin orientation. Without granting any deep insight into the microscopic picture, this model suggests that\nthe damping rate depends linearly on the electron-hole pairs lifetime which are created near the Fermi surface by magnetization\nprecession. In this paper we propose an alternative derivation of the Gilbert damping tensor within a mean-field approach\naccording to which we consider itinerant subsystem in the presence of nonequilibrium classical field mmm(rrr;t). Subject to the\nfunction mmm(rrr;t)is sufficiently smooth and slow on the scales determined by conduction electrons mean free path and scattering\nrate, the induced nonlocal spin polarization can be approached within a linear response, thus providing the damping parameter\ndue to the itinerant subsystem. In the following, we provide the derivation of a Kubo-St ˇreda formula for the components of\nthe Gilbert damping tensor and illustrate our approach for a two-dimensional Rashba ferromagnet, that can be modeled by\nthe interface between NM and FM layers. We argue that our theory can be further applied to identify properly the tensorial\nstructure of the Gilbert damping for more complicated model systems and real materials.\nMicroscopic framework\nConsider a heterostructure made of NM with strong spin-orbit interaction covered by FM layer as shown in Fig. 1, e.g., CoPt.\nIn general FMs belong to the class of strongly correlated systems with partially filled dorforbitals which are responsible\nfor the formation of localized magnetic moments. The latter can be described in terms of a vector field mmm(rrr;t)referred to as\nmagnetization, that in comparison to electronic time and length scales slowly varies and interacts with an itinerant subsystem.\nAt the interface (see Fig. 1) the conduction electrons of NM interact with the localized magnetic moments of FM via a certain\ntype of exchange coupling, sdexchange interaction, so that the Hamiltonian can be written as\nh=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM(rrr;t)+U(rrr); (1)\nwhere first two terms correspond to the Hamiltonian of conduction electrons, on condition that the two-dimensional momentum\nppp= (px;py) =p(cosj;sinj)specifies electronic states, mis the free electron mass, astands for spin-orbit coupling strength,\nwhile sss= (sx;sy;sz)is the vector of Pauli matrices. The third term in (1) is responsible for sdexchange interaction with the\nexchange field MMM(rrr;t) =Dmmm(rrr;t)aligned in the direction of magnetization and Ddenoting sdexchange coupling strength. We\nhave also included the Gaussian disorder, the last term in Eq. (1), which represents a series of point-like defects, or scatterers,\nhU(rrr)U(rrr0)i= (mt)\u00001d(rrr\u0000rrr0)with the scattering rate t(we set ¯h=1throughout the calculations and recover it for the final\nresults).\nSubject to the norm of the vector jmmm(rrr;t)j=1remains fixed, the magnetization, in broad terms, evolves according to (see,\ne.g., Ref.42),\n¶tmmm=fff\u0002mmm=gHHHeff\u0002mmm+csss\u0002mmm; (2)\nwhere fffcorresponds to so-called spin torques. The first term in fffdescribes precession around the effective magnetic field\nHHHeffcreated by the localized moments of FM, whereas the second term in (2) is determined by nonequilibrium spin density of\nconduction electrons of NM at the interface, sss(rrr;t). It is worth mentioning that in Eq. (2) the parameter gis the gyromagnetic\nratio, while c= (gmB=¯h)2m0=dis related to the electron g\u0000factor ( g=2), the thickness of a nonmagnetic layer d, with mBand\nm0standing for Bohr magneton and vacuum permeability respectively. Knowing the lesser Green’s function, G<(rrrt;rrrt), one\ncan easily evaluate nonequilibrium spin density of conduction electrons induced by slow variation of magnetization orientation,\nsm(rrr;t) =\u0000i\n2Tr\u0002\nsmG<(rrrt;rrrt)\u0003\n=Qmn¶tmn+:::; (3)\n2/8where summation over repeated indexes is assumed ( m;n=x;y;z). The lesser Green’s function of conduction electrons\ncan be represented as G<=\u0000\nGK\u0000GR+GA\u0001\n=2, where GK,GR,GAare Keldysh, retarded, and advanced Green’s functions\nrespectively.\nKubo-St ˇreda formula\nWe further proceed with evaluating Qmnin Eq. (3) that describes the contribution to the Gilbert damping due to conduction\nelectrons. In the Hamiltonian (1) we assume slow dynamics of the magnetization, such that approximation MMM(rrr;t)\u0019\nMMM+(t\u0000t0)¶tMMMwith MMM=MMM(rrr;t0)is supposed to be hold with high accuracy,\nH=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM+U(rrr)+(t\u0000t0)sss\u0001¶tMMM; (4)\nwhere first four terms in the right hand side of Eq. (4) can be grouped into the Hamiltonian of a bare system, H0, which\ncoincides with that of Eq. (1), provided by the static magnetization configuration MMM. In addition, the expression (4) includes the\ntime-dependent term V(t)explicitly, as the last term. In the following analysis we deal with this in a perturbative manner. In\nparticular, the first order correction to the Green’s function of a bare system induced by V(t)is,\ndG(t1;t2) =Z\nCKdtZd2p\n(2p)2gppp(t1;t)V(t)gppp(t;t2); (5)\nwhere the integral in time domain is taken along a Keldysh contour, while gppp(t1;t2) =gppp(t1\u0000t2)[the latter accounts for the\nfact that in equilibrium correlation functions are determined by the relative time t1\u0000t2] stands for the Green’s function of the\nbare system with the Hamiltonian H0in momentum representation. In particular, for the lesser Green’s function at coinciding\ntime arguments t1=t2\u0011t0, which is needed to evaluate (3), one can write down,\ndG<(t0;t0) =i\n2¥Z\n\u0000¥de\n2pZd2p\n(2p)2n\ngR\npppsm¶g<\nppp\n¶e\u0000¶gR\nppp\n¶esmg<\nppp+g<\npppsm¶gA\nppp\n¶e\u0000¶g<\nppp\n¶esmgA\npppo\n¶tMm; (6)\nwhere m=x;y;z, while gR,gA, and g<are the bare retarded, advanced, and lesser Green’s functions respectively. To derive the\nexpression (6) we made use of Fourier transformation gppp=Rd(t1\u0000t2)gppp(t1\u0000t2)eie(t1\u0000t2)and integration by parts.\nTo finally close up the derivation we employ the fluctuation-dissipation theorem according to which g<(e) = [gA(e)\u0000\ngR(e)]f(e), where f(e) = [eb(e\u0000m)+1]\u00001stands for the Fermi-Dirac distribution with the Fermi energy m. Thus, nonequi-\nlibrium spin density of conduction electrons (3) within linear response theory is determined by Qmn=Q(1)\nmn+Q(2)\nmn, where\nQ(1)\nmn=1\n4Trh\nsm¥Z\n\u0000¥de\n2pZd2p\n(2p)2n¶gR\nppp\n¶esngR\nppp\u0000gR\npppsn¶gR\nppp\n¶e+gA\npppsn¶gA\nppp\n¶e\u0000¶gA\nppp\n¶esngA\npppo\nf(e)i\n; (7)\nwhich involves the integration over the whole Fermi sea, and\nQ(2)\nmn=1\n4Trh\nsm¥Z\n\u0000¥de\n2p\u0012\n\u0000¶f(e)\n¶e\u0013Zd2p\n(2p)2n\ngR\npppsngR\nppp+gA\npppsngA\nppp\u00002gR\npppsngA\npppoi\n; (8)\nwhich selects the integration in the vicinity of the Fermi level. Generally, the form of Qmnbelongs to the class of Kubo-St ˇreda\nformula, and, in essence, represents the response to the external stimulus in the form of ¶tMn. We can immediately establish a\nquantitative agreement between the result given by Eq. (8) and the previous studies within a Kubo formalism40, 43–46which allow\na direct estimation within the framework of disordered alloys. Formally, the expression (7) corresponds to the so-called St ˇreda\ncontribution. Such a term was originally identified in Ref.47when studying quantum-mechanical conductivity. Notably, in\nEq. (7) each term represents the product of either retarded or advanced Green’s functions. In this case the poles of the integrand\nfunction are positioned on the same side of imaginary plane, making disorder correction smaller in the weak disorder limit (see,\ne.g., Ref.48). Meanwhile, having no classical analog this contribution appears to be important enough when the spectrum of the\nsystem is gapped and the Fermi energy is placed exactly in the gap47. It is worth mentioning that the contribution due to Eq. (7)\nhas never been discussed in this context before. In the meantime, Kubo-St ˇreda expression for the components of the Gilbert\ndamping tensor has been addressed from the perspective of first-principles calculations49and current-induced torques50.\n3/8Results and discussion\nLet us apply the formalism developed in the previous section to a prototypical model: we work out the Gilbert damping tensor\nfor a Rashba ferromagnet with the magnetization mmm=zzzaligned along the zaxis. In the limit of weak disorder the Green’s\nfunction of a bare system can be expressed as\ngR\nppp(e) =\u0000\ne\u0000H0\u0000SR\u0001\u00001=e\u0000eppp+id+a(sss\u0002ppp)z+(D+ih)sz\n(e\u0000eppp+id)2\u0000a2p2\u0000(D+ih)2; (9)\nwhere eppp=p2=(2m)is the electron kinetic energy. We put the self-energy SRdue to scattering off scalar impurities into Eq. (9),\nwhich is determined from SR=\u0000i(d\u0000hsz)(see, e.g., Ref.51). In particular, for jej>jDjwe can establish that d=1=(2t)\nandh=0 in the weak disorder regime to the leading order.\nWithout loss of generality, in the following we restrict the discussion to the regime m>jDj, which is typically satisfied with\nhigh accuracy in experiments. As previously discussed, the contribution owing to the Fermi sea, Eq. (7), can in some cases be\nignored, while doing the momentum integral in Eq. (8) results in,\n1\nmtZd2p\n(2p)2gR\nppp(e)sssgA\nppp(e) =D2\nD2+2ersss+Dd\nD2+2er(sss\u0002zzz)+D2\u0000er\nD2+2er(sss\u0002zzz)\u0002zzz; (10)\nwhere r=ma2. Thus, thanks to the factor of delta function d(e\u0000m) =\u0000¶f(e)=¶e, to estimate Q(2)\nmnat zero temperature one\nshould put e=min Eq. (10). As a result, we obtain,\nQ(2)\nmn=\u00001\n4pm\nD2+2mr0\n@2tmr D 0\n\u0000D 2tmr 0\n0 0 2 tD21\nA: (11)\nMeanwhile, to properly account the correlation functions which appear when averaging over disorder configuration one has\nto evaluate the so-called vertex corrections, which from a physical viewpoint makes a distinction between disorder averaged\nproduct of two Green’s function, hgRsngAidis, and the product of two disorder averaged Green’s functions, hgRidissnhgAidis, in\nEq. (8). Thus, we further proceed with identifying the vertex part by collecting the terms linear in dexclusively,\nGGGs=Asss+B(sss\u0002zzz)+C(sss\u0002zzz)\u0002zzz; (12)\nprovided A=1+D2=(2er),B= (D2+2er)Dd=(D2+er)2, and C=D2=(2er)\u0000er=(D2+er). To complete our derivation\nwe should replace snin Eq. (8) by Gs\nnand with the aid of Eq. (10) we finally derive at e=m,\nQ(2)\nmn=0\n@Qxx Qxy 0\n\u0000QxyQxx\n0 0\u0000mtD2=(4pmr)1\nA: (13)\nWe defined Qxx=\u0000mtmr=[2p(D2+mr)]andQxy=\u0000mD(D2+2mr)=[4p(D2+mr)2], which unambiguously reveals that\naccount of vertex correction substantially modifies the results of the calculations. With the help of Eqs. (3), (11), and (13) we\ncan write down LLG equation. Slight deviation from collinear configurations are determined by xandycomponents ( mxand\nmyrespectively, so that jmxj;jmyj\u001c1). The expressions (11) and (13) immediately suggest that the Gilbert damping at the\ninterface is a scalar, aG,\n¶tmmm=˜gHHHeff\u0002mmm+aGmmm\u0002¶tmmm; (14)\nwhere the renormalized gyromagnetic ratio and the damping parameter are,\n˜g=g\n1+cDQxy;aG=\u0000cDQxx\n1+cDQxy\u0019\u0000cDQxx: (15)\nIn the latter case we make use of the fact that mc\u001c1for the NM thickness d\u0018100mm — 100 nm. In Eq. (14) we have\nredefined the gyromagnetic ratio g, but we might have renormalized the magnetization instead. From physical perspective,\nthis implies the fraction of conduction electrons which become associated with the localized moment owing to sdexchange\ninteraction. With no vertex correction included one obtains\naG=mc\n2p¯htmrD\nD2+2mr; (16)\n4/8t=1ns\nt=10ns\nD=0.2meV\nD=0.3meV\nD=1meV\n501001502002503000.0000.0010.0020.0030.004\nT,KaGFigure 2. Gilbert damping, obtained from numerical integration of Eq. (8), shows almost no temperature dependence\nassociated with thermal redistribution of conduction electrons. Dashed lines are plotted for D=1meV for t=1andt=10ns,\nwhereas solid lines stand for D=0:2, 0:3, and 1 meV for t=100 ns.\nwhile taking account of vertex correction gives rise to a different result,\naG=mc\n2p¯htmrD\nD2+mr: (17)\nTo provide a quantitative estimate of how large the St ˇreda contribution in the weak disorder limit is, on condition that m>jDj,\nwe work out Q(1)\nmn. Using ¶gR=A(e)=¶e=\u0000[gR=A(e)]2and the fact that trace is invariant under cyclic permuattaions we conclude\nthat only off-diagonal components m6=ncontribute. While the direct evaluation results in Q(1)\nxy=3mD=[2(D2+2mr)]in the\nclean limit. It has been demonstrated that including scattering rates dandhdoes not qualitatively change the results, leading to\nsome smearing only52.\nInterestingly, within the range of applicability of theory developed in this paper, the results of both Eqs. (16) and (17)\ndepend linearly on scattering rate, being thus in qualitative agreement with the breathing Fermi surface model. Meanwhile, the\nlatter does not yield any connection to the microscopic parameters (see, e.g., Ref.53for more details). To provide with some\nquantitative estimations in our simulations we utilize the following set of parameters. Typically, experimental studies based on\nhyperfine field measurements equipped with DFT calculations54reveal the sdStoner interaction to be of the order of 0.2 eV ,\nwhile the induced magnetization of s-derived states equals 0.002–0.05 (measured in the units of Bohr magneton, mB). Thus,\nthe parameter of sdexchange splitting, appropriate for our model, is D\u00180.2–1 meV . In addition, according to first-principles\nsimulations we choose the Fermi energy m\u00183 eV . The results of numerical integration of (8) are presented in Fig. 2 for several\nchoices of sdexchange and scattering rates, t. The calculations reveal almost no temperature dependence in the region up to\nroom temperature for any choice of parameters, which is associated with the fact that the dominant contribution comes from the\nintegration in a tiny region of the Fermi energy. Fig. 2 also reveal a non-negligible dependence on the damping parameter with\nrespect to both Dandt, which illustrates that a tailored search for materials with specific damping parameter needs to address\nboth the sdexchange interaction as well as the scattering rate. From the theoretical perspective, the results shown in Fig. 2\ncorrespond to the case of non-interacting electrons with no electron-phonon coupling included. Thus, the thermal effects are\naccounted only via temperature-induced broadening which does not show up for m>jDj.\nConclusions\nIn this paper we proposed an alternative derivation of the Gilbert damping tensor within a generalized Kubo-St ˇreda formula.\nWe established the contribution stemming from Eq. (7) which was missing in the previous analysis within the linear response\ntheory. In spite of being of the order of (mt)\u00001and, thus, negligible in the weak disorder limit developed in the paper, it should\nbe properly worked out when dealing with more complicated systems, e.g., gapped materials such as iron garnets (certain half\nmetallic Heusler compounds). For a model system, represented by a Rashba ferromagnet, we directly evaluated the Gilbert\ndamping parameter and explored its behaviour associated with the temperature-dependent Fermi-Dirac distribution. In essence,\nthe obtained results extend the previous studies within linear response theory and can be further utilized in first-principles\ncalculations. We believe our results will be of interest in the rapidly growing fields of spintronics and magnonics.\n5/8References\n1.Žuti´c, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004).\n2.Bader, S. D. & Parkin, S. S. P. 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Rev.\nLett.117, 046601 (2016).\n52.Nunner, T. S., et al. Anomalous Hall effect in a two-dimensional electron gas. Phys. Rev. B 76, 235312 (2007).\n53.Eriksson, O., Bergman, A., Bergqvist, L. & Hellsvik, J. Atomistic spin dynamics: Foundations and applications (Oxford\nUniversity Press, Oxford, 2017).\n54.Brooks, M. S. S. & Johansson, B. Exchange integral matrices and cohesive energies of transition metal atoms. J. Phys. F:\nMet. Phys. 13, L197 (1983).\n7/8Acknowledgements\nA.A.P. acknowledges the support from the Russian Science Foundation Project No. 18-72-00058. O.E. acknowledges support\nfrom eSSENCE, the Swedish Research Council (VR), the foundation for strategic research (SSF) and the Knut and Alice\nWallenberg foundation (KAW). D.Y . acknowledges the support from the Russian Science Foundation Project No. 17-12-01359.\nAuthor contributions statement\nD.Y . conceived the idea of the paper and contributed to the theory. A.A.P. wrote the main manuscript text, performed numerical\nanalysis and prepared figures 1-2. M.I.B. and O.E. contributed to the theory. All authors reviewed the manuscript.\nAdditional information\nCompeting interests The authors declare no competing interests.\n8/8"    },    {        "title": "1106.4389v2.Effect_of_spin_diffusion_on_current_generated_by_spin_motive_force.pdf",        "content": "arXiv:1106.4389v2  [cond-mat.mes-hall]  21 Aug 2011Effect of spin diffusion on current generated by spin motive fo rce\nKyoung-Whan Kim1, Jung-Hwan Moon2, Kyung-Jin Lee2, and Hyun-Woo Lee1\n1PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang, 790-784, Korea\n2Department of Materials Science and Engineering, Korea Uni versity, Seoul 136-701, Korea\n(Dated: November 16, 2018)\nSpin motive force is a spin-dependent force on conduction el ectrons induced by magnetization\ndynamics. In order to examine its effects on magnetization dy namics, it is indispensable to take\ninto account spin accumulation, spin diffusion, and spin-fli p scattering since the spin motive force is\nin general nonuniform. We examine the effects of all these on t he way the spin motive force generates\nthechargeandspincurrentsinconventionalsituations, wh ere theconductionelectron spinrelaxation\ndynamics is much faster than the magnetization dynamics. Wh en the spin-dependent electric field\nis spatially localized, which is common in experimental sit uations, we find that the conservative part\nof the spin motive force is unable to generate the charge curr ent due to the cancelation effect by the\ndiffusion current. We also find that the spin current is a nonlo cal function of the spin motive force\nand can be effectively expressed in terms of nonlocal Gilbert damping tensor. It turns out that any\nspin independent potential such as Coulomb potential does n ot affect our principal results. At the\nlast part of this paper, we apply our theory to current-induc ed domain wall motion.\nPACS numbers:\nI. INTRODUCTION\nIn a ferromagnetic system, dynamics of space-time de-\npendent magnetizationvector M(x,t) is describedby the\nfollowing phenomenological equation [1–4]\n∂M\n∂t=−γM×Heff+α\nMsM×∂M\n∂t\n+µB\neMs(js·∇)M−βµB\neM2sM×(js·∇)M.(1)\nThis is the Landau-Lifshitz-Gilbert (LLG) equation gen-\neralized to include the spin-transfer torque terms (last\ntwo terms). Here, Heffis the functional derivative of en-\nergy with respect to M,αis Gilbert damping constant,\nMsis saturation magnetization, µBis Bohr magneton,\neis electron charge, βis nonadiabaticity, and jsis spin\npolarizedelectriccurrentgivenbydifferencebetweencur-\nrent by spin-up and spin-down electrons. LLG equation\ndescribes the dynamics of magnetization under applied\nelectromagnetic fields. On the other hand, there exists\na reciprocal process; temporal and spatial variation of\nmagnetization induces additional electromagnetic fields\non conduction electrons. These fields are spin-dependent\nand in general nonconservative. Thus the resulting spin-\ndependentmotiveforceiscalledspinmotiveforce(SMF).\nSMF is firstly predicted by Berger [5] and recently for-\nmulated by generalizing Faraday’s law [6]. It is also sug-\ngested [7] that SMF is also described by spin pumping\neffect[8]. Recently, SMFanditseffectisintensivelystud-\nied [7, 9–20] in this field.\nWithout any other perturbation, the explicit expres-\nsionsoftheinducedspinelectromagneticfieldsareknownas [7, 10–12, 19–21]\nE↑↓\ni=±/planckover2pi1\n2eM3s(∂tM×∂iM)·M, (2)\nB↑↓\ni=∓/planckover2pi1\n2eM3sǫijk(∂jM×∂kM)·M,(3)\nwhere↑and↓stand for spin-up and down electrons.\nThe fields generate spin-dependent Lorentz force F↑↓\ns=\ne(E↑↓\ni+v↑↓×B↑↓\ns). While the term “motive force” refers\ntoquantitiesofvoltagedimension,thetermSMFissome-\ntimes used to denote F↑↓\ns. In this paper, we adopt the\nlatter terminology. It is easily noticed that SMF is spin\ndependent, nonconservative, spatially varying, and local-\nized in general situations. As one can see from Eqs. (2)\nand (3), SMF is usually too small to be measured di-\nrectly. Another way is to study the effect of SMF on\nmagnetization dynamics. Since SMF induces additional\nspin current, additional spin-transfer torque arises and\nit changes LLG equation. Consequently, LLG equation\n(without applied spin current) is modified as [7]\n∂M\n∂t=−γM×Heff+1\nMsM×/parenleftbigg\nD·∂M\n∂t/parenrightbigg\n,(4)\nwhereDis modified damping tensor given by\nDij=αδij+η\nM4s/summationdisplay\nk(M×∂kM)i(M×∂kM)j,(5)\nandη=µB/planckover2pi1σ/2e2Ms. Here,σis electrical conductivity.\nHowever, the previous work has a crucial limitation\nthat spin density has been considered to be constant\nwhile spin density in reality is nonuniform since SMF is\nspatially varying. The main consequence of the nonuni-\nform spin density n↑↓(x,t) is diffusion current propor-\ntional to −∇n↑↓, which suppresses the effect of SMF.\nTherefore, a realistic model should take into account2\nspin accumulation (nonuniform spin density), diffusion\nand spin-flip scattering. The purpose of this paper is to\nfind the spin density n↑↓, diffusion current, total current\ninduced by SMF, and their effect on magnetization dy-\nnamics in the presence of spin accumulation, diffusion,\nand spin-flip scattering. As one shall see in Sec. III, the\nsolution of n↑↓in the most general situation is too com-\nplicated to study the effect on magnetization dynamics.\nTo obtain simple analytic expressions, we take an ap-\nproximation that spin-flip time is much shorter than the\ntime scale of magnetization dynamics. As a final com-\nment, our result does not assume any specific form of\nSMF. Thus, it remains valid for the modified SMF due\nto, for instance, nonadiabaticity [10], spin-orbit coupling\n[20], and other kinds of spin dependent electric field [22].\nSeveral previous works are closely related to our work.\nSpin drift-diffusion equation, which has similar form to\nour theory is suggested in Ref. [10]. And, the effect on\nspin and charge current is investigated from Boltzmann\nequation in Ref. [19]. We set our starting point as the\nequation of motion of conduction electrons in Ref. [1]\nto make our analysis consistent with previous theories in\nthis field. Different from the previous theories focusing\non 1D, we successfully generalized our result to 3D, and\nfound that nonconservative part of SMF plays a crucial\nrole in current in a higher dimensional system. In Sec.\nIII, we compare our result with the previous theory qual-\nitatively and quantitatively. In addition, we investigated\nthe effect of charge neutrality on our results. It turns\nout that charge neutrality potential does not change our\nprincipal results, charge current and spin current, even\nthough it changes charge density and spin density. Fur-\nthermore, we show that any spin independent potential\ncannot alter our principal results, either.\nThis paper is organized as follows. In Sec. II, we\nconstruct the spin drift-diffusion equation and introduce\nvariables. Then, we solve the equation in Sec. III, and\ndiscuss various implications. In Sec. IV, we apply our\nresult to current-induced domain wall (DW) motion and\nbriefly discuss the effect of spin diffusion. In Sec. V, we\ngeneralize our theory for general boundary condition and\nfor general spin indendendent potentials. Finally, in Sec.\nVI, there are concluding remarks. Technical details are\nin Appendices.\nII. MODEL\nA. Spin drift-diffusion equation\nTo construct the equation of n↑↓(x,t), we take the\nstarting point as the equation of spin density mof con-\nduction electrons [1],\n∂m\n∂t+∇·J=−1\nτexMsm×M−∝angb∇acketleftΓ∝angb∇acket∇ight.(6)\nHere,Jis spin current tensor, τex=/planckover2pi1/SJex, andSde-\nnotes the magnitude of spin of local magnetization. Theleft-hand side is based on the continuity equation. The\nfirst term on the right-hand side is the precession term\ndue to the exchange coupling between conduction elec-\ntrons and magnetization. ∝angb∇acketleftΓ∝angb∇acket∇ightincludes the effect of spin\nscattering processes. Here, the second rank tensor Jis\ndefined by\nJ=−µB\nejs⊗m\n|m|. (7)\nThe effect of the perpendicular component to Mof Eq.\n(6) is already investigated by Zhang and Li [1], and they\nfound the nonadiabatic term of LLG equation. In the\nabsence of spin accumulation, the magnitude of mis\nconstant, so it suffices to solve only the perpendicular\ncomponent of the equation. However, in the presence\nof the spin accumulation, the magnitude variation of m\nshould be also studied. We define spin number density\nns≡ |m|/µB. Taking care of the fact that nshas space-\ntime dependence, the parallel component of Eq. (6) to\nˆm≡m/|m|results in\n∂ns\n∂t+1\nµBˆm·∝angb∇acketleftΓ∝angb∇acket∇ight=1\ne∇·js. (8)\nIt is convenient to separate the variables to that of up\nand down electrons. ns=n↑−n↓andjs=j↑−j↓. Here,\nn↑↓andj↑↓denote spin number density of spin-up/down\nelectrons and charge current density generated by spin-\nup/down electrons, respectively. Equation (8) is nothing\nbut the continuity equation containing spin nonconserv-\ning processes described by Γ. To obtain independent\nequations of spin-up/down electrons, we use the follow-\ning continuity equation of total electron number density\n∂ne\n∂t=1\ne∇·je, (9)\nwherene=n↑+n↓is electron number density and je=\nj↑+j↓is charge current density. Combining Eqs. (8) and\n(9), one obtains\n∂n↑↓\n∂t±1\n2µBˆm·∝angb∇acketleftΓ∝angb∇acket∇ight=1\ne∇·j↑↓. (10)\nNote that Γrepresents spin-flip scattering processes. As\na simple model, we take the well-known form of spin-flip\nscattering,\n1\n2µBˆm·∝angb∇acketleftΓ∝angb∇acket∇ight=n↑\nτ↑−n↓\nτ↓, (11)\nwhereτ↑is characteristic time of the spin-flip scattering\nprocess from spin-up to -down state, and τ↓is similarly\ndefined. Then,\n∂n↑↓\n∂t+n↑↓\nτ↑↓−n↓↑\nτ↓↑=1\ne∇·j↑↓, (12)\nwhich is the spin drift-diffusion equation. Similar form\nof Eq. (12) was also suggested in Ref. [10].3\nFor simplicity, we may assume without losing general-\nitythat the SMF isturned onat t= 0 andthat, for t <0,\nthe system is in equilibrium. We set n↑↓(x,t= 0) =n↑↓\n0,\nwheren↑↓\n0is equilibrium electron density of spin up and\ndown at t <0. By definition, the equilibrium density\nn↑↓\n0is the equilibrium solution of Eq. (12) for t <0.\nInserting n↑↓(x,t <0) =n↑↓\n0to Eq. (12), one ob-\ntains an important constraint n↑\n0/τ↑=n↓\n0/τ↓. With\nthe help of this constraint, four variables n↑\n0,n↓\n0,τ↑and\nτ↓can be described by three variables, n↑\n0,n↓\n0, andτsf\n(τ−1\nsf=τ↑−1+τ↓−1). Then, Eq. (12) is rewritten with\nonly one spin-flip time τsf. In addition, current j↑↓can\nbe written as σ↑↓E↑↓\ns+eD↑↓∇n↑↓, whereσ↑↓andE↑↓\nsare\nrespectively the conductivity and SMF (divided by e) for\nspin-up/down electrons. Then, one straightforwardlyob-\ntains the final form of the equation of our model.\n∂n↑↓\n∂t−D↑↓∇2n↑↓+n↑\n0n↓\n0\nn↑\n0+n↓\n01\nτsf/parenleftigg\nn↑↓\nn↑↓\n0−n↓↑\nn↓↑\n0/parenrightigg\n=σ↑↓\ne∇·E↑↓\ns. (13)\nAs mentioned in Sec. I, we treat E↑↓\nsas nonconserva-\ntive, spatially varying fields. In addition, it is assumed\nthat spin dependence of Esis given by E↑\ns=−E↓\ns≡Es.\nSlight generalization of our theory at the final step al-\nlows to investigate the formula for E↑\ns∝negationslash=−E↓\ns. No other\nrestriction of E↑↓\nsis not assumed in order to obtain max-\nimally generalized result.\nAs suggested in Ref. [10], in realistic systems,\nthe Coulomb interaction should be taken into account.\nHence, oneintroducesCoulombpotential Vcandadditto\nthe spin motive force as E↑↓\ns→E↑↓\ns−∇Vc. The Coulomb\ninteraction strongly suppresses the charge accumulation.\nMathematically the interaction may thus be handled by\nimposing the charge neutrality constraint. We show in\nSec. V that chargeneutrality constraintchanges electron\ndensities, but not currents. Hence, the LLG equation is\nhardly affected by charge neutrality potential. For this\nreason, we do not take into account the Coulomb inter-\naction until Sec. V in order to show simple logical flow.\nNote that all variables in Eq. (13) are not indepen-\ndent. Einstein’s relation is given by σ↑↓=e2D↑↓N↑↓\nwhereN↑↓is density of states of spin-up/down elec-\ntrons at Fermi energy. Since N↑↓∝n↑↓\n0, one obtains\nσ↑/D↑n↑\n0=σ↓/D↓n↓\n0. This is one of the key constraints\nof our model.\nThe solution of Eq. (13) is very complicated as one\nshall see in Sec. III. To gain an insight, it is illustrative\nto assume that τsfis much smaller than the time scale\nof magnetization dynamics so Esis almost constant in\ntime scale within τsf. We found that, in this limit, the\nsolution is much simpler and it is easier to catch physical\nmeanings.B. Variable definitions and relations\nIn Sec. III, there appear several variables and quan-\ntities which have not been defined yet. To help readers,\nwe present definitions of them here, rather than Sec. III.\nSince Eq. (13) is coupled, it is convenient to solve it in\nmatrix form. Hence, we define spin accumulation vector,\nwhich is a column vector defined by\nN=/parenleftbigg\nn↑\nn↓/parenrightbigg\n. (14)\nSimilarly, we define current density and SMF vector.\nJ=/parenleftbigg\nj↑\nj↓/parenrightbigg\n, (15)\nE=/parenleftbigg\nE↑\ns\nE↓\ns/parenrightbigg\n=Es/parenleftbigg\n1\n−1/parenrightbigg\n. (16)\nEquations (14)-(16) are related by the following relation.\nJ=/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nE+e/parenleftbigg\nD↑0\n0D↓/parenrightbigg\n∇N.(17)\nThe first term in right-hand side corresponds to conven-\ntional electrical current and the second term corresponds\nto diffusion current.\nInstead of diffusion constants, it is more physical and\nintuitive to express results in terms of spin-flip length\nwhich is defined by λ↑↓\nsf=/radicalbig\nD↑↓τsf. The averaged spin\ndiffusion length is also defined by the conventional way\nλ−2\nsf=n↑\n0λ↑2\nsf+n↓\n0λ↓2\nsf\n(n↑\n0+n↓\n0)λ↑2\nsfλ↓2\nsf. (18)\nBy Einstein’s relation, Eq. (18) is equivalent to\nλ2\nsf=σ↑λ↓2\nsf+σ↓λ↑2\nsf\nσ↑+σ↓. (19)\nCombining Eqs. (18) and (19), λ↑↓\nsfis represented in\nterms of λsf.\nλ↑↓2\nsf=λ2\nsfσ\nσ↓↑n↓↑\n0\nn↑\n0+n↓\n0, (20)\nwhereσ=σ↑+σ↓is total electrical conductivity.\nConductivity polarization Pand density polarization\nPnare defined by\nP=σ↑−σ↓\nσ, (21)\nPn=n↑\n0−n↓\n0\nn↑\n0+n↓\n0. (22)\nWith these polarizations, σ↑↓andn↑↓\n0are represented\nin terms of σand (n↑\n0+n↓\n0) asσ↑↓= (1±P)σ/2 and\nn↑↓\n0= (1±Pn)(n↑\n0+n↓\n0)/2.4\nLastly, we use a mathematical convention that ˜A(k) is\nthe Fourier transform of a position dependent function\nA(x) with respect to x. That is,\n˜A(k)≡ F[A(x)](k) =1\n(2π)d/2/integraldisplay\nA(x)e−ik·xddx,(23)\nfor ad-dimensional system.\nIII. CHARGE AND SPIN CURRENTS IN THE\nPRESENCE OF SPIN DIFFUSION\nA. Solution of the spin drift-diffusion equation for\nlocalized electric field\nBefore solving Eq. (13) for general cases, we first solve\nthe equation for localized EssinceEsis localized in most\ncases. In Sec. V, we generalize our theory to include\nspatially extended Es.\nSinceEsislocalized,itispossibletotakeFouriertrans-\nform with respect to position. Then, ˜Eand˜Nare well-\ndefined localized functions of kexcept initial condition\npart. In addition, localized Esimplies that the bound-\nary condition is given by n↑↓(|x| → ∞,t) =n↑↓\n0because\nEs(|x| → ∞,t) = 0 does not affect spin density. After\nFourier transform, Eq. (13) is written as, in matrix form,\n∂˜N\n∂t+Ω\nτsf˜N=1\ne/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nik·˜E,(24)\nwhere\nΩ =\nλ↑2\nsfk2+n↓\n0\nn↑\n0+n↓\n0−n↑\n0\nn↑\n0+n↓\n0\n−n↓\n0\nn↑\n0+n↓\n0λ↓2\nsfk2+n↑\n0\nn↑\n0+n↓\n0\n.(25)\nEquation (24) is a first order ordinary differential equa-\ntion with respect to tand the initial condition is given\nby˜N(k,t= 0) = (2 π)d/2δ(k)(n↑\n0n↓\n0)T. The solution is\nsimply given by\n˜N(k,t) =e−Ωt/τsf˜N(k,0) (26)\n+/integraldisplayt\n0e−Ω(t−t′)/τsf/parenleftigg\nσ↑\ne0\n0σ↓\ne/parenrightigg\nik·˜E(k,t′)dt′.\nSince the first term of Eq. (26) represents the time vari-\nation of equilibrium number density, one can realize that\nthe term should be given by e−Ωt/τsf˜N(k,0) =˜N(k,0).\nMathematical derivation of this argument is given in Ap-\npendix A. The second term of Eq. (26) is almost impos-\nsible to take inverse Fourier transform. Hence, as men-\ntioned, we use an approximation that τsfis very small.\nIn this limit, Appendix B shows that\ne−Ωt/τsfΘ(t)≈τsfΩ−1δ(t), (27)\nwhere Θ( t) is Heaviside step function. By this approxi-\nmation, solution of the spin drift-diffusion equation Eq.(26) becomes\n˜N(k,t) =˜N(k,0)+τsf\neΩ−1/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nik·˜E(k,t).(28)\nThe inverse of Ω is explicitly given by\nΩ−1=1\ndetΩ\nλ↓2\nsfk2+n↑\n0\nn↑\n0+n↓\n0n↑\n0\nn↑\n0+n↓\n0\nn↓\n0\nn↑\n0+n↓\n0λ↑2\nsfk2+n↓\n0\nn↑\n0+n↓\n0\n,(29)\ndetΩ = λ↑2\nsfλ↓2\nsfk2/parenleftig\nk2+λ−2\nsf/parenrightig\n. (30)\nNow, excited charge density ∆ ne≡(n↑+n↓)−(n↑\n0+n↓\n0)\nand excited spin density ∆ ns≡(n↑−n↓)−(n↑\n0−n↓\n0) are\ngiven by,\n∆˜ne(k,t) =/parenleftbig1 1/parenrightbig˜N(k,t)−(2π)d/2δ(k)(n↑\n0+n↓\n0)\n=στsf\neλ2\nsf1−P2\n1−P2nPnk2+Pλ−2\nsf\nk2+λ−2\nsfik·˜Es(k,t)\nk2,(31)\n∆˜ns(k,t) =/parenleftbig1−1/parenrightbig˜N(k,t)−(2π)d/2δ(k)(n↑\n0−n↓\n0)\n=στsf\neλ2\nsf1−P2\n1−P2nk2+PnPλ−2\nsf\nk2+λ−2\nsfik·˜Es(k,t)\nk2(32)\nink-space.\nAt this stage, thereis no need to showcomplicatedreal\nspace expressions of the densities, because what affects\nLLG equation mainly is spin current. In the next subsec-\ntion, we find the expressions of charge and spin currents\nin bithk-space and real space.\nB. Charge and spin currents\nCharge and spin currents in the absence of spin diffu-\nsion are given by\nje=σ↑E↑\ns+σ↓E↑\ns=PσEs, (33)\njs=σ↑E↑\ns−σ↓E↑\ns=σEs. (34)\nIn this subsection, how the spin current and charge cur-\nrent generated by SMF is changed by spin diffusion from\nEqs. (33) and (34) is examined with the help of Eqs.\n(17) and (28). After some algebra,\n˜J=/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\n˜E+e/parenleftbigg\nD↑0\n0D↓/parenrightbigg\nik˜N\n=˜Es/parenleftbigg\nσ↑\n−σ↓/parenrightbigg\n+ik(ik·˜Es)\nk2/parenleftig\nk2+λ−2\nsf/parenrightig/parenleftbiggσ↑k2+Pσ↑λ−2\nsf\n−σ↓k2+Pσ↓λ−2\nsf/parenrightbigg\n.(35)5\nNow, the expressions of charge current ˜je=˜j↑+˜j↓and\nspin current ˜js=˜j↑−˜j↓are straightforward.\n˜je=Pσ/bracketleftigg\n˜Es+ik(ik·˜Es)\nk2/bracketrightigg\n, (36)\n˜js=P˜je+(1−P2)σ/parenleftigg\n˜Es+ik(ik·˜Es)\nk2+λ−2\nsf/parenrightigg\n.(37)\nWe present jsin terms of jein order for one to see easily\njs=jeforP=±1 ; for perfectly polarized electrons\nwithout spin-flip, the spin current should be the same\namount of the charge current.\nEquations(36)and (37) arethe principalresultsofthis\npaper. One can obtain d-dimensional real space expres-\nsions by taking inverse Fourier transform. This is one\nof the key advantages of our theory. Our result is easily\ngeneralizable to d-dimensional result. As one shall see,\nit turns out that the nonconservative part of Esplays a\ncrucial role in a higher dimensional system.\nFirst of all, we present 1D result. 1D real space ex-\npression of Eqs. (36) and (37) are\nje(x,t) = 0, (38)\njs(x,t) = (1−P2)/integraldisplay\ndx′e−|x−x′|/λsf\n2λsfσEs(x′,t).(39)\nOne can notice that the 1D charge current is perfectly\ncanceled by diffusion current for small spin-flip time\nlimit. This is natural in the sense that, for small spin-\nflip time, the system tends to be in equilibrium at each\ntimet. At equilibrium, the gradientofchemicalpotential\nvanishes, so does charge current. However, spin current\ndoes not vanish by this reason because of spin noncon-\nserving process. Due to spin diffusion, it is natural that\nthespincurrentbecomesnonlocalwith integrationkernel\nwidthλsf. Here, the factor (1 −P2) seems unexpected.\nThis factor comes from spin diffusion effect, and should\nexist regardless of diffusion length scale. It yields more\ncancelation for more polarized electrons. Eventually, for\nP=±1, spin current also vanishes, which is actually\nrequired since js=jein the limit P=±1.\nEquation(39) behavesquite differentlyfortwolimiting\ncases. Let λbe the characteristic length scale (such as\nDW width) of localized Es. Ifλsf≪λ,e|x−x′|/λsf≈\n2λsfδ(x−x′). Then,\njs(x,t)≈(1−P2)σEs(x,t), (40)\nwhich is local. Very short diffusion length cannot make\nthe spin current nonlocal. It is very interesting that (1 −\nP2)factordoesnotdisappeareventhoughdiffusioneffect\nis very small. The existence of diffusion makes (1 −P2)\nfactor regardless of how the effect is strong or weak. For\nλsf≫λ,\njs(x,t)≈σEs(1−P2)λ\n2λsfe−|x−X(t)|/λsf,(41)-50 0 500.00.51.01.52.02.53.0\n-100 -50 0 50 10001234(a)  (b)  \nFIG. 1: (color online) Results of micromagnetic simulation\nfor (a) charge current and (b) spin current induced by SMF\ngenerated from 1D DW oscillator, which is fixed at x= 0, but\nrotating with ω= 10GHz, in the presence of spin diffusion\nwithλsf= 0.5nm (black rectangle), 5 .0nm (red circle), and\n50nm (blue triangle). Solid line represents currents witho ut\ndiffusion. Here, DW width is set to be 10nm.\nwhereEsis averaged SMF and X(t) is the position of\nlocalized Es(such as DW position). One can see that\nthe current is highly suppressed by the factor λ/λsf. In\nthis highly diffusive regime, spin current is also highly\nsuppressed.\nThe main features of our result is similar to those of\nRef. [19], except for vanishing charge current. They\nclaim that charge current can exist in general, while we\nfind that Einstein’s relation prevents the existence of\ncharge current in 1D.\nWe confirmed our analytic result by comparing it with\nmicromagnetic simulation for a 1D DW oscillator. The\nresult is in Fig. 1. General features are the same as an-\nalytic result ; i) charge and spin currents are highly sup-\npressed by spin diffusion, ii) spin current becomes non-\nlocal, iii) charge current almost vanish independently of\ndiffusion length, and iv) spin current is more suppressed\nby larger diffusion length.\nNow, we generalize the results to higher dimension. In\n2D and 3D real spaces, Eqs. (36) and (37) are converted\nto\nj2D\ne=PσEs−Pσ\n2π∇/integraldisplay\nd2x′ln|x−x′|\nλsf∇′·Es,(42a)\nj3D\ne=PσEs+Pσ\n4π∇/integraldisplay\nd3x′∇′·Es\n|x−x′|, (42b)\nand\nj2D\ns=Pj2D\ne+(1−P2)σEs\n+(1−P2)σ\n2π∇/integraldisplay\nd2x′K0/parenleftbigg|x−x′|\nλsf/parenrightbigg\n∇′·Es,(43a)\nj3D\ns=Pj3D\ne+(1−P2)σEs\n+(1−P2)σ\n4π∇/integraldisplay\nd3x′e−|x−x′|/λsf\n|x−x′|∇′·Es,(43b)\nrespectively. Here, K0(x) is the zeroth order modified\nBessel function of the second kind. One can be uncom-\nfortable because j2D\neseems dependent on λsfwhile Eq.\n(36) is not. Here λsfis included because the argument6\nof logarithmic function should be dimensionless. In fact,\none can easily check that j2D\nedoes not depend on λsfby\nusing ln|x−x′|/λ′\nsf= ln|x−x′|/λsf+(constant) for any\npositiveλ′\nsf.\nFrom now on, we present only 3D expressionsbut omit\n2D, for simplicity. One can easily obtain 2D expressions\nby replacing the integral kernels 1 /|x−x′| → −2ln|x−\nx′|/λsfand exp( −|x−x′|/λsf)/|x−x′| →2K0/parenleftig\n|x−x′|\nλsf/parenrightig\n.\nThe reason why the expressions depend on dimension\nis nothing but the fact that the inverse Fourier trans-\nforms, which give integration kernels, depend on dimen-\nsion. Hence, essential physics are the same for 2D and\n3D except the mathematical expressions of the integra-\ntion kernels.\nOverallfeaturesofthe spincurrentinhigherdimension\nis similar to 1D, except for the existence of nonvanishing\nPjeterm. However, the main feature of jeis completely\ndifferent from 1D case. First of all, jedoes not vanish.\nWe argued qualitatively why the charge current vanishes\nin a 1D system by using chemical potential argument.\nHowever, in a higher dimensional system, chemical po-\ntential cannot be defined in general since Esis noncon-\nservative. Note that diffusion current D↑↓∇n↑↓is con-\nservative. Note also that nonconservative field cannot be\ncanceled by conservative field. This is why the charge\ncurrent exists in a higher dimensional system. The non-\nlocal term in Eq. (42) can be interpreted as Coulomb\npotential under charge density −∇ ·Es. The canceled\npart of the charge current is nothing but conservative\nCoulomb part of Es. Secondly, it is very interesting that\nEq. (42) is converted after some algebra to\nje=Pσ\n4π/integraldisplay\nd3x′∇′×(∇′×Es)\n|x−x′|. (44)\nNow one can notice the importance of nonconservative\npart ofEs(nonvanishing ∇ ×Es) and the dependence\nof this on the charge current. If ∇ ×Eshappens to be\nzero, the charge current also vanishes, and this is consis-\ntent to the chemical potential argument. Lastly, it is also\ninteresting that charge current does not depend on dif-\nfusion length. This is qualitatively understandable from\nthe fact that the effect of Esis maximally canceled by\ndiffusion current in small spin-flip time regime, regard-\nless of spin diffusion length. Here, maximal cancelation\nis slightly different from perfect cancelation in 1D case in\nthe sense that the (conservative) diffusion current cannot\ncancelPσEsperfectly in principle.\nIt might be ambiguous what the “conservative part”\nof a vector field is mathematically. Helmholtz’s theorem\nguarantees that a spatially localized vector field can be\nuniquely decomposed into conservative (curl-free) part\nand solenoidal (divergence-free) part. Note that the sec-\nondterminEq. (42)isexactlythe sameastheformulaof\n(negative of) conservative part of the Helmholtz decom-\nposition. Note also that the resulting total current [Eq.\n(44)] is divergence-free. Therefore, the charge diffusion\ncurrent and total charge current are respectively given\nFIG. 2: (color online) Results of the micromagnetic simu-\nlation for vortex resonant oscillation in 2D thin film. Here,\nthe vortex core size is about 5nm, the resonant frequency is\n605.5MHz, and P=Pn= 0.7. The size of arrows are log-\nscaled (size ∝ln(1+norm)). (a) Magnetization profile at the\ntime when all calculation was performed. The blue arrow\ndenotes the direction of vortex core motion. (b) SMF. (c)\nAccumulated charge density −e∆ne(divided by στsf/λ2\nsf).\nThe maximum value (white) is 1 .39×10−5V and the mini-\nmum value (black) is −1.62×10−5V. (d)∇ ·Es. The max-\nimum value (white) is 8 .43×1010V/m2and the minimum\nvalue (black) is −7.42×1010V/m2. (e) Charge diffusion cur-\nrent, which is the conservative part of SMF. (f) Total charge\ncurrent, which is the solenoidal part of SMF.\nby conservative part and solenoidal part of Helmholtz\ndecomposition. One shall see in Sec. V that this claim is\ngenerally valid for arbitrary boundary conditions.\nTo understand the maximal cancelation of the charge\ncurrent qualitatively, it would be very helpful to visualize\nthe Helmholtz decomposition of the charge current. We\nperformed a micromagnetic simulation for vortex reso-\nnant oscillation in 2D thin film. When the vortex DW\nwall [Fig. 2(a)] moves along the blue arrow, SMF is gen-\nerated as shown in Fig. 2(b). The spatial dependence of\nSMF induces charge accumulation as shown in Fig. 2(c).\nDue to the charge accumulation, charge diffusion current\nis generated as shown in Fig. 2(e). Figure 2(e) can be7\nqualitatively understood by Fig. 2(d). Recall that the\ndiffusion current is given by Coulomb field generated by\ncharge density −∇ ·Es. Thus, dipole-like nature of the\nchargedensity −∇·Es[Fig. 2(d)] implies dipole-like field\n[Fig. 2(e)]. Summing up Figs. 2(b) and 2(e), one obtains\ntotal current as Fig. 2(f). Note that the total current is\ndefinitely nonconservative (has finite curl). It is interest-\ningthatFig. 2(f)issimilartomagneticfieldgeneratedby\ntwo separate conducting wires. This infers the solenoidal\nnature of the total charge current. It is very interesting\nthat total chargecurrentbehavesasmagneticfield rather\nthan electric field.\nC. Magnetization dynamics and nonlocal damping\ntensor\nIt is also important to see how the magnetization dy-\nnamics is changed by our result. By analogue of Ref.\n[1], it is obvious that the modified LLG equation is de-\nscribed by Eq. (1), in which Eq. (43) are added to js\nterms, rather than Eq. (34). However, there exist two\nnontrivial features in the modified LLG equation.\nThe first one is spatial dependence of γandα. It is\nimportant to notice that γandαin Eq. (1) are renor-\nmalized parameters [1],\nγ=γ0/parenleftbigg\n1+nsµB\nMs1\n1+β2/parenrightbigg−1\n, (45)\nα=γ0\nγ/parenleftbigg\nα0+βnsµB\nMs1\n1+β2/parenrightbigg\n,(46)\nwhereγ0andα0are original parameters of the system.\nThere has not been any problem of this renormalization\nwithout spin accumulation, but nsis no longer constant\nin the presence of spin accumulation. Hence, γandα\ncannot be a simple constant in principle. At this stage, it\nis more convenient to write down LLG equation without\nparameter renormalization,\n/parenleftbigg\n1+1\n1+β2nsµB\nMs/parenrightbigg∂M\n∂t=−γ0M×Heff\n+/parenleftbigg\n1+β\nα01\n1+β2nsµB\nMs/parenrightbiggα0\nMsM×∂M\n∂t\n+µB\neMs(js·∇)M−βµB\neM2sM×(js·∇)M.(47)\nIn fact, the effect of nsvariation is small. Note that\nnsµB/Ms= (n↑\n0−n↓\n0)µB/Ms×∆ns/(n↑\n0−n↓\n0). Here, the\nfirst factor is of the order of 10−2and the second factor\nis first order in SMF. Then, the effect should be very\nsmall compared to ordinary first order effect of SMF. In\naddition, one shall see in Sec. IV, that symmetry can\nreduce the effect of ∆ nsin collective coordinate level. In\nthe case, the effect of ∆ nsvanishes through odd function\nintegration.\nThe next one is effective damping constant. Without\napplied electric field, the modified LLG equation is ob-\ntained by taking Gilbert damping as a damping tensorlike Eq. (4). Then, it is interesting to see how the damp-\ning tensor is generalized by spin diffusion effect. In the\npresence of spin diffusion, local damping tensor Dijbe-\ncomes nonlocal damping tensor Dij(x,x′). By using Eq.\n(43), it turns out that the damping tensor in Eq. (4) is\nmodified as\nDij(x,x′) =αδijδ(x−x′)+(1−P2)η\nM4se−|x−x′|/λsf\n2λsf\n×(M×∂xM)i(M′×∂xM′)j,(48)\nfor 1D and\nDij(x,x′) =αδijδ(x−x′)+η\nσM4sSkl(x,x′)\n×(M×∂kM)i(M′×∂lM′)j,(49)\nSij(x,x′) =σδijδ(x−x′)+P2σ\n4π∂i∂j1\n|x−x′|\n+(1−P2)σ\n4π∂i∂je−|x−x′|/λsf\n|x−x′|.(50)\nfor 3D. Here, M′=M(x′,t) and·is redefined as inner\nproduct with respectto both coordinatebasisand spatial\nbasis,\n[D ·f(x)]i=/summationdisplay\nj/integraldisplay\nddx′Dij(x,x′)fj(x′),(51)\nfor a vector field f. As a passing remark, the tensor Dis\nindeed a damping tensor in the sense that it decreasesto-\ntal magnetic energy. It can be demonstrated by showing\nenergy dissipation is negative,\ndE\ndt=−/integraldisplay/parenleftbigg\nHeff·∂M\n∂t/parenrightbigg\nddx <0.(52)\nThe mathematical details related to Eqs. (48)-(52) are\nin Appendix C.\nIV. EXAMPLE : DOMAIN WALL MOTION\nIn this section, we apply our theory to 1D DW mo-\ntion. We find equationofmotion ofcollectivecoordinates\n(X(t),φ(t)) of tail-to-tail transverse DW. Here, X(t) is\nDW position and φ(t) is tilting angle. Mathematical de-\ntails of obtaining the collective coordinate equation is\ndescribed in Appendix D.\nWithout SMF, DW motion is described by [3, 23–28]\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ, (53)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−γKd\nMssin2φ,(54)\nwhereb0\nJ=PjµB/eMsfor applied current j,Kdrepre-\nsents dipole field integration, and λis DW width. In the8\npresence of SMF, Eqs. (53) and (54) are modified as [29]\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ−2βη\n3λ2∂φ\n∂t, (55)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−2η\n3λ2∂φ\n∂t−γKd\nMssin2φ.(56)\nOne can expect that the effect of ηwill be suppressed by\nspin diffusion, so a renormalized parameter ˜ ηwill replace\nη. Thus, the expected equations of motion are\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ−2β˜η\n3λ2∂φ\n∂t, (57)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−2˜η\n3λ2∂φ\n∂t−γKd\nMssin2φ.(58)\nAs described in Appendix D, it turns out that Eqs. (57)\nand (58) are indeed valid, and the renormalized SMF\nparameter is given by\n˜η= (1−P2)F(ζ)η, (59)\nF(ζ) =−3\n2ζ−3\n2ζ2+3\n4ζ3/parenleftbiggΓ′(z)\nΓ(z)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=ζ\n2,(60)\nwhereζ=λ/λsfand Γ(z) is the Gamma function.\nThe renormalization is largely dependent on relative\nmagnitude of DW width and spin diffusion length. The\nasymptotic behavior of the function Fis given by\nF(ζ) =/braceleftigg\n3\n2ζ−3\n2ζ2+π2ζ3\n8+O(ζ4)ζ≪1,\n1−4\n5ζ2+O(ζ−4)ζ≫1.(61)\nForλ≪λsf, Eqs. (55) and (56) are reproduced except\n(1−P2) factor which should exist as already discussed.\nForλ≫λsf, the effect is highly suppressed by spin dif-\nfusion, so the overall effect of SMF goes as1\nλλsfrather\nthan1\nλ2.\nV. FURTHER GENERALIZATIONS\nA. Extended electric field - general boundary\ncondition\nIn the case that the magnetization dynamics is gen-\nerated by applied electric field, the electric field is no\nlonger localized. Hence, it is necessary to generalize our\ntheory to non-localized electric field ; Esdoes not vanish\nat|x| → ∞. In this case, ˜Edoes not give well-defined\nFouriertransform,butincludesdeltafunctionparts. Fur-\nthermore, in order to obtain Eq. (44), one obtains ad-\nditional boundary terms when integrating Eq. (42) by\nparts. Hence, the charge current may not be canceled by\nthe diffusion current even if the electric field is conserva-\ntive.\nThe problem can be treated by Green’s function\nmethod with given boundary condition, as described inthe last part of this section. However, it is hard to catch\nthe physical meaning, so we presentmore intuitive analy-\nsis for this case. It is very convenient to use the linearity\nof our theory. Suppose E↑↓can be decomposed into two\ncomponents E↑↓=E↑↓\n1+E↑↓\n2. Then, since Eqs. (13) and\n(17) are linear, the current j↑↓can also be decomposed\ninto two components j↑↓=j↑↓\n1+j↑↓\n2, where j↑↓\niis the\ncurrent generated by Ei.\nTo take advantage of this linearity, we decompose E\ninto three components as follows; any vector field E\ncan be uniquely decomposed into irrotational part E1,\nsolenoidal part E2, and boundary part E3,\nE↑↓=E↑↓\n1+E↑↓\n2+E↑↓\n3, (62)\n∇·E↑↓\n1∝negationslash= 0,∇×E↑↓\n1= 0,n·E↑↓\n1|x∈∂= 0,(63)\n∇·E↑↓\n2= 0,∇×E↑↓\n2∝negationslash= 0,n·E↑↓\n2|x∈∂= 0,(64)\n∇·E↑↓\n3= 0,∇×E↑↓\n3= 0,n·E↑↓\n3|x∈∂∝negationslash= 0.(65)\nHere,∂means boundary and ndenotes a normal unit\nvector perpendicular to the boundary. From given E↑↓,\none can obtain E↑↓\n3by solving Laplace equation with\nNeumann boundary condition and E↑↓\n1,2by Helmholtz’s\ntheorem. As previously mentioned, our result [Eq. (44)]\nindicates that the conservative source part E↑↓\n1cannot\ncontribute to the charge current while the nonconserva-\ntive solenoidal part E↑↓\n2can give rise to a nonvanishing\ncontribution σ↑E↑\n2+σ↓E↓\n2. [Eq. (42) or Eq. (77) for\ngeneral boundary condition]\nIn the presence of nonvanishing boundary condition, it\nis important to investigate the effect of E↑↓\n3to the spin\naccumulation. Fortunately, the source term of Eq. (13)\ndepends on the divergence of E↑↓\nsonly. Hence, E↑↓\n3can-\nnotcontribute tothe spin accumulationsince ∇·E↑↓\n3= 0.\nTherefore, E↑↓\n3canonlycontributetothecurrentsviathe\nfirst term in Eq. (17). Consequently, the expressions of\ncharge and spin current should be σ↑E↑\n3±σ↓E↓\n3added to\nEqs. (42) and (43) (or Eqs. (38) and (39)).\nIt can be a good example that a constant spin inde-\npendent electric field Eapp=Eappˆxis applied in a 1D\nwire. In this case, E↑↓\n3=Eappˆx. Hence, the charge and\nspin current should be σ↑Eapp±σ↓Eappadded to Eqs.\n(38) and (39), so\nje=σEapp, (66)\njs=PσEapp+(1−P2)/integraldisplay\ndx′e−|x−x′|/λsf\n2λsfσEs(x′,t).(67)\nIn this case, the charge current is not canceled by diffu-\nsion current, but only charge current generated by SMF\n(which is spatially localized) is canceled.\nB. Charge neutrality and other spin independent\npotentials\nThere is another important physical consequence of\nspin accumulation we have ignored. This is Coulomb9\npotential of accumulated electrons. Since Coulomb po-\ntential is in general strong, electron tends to make local\ncharge neutral. Hence, giving a charge neutrality con-\nstraint,\n∆n↑+∆n↓= 0 (68)\nis a good approximation. To do this, it is convenient\nto introduce charge neutrality potential Vcand replace\nE↑↓\ns→E↑↓\ns− ∇Vcas suggested in Ref. [10]. Then, the\nsolution of the spin drift-diffusion equation [Eq. (28)] is\nmodified by\n/parenleftbigg\n∆˜n↑\n∆˜n↓/parenrightbigg\n=τsf\neΩ−1/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nik·/parenleftbigg˜Es−ik˜Vc\n−˜Es−ik˜Vc/parenrightbigg\n,\n(69)\nwhereVcis found self-consistently to satisfy Eq. (68).\nAfter straightforward algebra, one obtains charge neu-\ntrality potential in terms of Es,\n˜Vc(k,t) =−Pik·˜Es\nk2−(Pn−P)ik·˜Es\nk2+λ−2\nsf,(70)\nVc(x,t) =−P\n4π/integraldisplay\nd3x′1\n|x−x′|∇·Es\n−Pn−P\n4π/integraldisplaye−|x−x′|/λsf\n|x−x′|∇·Es.(71)\nThe effect of this potential is twofold. Firstly, Vcgives\nadditional electrical current −σ↑↓∇Vc. Ink-space,\n∆˜j↑\nc(k,t) =−σ↑ikVc, (72)\n∆˜j↓\nc(k,t) =−σ↓ikVc. (73)\nSecondly, Vcaffects diffusion current through ∇n↑↓. Af-\nter some algebra, one obtains\n/parenleftbigg\n∆˜j↑\nD\n∆˜j↓\nD/parenrightbigg\n=ik/parenleftigg\nλ↑2\nsf0\n0λ↓2\nsf/parenrightigg\nΩ−1/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg/parenleftbigg\nk2Vc\nk2Vc/parenrightbigg\n=ikVc/parenleftbigg\nσ↑\nσ↓/parenrightbigg\n, (74)\nwhich exactly cancels Eqs. (72) and (73). Hence, there\nare no additional charge and spin currents attributed to\nVs.\nIt is very interesting that one does not need to assume\nanyspecific formof VctoreachEq. (74). The mathemat-\nical origin of the exact cancelation is that the additional\nforce is conservative (described by ik×(scalar)) and spin\nindependent. A spin independent potential generates ad-\nditional current, but spin accumulation adjusts fast to\nmake diffusion current cancel it. Of course, the modi-\nfied spin accumulation n↑↓affects LLG equation by Eq.\n(47), but weclaimedthatthisisignorable. Consequently,\nany spin independent potential cannot modify the main\nfeatures of our result. As another side remark, this can-\ncelation can be verified for exact Coulomb potential even\nwithout using the charge neutrality approximation Eqs.\n(68)-(71).C. Arbitrary boundary\nSince SMF is usually strongly localized, we have con-\nsidered an infinite boundary problem. However, for a\nfinite system, the expression of the charge and spin cur-\nrents should be slightly modified. Note that the key part\nof our theory is to find real space expression of Ω−1. (See\nEq. (28)). In real space, Ω is a differential operator,\nΩreal=\n−λ↑2\nsf∇2+n↓\n0\nn↑\n0+n↓\n0−n↑\n0\nn↑\n0+n↓\n0\n−n↓\n0\nn↑\n0+n↓\n0−λ↓2\nsf∇2+n↑\n0\nn↑\n0+n↓\n0\n.\n(75)\nHence, the problem is to find the inverse operator, i.e.,\nthe Green’s function corresponding Ω realat the given\ngeometry.\nIn order to find the Green’s function of Ω real, one\ncan get a hint from Eq. (29). Firstly, let GL(x,x′)\nandGH(x,x′) be respectively the Green’s function cor-\nresponding Laplacian ∇2and modified Helmholtz op-\nerator∇2−λ−2\nsffor the given geometry. In order to\nobtain the Green’s function, it is plausible to replace\n1/k2→ −GL(x,x′) and 1/(k2+λ−2\nsf)→ −GH(x,x′) in\nEq. (29). Then, one obtains\nGΩ(x,x′) =−1\nσ\nσ↑GL+σ↓GH\nλ↑2\nsfσ↑GL−σ↑GH\nλ↑2\nsf\nσ↓GL−σ↓GH\nλ↓2\nsfσ↓GL+σ↑GH\nλ↓2\nsf\n.(76)\nOnecanverifythis isindeedthe Green’sfunctionofΩ real\nby showing Ω realGΩ=δ(x−x′). Then, one straight-\nforwardly concludes that the generalized expressions of\ncharge and spin current [Eqs. (42)-(43)] are given by\nje=PσEs−Pσ∇/integraldisplay\nddx′GL(x,x′)∇′·Es,(77)\njs=Pje+(1−P2)σEs\n−(1−P2)σ∇/integraldisplay\nddx′GH(x,x′)∇′·Es.(78)\nForinfiniteboundarywithvanishingboundarycondition,\n4πGL=−1/|x−x′|and 4πGH=−e−|x−x′|/λsf/|x−x′|,\nso Eqs. (42)-(43) are reproduced.\nVI. CONCLUSION\nBy constructing spin drift-diffusion equation from the\nequation of motion of conduction electrons, we studied\nthe effect of SMF in the presence of spin accumulation,\nspin diffusion and spin flip scattering, whichwereignored\n[7] or considered only in 1D [19] in previous theories.\nIt turns out that, in realistic regime, the conservative\npart of charge current is canceled by diffusion current,\nand spin current becomes nonlocal. Consequently, the\nmagnetization dynamics is affected by nonlocal Gilbert\ndamping tensor, instead of the previously reported (lo-\ncal)Gilbert damping tensor. By calculatingspin-transfer10\ntorque generated by nonlocal spin current, we obtained\nthe explicit expressions of the nonlocal Gilbert damping\ntensor.\nDifferent from the previous work focusing on 1D [19],\nwe also obtained the results for 2D and 3D as well as 1D.\nIn a 1D system, the results of the previous theory are\nreproduced. It turnsout that Einstein’srelationprevents\nthe existence of charge current, but for parameter sets\nwhich do not satisfy the Einstein’s relation, the charge\ncurrent can be induced by the spin motive force. After\ngeneralizing the results to higher dimension, we find that\nthe nonconservative part of SMF plays an important role\nin charge and spin currents.\nAs an illustration of suppression of the effect of SMF,\nwe demonstrated equations of motion of collective coor-\ndinate of1Dcurrent-inducedDWmotion. We found that\nspin diffusion renormalizes SMF depending on the rela-\ntive magnitude of DW width and spin diffusion length.\nWe also investigated the system under spatially ex-\ntended (non-localized) electric field. In this case, it turns\nout that the spatially extended part of the electric field\ncan contribute to the charge current even though it is\nconservative. However, the existence of spatially ex-\ntended part of the electric field cannot alter the result\nthatthechargecurrent generated by (localized) SMF can-\nnot include conservative part.\nOur result is solid in the sense that our principal re-\nsults are not changed by any spin independent potential.\nA spin independent potential modifies current via addi-\ntional electric field, but spin density rapidly adjusts to\nmake diffusion current cancel it. Consequently, a spin\nindependent potential can modify spin density, but not\ncurrent.\nAcknowledgments\nThis work is financially supported by the NRF (2009-\n0084542, 2010-0014109, 2010-0023798), KRF (KRF-\n2009-013-C00019) and BK21. KWK acknowledges the\nfinancial support by TJ Park.\nAppendix A: Absence of Time Variation of\nEquilibrium Number Density\nIn this section, we show that e−Ωt/τsf˜N(k,0) does not\nhave time dependence, where Ω is given by Eq. (25) and\n˜N(k,0) = (2π)d/2δ(k)(n↑\n0n↓\n0)T. It suffices to calculate\ne−Ωt/τsfatk= 0 because of δ(k) factor. It is easy to\nshow that Ω is idempotent for k= 0. In other words,\nΩn|k=0= Ω|k=0. Then, for k= 0,\ne−Ωt/τsf=∞/summationdisplay\nn=0Ωn\nn!/parenleftbigg\n−t\nτsf/parenrightbiggn\n=I+∞/summationdisplay\nn=1Ω\nn!/parenleftbigg\n−t\nτsf/parenrightbiggn\n,\n=I+Ω(e−t/τsf−1). (A1)whereIis the identity matrix. Since\nΩ˜N(k,0) = (2π)d/2δ(k)Ω/parenleftbigg\nn↑\n0\nn↓\n0/parenrightbigg\n= 0 (A2)\nthe second term of Eq. (A1) vanishes after applied to\n˜N(k,0). Finally, one obtains e−Ωt/τsf˜N(k,0) =˜N(k,0).\nNote that the result is obtained without the approxi-\nmation that τsfis small.\nAppendix B: Derivation of Eq. (27)\nThe idea is based on the delta sequence\nlim\nn→∞ne−ntΘ(t) =δ(t), (B1)\nfor natural number n. This can be generalized as\nlim\nu→∞ne−uztΘ(t) =δ(t), (B2)\nwherezis a complex number satisfying ℜ[z]>0. This\ngeneralization is obvious in that uze−uztis localized at\nt= 0 and/integraltext∞\n−∞uze−uztΘ(t)dt= 1.\nWe now generalize this relation to matrices. For a\ndiagonalizable matrix Mwith eigenvalues λisatisfying\nℜ[λi]>0, we claim that\nlim\nu→∞uMe−uMtΘ(t) =δ(t)I. (B3)\nThe proof is simple. Since Mis diagonalizable, one can\nwrite\nuM=Q\nuλ10···\n0uλ2···\n.........\nQ−1,(B4)\nfor some Q. Then\nuMe−uMxΘ(x)\n=Q\nuλ1e−uλ1Θ(x) 0 ···\n0 uλ2e−uλ2Θ(x)···\n.........\nQ−1\n→Q\nδ(x) 0···\n0δ(x)···\n.........\nQ−1=δ(x)I, (B5)\nasu→ ∞.\nBy taking M= Ω and u= 1/τsf,\nlim\nτsf→+0Ω\nτsfe−Ωt/τsfΘ(t) =δ(t)I, (B6)\nso, for small τsf,\nΩ\nτsfe−Ωt/τsfΘ(t)≈δ(t)I, (B7)11\nwhich is exactly Eq. (27).\nAs a passing remark, one should check that the real\nparts of eigenvalues of Ω are really positive. It is guar-\nanteed by Tr Ω >0 and detΩ >0 for any nonzero k.\nHowever, the approximation Eq. (27) makes singularity\natk= 0. Fortunately, this singularity is removed by ik\nfactor in Eq. (26).\nAppendix C: Nonlocal damping tensor and energy\ndissipation\n1. 1D damping tensor\nStarting from\njs= (1−P2)/integraldisplay\ndx′e−|x−x′|/λsf\n2λsfσEs(x′,t),(C1)\nEs=/planckover2pi1\n2eM3s∂tM·(∂xM×M), (C2)\nspin-transfer torque driven by SMF is given by\nµB\neMsjs∂xM= (1−P2)η\nM3s∂xM\n×/integraldisplay\ndx′e−|x−x′|/λsf\n2λsf∂tM′·(∂xM′×M′)\n=M\nMs×/integraldisplay\ndx′/bracketleftigg\n(1−P2)η\nM4se−|x−x′|/λsf\n2λsf\n(M×∂xM)(M′×∂xM′)]·∂tM′.(C3)\nThen, comparing with Eqs. (4) and (51),\nDij(x,x′) =αδijδ(x−x′)+(1−P2)η\nM4se−|x−x′|/λsf\n2λsf\n×(M×∂xM)i(M′×∂xM′)j.(C4)\nIt is interesting to note that\nlim\nλsf→0Dij(x−x′) =δ(x−x′)/bracketleftbigg\nαδij+(1−P2)η\nM4s\n×(M×∂xM)i(M×∂xM)j].(C5)\nThis isexactlythe previousresultEq. (5) except (1 −P2)\nfactor, whichshouldexistregardlessofdiffusionstrength.\nNow, the remaining step is to calculate energy dissipa-\ntion. Energy dissipation is calculated by the integration\nenergy density dissipation −Heff·∂tM. From now on,\nthe subscription effis dropped until this section ends.\nEnergy dissipation by a nonlocal damping tensor Disgiven by\ndE\ndt=−/integraldisplay\ndxH·/bracketleftbiggM\nMs×D ·∂tM/bracketrightbigg\n=1\nMs/integraldisplay\ndx(M×H)·D ·∂tM\n≈ −γ\nMs/integraldisplay\ndx(M×H)·D·(M×H)\n=−γ\nMs/integraldisplay/integraldisplay\ndxdx′(M×H)iDij(x,x′)(M′×H′)j,(C6)\nup to first order. Here, H′=H(x′,t). The first term\nin Eq. (C4) gives definitely negative dE/dt. The second\nterm, which comes from SMF, gives\ndESMF\ndt=−(1−P2)ηγ\nMs/integraldisplay/integraldisplay\ndxdx′(H·∂xM)\n×e−|x−x′|/λsf\n2λsf(H′·∂xM′). (C7)\nIn order to show dE/dt < 0, it is sufficient to show that/integraltext/integraltext\ndxdx′f(x)e−a|x−x′|f(x′) is positive for real function\nf. This is verified by Parseval’s relation and convolution\ntheorem of Fourier transform.\n/integraldisplay/integraldisplay\ndxdx′f(x)e−a|x−x′|f(x′)\n=/integraldisplay\ndxf(x)∗/bracketleftbigg/integraldisplay\ndx′e−a|x−x′|f(x′)/bracketrightbigg\n=/integraldisplay\ndkF[f(x)](k)∗F/bracketleftbigg/integraldisplay\ndx′e−a|x−x′|f(x′)/bracketrightbigg\n(k)\n=√\n2π/integraldisplay\ndkF[f(x)](k)∗F[e−a|x|](k)F[f(x)](k)\n= 2a/integraldisplay\ndk|F[f(x)](k)|2\nk2+a2>0. (C8)\nThis implies dE/dt < 0.\n2. 3D damping tensor\nFrom now on, Einstein’s convention is used. Compo-\nnentwise expressions of spin current and SMF for a 3D\nsystem are\njs,i=σEs,i+P2σ\n4π∂i/integraldisplay\nd3x′∂′\njEs,j\n|x−x′|\n+(1−P2)σ\n4π∂i/integraldisplay\nd3x′e−|x−x′|/λsf\n|x−x′|∂′\njEs,j,(C9)\nEs,i=/planckover2pi1\n2eM3s∂tM·(∂iM×M). (C10)12\nBy integrating by parts in order to remove derivatives in\nfront ofEs,j,\njs,i=σEs,i−P2σ\n4π∂i/integraldisplay\nd3x′Es,j(x′)∂′\nj1\n|x−x′|\n−(1−P2)σ\n4π∂i/integraldisplay\nd3x′Es,j(x′)∂′\nje−|x−x′|/λsf\n|x−x′|,\n=σEs,i+P2σ\n4π/integraldisplay\nd3x′Es,j(x′)∂′\ni∂′\nj1\n|x−x′|\n+(1−P2)σ\n4π/integraldisplay\nd3x′Es,j(x′)∂′\ni∂′\nje−|x−x′|/λsf\n|x−x′|.(C11)\nHere, it is convenient to introduce nonlocal conductivity\ntensorS,\njs,i=/integraldisplay\nd3x′Sij(x,x′)Es,j(x′),(C12)\nSij(x,x′) =σδijδ(x−x′)+P2σ\n4π∂′\ni∂′\nj1\n|x−x′|\n+(1−P2)σ\n4π∂′\ni∂′\nje−|x−x′|/λsf\n|x−x′|.(C13)\nNow, spin-transfer torque driven by SMF is given by\nµB\neMs(js·∇)M\n=µB\neMs∂iM/integraldisplay\nd3x′·Sij(x,x′)Es,j(x′)\n=M\nMs×/bracketleftbiggη\nσM4s(M×∂iM)\n×/integraldisplay\nd3x′·Sij(x,x′)(M′×∂jM′)·∂tM′/bracketrightbigg\n.(C14)\nHence, the nonlocal damping tensor for 3D is given by\nDij(x,x′) =αδijδ(x−x′)+η\nσM4sSkl(x,x′)\n×(M×∂kM)i(M′×∂lM′)j.(C15)\nNow, we calculate energy dissipation. This is an ana-\nlogue of the previous section. After some algebra,\ndESMF\ndt=−γη\nσMs/integraldisplay/integraldisplay\nd3xd3x′Skl(x,x′)\n×(H·∂kM)(H′·∂lM′).(C16)\nIn order to show dESMF/dt <0, one should verify\n/integraldisplay/integraldisplay\nd3xd3x′(H·∂iM)Sij(x,x′)(H′·∂jM′)>0.(C17)\nNote that Sij(x,x′) is a function of ( x−x′). It is con-\nvenient to write Sij=Sij(x−x′) at this stage. Similar\nto the previous subsection, by using Parseval’s theorem\nand convolution theorem of Fourier transform,\n/integraldisplay/integraldisplay\nd3xd3x′(H·∂iM)Sij(x−x′)(H′·∂jM′)\n=/integraldisplay\nd3kF[H·∂iM](k)∗˜Sij(k)F[H·∂jM](k).(C18)Here,˜Sij(k) is given by\n˜Sij(k) =σ\n(2π)3/2/bracketleftigg\nδij−P2kikj\nk2−(1−P2)kikj\nk2+λ−2\nsf/bracketrightigg\n.\n(C19)\nNote that the integrand in Eq. (C18) is an expecta-\ntion value of 3 ×3 matrix ˜Sij(k) with respect to vector\nF[H·∂iM](k). Since ˜Sij(k) is a Hermitian matrix, the\nintegrand is always positive if the eigenvalues of ˜Sij(k)\nare positive. Recall that the eigenvalues of matrix kikj\nis given by (0 ,0,k2). The corresponding eigenvectors are\ndefinitely the eigenvectors of ˜Sij(k), so the eigenvalues\nof˜Sij(k) is given by/parenleftbigg\nσ\n(2π)3/2,σ\n(2π)3/2,σ\n(2π)3/2λ−2\nsf(1−P2)\nk2+λ−2\nsf/parenrightbigg\n,\nwhich are all positive. This proves dE/dt < 0.\nAppendix D: 1D DW motion in the presence of spin\ndiffusion\n1. Collective coordinate equation of 1D DW for\nspace-time dependent spin current and spin density\nThe starting equation is 1D version of Eq. (47).\n/parenleftbigg\n1+1\n1+β2nsµB\nMs/parenrightbigg∂M\n∂t=−γ0M×Heff\n+/parenleftbigg\n1+β\nα01\n1+β2nsµB\nMs/parenrightbiggα0\nMsM×∂M\n∂t\n+bJ∂M\n∂x−βbJM\nMs×∂M\n∂x, (D1)\nwherebJ=µBjs/eMs. The main difference from the\nconventional approach is that bJandnsare space-time\ndependent. The equation is rewritten as\nγ0M×Htot= 0, (D2)\nwhere\nγ0Htot=−/parenleftbigg\n1+1\n1+β2nsµB\nMs/parenrightbiggM\nM2s×∂M\n∂t+γ0Heff\n−/parenleftbigg\nα0+β\n1+β2nsµB\nMs/parenrightbigg1\nMs∂M\n∂t\n+bJM\nM2s×∂M\n∂x+βbJ\nMs∂M\n∂x. (D3)\nWe set the effective magnetic field as Heff=2A\nM2s∂2M\n∂x2+\nHKMx\nMsˆx+Hd, whereAandKdrepresents exchange cou-\npling and anisotropy, and Hdis dipole field. In addition,\nthe magnetization profile is set to be tail-to-tail trans-13\nverse wall,\nM=Ms(cosθ,sinθcosφ,sinθsinφ),(D4)\nsinθ= sech/parenleftbiggx−X(t)\nλ/parenrightbigg\n, (D5)\ncosθ= tanh/parenleftbiggx−X(t)\nλ/parenrightbigg\n, (D6)\nφ=φ(t), (D7)\nwhereX(t) is DW position and φ(t) is tilting angle.\nNote that Eq. (D2) implies Htot=aMfor some a.\nThen, one can define force density f=−Htot·∂M\n∂Xand\ntorque density τ=−Htot·∂M\n∂φ, which are identically\nzero. Then, collective coordinate equation is given by\ncalculating total force and total torque,\nF≡/integraldisplay\nfdx= 0, (D8)\nT≡/integraldisplay\nτdx= 0. (D9)\nEach equation implies respectively,\n−∂φ\n∂t+α\nλ∂X\n∂t+1\n1+β2/parenleftbigg\n−∂φ\n∂t+β\nλ∂X\n∂t/parenrightbigg\n×/integraldisplay∆nsµB\nMssech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ\n=−βb0\nJ\nλ−β\nλ/integraldisplay\nbs\nJsech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ,(D10)\n1\nλ∂X\n∂t+α∂φ\n∂t+1\n1+β2/parenleftbigg1\nλ∂X\n∂t+β∂φ\n∂t/parenrightbigg\n×/integraldisplay∆nsµB\nMssech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ\n=−b0\nJ\nλ−/integraldisplaybs\nJ\nλsech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ−γKd\nMssin2φ.(D11)\nwhereb0\nJis space-time independent part of bJwhich\ncomes from applied spin current, bs\nJ=bJ−b0\nJ,Kdcor-\nresponds integration of dipole field. Here, αandγare\nrenormalized parameter by the same way with n↑↓\n0. One\ncan check that Eqs. (D10) and (D11) reproduces Eqs.\n(53) and (54) if ∆ ns= 0 and bs\nJ= 0.\n2. 1D DW motion in the presence of SMF and spin\ndiffusion\nNow, we apply\nbs\nj=µBσ\neMs1−P2\n2λsf/integraldisplay\ndx′e−|x−x′|/λsfEs(x′,t),(D12)\nEs=/planckover2pi1\n2eλsech2/parenleftbiggx−X\nλ/parenrightbigg∂φ\n∂t, (D13)\n∆˜ns=στsf\neλ2\nsf1−P2\n1−P2nk2+PnPλ−2\nsf\nk2+λ−2\nsfik˜Es\nk2.(D14)to Eqs. (D10) and (D11).\nFirst, we calculate the integral/integraltext∆nssech2/parenleftbigx−X\nλ/parenrightbig\ndx\nwhichcorrespondstotheeffectofspindensity. Notethat\n∆nsis an odd function of x−X. Hence, the integrand\nis an odd function so the integral vanishes.\nThe next step is to calculate/integraltextbs\nJ\nλsech2/parenleftbigx−X\nλ/parenrightbigdx\n2λ. Af-\nter some algebra,\n/integraldisplaybs\nJ\nλsech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ\n=(1−P2)ηζ\n4λ2∂φ\n∂t/integraldisplay/integraldisplay\ne−ζ|u−u′|sech2usech2u′dudu′\n=(1−P2)ηζ\n4λ2∂φ\n∂t/integraldisplayπζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk\n=(1−P2)ηζ\n4λ2∂φ\n∂t/integraldisplay∞\n02πζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk(D15)\nwhereζ=λ/λsf,u= (x−X)/λ, andu′= (x′−X)/λ.\nParseval’s relation and convolution theorem of Fourier\ntransform is used at the third step. By using the identity\ncsch2/parenleftbiggkπ\n2/parenrightbigg\n= 4∞/summationdisplay\nn=1ne−nπk,(D16)\nthe integral can be expressed by Laplace transform L.\n/integraldisplay∞\n02πζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk=∞/summationdisplay\nn=1L/bracketleftbigg8πζnk2\nk2+ζ2/bracketrightbigg\n(nπ)\n=∞/summationdisplay\nn=1/integraldisplay∞\n016\nnπsinnπζu\n(u+1)3du.(D17)\nRecall the Fourier series of a sawtooth function\n∞/summationdisplay\nn=116\nnπsinnπx= 8(2n+1−x),(D18)\nfor 2n < x < 2n+2. Then the integral is given by\n∞/summationdisplay\nn=1/integraldisplay∞\n016\nnπsinnπζu\n(u+1)3du\n=∞/summationdisplay\nn=0/integraldisplay2n+2\nζ\n2n\nζ8(2n+1−uζ)\n(1+u)3du\n=−4−4ζ+2ζ2∞/summationdisplay\nn=01\n(n+ζ/2)2.(D19)\nBy using the relation\ndn\ndznΓ′(z)\nΓ(z)= (−1)n+1n!∞/summationdisplay\nk=11\n(z+k)n+1,(D20)14\none obtains the integral in closed form as\n8\n3F(ζ)\nζ≡/integraldisplay∞\n02πζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk\n=−4−4ζ+2ζ2/parenleftbiggΓ′(z)\nΓ(z)/parenrightbigg′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=ζ\n2\n=−4−4ζ+2ζ2Γ′′/parenleftig\nζ\n2/parenrightig\nΓ/parenleftig\nζ\n2/parenrightig\n−Γ′/parenleftig\nζ\n2/parenrightig2\n/parenleftig\nζ\n2/parenrightig2.(D21)\nThe prefactor 8 /3ζis to make lim λsf→0F(ζ) = 1. There-\nfore, we finally obtain\n/integraldisplaybs\nJ\nλsech2/parenleftbiggx−X\n2λ/parenrightbiggdx\n2λ= (1−P2)2η\n3λ2F(ζ)∂φ\n∂t,\n(D22)and, consequently,\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ−2β˜η\n3λ2∂φ\n∂t, (D23)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−2˜η\n3λ2∂φ\n∂t−γKd\nMssin2φ,(D24)\nwhereηis renormalized parameter by spin diffusion de-\nfined as ˜η= (1−P2)F(ζ)η.\n[1] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[2] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eu-\nrophys. Lett. 69, 990 (2005).\n[3] G. Tatara, T. Takayama, H. Kohno, J. Shibata, Y.\nNakatani, and H. Fukuyama, J. Phys. Soc. Jpn. 75,\n064708 (2006).\n[4] M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zang-\nwill, Phys. Rev. B 75, 214423 (2007).\n[5] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[6] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[7] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[9] W. M. Saslow, Phys. Rev. B 76, 184434 (2007).\n[10] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[11] J. Shibata and H. Kohno, Phys. Rev. Lett. 102, 086603\n(2009).\n[12] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79,\n014402 (2009).\n[13] R. A. Duine, Phys. Rev. B 79, 014407 (2009).\n[14] J. Ohe, S. E. Barnes, H.-W. Lee, and S. Maekawa, Appl.\nPhys. Lett. 95, 123110 (2009).\n[15] J. Ohe and S. Maekawa, J. Appl. Phys 105, 07C706\n(2009).\n[16] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q.Niu, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102,\n067201 (2009).\n[17] P. N. Hai, S. Ohya, M. Tanaka, S. E. Barnes, and S.\nMaekawa, Nature 458, 489 (2009).\n[18] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 81,\n060404(R) (2010).\n[19] S. S.-L. Zhang and S. Zhang, Phys. Rev. B 82, 184423\n(2010).\n[20] K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee (un-\npublished).\n[21] G. E. Volovik, J. Phys. C 20, L83 (1987).\n[22] E. Ya. Sherman, A. Najmaie, H. M. van Driel, A. L.\nSmirl, and J. E. Sipe, e-print arXiv:cond-mat/0606725.\n[23] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[24] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[25] A. P. Malozemoff and J. C. Slonczewski, Magnetic Do-\nmains Walls in Bubble Materials (Academic, New York,\n1979).\n[26] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[27] S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W.\nLee, Appl. Phys. Lett. 92, 202508 (2008)\n[28] J. Ryu and H.-W. Lee, J. Appl. Phys. 105, 093929\n(2009).\n[29] S.-I. Kim, J.-H. Moon, W. Kim, and K.-J. Lee, Curr.\nAppl. Phys. 11, 61 (2011)."    },    {        "title": "1206.4819v2.Fast_domain_wall_propagation_in_uniaxial_nanowires_with_transverse_fields.pdf",        "content": "arXiv:1206.4819v2  [cond-mat.mtrl-sci]  18 Aug 2013Fast domain wall propagation in uniaxial nanowires with tra nsverse fields\nArseni Goussev1,2, Ross G. Lund3, JM Robbins3, Valeriy Slastikov3, Charles Sonnenberg3\n1Department of Mathematics and Information Sciences,\nNorthumbria University, Newcastle Upon Tyne, NE1 8ST, UK\n2Max Planck Institute for the Physics of Complex Systems,\nN¨ othnitzer Straße 38, D-01187 Dresden, Germany\n3School of Mathematics, University of Bristol, University W alk, Bristol BS8 1TW, United Kingdom\n(Dated: May 14, 2018)\nUnder a magnetic field along its axis, domain wall motion in a u niaxial nanowire is much slower\nthan in the fully anisotropic case, typically by several ord ers of magnitude (the square of the di-\nmensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse\nto the wire, this behaviour is dramatically reversed; up to a critical field strength, analogous to the\nWalker breakdown field, domain walls in a uniaxial wire propa gate faster than in a fully anisotropic\nwire (without transverse field). Beyond this critical field s trength, precessional motion sets in, and\nthe mean velocity decreases. Our results are based on leadin g-order analytic calculations of the\nvelocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation.\nPACS numbers: 75.75.-c, 75.78.Fg\nIntroduction\nThe dynamics of magnetic domain walls in ferromag-\nnetic nanowiresunder external magnetic fields [1–12] and\nspin-polarised currents [12–20] is a central problem in\nmicromagnetics and spintronics, both as a basic physical\nphenomenonaswellasacornerstoneofmagneticmemory\nand logic technology [3, 16–18]. From the point of view\nof applications, it is desirable to maximise the domain\nwall velocity in order to optimise switching and response\ntimes.\nPartly because of fabrication techniques, attention has\nbeen focused on nanowires with large cross-sectional as-\npect ratio, typically of rectangular cross-section. In this\ncase, even if the bulk material is isotropic (e.g., permal-\nloy), the domain geometry induces a fully anisotropic\nmagnetic permeability tensor, with easy axis along the\nwire and hard axis along its shortest dimension [21, 22].\nNanowires with uniaxial permeability, characteristic of\nmoresymmetricalcross-sectionalgeometries(e.g., square\nor circular), have been less studied [23–25]. Here we in-\nvestigate domain wall (DW) motion in uniaxial wires in\nthe presence of transverse fields. We show that the DW\nvelocityinuniaxialwiresdependsstronglyonthelongitu-\ndinal applied field H1, increasing with H1up to a certain\ncritical field and thereafter falling off as precessional mo-\ntion sets in. We employ a systematic asymptotic expan-\nsion scheme, which differs from alternative approaches\nbased on approximate dynamics for the DW centre and\norientation; a detailed account of this scheme, also in-\ncluding anisotropy and current-induced torques, will be\ngiven separately [29] .\nWe employ a continuum description of the magnetisa-\ntion. For a thin nanowire, this is provided by the one-\ndimensionalLandau-Lifshitz-Gilbert(LLG)equation[22,26–28], which we write in the non-dimensionalised form\n˙M=γM×H−αM×(M×H). (1)\nHereM(x,t) is a unit-vector field specifying the orien-\ntation of the magnetisation, which we shall also write\nin polar form M= (cosΘ ,sinΘcosΦ ,sinΘsinΦ). The\neffective magnetic field, H(m), is given by\nH=Am′′+K1m1ˆx−K2m2ˆy+Ha.(2)\nHereAis the exchange constant, K1is the easy-axis\nanisotropy, K2>0 is the hard-axis anisotropy, Hais\nthe applied magnetic field (taken to be constant), γis\nthe gyromagnetic ration, and αis the Gilbert damping\nparameter. For convenience we choose units for length,\ntime and energy so that A=K1=γ= 1.Domains cor-\nrespond to locally uniform configurations in which Mis\naligned along one of the local minima, denoted m+and\nm−, of the potential energy\nU(m) =−1\n2(m2\n1−K2m2\n2)−m·Ha.(3)\nTwo distinct domains separated by a DW are described\nby the boundary conditions M(±∞,t) =m±.\nFor purely longitudinal fields Ha=H1ˆxand forH1\nbelowtheWalkerbreakdownfield HW=αK2/2,theDW\npropagates as a travelling wave [1], the so-called Walker\nsolution Θ( x,t) =θW(x−VWt), Φ(x,t) =φW, whereθW\nandφWare given by\nθW(ξ) = 2tan−1(e−ξ/γ),sin2φW=H1/HW.(4)\nThe width of the DW, γ, is given by γ= (1 +\nK2cos2φW)−1/2, and the velocity is given by\nVW=−γ(α+1/α)H1. (5)\nForH1> HW, the DW undergoes non-uniform preces-\nsion and translation, with mean velocity decreasing with2\nH1[1, 5, 6, 9]. The effects of additional transverse fields\nhave been examined recently [7, 11].\nIf the cross-sectional geometry is sufficiently symmet-\nrical (e.g., square or circular), the permeability tensor\nbecomes uniaxial, so that K2= 0 [21, 22]. The dynamics\nin this case is strikingly different. The LLG equation has\nan exact solution, Θ( x,t) =θ0(x−VPt), Φ(x,t) =−H1t,\nin which the DW propagates with velocity\nVP=−αH1 (6)\nand precesses about the easy axis with angular velocity\n−H1[23, 24]. The precessing solution persists for all H1\n– there is no breakdown field – but becomes unstable for\nH1/greaterorsimilar1/2 [25].\nForH1< HW, the ratio VW/VPisgiven by γ(α−2+1).\nFor typical values of α(0.01 – 0.1), the uniaxial velocity\nVPis less than the fully anisotropic velocity VWby sev-\neral orders of magnitude. As we show below, applying a\ntransverse field H2>0 to a uniaxial wire dramatically\nchanges its response to an applied longitudinal field H1.\nThe transverse field, analogous to hard-axis anisotropy,\ninhibits precession and facilitates fast DW propagation.\nForH1lessthanan H2-dependentcriticalfield H1c, given\nin the linear regime by (29) below, there appears a trav-\nelling wave, while for H1> H1c, there appears an oscil-\nlating solution, as in the Walker case. The DW velocity\nof travelling wave exceeds that of oscillating solution.\nVelocity of travelling wave\nWe first obtain a general identity, of independent in-\nterest, which relates the velocity of a travelling wave\nM(x,t) =m(x−Vt) (assuming one exists) to the change\nin potential energy across the profile (for zero transverse\nfield, this coincides with results of [1] and [10]). Noting\nthat˙M=−Vm′, we take the squareof (1) and integrate\nover the length of the wire to obtain\nV2||m′||2= (1+α2)||m×H||2. (7)\nHere we use the notation\n||u||2=/angbracketleftu,u/angbracketright,/angbracketleftu,v/angbracketright=/integraldisplay∞\n−∞u·vdx(8)\nfor theL2-norm and inner product of vector fields (anal-\nogous notation for scalar fields is used below). Next, we\ntake the inner product of (1) with Hto obtain\nV/angbracketleftm′,H/angbracketright=−α||m×H||2. (9)\nNoting that m′·H=/parenleftbig1\n2m′·m′−U(m)/parenrightbig′, we combine\n(7) and (9) to obtain\nV=1\n2(α+1/α)||m′||−2(U(m−)−U(m+)).(10)The identity (10) has a simple physical interpretation;\nthe velocity is proportional to the potential energy dif-\nference across the wire, and inversely proportional to the\nexchange energy of the profile.\nFrom now on, we consider the uniaxial case K2= 0\nandappliedfieldwithlongitudinalandtransversecompo-\nnentsH1,H2>0 (by symmetry, we can assume H3= 0)\nwith|Ha|<1. An immediate consequence of (10) is\nthat, in the uniaxial case, the velocity must vanish as\nH1goes to zero. For when H1= 0, the local minima\nm±are related by reflection through the 23-plane, and\nU(m+) =U(m−).\nSmall transverse field\nIn order to understand travelling wave and oscillating\nsolutions as well as the transition between them, we first\ncarry out an asymptotic analysis in which both H1and\nH2are regarded as small, writing H1=ǫh1,H2=ǫh2\nand rescaling time as τ=ǫt(a systematic treatment in-\ncludingcurrent-inducedtorqueswillbegivenin[29]). We\nseek a solution of the LLG equation (1) of the following\nasymptotic form:\nΘ(x,t) =θ0(x,τ)+ǫθ1(x,τ)+..., (11)\nΦ(x,t) =φ0(x,τ)+ǫφ1(x,τ)+... (12)\nIt is straightforward to check that the boundary condi-\ntions, namely that mapproach distinct minima of Uas\nx→ ±∞, imply that\nm(±∞,τ) = (±1,ǫh2,0)+O(ǫ2).(13)\nThe leading-order equations for Θ and Φ become\nθ0,xx−1\n2(1+φ2\n0,x)sin2θ0= 0, (14)\n/parenleftbig\nsin2θ0φ0,x/parenrightbig\nx= 0. (15)\nThe only physical (finite-energy) solutions of (14) and\n(15) consistent with the boundary conditions (13) are of\nthe form\nφ0(x,τ) =φ0(τ) (16)\nθ0(x,τ) = 2arctanexp( −(x−x∗(τ))),(17)\nwhereφ0andx∗respectivelydescribetheDWorientation\nand centre, and are functions of τalone. It is convenient\nto introduce a travelling coordinate ξ=x−x∗(τ) and\nrewrite the ansatz (11)–(12) as\nΘ(x,t) =θ0(ξ,τ)+ǫθ1(ξ,τ)+..., (18)\nΦ(x,t) =φ0(ξ,τ)+ǫφ1(ξ,τ)+... (19)\nTo obtain equations for φ0(τ) andx∗(τ) we must pro-\nceed to the next order. It is convenient to introduce new3\nvariables at order ǫwhich, in light of the boundary con-\nditions (13), vanish at x=±∞, as follows:\nΘ1:=θ1−h2cosφ0cosθ0, (20)\nu:=φ1sinθ0+h2sinφ0. (21)\nThese satisfy the linear inhomogeneous equations\nLΘ1=f, (22)\nLu=g. (23)\nHereLis the self-adjoint Schr¨ odinger operator given by\nL=−∂2\n∂ξ2+W(ξ), (24)\nwhere\nW=θ′′′\n0\nθ′\n0= 1−2sech2ξ, (25)\nandf(ξ,τ) andg(ξ,τ) are given by\nf= (1+α2)−1sinθ0(−α˙x∗−˙φ0)−h1sinθ0,\ng= (1+α2)−1sinθ0(˙x∗−α˙φ0)+2h2sin2θ0sinφ0.\n(26)\nTheDWposition x∗andorientation φ0aredetermined\nfrom the solvability conditions for (22) – (23). According\nto the Fredholm alternative, given a self-adjoint opera-\ntorLonL2(R), a necessary condition for the equation\nLΘ1=fto have a solution Θ 1is thatfbe orthogonal to\nthe kernel of L. If this is the case, a sufficient condition is\nthatthespectrumof Lisisolatedawayfrom0. From(24)\nand (25) it is clear that θ′\n0belongs to the kernel of L, and\nsince the eigenvalues of a one-dimensional Schr¨ odinger\noperator are nondegenerate, it follows that θ′\n0spans the\nkernel of L. Moreover, since W(ξ)→1 asξ→ ±∞, it\nfollowsthat the spectrum of Lis discrete near0. (In fact,\nWis a special case of the exactly solvable P¨ oschl-Teller\npotential, but we won’t make use of this fact.) Requiring\nfandgin (22) and (23) to be orthogonal to θ′\n0and not-\ning that /angbracketleftθ′\n0,θ′\n0/angbracketright= 2,/angbracketleftθ′\n0,sinθ0/angbracketright=−2,/angbracketleftθ′\n0,1/angbracketright=−π, and\n/angbracketleftθ′\n0,cosθ0/angbracketright= 0, we obtain the following system of ODEs\nforφ0andx∗:\n˙φ0=−h1−απ\n2h2sinφ0, (27)\n˙x∗=−αh1+π\n2h2sinφ0. (28)\nTravelling wave solutions appear provided (27) has\nfixed points; this occurs for h1below a critical field h1,c\ngiven by\nh1,c=απh2\n2, (29)The velocity and orientation of the travelling wave are\ngiven by\n˙x∗=−/parenleftbigg\nα+1\nα/parenrightbigg\nh1, (30)\nsinφ0=−h1\nh1,c. (31)\nThere are two possible solutions for φ0∈[0,2π), only\none of which is stable. Oscillating solutions appear for\nh1> h1c, and are given by\nh1tan1\n2φ0=−h1,c−/radicalBig\nh2\n1−h2\n1,ctan/parenleftBig\n1\n2/radicalBig\nh2\n1−h2\n1,cτ/parenrightBig\n(32)\nwith the period T= 2π//radicalBig\nh2\n1−h2\n1,c. The mean preces-\nsional and translational velocities are obtained by aver-\naging over a period, with result\n/angbracketleftBig\n˙φ0/angbracketrightBig\n=−sgn(h1)/radicalBig\nh2\n1−h2\n1,c, (33)\n/angbracketleft˙x∗/angbracketright=−/parenleftbigg\nα+1\nα/parenrightbigg\nh1+1\nαsgn(h1)/radicalBig\nh2\n1−h2\n1,c.(34)\nNote that for h1=h1,c, (34) coincides with the travelling\nwave velocity (30), whereas for h1≫h1,c, (34) reduces\nto the velocity of the precessing solution given by (6).\nThe behaviour is similar in many respects to the\nWalker case (i.e., K2/negationslash= 0 and H2= 0). Here, the trans-\nverse field rather than hard-axis anisotropy serves to ar-\nrest the precession of the DW (provided the longitudinal\nfield is not too strong). There are differences as well;\nin the transverse-field case there is just one stable trav-\nelling wave, whereas in the Walker case there are two.\nAlso, in the transverse-field case the asymptotic value of\nthe magnetisation has a transverse component, whereas\nin the Walker case it has none.\nModerate transverse field\nWe can extend the travelling wave analysis to the\nregime where H2is no longer regarded as small. We\ncontinue to regard H1as small, writing H1=ǫh1and\nV=ǫv, and expand the travelling wave ansatz Θ( x,t) =\nθ(x−Vt), Φ(x,t) =φ(x−Vt) to first order in ǫ, writing\nθ=θ0+ǫθ1,φ=φ0+ǫφ1. Substituting into the LLG\nequation, we obtain the O(ǫ0) equations\nθ′\n0= (H2−sinθ0), φ0= 0, (35)\nwith boundary conditions sin θ0±=H2,θ0+> π/2 and\nθ0−< π/2. Thus, for H2=O(ǫ0), azimuthal symmetry\nisbrokenatleadingorder,andthestaticprofileisparallel\nto the transversefield (the alternativesolution with φ0=\nπis unstable). The solution of (35) is given by\ntanθ0\n2=κ\nH2tanh/bracketleftbigg\ntanh−1/parenleftbiggH2−1\nκ/parenrightbigg\n−κ\n2ξ/bracketrightbigg\n+1\nH2,\n(36)4\nwhereκ=/radicalbig\n1−H2\n2.\nAt order ǫwe obtain the linear inhomogeneous equa-\ntions\nLθ1=α\n1+α2vθ′\n0−h1sinθ0, (37)\nMφ1=1\n1+α2v(cosθ0)′, (38)\nwhere\nL=−d2\ndξ2+θ′′′\n0\nθ′\n0, M=−d\ndξsin2θ0d\ndξ+H2sinθ0.(39)\nHereθ0is given by (36), and θ1,φ1are requiredto vanish\nasξ→ ±∞. As above, the Fredholm alternative implies\nthat the right-hand side of (37) must be orthogonal to θ′\n0\nin order for a solution to exist. Calculation yields\nV=−/parenleftbigg\nα+1\nα/parenrightbigg\n(1−(H2/κ)cos−1H2)−1H1.(40)\nForH2= 0, this coincides with (30); thus, (40) gives\nH2-nonlinear corrections to the velocity. Moreover, it is\nstraightforward to show that (40) is consistent with the\ngeneral identity (10). Finally, one can also show that\nMhas trivial kernel with spectrum bounded away from\nzero, so that (38) is automatically solvable.\nItisinterestingtocomparetheDWvelocitywithtrans-\nverse field to the Walker case. From (5) and (40),\nVW/V=γ(1−(H2/κ)cos−1H2)<1.(41)\nThus, to leading orderin H1, the DWvelocity in a uniax-\nial wire with transverse field exceeds the Walker velocity.\nNumerical results below establish that this continues to\nhold asH1approaches the critical field H1c.\nNumerical results\nToverifyouranalyticalresults, wesolvetheLLGequa-\ntion (1) using a finite-difference scheme on a domain\n−L≤x≤LwhereL= 100 (the DW has width of\norder 1). Neumann boundary conditions, m′= 0, are\nmaintained at the endpoints. The damping parameter α\nis taken to be 0 .1 throughout. As initial condition we\ntake the stationary profile, with θ0given by (36) and\nφ0= 0. After an initial transient period, during which\nthe asymptotic values of matx→ ±Lconverge to m±,\na stable solution emerges, in which the DW propagates\nwith a characteristic mean velocity V. (For convenience,\nwehavetaken H1<0, sothat Vis positive.) In Figure1,\nnumerically computed values of Vare plotted as a func-\ntion of|H1|for three fixed values of the transverse field:\nH2= 0.2,H2= 0.1, and the limiting case H2= 0, where\nthe dynamics is given by the precessing solution. There\nis good quantitative agreement with the analytic results\nfor small transverse fields, (30), for |H1|< H1,c, and(34), for |H1|> H1,c, In Figure 2, the analytic expres-\nsions for the velocity for small and moderate transverse\nfields are compared to numerical results for H2= 0.2 and\n|H1| ≪H1c. The moderate-field expression (40), which\ndepends nonlinearly in H2, gives excellent agreement for\nsmalldrivingfields. Fornonzero H2, the velocityexhibits\na peak at a critical field |H1c|, which depends on H2.\n0 0.1 0.2 0.3 0.400.050.10.150.20.250.30.35\n|H1|V\n  \nH2= 0.2\nH2= 0.1\nH2= 0\nFIG. 1: Average DW velocity Vas a function of the driving\nfield|H1|for three values of the transverse field H2. The an-\nalytic formulas (solid curves) (30), for |H1|< H1,c, and (34),\nfor|H1|> H1,c, are plotted against numerically computed\nvalues (open circles). For H2= 0, the analytic formula is\nexact.\n00.005 0.01 0.015 0.02 0.02500.050.10.150.20.250.3\n|H1|V\n  \nSmall −H2theory\nModerate −H2theory\nNumerics\nFIG. 2: DW velocity Vas a function of the driving field\n|H1|forH2= 0.2. The expressions for small-transverse field\n(30) (red curve) and moderate-transverse field (40) (light b lue\ncurve) are plotted against numerically computed values (op en\ncircles).\nFigure 3 shows the dependence of the critical field |H1,c|5\nonH2, in close agreement with the analytic result (29).\n0 0.05 0.1 0.15 0.200.0050.010.0150.020.0250.030.035\nH2|H1,c|\nFIG. 3: The critical driving field |H1,c|as a function of the\ntransverse field H2. A linear fit (blue curve) through the nu-\nmerically computed data (blue diamonds) is plotted alongsi de\nthe analytical result (29) (red curve).\nFIG. 4: The magnetization distribution, θ(x,t) andφ(x,t),\nfor two values of the driving field: H1=−0.01 in figures\n(a) and (b), and H1=−0.05 in figures (c) and (d). The\ntransverse field is taken as H2= 0.1 throughout.\nAs in the Walker case, the properties of the propagat-\ning solution are qualitatively different for driving fields\n|H1|below and above the critical field. This is confirmed\nin Figure 4, which shows contour plots of the magne-\ntization in the ( x,t)-plane. Figs. 4(a) and 4(b), where\nH1=−0.01, exemplify the case |H1|<|H1c|. The mag-\nnetisation evolves as a fixed profile translating rigidly\nwith velocity V. For|H1|>|H1c|, as exemplified by\nFigs. 4(c) and 4(d), where H1=−0.05. the magnetiza-\ntion profile exhibits a non-uniform precession as it prop-agates along the nanowire, with mean velocity in good\nagreement with (34).\nSummary\nWe haveestablished, both analyticallyin leading-order\nasymptotics and numerically, the existence of travelling\nwaveandoscillatingsolutionsoftheLLGequationinuni-\naxial wires in applied fields with longitudinal and trans-\nverse components. We have obtained analytic expres-\nsions for the velocity, (30) and (40), and for the critical\nlongitudinal field, (29), above which the travelling wave\nsolution ceases to exist. We have also obtained the mean\nprecessional and linear velocities (33) and (34) for oscil-\nlating solutions. The analytic results are confirmed by\nnumerics.\n[1] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[2] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu,\nT. Shinjo, Phys. Rev. Lett. 92077205 (2004).\n[3] D.A. Allwood, G. Xiong, C.C. Faulkner, D. Atkinson,\nD. Petit and R.P. Cowburn, Science 309, 1688 (2005).\n[4] R.P. Cowburn, Nature (London) 448, 544 (2007).\n[5] G.S.D Beach, C. Nistor, C. Knutson, M. Tsoi, and\nJ.L. Erskine, Nature Mater. 4, 741(2005).\n[6] J. Yang, C. Nistor, G.S.D. Beach, and J.L. Erskine,\nPhys. Rev. B 77, 014413 (2008).\n[7] M.T. Bryan, T. Schrefl, D. Atkinson, D.A. Allwood,\nJ. Appl. Phys. 103, 073906 (2008).\n[8] O.A. Tretiakov, D. Clarke, Gia-Wei Chern, Ya. B. Baza-\nliy and O. Tchernyshyov, Phys. Rev. Lett. 100127204\n(2008).\n[9] X.R. Wang, P. Yan, J. Lu, Europhys. Lett. 86, 67001\n(2009).\n[10] X.R. Wang, P. Yan , J. Lu, C. He, Ann. Phys. 324, 1815–\n1820 (2009).\n[11] J. Lu and X.R. Wang, J. Appl.Phys. 107, 083915 (2010).\n[12] A. Mougin, M. Cormier, J.P. Adam, P.J. Metaxas and\nJ. Ferr´ e, Europhys. Lett. 78, 57007 (2007).\n[13] Z. Li and S. Zhang, Phys. Rev. Lett. 92207203 (2004).\n[14] A. Thiaville, Y. Nakatani,, J. Miltat and Y. Suzuki, Eu-\nrophys. Lett. 69, 990 (2005).\n[15] G.S.D Beach, C. Knutson, C. Nistor, M. Tsoi, and\nJ.L. Erskine, Phys. Rev. Lett. 97057203 (2006)\n[16] S. S. P. Parkin, M. Hayashi and L. Thomas, Science 320,\n190 (2008).\n[17] M. Hayashi, L. Thomas, R. Moriya, C. Rettner and\nS. S. P. Parkin, Science 320, 209 (2008).\n[18] L. Thomas, R. Moriya, C. Rettner, S. and S. P. Parkin,\nScience330, 1810 (2010).\n[19] O.A. Tretiakov and Ar. Abanov,\nPhys. Rev. Lett. 105157201 (2010).\n[20] O.A. Tretiakov, Y. Liu and Ar. Abanov,\nPhys. Rev. Lett. 108247201 (2012).\n[21] V. Slastikov and C. Sonnenberg, IMA J. Appl. Math. 77\nno. 2, 220 (2012)6\n[22] A. Hubert and R. Sch¨ afer, Magnetic Domains: The\nAnalysis of Magnetic Microstructures (Springer, Berlin,\n1998).\n[23] Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104,\n037206 (2010).\n[24] A. Goussev, J.M. Robbins, V. Slastikov, Phys. Rev. Lett .\n104, 147202 (2010).\n[25] Y. Gou, A. Goussev, J. M. Robbins, V. Slastikov, Phys.\nRev. B84, 104445 (2011)[26] L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietu-\nnion8, 153 (1935).\n[27] T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans.\nMag.40, 3443 (2004).\n[28] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys.\nRep.194, 117 (1990).\n[29] A. Goussev, R. Lund, J.M. Robbins, C. Sonnenberg,\nV. Slastikov, in preparation."    },    {        "title": "1207.0310v1.Establishing_micromagnetic_parameters_of_ferromagnetic_semiconductor__Ga_Mn_As.pdf",        "content": "Establishing micromagnetic parameters of ferromagnetic\nsemiconductor (Ga,Mn)As\nP. N\u0014 emec,1V. Nov\u0013 ak,2N. Tesa\u0014 rov\u0013 a,1E. Rozkotov\u0013 a,1H. Reichlov\u0013 a,2, 1\nD. Butkovi\u0014 cov\u0013 a,1F. Troj\u0013 anek,1K. Olejn\u0013 \u0010k,2P. Mal\u0013 y,1R. P. Campion,3\nB. L. Gallagher,3Jairo Sinova,4, 2and T. Jungwirth2, 3\n1Faculty of Mathematics and Physics, Charles University in Prague,\nKe Karlovu 3, 121 16 Prague 2, Czech Republic\n2Institute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n3School of Physics and Astronomy, University of Nottingham,\nNottingham NG7 2RD, United Kingdom\n4Department of Physics, Texas A&M University,\nCollege Station, TX 77843-4242, USA\n(Dated: November 20, 2018)\nPACS numbers: 75.50.Pp,75.30.-m,75.70.Ak\n1arXiv:1207.0310v1  [cond-mat.mtrl-sci]  2 Jul 2012(Ga,Mn)As is at the forefront of research exploring the synergy of magnetism\nwith the physics and technology of semiconductors, and has led to discover-\nies of new spin-dependent phenomena and functionalities applicable to a wide\nrange of material systems. Its recognition and utility as an ideal model material\nfor spintronics research has been undermined by the large scatter in reported\nsemiconducting doping trends and micromagnetic parameters. In this paper\nwe establish these basic material characteristics by individually optimizing the\nhighly non-equilibrium synthesis for each Mn-doping level and by simultane-\nously determining all micromagnetic parameters from one set of magneto-optical\npump-and-probe measurements. Our (Ga,Mn)As thin-\flm epilayers, spannig\nthe wide range of accessible dopings, have sharp thermodynamic Curie point\nsingularities typical of uniform magnetic systems. The materials show system-\natic trends of increasing magnetization, carrier density, and Curie tempera-\nture (reaching 188 K) with increasing doping, and monotonous doping depen-\ndence of the Gilbert damping constant of \u00180:1\u00000:01 and the spin sti\u000bness of\n\u00182\u00003 meV nm2. These results render (Ga,Mn)As well controlled degenerate\nsemiconductor with basic magnetic characteristics comparable to common band\nferromagnets.\nUnder equilibrium growth conditions the incorporation of magnetic Mn ions into III-\nAs semiconductor crystals is limited to approximately 0.1%. To circumvent the solubility\nproblem a non-equilibrium, low-temperature molecular-beam-epitaxy (LT-MBE) technique\nwas employed which led to \frst successful growths of (In,Mn)As and (Ga,Mn)As ternary\nalloys with more than 1% Mn and to the discovery of ferromagnetism in these materials.1{6\nThe compounds qualify as ferromagnetic semiconductors to the extent that their magnetic\nproperties can be altered by the usual semiconductor electronics engineering variables, such\nas doping, electric \felds,7{12or light.13{27By exploiting the large spin polarization of car-\nriers and low saturation moment in (Ga,Mn)As and building on the well established het-\nerostructure growth and microfabrication techniques in III-V semiconductors, (Ga,Mn)As\nhas been extensively used for spintronics research of direct and inverse magneto-transport\nphenomena.28{37Besides the more conventional spintronic e\u000bects based on Mott's two-spin-\nchannel model of conduction in ferromagnets, (Ga,Mn)As has become particularly fruitful\nfor exploring the second, more physically intriguing spintronics paradigm based on Dirac's\n2spin-orbit coupling.34,38{46\nThe apparent potential of (Ga,Mn)As to become the test-bed model material for many\nlines of spintronics research has been hindered by the large scatter in reported semiconduct-\ning doping trends and micromagnetic parameters. Our strategy to tackle this problem begins\nfrom the synthesis of a set of (Ga,Mn)As materials spanning a wide range of Mn dopings. Be-\ncause of the highly non-equilibrium nature of the heavily-doped ferromagnetic (Ga,Mn)As,\nthe growth and post-growth annealing procedures have to be individually optimized for each\nMn-doping level in order to obtain \flms which are as close as possible to idealized uniform\n(Ga,Mn)As mixed crystals with the minimal density of compensating and other uninten-\ntional defects. An extensive set of characterization measurements has to accompany the\nsynthesis to guarantee that the materials show systematic doping trends; monitoring the\nthermodynamic Curie point singularities is essential for assuring the uniformity and high\nmagnetic quality of the materials.10,47{49When omitting the above procedures,50extrinsic\nimpurities and sample inhomogeneities can yield non-systematic doping trends and conceal\nthe intrinsic magnetic properties of (Ga,Mn)As.\nThe focus of the work presented in this paper is on the systematic study of the Gilbert\ndamping and spin sti\u000bness constants which, together with magnetic anisotropy \felds, rep-\nresent the basic micromagnetic parameters of a ferromagnet. A more than an order of mag-\nnitude experimental scatter and a lack of any clear trend as a function of Mn-doping can be\nfound in the literature for the Gilbert damping and spin-sti\u000bness constants.51{60(See Supple-\nmentary information for a detailed discussion of previous experimental works.) This re\rects\npartly the issues related to the control and reproducibility of the synthesis of (Ga,Mn)As\nand partly the di\u000eculty with applying common magnetic characterization techniques, such\nas neutron scattering, to the thin-\flm dilute-moment (Ga,Mn)As samples. Hand-in-hand\nwith the optimization of the material synthesis we have developed experimental capabilities\nbased on the magneto-optical (MO) pump-and-probe method which allow us to simultane-\nously determine the magnetic anisotropy, Gilbert damping, and spin sti\u000bness constants from\none consistent set of measured data. Our results are summarized in Fig. 1. The Curie point\nsingularity in the temperature derivative of the resistivity d\u001a=dT measured throughout the\nseries of optimized ferromagnetic (Ga,Mn)As samples with metallic conduction is shown in\nFig. 1a. The data span the nominal doping range from x\u00191.5 to 13% and corresponding\nCurie temperatures from Tc= 29 to 188 K, and illustrate the high quality of all the epilayers\n30.8 1.0 1.20.40.60.81.0\n01 0 0 2 0 00246dU/dT ( 10\u00105:cm/K)\nT (K)× 0.5\n(dU/dT)*\nT*13% 5.2%1.5%\n3.010a\nbc\nd2468 1 0-200-150-1000255075Ki (mT)\nx (%)Ku\nKc\nKout\n1.01.52.02.5\n2468 1 00246810D (10-2)\nx (%)\nD (meV.nm2)\n05 0 0 1 0 0 0 1 5 0 0 2 0 0 09% 5.2% 3.8% MO signal (rel. units)\nTime delay (ps)2.5%\nFig. 1FIG. 1: Micromagnetic parameters of optimized epilayers of ferromagnetic (Ga,Mn)As.\na, Examples of sharp Curie point singularities in the temperature derivative of the resistivity in the\nseries of optimized ferromagnetic (Ga,Mn)As epilayers with metallic conduction; Tcmonotonously\nincreases with increasing nominal Mn doping between 1.5 and 13%. Inset shows d\u001a=dT normalized\nto its peak value with the temperature axis normalized to Tc.b,Examples of oscillatory parts of\nMO signals measured in 18 nm thick (Ga,Mn)As epilayers with the depicted nominal Mn doping\nfor external magnetic \feld \u00160Hext= 400 mT applied along the [010] crystallographic direction;\nthe curves are normalized and vertically o\u000b-set for clarity. c,Dependence of anisotropy constants\non nominal Mn doping. d,Dependence of the Gilbert damping constant \u000band the spin sti\u000bness\nconstantDon nominal Mn doping.\nwithin the series. Examples of the measured magnetization precession signals by the MO\npump-and-probe method are shown in Fig. 1b. From these time-dependent magnetization\nmeasurements we obtained the magnetic anisotropy constants Ki, Gilbert damping constant\n4\u000b, and spin sti\u000bness constant Dwhich are summarized in Figs.1c,d. We now proceed to the\ndetail discussion of our experimental techniques and the discussion of the measured results\nin the context of physics of degenerate semiconductors and band ferromagnets.\nOptimization of the (Ga,Mn)As synthesis. Our (Ga,Mn)As layers were grown at\nthe growth rate of approximately 0.2 monolayers/second. The Mn \rux, and hence the nom-\ninal Mn doping x, was determined by measuring the ratio of the beam equivalent pressures\n(BEP) of Mn and Ga sources before each growth. The Mn content was cross-checked by\nsecondary ion mass spectroscopy (SIMS) and by comparing the growth rates of GaAs and\n(Ga,Mn)As measured by the oscillations of the re\rection high-energy electron di\u000braction\n(RHEED).\nThere are two critical growth parameters of (Ga,Mn)As: the substrate temperature,\nand the As-to-(Ga+Mn) \rux ratio. At the typical temperatures of \u0018200\u000eC neither an\noptical pyrometer nor a radiatively coupled temperature sensor are applicable. Instead,\nwe used the GaAs band-edge spectrometer to measure the substrate temperature and the\npredictive substrate heater control to stabilize the temperature during the growth. For a\ngiven As:(Ga+Mn) ratio the substrate temperature fully determines the growth regime: the\ngrowth proceeds two-dimensionally at low temperatures, and turns irreversibly into the 3D\ngrowth mode when a critical temperature is exceeded. The scatter of the critical substrate\ntemperature for given xand As:(Ga+Mn) ratio is remarkably small, typically less than\n2\u000eC. In excess As \rux the 2D/3D transition occurs at higher temperature. The highest\nquality samples are grown in a narrow window of the 1:1 stoichiometric As:(Ga+Mn) ratio\nand at the substrate temperature approaching as close as possible from below the 2D/3D\ncritical temperature for given x. The As:(Ga+Mn) ratio was adjusted by the As-cell valve,\nand calibrated using the As-controlled RHEED oscillations. In insets of Fig. 2a we show\nexamples of RHEED patterns for the x= 7% nominally doped (Ga,Mn)As material grown\nat stoichiometric 1:1 ratio of As:(Ga+Mn) for substrate temperature of 225 K which is\nabove the 3D/2D boundary and 210 K which is below the boundary. The optimal growth\ntemperature for this doping is 215 K. In the main panel of Fig. 2a we plot the optimal\ngrowth temperature as a function of nominal Mn doping, showing the rapidly decreasing\ngrowth temperature trend.\nThe next important factor determining the quality of the resulting (Ga,Mn)As materials\nare post-growth annealing conditions. In Fig. 2b we show the dependence of the Curie tem-\n5000.51.0U/dT)*50 100 150 200 250246U (10-3:cm)\nT (K)annealing time\n160 °C\n160170180190TC (K)140°C\n180°Cd\nb ea\n02468 1 0 1 2 1 4100150200250300350\nGrowth T (°C)\nx (%)\n50 100 150 200 2500204060M (emu/cm3)\nT (K)50 100 150 200 250-0.50.0(dUT (K)11 014015016015 nm\nAnnealing time (h)160°C\n01 0 2 0 3 070125150175 200 °C\n100 nmTC (K)\nAnnealing time (h)25 nmc f\nFig. 2FIG. 2: Optimization of the (Ga,Mn)As synthesis. a , Optimal growth temperature as a\nfunction of the nominal Mn doping. Insets show examples of RHEED images of the 2D growth\nat 210\u000eC (lower inset) and 3D growth at 225\u000eC (upper inset) of the 7% Mn-doped (Ga,Mn)As.\nb, Dependence of the Curie temperature on the annealing time for three di\u000berent annealing tem-\nperatures in a 15 nm thick (Ga,Mn)As epilayer with 13% nominal Mn doping. c, Dependence of\nthe Curie temperature on the annealing time for the annealing temperature of 200\u000eC in a 100 nm\nthick (Ga,Mn)As epilayer with 13% nominal Mn doping, and in the same epilayer thinned down to\n25 nm by wet etching. d{f, Temperature dependencies of resistivity \u001a,d, temperature derivative\nof the resistivity d\u001a=dT ,e, and remnant magnetization M,f, in a 20 nm thick (Ga,Mn)As epilayer\nwith 13% nominal Mn doping at successive annealing times at the optimal annealing temperature\nof 160\u000eC for this doping.\nperatureTcon the annealing time for three di\u000berent annealing temperatures for the record\nTc= 188 K sample with nominal 13% Mn doping and \flm thickness 15 nm. These curves\n6illustrate the common trend in annealing (at temperatures close to the growth temperature)\nsuggesting the presence of competing mechanisms. One mechanism yields the increase of Tc\nand is ascribed in a number of reports to the removal of charge and moment compensating\ninterstitial Mn impurities (see e.g. the detailed annealing study in Ref. 10). The removal is\nslowed down by the growth of an oxide surface layer during annealing10and an additional\nmechanism can eventually yield reduction of Tcafter su\u000eciently long annealing times, de-\npending on the annealing temperature. The origin of this detrimental mechanism may be in\nMn clustering or in the competition between the non-equilibrium (Ga,Mn)As phase and the\nequilibrium MnAs second phase. Because of the competing mechanisms, the absolutely high-\nest Curie temperature for the given nominal doping is achieved at intermediate annealing\ntemperature and time, as illustrated in Fig. 2b.\nThe remaining critical parameter of the synthesis is the epilayer thickness. For a given\nnominal doping, the highest attainable Tcis reached only in thin \flms, typically thinner\nthan\u001850 nm. In Fig. 2c we illustrate the importance of the \flm thickness for obtaining\nhigh quality (Ga,Mn)As materials. A 100 nm thick \flm is grown with nominal 13% doping\nand, unlike the thin record Tc\flm discussed above, here the maximum Tcachieved by\nannealing is only about 140 K. However, if the same \flm is thinned down (to e.g. 25 nm)\nby wet etching and annealed at the same conditions, the achieved Curie temperatures are\nsigni\fcantly higher.\nAn increase of Tcis not the only parameter followed to ascertain that a sample is of\nhigh quality. A key characterization tool are the thermodynamic Curie point singularities.47\nThis is illustrated in Figs. 2d-f where we compare resistivity and magnetization measured\nat increasing time steps during the optimizing annealing procedure. The development of\nsharply vanishing magnetization M(T) atTcand the onset of the singularity in d\u001a=dT are\nwell correlated with increasing Tcand conductivity within the annealing sequence.\nAfter \fnding the optimal growth and post-growth conditions for each individual nominal\ndoping we obtained a series of samples spanning the wide range of Mn dopings. The samples\ncan be divided into several groups: at nominal dopings below \u00180:1% the (Ga,Mn)As ma-\nterials are paramagnetic, strongly insulating, showing signatures of the activated transport\ncorresponding to valence band { impurity band transitions at intermediate temperatures,\nand valence band { conduction band transitions at high temperatures (see Fig. 3a).61,62For\nhigher nominal dopings, 0 :5.x.1:5%, no clear signatures of activation from the valence\n72468 1 010-2100102V (:-1cm-1)\n1000/T (K-1)0.05%1%2%7%\n~ 100 meV\nEg/20123p (1021cm-3)\n \n200a\nbc\nd120255075100\n Msat(emu/cm3)NMn (1021cm-3)\n02468 1 0 1 2 1 4050100150200TC (K)\nx (%)\nFig. 3be02468 1 0 1 2 1 410-2100102V (:-1cm-1)\nx (%)FIG. 3: Doping trends in the series of optimized (Ga,Mn)As epilayers. a , Temperature\ndependence of the conductivity \u001b(T) of optimized (Ga,Mn)As epilayers with depicted nominal\nMn doping. Dashed lines indicate the activated parts of \u001b(T) of the insulating paramagnetic\n(Ga,Mn)As with 0.05% Mn doping, corresponding to the Mn acceptor level and the band gap,\nrespectively. b-e, Conductivity, b, hole density, c, saturation magnetization and corresponding\nMn moment density, d, and Curie temperature, e, as a function of the nominal Mn doping in the\nseries of optimized (Ga,Mn)As epilayers.\nband to the impurity band are seen in the dc transport, con\frming that the bands start to\noverlap and mix, yet the materials remain insulating.61,62Atx\u00191:5%, the low-temperature\nconductivity of the \flm increases abruptly by several orders of magnitude (see Fig. 3b), and\nthe system turns into a degenerate semiconductor.61,62The onset of ferromagnetism occurs\nalready on the insulating side of the transition at x\u00191% and the Curie temperature then\nsteadily increases with increasing nominal Mn doping up to \u001913%. The hole concentration\npcan be measured by the slope of the Hall curve at high \felds (see Supplementary infor-\nmation) with an error bar due to the multi-band nature estimated to \u001820%.63Within this\n8uncertainty, the overall trend shows increasing pwith increasing doping in the optimized\nmaterials, as shown in Fig. 3c. Similarly, the saturation moment and Tcsteadily increase\nwith increasing nominal doping up to x\u001913%, as shown in Figs. 3d,e. Assuming 4.5 \u0016B\nper Mn atom64we can estimate the density NMnof uncompesated Mn Gamoments from the\nmagnetization data (see left y-axis in Fig. 3d). An important conclusion can be drawn when\ncomparing this estimate with the hole density estimated from the Hall resistance. Since\nthere is no apparent de\fcit of pcompared to NMn, and since the interstitial Mn impurity\ncompensates one local moment but two holes we conclude that interstitial Mn is completely\n(within the experimental scatter) removed in our optimally annealed epilayers. Hence, our\nseries of optimized (Ga,Mn)As materials have reproducible characteristics, showing an over-\nall trend of increasing saturation moment with increasing x, increasing Tc(reaching 188 K),\nand increasing hole density. The materials have no measurable charge or moment compen-\nsation of the substitutional Mn Gaimpurities and have a large degree of uniformity re\rected\nby sharp Curie point singularities.\nDetermination of the micromagnetic parameters. We now proceed to the de-\ntermination of the magnetic anisotropy, Gilbert damping, and spin sti\u000bness constants of\nour (Ga,Mn)As epilayers from the MO time-resolved measurements of the magnetization\nprecession. In the MO pump-and-probe experiments, we used a femtosecond titan sapphire\nlaser that was spectrally tuned to 1.64 eV, i.e., above the band gap of GaAs. The possibility\nto excite and detect precession of ferromagnetic Mn moments in (Ga,Mn)As by this method\nhas been extensively discussed in previous MO studies.16{27All experiments presented be-\nlow were preformed at temperature of approximately 15 K in re\rection geometry. External\nmagnetic \felds up to 550 mT were applied in the [010] and [110] crystallographic directions.\nThe intensity of the pump pulse was \u001830\u000040\u0016Jcm\u00002, with the pump to probe intensity\nratio\u001820 : 1\u000010 : 1. The penetration depth of the laser beam ( \u0018600 nm) safely exceeds\nthe thickness of the studied (Ga,Mn)As epilayers.\nThe anisotropy constants, shown in Fig. 1c, where obtained combining three complemen-\ntary measurements. In the \frst experiment we measured the external magnetic \feld Hext\ndependence of the precession frequency fof the time resolved MO signal. In the studied\n(Ga,Mn)As/GaAs epilayers, the internal magnetic anisotropy \felds are dominated by three\ncomponents. The out-of-plane component Koutis a sum of the thin-\flm shape anisotropy\nand the magnetocrystalline anisotropy due to the compressive growth strain in (Ga,Mn)As.\n9The cubic magnetocrystalline anisotropy Kcre\rects the zinc-blende crystal structure of the\nhost semiconductor. The additional uniaxial anisotropy component along the in-plane di-\nagonalKuis not associated with any measurable macroscopic strain in the epilayer and is\nlikely of extrinsic origin. The precession frequency is given by,\nf=g\u0016B\nhq\u0000\nHextcos('\u0000'H)\u00002Kout+Kc(3 + cos 4')=2 + 2Kusin2('\u0000\u0019=4) + \u0001Hn\u0001\n\u0002p\n(Hextcos('\u0000'H) + 2Kccos 4'\u00002Kusin 2'+ \u0001Hn); (1)\nwheregis the Land\u0013 e g-factor of Mn moments, \u0016Bthe Bohr magneton, 'and'Hare the\nin-plane magnetization and external magnetic \feld angles measured from the [100] crystal\naxis, and \u0001 Hnis the shift of the resonant \feld for the higher index nspin wave modes\nwith respect to the n= 0 uniform precession mode. In order to uniquely determine the\nanisotropy constants, the \feld-dependent precession frequency measurements were comple-\nmented by MO experiments with variable polarization angle of the probe beam. The latter\nmeasurements allow us to precisely determine the angle of the equilibrium easy axis of the\nmagnetization (see Supplementary information).26,27Finally, we con\frmed the consistency\nof the obtained anisotropy constants by performing static measurements of magnetization\nhysteresis loops by the superconducting quantum interference device (SQUID). Results for\nferromagnetic materials from our series of optimized (Ga,Mn)As epilayers are summarized in\nFig. 1c. Note that the values of KoutandKcfor the given Mn-doping are well reproducible in\nmaterials whose synthesis yields the same optimized values of the basic structural, magnetic\nand transport properties. For the Kuconstant, variations in the width of the optimized\nthin (Ga,Mn)As \flms or of other otherwise insigni\fcant changes of the growth or annealing\nconditions may yield sizable changes of Ku. This con\frms the presumed subtle extrinsic\nnature of this magnetic anisotropy component.\nThe sign of Koutimplies that all studied (Ga,Mn)As/GaAs materials are in-plane fer-\nromagnets. The competing magnitudes of KcandKuand the di\u000berent doping trends of\nthese two in-plane magnetic anisotropy constants (see Fig. 1c) are therefore crucial for the\nmicromagnetics of the materials. The biaxial anisotropy Kcdominates at very low dopings\nand the easy axis aligns with the main crystal axis [100] or [010]. At intermediate dopings,\nthe uniaxial anisotropy Kuis still weaker but comparable in magnitude to Kc. In these\nsamples the two equilibrium easy-axes are tilted towards the [1 \u001610] direction and their angle\nis sensitive to small changes of external parameters such as temperature. This allows for\n10exciting the magnetization precession by laser pulses in the pump-and-probe MO experi-\nments. At very high dopings, the uniaxial anisotropy dominates and the system has one\nstrong easy-axis along the [1 \u001610] in-plane diagonal. In the low-doped and high-doped samples\nwith very stable easy-axes aligned with one of the main crystal directions the dynamical MO\nexperiments become unfeasible.\nThe Gilbert damping constant \u000b, shown in Fig. 1d, is obtained by \ftting the measured\ndynamical MO signal to Landau-Lifshitz-Gilbert (LLG) equations using the experimentally\nobtained magnetic anisotropy constants. The high accuracy of the LLG \fts is demonstrated\nin Figs. 4a,b on data measured in a x= 5:2% doped sample. The obtained dependence of \u000b\non the external magnetic \feld applied along the [010] and [110] directions is shown in Fig. 4c.\nAt smaller \felds, \u000bis not constant and shows a strong anisotropy with respect to the \feld\nangle. When plotted as a function of frequency, however, the dependence on the \feld-angle\ndisappears, as shown in Fig. 4d. Analogous results are obtained for the entire series of the\noptimized materials. We can therefore conclude that the apparent anisotropy of \u000bcan in\nour materials be ascribed fully to the \feld-angle dependence via the precession frequency. In\nall our studied materials, the frequency-independent Gilbert damping constant is isotropic\nand can be accurately determined from MO data with precession frequencies f&15 GHz.\nWe point out that in ferromagnetic resonance (FMR) experiments, the measurement fre-\nquency was limited to two values, f= 9 and 35 GHz which even in the optimized (Ga,Mn)As\nmaterials is not su\u000ecient to reliably separate the intrinsic Gilbert damping constant from\nthe inhomogeneous broadening of the FMR line-width. The dynamical MO measurements,\non the other hand, span a large enough range of frequencies and allow us to extract a con-\nsistent set of frequency-independent values of \u000bfor our series of optimized ferromagnetic\n(Ga,Mn)As materials. We \fnd a systematic doping trend across the series in which the\nGilbert constant decreases from \u00180:1 to 0:01 when the nominal Mn doping increases from\n\u00182% to 5% and then remains nearly constant (see Fig. 1d). The magnitudes of \u000band the\ndoping dependence are consistent with Gilbert damping constants in conventional transi-\ntion metal ferromagnets. In metals, \u000btypically increases with increasing resistivity and is\nenhanced in alloys with enhanced spin-orbit coupling.65{67Similarly, in our measurements\nin (Ga,Mn)As, the increase of \u000bcorrelates with a sizable increase of the resistivity in the\nlower Mn-doped samples. Also, the spin-orbit coupling e\u000bects tend to be stronger in the\nlower doped samples with lower \flling of the valence bands and with the carriers closer to\n1166[110]05 0 0 1 0 0 0 1 5 0 0 2 0 0 0-50050MO signal ( µrad)\nTime delay (ps)Hext || [110]\n05 0 0 1 0 0 0 1 5 0 0 2 0 0 0-10010MO signal ( µrad)\nTime delay (ps)Hext || [010]a b\nc d\n05 1 0 1 5 2 0024D (10-2)\nf (GHz)[010], [110]02 0 0 4 0 0 6 0 0024D (10-2)\nµ0Hext (mT)[010]\nFig. 4FIG. 4: Determination of the Gilbert damping constant from MO experiments. a,b,\nOscillatory part of the MO signal (points) measured in a 18 nm thick epilayer with 5.2% nominal Mn\ndoping for external magnetic \feld \u00160Hext= 100 mT applied along the crystallographic directions\n[010] and [110]; lines are \fts by the LLG equation. c,Dependence of the Gilbert damping on\nexternal magnetic \feld applied along the [010] and [110] crystallographic directions. d,Dependence\nof the Gilbert damping on the precession frequency.\nthe metal-insulator transition.68Theory ascribing magnetization relaxation to the kinetic-\nexchange coupling of Mn moments with holes residing in the disordered, exchange-split, and\nspin-orbit-coupled valence band of (Ga,Mn)As yields a comparable range of values of \u000bas\nobserved in our measurements.51\nSimilar to the Gilbert constant, there has been a large scatter60in previous reports of\nexperimental values of the spin-sti\u000bness in (Ga,Mn)As inferred from FMR,53{56magneto-\noptical studies,59and from complementary static magnetization and domain structure\n12measurements.57,58We attribute the lack of a consistent picture obtained from these measure-\nments to sample inhomogeneities and extrinsic defects in the studied (Ga,Mn)As epilayers\nwith thicknesses typically exceeding 100 nm and to experimental data which allowed only\nan indirect extraction of the spin sti\u000bness constant. The MO pump-and-probe technique\nutilized in our work allows in principle for the direct measurement of the spin sti\u000bness,\nhowever, one has to \fnd the rather delicate balance between thin enough epilayers to avoid\nsample inhomogeneity and thick enough \flms allowing to observe the higher-index Kittel\nspin-wave modes69of a uniform thin-\flm ferromagnet. For these modes, the spin-sti\u000bness\nparameterDis directly obtained from the measured resonant \felds,\n\u0001Hn\u0011H0\u0000Hn=Dn2\nL2\u00192\ng\u0016B; (2)\nwhereLis the thickness of the ferromagnetic \flm. The MO pump-and-probe technique\nhas the key advantage here that, unlike FMR, it is not limited to odd index spin wave\nmodes.69The ability to excite and detect the n= 0, 1, and 2 resonances is essential for the\nobservation of the Kittel modes in our optimized (Ga,Mn)As epilayers whose thickness is\nlimited to \u001850 nm.\nIn Fig. 5a we show an example of the time dependent MO signal measured in a 48 nm\nthick optimized epilayer with 7% nominal Mn doping. Three spin wave resonances (SWRs)\nare identi\fed in the sample with frequencies f0,f1, andf2, as shown in Figs. 5b,c. The asso-\nciation of these SWRs with the Kittel modes, described by Eq. (2), is based on experiments\nshown in Figs. 5b-e. In Fig. 5c we plot the dependence of the three detected precession\nfrequencies on the external magnetic \feld applied along the [010] and [110] crystal axes.\nAt saturation \felds, which for the 7% Mn-doped sample are &70 mT, the equilibrium\nmagnetization vector is aligned with Hextand Eq. (1) with '='Hcan be used to \ft the\ndata. We emphasize that all six displayed dependences fn(Hext) forn= 0, 1, and 2, and\n'H= 45\u000eand 90\u000ecan be accurately \ftted by one set of magnetic anisotropy constants. We\ncan therefore use Eq. (2) to convert the measured frequency spacing of individual SWRs\nto \u0001Hn. In Fig. 5d we show that \u0001 Hnin our optimized epilayers is proportional to n2as\nexpected for the Kittel modes in homogeneous \flms.\nThe magnetic homogeneity and the applicability of Eq. (2) in our epilayers is further\ncon\frmed by the following experiments: We prepared three samples by etching the original\n48 nm thick (Ga,Mn)As \flm down to the thicknesses of 39, 29 and 15 nm, respectively.\n1301 0 0 2 0 0 3 0 0051015f2f1f (GHz)\nµ0Hext (mT)f0\nHext || [010]; [110]\n           \n           \n           f0f05 0 0 1 0 0 0 1 5 0 0 2 0 0 0-50050MO signal ( µrad)\nTime delay (ps)48 nm\n200300400'Hn (mT)a c\nb d\n05 1 0 1 5 2 0FFT (arb. units)\nf (GHz)48 nm\n39 nm\n29 nm\n15 nmf1\nf2012340100µ0'n2\nFig. 50.6 0.9 1.280160240µ0'H1 (mT)\n1/L2 (10-3 nm-2)eFIG. 5: Determination of the spin sti\u000bness constant from MO experiments. a, Oscillatory\npart of the MO signal (points) measured in a 48 nm thick epilayer with 7% nominal Mn doping\nfor external magnetic \feld \u00160Hext= 20 mT applied along the [010] crystallographic direction;\nline is a \ft by a sum of three damped harmonic functions. b,Fourier spectra of oscillatory MO\nsignals (points) measured for \u00160Hext= 20 mT applied along the [010] crystallographic direction\nin samples prepared by etching from the 48 nm thick epilayer. The curves are labeled by the \flm\nthicknesses, normalized, and vertically o\u000b-set for clarity; lines are \fts by a sum of Lorentzian peaks.\nc,Dependence of the measured precession frequency (points) on the magnetic \feld for two di\u000berent\norientations of the \feld in the 48 nm thick epilayer; lines are \fts by Eq. (1). d,Dependence of the\nmeasured mode spacing on square of the mode number in the 48 nm thick epilayer. e,Dependence\nof the spacing between the two lowest modes (\u0001 H1) on the \flm thickness. Lines in dandeare\n\fts by Eq. (2) with spin sti\u000bness D= 2:43 meV nm2.\n14As seen in Fig. 5b, the frequency f0is independent of the \flm thickness which con\frms\nthat it corresponds to the uniform precession mode and that the \flm is homogeneous, i.e.,\nthe magnetic anisotropy constants do not vary across the width of the (Ga,Mn)As epilayer.\nThe spacing \u0001 H1shown in Fig. 5e scales as L\u00002and the values of Dextracted from the\nn-dependence of the resonant \feld spacings in the L= 48 nm epilayer (see Fig. 5d) and from\ntheL-dependence of \u0001 H1(see Fig. 5e) give the same D= 2:43\u00060:15 meVnm2. Identical\nvalue of the spin sti\u000bness was also obtained from measurements in an epilayer grown with\nthe same doping and thickness of 18 nm in which we detected the frequencies f0andf1\nand applied Eq. (2). These measurements con\frm the reliability of extracted values of the\nspin sti\u000bness. We note that the SWR frequencies are determined with high accuracy in\nour measurements and that the indicated error bars in Fig. 1d re\rect the uncertainty of\nthe \flm thickness. As shown in Fig. 1d, we observe a consistent, weakly increasing trend\ninDwith increasing doping and values of Dbetween 2 and 3 meVnm2in the studied\nferromagnetic samples with nominal doping 3.8-9%. (Note that apart from the di\u000eculty\nof exciting magnetization precession in the very low and high-doped samples with stable\neasy-axes, the measurements of Dwere unfeasible on the lower doping side of the series\nbecause of the increasing damping and the corresponding inability to detect the higher SWR\nmodes.) Similar to the Gilbert damping constant, our measured spin sti\u000bness constant in the\noptimized (Ga,Mn)As epilayers is comparable to the spin sti\u000bness in conventional transition\nmetal ferromagnets.70\nWe remark, that we tested the inapplicability of the SWR experiments for the direct\ndetermination of the spin sti\u000bness in thick non-uniform materials. In the Supplementary\ninformation we show measurements in \u0018500 nm thick as-grown and annealed samples with\n7% nominal Mn-doping. The Curie temperatures of \u001860 and 90 K can be inferred only\napproximately from smeared out singularities in d\u001a=dT andM(T) and are signi\fcantly\nsmaller than Tcin the thin optimized epilayers with the same nominal doping. The \flms\nare therefore clearly inhomogeneous and contain compensating defects. Because of the large\nthickness of the epilayers we observe up to \fve SWR modes, however, consistent with the\ninhomogeneous structure of the \flms, the corresponding \u0001 Hndo not show the quadratic\nscaling with nof the Kittel modes of Eq. (2).\nIn the experiments discussed above we have established the systematic semiconducting\ndoping trends and basic magnetic characteristics of epilayers which have been optimized to\n15represent as close as possible the intrinsic properties of idealized, uniform and uncompen-\nsated (Ga,Mn)As. Our study supports the overall view of (Ga,Mn)As as a well behaved\nand understood degenerate semiconductor and band ferromagnet and, therefore, an ideal\nmodel system for spintronics research. We conclude in this paragraph by commenting on\nthe implications of systematic studies of optimized (Ga,Mn)As materials in the context of\nthe recurring alternative proposal of an intricate impurity band nature of conduction and\nmagnetism of (Ga,Mn)As.71In the impurity band picture, the Fermi level in materials with\n\u00181021cm\u00003Mn-acceptor densities is assumed to reside in a narrow impurity band detached\nfrom the valence band, i.e., the band structure keeps the form closely reminiscent of a sin-\ngle isolated Mn Gaimpurity level. Previously, the systematic measurements of the infrared\nconductivity on the extensive set of optimized materials49disproved one of the founding\nelements of the impurity band picture which was the red-shift of the mid-infrared peak with\nincreasing doping.72In the systematic measurements in Ref. 49, the mid-infrared peak was\nobserved to blue-shift49,73and experimentalists focusing on the infrared spectroscopy49,73,74\nreached the consensus that the valence and impurity bands are merged in the highly doped\nferromagnetic (Ga,Mn)As materials. The large values of the spin sti\u000bness of the order\nmeVnm2, experimentally determined in the present work, are consistent with model Hamil-\ntonian and ab initio calculations60,75{77which all consider or obtain the band structure of the\nferromagnetic (Ga,Mn)As with merged valence and impurity bands.62On the other hand,\nfor carriers localized in a narrow impurity band the expected spin sti\u000bness would be small\nin a dilute moment system like (Ga,Mn)As, in which the magnetic coupling between remote\nMn moments is mediated by the carriers.78By recognizing that the bands are merged, the\ndistinction between a \"valence\" and \"impurity\" band picture of ferromagnetic (Ga,Mn)As\nbecomes mere semantics with no fundamental physics relevance. 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Nature Materials 4, 195 (2005). arXiv:cond-mat/0503185.\nAcknowledgment\nWe acknowledge theoretical assistence of Pavel Motloch and support from EU ERC Ad-\nvanced Grant No. 268066 and FP7-215368 SemiSpinNet, from the Ministry of Education of\nthe Czech Republic Grants No. LM2011026, from the Grant Agency of the Czech Republic\nGrant No. 202/09/H041 and P204/12/0853, from the Charles University in Prague Grant\nNo. SVV-2012-265306 and 443011, from the Academy of Sciences of the Czech Repub-\nlic Preamium Academiae, and from U.S. grants onr-n000141110780, NSF-MRSEC DMR-\n0820414, NSF-DMR-1105512. .\n22 1 Establishing micromagnetic parameters of ferromagnetic \nsemiconductor (Ga,Mn)As : Supplementary information  \n \nP. Němec,1 V. Novák,2 N. Tesařová,1 E. Rozkotová,1 H. Reichlová2,1, D. Butkovičová1, \nF. Trojánek,1 K. Olejník,2 P. Malý,1 R. P. Campion,3 B. L. Galla gher,3 Jairo Sinova,4,2 and \nT. Jungwirth2,3 \n \n1 Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3,  \n121 16 Prague 2, Czech Republic  \n2 Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 16253 Praha 6, Czech Republic  \n3 School of  Physics and Astronomy, University of Nottingham, Nottingham NG72RD,  \nUnited Kingdom  \n4 Department of Physics, Texas A&M University, College Station, TX 77843 -4242, USA  \n \nEXPERI MENTS PRECEEDING OUR DETERMINATION OF MIC ROMAGNETIC \nPARAMETERS OF (GA,MN )AS \n Magn etic anisotropy fields, Gilbert damping  constant  and spin -stiffness are the  basic \nparameters of a ferromagnet which determine its  micromagnetic properties. The anisotropy \nfields are as sociated with the energy required to coherently rotate  magnetic moments of the \nentire ferromagnet. They can  be determined in a straightforward way in magnetization  or \nmagneto -transport  measurements from external mag netic fields required to reorient the \nmagnetization of a  ferromagnetic sample, or in magnetization dynamics ex periments from the \nfield-dependent resonant frequencies  [1, 2, 3, 4]. \n Gilbert damping characterizes  dissipative  process es that drive  the magnetization  \nmotion towards an equilibrium  state.  This phenomenon is usually investigated by the \nfrequency -domain -based ferromagnetic resonance (FMR) experiment where  the \nphenomenological Gilbert damping coefficient  is deduced  from the resonance peak \nlinewidth  [3, 4]. The experimentally measured FMR linewidths contain not only the \nfrequency -dependent linewidth due to  the Gilbert damping but also the frequency -\nindependent inhomogeneous linewidth broadening [ 3, 4]. To separate them, it is necessary to \nmeasure the linewidth s at several microwave frequencies  [3, 4]. In FMR t hese frequencies are \ngiven by the resonant -cavity fre quency  that significantly complicates the frequency change .  2 Therefore, the experiments are usually performed at only two different frequencies (typically, \n9 and 35 GHz [ 4]) that makes the corresponding separation of the individual components in \nthe measure d signal rather questionable. Alternatively ,  can be determined from the time-\ndomain based magneto -optical pump -and-probe experiment by fitting the damping of the \nmeasured oscillatory data by Landa u-Lifshitz -Gilbert  equation [ 5, 6]. However, to obtain the \nGilbert damping coefficient  from the measured value of  it is necessary to take into account \na realistic magnetic anisotropy of the investigated samples (see below). Moreover, also the \nfrequency dependence of  has to be measured for a separation of the intrinsic value of the \nGilbert damping coefficient  from the inhomogeneous parts of  . The absence of these two \nrequirements  and the un -optimized magnetic properties of the investigated samples led to  a \nlarge scatter in the deduced values of  for Ga 1-xMn xAs with a different Mn content x: The \nincrease of  from  0.02 to  0.08 for the increase of x from 3.6% to 7.5% was reported in \nRef. 5. On the contrary, in Ref. 6 the values of  from 0.06 to 0.19 – without any apparent \ndoping trend – were observed for x from 2% to 11%.  \nThe spin -stiffness is associated with the exchange energy of non -uniform local \ndirections of the magnetization, in particular with the energy of small wave -vector spin -wave \nexcitations of the ferromagnet. Considering a specific model of the rmodynamic properties of \nthe studied ferromagnet, the spin -stiffness can be indirectly inferred from the measured \ntemperature dependence of magnetization  [7], Curie temperature  [7], or domain wall width  \n[8]. The direct determination of the spin -stiffness f rom magnetization dynamics experiments \nis significantly more challenging than in the case of the magnetic anisotropy fields  [9-13]. The \nlow-energy non -uniform collective excitations of the system can be strongly affected by \ninhomogeneities or surface prope rties of the ferromagnet for which specific models have to be \nassumed in order to extract the spin -stiffness constant from the measured data. An exception \nare the Kittel spin -wave modes of a uniform thin -film ferromagnet for which the spin -stiffness \nparame ter D is directly obtained from the measured resonant f ields (see below) . To date, spin -\nwave resonance measurements of (Ga,Mn)As have been reported on > 100 nm thick epilayers  \n[9-12]. The Kittel modes with Hn ~ n2 were observed only in a 120 nm thick, 8% Mn doped \n(Ga,Mn)As for magnetic fields applied close to the magnetic easy -axis [11]. Measurements of \nthe same sample in other field orientations showed different trends which indicated the \npresence of strong inhomogeneities and surface dependent effects  [11]. A non -Kittel -like \nlinear or sublinear dependence of the resonant fields on the mode index has been reported also \nin the other ferromagnetic resonance measurements of thick (Ga,Mn)As epilayers  [9-12]. In  3 complementary studies of the magnetization dynami cs induced and detected  by magneto -\noptical pump -and-probe measurements, only two resonant frequencies were identi fied [13]. \nBased on the theoretical modeling, they were not ascribed to the Kittel modes but rather to \ncoupled bulk -surface modes which again m ade the extraction of the spin -stiffness constant \ndependent on the considered model of bulk and surface properties of the studied sample  [13]. \nThe extracted values of the spin -stiffness from all available magnetic resonance data in \n(Ga,Mn)As materials, com plemented by values inferred from magnetization and domain \nstudies  [7, 8], are scattered over more than an order of magnitude and show no clear trend as a \nfunction of Mn -doping or other material parameters of the (Ga,Mn)As ferromagnetic \nsemiconductor  [14]. \nIn this Supplementary material we show how we are able to deduce from a single  \nmagneto -optical pump -and-probe experiment all these micromagnetic parameters. In \nparticular, the anisotropy fields can be determined from the dependence of the precession \nfrequ ency on the external magnetic filed combined with the probe -polarization dependence of \nthe precession signal amplitude. The Gilbert damping constant can be deduced from the \nprecession signal dampin g. Finally, the spin stiffness can be obtained from the mut ual spacing \nof the precession modes  which are present in the measured oscillatory magneto -optical signal.  \n  \nSAMPLES  \nThe time -resolved magneto -optical experim ents described  below were performed in a \nlarge set of optimized  (Ga,Mn)As epilayers  whose selected properties are described in detail \nin the main paper.  In Fig. 1 we show results of the Hall effect measurements at 4.2 K. For this \npurpose the samples were lithographically patterned into Hall -bars of 60 m width. It can be \nseen in the figure that the Hall signal is affected by longitudinal magnetoresistance Rxx of the \nsamples, especially at low dopings. Therefore, we extracted p from high field data and by \nfitting the measured transversal resistance Rxy by \n \n Rxy = B/(epd) + k 1Rxx + k 2Rxx2       (1) \n \nwhere d is the sample thickness and k1 and k2 are fitting constants reflecting the anomalous \nHall effect and possible imperfections in the geometry of the Hall bars. We also emphasize \nthat, apart from the commo n experimental scatter and from the corrections due to the non -zero \nmagnetoresistance and due to the anomalous Hall effect, the carrier density can in principle be  4 inferred only approximately from the slope of the Hall curve in a multi -band, spin -orbit \ncoupled exchange -split system such as the (Ga,Mn)As. The error bar due to the multi -band \nnature is estimated to  be 20% [ 15]. Due to these uncertainties we can only make semi -\nquantitative conclusions based on the measured Hall effect hole densities.  \n0 5 10 15 200.800.850.900.951.00\n 1.5%\n 2%\n 3%\n 5.2%\n 13%RXX / RXX(0)\n0Hext (T)1\n234\n \n0 5 10 15 20020406080\n 1.5%     2%\n 3%        5.2%\n 13%RXY()\n0Hext (T)1\n234  \n \nFig. 1. (a) Longitudinal resistances Rxx [normalized to Rxx(0)], and (b) transversal (Hall) resistances  Rxy as a \nfunction of normal magnetic field μ0Hext measured in (Ga,Mn)As epilayer with depicted Mn concentration x; \nsample s temperature 4.2 K.  \n \n For a n evaluation of material parameters from an experimentally measured data (e.g., \nfor an evaluation of the hole densities from the measured transversal resistances  which is \ndescribed above)  it is necessary to know the (Ga,Mn)As epilayer thicknesses. However, \naccurate determination of layer thicknesses is a nontrivial task in case of thin (Ga,Mn)As \nlayers. Some standard techniques (e.g. , X-ray reflectivity  or optical ellipsometry) are \ninapplicable due to the weak contrast between the (Ga,Mn)As layer and the GaAs substrate, \nor unknown optical parameters. The relative accuracy of other common techniques (e.g. , of \nX-ray diffraction) does not exceed 10% because  of the small thickness of the measured layer. \nTherefore, we used a thickness estimation based on the following quantities: (i) the growth \ntime and the growth rate of the GaAs buffer layer measured by the RHEED oscillations \n(typical accuracy of ±3%); (ii) increase in the growth rate by adding the known Mn -flux \nmeasured by the beam -flux monitor relatively to the Ga flux (typical accuracy of ±5% of the \nMn vs. Ga flux ratio); (iii) reduction of thickness by the native oxidation ( -1.5 nm ± 0.5 nm); \n(iv) reducti on of thickness by thermal oxidation ( -1.0 nm ± 0.5 nm). Relative accuracy of \nsteps (i) and (ii) was verified on separate calibration  growths of (Ga,Mn)As on AlAs, where \nan accurate X-ray reflectivity method to measure the (Ga,Mn)As layer thickness could b e \nused. Typical thicknesses of the native and the thermal oxides in steps (iii) and (iv) were  5 determined by XPS. The resulting total accuracy of the (Ga,Mn)As layer  thickness \ndetermination  is thus 3% (relative random error) and 1 nm (systematic error) . \n \nEXPERIMENTAL DETAILS  ABOUT MAGNETO -OPTICAL EXPERIMENTS  \nWe investigated laser -pulse induced dynamics of magnetization by a pump -and-probe \nmagneto -optical (MO) technique.  A schematic diagram of the experimental set -up is shown \nin Fig. 2.  The output of a femt osecond laser is divided into a strong pump pulse and a weak \nprobe pulse that are focused to a same spot on the sample. Laser pulses, with the time width \nof 200 fs and the repetition rate of 82 MHz, were tuned to 1.64 eV, i.e. above the \nsemiconductor band gap, in order to excite magnetization dynamics by photon absorption. \nThe pump pulses were usually circularly polarized (with a helicity controlled by a wave plate) \nand the probe pulses were linearly polarized. The measured magneto -optical signals \ncorrespon d to the probe polarization rotation induced by the pump pulses (see Fig. 2). The \nexperiment was performed close to the normal incidence geometry ( θi = 2° and 8° for pump \nand probe pulses, respectively) with a sample mounted  in a cryostat, which was placed \nbetween the poles of an electromagnet. All the experimental data in this Supplementary \nmaterial were measured at temperature of 15 K , at pump exci tation intensity 30  -40 J.cm-2, \nand they correspond to the helicity -independent part of the measured signal [ 16]. The external \nmagnetic field Hext was applied in the sample plane at an angle H with respect to the [100] \ncrystallographic direction in the s ample plane (see Fig. 2). Prior to all time -resolved \nexperiments, we always prepared the magnetization in a well-defined state by first applying a \nstrong saturating magnetic field at an angle H and then reducing it to the desired magnitude \nof  Hext. \n \nFig. 2.  Schematic diagram of the experimental set -up for a detection of the magnetization precession induced by \nan impact of the circularly polarized femtosecond laser pump pulse  in (Ga,Mn)As . Rotation of the polarization \nplane of reflected linearly polarize d probe pulses is measured as a function of the time delay  ∆t between pump \nand probe pulses. The orientation of magnetization in the sample is described by the polar angle  and azimuthal \nangle . The external magnetic field Hext is applied in the sample plane at an angle H.  6  \nThere are several microscopic mechan isms that can lead to a precession of \nmagnetization due to the impact of pump laser pulse . In particular, very recently we reported \non the precession of magnetization due to optical spin -transfer torque (OSTT)  [16] and optical \nspin-orbital torque (OS OT) [ 17]. However, the most common mechanism, which is \nresponsible for the oscillatory MO signals measured in the majority of (Ga,Mn)As samples at \nlow excitation intensities, is the change of the sample magnetic anisotropy due to the pump -\ninduced temperature inc rease [17] that is schematically shown in Fig. 3. Before an impact of \nthe pump pulse the magnetization points to the easy axis direction [see Fig. 3(a)]. Absorption \nof the laser pulse leads to a photo -injection of electron -hole pairs. The subsequent fast \nnonradiative recombination of photo -injected electrons induces a transient increase of the \nlattice temperature (within tens of picoseconds after the impact of the pump pulse). The laser -\ninduced change of the lattice temperature  then leads to a change of the  easy axis position  [17]. \nAs a result , magnetization starts to follow the easy axis shift by the precessional motion  [see \nFig. 3(b)]. Finally, dissipation of the heat and recombination of the excess holes lead to the \nreturn of the easy axis to the equilibr ium position and the precession of magnetization is \nstopped by the Gilbert damping  [see Fig. 3(c)]. The most important point from the perspective \nof the present paper is that the precession of magnetization induced by the laser pulses is \ndetermined by the magnetic anisotropy of the sample which makes this method an all -optical \nanalog to FMR [ 18]. \n \n \nFig. 3. Schematic illustration of the thermal laser pulse -induced precession of magnetization.  (a) In the \nequilibrium , the magnetization points to the easy axis  direction, which is located in the sample plane at azimuthal \nangle . (b) Impact of a pump pulse induces a transient increase of the lattice temperature that lead s to a change \nof the easy axis position and, consequently, to the precession of magnetization . (c) Dissipation of the heat lead s \nto the return of the easy axis to the equilibrium position. Simultaneously with this, the precession of \nmagnetization is stopped by the Gilbert damping.  \n  7 ANALYTICAL DESCRIPTION OF MAGNE TIZATION DYNAMICS IN  (GA,MN)AS  \nThe dynamics of magnetization is described by the Landau -Lifshitz -Gilbert (LLG) \nequation. We used LLG equation in spherical coordinates where the time evolution of \nmagnetization magnitude Ms and orientation, which is characterized by the polar  and \nazimuthal  angles, is given by:  \n \n0dtdMs\n ,           (2) \n \nsin 12BAM dtd\ns\n,       (3) \n\n  \nsin sin 12BAM dtd\ns\n,       (4) \n \nwhere  is the Gilbert damping coefficient . The gyromagnetic ratio \n/Bg , where g = 2 \nis the Land é g-factor of Mn moments, B the Bohr magneton, and \n is the reduced Planc k \nconstant . Functions\nddFA  and \nddFB  are the derivatives of the energy density \nfunctional F with respect to  and , respectively. We expressed F in a form  [2]: \n \n\n   \n\n\n\n    \n\n\nH H H extu\nout c\nHKK KMF\n  \ncos sin sin cos cos2sin1 sin2cos cos sin2sin41sin2 2 2 2 2 2\n, (5) \n \nwhere Kc, Ku and Kout are constants that characterize the cubic, uniaxial and out -of-plane \nmagnetic anisotropy  fields in (Ga,Mn)As , respectively,  and Hext is the external magnetic field \nwhose ori entation is given by the angles H and H. For a small deviations  and  from the \nequilibrium values 0 and 0, the solution of Eqs. ( 3) and ( 4) can be written in a form  \n \n      tf eA ttkD2cos0\n,       (6) \n      tf eA ttkD2cos0\n,       (7) \n \nwhere the constants A (A) and  θ () describe the initial amplitude and phase of θ (), \nrespectively. The precession frequency f and oscillation damping rate kD are given by   8  \n\n   \n \n   \n    \n\n\n\n\n\n    \n\n\n \n   \n\n\n \n\n\n22\n22\n2sin3124cos53812 cos2sin2 4cos2 cos4sin2 4cos322 cos2sin2 4cos2 cos4sin224cos32 cos\n21\n      \n  \n\n\nu\nc out H extu c H extuc\nout H extu c H extuc\nout H ext\nB\nKK K HK K HKKK HK K HKKK H\ngf\nh\n, \n            (8) \n        2sin31 4cos5322 cos 2122 uc\nout H extB\nD KKK Hgk\n. (9) \n  \nIn our case , the investigated  (Ga,Mn)As epilayers are in -plane magnets (i.e., θ  π/2), the \nexternal magnetic field is applied in the sample plane (i.e., θH  π/2), and the precession \ndamping is relatively slow (i.e., 2  0) which yields  \n \n \n\n      \n2sin2 4cos2 cos4sin224cos32 cos2\nu c H extuc\nout H ext B\nK K HKKK H\nhgf\n  \n\n\n  , (10) \n \n         2sin31 4cos5322 cos 22uc\nout H extB\nD KKK Hgk\n. (11) \n \nEq. ( 10) express the sensitivity of the magnetization precession frequency to the magnetic \nanisotropy of the material that is a well -known effect which form the basis for the \ninterpretation of  FMR [3]. More interestingly, Eq. (11) shows that the precession damping kD, \nwhich is measured experimentally, depends not only on the Gilbert damping parametr   but \nalso on the sample anisotropy and on the mutual orientation of the external magnetic field and \nthe magnetization .   \n We note  that in previously reported magneto -optical pump -and-probe experiments [ 5, \n6, 19-23] the measured  experimental data were modeled by LLG equation in the form  \n  9 \n\n\n\n\n \n\n  \ndttMdtMMt HtMdttMd\nseff)()( )( )()( ,    (12) \n \nwhere Heff is the effective magnetic field . However, in (Ga,Mn) As the magnetic anisotropy is \nrather complex  and, therefore , modeling of MO signals by LLG in this form does not provide \nrealistic values of  because it is not possible to disentangle the effect  of magnetic anisotropy \nfrom  [see Eq. ( 11)]. We believe tha t this is one of the reason s why the dependence of  on \nMn concentration was so different in Ref. 5 and Ref 6. Similarly, the change of magnetic \nanisotropy of (Ga,Mn)As during the deposition of metal overlayer could be partially \nresponsible for the changes  of   that were reported in Ref. 23.  \n  \nEVALUATION  OF MAGNETIC ANISOTRO PY  \n  The dependence of the precession frequency on the magnetic anisotropy fields and on \nthe magnitude and orientation of external magnetic field [cf. Eq. (10)] enables to evaluate the  \nmagnetic anisotropy from the experimentally me asured precession frequencies very similarly \nas in the case of FMR [ 3]. In particular, for a sufficiently strong external magnetic field  = \nH and the following equations can be used to fit the precession fre quencies measured   \na) for Hext along  the [110] crystallographic direction (i.e., H = /4): \n  u c ext c out extBK K HK K Hhgf 2 2 2   \n     (13) \n \nb) for Hext along  the [010] crystallographic direction (i.e., H = /2): \n c ext u c out extBK HK K K Hhgf 2 2 2   \n     (14) \n \nc) for Hext along  the [-110] crystallographic direction (i.e., H = 3/4): \n  u c ext u c out extBK K HK K K Hhgf 2 2 2 2   \n    (15) \n \nAs an example, in Fig. 4 we show the measured dependences f (Hext) and their fits for two \norientations of Hext. To increase the precision of the magnetic anisotropy deter mination even \nfurther, for all the investigated samples we supplemented this method by two additional  10 experimental techniques that provide information about the samples magnetic anisotropy  – \nnamely, the probe -polarization dependence of the MO precession si gnal amplitude and \nSQUID  magnetometry . \n0 100 200 300 400 500 60005101520[010]f (GHz)\n0Hext (mT)[110]1\n234\n \nFig. 4. Dependence of the precession frequency f on external magnetic field Hext applied along the [010] and \n[110] crystallographic directions in Ga1-xMn xAs epilayer with x = 5.2% (points);  the lines are fits by Eqs. ( 14) \nand ( 13), respectively , with Kc = 31 mT, Ku = 27.5 mT, and Kout = -190 mT . \n \nIn (Ga,Mn)As there two MO effects that are responsible for the measured rotation of \nthe polarization plane  of the reflected linearly polarized light  at normal incidence [ 24]. The \nfirst of the MO effects is the well -known polar Kerr effect (PKE), where  occurs due to the \ndifferent index of refraction for  + and  - circularly polarized light propagating parall el to \nthe direction of magnetization  M. The p olarization rotation  due to PKE is proportional to the \nprojection of magnetization to the direction of light propagation , it is linear in magnetization \n(i.e., its sign is changed when the direction of magnetizat ion is reversed) , and it is independent \non the orientation of the input linear polarization  (see Fig. 5(b) for the angle definition ) \n[24]. The second MO effect is the magnetic linear dichroism (MLD), which originates from \ndifferent absorption (reflection ) coefficient for light linearly polarized parallel and \nperpendicular to M, that occurs if the light propagates perpendicular to the direction of \nmagnetization M. The p olarization rotation  due to MLD is proportional to the projection of \nmagnetization to th e direction perpendicular to the direction of light propagation, it is \nquadratic in magnetization (i.e., its sign is not changed when the direction of magnetization is \nreversed) and it varies as sin(2 β) [24]. In Fig.  5(a) we show  the MO signals measured by \nprobe pulses with different orientations  for identical pumping conditions. The measured \ndynamical MO signal δMO , which  is a function of the time delay between pump and probe \npulses t and the probe  polarization orientation  β, can be fitted well by the phenomenological \nequation [ 24],  11  \n   p Gt t\ne C eΦtf A t MO       2cos ,\n,    (16)  \n \nwhere A and C are the amplitudes of the oscillatory and pulse function, respectively, f is the \nferromagnetic moment precession freque ncy, Φ is the phase factor, τG is the Gilbert damping \ntime, and τp is the pulse function decay time. All the measured data in Fig. 5(a) can be fitted \nwell by Eq. ( 16) with a one set of parameters f, τG and τp. The dependence A() obtained by  \n0 500 1000 1500-40-2002040MO signal ( rad)\nTime delay (ps)(a)  = +30 deg\n = +120 deg = +75 deg\n = -15 deg\n     \n  \nFig. 5. (a) Dynamics of the MO signal measured by probe pulses with different probe polarization orientations  \nin (Ga,Mn)As epilayer with x = 5.2% for 0Hext = 0 mT (points); lines are fits by Eq. ( 16) with parameters f = 3.2 \nGHz, τG = 360 ps and τp = 1050 ps. (b ) Definition of the angle  that describes the orientation of the probe \npolarization plane E.  \n-45 0 45 90 13501020\nAMLDAPKEA (rad)\n (deg)0\n \nFig. 6.  Probe -polarization dependence of the oscillatory part A of the MO signal that was obtained by fi tting the \ndynamics shown in Fig. 5(a) by Eq.  (16); the values of A at time delay of 200 ps are shown (points). Lines are \nfits of A() by a sum of a polarization -independent signal due to PKE and a polarization -dependent signal due to \nMLD (Eq. (2) in Ref. 24). The vertical arrow depicts the deduced easy axis position in the sample without the \npump pulse, 0. \n \nthis fitting procedure is displayed in Fig. 6. The position of the maximum in the dependence \nA() at   120° corresponds to the equilibrium position o f the easy axis in the sample - i.e., \nthe in -plane position of magnetization without the pump pulse 0 [24]. The position of the \neasy axis in the sample plane is given by the relative magnitude of the cubic ( Kc) and uniaxial  12 (Ku) anisotropy fields. Therefo re, by measuring 0 without external magnetic field applied,  we \nare directly measuring  the ratio  Kc / Ku. \nThe in -plane anisotropy constants can be obtained also from magnetization loops \nmeasured by SQUID magnetometry. For any external magnetic field the or ientation of \nmagnetization is determined by the minimum of the energy [cf. Eq. ( 5)]. If the orientation of \nmagnetization as function of external magnetic field is known, the projection of the \nmagnetization into the measurement axis can be easily numericall y evaluated for every point \nof the magnetization loop. To obtain the anisotropy constants, we fitted the experimental data \nmeasured by SQUID until we obtained the best agreement between the data and the calculated \nmagnetization – see Fig. 7. It is worth no ting that this model does not describe the switching \nmechanism (governed by the domain wall physics which is not treated in our single domain \ndescription), so the parts of hysteresis loops containing the switching were not used in the \nanalysis. Moreover, i n the case of uniaxial systems (K u>K c) an analytical expression for the \nmagnetization measured along the hard axis can be utilized to analyze the data [ 1]. \n-400 -200 0 200 400-10-50510Magnetization (10-6 emu)\n0Hext (mT)1\n234\n \nFig. 7. Evaluation of the magnetic anisotropy from the SQUID magnetometr y. The SQUID measurement along \n[110] crystallographic direction in (Ga,Mn)As epilayer with x = 5.2% (points) is compared with the calculated \nmagnetization projection  for Kc = 31 mT and Ku = 27.5 mT  (line).  \n \nTo sum up, by a simultaneous fitting of the measu red dependence of the precession \nfrequency on an external magnetic field (Fig. 4), of the MO signal precession amplitude on a  \nprobe -polarization  (Fig. 6), and  of the data measured by  SQUID magnetometry  (Fig. 7) we \nevaluated very precisely the magnetic anis otropy f or all the investigated samples . The \nexample of the obtained in -plane angular dependence of the free energy in (Ga,Mn)As \nepilayer with Mn concentration x = 5.2% is shown in Fig. 8.  13 \n0 45 90 135 180-2-101\n120 130 140 150-1.98-1.96 Free energy F (arb. units)\n (deg)\n[-100][-110][010][110][100]\nF\n (deg) \nFig. 8. In-plane angular dependence of the free energy [Eq. ( 5)] in (Ga,Mn)As epilayer with x = 5.2% ; anisotropy \nfields Kc = 31 mT and  Ku = 27.5 mT . \n \nDETERMINATION OF GIL BERT DAMPING COEFFIC IENT  \n For numerical modeling of the measured MO data, we first computed from the LLG \nequatio n (Eqs. ( 3) and (4) with the measured magnetic anisotropy  fileds ) the time -dependent \ndeviations of the spherical angles [ (t) and (t)] from the corresponding equilibrium values \n(0, 0). Then we calculated how such changes of  and  modify the static magneto -optica l \nresponse of the sample MOstat, which is the signal that we detect experimentally  [24]: \n \n     0\n00 2sin2 2cos2 ,MLD s MLD PKEPMt MPt Pt t MO\n.  (17) \n \nThe first two terms in Eq. ( 17) are connected with the out-of-plane and in -plane movement of \nmagnetization , and the last term describe s a change of the static magneto -optical response of \nthe sample due to the laser -pulses induced demagnetization [ 24]. PPKE and  PMLD are MO \ncoefficients that describe the MO response of the sample which we measured independently \nin a static MO experiment f or all the samples – see Fig. 9 for MO spectra measured in s ample \nwith x = 5.2%.   14 \n1.2 1.4 1.6 1.8 2.0 2.2 2.4-4-20246\n5laserMLDStatic MO signal (mrad)\nEnergy (eV)PKE\n1\n234 \nFig. 9. Spectral dependence of static PKE and MLD  in (Ga,Mn)As epilayer with x = 5.2% , the arrow indicate the \nspectral position of the laser pulses  used in the time -resolved experiment shown in Fig. 5 and Fig. 10; note that \nthe data for MLD are multiplied by 5 for clarity.  \n \n The example s of the fitting of the dynamical MO optical data are shown in Fig. 10. \nThe measured data can be fitted well by LLG  for time delays longer than  150 ps, which is a \ntime that it takes to establish the quasi -equilibrium conditions in the sample. We stress that \nthe only fitting parameters in our modeling are the Gilbert damping coefficient , the initial \ndeviation of the  spherical angles from the corresponding equilibrium values, and the \nparameters describing the in -plane movement of the easy axis and the demagnetization signal, \nwhich are apparent as the non -oscillatory signal in the measured dynamics [ 24]. The obtained \ndependence of  on Hext is shown in Fig. 11(a) for two different orientations of Hext. For Hext \napplied along the [010] direction,  decreases monotonously with Hext. On the contrary, for \nHext applied along [110] direction,  is a non -monotonous  function of  Hext reaching a similar \nvalues of  for 0 mT and 100 mT. However, this non -monotonous dependence is a \nconsequence of the field -induced frequency decrease (see Fig. 4) when the magnetic field is \napplied along the magnetically hard [110] direction (see Fig.  8). When  is plotted as a \nfunction of the precession frequency (rather than the external field) we do not observe any \nsignificant difference between the different crystallographic directions – see Fig. 11(b). A \nfield dependent damping parameter was repor ted in various magnetic ma terials and a variety \nof underl ying mechanisms responsible for it were suggested as an explanation [ 25-29]. We \nnote that t he damping parameter  extracted from the fits should be regarded as  a \nphenomenolo gical parameter that accou nts for combined effect s of a (frequency independent) \nintrinsic Gilbert damping, an inhomogeneous broadening, a two magnon scattering, and \nvarious propagation spin wave processes resulting from the nonuniform spatial profile of the \nexcited precession.  We a lso note that the rate of decrease of  with f is sample dependent and,  15 therefore, we show in the main paper the doping dependence of the frequency -independent \npart of . \n0 500 1000 1500 2000-40-30-20-100MO signal ( rad)\nTime delay (ps)(a) 10 mT1\n234\n \n0 500 1000 1500 2000-20-15-10-50MO signal ( rad)\nTime delay (ps)(b) 400 mT1\n234  \nFig. 10. Dynamics of the MO s ignal measured for  external magnetic field  (a) 0Hext = 10 mT and (b) 0Hext = \n400 mT applied along the [010] crystallographic direction in (Ga,Mn)As epilayer x = 5.2%  (points); lines are fits \nby LLG . \n0 200 400 6000246(a) (10-2)\n0Hext (mT)[110]\n[010]\n \n0 5 10 15 200246(b) (10-2)\nf (GHz)[010], [110]  \nFig. 11. (a) Dependence of the Gilbert damping coefficient α on external magnetic field Hext applied along the \n[010] and [110] crystallographic directions in (Ga,Mn)As epilayer with x = 5.2% . (b) Same data as in ( a) but as a \nfunction of the precession fre quency  f. \n \nDETERMINATION OF SPI N STIFFNESS  \nAs we show in Fig. 5 of the main paper, we observed more than one precession mode \nin (Ga,Mn)As epilayers with a sufficient thickness.  These precession modes are the spin \nwave resonances (SWRs) – i.e., spin waves (or magnons) that are selectively amplified by \nfulfilling the boundary conditions of the thin magnetic film [ 18, 30]. Up to now, SWRs in \n(Ga,Mn)As were investigated mainly in a frequency -domain where they are apparent as \nmultiple absorption peaks in the FM R spectra  [3, 10 - 12]. The existence of multiple  16 resonances in FMR reveal that there exist several external magnetic fields at which the \nLarmor precession frequency in the sample coincides with the microwave frequency. The \nresonant field for the n-th mode  (Hn) is obtained by solving the LLG  equation with a term \ncorresponding to exchange interactions in the material and by considering the appropriate \nboundary condition  [11]. In homogeneous thin films with a thickness L, only the \nperpendicular standing waves  with a wave vector k fulfilling the resonant condition \nn kL\nare amplified; the mode with n = 0 denotes the uniform magnetization precession \nwith zero k vector. In principle, there exist two symmetric  boundary conditions which  are \nschemat ically illustrated in Fig. 12. The position of n-th SWR mode in the FMR spectrum Hn \nis given by the Kit tel relation [ 11] and the following equation applies  \n \n22\n2\n0L gDn H H H\nBn n\n\n ,       (18) \n \nwhere n is an integer, D is the exchange spin stiffness cons tant, B is the Bohr magneton, g is \nthe g -factor, and L is the sample thickness  \n \n \n              \n  \n    \nFig. 12. Spin wave resonances in homogeneous thin magnetic films with a thickness L that have a node (a) or \nmaximum (b) at the surface; n is the mode numb er. \n \nIn FMR only the modes with odd n are observed  [11] and the corresponding resonant \nfields are smaller than that of the uniform magnetization precession (i.e., Hn > 0). In the \nmagneto -optical pump -and-probe experiment , the external magnetic field is ke pt constant \nduring the measurement of any dynamical MO trace. Consequently, the SWRs are apparent as \nadditional frequencies that are larger than that of the uniform magnetization precession. \nUltrafast optical pulses also excite all resonant modes without a ny k selectivity  [18, 30]. \nConsequently, for a homogeneous magnetic film with a given thickness, a higher number  of \nSWRs is detectable in the MO dynamical traces than in the FMR spectra. This is particularly \nimportant for (Ga,Mn)As that is magnetically hom ogeneous only when prepared in a form of  17 rather thin films and, therefore, where only a limited number of SWRs is present within a \ndetectable range of the precession frequencies. For an external magnetic field Hext applied in \nthe sample plane, the angular frequency of the n-th SWR mode  fn is given by  [3, 31] \n \n\n   n u c H extn uc\nout H ext B\nn\nH K K HH KKK H\nhgf\n   \n\n\n \n   \n2sin2 4cos2 cos4sin224cos32 cos2\n, \n           (19) \n \nwhich enables to convert the experimentally measured frequency spacing of individual modes \nto the field differences Hn from which the magnitude of the spin s tiffness D can be evaluated \nusing Eq. ( 18) (see Fig. 5 in the main paper) .  \nAs we illustrate  in the following  chapter, t he magnetic homogenei ty of the \ninvestigated epilayer is  absolutely essential for a correct determination of D from the \nmeasured SWR spac ing. Therefore , the experimental results obtained  in samples that had \nbeen prepared by etching the original 48 nm thick (Ga,Mn)As epilayer  down to the thickness \n39, 29 a nd 15 nm are of fundamental importance.  In Fig. 13 we show  the corresponding FFT \nspectr a of the measured oscillatory MO signal.  Clearly, the frequency f0 of the lowest SWR \ndoes not depend on the film thickness. This confirms that the lowest observed SWR really \ncorresponds to the uniform precession of magnetization and, moreover, it proves th at this film \nis magnetically homogeneous. Also the spacing H1 shows the expected [see Eq. (2)] linear \ndependence on n2 and 1/ L2 (see Fig. 5 in the main paper) that enables a reliable determination \nof the value of D. In Fig. 14 we compare the experimental data for 48 nm and 15 nm thick \nepilayers from which the FFT spectra depicted in Fig.  13 were computed. Clearly, the  etching \nof the sample from 48 nm to 15 nm not only suppressed the higher  SWRs, which is apparent \nfrom the purely sinusoidal shape of the dat a for the 15 nm film , but it also increased the \nprecession damping, which is probably a consequence of a slight variation of the etched film  \nthickness within the laser spot size of 25 m. These data illustrate that the magneto -optical \npump -and-probe experi ment is a very sensitive diagnostic tool not only of the magnetic but \nalso of the structural quality  of thin magnetic films.   18 \n0 5 10 15 20FFT (arb. units)\nf (GHz)48 nm\n39 nm\n29 nm\n15 nmf0\nf1\nf2 \nFig. 13. Fourier spectra of oscillatory MO signals measured for 0Hext = 20 mT applied along the [010] \ncrystallographic direction in samples prepared by etching from 48 nm thick (Ga,Mn)As epilayer with x = 7% \n(points), the curves are labeled by the film thicknesses , normalized  and vertically shifted for clarity; the lines are \nfits by a sum of Lorentzian pea ks. \n0 500 1000 1500 200048 nmMO signal (arb. units)\nTime delay (ps)15 nm1\n234\n \nFig. 14. Comparison of oscillatory parts of MO signals measured in the original 48 nm thick epilayer and in the \nepilayer that was etched down to 15 nm; the curves are normalized and  vertically shifted for clarity. \nExperiment al conditions are described in Fig. 13. \n \nDEMONSTRATION OF INA PPLICABILITY OF SPIN STIFFNESS \nMEASUREMENT  IN THICK (GA,MN)AS EPILAYERS  \nFinally, we illustrate the significance of the film magnetic homogeneity for a correct \nevaluation of the spin stiffness. F or this purpose we selected a 500 nm thick (Ga,Mn)As \nepilayer with 7% Mn (i.e, a sample with the same nominal Mn doping as the one used in \nexperiments depicted in Fig. 13 and Fig. 14). In Fig. 15 we show the temperature dependent \nmagnetization projections to several crystallographic directions  measured in the as -grown and \nannealed sample s. In the as -grown sample , the temperature dependence of magnetization \nprojections is strongly non -monotonous [see Fig. 15(a)]. Moreover,  the Curie temperature  Tc  19 is only  60 K that is very low for a material with 7% Mn. This is a consequence of a high \nconcentration of unintentional interstitial Mn impurities in the sample that compensate both \nthe local moment and the holes produced by substitutional Mn atoms  [32]. The amoun t of \ninterstitial Mn impurities in the sample can be reduced by a thermal annealing  [33]. However, \neven very long annealing times are not sufficient for obtaining a high quality sample from the \nthick epilayer due to the formation of the surface oxide that controls the outdiffusion of \ninterstitial Mn impurities  [33]. Therefore, the 40 h long annealing at 200 °C led to an increase \nof Tc but only to 90 K, which is still substantially lower than Tc  150 K observed in thin \nsamples with the same nominal concentr ation of Mn. Simultaneously, the temperature \ndependence of magnetization does not show the expected sharply vanishing magnetization at \nTc (cf. Fig. 2 in the main paper  for the data in optimized epilayers ).  \n0 25 50 75 10005101520\n [110]\n [-110]\n [010]Magnetization (emu/cm3)\nTemperature (K)(a) as-grown\n1\n234\n \n0 25 50 75 10005101520\n [110]\n [-110]\n [010]Magnetization (emu/cm3)\nTemperature (K)(b) annealed\n1\n234  \nFig. 15. Temperature dependence of the magnetization projections to different crystallographic directions \nmeasured by SQUID  in 500 nm thick (Ga,Mn)As epilayer with x = 7%. (a)  As-grown sample. (b) Sample  \nannealed for 40 hours at 200°C . \n0 50 100 150 200 2506789 (10-3cm)\nTemperature (K)(a)\n1\n234\n \n0 50 100 150 200 250024d/dt (10-5cm/K)\nTemperature (K)(b)\n1\n234  \nFig. 16. Temperature dependence of the resistivity  (a) and its temperature derivative d/dT (b) measured in \n500 nm thick (Ga,Mn)As epilayer with x = 7% annealed for 40 hours at 200°C.  \n \nIn Fig. 16 we show the t emperature dependence of the resistivity and its temperature \nderivative measured in the annealed sample. Clearly, there is no sharp Curie point singularity  20 in the temperature derivative of the resistivity which is the fingerprint of a high magnetic \nquality of (Ga,Mn)As epilayer (cf. Fig. 1(a) in the main paper)  \nIn Fig. 17 we show the time -resolved magneto -optical signals measured in this 500 nm \nthick epilayer. In the as -grown sample two precession modes can be identified . In the \nannealed sample the improved magnetic quality leads to a strong suppression of the \nmagnetization precession damping with respect to that observed in the as -grown sample. For \nexample, the data shown in Fig. 17(a) and (b) for the lowest modes correspond to damping \ntimes of 210 ps and 46 0 ps for the as -grown and annealed sample, respectively. In addition, \nthe annealing led to a considerable increase of the number of observed SWR modes in the \nmeasured TRMO signal. However, their identification is a rather complicated task.  \n-60-30030\n0204060\n0 1000 2000 3000-50050100\n0 500 1000-60060120  (a) as-grown\n  (b) annealedMO signal  ( rad)\nTime delay (ps)MO signal  ( rad)\nTime delay (ps)\n \nFig. 17. Time -resolved magneto -optical signals (points) measured in as -grown (a) and annealed (b) 500 nm thick \n(Ga,Mn)As epilayer with x = 7%; note the different  x-scales in (a) and (b). The lines in the upper parts of the \nfigures are a sum of  damped harmonic functions and the corresponding precession modes are plotted in the lower \nparts of the figures.  External magnetic field of 10  mT was applied along the [010] crystallographic direction.  \n \nIn Fig.  18 we show the FFT spectrum of the oscillato ry MO signal s measured in the \nannealed sample for external magnetic fields  of 10 mT and 20 mT . Even though the magnetic \nfield change was rather  small, the FFT spectra were changed dramatically. In particular, at 10 \nmT there are 3 peaks with comparable inte nsities (and 5 peaks in total) while at 20 mT there \nis only 1 strong peak (and 4 peaks in total).  In Fig. 19 we show the dependence of the \nfrequency of SWR modes on the external magnetic field – at least for the first sight , it is not \napparent how to assig n the observed modes to mode numbers defined by Eq. ( 18), which is \nthe basic requirement for an evaluation of the spin stiffness from the measured data .  \n  21 \n0 5 10FFT (arb. units)\nf (GHz)(a) 10 mT\n1\n234\n0 5 10FFT (arb. units)\nf (GHz)(b) 20 mT\n1\n234 \nFig. 18. Fourier spectrum of the oscillatory p art of the MO signal measured in the annealed sample for external \nmagnetic fields of 10  mT (a) and 20 mT (b) applied along the [010] crystallographic direction (points); the red \nline is a fit by a sum of Lorentzian peaks (green lines) and the arrows indica te positions of the peak frequencies.  \n \n0 10 20 30 40 5002468fn (GHz)\n0Hext (mT)1\n234\n \nFig. 19. Dependence of the precession frequency fn on Hext measured in the annealed sample for external \nmagnetic field applied along the [010] crystallographic direction.  \n \nIn Fig. 20 (a) we  show a plausible assignment  of the measured frequencies to four  \nSWRs described in the previous chapter  and one non -propagating surface mode [ 11]. We note \nthat t he identification  of the lowest mode  for fields below 15 mT as the surface mode is based \non the  analysis reported in Ref. 11 – in particular, due to the observations that this mode is \napparent  only at certain external magnetic fields and that it has a smaller amplitude that the \none assigned to the homogeneous precession [see Fig. 18(a)]. Following t he analysis reported \nin the previous chapter , we can now proceed to the evaluation of the spin stiffness. In Fig.  \n20 (b) the deduced values of Hn are plotted as a function of n2. The observed mode spacing \ndeviates significantly from that expected for SWR s in a magnetically homogeneous film  [see \nEq. ( 18)] which is another fingerprint of the mag netic inhomogeneity in this 500  nm thick \nepilayer [9-12]. Consequently, despite a large number of SWRs detected in this sample,  they \ncannot be used for a direct determination of the spin stiffness. \n  22 \n0 10 20 30 40 5002468\nSMn = 3\nn = 2\nn = 1\nn = 0(a)fn (GHz)\n0Hext (mT)1\n234\n0 2 4 6 8 10050100150\n(b)Hn (mT)\nn21\n234 \nFig. 20. (a) Dependence of the precession frequency fn on Hext measured in the annealed sample for external \nmagnetic field applied along the [010] crystallographic dir ection re-plotted from Fig. 19 with the depicted \nassignment of precession frequencies to the individual SWRs  and to the surface mode , SM (points ). Lines are fits \nby Eq. ( 19). (b) Dependence of mode spacing Hn on square of the mode number n (points), line is the \ntheoretical dependence Hn ~ n2. \n \nREFERENCES  \n[1] Wang, K. -Y. et al.  Spin Reorientation Transition in Single -Domain (Ga,Mn)As.  \nPhys. Rev. Lett . 95, 217204 (2005) . \n[2] Zemen, J.,  Kucera, J.,  Olejnik ,  K.  &  Jungwirth,  T. Magnetocrystalline anisotropies in \n(Ga,Mn)As: Systematic theoretical study and comparison with experiment.  Phys. Rev. B  80, \n155203 (2009).  arXiv:0904.0993  \n[3] Liu, X. & Furdyna, J. K. Ferromagnetic resonance in Ga 1−xMn xAs dilute ma gnetic \nsemiconductors. J. Phys. Cond. Matter.  18, 245-279 (2006).  \n[4] Khazen, Kh. et al . Anisotropic magnetization relaxation in ferromagnetic Ga 1−xMn xAs \nthin films. Phys. Rev. B 78, 195210 (2008).  \n[5] Qi, J. et al.  Ultrafast laser -induced coherent spin dy namics in ferromagnetic \nGa1-xMn xAs/GaAs structures. Phys. Rev. B  79, 085304 (2009 ). \n[6] Kobayashi, S. , Hashimoto, Y. and Munekata , H. 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K., Grimsditch, M., and Merlin, R. \nUltrafast op tical study of magnons in the ferromagnetic  semiconductor GaMnAs . \nSuperlatt. Microstruct.  41, 372 (2007).  \n[32] Jungwirth, T. et al.  Systematic study of Mn -doping trends in optical properties of \n(Ga,Mn)As . Phys. Rev. Lett.  105, 227201 (2010) and its Supplem entary material ,  \narXiv: 1007.4708.  \n[33] Olejník, K., Owen, M. H. S. , Novák, V., Mašek, J., Irvine, A. C. , Wunderlich, J., and \nJungwirth, T. Enhanced annealing, high Curie temperature, and low -voltage gating in \n(Ga,Mn)As:  A surface oxide control study . Phys. Rev. B  78, 054403 (2008),  \narXiv: 0802.2080.  47"    },    {        "title": "1907.02041v3.Anisotropy_of_spin_transfer_torques_and_Gilbert_damping_induced_by_Rashba_coupling.pdf",        "content": "arXiv:1907.02041v3  [cond-mat.mes-hall]  7 Dec 2020Anisotropy of spin-transfer torques and Gilbert damping in duced by Rashba coupling\nI.A. Ado,1P.M. Ostrovsky,2,3and M. Titov1,4\n1Radboud University, Institute for Molecules and Materials , NL-6525 AJ Nijmegen, The Netherlands\n2Max Planck Institute for Solid State Research, Heisenbergs tr.1, 70569 Stuttgart, Germany\n3L.D. Landau Institute for Theoretical Physics RAS, 119334 M oscow, Russia\n4ITMO University, Saint Petersburg 197101, Russia\nSpin-transfer torques (STT), Gilbert damping (GD), and effe ctive spin renormalization (ESR) are\ninvestigated microscopically in a 2D Rashba ferromagnet wi th spin-independent Gaussian white-\nnoise disorder. Rashba spin-orbit coupling-induced aniso tropy of these phenomena is thoroughly\nanalysed. For the case of two partly filled spin subbands, a re markable relation between the\nanisotropic STT, GD, and ESR is established. In the absence o f magnetic field and other torques\non magnetization, this relation corresponds to a current-i nduced motion of a magnetic texture with\nthe classical drift velocity of conduction electrons. Fina lly, we compute spin susceptibility of the\nsystem and generalize the notion of spin-polarized current .\nPossibility to efficiently manipulate magnetic order by\nmeansofelectriccurrenthasgainedalotofattentionover\nthe past decades1,2. Potential applications include race\ntrackmemory3,4, spin torquemagnetization switching5,6,\nskyrmion-based technology7,8, and other promising con-\ncepts. Spintronic logic and memory devices based on\ncurrent-driven magnetization dynamics are believed to\nachieve high speed, low volatility, outstanding durabil-\nity, and low material costs with promises to outperform\ncharge-trapping solid-state memory devices9.\nIn the light of recent detection of fast domain wall\n(DW) motion in magnetic films10,11and predictions of\nevenhigherDWvelocitiesinantiferromagnets12, current-\ninduced dynamics of domain walls, skyrmions, and other\nmagnetic textures remain an important research subject\nin the field of spintronics. Such dynamics is mainly de-\ntermined by the interplay of the two phenomena: Gilbert\ndamping (GD) and spin torques13–16.\nIn the absence of spin-orbit coupling (SOC), spin\ntorques emerge only in the systems with nonuniform\nmagnetization profiles and are most often referred to as\nspin-transfer torques (STT). At the same time, the clas-\nsification of spin torques usually gets more complicated\nif coupling between spin and orbital degrees of freedom\nbecomes pronounced. Moreover, the debate on the mi-\ncroscopic origin of spin torques in the latter case remains\nongoing17,18. Below, we regard STT, in the continuum\nlimit, as a contribution to the total torque on magnetiza-\ntion that is linearwith respect to both the electricfield E\nand the first spatial derivatives of the unit vector of mag-\nnetization direction n. We note that, in the absence of\nSOC, physics of STT is well understood15,16.\nIn a similar fashion, Gilbert damping may be gener-\nally associated with the terms of the Landau-Lifshitz-\nGilbert (LLG) equation that are odd under time reversal\nand linear with respect to the time derivative of n. In\nthe most simplistic approach, GD is modeled by a sin-\ngle phenomenological term αn×∂tnthat corresponds to\n“isotropic” damping.\nHowever, it has been known for quite a while that GD\nmay exhibit anisotropic behaviour19–27. Or, to be more\nprecise, that the scalar damping constant α, in general,should be replaced by a damping matrix with the com-\nponents depending on the orientation of n. These two\nmanifestations of anisotropy may be referred to as rota-\ntional and orientational anisotropy, respectively22. Ex-\nperimental observation of the orientational anisotropy\nof Gilbert damping has been reported very recently in\na metal ferromagnet (FM)/semiconductor interface of\nFe/GaAs(001)28and in epitaxial CoFe films29. The au-\nthors of Ref. [28] argued that the measured anisotropy\nrooted in the interplay of interfacial Rashba and Dressel-\nhaus spin-orbit interaction.\nGiven the equal importance of GD and STT in the\ncontext of current-induced magnetization dynamics and\nthe significant progressmade in the understanding of the\nanisotropic nature of Gilbert damping, we find it surpris-\ning that the anisotropyof spin-transfer torques has so far\nonly been addressed phenomenologically24,30.\nIn the present paper, we consider a 2D Rashba FM\nwithspin-independent electronscattering. Amicroscopic\nanalysis, performed for an arbitrarymagnetization direc-\ntion, allows us to quantify the rotational as well as the\norientationalanisotropyofboth STT and GD induced by\nRashba SOC. Our results indicate that, for a Rashba FM\nsystem, spin-transfer torques TSTTand Gilbert damp-\ningTGDentering the LLG equation\n∂tn=γn×Heff+TSTT+TGD+... (1)\nnaturally acquire the following forms:\nTSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],(2a)\nTGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],(2b)\nwhereξi=ξi(n), the operator ∂v= (vd·∇) is expressed\nvia the classical electron drift velocity vd=eE/planckover2pi1τ/m,\nandn/bardbl/⊥stands for the in-plane/perpendicular-to-the-\nplane component of the vector field n:\nn=n/bardbl+n⊥,n⊥=eznz=ezcosθ.(3)\nFor convenience, we have included the term ξ0∂tninto\nthe definition of TGD. This term, being even under time\nreversal, leads to a renormalization of spin in the LLG2\nequation16and does not contribute to damping. In what\nfollows, we refer to such renormalization as effective spin\nrenormalization (ESR).\nThe rotational and orientational anisotropy arising in\nEqs. (2) appear to be a natural consequence of the fact\nthat the Rashba spin-orbit interaction singles out the di-\nrection perpendicular to the electron 2D plane. The ori-\nentationalanisotropyofthedimensionlessfunctions ξi(n)\nis determined by all space symmetries of the system and,\nfor a general Rashba FM, may turn out to be rather\ncomplex. However, for the particular interface model of\ntheC∞vsymmetry class, which we consider below, one\nsimply finds ξi=ξi(n2\nz).\nBefore we proceed, let us describe at least two impor-\ntant outcomes of Eqs. (2). First, according to the usual\nconvention, STT consist of two contributions: the adi-\nabatic torque ∝(js·∇)nand the nonadiabatic torque\n∝n×(js·∇)n, wherejsdenotes a spin-polarized cur-\nrent. For vanishing SOC, the adiabatictorque has aclear\nphysical meaning. As far as spins of conduction elec-\ntrons adiabatically follow local magnetization direction,\nthe corresponding change of their angular momentum is\ntransferred to the magnetic texture. Since ↑and↓spins\npoint in the opposite directions along n, the transfer rate\nis proportional to ( js·∇)n, where js=j↑−j↓. In\nthe presence of SOC, however, conduction spins are no\nlonger aligned with the direction of nand, thus, the en-\ntire concept of spin-polarized current becomes somewhat\nvague. For the particular Rashba model, our results re-\nveal an important relation between the adiabatic torque\nand ESR, providing steps toward better understanding\nof the former for systems with SOC.\nAnother remarkable property of Eqs. (2) is a simple\nand exact relation between the nonadiabatic torque and\nGD, which has an important implication for current-\ninduced motion of magnetic textures (e.g., domain walls\nor skyrmions). Indeed, by transforming Eq. (1) into the\nmoving reference frame31r′=r−vdt, one immedi-\nately observes that both components of the nonadiabatic\ntorqueareexactlycancelledbythe correspondingGilbert\ndamping terms. Therefore, if the effect of other driving\ntorques on the motion of a magnetic texture is negligible,\nthen its terminal velocity, in the moving reference frame,\nshall vanish for mediate currents32,33(in the absence of\nmagnetic field). This implies that, in the laboratory ref-\nerence frame, the texture moves with the universal elec-\ntron drift velocity vd. Certainly, in the presence of, e.g.,\nspin-orbit torques, which can assist motion of domain\nwalls and skyrmions10,34, the resulting dynamics might\ndiffer. In any case, the analysisofsuch dynamics can still\nbe performed in the moving reference frame, where the\neffect of the nonadiabatic spin-transfer torque is conve-\nniently absent.\nHaving outlined our main results, we skip further dis-\ncussion until Sec VII. The rest of the paper is organized\nas follows. In Sec. I we introduce the model and use\nan expansion in spatial gradients to reduce the analysis\nto a study of a homogeneous system. Self-energy andKubo formulas are addressed in Sec. II. A general re-\nlation between STT, GD, and ESR (in the considered\nmodel) is obtained in Sec. III, while in Sec. IV we estab-\nlish the exact vector structures of these quantities. Some\nanalytical insight into our general results is provided in\nSec. V and Sec. VI. An extensive Discussion of Sec VII\nis followed by Conclusions (and seven Appendices).\nI. MODEL\nA. Generalized torque in s-dmodel\nIn whatfollows, weadoptthe ideologyofthe s-dmodel\nby performing a decomposition of a FM into a system of\nlocalized spins Siand a system of noninteracting con-\nduction electrons. Despite being rather simplistic, this\napproach has proven to describe very well the key prop-\nerties of current-induced magnetization dynamics in fer-\nromagnetic systems35–38.\nIf the value of |Si|=Scan be assumed sufficiently\nlarge, then it is natural to treat the localized spins clas-\nsically by means of the unit vector n(ri) =Si/S, which\npoints in the opposite to local magnetization direction.\nIn this case, the s-d-like local exchange interaction be-\ntween the localized spins and conduction electrons is\ngiven, in the continuum limit, by\nHsd=JsdSn(r,t)·σ, (4)\nwithJsdquantifying the strength of the exchange and\nPauli matrices σrepresenting the spins of conduction\nelectrons.\nIt is known16that interaction of the form of Eq. (4),\nleads to the following LLG equation for the dynamics of\nthe vector n:\n∂tn=γn×Heff+JsdA\n/planckover2pi1[s(r,t)×n(r,t)],(5)\nwhereγisthebaregyromagneticratio, Heffdescribesthe\neffective magnetic field, Adenotes the areaof the magnet\nunit cell, and s(r,t) stands for the nonequilibrium spin\ndensity of conduction electrons39. The second term on\nthe right hand side of Eq. (5) represents the generalized\ntorque on magnetization\nT=JsdA\n/planckover2pi1[s(r,t)×n(r,t)]. (6)\nAssuming slow dynamics of n(r,t) on the scale of elec-\ntron scattering time and smoothness of magnetization\nprofile on the scale of electron mean free path, one may\nexpand the generalized torque in time and space gradi-\nents ofn. In this paper, we consider twoparticularterms\nof such expansion,\nT=TSTT+TGD+..., (7)\nignoring all other contributions (such as, e.g., spin-orbit\ntorques). In Eq. (7) and below, we identify spin-transfer3\ntorquesTSTTas a double response of Tto the electric\nfieldEandtothespatialgradientsof n, whiletheGilbert\ndamping vector TGD(which also includes the ESR term)\nis defined as a response to the time derivative of n,\nTSTT\nα=/summationdisplay\nβγδTSTT\nαβγδEβ∇γnδ, (8a)\nTGD\nα=/summationdisplay\nδTGD\nαδ∂tnδ. (8b)\nMicroscopic analysis of the tensors TSTTandTGDis the\nmain subject of the present work.\nB. Single particle problem\nAccording to Eqs. (8), the vectors TSTTandTGDrep-\nresent linear response to the time derivative of magne-\ntization direction and to the time derivative of vector\npotential, respectively. Hence, computation of both vec-\ntors can be performed with the help of Kubo formulas\nthat make use of Green’s functions of the correspond-\ning time-independent problem. We choose the latter to\noriginate in the 2D Rashba model40with the effective\ns-d-type term of Eq. (4),\nH=p2/2m+αR[p×σ]z+JsdSn(r)·σ,(9)\nwhereαRcharacterizes the strength of Rashba coupling\nandmis the effective electron mass.\nThe Hamiltonian of Eq. (9) should be supplemented\nwith a momentum relaxation mechanism since both STT\nandGDtensors,similarlytotheconductivitytensor,con-\ntain essentially dissipative components. We assume that\nmomentum relaxation in the system is provided by scat-\ntering on a spin-independent Gaussian white-noise dis-\norder potential Vdis(r). Thus, the full Hamiltonian of a\nsingle conduction electron reads\nHdis=H+Vdis(r), (10)\nwhere the disorder potential is characterized by the zero\naverage∝an}b∇acketle{tVdis(r)∝an}b∇acket∇i}ht= 0 and the pair correlator\n∝an}b∇acketle{tVdis(r)Vdis(r′)∝an}b∇acket∇i}ht= (/planckover2pi12/mτ)δ(r−r′).(11)\nThe angular brackets in Eq. (11) stand for the averaging\nover the disorder realizations, τis the mean scattering\ntime measured in the inverse energy units.\nOne can readily observe from Eq. (6) that the general-\nized torque Tcan be understood as a spatial density of a\ndisorder-averagedmeanvalueoftheoperator( JsdA//planckover2pi1)ˆT,\nwhere we refer to\nˆT=σ×n(r), (12)\nas the dimensionless torque operator.C. Expansion in spatial gradients\nComputation of STT involves the expansion of the\nHamiltonian Hof Eq. (9) and the corresponding Green’s\nfunction\nGR,A= (ε−H±i0)−1(13)\nin the first spatial gradients of nup to the linear terms.\nWe obtain the latter utilizing the Taylor expansion\nn(r) =n(r∗)+/summationdisplay\nγ(r−r∗)γ∇γn(r∗),(14)\nat some particular point r∗.\nWith the help of Eq. (14), Hcan be, then, approxi-\nmated as\nH=H+JsdS/summationdisplay\nγ(r−r∗)γ∇γn(r∗)·σ,(15)\nwhere the Hamiltonian\nH=p2/2m+αR[p×σ]z+JsdSn(r∗)·σ(16)\ndescribes the homogeneouselectronic system with a fixed\ndirection of magnetization set by n(r∗).\nSimilarly, we approximate the Green’s function GR,A,\nemploying the Dyson series\nGR,A(r,r′) =GR,A(r−r′)+JsdS/integraldisplay\nd2r′′GR,A(r−r′′)\n×/bracketleftig/summationdisplay\nγ(r′′−r∗)γ∇γn(r∗)·σ/bracketrightig\nGR,A(r′′−r′) (17)\nand the Green’s function\nGR,A= (ε−H±i0)−1(18)\nthat corresponds to the homogeneous system. Note that,\nin Eq. (17), we kept only the terms that are linear in the\ngradients of n, as prescribed.\nD. Spectrum of the homogeneous system\nThe spectrum of Hincorporates two spectral branches\nε±(p) =p2/2m±/radicalig\n∆2\nsd+(αRp)2−2ςαR∆sdpsinθsinϕ,\n(19)\nwhere the angle θstands for the polar angle of nwith\nrespecttothe zaxis[seealsoEq.(3)], while ϕisthe angle\nbetween the momentum pand the in-plane component\nof the vector n:ϕ=φp−φn. We have also introduced\nthe notations\n∆sd=|Jsd|S, ς = signJsd, (20)\nwhere ∆ sdhas a meaning of half of the exchange\ninteraction-induced splitting (in the absence of SOC).4\nFIG. 1. Guide for an eye: spectrum of the homogeneous sys-\ntem of conduction electrons with a fixed direction of magne-\ntization. Note that the actual spectrum is not isotropic, an d\nthe two subbands may even touch each other. We restrict the\nanalysis to the case of ε >∆sd. For the latter, both subbands\nare always partly filled.\nIf the chemical potential εexceeds the value of ∆ sd,\nboth subbands are always partly filled41. Below, we fo-\ncus solely on the latter case, which is schematically illus-\ntrated in Fig. 1. Note that the spectrum is not isotropic.\nMoreover, for finite values of sin θ, separation of the\ntwo subbands diminishes and they may even touch each\nother.\nIn what follows, we also find it convenient to intro-\nduce the energy scale ∆ so=|αR|√\n2mε, which is equal\nto half of the spin-orbit coupling-induced splitting of the\nbranches (for vanishing ∆ sd).\nE. Roots of dispersion relation\nNow let us analyze the roots of the dispersion of\nEq. (19). Using, for example, Ref. [42], one can show\nthat, under the assumption ε >∆sd, the quartic func-\ntion (ε+(p)−ε)(ε−(p)−ε) of the absolute value of mo-\nmentum palways has four real roots: two positive and\ntwonegative. The former twodefine the angle-dependent\nFermi momenta p±corresponding to ε±branches. The\nfour roots are distinct in all cases, except one. Namely,\nwhenn⊥= 0 (i.e., when sin θ= 1) and ∆ so= ∆sd, the\nsubbands touch each other. We will not consider this\nparticular case.\nUsing the notation p±,negfor the negative roots, we\nhave\np−> p+>0> p+,neg> p−,neg, (21)\nwhere\np∓=1\n2/parenleftbigg√\n2u±/radicalig\n−2u−2q−r/radicalbig\n2/u/parenrightbigg\n,(22a)\np±,neg=1\n2/parenleftbigg\n−√\n2u±/radicalig\n−2u−2q+r/radicalbig\n2/u/parenrightbigg\n,(22b)\nu >0 is the largest root of the resolvent cubic\nu3+qu2−(s−q2/4)u−r2/8, (23)while the parameters q,s, andrare given by\nq=−4m(ε+mα2\nR), s= (2m)2(ε2−∆2\nsd),(24a)\nr= 8m2αRς∆sdsinθsinϕ. (24b)\nIt is straightforward to see, from Eqs. (24), that the\ndependence on the momentum angle enters Eq. (23) only\nvia the parameter r2. As a result, the quantity umay\nonly depend on sin2ϕand other parameters of the model\nthat areϕindependent. This will play an important role\nbelow.\nForαR= 0 (vanishing SOC), ∆ sd= 0 (nonmagnetic\nlimit), or n=n⊥(perpendicular-to-the-plane magneti-\nzation) situation with the rootsbecomes less complex. In\nthese cases, ( ε+(p)−ε)(ε−(p)−ε) is biquadratic (with\nrespect to p) andp±=−p±,neg, as one can also see di-\nrectly from Eqs. (22). Furthermore, the Fermi momenta\np±, then, are angle independent, while their values yield\nthe relations\np2\n±= 2m[ε∓∆sd],forαR= 0,(25a)\np2\n±= 2m/bracketleftbig\nε+mα2\nR∓λ(0)/bracketrightbig\n,for ∆sd= 0,(25b)\np2\n±= 2m/bracketleftbig\nε+mα2\nR∓λ(∆sd)/bracketrightbig\n,forn=n⊥,(25c)\nwhereλ(Υ) =/radicalbig\nΥ2+2εmα2\nR+m2α4\nR.\nII. DISORDER AVERAGING\nHaving analysed the spectrum of the “clean” homoge-\nneous system, we can proceed with the inclusion of the\ndisorder. In what follows, we assume ε0τ≫1, where\nε0is the difference between the Fermi energy εand the\nclosest band edge. We start with a calculation of the\nself-energy in the first Born approximation.\nA. Self-energy\nAccording to Eq. (11), the self-energy is defined as\nΣR,A(r) = (/planckover2pi12/mτ)GR,A(r,r), (26)\nwith the Green’s function GR,Aof Eq. (13). It should be\nexplicitly pronounced that ΣR,A(r) may have a spatial\ndependenceoriginatinginthespatialdependenceof n(r).\nHowever, as we are about to see, the first spatial gradi-\nents of magnetization do not affect the self-energy in the\nmodel under consideration.\nDisregarding the “real” part of the self-energy that\nshould be included in the renormalized value of the\nchemical potential, we focus only on the calculation of\nImΣ(r) =−i[ΣR(r)−ΣA(r)]/2. By substituting the\nexpansion of Eq. (17) into Eq. (26), switching to mo-\nmentum representation, and symmetrizing the result we\nobtain\nImΣ(r) = Σ(0)+/summationdisplay\nγδ/braceleftig\n(r−r∗)γΣ(1)\nδ+Σ(2)\nγδ/bracerightig\n∇γnδ(r∗),\n(27)5\nwith\nΣ(0)=1\n2imτ/integraldisplayd2p\n(2π)2/parenleftbig\nGR−GA/parenrightbig\n,(28a)\nΣ(1)\nδ=ς∆sd\n2imτ/integraldisplayd2p\n(2π)2/parenleftig\nGRσδGR−GAσδGA/parenrightig\n,(28b)\nΣ(2)\nγδ=ς∆sd/planckover2pi1\n4mτ/integraldisplayd2p\n(2π)2/parenleftig\nGRσδGRvγGR−\nGRvγGRσδGR+h.c./parenrightig\n,(28c)\nwhere “h.c.” denotes Hermitian conjugate, GR,Ais the\nGreen’s function of Eq. (18) in momentum representa-\ntion,\nGR,A=ε−p2/2m+αR[p×σ]z+ς∆sdn(r∗)·σ\n(ε−ε+(p)±i0)(ε−ε−(p)±i0),(29)\nandv=∂H/∂pis the velocity operator. In Eqs. (28),\nΣ(0)defines the scattering time (for uniform magneti-\nzation), Σ(1)corresponds to the renormalization of the\ngradient term on the right hand side of Eq. (15), while\nΣ(2)determinesthe possible dependence ofthe scattering\ntime on the first spatial gradients of magnetization.\nTo proceed, we take advantage of the additional sym-\nmetrization of the integrands with respect to the trans-\nformation43ϕ→π−ϕand observe that, in the first\nBorn approximation, integration over the absolute value\nof momentum, in Eqs. (28), is reduced to a calculation\nof residues at p=p±. Using Eqs. (22), we, then, get\nΣ(0)=−1\n2τ/integraldisplay2π\n0dϕ\n2π/bracketleftbig\n1+rW1+rW2n(r∗)·σ\n+W3n/bardbl(r∗)·σsinϕ/bracketrightbig\n,(30)\nwhereWi=Wi/parenleftbig\nr2,u(r2)/parenrightbig\nare some functions of the pa-\nrameterr2andϕ-independent parameters of the model.\nSincer∝sinϕand, obviously, all integrals of the form/integraltext2π\n0W(sin2ϕ)sinϕdϕvanish for arbitrary function W,\nwe obtain a particularly simple result for the constant\npart of the self-energy,\nΣ(0)=−1/2τ. (31)\nSimilar, but more lengthy, analysis shows that each\ncomponent of Σ(1)and Σ(2)is equal to zero. Therefore,\nthereexistsnorenormalizationofthegradienttermofthe\nHamiltonian Has well as no scattering time dependence\non the first magnetization gradients. The self-energy, in\nthe first Born approximation, is found as\nΣR,A(r) =∓i/2τ. (32)\nB. Kubo formula for STT\nAs was outlined in Sec. IB, the generalized torque\nT(r0) of Eq. (6), at a certain position r0in space, is de-\nfined as a disorder-averaged mean value of the operatorFIG.2. Diagrammatic representationoftheSTTtensor TSTT\nαβγδ\nof Eq. (34). Solid lines correspond to the disorder-average d\nGreen’s functions gR,A. Vertex corrections (impurity ladders)\nare represented by green fillings.\n(JsdA//planckover2pi1)δr0ˆT, whereδr0=δ(r−r0). At zero tempera-\nture, thelinearresponse44ofTα(r0)tothezerofrequency\nelectric field Eis given by the standard Kubo expression\ne/planckover2pi1\n2πJsdA\n/planckover2pi1/angbracketleftig\nTr/bracketleftig\nGAδr0ˆTαGRv/bracketrightig\nE/angbracketrightig\n,(33)\nwherev=∂H/∂pis the velocity operator, Tr stands for\nthe operator trace, and angular brackets represent the\ndisorder averaging.\nFrom Eq. (33), we can further deduce the Kubo for-\nmula for spin-transfer torques. In order to do that, we\nsubstitutetheexpansionofEq.(17)intoEq.(33)andcol-\nlect all terms proportional to ∇γnδ(r∗). Then we switch\ntomomentum representationandperformspatialaverag-\ning of torque on the scale of transport mean free path in\nthe vicinity of r=r0. In the noncrossingapproximation,\nthis leads to the general formula for the STT tensor,\nTSTT\nαβγδ=e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\n×itr/bracketleftig\ngAσδgAvγgAˆTvc\nαgRvvc\nβ−h.c./bracketrightig\n,(34)\nwhere the superscript “vc” marks the vertices corrected\nwith the impurity ladders, the notation tr refers to the\nmatrix trace, and\ngR,A=∝an}b∇acketle{tGR,A∝an}b∇acket∇i}ht= (ε−H±i/2τ)−1(35)\nis the disorder-averaged Green’s function of the homoge-\nneous system. In Eq. (35), we have used the result for\nthe self-energy obtained in Sec. IIA.\nThe expression of Eq. (34) is represented diagrammat-\nically in Fig. 2. We note that similar diagrams have been\nused in Ref. [45] to compute STT in a 3D FM, in the\nabsence of SOC, and in Ref. [46] to study STT for the\nmodel of massive Dirac fermions.\nC. Kubo formula for GD and ESR\nSimilarly, from the zero frequency linear response44of\nTα(r0) to the time derivative of n,\nJsdS/planckover2pi1\n2πJsdA\n/planckover2pi1/angbracketleftig\nTr/bracketleftig\nGAδr0ˆTαGRσ/bracketrightig\n∂tn/angbracketrightig\n,(36)6\nonemayderivethe formulaforthe GDtensorofEq.(8b),\nTGD\nαδ=∆2\nsdA\n2π/planckover2pi12S/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAˆTvc\nαgRσδ/bracketrightig\n,(37)\nwhere, according to the definition of TGD, spatial depen-\ndence of nis completely disregarded.\nNote that n,∇γnδ, and∂tnin Eqs. (8), (34), and (37)\nare all taken at r=r0. From now on, we consistently\nomit the argument of all these functions.\nD. Relation between TGDand vertex corrections\nto the torque operator ˆT\nVertex corrected torque operator that enters both\nEqs. (34) and (37) can be expressed with the help of\nvertex corrected Pauli matrices. One can infer the latter\nfrom the “matrix of one dressing” M, whose elements\nMij=1\n2mτ/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAσigRσj/bracketrightig\n(38)\narethe coordinates(in the basis {σx,σy,σz}) ofthe oper-\natorσidressed with a single impurity line. We note that,\nin the model considered, vertex corrected Pauli matrices\nσvc\niappear to have zero trace if ε >∆sd. This is a direct\nconsequence of the fact that the self-energy in Eq. (32) is\nscalar. Hence, {σx,σy,σz}is, indeed, a proper basis for\nthe operators σvc\ni.\nMatrixrepresentationofthe operator ˆT=σ×n, with\nrespect to this basis, is defined as\nˆTi=/summationdisplay\njUijσj, U=\n0nz−ny\n−nz0nx\nny−nx0\n.(39)\nSince, obviously,\nˆTvc\ni=/summationdisplay\njUijσvc\nj, (40)\nwe can see, from Eq. (38), that the geometric series\nT=U(M+M2+···) =UM(I−M)−1,(41)\nprovides the matrix representation of vertex corrections\nto the torque operator. Moreover, from Eq. (37), it is\nevident that the GD tensor is, in fact, determined by the\nsame matrix T,\nTGD\nαδ=∆2\nsdAmτ\nπ/planckover2pi12STαδ. (42)\nE. Crossing diagrams\nIt has been demonstrated recently that the diagrams\nwith two crossing impurity lines may contribute to such\nquantitiesasthe anomalousHall effect47–49, the spinHalleffect50, and the Kerr effect51in the same leading or-\nder with respect to the small parameter ( ε0τ)−1, as the\nconventionalnoncrossingapproximationdoes. Scattering\nmechanisms associated with these diagrams, in general,\nshould affect spin torques and damping as well.\nIn the presentstudy we, however,completely disregard\nthe crossing diagrams, as being significantly more diffi-\ncult to calculate. At the same time, preliminary anal-\nysis shows that the related additional contributions to\nSTT, GD, and ESR are parametrically different from the\npresent resultsand that, for ε≫∆sd, they are negligible.\nIII. RELATION BETWEEN STT, GD, AND ESR\nA. Symmetrization of STT diagrams\nCalculation of spin-transfer torques can be performed\nwith the help of Eq. (34) directly. Such brute-force cal-\nculation has been originally performed by us. We have,\nhowever, subsequently found a shortcut that makes it\npossible not only to obtain the same results in a much\nmore concise manner but also to establish a general re-\nlation between TSTTandTGDtensors. This alternative\napproach takes a reformulation of the result of Eq. (34)\nin a more symmetric form.\nWe apply the identity gAvγgA=∂gA/∂pγin Eq. (34)\nand perform integration by parts. Then, we take a half-\nsum of the result obtained and the original expression of\nEq. (34). This leads to the formula\nTSTT\nαβγδ=δTSTT\nαβγδ+e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\n×i\n2tr/bracketleftbigg\n−gAσδgAˆTvc\nαgR∂vvc\nβ\n∂pγ−h.c./bracketrightbigg\n,(43)\nwhere the first term on the right-hand side\nδTSTT\nαβγδ=e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\ni\n2tr/bracketleftig\ngAσδgAvγgAˆTvc\nαgRvvc\nβ−gAvγgAσδgAˆTvc\nαgRvvc\nβ\n−gAσδgAˆTvc\nαgRvγgRvvc\nβ−h.c./bracketrightig\n.(44)\nis illustrated schematically in Fig. 3 by a group of en-\ncircled diagrams. The remaining two diagrams in Fig. 3\ncorrespond to the second term on the right-hand side of\nEq.(43). We will seebelowthat, in fact, the entiretensor\nδTSTTdoes vanish.\nB. Relation between TSTTand vertex corrections\nto the torque operator ˆT\nAs was argued in Ref. 52 on the basis of perturbative\nexpansions, the velocity operator v=p/m−αR[ez×σ],7\nFIG. 3. Another diagrammatic representation of the STT tens orTSTT\nαβγδ, given by Eq. (43). Six diagrams encircled by the\ndashed line define the δTSTT\nαβγδtensor of Eq. (44) that vanishes for any direction of nprovided ε >∆sd. Solid lines correspond\nto the disorder-averaged Green’s functions gR,A. Vertex corrections (impurity ladders) are represented by green fillings.\ncorrectedbyanimpurityladder,hasaparticularlysimple\nform in the present model,\nvvc=p/m. (45)\nA formal proof of this statement that does not refer to\nany perturbative expansion is presented in Appendix A.\nInterestingly, Eq. (45) also allows to make a spin-orbit\ntorque (SOT) calculation extremely concise. We provide\na brief discussion of this matter in the same Appendix A.\nIt is important that the momentum operator p, as well\nasvvc, commutes with the Green’s function gR,A. In Ap-\npendix B, we demonstrate that this is sufficient for the\nentire tensor δTSTTto vanish. As a result, TSTTis de-\ntermined by the second term on the right hand side of\nEq. (43) alone. Computation of the this term is facili-\ntated by the relation\n∂vvc\nβ/∂pγ=δβγ/m, (46)\nwhereδq1q2is Kronecker delta. With the help of the\nabove, the STT tensor of Eq. (43) readily simplifies to\nTSTT\nαβγδ=δβγe∆2\nsdA\n2π/planckover2pi1Sm/integraldisplayd2p\n(2π)2\n×i\n2tr/bracketleftig\n−gAσδgAˆTvc\nαgR−h.c./bracketrightig\n,(47)\nsince, as we have mentioned, δTSTT= 0.\nEmploying the Hilbert’s identity for the Green’s func-\ntions of Eq. (35),\ngA−gR=gR(i/τ)gA, (48)\nwe can further reduce44Eq. (47) to the formula\nTSTT\nαβγδ=δβγe∆2\nsdAτ\n2π/planckover2pi1Sm/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAˆTvc\nαgRσδ/bracketrightig\n,(49)which resembles very closely the formula of Eq. (37) for\nthe GD tensor. The result of Eq. (49) can also be ex-\npressed in terms of the matrix Tas\nTSTT\nαβγδ=δβγe∆2\nsdAτ2\nπ/planckover2pi1STαδ, (50)\nwhere we have again used the argumentationof Sec. IID.\nC. Relation between TSTTandTGD\nIt can now be seen that both TSTTandTGDvectors\nturn out to be fully defined by the matrix of vertex cor-\nrectionsTto the torque operator. Moreover, comparison\nof Eq. (42) and Eq. (50) reveals a remarkable direct con-\nnection between the STT and GD tensors,\nTSTT\nαβγδ=δβγe/planckover2pi1τ\nmTGD\nαδ, (51)\nwhich is one of the central results of the paper.\nAccordingtothe definitionsofEqs.(8), the established\nrelation between the two tensors indicates that all quan-\ntities of interest (STT, GD, and ESR) may be related to\nthe action of a single linear operator Ξ,\nTSTT= Ξ[∂vn],TGD= Ξ[∂tn],(52)\non one of the vectors, ∂vnor∂tn. We remind here the\nshort-handed notations for the directional spatial deriva-\ntive53∂v= (vd·∇) and for the classical drift velocity of\nconduction electrons vd=eE/planckover2pi1τ/m.\nThe matrix of the operator Ξ coincides with the ma-\ntrixTGD, being also proportional to the matrix T[see\nEqs. (8b), and (42)]. In the next section we obtain the\ngeneral form of the latter and then use it to derive the\nexact vector forms of TSTTandTGD.8\nIV. VECTOR FORMS\nA. Matrix gauge transformation\nIn order to establish the structure of the operator Ξ,\nit should be first noted that the constraint n2≡1 is\nresponsibleforanessentialfreedominthedefinition of T.\nFor an arbitrary operator of differentiation ∂, we have\n1\n2∂n2=/summationdisplay\nδnδ∂nδ= 0. (53)\nTherefore, the left hand sides of\nTSTT\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδTαδ∂vnδ,(54a)\nTGD\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδTαδ∂tnδ,(54b)\nremain invariant under the addition of the matrix row\nR= (nx,ny,nz), with an arbitrary coefficient, to any of\ntherowsofthematrix T. Inotherwords,thetransforma-\ntionT → T Xdoes not change TSTTandTGD, provided\nTX=T+XR, (55)\nwith any matrix column X= (X1,X2,X3)T.\nB. Vector structure of TSTTandTGD\nThe matrix Tis defined in Eq. (41) with the help of\nthe matrix M. The latter is determined by the disorder-\naveraged Green’s function which, in momentum repre-\nsentation, takes the form\ngR,A=ε±i/2τ−p2/2m+αR[p×σ]z+ς∆sdn·σ\n(ε−ε+(p)±i/2τ)(ε−ε−(p)±i/2τ).\n(56)\nUsing Eq. (56), one can prove that M, in general, is\nexpressed as a linear combination of six matrices,\nI, P, U, U2, P UP, P U2P, (57)\nwhereUis introducedin Eq.(39) and P= diag(1 ,1,0)is\na diagonal matrix. In Appendix C, we demonstrate how\nthe components of this decomposition can be calculated\nforn∝ne}ationslash=n⊥.\nThen, in Appendix D, we show that any power of M\nretains the same structure. It immediately follows that\nthe matrix T=U(M+M2+···) can be represented as\nT=c1U+c2UP+c3U2+c4U3+c5UP UP+c6UP U2P,\n(58)\nwhereciare some dimensionless scalar functions.\nThe representation of Eq. (58) can be substantially\nsimplified with the use of the matrix gauge transforma-\ntion described in the previous section. Namely, by takingadvantage of the directly verifiable relations\nU2=RTR−I, U3=−U, (59a)\nUP UP= (I−P)RTR−n2\nzI, (59b)\nUP U2P=UPRTR−UP+n2\nzU(I−P) (59c)\nwe find that the choice of the gauge\n/tildewideX=−[c3I+c5(I−P)+c6UP]RT,(60)\nfor the transformation T → T /tildewideX≡/tildewideT, leads to\n/tildewideT=t0I+t/bardblUP+t⊥U(I−P),(61)\nor, more explicitly, to\n/tildewideT=\nt0nzt/bardbl−nyt⊥\n−nzt/bardblt0nxt⊥\nnyt/bardbl−nxt/bardblt0\n, (62)\nwhere the quantities tiare related to the matrix Tby\nmeans of the relations\nt0=−c3−c5n2\nz, (63a)\nt/bardbl=c1+c2−(c4+c6), (63b)\nt⊥=c1−c4+c6n2\nz. (63c)\nReplacing Twith/tildewideTin Eqs. (54),\nTSTT\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδ/tildewideTαδ∂vnδ,(64a)\nTGD\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδ/tildewideTαδ∂tnδ,(64b)\nwe observe that the operator Ξ in Eq. (52) is represented\nby three dimensionless quantities ξ0,ξ/bardbl,ξ⊥, such that\nξi=∆2\nsdAmτ\nπ/planckover2pi12Sti, (65)\nwhile the vector structure of TSTTandTGDis, indeed,\nprovided by the formulas\nTSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],\nTGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],\nannouncedin the introductorypart. With someremarks,\nthey remain valid for n=n⊥as well. We consider this\nspecific case separately, in Sec. VB.\nIn the next section, we derive closed-form results for\nξ0,ξ/bardbl, andξ⊥, in two particular regimes. Afterwards, we\nfind asymptotic expansions of these functions in either\nsmallαRor in small ∆ sd. All the obtained results are\ncollected in Table I and represented in Fig. 4 alongside\nwith the corresponding numerical curves.9\nV. CLOSED-FORMS\nThe analysis of TSTTandTGDtensors, as has been\npointed out, reduces to integration in Eq. (38) and sub-\nsequent matrix arithmetics. Unfortunately, for arbitrary\ndirection of magnetization, the results cannot be ex-\npressed in terms of elementary functions. For example,\nforn⊥= 0, Eq. (38) already involves elliptic integrals.\nThe complexity is caused, primarily, by the angle de-\npendence of the dispersion relation roots p±,p±,negof\nEqs. (22). Additional complications arise due to the fact\nthat all four roots are distinct.\nOn the other hand, if the parameter rdefined in\nEq. (24b) vanishes, then the angle dependence of p±,\np±,negis absent and, furthermore, p±=−p±,neg(see\nalso Sec. IE). In this case, angle integration in Eq. (38)\nis trivial, while integration over the absolute value pof\nmomentum can be replaced with an integration over p2.\nFor such integrals, we can extend the integration contour\nto−∞and close it through the upper half-plane. Then\nthe value of the integral is given by a sum of residues at\nthep2\n±poles of Eqs. (25) that acquire finite imaginary\nparts due to a ε→ε+i/2τshift.\nHence, computation of the matrix Mis straightfor-\nward when αR= 0, ∆ sd= 0, orn=n⊥. In this section,\nwe calculate ξ0,ξ/bardbl, andξ⊥, for the first and third cases.\nIn the next section, we use the first two cases as reference\npoints for perturbative analysis of these functions.\nA. Vanishing spin-orbit coupling\nWewillstudythecaseof αR= 0first. Inthe absenceof\nSOC, conservation of spin brings a technical difficulty to\nthe calculation of T. Namely, at zero frequency and zero\nmomentum, the matrix of disorder-averaged advanced-\nretarded spin-spin correlators M(I− M)−1that enters\nEq. (41) cannot be finite. Indeed, using the formulas of\nAppendix C with αR= 0, one finds\nM=I−2ςτ∆sd\n1+(2τ∆sd)2U(I−2ςτ∆sdU),(66)\nso thatI− Mis proportional to U. But det U= 0\nand, therefore, M(I− M)−1=∞. Physically, this di-\nvergence is caused by the absence of linear response of\nelectron spins polarized along nto time-dependent ho-\nmogeneous perturbations of Jsd(cf. Sec. 8.3 in Ref. 54).\nNevertheless, even in the limit of zero momentum and\nzero frequency, STT, GD, and ESR remain finite, since\nthe series\nT=UM+UM2+UM3+... (67)\nactually converges.\nThe sum in Eq. (67) is most easily calculated in the\ndiagonal representation of U,\nU=VUdiagV†, U diag= diag(i,−i,0),(68)which is defined by the unitary matrix\nV=\niny−nxnz√\n2(n2x+n2y)−iny+nxnz√\n2(n2x+n2y)nx\n−inx+nynz√\n2(n2x+n2y)inx−nynz√\n2(n2x+n2y)ny√\nn2x+n2y√\n2√\nn2x+n2y√\n2nz\n.(69)\nIntroducing MU=V†MVand making use of the rela-\ntionUdiag=UdiagP, to take care of the potential diver-\ngence, we can rewrite Eq. (67) as\nT=VUdiag(PMU+PM2\nU+PM3\nU+...)V†,(70)\nwhere, according to Eqs. (66) and (68),\nPMk\nU= diag/parenleftig\n[1+2iςτ∆sd]−k,[1−2iςτ∆sd]−k,0/parenrightig\n.\n(71)\nSummation in Eq. (70) is trivially performed, leading to\nT=−ς\n2τ∆sdVU2\ndiagV†=−ς\n2τ∆sdU2=\nς\n2τ∆sd/parenleftbig\nI−RTR/parenrightbig\n=/tildewideT −/tildewideXR,(72)\nwhere/tildewideT= (ς/2τ∆sd)Irepresents the gauge of Eq. (61)\nand we have used the first identity of Eq. (59a).\nThe above result clearly corresponds to t0=ς/2τ∆sd\nandt/bardbl=t⊥= 0, or\nξ0=ς∆sdAm\n2π/planckover2pi12S, ξ /bardbl=ξ⊥= 0. (73)\nHence, Gilbert damping and the nonadiabatic spin-\ntransfertorqueareboth absentwhen αR= 0, asit should\nbe in the model with no SOC, spin-dependent disorder,\nor other sources of spin relaxation.\nThe parameter ξ0defines the effective spin renormal-\nization(duetoconductionelectrons)intheLLGequation\nas16ξ0=−δSeff/S. In fact, for αR= 0, the effective spin\nrenormalization coincides with actual spin renormaliza-\ntion. Indeed, without SOC, all electrons are polarized\nalong±n, and, for the calculation of the total electron\nspin in a unit cell,\nδS=δS↑−δS↓=ς\n2(N+−N−) =\nςA\n8π2/planckover2pi12\n/integraldisplay\nε+(p)≤εpdpdφ p−/integraldisplay\nε−(p)≤εpdpdφ p\n,(74)\none may use ε±(p)≤ε⇔p2≤2m(ε∓∆sd) to obtain\nδS=−ς∆sdAm\n2π/planckover2pi12. (75)\nThus,δS=−ξ0S=δSeffin this case.\nIn Appendix E, we compute spin susceptibility of the\nsystem for αR∝ne}ationslash= 0 and demonstrate that the spin renor-\nmalization does not depend on the SOC strength. At the10\nsametime, the effectivespinrenormalizationdoes. More-\nover, the identity δSeff=δSis, in fact, a very specific\ncase. It holds either for vanishing spin-orbit interaction,\noratsomeparticularvalueof∆ so≈∆sd, asonecanlearn\nfrom Table I and Fig. 4 (we recall that ∆ so=|αR|√\n2mε\ncharacterizes the SOC-induced splitting of the spectral\nbranches).\nB. Perpendicular-to-the-plane magnetization\nNow we turn to the n=n⊥regime. The formulas of\nAppendix C arenot applicable in this case. Nevertheless,\none can perform the integration in Eq. (38) directly, uti-\nlizing the expression for the Green’s function of Eq. (56)\nwith sinθ= 0 (and n=ezcosθ). It follows that\nM=/bracketleftbig\n1+4τ2(∆2\nsd+∆2\nso)/bracketrightbig−1/parenleftig/bracketleftbig\n1+2(τ∆so)2/bracketrightbig\nP\n+/bracketleftbig\n1+4(τ∆sd)2/bracketrightbig\n(I−P)−2ςτ∆sdP UP/parenrightig\n(76)\nand, after some arithmetic,\nT=ς\n2τ∆sd/bracketleftigg\n1−/parenleftbig\nτ∆2\nso/parenrightbig2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\nP\n+1\n2/bracketleftigg\n∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\nP UP.(77)\nSubstitution of this result into Eqs. (54) shows that, in\nthis case, both TSTTandTGDare represented as linear\ncombinations of two vector forms: ∂n/bardblandn⊥×∂n/bardbl.\nSincen=n⊥and, thus, ∂n⊥= 0, the coefficients in\nfront of these forms should be recognized as t0andt/bardbl,\nrespectively. With the help of Eq. (75), we, therefore,\nfind\nξ0=−δS\nS/bracketleftigg\n1−/parenleftbig\nτ∆2\nso/parenrightbig2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n,(78a)\nξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n.(78b)\nFor a fixed n=n⊥, however, one cannot directly de-\nfineξ⊥. Indeed, the latter function, in this case, is a\nprefactor in front of the vanishing vector form n×∂n⊥\nand, in principle, can be even taken arbitrary. The only\nway to assign a clear meaning to ξ⊥, here, is to consider\nitsasymptoticbehaviouratsmallvaluesofsin θ. Namely,\none should expand the integrands in Eq. (38) up to sin2θ\nand, after the integration, compute the coefficients of the\ndecomposition of Eq. (58) with the same accuracy. Ap-\nplication of a sin θ→0 limit in Eq. (63c), afterwards,\nwill lead to\nξ⊥=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n1\n2∆2\nso/bracketleftbig\n1+(2τ∆sd)2/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n.(79)One may use Eqs. (78b) and (79) to evaluate the\nstrengthoftherotationalanisotropyofGDandthenona-\ndiabatic STT, given n≈n⊥. We see, for example, that,\nfor small sin θ, the ratio\nξ/bardbl/ξ⊥= 2+∆2\nso\n∆2\nsd+(1/2τ)2+O(sin2θ),(80)\nexceeds 2, making the rotational anisotropy considerable\neven if SOC is weak. At the same time, for strong spin-\norbit coupling, ξ/bardblcan potentially be orders of magnitude\nlarger than ξ⊥(see also Fig. 4).\nFor the perpendicular-to-the-plane magnetization, GD\nwas analyzed previously in Ref. [55] under an additional\nassumption of large chemical potential. Our result for\nthe Gilbert damping coefficient ξ/bardbl, given by Eq. (78b),\ncoincides with the expression on the right hand side of\nEq. (25) of Ref. [55], to an overall factor that we were\nunable to identify (most likely, it is equal to 4). The\nτ→ ∞limit of the same expression was derived recently\nin Ref. [56] (with another overall factor). This paper also\nmentions the role of the diagonal terms of the GD tensor\non ESR.\nA separate study of the nonadiabatic STT (also lim-\nited to the n=n⊥case) was reported in Ref. [57]. As\nwe have shown above, this torque should be fully de-\ntermined by the very same function ξ/bardblas is GD. The\nauthors, however, ignored vertex corrections, and, as it\nseems, overlooked this fact. In any case, their results\ndiffer from those of Eq. (78b).\nVI. ASYMPTOTIC EXPANSIONS\nWe proceed with a calculation of the ξiexpansions\nin either small αRor small ∆ sd. To perform such cal-\nculation, one should expand the integrands in Eq. (38)\nor, alternatively, in Eqs. (C2), with respect to the corre-\nspondingvariable. Thentheresultcanbeintegratedover\nthe poles, provided by Eqs. (25a) and (25b), respectively\n(whereεshould be replaced with ε+i/2τ).\nA. Weak spin-orbit coupling\nKeeping the notation of Sec. VA for the matrices M\nandTin the absence of SOC, below we use the symbols\nδMandδTtorepresenttherespectivecontributionspro-\nvided by finite αR.\nSinceδM ∝ne}ationslash= 0, the result of matrix inversion in\nT+δT=U(M+δM)(I−M−δM)−1(81)\nis finite, making the analysis straightforward yet rather\ncumbersome. Retaining only proportionalto α2\nRterms in\nδM(see Appendix F for explicit formulas), we obtain\nδT=δc2P+δc3U+δc4U2+..., (82)11\nξ0/(−δS\nS) orδSeff/δS ξ/bardbl/(|δS\nS|τ∆sd) ξ⊥/(|δS\nS|τ∆sd)\nαR= 0 1 0 0\nO(∆2\nso)1+2(τ∆so)2\n1+(2τ∆sd)21−n2\nz\n1+n2z(∆so/∆sd)2\n1+(2τ∆sd)2/bracketleftbigg\n(2τ∆sd)2+2\n1+n2z/bracketrightbigg(∆so/∆sd)2\n1+(2τ∆sd)21+(2nzτ∆sd)2\n1+n2z\n∆sd→0/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftbigg\n4n2\nz+1+n2\nz\n2(τ∆so)2/bracketrightbigg\n2+1\n(τ∆so)21\n2(τ∆so)2\nn=n⊥1−(τ∆2\nso)2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)21\n2∆2\nso/bracketleftbig\n1+(2τ∆sd)2/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2\nTABLE I. Closed-form results and asymptotic expansions for the dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic\nspin-transfer torques, Gilbert damping, and effective spin renormalization. The results are expressed in terms of the e nergy\nscales ∆ sd=|Jsd|Sand ∆ so=|αR|√\n2εmthat describe, respectively, the exchange and spin-orbit- induced splitting. The second\nrow shows the expansion up to the second order in ∆ so. The third row provides the leading order terms of the expans ion with\nrespect to small ∆ sd. Spin renormalization is defined in Eq. (75) by δS=−JsdSAm/2π/planckover2pi12.\nwhere dots represent terms that do not contribute to the\nδ/tildewideTgauge in the α2\nRorder and\nδc2=∆2\nso\n2∆2\nsd1\n1+n2z, (83a)\nδc3=−τ∆2\nso\nς∆sd/bracketleftig\n1+(2τ∆sd)2/bracketrightig1−n2\nz\n1+n2z,(83b)\nδc4=−∆2\nso\n2∆2\nsd1+(2nzτ∆sd)2\n1+(2τ∆sd)21\n1+n2z.(83c)\nThen, utilizingEqs.(63)with cireplacedby δci,wearrive\nat the second-orderexpansionsin small SOC strength for\nthe functions ξi. Those are collected in the second row\nof Table I.\nWe may again use the obtained results to quantify the\nrotational anisotropy of GD and the nonadiabatic STT\nby computing the ratio\nξ/bardbl/ξ⊥= 2+1−n2\nz\nn2z+1/(2τ∆sd)2+O(∆2\nso).(84)\nFor weak spin-orbit coupling, the rotational anisotropy\nis minimal when magnetization is perpendicular to the\nplane and increases for the magnetization approaching\nthe in-plane direction.\nWe also note that the asymptotic expansions up to the\norderα2\nRallowustoestimatetheorientationalanisotropy\nofξi. Employing the notation ξi=ξi(n2\nz), we find\nξ0(0)−ξ0(1) =2(τ∆so)2\n1+(2τ∆sd)2, (85a)\nξ/bardbl(0)−ξ/bardbl(1) =1\n1+(2τ∆sd)2∆2\nso\n∆2\nsd, (85b)\nξ⊥(0)−ξ⊥(1) =1−(2τ∆sd)2\n1+(2τ∆sd)2∆2\nso\n2∆2\nsd,(85c)\nfor weak SOC. Clearly, ξ0andξ/bardblare both maximal for\nn⊥= 0. On the other hand, the expression on the righthand side of Eq. (85c) can change sign, depending on the\nvalue of τ∆sd. Therefore, the orientational anisotopy of\nξ⊥in a “clean” system ( τ∆sd≫1) differs from that in a\n“dirty” one (Fig. 4 corresponds to the case of a “clean”\nsystem).\nInterestingly, at αR= 0 the matrix function δTturns\nout to be discontinuous. Namely, its elements have fi-\nnite limits for αR→0. This discontinuity has, however,\nno physical consequences, since the matrix δTitself is\nnot gauge invariant. In the δ/tildewideTgauge, the discontinuity\nis removed and, thus, it does not affect the physically\nrelevant quantities ξ0,ξ/bardbl, andξ⊥. This property demon-\nstrates the importance of full analysis of all components\nof the STT and GD tensors.\nB. Weak exchange interaction\nUp to the linear order in ∆ sd, we have\nM=I+2(τ∆so)2P\n1+4(τ∆so)2−2ςτ∆sdU+4(τ∆so)2P UP\n[1+4(τ∆so)2]2.\n(86)\nThis corresponds to the following coefficients of the de-\ncomposition of Eq. (58),\nc1=1\n4(τ∆so)2, c2= 1+1\n4(τ∆so)2, (87a)\nc3=−ςτ∆sd\n4(τ∆so)4, c4= 0, (87b)\nc5=−ςτ∆sd/bracketleftbig\n1+8(τ∆so)2/bracketrightbig\n4(τ∆so)4, c6= 0.(87c)\nSubstituting the latter expressions into Eqs. (63), one\nobtains the leading-order contributions to ξiin the limit\nof small ∆ sd. The respective results are presented in the\nthird row of Table I. Using them, we can find yet another\nexpression for the ratio\nξ/bardbl/ξ⊥= 2+(2τ∆so)2+O(∆2\nsd).(88)12\nFIG. 4. Dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic spin-transfer torques, Gilbert dam ping, and effective spin\nrenormalization as functions of the spin-orbit coupling st rengthαRfor four different polar angles of magnetization ( nz= cosθ).\nThe notations coincide with those of Table I. We use the dimen sionless combinations ετ= 50,τ∆sd= 10. Since for θ= 0 it is\nimpossible to compute ξ⊥numerically, only analytical result is shown. The O/parenleftbig\n1/∆4\nso/parenrightbig\nexpansion is addressed in Appendix G.\nRemarkably, the rotational anisotropy of GD and the\nnonadiabatic STT, ξ/bardbl/ξ⊥= 2, persists to both limits\n∆sd≪∆so≪1/τand ∆ so≪∆sd≪1/τ,(89)\nin which the Fermi surfaces defined in Eq. (19) are not\nonly essentially isotropic but, at the same time, do get\nstrongly broadened by the disorder (the broadening 1 /τexceeds the splitting of the subbands).\nIt is also interesting to mention that, for small values\nof ∆sd, the nonadiabatic spin-transfer torque dominates\nover the adiabatic one: ξ/bardbl,⊥/ξ0∝1/∆sd. This agrees\nwiththeintuitivelogicthat, foraweakexchangebetween\nconduction and localized spins, the former would rather\nnot adiabatically follow the direction of the latter.13\nVII. DISCUSSION\nA. Role of vertex corrections\nWe would like to begin this final section by stressing\nthat it is the accurate consideration of vertex corrections\nthat is responsible for the established vector structures\nof anisotropic STT, GD, and ESR, as well as for the\nrelation between them. Practically none of this would be\nseen from an uncontrolled analysis that ignores vertex\ncorrections.\nFor example, if one does not apply the disorder dress-\ning to the current vertex v, the relation of Eq. (50) will\nno longer be valid. Instead, the STT tensor, in this case,\nwill contain 18 additional nonzero components of differ-\nent symmetries, which one might by mistake interpret as\nphysical torques.\nB. Renormalization of spin\nIn Sec. VA, we have demonstrated that, in the limit\nof vanishing SOC, the ESR factor δSeff=−ξ0Sdoes\ncoincide with the actual total electron spin in a unit cell\nδS=−JsdSAm/2π/planckover2pi12. On the other hand, this equality\nbreaksdown forfinite αR, and the ratio δSeff/δSstarts to\ndepend on all of the parameters of the system, including\nscattering time (see Table I and Fig. 4).\nForlargevaluesofspin-orbit-inducedsplitting∆ so, the\nquantity ξ0(which determines ESR) understandably de-\ncays due to the effective randomization of the electron\nspin direction induced by SOC. What is, however, rather\ninteresting, is that, for relatively small values of αR, the\nESRfactor δSeffexceedsδS,reachingthemaximumvalue\nat ∆so≈∆sd. We do not have an intuitive explanation\nfor such behaviour.\nC. LLG equation\nIt is instructive to compare the microscopic LLG\nEq.(1)toitsconventionalphenomenologicalcounterpart.\nIn the absence of spin-orbit, thermal, and other torques\nthat we do not consider in this study, the latter equation\nreads\n∂tn=γn×Heff+(js·∇)n\n−α[n×∂tn]−β[n×(js·∇)n],(90)\nwhere the vector quantity jsis interpreted as the phe-\nnomenological spin-polarized current, while the param-\netersαandβdefine Gilbert damping and the nonadi-\nabatic spin-transfer torque, respectively. The latter is\nalsocommonlyreferredtoasthe β-torque. Theadiabatic\nspin-transfertorqueisrepresentedbytheterm( js·∇)n,\nwhileHeffstands for effective field contributions.\nFirst, taking into account Eqs. (2), we can rewrite the\nmicroscopic LLG Eq. (1) in a form which is similar tothat of Eq. (90),\n∂tn= ¯γn×Heff+(js·∇)n\n−α/bardbl[n×∂tn/bardbl]−β/bardbl[n×(js·∇)n/bardbl]\n−α⊥[n×∂tn⊥]−β⊥[n×(js·∇)n⊥],(91)\nwhere\njs=vdξ0\n1−ξ0=−vdδSeff\nS+δSeff,(92a)\nα/bardbl,⊥=ξ/bardbl,⊥\n1−ξ0, β/bardbl,⊥=ξ/bardbl,⊥\nξ0,¯γ=γ\n1−ξ0(92b)\nand each of the quantities js,α/bardbl,⊥,β/bardbl,⊥, ¯γdepend on\nthe orientation of the vector n. For the particular 2D\nRashba FM model system considered in this paper,\njs=js(n2\nz), α /bardbl,⊥=α/bardbl,⊥(n2\nz),(93a)\nβ/bardbl,⊥=β/bardbl,⊥(n2\nz),¯γ= ¯γ(n2\nz). (93b)\nWe see that the microscopic LLG Eq. (91) is essentially\nanisotropic, in contrast with the phenomenological LLG\nEq. (90). Namely, the coefficients αandβgot split into\ntwo components each. Moreover, the new coefficients\nα/bardbl,⊥andβ/bardbl,⊥as well as the other parametersof the LLG\nequation became dependent on the direction of magneti-\nzation. We note that the splitting of the GD coefficient α\nhas been reported, for a Rashba FM, in Ref. [58].\nNext, let us comment on the microscopic definiton of\nthe spin-polarized current formulated in Eq. (92a). Nor-\nmally, ifspins ofconductionelectrons(travellingwith the\ncharacteristic velocity v) adiabatically follow the direc-\ntion ofn, one assumes js=−vδS/(S+δS), where δS\nis a contribution from conduction electrons to the total\nspin of the system. In this case, Eq. (90) can be simply\nviewedasamanifestationofthetotalangularmomentum\nconservation (for n×Heff= 0),\n(S+δS)∂tn+δS(v·∇)n= 0. (94)\nwhere−δS(v·∇)nis the rate of angular momentum\ntransfer from conduction to total spin.\nThe definition of the vector quantity js, given by\nEq. (92a), provides a perfect generalization of the above\nlogic for a system with finite Rashba SOC. Indeed, con-\nduction spins no longer follow the direction of n(due to,\ne.g., nonzero damping). Nevertheless, −δSeff(vd·∇)n\nstill has a meaning of the rate of “angular momentum\ntransfer” from the effective conduction spin δSeffto the\ntotalS+δSeff. Importantly, it was a fully controllable\naccurate microscopic treatment of the problem that led\nus to Eq. (92a). (We identified the drift velocity vd\nas a “proportionality coefficient” between the STT and\nGD tensors and observedthat the adiabatic spin-transfer\ntorque and ESR are described by the same quantity ξ0.)\nFinally, for the sake of historical integrity, let us also\nmention that the equalities α/bardbl=β/bardblandα⊥=β⊥, in\nthis system, are equivalent59to the relation\nδSeff=−S/2, (95)\nwhich appears to be rather unphysical.14\nD. Material derivative and moving reference frame\nIn the presence of the anisotropic STT and GD of\nEqs. (2), it is natural to analyse the microscopic LLG\nEq. (1) in such a frame, where the effect of the nonadia-\nbaticspin-transfertorqueisabsent. Namely, inthe frame\nthat moves with the classical drift velocity of conduction\nelectrons vd. One may use a nice analogy to continuum\nmechanics as an illustration of this fact.\nIndeed, despite the essentially anisotropic character of\nbothTSTTandTGD, their sum is conveniently expressed\nin the LLG Eq.(1) viathe operatorofmaterialderivative\nDt=∂t+(vd·∇) as\n(1−ξ0)Dtn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig\nn×Dtn/bardbl/bracketrightbig\n−ξ⊥[n×Dtn⊥]+...,(96)\nwhere we have moved the term ξ0Dtnto the left hand\nside and added ( vd·∇)nto both sides. By considering\nconduction electrons as a “fluid” flowing with the drift\nvelocity vd, one may interpret the material derivatives\nof Eq. (96) as the change rates of components of nthat\nare associated with the electronic “fluid parcels”. Thus,\nin the moving (“flowing”) frame, r′=r−vdt, the ma-\nterial derivatives Dtare automatically replaced31by the\nordinary time derivatives ∂t.\nIn other words, in the movingreferenceframe, Eq. (96)\ntakes the form of the LLG equation\n(1−ξ0)∂tn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig\nn×∂tn/bardbl/bracketrightbig\n−ξ⊥[n×∂tn⊥]+...(97)\nthat comprises the analogue of the adiabatic torque\n(vd·∇)n, two components of damping, and (repre-\nsentedherebydots)allotherpossibletorques. As longas\nthe latterareabsent, the dynamicsofamagnetictexture,\ngoverned by such equation (under mediate currents and\nin the absence of magnetic field), is likely to be a motion\nwith zero terminal velocity (as it is32,33, in the isotropic\ncase, for domain walls). For a general situation, current-\ninduced magnetic dynamics can differ significantly. Nev-\nertheless, it should still be more convenient to perform\nthe analysis once the effect of the nonadiabatic STT has\nbeen accounted for by switching to the “flowing” frame.\nInterestingly, any “propagating” texture of the form\nn(r,t)=ζ(r−vdt)=ζr(t)nullifiesthesum TSTT+TGD.\nHence, for such textures, the LLG Eq. (1) reads\ndζr/dt=γζr×Heff+..., (98)\nwherercan be regardedas aparameter. Ifone takesinto\naccountonlyspin-transfertorquesandfieldlike spin-orbit\ntorque, solutions of this equation will have an oscillatory\ncharacter. Note that Eq. (98) is different from the LLG\nequation\n0 =γζr×Heff (99)\nthat describes the uniform motion of the ground state in\nthe presence of the Galilean invariance [the case α=β\nin Eq. (90)]13,15,16,60.E. Response to electric current\nSo far, we have computed spin-transfer torques as a\nlinear response of the system to the external electric field\nE. In experiment, however, it is not the electric field\nbut rather the electric current jwhich is externally ap-\nplied. To relate spin torques to the latter, one should\ncompute the conductivity tensor ˆ σand, afterwards, use\nthe identity\nE= ˆσ−1j (100)\nto replace Ewithj. Importantly, the conductivity ten-\nsor has to be computed up to the linear order in first\nmagnetization gradients ∇αnβ.\nF. Relation to Edelstein effect\nIt is worth noting that some of our results can be inde-\npendently benchmarked. As it was suggested in Ref. 36,\nthereexistsaconnectionbetweensomeparticularpairsof\nquantities in the model of Eq. (9), as, e.g., between the\nDzyaloshinskii-Moriya interaction strength and the ex-\nchange stiffness, or between spin-orbit torques and spin-\ntransfer torques. The latter relation is relevant to our\nstudy.\nA general interpretation of the approach described in\nRef. 36 would be the following. Suppose there exists\na quantity F(αR) which, for the model with αR= 0,\ndepends on the gradients of n, such that\nF(0) =F(∇xn,∇yn). (101)\nThen, up to the linear order with respect to αR, one\nwould obtain61\nF(αR) =F(0)+αR/bracketleftbigg∂\n∂αRF(/tildewide∇xn,/tildewide∇yn)/bracketrightbigg\nαR=0,(102)\nwhere\n/tildewide∇in=∇in+2mαR\n/planckover2pi1[n×[ez×ei]].(103)\nLet us now choose three functions Fi(αR) to be the\ncomponents of the vector TSTT. Using the expression\nfor the quantity ξ0in the limit αR= 0 (see Table I), we\ncan write\nTSTT=eA\n2π/planckover2pi1Jsdτ(E·∇)n. (104)\nFrom Eq. (102) we, then, find another contribution to\nthe generalized torque in the ∝αRorder\nTSOT=2mαR\n/planckover2pi1eA\n2π/planckover2pi1Jsdτ[n×[ez×E]],(105)\nwhichis preciselythe expressionfor the Edelsteineffect62\ninaformofafieldliketorqueonmagnetization. Inasimi-\nlarway,vanishingofthefunctions ξ/bardblandξ⊥whenαR= 015\ncan be translated into the absence52of the antidamping\nSOT in the model of Eq. (9).\nTheresultofEq.(105)coincideswith thedirectderiva-\ntion of SOT, for the model of Eq. (9), that has been re-\nported previously52. A more compact and accurate form\nof this derivation is also presented in Appendix A. Such\nindependent consistency check adds to the credibility of\nour results.\nCONCLUSIONS\nWe have presented a thorough microscopic analysis of\nSTT, GD, and ESR, for the particular 2D FM system\nwith Rashba spin-orbit coupling and spin-independent\nGaussian white-noise disorder. Assuming arbitrary di-\nrection of magnetization, we have established the ex-\nact relation between these effects. We have intro-\nduced the notion of the matrix gauge transformation\nfor magnetization-dependent phenomena and used it to\nexpress spin-transfer torques, Gilbert damping, and ef-\nfective spin renormalization in terms of meaningful vec-\ntor forms. The latter allowed us to quantify the SOC-\ninduced anisotropy of the former. We have analysed,\nboth analytically and numerically, three dimensionless\nfunctions that fully define anisotropic STT, GD, and\nESR. We have also generalized the concept of spin-\npolarized current, computed spin susceptibility of the\nsystem, and obtained a number of other results.\nIt would be an interesting challenge to observe the\nanisotropy of STT experimentally. It might be possi-\nble to do this by measuring current-induced corrections\nto the magnon spectrum asymmetry that is normally as-\nsociated with the Dzyaloshinskii-Moriya interaction. We\nalso believe that, to some extent, the anisotropy of STT\nand GD might explain the differences in dynamics of do-\nmainwalls(andskyrmions)with differentcharacteristics.\nACKNOWLEDGMENTS\nWe would like to thank Jairo Sinova for pointing out a\nnumber of flaws in the original version of the manuscript.\nWe are also grateful to Artem Abanov, Arne Brataas,\nSergey Brener, Ivan Dmitriev, Rembert Duine, Olena\nGomonay, Andrew Kent, Alessandro Principi, Alireza\nQaiumzadeh, and Yaroslav Tserkovnyak for helpful dis-\ncussions. This research was supported by the JTC-\nFLAGERAProjectGRANSPORTandbytheDutch Sci-\nence Foundation NWO/FOM 13PR3118. M.T. acknowl-\nedges the support from the Russian Science Foundation\nunder Project 17-12-01359.Appendix A: Vertex corrections to velocity\noperator; spin-orbit torque\nIn order to compute vertex corrections to the velocity\noperator v=p/m−αR[ez×σ], we first apply a single\nimpurity line to the scalar part of the latter,\n(p/m)1×dr=1\nmτ/integraldisplayd2p\n(2π)2gR(p/m)gA.(A1)\nDuetothefactthatthemomentumoperator pcommutes\nwith the Green’s functions gR,A, the above relation can\nbe equivalently written as\n(p/m)1×dr=i\nm/integraldisplayd2p\n(2π)2(p/m)/parenleftbig\ngR−gA/parenrightbig\n,(A2)\nwhere we have used the Hilbert’s identity of Eq. (48).\nThe subsequent analysis follows the route of Sec. IIA.\nIntegration over the absolute value of momentum in\nEq. (A2) is performed by computing residues at p=p±.\nSymmetrization of the obtained result, with respect to\nthe transformation43ϕ→π−ϕ, leads to\n(p/m)1×dr=/integraldisplay2π\n0dϕ\n2π/parenleftig\nαR(1+rW4)[ez×σ]\n+(αR+rW5)/braceleftbig\nn/bardbl[σ×n]z−/parenleftbig\nn/bardbl·σ/parenrightbig\n[ez×n]/bracerightbig\ncos2ϕ\n+(W6+W7n·σ)[ez×n]sinϕ/parenrightig\n,(A3)\nwhereWi=Wi/parenleftbig\nr2,u(r2)/parenrightbig\nare some functions of the pa-\nrameterr2andϕ-independent parameters of the model.\nAgain, all terms that contain Wivanish identically after\nintegration over the angle and we conclude that\n(p/m)1×dr=αR[ez×σ]. (A4)\nNext, we observe that the corrected by an impurity\nladder velocity operator vvccan be recast in the form\nvvc=/braceleftbig\np/m−αR[ez×σ]/bracerightbigvc=\np/m+/braceleftbig\n(p/m)1×dr−αR[ez×σ]/bracerightbigvc.(A5)\nAccording to Eq. (A4), expression inside the brackets on\nthe second line vanishes, leading us to the desired result,\nvvc=p/m, (A6)\nwhich coincides with Eq. (45) of the main text. Note\nthat, since the momentum operator commutes with the\nGreen’s functions, Eq. (A6) determines both advanced-\nretardedand retarded-advancedvertexcorrectionsto the\nvelocity operator.\nOne immediate consequence of Eqs. (A4) and (A6) is\na trivial form of spin-orbit torque in the considered inter-\nface Rashba model. Indeed, it was conjectured in Ref. 52\nthat the antidamping SOT, in this model, is identically16\nabsent, while the field-like SOT is entirely isotropic. To\nprove the conjecture, we use the Kubo formula for SOT\nTSOT=eJsdA\n2π/planckover2pi12/integraldisplayd2p\n(2π)2tr/braceleftig\nˆTgR(vvc·E)gA/bracerightig\n.(A7)\nSubstituting vvc=p/mand using Eq. (A1), we immedi-\nately find\nTSOT=eJsdAmτ\n2π/planckover2pi12tr/braceleftig\nˆT/parenleftig\n(p/m)1×dr·E/parenrightig/bracerightig\n,(A8)\nFinally, with the help of Eqs. (12) and (A4), we obtain\nthe expression for spin-orbit torque,\nTSOT=eJsdAmτα R\n2π/planckover2pi12tr/braceleftbig\n[σ×n]([ez×σ]·E)/bracerightbig\n=\neJsdAmτα R\nπ/planckover2pi12[n×[ez×E]],(A9)\nwhich coincides with that of Eq. (105), as expected.\nAppendix B: Vanishing of δTSTT\nWe will now prove that the absence of the spin compo-\nnent inthe vertexcorrectedvelocityoperator vvcnullifies\nthe contribution δTSTTto the STT tensor of Eq. (44).\nUsing cyclic permutations under the matrix trace and\nthe fact that vvc=p/mcommutes with any function of\nmomentum, one can rewrite Eq. (44) as\nδTSTT\nαβγδ=−e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2pβτ\n2mtr[Λ1+Λ2] (B1)\nwith\nΛ1=/parenleftig\nvγgAˆTvc\nαgRσδ−σδgAˆTvc\nαgRvγ/parenrightiggRgA\niτ,(B2a)\nΛ2=/parenleftig\nσδgAvγgAˆTvc\nα−vγgAσδgAˆTvc\nα/parenrightiggRgA\niτ\n−/parenleftig\nˆTvc\nαgRvγgRσδ−ˆTvc\nαgRσδgRvγ/parenrightiggRgA\niτ.(B2b)\nIn Eq. (B2a), we employ the Hilbert’s indentity of\nEq. (48) to replace the factor gRgA/iτwithgR−gAand\nagain use cyclic permutations to obtain\nΛ1=ˆTvc\nαgRσδgRvγgA−ˆTvc\nαgRvγgRσδgA\n−ˆTvc\nαgRσδgAvγgA+ˆTvc\nαgRvγgAσδgA.(B3)\nA similar procedure is performed to simplify the expres-\nsion for Λ 2. We note, however, that terms with only re-\ntarded or only advanced Green’s functions, in Eq. (B2b),\nshould be disregarded44. Hence, gRgA/iτis replaced\nwithgRin the first line of Eq. (B2b) and with −gAin\nthe second line. After moving the torque operator to the\nfirst place in each term,\nΛ2=ˆTvc\nαgRσδgAvγgA−ˆTvc\nαgRvγgAσδgA\n+ˆTvc\nαgRvγgRσδgA−ˆTvc\nαgRσδgRvγgA,(B4)\nwe conclude that Λ 1+Λ2= 0 and, therefore, δTSTT= 0\nas well.Appendix C: Structure of M\nUsing Green’s function of Eq. (56) we compute the\nmatrix trace in Eq. (38) and further symmetrize the in-\ntegrandswith respect to the transformation43ϕ→π−ϕ.\nThis results in the decomposition\nM=γ1I+γ2P+γ3U+γ4U2+γ5P UP+γ6P U2P(C1)\nwhere the coefficients are given in the integral form,\nγ1= 2/bracketleftig/parenleftig\n∆2\nsd+|ε+i/2τ|2/parenrightig\nI0−2(ε+δso)I1+I2/bracketrightig\n,\n(C2a)\nγ2=−4/bracketleftigg\n2δson2\nz\n1−n2zI1+/parenleftbig\n1+n2\nz/parenrightbig\nς∆sd/radicalbig\n1−n2zJ1−1+n2\nz\n1−n2zJ2/bracketrightigg\n,\n(C2b)\nγ3=−2\nτ/bracketleftigg\nς∆sdI0−1/radicalbig\n1−n2zJ1/bracketrightigg\n, (C2c)\nγ4= 4ς∆sd/bracketleftigg\nς∆sdI0−1/radicalbig\n1−n2zJ1/bracketrightigg\n,(C2d)\nγ5=−2\nτ/radicalbig\n1−n2zJ1, (C2e)\nγ6=−4/bracketleftigg\n2δso\n1−n2zI1+ς∆sd/radicalbig\n1−n2zJ1−2\n1−n2zJ2/bracketrightigg\n,(C2f)\nwithδso=mα2\nRand\nIk=/integraldisplayd2p\n(2π)2(2mτ)−1/parenleftbig\np2/2m/parenrightbigk\n|ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2,\n(C3a)\nJk=/integraldisplayd2p\n(2π)2(2mτ)−1(αRpsinϕ)k\n|ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2.\n(C3b)\nSomeofEqs.(C2)formallybecomeinvalidwhen n=n⊥.\nHowever,structureof MandTintherespectivecasewas\nanalysed directly in Sec. VB.\nAppendix D: Structure of Mk\nWe have already demonstrated that\nM ∈spanL,L={I, P, U, U2, P UP, P U2P},(D1)\nLet us now prove that any natural power of Mbelongs\nto the same linear span,\nMk∈spanL,∀k∈N. (D2)\nTheoperationofmatrixproduct, byitself, isnotclosed\non spanL. Moreover, 14 of 36 elements of L×Ldo not17\nbelong to span L. On the other hand, a combination of\ntwo such elements (matrices P UandUP),\nP U+UP={P,U}=U+P UP, (D3)\nobviously does. Similarly, the remaining 12 “unsuit-\nable” elements of L × Ldo form 6 pairs, such that\nthe corresponding anticommutators (namely, {P,U2},\n{P UP,U},{P U2P,U},{P UP,U2},{P U2P,U2}, and\n{P UP,P U2P}) belong to span L.\nIn general, the following statement holds: operation of\nmatrix anticommutation sends elements of L × Lto a\nlinear span of L,\n{,}:L×L → spanL. (D4)\nTaking into account the fact that anticommutator is a\nbilinear map, we deduce from Eq. (D4):\n{,}: spanL×spanL →spanL.(D5)\nFinally, since for arbitrary kwe have\nMk=1\n2{M,Mk−1}, (D6)\nthe desired result, Mk∈spanL, is proven by induction.\nAppendix E: Spin susceptibility in the presence of\nSOC\nIn this Appendix, the total spin δSof conduction elec-\ntronsin a unit cell ofthe area Ais computed fora general\ncase ofαR∝ne}ationslash= 0. We use the following standard definition:\nδS=A\n2πi/integraldisplay\ndǫf(ǫ)/integraldisplayd2p\n(2π/planckover2pi1)2tr/bracketleftigσ\n2/parenleftbig\nGA−GR/parenrightbig/bracketrightig\n,(E1)\nwherefstands for the Fermi-Dirac distribution,\nf(ǫ) = (1+exp[( ǫ−ε)/T])−1, (E2)\nandGA,Rrefers to the momentum-dependent Green’s\nfunction of Eq. (29). We will first consider the in-plane\ncomponent of δS.\nMatrix trace calculation followed by an integration\noverǫ, in Eq. (E1), gives\nδSx=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdnx−αRpy\nε+(p)−ε−(p)(f+−f−),(E3a)\nδSy=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdny+αRpx\nε+(p)−ε−(p)(f+−f−),(E3b)\nwheref±=f(ε±(p)). It is convenient to introduce the\nquantity δS+=δSx+iδSy. For the latter, we find\nδS+=A\n4αR/integraldisplayd2p\n(2π/planckover2pi1)2(f+−f−)\n×/parenleftbigg\ni∂\n∂px−∂\n∂py/parenrightbigg\n[ε+(p)−ε−(p)],(E4)where we took advantage of the fact that the fractions\nin Eqs. (E3) can be expressed as the derivatives with\nrespect to the components of momentum. In the zero-\ntemperaturelimit, onecanuseGreen’stheoremtoreduce\nthe double integrals in Eq. (E4) to the integrals over the\nclosed curves C±={p|ε±(p) =ε},\nδS+=δS+\n++δS−\n+, (E5a)\nδS±\n+=±A\n4αR/integraldisplay\nC±dpx+idpy\n(2π/planckover2pi1)2[ε+(p)−ε−(p)].(E5b)\nNext, we follow the approach used by K.-W. Kim et al.\nin Ref. 41. Using the variable w=px+ipyand the\nrelationε±(p) =p2/2m±[ε+(p)−ε−(p)]/2, we find\nδS±\n+=A\n16π2/planckover2pi12αR/integraldisplay\nC±dw/parenleftbigg\n2ε−w∗w\nm/parenrightbigg\n,(E6)\nwherew∗w=p2andC±={w|ε±(w,w∗) =ε}are now\nregarded as contours in the complex w-plane. Since the\ncontours are closed, Eq. (E6) is further simplifed to\nδS±\n+=−A\n16π2/planckover2pi12mαR/integraldisplay\nC±dww∗w. (E7)\nIn order to perform integration in Eq. (E7), we solve\nthe equation ε±(w,w∗) =εforw∗and express the result\nas a function of w∈C±,\nw∗=2m\nw2/parenleftig\nw/bracketleftbig\nε+mα2\nR/bracketrightbig\n−imαRς∆sdn+±√\nR/parenrightig\n,(E8)\nwheren+=nx+inyandRis a cubic function of w.\nDifferent signs in front of the square root in Eq. (E8)\ncorrespond to two different functions w∗=w∗\n±(w) of\nw∈C±, respectively. We do not specify which sign\ncorrespondsto which function. Such ambiguity, however,\ndoes not affect the final result for δS+. Indeed, it can\nbe proven41that all three zeroes of Rare of the form\nwk=irkn+with real rk. Then, from the general relation\n[ε−ε+(w,w∗)][ε−ε−(w,w∗)] =−R\n+/parenleftbiggw∗w\n2m−/bracketleftbig\nε+mα2\nR/bracketrightbig\n+imαRς∆sdn+\nw/parenrightbigg2\n,(E9)\nwe learn that\n[ε−ε+(wk,w∗\nk)][ε−ε−(wk,w∗\nk)]≥0 (E10)\nand, thus, ε−(wk,w∗\nk)< ε⇒ε+(wk,w∗\nk)≤ε. Hence,\nall the singularities of w∗\n−that lie inside the contour C−\nare, in fact, located inside or, at most, on the contour\nC+(note that C+is inside C−). Disregarding the case63\nwk∈C±and using Cauchy integral theorem, we can\nshrink64C−in Eq. (E7) to obtain\nδS+=−A\n16π2/planckover2pi12mαR/integraldisplay\nC+dw/parenleftbig\nw∗\n++w∗\n−/parenrightbig\nw,(E11)18\nso that the terms ±√\nR, in Eq. (E8), do not contribute\ntoδS+. The only remaining singularity of the integrand\nis located at the origin and, by the residue theorem,\nδS+=−ς∆sdAm\n2π/planckover2pi12n+orδS/bardbl=−ς∆sdAm\n2π/planckover2pi12n/bardbl,(E12)\nwhich completes the computation of the in-plane compo-\nnent ofδS.\nIn order to calculate δSz, it is useful to introduce the\n“magnetization”vector M=ς∆sdn. Intermsof M,one\ncan straightforwardlyestablish the “thermodynamic” re-\nlationδSi=∂Ω/∂Mi, where Ω has a meaning of the\nelectronic grand potential in a unit cell,\nΩ =−TA\n2πi/integraldisplay\ndǫg(ǫ)/integraldisplayd2p\n(2π/planckover2pi1)2tr/bracketleftbig\nGA−GR/bracketrightbig\n,(E13a)\ng(ǫ) = log(1+exp[( ε−ǫ)/T]). (E13b)\nWe further note that, according to Eq. (E12), δSxand\nδSydo not depend on Mz. Therefore, equating the sec-\nond derivatives, we find\n∂δSz\n∂Mα=∂2Ω\n∂Mα∂Mz=∂δSα\n∂Mz= 0,(E14)\nwhereα=x,y. As a result, δSzdoes not depend on Mx\nandMyand, thus, can be computed for Mx=My= 0\n(or, equivalently, for nx=ny= 0).\nFrom Eq. (E1) we obtain\nδSz=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdnz\nε+(p)−ε−(p)(f+−f−),(E15)which, for nx=ny= 0, can be integrated over the mo-\nmentum angle with the result\nδSz=Aς∆sdnz\n4π/planckover2pi12∞/integraldisplay\n0pdpf+−f−/radicalig\n∆2\nsd+(αRp)2.(E16)\nAtzerotemperature, theintegrationdomaininEq.(E16)\nis reduced to a finite interval p+< p < p −, wherep±are\ngiven by Eq. (25c). After some algebraic practice, we\nfinally arrive at\nδSz=Aς∆sdnz\n4π/planckover2pi12α2\nR/radicalig\n∆2\nsd+(αRp)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglep−\np+=−ς∆sdAm\n2π/planckover2pi12nz.\n(E17)\nCombining the results of Eqs. (E12) and (E17) into a\nsingle vector form\nδS=−ς∆sdAm\n2π/planckover2pi12n, (E18)\nweseethat, onaverage,evenforfinite valuesofspin-orbit\ncoupling strength αR, spins of conduction electrons, in\ntheequilibrium, arealignedwiththe localmagnetization.\nMoreover, the spin susceptibility tensor is fully isotropic\nand is expressed by a single scalar parameter\nδS=−|δS|=−ς∆sdAm\n2π/planckover2pi12, (E19)\nwhich coincides with that given by Eq. (75) of the main\ntext.\nAppendix F: Expansion of Mup toα2\nR\nExpansion of Eqs. (C2) up to α2\nR= ∆2\nso/2εmprovides us with the coefficients\nδγ1=−/bracketleftbigg2τ∆so\n1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig\n1+(2nzτ∆sd)2/bracketrightbig\n, δγ 2= 2/bracketleftbiggτ∆so\n1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig\n1−(1+2n2\nz)(2τ∆sd)2/bracketrightbig\n,(F1a)\nδγ3=/bracketleftbigg4τ∆so\n1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2\n1+(2τ∆sd)2ςτ∆sd, δγ 4=−2/bracketleftbigg4τ2∆so∆sd\n1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2\n1+(2τ∆sd)2, (F1b)\nδγ5=−2/bracketleftbigg2τ∆so\n1+(2τ∆sd)2/bracketrightbigg2\nςτ∆sd, δγ 6=−/bracketleftbigg4τ2∆so∆sd\n1+(2τ∆sd)2/bracketrightbigg2\n(F1c)\nof the decomposition that we refer to in Sec. VIA: δM=δγ1I+δγ2P+δγ3U+δγ4U2+δγ5P UP+δγ6P U2P.\nAppendix G: O(1/∆4\nso)expansion of ξi(limit of strong SOC)\nThe quantities ξiare shown in the plots of Fig. 4 as functions of the spin-orbit coupling strength αR(while keeping\nbothmandεconstant). Therefore, the right “tails” of the curves can be pro perly fit using the asymptotic expansion\nwith respect to the parameter 1 /∆so. Such expansion can be obtained indirectly, from the expansion in sm all ∆sd.\nBelow, for consistency with the results of Sec. VIB, we list all the co ntributions to ξithat do not exceed the fourth19\norder in 1 /∆so,\nξ0=−δS\nS/bracketleftigg/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n4n2\nz+1+n2\nz\n2(τ∆so)2/bracketrightigg\n+6/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−3n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n, (G1a)\nξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n2+1\n(τ∆so)2−/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n4n2\nz−1−7n2\nz\n(τ∆so)2/bracketrightigg\n−4/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−3n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n, (G1b)\nξ⊥=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n1\n2(τ∆so)2+/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n2n2\nz+1−5n2\nz\n2(τ∆so)2/bracketrightigg\n+2/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−5n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n. 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Lett. 95, 107204 (2005).\n61According to the definition of Eq. (9), the spin-orbit cou-\npling term has an opposite sign as compared to that used\nin Ref. 36.\n62V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n63The conditions wk∈C±can only be fulfilled for some\nparticular values of ε. SinceδSis a continuous function\nofε, one may just ignore such values.\n64See Ref. 41 for important details on branch cuts."    },    {        "title": "2205.09913v1.Effects_of_Crystalline_Disorder_on_Interfacial_and_Magnetic_Properties_of_Sputtered_Topological_Insulator_Ferromagnet_Heterostructures.pdf",        "content": "Effect s of Crystal line Diso rder on Interfacial and \nMagnetic  Properties of Sputter ed Topological \nInsulator/Ferromagnet  Heterostructures  \n1Nirjhar Bhattacharjee,3Krishnamurthy Mahalingam , 2Adrian Fedorko, 1Alexandria Will -Cole,  \n1Jaehyeon Ryu, 3Michael Page, 3Michael McConney,  1†Hui Fang, 2Don Heiman, 1Nian Xiang \nSun* \n1 Northeastern University, Department of Electrical and Computer Engineering , Boston MA \n02115  \n2 Northeastern University, Department of Physics , Boston MA 02115  \n3Air Force Research Laboratory, Nano -electronic Materials Branch, Wright Patterson Air Force \nBase, OH 05433  \nKEYWORDS. Topological Insulator, Ferromagnet, Interface, Spin Pum ping, Spin Orbit Torque, \nAntiferromagnet  \nABSTRACT . Thin film s of Topological insulators  (TIs)  coupled with ferromagnets  (FMs) are \nexcellent  candidates for energy -efficient spintronics devices.  Here, the effect of crystalline \nstructural disorder of TI on interfac ial and magnetic properties of  sputter -deposited  TI/FM , \nBi2Te3/Ni 80Fe20, heterostructures is reported . Ni and a small er amount  of Fe from Py was found to diffuse  across the interface  and react with  Bi2Te3. For highly crystalline  c-axis oriented  Bi2Te3 \nfilms , a giant enhancement in Gilbert damping  is observed , accompanied by an effective  out-of-\nplane magnetic anisotropy  and enhanced damping -like spin -orbit torque  (DL-SOT) , possibly  due \nto the topological surface states  (TSS)  of Bi2Te3. Further more , a spontaneous exchange bias is \nobserved in hysteresis loop measurements at low temperatures . This is because of an  \nantiferromagnetic  topological interfac ial layer  formed by reaction of the diffused  Ni with Bi2Te3 \nwhich  couples with the FM , Ni80Fe20. For increas ing disorder of Bi 2Te3, a significant  weakening  \nof exchange interaction in the AFM interfacial layer  is found . These experimental results  Abstract \nlength is one paragraph.  \n \n1. INTRODUCTION  \nTopological insulators (TIs) of the (Bi,Sb) 2(Te,Se) 3 family of compounds are van der Waals (vdW) \nchalcogenide materials with tetradymite structures. TIs possess large spin -orbit coupling (SOC) \nresulting in  dissipationless surface conducting st ates – topological surface states ( TSS) [1-3]. \nIntroducing magnetic order in TIs leads to gap opening in the TSS bands and possibility of \ndissipationless quantum anomalous Hall (QAH) and axion insulator states [ 4-21]. Stimulated by \nthese remarkable  material properties, TIs are regarded as  promising candidates for realization of \nenergy efficient spintronic devices. TIs possess highly reactive surfaces , thus making them  \nsusceptible to formation of interfacial phases  when coupled with metalli c films [ 22-25]. Because \nof their composition, t hese interfacial layers  have the potential for hosting fascinating topological \nmagnetic phases [ 25]. The m ajority of reported experiment s have studied TIs grown from \nMolecular Beam Epitaxy (MBE) [22-24], whic h is a standard  technique for growing high -quality , \ncrystalline -ordered thin films . However, MBE  suffers from low throughput and is constrained by sample dimensions , making it incompatible for integration in industrial CMOS processes. \nMagnetron sputtering on the other hand is the semiconductor industry’s  accepted  thin film \ndeposition technique because of its advantage of high throughput  and large area film growth. \nSputtering also allows easy deposition of TIs with varying crystalline disorder [25 -29]. This \ncontrollability  opens up the possibility of exploration of their disorder -dependent electronic and \nmagnetic properties .  \nRecently, the topological antiferromagnetic (AFM) compound  NiBi 2Te4 was discovered in the \ninterface of highly c-axis-oriente d sputtered Bi2Te3/Ni80Fe20 heterostructures [25]. Ni from the \nNi80Fe20 (Py) layer diffuses and reacts with Bi 2Te3 layer, and the reaction is promoted by the \ndelocalized TSS electrons  [23-25]. Also, recent experiment s have  show n the  presence of TSS even \nin amorphous Bi2Se3 [30]. In this work , the effects of crystalline structural disorder on the interface \nand magnetic properties  of Bi 2Te3/Py heterostructures  are investigated . The m agnetic  species , \nlargely Ni and small er amounts of Fe , are found to diffuse across the interface into Bi 2Te3, resulting \nin a magnetic  interfacial layer. For increas ing c-axis-oriented texture of Bi 2Te3, increasing  amount s \nof diffused magnetic  species were found to react with Bi 2Te3, which also leads  to enhanced  \nmagnetic  propert ies. This phenomenon  was identified in  room tem perature  hysteresis loop \nmeasurements  of the magnetic moment versus applied magnetic field , m(H), for the Bi 2Te3/Py \nsamples compared to a Py control sample.  As a result of the diffusion  of the magnetic species (Ni, \nFe) and reaction with Bi 2Te3, the saturation magnetic moment  (m) is reduced by  Δm in the \nBi2Te3/Py compared to Py  samples  suggesting change in valence state of the magnetic species . \nThe values of Δm becomes smaller  for increasing  disorder  in Bi 2Te3 suggesting lesser reaction \nbetween diffused Ni, Fe and Bi 2Te3. Further, a giant enhancement in Gilbert damping , an out-of-\nplane canting of magnetization  and enhanced DL -SOT  were  observed  in samples with highly c-axis oriented TI . However, with  significantly reduced  crystallinity, surprisingly the granular \nBi2Te3 samples had a comparabl e enhanced spin -charge conversion efficiency as samples with \nhighly c-axis-oriented Bi 2Te3, possibly due to the  quantum confinement effect in smaller crystallite \ngrains [ 26,27 ]. Low-temperature m(H) and m(T) measurements revealed an AFM ordered phase \nin the predominantly Ni-diffused Bi2Te3 interface from the  formation of the topological AFM \ncompound  NiBi 2Te4 [25]. Interestingly, the strength of the exchange interaction of the  interfacial  \nAFM phase , as monitore d by the exchange bias,  was found to weaken significantly with increase \nin disorder of the Bi 2Te3 layer. These results indicate  strong topological property of TI s with high \ncrystalline c-axis-oriented growth , which weakens  considerably with in creasing crystalline \ndisorder . These experimental results  show the possibility of tailoring topological properties of TIs \nby control of crystalline structural disorder .  \n2. EXPERIMENTAL RESULTS AND DISCUSSIONS  \n \nFigure 1.  a) Schematic model of three  quintuple Bi2Te3 unit cell s.  Cross -sectional HRTEM images \nshowing structural disorder in b) GBT, c) DBT and d) CBT samples. e) XRD data for the GBT, \nDBT and CBT samples. Inset: normalized plots of symmetric XRD data. f) XRR plots and \ntheoretical fitting for GBT, DBT and CBT samples used for characterization of thickness and \nsurface roughness. The data for CBT samples are similar to the ones in ref [25] . \n2.1. Crystalline Structure Properties of Sputter -deposited Bi 2Te3. Samples of 30 nm  Bi2Te3 \nwith varying crystalline disorder , (1) granular  (GBT), (2) randomly oriented polycrystalline \ndisordered  (DBT) , and (3) highly c-axis-oriented crystalline  (CBT) were grown using RF \nmagnetron sputtering  on amorphous thermally oxidized Si substrates  (see Supporting Information  \nSection S1 for grain size characterization ). Crystalline structural propert y of the Bi 2Te3 samples  \nwere verified using  X-ray diffraction (XRD ) and high -resolution transmission electron microscope \n(HRTEM) i maging measurements , as shown in Fig. 1a,b,d. The GBT samples did not show  any \nsignificant diffraction peaks in the XRD  measurement , suggesting a high amorphous content. \nFurther, HRTEM images of the GBT, DBT and CBT samples shown in Fig. 1a verifies the \ngranular, randomly oriented vdW domains and high c-axis-oriented layered structure , respectively. \nThe thickness and surface roughness  of the samples  were characterized using  X-ray reflectometry \n(XRR ) measurements , as shown in Fig. 1e. From the fitting of XRR data , thickness of ~30 nm was \nobtained for all three samples. The fits to the XRR data also revealed surface roughness of  0.7 nm, \n1.7 nm and 1.0 nm for the GBT, DBT and CBT samples , respectiv ely, which are typical surface \nroughness values for sputter -deposited thin films , confirming growth of high -quality TI films .  \n \n \n  \nFigure 2.  Cross -sectional HRTEM images of a) highly amorphous GBT/Py, b) disordered DBT/Py  \nwith randomly oriented vdW domains  and c) highly c-axis oriented CBT/Py  (similar to ref [25]). \nThe yellow dashed lines mark the approximate interface between the Bi 2Te3 and Py layers.  Atomic \n% (At %) characterized using EDS for d) GBT/Py, e) DBT/Py and f) CBT/Py samples. The \ninterface layers are highlighted in green and blue colors for the Ni -Bi2Te3 and Py*, respectively. \ng) Schematic of the Bi 2Te3/Py samples with the layers marked. h) m(H) measurements for IP \norientation at room temperature showing loss of moments in the Bi 2Te3/Py samples due to \ninterfacial Ni and Fe diffusion and reaction with Bi 2Te3. Inset: comparison of % loss of moments \n(Δm%) in GBT/Py, DBT/Py and CBT/Py samples compared to the control Py sample.  The data  \npresented  for the CBT sample are similar to the ones in ref [25].  \n2.2. Morphology of Interfacial Layer formed by Ni Diffusion  into Bi 2Te3. Heterostructure \nsamples of GBT/Py, DBT/Py and CBT/Py were grown where the thickness of the layers was \nmaintained at 30 nm and 20 nm respectively for Bi 2Te3 and Py respectively. HRTEM imaging and \nenergy  dispersive X -ray spectroscopy (EDS) measurements were performed to characterize the \nmorphology and stoichiometric composition a long the cross section of the samples , as shown in \nFigures  2a-f (see Supporting Information  Table S1 for average atomic % ). The HRTEM images in \nFigures 2 a -c clearly show a highly amo rphous nature of GBT, randomly oriented vdW layered \ncrystalline domains in DBT and highly oriented vdW layers in the CBT layers. A closer \nexamination of the interfaces of the heterostructures also reveal a rougher interface in the \ndisordered GBT/Py and DBT /Py samples compared to the CBT/Py sample. The EDS cross -\nsection al profiles of atomic % of elements in Figures 2d -f show  a significant  diffusion of Ni (and \nsmaller amounts of Fe) across the TI/FM interface into the Bi 2Te3 layers , forming an inter facial \nlayer denoted as  Ni-Bi2Te3. In general, the Ni and Fe have a large gradient over a Bi2Te3 distance \nof 5 to 14 nm, where the Ni averages 30 to 40 %, while the Fe diffusion is much smaller  in the \nGBT /Py and CBT/Py samples.  The disordered GBT/Py and DBT/Py samples also have ~3% of Ni  \ndiffused  throughout the thickness of the Bi 2Te3 layer. However, t he predominantly Ni-Bi2Te3 layer  \nin the  highly  ordered CBT/Py sample  appears to  act as a barrier . This prevents the  diffusion of Ni further into the Bi 2Te3 bulk. The formation of the Ni -Bi2Te3 layer is likewise accompanied by a \nthin Fe -rich region in the intermediate Py layer (marked Py*). It is also noted that the moderately -\ndisordered DBT/Py sample which has randomly oriented vdW polycrystalline domains has \ndeveloped  a much higher Ni and Fe diffusion of ~47% and ~9%  at the interface , respectively. The \nFe diffusion, however, is only ~3 -4% at the interface in the GBT/Py and CBT/Py samples.  \nThe diffusion of Ni into high -quality CBT Bi 2Te3 was previously shown to result from  solid -\nstate reactions leading to the formation of Ni -Te bonds and formation of the topological AFM \ncompound, NiBi 2Te4 [25].  Similar to that study, t he room temperature m(H) measurements can \nbe used here as an indicator of the reaction of Ni with Bi2Te3 that is promoted  by the delocalized \nTSS electrons . As shown in Figure  2h, all the Bi2Te3/Py samples  show a clear decrease in \nsaturation magnetic moment for increasing disorder . This reduction in moments  result s from \nchange in valence state of the reacting magnetic species . This loss of saturation moment  is \ncompared to a control sample of Py  by Δm. The Δm% values were found to be 13%, 3 7% and 41% \nfor the GBT, DBT and CBT samples, respectively. This clear  enhancement  in the  loss of moment s \nwith crystalline order  and hence reactivity of Ni with Bi 2Te3 [25] is due to  strengthening  of TSS \nwith increas ing crystallinity  of Bi 2Te3.  \n2.3. Disorder Effects o n Room -Temperature Magnetic Properties of Bi2Te3/Py. To investigate \nthe effect s of disorder, m(H) hysteresis loop, and ferromagnetic resonance (FMR) were performed \non the three types of samples, highly disordered GBT/Py, moderately disordered DBT/Py and \nhighly ord ered CBT/Py. First, m(H) hysteresis loop measurements were performed on the samples , \nwith the magnetic field oriented in -plane (IP) and out -of-plane (OP) relative to  the film plane , as \nshown in Figures 3 a -c. For increasing c-axis-oriented growth of  the Bi2Te3 layer , the saturation \nfield, Hs measured in the IP and OP configurations show an increasing and decreasing trend , respectively . Also,  the ratio of remanence  to saturation  magnetization  (Mr/Ms) in IP m(H) loop \ndecreases  for increasing crystalline order of Bi 2Te3. These trends indicate a n increase  in effective \nOP magnetic easy-axis with increased c-axis-oriented texture of Bi 2Te3 in the Bi 2Te3/Py \nheterostructure samples . This enhanced OP magnetic anisotropy  is a characteristic of interaction \nof the magnetic moments with large  SOC  in the  interfaces  [31]. A large OP anisotropy has been  \npreviously  predicted and observed in other  TI/FM -based materials systems [32,33 ] (also see \nSupporting Information  Section S4 ). As shown in Figure  3b, the OP m(H) loops for the Bi 2Te3/Py \nsamples also exhibit a smaller hysteresis -loop in the low -field region s. These smaller components \nof m(H) loop are more prominent in the DBT/Py and CBT/Py samples , which otherwise exhibit a \nlower OP easy-axis of magnetic moments compared to the highly c-axis-oriented CBT/Py sample . \nThe DBT/Py sample also had an unusually large coercive field ( Hc) resulting from the randomly \noriented vdW layered crystalline domains , as observed in both the IP and OP m(H) loop \nmeasurements in Figures  3a,b. These effects in the samples with highly disordered TIs are possibly \npresent due to disordered magnetic texture that emerg e in the ir relatively  rougher interfaces with \na net OP component.  \nFurther  information was obtained using  ferromagnetic resonance (FMR) measurements to \nunderstand the changes in magnetization dynamic s with changes in TI disorder . The FMR \nlinewidth ( ΔH) and resonance field ( Hres) were extracted from the FMR sign al at different constant \nfrequencies  (fres) (Supporting Information  Section S3 ). The Gilbert damping parameter  (𝛼) was \nextracted  by fitting a straight line to the FMR linewidth versus frequency plot  using the equation , \n𝛥𝐻=𝛥𝐻0+𝛾\n2𝜋𝛼𝑓𝑟𝑒𝑠. Here , ΔH 0 is the inhomogeneous linewidth and 𝛾 is the gyromagnetic ratio . \nAs shown in Figure  3d, the values of  𝛼 extracted  for the Py, GBT/Py, DBT/Py and CBT/Py \nsamples  are 0.0053, 0.0076, 0.0089 and 0.0123 , respectively.  This shows a progressive increase in 𝛼 with increase in crystallite grain size of Bi 2Te3, and a giant enhancement  when  the Bi2Te3 layer \nis highly c-axis oriented  (summarized in Figure 3f) . This effect was also observed in other \nBi2Te3/FM heterostructure materials (Supporting Information  Section S 5), which signals  a large \nenhancement in SOC and presence of robust  TSS in highly c-axis-oriented Bi 2Te3. In addition , the \neffective magnetization, 4π Meff, and the perpendicular magnetic anisotropy (PMA) field, 𝐻⊥, were \nextracted by fitting the modified Kittel equation  to the fres versus Hres plots shown in Fig ures 3e,f , \n𝑓𝑟𝑒𝑠=𝛾\n2𝜋√(𝐻𝑟𝑒𝑠+𝐻𝑎)(𝐻𝑟𝑒𝑠+𝐻𝑎+4𝜋𝑀𝑒𝑓𝑓), where  4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻⊥ and 𝐻𝑎 is the \nuniaxial anisotropy field . The 4𝜋𝑀𝑠 values were found to be 15.2, 14 .7, 14. 3 and 14.1 kOe , \nrespectively , for Py, GBT /Py, DBT/Py and CBT/Py samples . The decrease  in 4𝜋𝑀𝑠 for increasing \ncrystalline order  demonstrates  the reduction  in saturation magnetization due to interfacial diffusion \nof Ni  and Fe from Py into Bi 2Te3 [25]. Also , the 𝐻⊥ values increased from 4.35, 5.07, 5.54 and \n5.79 kOe, for the Py, GBT/Py , DBT/Py and CBT/Py sample s, respectively. This enhancement  in \n𝐻⊥ supports the m(H) results  that show an increase in effective OP anisotropy  with increasing  c-\naxis-oriented texture of Bi 2Te3. The magnetic properties measured using m(H) loops and FMR are \nsummarized in Table 1. \nThe enhancement in α for increasing c-axis texture of Bi 2Te3 can be attributed to a large spin -\npumping  effect  modeled  by the spin -mixing conductance, 𝑔↑↓=4𝜋𝑀𝑠𝑡𝐹𝑀𝛥𝛼\nℏ𝛾, where tFM is the \nthickness of the FM layer  and 𝛥𝛼 is the enhancement in Gilbert damping  and ℏ is the reduced \nPlank’s constant . The resulting  𝑔↑↓ values increased with increasing crystalline c-axis orientation \nof Bi 2Te3 and, were 2.10×10-18, 2.38×10-18 and 4.08×10-18 m-2 for the GBT/Py, DBT/Py and \nCBT/Py samples , respectively . The 𝑔↑↓ values were calculated  assuming the Gilbert damping \nenhancement  in the Bi 2Te3/Py samples  are entirely due to sp in-pumping . The loss of magnetization \nfrom  interfacial diffusion  of Ni from Py  and spin -memory loss due to interfacial proximity -induced magnetization [ 13-19] may also play a role in the enhancement of α in Bi 2Te3/Py samples . But \nthese contributions to wards  enhancement in α could  not be isolated  because of complexity  in these \nheterostructure material systems. However, the large enhancement in 𝛼 with highly c-axis-oriented \nTIs is also observed in other TI/FM materials  systems  [27,33 -37], including those which do not  \nshow  interfacial diffusion  (see Supporting Information  Section S4) . This suggests that for highly  \ncrystalline  oriented TI s, the TI/FM heterostructures experience a giant  enhancement  in spin -\npumping  predominantly from  the presence of robust TSS. The reduction in magnetization of the \nPy layer because of diffusion of Ni and reacti on with Bi2Te3 was previously shown [ 25]. These \nresults provide strong evidence of enhancement  in SOC strength and topological properties in \nhighly c-axis-oriented TI samples  compared to disordered TIs .  \n \nFigure 3.  Normalized m(H) loops measured : a) in-plane and b) out-of-plane for the GBT/Py, \nDBT/Py and CBT/Py samples. Inset: Expanded low -field regions showing enhanced Hc for the \nBi2Te3/Py samples compared to the Py control sample. c) Comparison of the saturation fields (IP \nand OP) and Mr/Ms ratio clearly highlight ing an increase in OP anisotropy with crystalline c-axis \norientation of Bi 2Te3. d) FMR linewidth versus frequency for extraction of α. e) FMR resonance \nfrequency versus field for extracting 𝐻⊥ and 4πM eff. f) Visual comparison of α and 𝐻⊥ extracted \nfrom d and e respectively. All measurements here were performed at 300 K.  \nTable 1. Summar y of room -temperature magnetic properties of the GBT/Py, DBT/Py and CBT/Py \nheterostructure samples.   \nSample  Hs OOP  Hs IP Mr/Ms 4πMs 𝑯⊥-FMR  \nα \n(kOe)  (kOe)  (%) (kOe)  (kOe)  \nPy 11.9 0.95 96 1.52 4.35 0.0053  \nGBT/Py  11.2 1.22 96 1.47 5.07 0.0075  \nDBT/Py  9.7 2.01 90 1.43 5.54 0.0089  \nCBT/Py  8.1 3.05 84 1.41 5.78 0.0123  \n \n \nFigure 4 . a) Schematic for ST -FMR experimental setup  with LIA and phase -locked RF current \nsource . The brown and dark blue arrows signify up -spin and down -spin states , respectively. ST -\nFMR data and Lorentzian fitting for : b) GBT/Py, c) DBT/Py , and d) CBT/Py samples , measured \nat 4 GHz frequency. e) DL-SOT (blue) and Oersted  plus FL-SOT for the GBT/Py, CBT/Py and \nDBT/Py samples extracted from b -d. f) 𝜃𝑆𝐻 of the GBT/Py, CB T/Py and DBT/Py samples.  Inset: \n𝜎𝑆𝐻 of the GBT/Py, CBT/Py and DBT/Py samples . \n2.4. Spin -Orbit Torque Properties of CBT/Py, DBT/Py and GBT/Py Samples . The spin-orbit \ntorque ( SOT ) characteristics were extracted from the symmetric and antisymmetric components  \nof the fitted Lorentzian , as shown in Fig ures 4b-d, using the equations  [43], 𝑉𝑠=\n−𝐼𝑟𝑓𝛾cos 𝜃𝐻\n4𝑑𝑅\n𝑑𝜃𝐻[𝜏𝐷𝐿1\n𝛥𝐹𝑠𝑦𝑚] and 𝑉𝐴=−𝐼𝑟𝑓𝛾cos 𝜃𝐻\n4𝑑𝑅\n𝑑𝜃𝐻𝜏⊥(1+𝜇0𝑀𝑒𝑓𝑓\n𝐻𝑒𝑥𝑡)1\n2\n𝛥𝑓𝑟𝑒𝑞𝐹 𝑎𝑠𝑦𝑚 ] (see Supporting \nInformation  Section s 3-5). Here , 𝐼𝑟𝑓 is the RF current injected, 𝜃𝐻 is the in -plane angle of the \nexternal DC field relative to the  injected  RF current, 𝑑𝑅\n𝑑𝜃𝐻 is the derivative of the anisotropic \nmagnetoresistance (AMR) relative to 𝜃𝐻, 𝛥𝑓𝑟𝑒𝑞 is the linewidth in the frequency domain, 𝜏𝐷𝐿 is \nthe damping -like DL -SOT,  𝜏⊥, includes  a combination the Oersted field torque  and the field -like \nSOT (FL -SOT) , 𝛾 is the gyromagnetic rati o, 𝜇0 is the permeability of vacuum  and 𝛥𝐻 is the \nlinewidth of the FMR signal . The 𝜏DL correspond s to the symmetric  component of the Lorentzian, \nwhile the 𝜏⊥ correspond to the  antisymmetric components of the Lorentzian function  [43]. The \nDL-SOT was the largest in the CBT/Py sample with a value of 0.1 6 Oe,  compared to 0.1 1 Oe and \n0.10 Oe in GBT/Py and DBT/Py, respectively, as shown in Fig ure 4e. The spin-Hall conductivity, \n𝜎𝑆𝐻, which measures the spin current, 𝐽𝑠 generated from electric field , E across the STFMR device  \nis given by 𝜎𝑆𝐻=𝐽𝑠\n𝐸=𝜏𝐷𝐿𝑀𝑠𝑡𝐹𝑀\n𝐸. The average 𝜎𝑆𝐻 values  shown in Figure 4f were calculated to be 8.1×104ℏ\n2𝑒, 5.5×104ℏ\n2𝑒 and 8.1×104ℏ\n2𝑒Ω−1𝑚−1, for GBT/Py, DBT/Py, and CBT/Py samples, \nrespectively.  Assuming negligible FL -SOT, t he spin -charge current conversion efficiency \nmeasured by the spin -Hall angle  given by , 𝜃𝑆𝐻= =(𝑉𝑠\n𝑉𝑎)(𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑡𝑇𝐼\nℏ)√1+(4𝜋𝑀𝑒𝑓𝑓\n𝐻𝑒𝑥𝑡)  (see \nSupporting Information  Section S5) . The values of 𝜃𝑆𝐻 are calculated as 0.69, 0.51 and 0.65 for \nthe GBT/Py, DBT/Py and CBT/Py samples , respectively . The 𝜎𝑆𝐻 and 𝜃𝑆𝐻 are presented in Figure \n4f which  follow similar trends as expected . The GBT/Py and  CBT/Py sample s have a much larger \n𝜎𝑆𝐻 and 𝜃𝑆𝐻 compared to the disordered polycrystalline DBT/Py sample. This  points towards a \nreduction in charge -spin current conversion efficiency with degradation in crystalline ordering  \npossibly due to scattering of spin current in the randomly oriented crystalline TI domains . The \nGBT/Py sample , however , regains the spin -charge conversion efficiency possibly because of \nquantum confinement effect in the smaller grain size of the GBT sample [26, 27] . The symmetric \nLorentzian in ST -FMR also includes a contribution from spin-pumping due the inverse spin -Hall \neffect ( ISHE ), which results in the 𝛼 enhancement  due to spin -pumping , as shown in the Figure \n3d of the main text . However, because of the complex ity of the interface in the Bi 2Te3/Py samples, \nISHE contribution to the symmetric component of FMR spectra could not be accurately isolated \nfrom the DL -SOT. Furthermore, t he contribution of spin -pumping in Py -based heterostructures \nhas been shown to be muc h smaller than the AMR component [44,45]. H ence, the ISHE \ncomponent s can be safely neglected  from the  calculations of DL -SOT  and 𝜃𝑆𝐻 for comparison of \nthe samples .  \nFigure 5.  a) Schematic of the AFM -FM heterostructure materials system and exchange bias in \nm(H) loops. b) m(H) loops measured at 6 K under ZFC condition of the GBT/Py, DBT/Py and \nCBT/Py  [25] samples showing spontaneous exchange bias.  The size of the arrows qualitative ly \nindicates the magnitude of shifts in the exchange bias.  c) m(T) measurements of the GBT/Py, \nDBT/Py and CBT/Py  samples and their derivatives for characterization of TN. d) Exchange bias \nand TN values of the GBT/Py, DBT/Py and CBT/Py samples extracted fro m b and c.  The m(H) and \nm(T) data presented for the CBT sample are similar to the ones in ref [25].  \n2.5. Effect of Crystalline Disorder in  Interfacial Topological AFM  Phase . Creation of an  \ninterfacial AFM -ordered layer was reported  in the interface of  highly c-axis-oriented Bi 2Te3/Py \nheterostructure s [25]. That AFM ordering in the interfacial layer was found to exist  because of  the \npresence of the topological AFM  compound, NiB i2Te4. Here, t he effect of crystalline  disorder of \nBi2Te3 on the AFM propert y of the Ni /Fe-diffused Bi2Te3 interface  was also  studied using  zero-\nfield-cooled (ZFC) , m(H) and m(T) measurements  at low temperatures  as shown in Figure 5 . \nWhereas  the m(H) measurements  performed at 300 K are well-centered along the H-field axis, the \nm(H) loops measured at 6 K shown in Figure 5 b are significantly shifted off -center. This shift in \nthe magnetic hysteresis loop is characteristic of spontaneous exchange bias that arises from an \ninterfacial AFM -FM interaction given by 𝐻𝐸𝐵=𝐽𝑒𝑥\n4𝜋𝑀𝑠𝑡𝐹𝑀 [38-42], as illustrated in Figure 5 a. Here , \nJex is the interfacial AFM -FM exchange energy and tFM is the thickness of the ferromagnetic layer. \nAs shown in Fig ures 4b,d, the CBT/Py with highly c-axis-oriented crystalline texture d Bi2Te3 has \nthe largest exchange bias of HEB = 83 Oe, while  the DBT/Py sample with randomly oriented \npolycrystalline grains o f Bi 2Te3 has a slightly reduced  exchange bias of HEB = 73 Oe. The exchange \ninteraction strength in AFM materials is related to the Néel  temperature , TN which were determined \nusing ZFC m(T) measurements [39,41 ] at a constant field of 50 Oe , as shown in Figure 5 c. The \nCBT/Py and DBT/Py samples also show high values of TN = 63 and 60 K , respectively. It must \nalso be noted that the large exchange bias and high TN in the disordered DBT/Py sample should  \nalso have significant contributions from th e larger Ni and Fe interface concentration of 46% and \n9% respectively, possibly causing  the HEB and TN to be comparable to the highly c-axis-oriented \nCBT /Py sample . The DBT/Py sample also shows a secondary magnetic phase at 38 K also as \nobserved from the smaller peak in the d m/dT plot in Figure 5c  which can also influence the HEB \nand Hc in the sample  at measurement temperature of 6 K. This possibly emerges due to the \nrandomly oriented vdW crystalline domains affecting exchange interaction between the interfacial \nmagnetic species. However, the highly disordered GBT/Py sample with granular Bi 2Te3 had a \nlarge  reduction in exchange bias to HEB = 12 Oe and TN = 20 K , clearly showing reduction in \nexchange in teraction strength in the AFM interfac ial layer . This follows from the much lower Δm and hence lesser reaction between the diffused Ni and Bi 2Te3 in the GBT/Py sample compared to \nthe DBT/Py and CBT/Py samples.  In addition to the spontaneous exchange bias, the m(H) \nmeasurements also show a characteristic enhancement in coercive field ( Hc) in all the Bi 2Te3/Py \nsamples , as shown in Figure 5 b, due to f rustrated magnetic moments at the interface [ 40]. These \nresults indicate the persistence of exchange properties even in highly disordered TIs.  \n \n3. CONCLUSION  \nInterfacial and magnetic properties of sputtered -deposited  TI/FM Bi 2Te3/Py heterostructures were \nstudied for varying crystalline structural disorder of the Bi2Te3. An interface layer was found to \nform because of diffusion of Ni and small er amounts of Fe into Bi 2Te3. The diffused Ni and Fe \nwere found to undergo solid -state che mical reactions with Bi 2Te3 promoted by the TSS electrons. \nWith increas ing crystall ine c-axis-oriented texture  of the Bi 2Te3, the strengthening  of topological \nproperty of Bi 2Te3 led to an enhanced  reaction between the diffused species and Bi 2Te3, which \nwas observed by a larger loss of Py magnetic moment. Increase in crystalline c-axis orientation of \nBi2Te3 also resulted in a notable  increase in OP magnetic anisotropy , Gilbert damping and spin-\norbit torque  as observed from m(H) loop and FMR measurements. Interestingly, polycrystalline \ndisordered Bi 2Te3 sample had a reduced charge -spin current conversion efficiency  possibly \nbecause of scattering of spins from polycrystalline grain boundaries . Whereas  the samples with \ngranular and  highly c -axis-oriented Bi 2Te3 had a comparable charge -spin current conversion \nefficiencies , which possibly resulted from quantum confinement effect  in smaller crystalline grains  \nand strong TSS , respectively.  As such, t his is expected to result in  reduced spin-charge conversion \nefficiency . Further more , low temperature magnetization measurements showed surprising \nresilience of the topological property of Bi 2Te3 as the AFM order persisted even in highly disordered granular Bi 2Te3/Py heterostructures. However,  for this sample t he exchange interaction \nstrength of the interfacial AFM phase was  found to weaken significantly with the increase in \nstructural disorder of Bi 2Te3. This was verified by degradation in HEB and TN with increase in \ndisorder of the Bi2Te3. These results open the path for further exploration of crystalline disorder \nin TIs and TI/FM interfaces . These thin film heterostructures  can be easily grown using a CMOS -\ncompatible sputtering process that can lead  to realization of energy efficient spintronic devices.  \n4. EXPERIMENTAL SECTION  \n4.1. Material Growth.  Bi2Te3 thin films of thickness 30 nm were grown by  co-sputtering a \ncomposite Bi 2Te3 target with Te target , using RF magnetron sputtering at 90  W and 20  W, \nrespectively , with 4  mTorr Ar pressure on thermally oxidized Si substrate s. The base pressure of \nthe sputtering chamber was ~ 8×10-8 Torr. The GBT, DBT and CBT samples were grown with \nsubstrate maintained at 20 °C , 160 °C and 250°C , respectively. The 160°C DBT and 250°C CBT \nsamples were further annealed at the growth temperatures inside the PVD process chamber in 45 \nmTorr pressure in Ar environment for 25 minutes.  The CBT samples were gown using the same \nmethod as ref [25].  The sampl es were capped with 2  nm Al at room temperature before breaking \nvacuum  which oxidizes to AlO x on exposure to atmosphere . For the magnetic and ST -FMR \nexperiments, 20 nm Py and 3 nm TiO x capping w ere deposited at room  temperature after deposition \nof Bi 2Te3. \n4.2. XRD Characterization.  X-ray diffraction was collected using a background -free, highly \ncollimated beam of Cu -Kα1 radiation (wavelength λ = 1.54056 Å) . The X -rays were captured  by \na 2D charged -coupled device (CCD). The Bragg reflections were  indexed acc ording to the Bi 2Te3 \nbulk hexagonal unit cell, as indicated by ( h, k, -(h+k), l) where h, k, and l are the Miller indices  \n[25].  4.3. TEM and XEDS Characterization.  Samples for TEM investigations were prepared by \nfocused ion beam milling (FIB) using a Ga+ ion source. Prior to TEM observation an additional \ncleaning procedure was performed by Ar -ion milling to reduce a surface amorphous layer and \nresidual Ga from  the FI B process. The TEM observations were performed using a Talos 200 -FX \n(ThermoFiszher Scientific Inc.) TEM operated at an acceleration voltage of 200 kV  [25]. EDS \nmeasurements were performed using a ChemiSTEM (ThermoFisher Scientific) and processing of \nthe sp ectra was performed using Esprit 1.9 (Brucker Inc.) software  [25].  \n4.4. FMR Measurements.  FMR measurements of α, 4𝜋𝑀𝑒𝑓𝑓, and Ha were performed using a \nspin-torque FMR (ST -FMR) experimental setup. The analysis of the experiment is explained in \nSupporting Information  Section S4. RF current is provided by a HP 8350  RF source. A SR830 \nlock-in amplified (LIA) provides reference low -frequ ency AC for modulation phase -locked with \nthe RF current. The LIA was used for detection of ST -FMR signal. The bias DC field is provided \nby an Fe -core electromagnet on a rotating stage with precise angular control. The reported ST -\nFMR experiment was perform ed at a 45° angle of the microstrip relative to  the DC bias field. \nControl of the experiment and data acquisition was done using NI LabVIEW. The S11, S12 and \nimpedance values were used to calculate the RF current,  and the E-field using vector network \nanaly zer (VNA) . The total power lost in the electrical components , such as wires and connectors , \nwere measured to be ~60% , hence, 40% of 8 dbm power was used for RF current calculations.  \nFMR  characteristics in the Bi 2Te3/Py heterostructure samples  were also extracted from the ST -\nFMR experiment spectra as shown in Figure S2 . The devices were patterned using  ion-milling \nprocess for the DBT/Py and C BT/Py samples  and using lift -off process for the Py and GBT/Py \nsamples . The FMR  characteristics were ext racted by fitting Lorentzian functions to the spectra  \n[43], as shown in Figures S2a-d, using the equation , 𝑉𝑚𝑖𝑥 =𝑉𝑆𝐹𝑠𝑦𝑚+ 𝑉𝐴𝐹𝑎𝑠𝑦𝑚 , which clearly shows broadening of FMR linewidth progressively from Py, GBT/Py, DBT/Py to CBT/Py \nsamples . Here,  𝑉𝑚𝑖𝑥 is the DC voltage output recorded in the LIA , 𝐹𝑠𝑦𝑚 = Δ𝐻2\n(𝜇0𝐻−𝜇0𝐻𝑟𝑒𝑠)2+Δ𝐻2 and \n𝐹𝑎𝑠𝑦𝑚 = 𝛥𝐻(𝜇0𝐻−𝜇0𝐻𝐹𝑀𝑅 )\n(𝜇0𝐻−𝜇0𝐻𝑟𝑒𝑠)2+𝛥𝐻2 are the symmetric and antisymmetric components of the Lorentzian \nfunction , 𝛥𝐻 is the linewidth of the FMR signal , and Hres is the FMR field.  The ST -FMR \nmeasurements were performed at 45° angle relative to the external applied magnetic field. T he \nvalues of 𝛥𝐻, 𝐻𝐹𝑀𝑅, 𝑉𝑆 and 𝑉𝐴 were extracted by fitting the ST -FMR signal using above equation  \nfor the analysis of  𝛼, 4𝜋𝑀𝑠, 𝐻⊥ and 𝜃𝑆𝐻 reported in the main text  [25]. \n4.5. Hysteresis Loop Measurements.  Magnetization m(H) and m(T) measurements were obtained \nusing a  Quantum Design MPMS XL -7 superconducting quantum interference device (SQUID) \nmagnetometer  [25]. Hysteresis loop m(H) measurements were carried out at various temperatures \nbetween 6 and 300 K. The ZFC m(T) measurements were obtained while increasing the \ntemperature in an applied f ield of 5 0 Oe . Room temperature m(H) measurements were taken using \na vibrating sample magnetometer (VSM).  \n \nPresent Addresses  \n†Dartmouth College , Thayer School of Engineering, Hanover, NH 03755  \nNotes  \nThe authors declare no competing financial interests.  \nAuthor Contributions  \nThe manuscript was written throug h contributions of all authors. All authors have given approval \nto the final version of the manuscript.   ACKNOWLEDGEMENT  \nWe thank Charles Settens  and MIT, Materials Research Laboratory for their help with XRD \nmeasurements.  We thank Neville Sun and Mehdi Nasrollahpourmotlaghzanjani for help with VNA \nmeasurements.  We also thank Ivan Lisenkov for his valuable input in understanding FMR \nexperiments.  Certain commercial equipment s are identified in this paper to foster understanding , \nbut such identification does not imply recommendation or endorsement by Northeastern University  \nand AFRL.  \nFunding Sources  \nThis work is partially supported by the U.S Army under grant no. W911NF20P0009, the NIH \nAward UF1NS107694 and by the NSF TANMS ERC Award 1160504. The work of DH and AF \nwas partially supported by the National Science Foundation grant DMR -1905662 and the Air  \nForce Office of Scientific Research award FA9550 -20-1-0247. The work of KM was supported by \nAir Force Research Laboratory under AFRL/NEMO contract: FA8650 -19-F-5403 TO3. Studies \nemploying the Titan 60 -300 TEM was performed at the Center for Electron Micro scopy and \nAnalysis (CEMAS) at The Ohio State University with support through Air Force contract FA8650 -\n18-2-5295.  \nABBREVIATIONS  \nAFM, Antiferromagnet; CBT, c -axis oriented Bi2Te3; DBT, Disordered Bi2Te3; DL -SOT, \nDampin -like spin orbit torque; EDS, Energy -dispersive X -ray spectroscopy; FM, Ferromagnet; \nFMR, Ferrommagnetic resonance; GBT, Granular Bi2Te3; HRTEM, High resolution \ntransmission electron microscopy; IP, In -plane; MBE, Molecular beam epitaxy; OP, Out -of-\nplane; QAH, Quantum anomalous hall; RF, Radio  frequency; SOC, Spin orbit coupling; TI, Topological insulator; TSS, Topological surface states; XRD, X -ray diffraction; XRR, X -ray \nreflectometry; ZFC, Zero field cooled . \nREFERENCES  \n1. H Zhang, C.X. Liu, X.L. Qi, X. Dai, Z. Fang and S.C. Zhang, Topologica l insulators in  \nBi2Se3, Bi2Te3 and Sb 2Te3 with a single Dirac cone on the surface. Nat . 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Valenzuela , Control of Spin –Orbit Torques by Interface Engineering in \nTopological In sulator Heterostructures , Nano Lett., 20, 8, 5893 (2020 ). \n \n \n Supplementary Materials : Supporting Information for Effects of Crystalline Disorder on \nInterfacial and Magnetic Properties of Sputtered Topological Insulator/Ferromagnet \nHeterostructures  \nS1. Grain size calculations of Bi 2Te3 Samples  \n \nFigure B1. a) GIXD plots for GBT, DBT and CBT samples. Representative Gaussian fitting of the \nGIXD data for b) GBT, c) DBT and d) CBT samples, used for extracting the FWHM.]  \nAs shown in Fig. S1a, grazing -angle XRD (GIXD) measured at a small incidence angle (~1º) \nclearly show the presence of comparable diffraction peak intens ities from ( hkl) = (00l), (10l) and \n(11l) orientations in both the GBT and DBT samples. The comparable intensities suggest the \nsamples are disordered polycrystalline in nature, where h, k and l are the miller indices. The CBT \nsample had clear ( 00l) orienta tion that was determined from both symmetric -XRD (main text Fig. \n1e) and GIXD measurements, showing that the CBT thin film is highly textured and oriented along \nthe crystalline c-axis. Crystallite grain sizes for the sputter -grown Bi 2Te3 samples were deter mined \nusing the Scherrer equation, 𝑑𝑔𝑟𝑎𝑖𝑛 =𝐾𝜆\n𝐹𝑊𝐻𝑀  𝑐𝑜𝑠𝜃 . Here, dgrain is the grain size, K is the shape factor \nassumed to be 0.9, λ=1.54 Å is the wavelength of the X -ray source, θ is the Bragg angle and \nFWHM is the full -width at half -maximum of the peaks. The FWHM parameters of the peaks are \nextracted by fitting a Gaussian function to the raw data for the GIXD peaks as shown in Fig. S1. \nThe average calculated grain sizes for the sa mples are 8.2 nm, 16.4 nm and 18.7 nm for GBT, DBT \nand CBT, respectively.  \nS2. Summary of Cross -Sectional Atomic% in GBT/Py, DBT/Py and CBT/Py Samples  \nTable S1. Average atomic % of elements along the cross section of the GBT/Py, DBT/Py and \nCBT/Py samples, measured using cross -sectional EDS.  \nSample  Element  Bi2Te3 Ni-Bi2Te3 Py* Py \nGBT/Py  Bi 39 20 2 0 \nTe 57 39 6 0 \nNi 4 38 68 81 \nFe 0 3 25 19 \nDBT/Py  Bi 39 18 2 0 \nTe 57 26 4 0 \nNi 3 47 74 81 \nFe 0 9 19 19 \nCBT/Py  Bi 39 22 3 0 \nTe 60 34 5 0 \nNi 0 39 70 81 \nFe 0 4 22 19 \n S3. FMR Spectra for Bi 2Te3/Py Samples  \n \nFigure B2. Representative FMR spectra for the a) Py, b) GBT/Py, c) DBT/Py and d) CBT/Py \nsamples, measured using ST -FMR experimental setup for extracting Gilbert damping and \ninterfacial magnetic anisotropy field.  \nFMR characteristics in the Bi 2Te3/Py heterostructure samples were extracted from fitting \nLorentzian functions to the ST -FMR experiment spectra as shown in Fig. S2. The devices were \npatterned using ion -milling process for the DBT/Py and CBT/Py samples and using lift -off process \nfor the Py and GBT/Py samples (see Methods for details).  \nS4. Calculation of Anisotropic Magnetoresistance  in Bi2Te3/Py Samples  \n \nFigure B3. Angular dependence of resistance of a) GBT/Py, b) DBT/Py and c) CBT/Py relative to \nthe in -plane external field of 200 Oe, which was used to calculate AMR. The blue dots represent \nraw data, and the blue dashed curve is the theoretical fit.  \nAnisotropic magnetoresistance (AMR), which is the angular dependenc e of the resistivity of the \nmaterial in an applied DC magnetic field, is the primary component of the ST -FMR signal. The \nAMR of a rectangular microstrip can be characterized by in -plane angular -dependent measurement \nof resistance and is given by, 𝑅=𝑅𝐴𝑀𝑅cos2𝜃𝐻+𝑅0. Here, 𝑅 is the total resistance of the \nmicrostrip, 𝑅𝐴𝑀𝑅 is the AMR component of the resistance and 𝑅0 is the resistance at 90° angle. \nFigure S2 shows the resistance measured for the Bi2Te3/Py samples GBT, DBT and CBT at various \nangles  in an in -plane magnetic field of 200 Oe. From fitting resistance measurements with the \nabove equation, we obtained the average AMR values for the samples as 0.93 Ω, 0.57 Ω and 0.72 \nΩ, respectively. Further, taking a derivative of the above relation, we ob tain the 𝑑𝑅/𝑑𝜃𝐻 plots that \nare used in Eq. (3) in the main text for calculation of SOT field values using ST -FMR.  \n \nS5. Spin -Hall Angle Analysis of Bi 2Te3/Py Samples  \n \nFigure S4. Trend of a) θ SH calculated using the Vs -only method, b) θ SH  calculated using the V s/Va \nmethod and c) 𝜎𝑆𝐻 for the GBT/Py, DBT/Py and CBT/Py samples.  \nSpin-Hall angle ( θSH) is the standard measure of spin -charge conversion efficiency in FM/normal \nmetal (NM) heterostructure systems (Wang et al., J. Phys. D: Appl. Phys. 51, 273002, 2018). In \nmost cases, the θSH can be accurately determined from ST -FMR experiments using the Vs -only \nmethod, 𝜃𝑆𝐻=𝜎𝑆𝐻\n𝜎𝑥𝑥, where 𝜎𝑥𝑥 is the longitudinal conductivity of the NM thin film. The average \n𝜎𝑆𝐻 values shown in Figure S4c are 8.1×104ℏ\n2𝑒, 5.5×104ℏ\n2𝑒 and 8.1×104ℏ\n2𝑒Ω−1𝑚−1, for \nGBT/Py, DBT/Py, and CBT/Py samples, respectively. T he determination of 𝜎𝑥𝑥 for the GBT, DBT \nand CBT thin films (with AlO x cap) without Py deposition was found to be 2.80×10−4Ω−1𝑚−1, \n1.72×10−4Ω−1𝑚−1, and 4.08×10−4Ω−1𝑚−1, respectively.  Using the above relation, average \nθSH  values obtained were 3.68, 3.18 and 1.98 for the GBT/Py, DBT/Py and CBT/Py, respectively, \nas shown in Fig. S4a. However, the Vs -only method assumes no interfacial diffusion, which leads \nto a non -physical result in th ese TI/FM materials systems. The resistance in the TI layer is certainly \naltered by the diffusion of Ni and Fe from Py and reaction with Bi 2Te3. Moreover, the trend of θSH  \nobtained from the Vs -only method does not follow the 𝜎𝑆𝐻 trend, which also means  the θSH values \nobtained using the Vs -only method is incorrect. For a better estimation of the θSH  parameter in \nthese TI/FM systems with a complex interfacial morphology, the Vs/Va method given by, 𝜃𝑆𝐻=\n(𝑉𝑠\n𝑉𝑎)(𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑡𝑇𝐼\nℏ)√1+(4𝜋𝑀𝑒𝑓𝑓\n𝐻𝑒𝑥𝑡), should be a more reliable method. This method assumes the \nantisymmetric component is completely from the Oersted field. Given the comparable values of \n𝜏⊥ for the GBT/Py, DBT/Py and CBT/Py (see Main Text Figure 4e), the FL -SOT can be safely \nassume d to be negligibly small compared to Oersted torque.  Using the Vs/Va method, the average \nθSH  for the GBT/Py, DBT/Py and CBT/Py were 0.69, 0.51 and 0.65, respectively, as shown in Fig. \nS4b. As expected, values also follow the trend of 𝜎𝑆𝐻 for the sampl es. This clearly suggests \nreduction in spin -charge conversion efficiency for the polycrystalline disordered TI sample \nDBT/Py.  \nS6. Crystalline Disorder -Dependent Spin -Orbit Coupling Effects in TI/FM Heterostructures  \nS6.1. Crystalline Disorder -Dependent Magnetic Properties of Bi 2Te3/CoFeB  \nThe crystalline disorder -dependent, room -temperature magnetic properties of TI/FM were also \nstudied for Bi 2Te3/CoFeB (CFB) heterostructure samples, as shown in Fig. S4. The FMR spectra \nshow a giant enhancement in α and 𝐻⊥ for the CBT/CFB samples with high c-axis orientation. The \nmeasurements were performed using a broad -band FMR setup for comparison with the Bi 2Te3/Py \nsamples. The extracted α values from FMR measurements in Fig. S4b were found to be 0.015, \n0.026 and 0.283 for CFB, GBT/CFB and CBT/CFB, respectively. These increasing values for \nincreasing order confirm the giant enhancement in SOC and strong TSS eff ects in highly c-axis \noriented TI compared to highly disordered TI materials. Further, Kittel equation fits to the \nresonance frequency versus field in Fig. S4c give estimates for 𝐻⊥ of -13.2 Oe,  100.7 Oe and 1485 \nOe, for the respective sample revealing a  giant enhancement in PMA field for the highly c-axis \noriented Bi 2Te3 sample. This clearly shows a large enhancement in interfacial PMA field in highly c-axis oriented CBT/CFB samples compared to a higher in -plane anisotropy of the disordered \nGBT/CFB sampl es. The FMR measurements are also supported by the increase in Mr/Ms ratio in \nthe m(H) measurements, shown in Fig. S4d. The Mr/Ms values are 86%, 70% and 66% for the \nCFB, GBT/CFB and CBT/CFB samples, respectively, which shows an enhanced OP magnetic \nanisot ropy for the highly oriented CBT/CFB sample compared to the disordered GBT/CFB \nsample. Table S2 summarizes the magnetic properties of the Bi 2Te3/CoFeB samples obtained from \nthe FMR and m(H)  loop measurements.  \n \nFigure B5. a) Representative normalized FMR spectra for CFB, GBT/CFB and CBT/CFB at 6 \nGHz showing broadening of linewidth. b) FMR linewidth versus frequency plots and linear fit for \nextracting Gilber t damping in GBT/Py, CBT/Py and DBT/Py samples. c) FMR frequency versus \nfield used for extracting 4𝜋𝑀𝑠 and 𝐻⊥ in GBT/Py, CBT/Py and DBT/Py samples. d) m(H) \nmeasurement for the GBT/Py, CBT/Py and DBT/Py samples.  \nTable S2. Magnetic Properties of Bi 2Te3/CoFeB Samples  \nSample  𝜶 𝟒𝝅𝑴𝒔 \n(kOe)  𝑯⊥ \n(Oe)  Mr/Ms (%) \nCFB  0.01484  16.01  -13.17  86 \nGBT/CFB  0.02604  15.89  100.74  70 \nCBT/CFB  0.28336  14.51  1485.39  66 \n \nS6.2. Crystalline Disorder -Dependent Magnetic Properties of YIG/Bi 2Te3 \nThe effect of TSS for giant enhancement in α was verified using Y 3Fe5O12 (YIG) as the FM layer. \nDisordered GBT and highly c-axis oriented CBT thin films of thickness 30 nm were grown on \nGGG (111)/YIG (444) (70 nm) substrates using the same deposition condi tions mentioned in the \nmain text. YIG is a chemically stable rare -earth garnet material, and because it is an oxide \ncompound it is not expected to experience diffusion across the interface with TI (Chen et al., Appl. \nPhys. Lett. 114, 031601, 2019). This wa s verified by m(H) measurements of  YIG and YIG/GBT \nsamples that did not show any reduction in magnetic moment, as shown in Fig. S4c, unlike with \nPy. FMR measurements for the YIG/Bi 2Te3 samples revealed a similar trend as the Bi 2Te3/FM \nsamples and showed a  giant enhancement in FMR linewidth and α for YIG/GBT heterostructures, \nas shown in Figs. S5b,d. The α values extracted from the linear fitting of Δ𝐻 versus resonance \nfrequency plots in Figure S3d are 8.43×10−4, 2.83×10−3 and 7.09×10−3 for increasing \ncrystallinity. This clearly shows a giant enhancement in Gilbert damping in the YIG/CBT \nheterostructure sample as a result of the TSS in the highly c-axis oriented Bi 2Te3. Fitting the Kittel equation also revealed changes in 𝛾\n2𝜋 values fr om 2.81 for YIG to 2.83 and 2.84 for YIG/GBT and \nYIG/CBT samples, respectively. These changes in 𝛾\n2𝜋 values provide strong evidence of interaction \nbetween the TSS and magnetic moments in YIG that are enhanced in the YIG/CBT sample (Liu \net al., Phys. Rev.  Lett. 125, 017204, 2020). Table S3 summarizes the magnetic properties of the \nYIG/Bi 2Te3 samples obtained from FMR measurements.  \n \nFigure B6. a) XRD data for sputtered CBT grown on YIG showing strong c -axis oriented texture. \nb) FMR spectra of YIG/Bi 2Te3 samples measured at 6 GHz showing enhancement in linewidth \nafter deposition of GBT and CBT. c) FMR frequency versus resonance field plots and Kittel \nequation fitting of YIG, YIG/GBT and YIG/CBT samples. d) FMR linewidth versus frequency plots \nof YIG, YIG/ GBT and YIG/CBT samples and linear fitting.  \nTable S3. Magnetic Properties of YIG/Bi 2Te3 Samples.  \nSample  𝜸\n𝟐𝝅 𝜶 𝟒𝝅𝑴𝒔 (kOe)  \nYIG 2.81 0.000843  1.89 \nGBT/YIG  2.83 0.00283  1.89 \nCBT/YIG  2.84 0.00710  1.89 \n "    },    {        "title": "2105.03576v1.A_second_order_numerical_method_for_Landau_Lifshitz_Gilbert_equation_with_large_damping_parameters.pdf",        "content": "A SECOND-ORDER NUMERICAL METHOD FOR\nLANDAU-LIFSHITZ-GILBERT EQUATION WITH LARGE\nDAMPING PARAMETERS\nYONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAbstract. A second order accurate numerical scheme is proposed and imple-\nmented for the Landau-Lifshitz-Gilbert equation, which models magnetization\ndynamics in ferromagnetic materials, with large damping parameters. The\nmain advantages of this method are associated with the following features:\n(1) It only solves linear systems of equations with constant coe\u000ecients where\nfast solvers are available, so that the numerical e\u000eciency has been greatly im-\nproved, in comparison with the existing Gauss-Seidel project method. (2) The\nsecond-order accuracy in time is achieved, and it is unconditionally stable for\nlarge damping parameters. Moreover, both the second-order accuracy and the\ngreat e\u000eciency improvement will be veri\fed by several numerical examples\nin the 1D and 3D simulations. In the presence of large damping parameters,\nit is observed that this method is unconditionally stable and \fnds physically\nreasonable structures while many existing methods have failed. For the do-\nmain wall dynamics, the linear dependence of wall velocity with respect to the\ndamping parameter and the external magnetic \feld will be obtained through\nthe reported simulations.\n1.Introduction\nFerromagnetic materials are widely used for data storage due to the bi-stable\nstates of the intrinsic magnetic order or magnetization. The dynamics of magneti-\nzation has been modeled by the Landau-Lifshitz-Gilbert (LLG) equation [9,13]. In\nparticular, two terms are involved in the dynamics of the LLG equation: the gyro-\nmagnetic term, which is energetically conservative, and the damping term, which\nis energetically dissipative.\nThe damping term is important since it strongly a\u000bects the energy required and\nthe speed at which a magnetic device operates. A recent experiment on a magnetic-\nsemiconductor heterostructure [25] has indicated that the Gilbert damping constant\ncan be adjusted. At the microscopic level, the electron scattering, the itinerant\nelectron relaxation [11], and the phonon-magnon coupling [16, 17] are responsible\nto the damping, which can be obtained from electronic structure calculations [19].\nFor the application purpose, tuning the damping parameter allows one to optimize\nthe magneto-dynamic properties in the material, such as lowering the switching\ncurrent and increasing the writing speed of magnetic memory devices [23].\nWhile most experiments have been devoted to small damping parameters [4,14,\n22], large damping e\u000bects are observed in [10,18]. The magnetization switching time\nDate : May 11, 2021.\n2010 Mathematics Subject Classi\fcation. 35K61, 65N06, 65N12.\nKey words and phrases. Micromagnetics simulations, Landau-Lifshitz-Gilbert equation,\nsecond-order method, large damping parameter.\n1arXiv:2105.03576v1  [physics.comp-ph]  8 May 20212 Y. CAI, J. CHEN, C. WANG, AND C. XIE\ntends to be shorter in the presence of the large damping constant [18]. Extremely\nlarge damping parameters ( \u00189) are presented in [10].\nThe LLG equation is a vectorial and nonlinear system with the \fxed length of\nmagnetization in a point-wise sense. Signi\fcant e\u000borts have been devoted to design\ne\u000ecient and stable numerical methods for micromagnetics simulations; see [6, 12]\nfor reviews and references therein. Among the existing numerical works, semi-\nimplicit schemes have been very popular since they avoid a complicated nonlinear\nsolver while preserving the numerical stability; see [2, 7, 24], etc. In particular,\nthe second-order accurate backward di\u000berentiation formula (BDF) scheme is con-\nstructed in [24], with a one-sided interpolation. In turn, a three-dimensional lin-\near system needs to be solved at each time step, with non-constant coe\u000ecients.\nMoreover, a theoretical analysis of the second order convergence estimate has been\nestablished in [5] for such a BDF2 method. As another approach, a linearly implicit\nmethod in [2] introduces the tangent space to deal with the length constraint of\nmagnetization, with the \frst-order temporal accuracy. As a further extension, high-\norder BDF schemes have been constructed and analyzed in a more recent work [1].\nAn unconditionally unique solvability of the semi-implicit schemes has been proved\nin [1,5], while the convergence analysis has required a condition that the temporal\nstep-size is proportional to the spatial grid-size. However, an obvious disadvantage\nhas been observed for these semi-implicit schemes: the vectorial structure of the\nLLG equation leads to a non-symmetric linear system at each time step, which\ncannot be implemented by an FFT-based fast solver. In fact, the GMRES is often\nused, while its e\u000eciency depends heavily on the temporal step-size and the spatial\ngrid-size, and extensive numerical experiments have indicated much more expensive\ncomputational costs than standard Poisson solvers [24].\nThe Gauss-Seidel projection method (GSPM) is another popular set of numerical\nalgorithms since only linear systems with constant coe\u000ecients need to be solved at\neach time step [8,15,21]. This method is based on a combination of a Gauss-Seidel\nupdate of an implicit solver for the gyromagnetic term, the heat \row of the harmonic\nmap, and a projection step to overcome the sti\u000bness and the nonlinearity associated\nto the LLG equation. In this numerical approach, the implicit discretization is only\napplied to the scalar heat equation implicitly several times; therefore, the FFT-\nbased fast solvers become available, due to the symmetric, positive de\fnite (SPD)\nstructures of the linear system. The original GSPM method [20] turns out to be\nunstable for small damping parameters, while this issue has been resolved in [8] with\nmore updates of the stray \feld. Its numerical e\u000eciency has been further improved\nby reducing the number of linear systems per time step [15]. One little de\fciency\nof GSPM is its \frst-order accuracy in time.\nMeanwhile, in spite of these improvements, the GSPM method is computation-\nally more expensive than the standard Poisson solver, because of the Gauss-Seidel\niteration involved in the algorithm. An additional de\fciency of the GSPM is its\n\frst-order accuracy in time. Moreover, most of the above-mentioned methods have\nbeen mainly focused on small damping parameters with the only exception in a\ntheoretical work [1]. In other words, there has been no numerical method designed\nspeci\fcally for real micromagnetics simulations with large damping parameters. In\nthis paper, we propose a second-order accurate numerical method to solve the LLG\nequation with large damping parameters, whose complexity is also comparable toA SECOND-ORDER METHOD FOR LLG EQUATION 3\nsolving the scalar heat equation. To achieve this goal, the LLG system is refor-\nmulated, in which the damping term is rewritten as a harmonic mapping \row. In\nturn, the constant-coe\u000ecient Laplacian part is treated by a standard BDF2 tem-\nporal discretization, and the associated dissipation will form the foundation of the\nnumerical stability. Meanwhile, all the nonlinear parts, including both the gyro-\nmagnetic term and the remaining nonlinear expansions in the damping term, are\ncomputed by a fully explicit approximation, which is accomplished by a second\norder extrapolation formula. Because of this fully explicit treatment for the nonlin-\near parts, the resulting numerical scheme only requires a standard Poisson solver at\neach time step. This fact will greatly facilitate the computational e\u000borts, since the\nFFT-based fast solver could be e\u000eciently applied, due to the SPD structure of the\nlinear system involved at each time step. In addition, the numerical stability has\nbeen demonstrated by extensive computational experiments, and these experiments\nhas veri\fed the idea that the dissipation property of the heat equation part would\nbe able to ensure the numerical stability of the nonlinear parts, with large damping\nparameters.\nThe rest of this paper is organized as follows. In section 2, the micromagnetics\nmodel is reviewed, and the numerical method is proposed, as well as its comparison\nwith the GSPM and the semi-implicit projection method (SIPM). Subsequently,\nthe numerical results are presented in section 3, including the temporal and spa-\ntial accuracy check in both the 1D and 3D computations, the numerical e\u000eciency\ninvestigation (in comparison with the GSPM and SIPM algorithms), the stability\nstudy with respect to the damping parameter, and the dependence of domain wall\nvelocity on the damping parameter and the external magnetic \feld. Finally, some\nconcluding remarks are made in section 4.\n2.The physical model and the numerical method\n2.1.Landau-Lifshitz-Gilbert equation. The LLG equation describes the dy-\nnamics of magnetization which consists of the gyromagnetic term and the damping\nterm [3,13]. In the nondimensionalized form, this equation reads as\nmt=\u0000m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b) (2.1)\nwith the homogeneous Neumann boundary condition\n(2.2)@m\n@\u0017\f\f\f\n@\n= 0;\nwhere \n is a bounded domain occupied by the ferromagnetic material and \u0017is unit\noutward normal vector along @\n.\nIn more details, the magnetization m: \n\u001aRd!R3;d= 1;2;3 is a three-\ndimensional vector \feld with a pointwise constraint jmj= 1. The \frst term on the\nright-hand side in (2.1) is the gyromagnetic term and the second term stands for\nthe damping term, with \u000b>0 being the dimensionless damping coe\u000ecient.\nThe e\u000bective \feld he\u000bis obtained by taking the variation of the Gibbs free energy\nof the magnetic body with respect to m. The free energy includes the exchange\nenergy, the anisotropy energy, the magnetostatic energy, and the Zeeman energy:\n(2.3)F[m] =\u00160M2\ns\n2\u001aZ\n\n\u0000\n\u000fjrmj2+q\u0000\nm2\n2+m2\n3\u0001\n\u00002he\u0001m\u0000hs\u0001m\u0001\ndx\u001b\n:4 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTherefore, the e\u000bective \feld includes the exchange \feld, the anisotropy \feld, the\nstray \feldhs, and the external \feld he. For a uniaxial material, it is clear that\nhe\u000b=\u000f\u0001m\u0000q(m2e2+m3e3) +hs+he; (2.4)\nwhere the dimensionless parameters become \u000f=Cex=(\u00160M2\nsL2) andq=Ku=(\u00160M2\ns)\nwithLthe diameter of the ferromagnetic body and \u00160the permeability of vacuum.\nThe unit vectors are given by e2= (0;1;0),e3= (0;0;1), and \u0001 denotes the\nstandard Laplacian operator. For the Permalloy, an alloy of Nickel (80%) and\nIron (20%), typical values of the physical parameters are given by: the exchange\nconstantCex= 1:3\u000210\u000011J/m, the anisotropy constant Ku= 100 J/m3, the sat-\nuration magnetization constant Ms= 8:0\u0002105A/m. The stray \feld takes the\nform\nhs=1\n4\u0019rZ\n\nr\u00121\njx\u0000yj\u0013\n\u0001m(y)dy: (2.5)\nIf \n is a rectangular domain, the evaluation of (2.5) can be e\u000eciently done by the\nFast Fourier Transform (FFT) [20].\nFor brevity, the following source term is de\fned\nf=\u0000Q(m2e2+m3e3) +hs+he: (2.6)\nand the original PDE system (2.1) could be rewritten as\nmt=\u0000m\u0002(\u000f\u0001m+f)\u0000\u000bm\u0002m\u0002(\u000f\u0001m+f): (2.7)\nThanks to point-wise identity jmj= 1, we obtain an equivalent form:\n(2.8)mt=\u000b(\u000f\u0001m+f) +\u000b\u0000\n\u000fjrmj2\u0000m\u0001f\u0001\nm\u0000m\u0002(\u000f\u0001m+f):\nIn particular, it is noticed that the damping term is rewritten as a harmonic map-\nping \row, which contains a constant-coe\u000ecient Laplacian di\u000busion term. This fact\nwill greatly improve the numerical stability of the proposed scheme.\nFor the numerical description, we \frst introduce some notations for discretization\nand numerical approximation. Denote the temporal step-size by k, andtn=nk,\nn\u0014\u0004T\nk\u0005\nwithTthe \fnal time. The spatial mesh-size is given by hx=hy=hz=\nh= 1=N, andmn\ni;j;`stands for the magnetization at time step tn, evaluated at the\nspatial location ( xi\u00001\n2;yj\u00001\n2;z`\u00001\n2) withxi\u00001\n2=\u0000\ni\u00001\n2\u0001\nhx,yj\u00001\n2=\u0000\nj\u00001\n2\u0001\nhyand\nz`\u00001\n2=\u0000\n`\u00001\n2\u0001\nhz(0\u0014i;j;`\u0014N+ 1). In addition, a third order extrapolation\nformula is used to approximate the homogeneous Neumann boundary condition.\nFor example, such a formula near the boundary along the zdirection is given by\nmi;j;1=mi;j;0;mi;j;N +1=mi;j;N:\nThe boundary extrapolation along other boundary sections can be similarly made.\nThe standard second-order centered di\u000berence applied to \u0001 mresults in\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\nh2x\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\nh2y\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\nh2z;A SECOND-ORDER METHOD FOR LLG EQUATION 5\nand the discrete gradient operator rhmwithm= (u;v;w )Treads as\nrhmi;j;k=2\n64ui+1;j;k\u0000ui\u00001;j;k\nhxvi+1;j;k\u0000vi\u00001;j;k\nhxwi+1;j;k\u0000wi\u00001;j;k\nhxui;j+1;k\u0000ui;j\u00001;k\nhyvi;j+1;k\u0000vi;j\u00001;k\nhywi;j+1;k\u0000wi;j\u00001;k\nhyui;j;k +1\u0000ui;j;k\u00001\nhzvi;j;k +1\u0000vi;j;k\u00001\nhzwi;j;k +1\u0000wi;j;k\u00001\nhz3\n75:\nSubsequently, the GSPM and the SIPM numerical methods need to be reviewed,\nwhich could be used for the later comparison.\n2.2.The Gauss-Seidel projection method. The GSPM is based on a combi-\nnation of a Gauss-Seidel update of an implicit solver for the gyromagnetic term,\nthe heat \row of the harmonic map, and a projection step. It only requires a series\nof heat equation solvers with constant coe\u000ecients; as a result, the FFT-based fast\nsolvers could be easily applied. This method is \frst-order in time and second-order\nin space. Below is the detailed outline of the GSPM method in [8].\nStep 1. Implicit Gauss-Seidel:\ngn\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(mn\ni+ \u0001tfn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(m\u0003\ni+ \u0001tf\u0003\ni); i= 1;2; (2.9)\n(2.10)0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3m\u0003\n1\u0000g\u0003\n1mn\n3)\nmn\n3+ (g\u0003\n1m\u0003\n2\u0000g\u0003\n2m\u0003\n1)1\nA:\nStep 2. Heat \row without constraints:\n(2.11) f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +h\u0003\ns+he;\n(2.12)0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA=0\n@m\u0003\n1+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n1+f\u0003\n1)\nm\u0003\n2+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n2+f\u0003\n2)\nm\u0003\n3+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n3+f\u0003\n3)1\nA:\nStep 3. Projection onto S2:\n(2.13)0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003\u0003j0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA:\nHerem\u0003denotes the intermediate values of m, and stray \felds hn\nsandh\u0003\nsare\nevaluated at mnandm\u0003, respectively.\nRemark 2.1. Two improved versions of the GSPM have been studied in [15], which\nturn out to be more e\u000ecient than the original GSPM. Meanwhile, it is found that\nboth improved versions become unstable when \u000b > 1, while the original GSPM\n(outlined above) is stable even when \u000b\u001410. Therefore, we shall use the original\nGSPM in [8]for the numerical comparison in this work.6 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n2.3.Semi-implicit projection method. The SIPM has been outlined in [5,24].\nThis method is based on the second-order BDF temporal discretization, combined\nwith an explicit extrapolation. It is found that SIPM is unconditionally stable and\nis second-order accurate in both space and time. The algorithmic details are given\nas follows.\n(2.14)8\n>>>>>><\n>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0000\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0001\n\u0000\u000b^mn+2\nh\u0002\u0010\n^mn+2\nh\u0002(\u000f\u0001h~mn+2\nh+^fn+2\nh)\u0011\n;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;\nwhere ~mn+2\nhis an intermediate magnetization, and ^mn+2\nh,^fn+2\nhare given by the\nfollowing extrapolation formula:\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh;\nwithfn\nh=\u0000Q(mn\n2e2+mn\n3e3) +hn\ns+hn\ne. The presence of cross product in the\nSIPM yields a linear system of equations with non-symmetric structure and vari-\nable coe\u000ecients. In turn, the GMRES solver has to be applied to implement this\nnumerical system. The numerical evidence has revealed that, the convergence of\nGMRES solver becomes slower for larger temporal step-size kor smaller spatial\ngrid-sizeh, which makes the computation more challenging.\n2.4.The proposed numerical method. The SIPM in (2.14) treats both the\ngyromagentic and the damping terms in a semi-implicit way, i.e., \u0001 mis computed\nimplicitly, while the coe\u000ecient functions are updated by a second order accurate,\nexplicit extrapolation formula. The strength of the gyromagnetic term is controlled\nby \u0001m+fsince the length of mis always 1. Meanwhile, the strength of the\ndamping term is controlled by the product of \u0001 m+fand the damping parameter\n\u000b. For small \u000b, say\u000b\u00141, it is reasonable to treat both the gyromagentic and\nthe damping terms semi-implicitly. However, for large \u000b, an alternate approach\nwould be more reasonable, in which the whole gyromagentic term is computed by\nan explicit extrapolation, while the nonlinear parts in the damping term is also\nupdated by an explicit formula, and only the constant-coe\u000ecient \u0001 mpart in the\ndamping term is implicitly updated. This idea leads to the proposed numerical\nmethod. To further simplify the presentation, we start with (2.8), and the numerical\nalgorithm is proposed as follows.\n(2.15)8\n>>>>>>>>>><\n>>>>>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0010\n\u000f\u0001h^mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000fjrh^mn+2\nhj2\u0000^mn+2\nh\u0001^fn+2\nh\u0011\n^mn+2\nh;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;A SECOND-ORDER METHOD FOR LLG EQUATION 7\nwhere\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh:\nTable 1 compares the proposed method, the GSPM and the SIPM in terms\nof number of unknowns, dimensional size, symmetry pattern, and availability of\nFFT-based fast solver of linear systems of equations, and the number of stray \feld\nupdates. At the formal level, the proposed method is clearly superior to both the\nGSPM and the SIPM algorithms. In more details, this scheme will greatly improve\nthe computational e\u000eciency, since only three Poisson solvers are needed at each\ntime step. Moreover, this numerical method preserves a second-order accuracy in\nboth space and time. The numerical results in section 3 will demonstrate that the\nproposed scheme provides a reliable and robust approach for micromagnetics simu-\nlations with high accuracy and e\u000eciency in the regime of large damping parameters.\nTable 1. Comparison of the proposed method, the Gauss-Seidel\nprojection method, and the semi-implicit projection method.\nProperty or number Proposed method GSPM SIPM\nLinear systems 3 7 1\nSize N3N33N3\nSymmetry Yes Yes No\nFast Solver Yes Yes No\nAccuracy O(k2+h2)O(k+h2)O(k2+h2)\nStray \feld updates 1 4 1\nRemark 2.2. To kick start the proposed method, one can apply a \frst-order al-\ngorithm, such as the \frst-order BDF method, in the \frst time step. An overall\nsecond-order accuracy is preserved in this approach.\n3.Numerical experiments\nIn this section, we present a few numerical experiments with a sequence of damp-\ning parameters for the proposed method, the GSPM [8] and the SIPM [24], with\nthe accuracy, e\u000eciency, and stability examined in details. Domain wall dynamics\nis studied and its velocity is recorded in terms of the damping parameter and the\nexternal magnetic \feld.\n3.1.Accuracy and e\u000eciency tests. We set\u000f= 1 andf= 0 in (2.8) for conve-\nnience. The 1D exact solution is given by\nme= (cos(X) sint;sin(X) sint;cost)T;\nand the corresponding exact solution in 3D becomes\nme= (cos(XYZ ) sint;sin(XYZ ) sint;cost)T;\nwhereX=x2(1\u0000x)2,Y=y2(1\u0000y)2,Z=z2(1\u0000z)2. In fact, the above exact\nsolutions satisfy (2.8) with the forcing term g=@tme\u0000\u000b\u0001me\u0000\u000bjrmej2+me\u0002\n\u0001me, as well as the homogeneous Neumann boundary condition.8 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFor the temporal accuracy test in the 1D case, we \fx the spatial resolution\nash= 5D\u00004, so that the spatial approximation error becomes negligible. The\ndamping parameter is taken as \u000b= 10, and the \fnal time is set as T= 1. In the 3D\ntest for the temporal accuracy, due to the limitation of spatial resolution, we take\na sequence of spatial and temporal mesh sizes: k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor the \frst-order method and k=hx=hy=hz=h= 1=N0for the second-\norder method, with the variation of N0indicated below. Similarly, the damping\nparameter is given by \u000b= 10, while the \fnal time Tis indicated below. In turn,\nthe numerical errors are recorded in term of the temporal step-size kin Table 2. It\nis clear that the temporal accuracy orders of the proposed numerical method, the\nGSPM, and the SIPM are given by 2, 1, and 2, respectively, in both the 1D and\n3D computations.\nThe spatial accuracy order is tested by \fxing k= 1D\u00005,\u000b= 10,T= 1 in 1D\nandk= 1D\u00003,\u000b= 10,T= 1 in 3D. The numerical error is recorded in term of\nthe spatial grid-size hin Table 3. Similarly, the presented results have indicated\nthe second order spatial accuracy of all the numerical algorithms, including the\nproposed method, the GSPM, and the SIPM, respectively, in both the 1D and 3D\ncomputations.\nTo make a comparison in terms of the numerical e\u000eciency, we plot the CPU time\n(in seconds) vs. the error norm kmh\u0000mek1. In details, the CPU time is recorded\nas a function of the approximation error in Figure 1a in 1D and in Figure 1b in\n3D, with a variation of kand a \fxed value of h. Similar plots are also displayed in\nFigure 1c in 1D and Figure 1d in 3D, with a variation of hand a \fxed value of k. In\nthe case of a \fxed spatial resolution h, the proposed method is signi\fcantly more\ne\u000ecient than the GSPM and the SIPM in both the 1D and 3D computations. The\nSIPM is slightly more e\u000ecient than the GSPM, while such an advantage depends\non the performance of GMRES, which may vary for di\u000berent values of kandh. In\nthe case of a \fxed time step size k, the proposed method is slightly more e\u000ecient\nthan the GSPM, in both the 1D and 3D computations, and the GSPM is more\ne\u000ecient than the SIPM.\n3.2.Stability test with large damping parameters. To check the numerical\nstability of these three methods in the practical simulations of micromagnetics with\nlarge damping parameters, we consider a thin \flm of size 480 \u0002480\u000220 nm3with\ngrid points 100\u0002100\u00024. The temporal step-size is taken as k= 1 ps. A uniform\nstate along the xdirection is set to be the initial magnetization and the external\nmagnetic \feld is set to be 0. Three di\u000berent damping parameters, \u000b= 0:01;10;40,\nare tested with stable magnetization pro\fles shown in Figure 2. In particular, the\nfollowing observations are made.\n\u000fThe proposed method is the only one that is stable for very large damping\nparameters;\n\u000fAll three methods are stable for moderately large \u000b;\n\u000fThe proposed method is the only one that is unstable for small \u000b.\nIn fact, a preliminary theoretical analysis reveals that, an optimal rate convergence\nestimate of the proposed method could be theoretically justi\fed for \u000b>3. Mean-\nwhile, extensive numerical experiments have implied that \u000b > 1 is su\u000ecient to\nensure the numerical stability in the practical computations.A SECOND-ORDER METHOD FOR LLG EQUATION 9\nTable 2. The numerical errors for the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nh= 5D\u00004; Right: 3D with k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor GSPM and k=hx=hy=hz=h= 1=N0for the proposed\nmethod and SIPM, with N0speci\fed in the table.\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.459D-4 5.226D-4 5.588D-4 1/20 6.171D-4 4.240D-4 4.246D-4\n2.0D-2 1.147D-4 1.345D-4 1.436D-4 1/24 4.381D-4 3.010D-4 3.014D-4\n1.0D-2 2.899D-5 3.402D-5 3.631D-5 1/28 3.268D-4 2.245D-4 2.248D-4\n5.0D-3 7.192D-6 8.529D-6 9.119D-6 1/32 2.531D-4 1.739D-4 1.741D-4\n2.5D-3 1.699D-6 2.321D-6 2.518D-6 1/36 2.017D-4 1.386D-4 1.387D-4\norder 2.007 1.961 1.957 { 1.902 1.903 1.903\n(a)Proposed method\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=h2k\u0001k1k\u0001k 2k\u0001kH1\n2.5D-3 2.796D-4 2.264D-4 1.445D-3 1/36 4.194D-4 2.683D-4 2.815D-4\n1.25D-3 1.425D-4 1.174D-4 7.720D-4 1/64 2.388D-4 1.399D-4 1.500D-4\n6.25D-4 7.170D-5 5.940D-5 4.026D-4 1/144 1.069D-4 6.106D-5 6.736D-5\n3.125D-4 3.591D-5 2.971D-5 2.069D-4 1/256 6.021D-5 3.442D-5 3.860D-5\n1.5625D-4 1.799D-5 1.488D-5 1.054D-4 1/400 3.855D-5 2.208D-5 2.501D-5\norder 0.991 0.984 0.945 { 0.992 1.032 1.000\n(b)GSPM\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.315D-4 5.111D-4 8.774D-4 1/20 6.170D-4 4.240D-4 4.249D-4\n2.0D-2 1.128D-4 1.334D-4 2.255D-4 1/24 4.380D-4 3.010D-4 3.016D-4\n1.0D-2 2.872D-5 3.399D-5 5.706D-5 1/28 3.268D-4 2.245D-4 2.251D-4\n5.0D-3 7.174D-6 8.552D-6 1.433D-5 1/32 2.531D-4 1.739D-4 1.743D-4\n2.5D-3 1.721D-6 2.333D-6 3.784D-6 1/36 2.017D-4 1.386D-4 1.389D-4\norder 1.991 1.951 1.969 { 1.902 1.903 1.902\n(c)SIPM\nUnder the same setup outlined above, we investigate the energy dissipation of\nthe proposed method, the GSPM, and the SIPM. The stable state is attainable at\nt= 2 ns, while the total energy is computed by (2.3). The energy evolution curves\nof di\u000berent numerical methods with di\u000berent damping parameters, \u000b= 2;5;8;10,\nare displayed in Figure 3. One common feature is that the energy dissipation rate\nturns out to be faster for larger \u000b, in all three schemes. Meanwhile, a theoretical\nderivation also reveals that the energy dissipation rate in the LLG equation (2.1)\ndepends on \u000b, and a larger \u000bleads to a faster energy dissipation rate. Therefore,\nthe numerical results generated by all these three numerical methods have made a\nnice agreement with the theoretical derivation.10 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTable 3. The numerical errors of the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nk= 1D\u00005; Right: 3D with k= 1D\u00003.\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(a)Proposed method\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.244D-3 1/2 4.256D-3 2.470D-3 2.470D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.810D-4 5.589D-4 5.744D-4\n1.0D-2 4.619D-4 4.622D-4 5.158D-4 1/8 2.447D-4 1.388D-4 1.423D-4\n5.0D-3 1.153D-4 1.156D-4 1.302D-4 1/16 6.103D-5 3.468D-5 3.613D-5\norder 2.000 2.000 1.995 { 2.037 2.047 2.030\n(b)GSPM\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(c)SIPM\nMeanwhile, we choose the same sequence of values for \u000b, and display the energy\nevolution curves in terms of time up to T= 2 ns in Figure 4. It is found that the\nproposed method have almost the same energy dissipation pattern with the other\ntwo methods for moderately large damping parameters \u000b= 2;5;8. In the case of\n\u000b= 10, the SIPM has a slightly di\u000berent energy dissipation pattern from the other\ntwo numerical methods.\n3.3.Domain wall motion. A Ne\u0013 el wall is initialized in a nanostrip of size 800 \u0002\n100\u00024 nm3with grid points 128 \u000264\u00024. An external magnetic \feld of he= 5 mT\nis then applied along the positive xdirection and the domain wall dynamics is\nsimulated up to 2 ns with \u000b= 2;5;8. The corresponding magnetization pro\fles are\nvisualized in Figure 5. Qualitatively, the domain wall moves faster as the value of\n\u000bincreases. Quantitatively, the corresponding dependence is found to be linear;\nsee Figure 6. The slopes \ftted by the least-squares method in terms of \u000bandhe\nare recorded in Table 4.A SECOND-ORDER METHOD FOR LLG EQUATION 11\n10-610-510-410-3100101102\nProposed method\nGSPM\nSIPM\n(a)Varyingkin 1D up to\nT= 1\n1.8 2 2.2 2.4 2.6 2.8 3 3.2\n10-7101102103\nProposed method\nGSPM\nSIPM(b)Varyingkin 3D up to T=\n0:1\n10-510-410-310-2101102103\nProposed method\nGSPM\nSIPM\n(c)Varyinghin 1D up to\nT= 1\n10-410-310-210-1100101102103\nProposed method\nGSPM\nSIPM(d)Varyinghin 3D up to\nT= 1\nFigure 1. CPU time needed to achieve the desired numerical ac-\ncuracy, for the proposed method, the GSPM and the SIPM, in\nboth the 1D and 3D computations. The CPU time is recorded as\na function of the approximation error by varying korhindepen-\ndently. CPU time with varying k: proposed method <SIPM<\nGSPM; CPU time with varying h: proposed method /GSPM<\nSIPM.\n4.Conclusions\nIn this paper, we have proposed a second-order accurate numerical method to\nsolve the Landau-Lifshitz-Gilbert equation with large damping parameters. For the\nnumerical convenience, the LLG system is reformulated so that in which the damp-\ning term is rewritten as a harmonic mapping \row .This numerical scheme is based on\nthe second-order backward-di\u000berentiation formula approximation for the temporal\nderivative, combined with an implicit treatment of the constant-coe\u000ecient di\u000busion\nterm, and the fully explicit extrapolation approximation of the nonlinear terms, in-\ncluding the gyromagnetic term and the nonlinear part of the harmonic mapping\n\row. Thanks to the large damping parameter, the proposed method is veri\fed12 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFigure 2. Stable structures in the absence of magnetic \feld at\n2 ns when\u000b= 0:01;10;40. The color denotes the angle between the\n\frst two components of the magnetization vector. Top: Proposed\nmethod; Middle: GSPM; Bottom: SIPM. Left: \u000b= 40; Middle:\n\u000b= 10; Right: \u000b= 0:01.\n(a)Proposed\n (b)GSPM\n (c)SIPM\nFigure 3. Energy evolution curves of three numerical methods,\nwith di\u000berent damping constants, \u000b= 2;5;8;10, up tot= 2 ns in\nthe absence of external magnetic \feld. Left: Proposed numerical\nmethod; Middle: GSPM; Right: SIPM. One common feature is\nthat the energy dissipation rate is faster for larger \u000b, which is\nphysically reasonable.A SECOND-ORDER METHOD FOR LLG EQUATION 13\n(a)\u000b= 2\n (b)\u000b= 5\n(c)\u000b= 8\n (d)\u000b= 10\nFigure 4. Energy evolution curves in terms of time, for the nu-\nmerical results created by three numerical methods up to t= 2 ns\nin the absence of external magnetic \feld for (a) \u000b= 2, (b)\u000b= 5,\n(c)\u000b= 8, and (d) \u000b= 10. The energy dissipation pattern of the\nproposed method is consistent with the other two methods for (a),\n(b), and (c), and the SIPM has a slightly di\u000berent energy dissipa-\ntion pattern from the other two methods for (d).\nto be unconditionally stable. The proposed method is much more e\u000ecient than\nother semi-implicit schemes since only symmetric, positive de\fnite linear systems\nof equations with constant coe\u000ecients need to be solved. Meanwhile, the proposed\nmethod is more accurate than the standard Gauss-Seidel projection method, due\nto its second-order accuracy in time. Numerical results in 1D and 3D are pro-\nvided to demonstrate the accuracy and the e\u000eciency of the proposed numerical\nmethod. In addition, micromagnetics simulations using the proposed method have\nprovided physically reasonable structures and captured the linear dependence of\nthe domain wall velocity with respect to the damping parameter. Therefore, the\nproposed method could be e\u000eciently used for challenging practical simulations of\nmicromagnetics with large damping parameters.14 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n(a)Magnetization for initial state\n (b)Magnetization with \u000b= 2 at\n2 ns\n(c)Magnetization with \u000b= 5 at\n2 ns\n(d)Magnetization with \u000b= 8 at\n2 ns\nFigure 5. Magnetization pro\fles of Ne\u0013 el wall motion in the pres-\nence of a magnetic \feld he= 5 mT, with \u000b= 2;5;8 at 2 ns for\nthe proposed numerical method. The in-plane arrow denotes the\n\frst two components of the magnetization vector. The wall moves\nfaster for larger values of \u000band its velocity depends linearly on \u000b.\nFigure 6. Linear dependence of the wall velocity with respect to\nthe damping parameter \u000b(left) and the external magnetic \feld he\n(right).\nAcknowledgments\nThis work is supported in part by the grants NSFC 11971021 (J. Chen), NSF\nDMS-2012669 (C. Wang), NSFC 11771036 (Y. Cai).A SECOND-ORDER METHOD FOR LLG EQUATION 15\nTable 4. Linear dependence of the domain wall velocity Vin\nterms of the external magnetic \feld heand the damping parameter\n\u000b.\n\u000bV(m/s) he(mT)5 6 7 8 9 10 Slope\n3 76 91 109 123 139 154 1.024\n4 105 118 139 157 179 196 0.928\n5 129 145 169 192 217 244 0.932\n6 153 169 200 227 256 286 0.927\n7 177 196 232 263 294 333 0.927\n8 200 222 263 303 333 385 0.954\n9 230 250 294 345 385 435 0.954\n10 253 270 323 370 417 476 0.943\nSlope 0.984 0.910 0.910 0.933 0.917 0.950 {\nReferences\n[1] G. Akrivis, M. Feischl, B. Kov\u0013 acs, and C. Lubich, Higher-order linearly implicit full dis-\ncretization of the Landau-Lifshitz-Gilbert equation , Math. 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Gao, Optimal error estimates of a linearized Backward Euler FEM for the Landau-Lifshitz\nequation , SIAM J. Numer. Anal. 52(2014), no. 5, 2574{2593.\n[8] C.J. Garc\u0010a-Cervera and W. E, Improved Gauss-Seidel projection method for micromagnetics\nsimulations , J. Comput. Phys. 171(2001), no. 1, 357{372.\n[9] T.L. Gilbert, Phys. Rev. 100(1955), 1243. [Abstract only; full report, Armor Research Foun-\ndation Project No. A059, Supplementary Report, May 1, 1956 (unpublished)].\n[10] T.L. Gilbert and J.M. Kelly, Anomalous rotational damping in ferromagnetic sheets , Armour\nResearch Foundation of Illinois Institute of Technology (1955). (unpublished).\n[11] B. Heinrich, D. Fra\u0013 \u0010tov\u0013 a, and V. Kambersk\u0013 y, The in\ruence of s-d exchange on relaxation of\nmagnons in metals , Phys. Stat. Solidi B-basic Solid Stat. Phys. 23(1967), 501{507.\n[12] M. Kruz\u0013 \u0010k and A. Prohl, Recent developments in the modeling, analysis, and numerics of\nferromagnetism , SIAM Rev. 48(2006), no. 3, 439{483.\n[13] L.D. Landau and E.M. Lifshits, On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies , Phys. Z. Sowjet. 63(1935), no. 9, 153{169.\n[14] D.M. Lattery, D. Zhang, J. Zhu, X. Hang, J. Wang, and X. Wang, Low Gilbert damping\nconstant in perpendicularly magnetized W/CoFeB/MgO \flms with high thermal stability ,\nSci. Rep. 8(2018), 13395.\n[15] P. Li, C. Xie, R. Du, J. Chen, and X. Wang, Two improved Gauss-Seidel projection methods\nfor Landau-Lifshitz-Gilbert equation , J. Comput. Phys. 401(2020), 109046.\n[16] T. Nan, Y. Lee, S. Zhuang, Z. Hu, J. Clarkson, X. Wang, C. Ko, H. Choe, Z. Chen, D. Budil,\nJ. Wu, S. Salahuddin, J. Hu, R. Ramesh, and N. Sun, Electric-\feld control of spin dynamics\nduring magnetic phase transitions , Sci. Adv. 6(2020), no. 40, eabd2613.16 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n[17] H. Suhl, Theory of the magnetic damping constant , IEEE Trans. Magn. 34(1998), 1834{\n1838.\n[18] T. Tanaka, S. Kashiwagi, Y. Otsuka, Y. Nozaki, Y. Hong, and K. Matsuyama, Microwave-\nassisted magnetization reversal of exchange-coupled composite nanopillar with large Gilbert\ndamping constant , IEEE Tran. Magn. 50(2014), 1{3.\n[19] H. Tang and K. Xia, Gilbert damping parameter in MgO-based magnetic tunnel junctions\nfrom \frst principles , Phys. Rev. Applied 7(2017), 034004.\n[20] C. Wang and J.-G. Liu, Convergence of gauge method for incompressible \row , Math. Comp.\n69(2000), 1385{1407.\n[21] X. Wang, C.J. Garc\u0013 \u0010a-Cervera, and W. E, A Gauss-Seidel projection method for micromag-\nnetics simulations , J. Comput. Phys. 171(2001), no. 1, 357{372.\n[22] R. Weber, D. Han, I. Boventer, S. Jaiswal, R. Lebrun, G. Jakob, and M. Kl aui, Gilbert\ndamping of CoFe-alloys , J. Phys. D 52(2019), 325001.\n[23] D. Wei, Micromagnetics and recording materials , Springer Briefs in Apllied Sciences and\nTechnology, Springer Berlin Heidelberg, 2012.\n[24] C. Xie, C.J. Garc\u0013 \u0010a-Cervera, C. Wang, Z. Zhou, and J. Chen, Second-order semi-implicit\nmethods for micromagnetics simulations , J. Comput. Phys. 404(2020), 109104.\n[25] D. Zhang, M. Li, L. Jin, C. Li, Y. Rao, X. Tang, and H. Zhang, Extremely large magnetiza-\ntion and Gilbert damping modulation in NiFe/GeBi bilayers , ACS Appl. Electron. Mater. 2\n(2020), no. 1, 254{259.\nSchool of Mathematical Sciences, Beijing Normal University, Beijing, China.\nEmail address :yongyong.cai@bnu.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzhou, China.\nEmail address :jingrunchen@suda.edu.cn\nMathematics Department, University of Massachusetts, North Dartmouth, MA 02747,\nUSA.\nEmail address :cwang1@umassd.edu\nSchool of Mathematical Sciences, Soochow University, Suzhou, China.\nEmail address :20184007005@stu.suda.edu.cn"    },    {        "title": "0912.5521v1.Spin_torque_and_critical_currents_for_magnetic_vortex_nano_oscillator_in_nanopillars.pdf",        "content": "1 \n  \nSpin torque and critical currents for magnet ic vortex nano-oscillator in nanopillars \n \nKonstantin Y. Guslienko1,2*, Gloria R. Aranda1, and Julian M. Gonzalez1 \n \n1Dpto. Fisica de Materiales, Universidad de l Pais Vasco, 20018 Donostia-San Sebastian, Spain  \n2IKERBASQUE, the Basque Foundation for Science, 48011 Bilbao, Spain \n \n \n   We calculated the main dynamic parameters of the spin polarized current  induced magnetic vortex \noscillations in nanopillars, such as the range of current density, wh ere a vortex steady oscillation state \nexists, the oscillation frequency and orbit radius. We accounted for both the non-linear vortex frequency \nand non-linear vortex damping. To describe the vortex excitations by the spin polarized current we used \na generalized Thiele approach to motion of the vortex  core as a collective coordinate. All the results are \nrepresented via the free layer sizes, saturation magnetiza tion, Gilbert damping and the degree of the spin \npolarization of the fixed layer. Pr edictions of the developed model can be checked experimentally.  \n  \nKey words: spin polarized current, magnetic nanopillar, nano-oscilla tors, magnetic vortex \n \n*Corresponding author. Electronic mail: sckguslk@ehu.es  2 \n      Now excitations of the microwave oscillatio ns in magnetic nanopilla rs, nanocontacts and tunnel \njunctions by spin polarized curren t as well as the current induced domain wall motions in nanowires are \nperspective applications of spintronics.1 A general theoretical approach to microwave generation in \nnanopillars/nanocontacts driven by spin-polarized current based on the universal model of an auto-\noscillator with negative damping and nonlinear frequency shift was de veloped recently by Slavin and \nTiberkevich [see Ref. 2 and references therein]. Th e model was applied to the case of a spin-torque \noscillator (STO) excited in a uniformly magnetized free layer  of nanopillar,  and explains the main \nexperimentally observed effects such as the power and frequency of the gene rated microwave signal. \nHowever, the low generated power ~ 1 nW of such STO prevents their practic al applications. Recently  \nextremely narrow linewidth of 0.3 MHz and relatively high generated power was detected for the \nmagnetic vortex (strongly non-uniform st ate) nano-oscillat ors in nanopillars.3 The considerable \nmicrowave power emission from a vortex STO in magnetic tunnel junctions was observed.4 It was \nestablished that the permanent perpendicular to the plane (CPP) spin polarized current I can excite \nvortex motion in free layer of th e nanopillar if the current intens ity exceeds some critical value, Ic1.5 \nThen, in the interval Ic1 <I <Ic2 (Ic2 is a second critical current) th ere is microwave generation at a \nfrequency which smoothly increase with the current increasing.6 This frequency corresponds to the \nvortex oscillations with a stationary orbit determ ined by the current value. If the current exceeds a \ncritical value Ic27,8 the vortex steady state is not stable anym ore, presumably because the vortex reaches a \ncritical velocity9 and reverses its core. The vortex with op posite core cannot be excited for the given \ncurrent sign I and the oscillations stop. Such  excitation scheme is different  from the current-in-plane \n(CIP) case, where one needs to apply a.c. CIP of about the resonance frequency to excite the vortex \nmotion.10 Low value of Ic1 and high value of Ic2 make the vortex CPP nano-os cillators attractive for \napplications as microwave devices. However, the calcu lations of the spin torque term (ST force) gave \ncontradictory results for the ST force and Ic1. The standard STO approach of Ref. 2 is not applicable to 3 \n the vortex dynamics due to specific damping term. The problem was reduced to  the problem of vortex \ncore reversal in the perpe ndicular to the layer plane  magnetic field in Ref. 7 but vortex steady state \ndynamics was not accounted. Using the Thiele approach  the ST force was calculated in Refs. 5, 8 which \ndiffers in 2 times from one calculate d form the energy dissipation balance.6 Non-linearity of the main \ngoverning parameters and the Oersted field of curre nt were not accounted or accounted incorrectly. The \ncritical current Ic2 has not yet been calculated.  \n     In this Letter we present a simple and effectiv e approach to calculate the ST force and the critical \ncurrents of the vortex STO in nanopillar. The appro ach is based on the Thiele formulation of the non-\nuniform magnetization dynamics11 and conception of the linear spin excitations.12 The nanopillar device \nconsists of two ferromagnetic layers , typically FeNi or Co and a nonma gnetic metallic spacer, typically \nCu, arranged in a vertical stack (Fig . 1). Magnetization of one layer is fixed (this layer is the so called \npolarizer), whereas the magnetization of  the second layer of the nanopillar ()t,rM  is free to rotate. The \ncurrent of spin polarized elec trons transfers some torque sτ from the polarizer, which excites \nmagnetization dynamics of the free layer. We st art from the Landau-Lifshi tz equation of motion \ns LLG eff τ mm Hm m γ α γ +× + × −= \u0005 \u0005 , where sM/Mm= , Mss  is the saturation magnetization, γ>0 is the \ngyromagnetic ratio,  Heff is the effective field, and LLGα  is the Gilbert damping. We use the ST term in \nthe form suggested by Slonczewski,13 () Pm mτ × × =Jsσ , where ()sLMe2/η σ== , η is the current \nspin polarization ( η=0.2 for FeNi), e is the electron charge, L is the free layer (dot) thickness, J is the \ncurrent density, and z PP=  is the unit vector of the polarizer magnetization ( P=+1/-1). We assume the \npositive vortex core polarization p=+1, P=+1 and define the current (flow of the positive charges) as \npositive I>0 when it flows from the polarizer to free layer.  The spin polarized curr ent can excite a vortex \nmotion in the free layer if only IpP > 0 (only the electrons bringing  a magnetic moment from the \npolarizer to free layer opposite to the core polarization can excite a vortex motion). Except p, the vortex 4 \n is described by its core position in the free layer, X=(X,Y), and chirality C=±1 .14 Let denote the \nSlonczewski´s energy density which correspond s to the spin polarized current as sw. Then, using the \nThiele approach and the ST field Pm m × =∂ ∂ a ws/ , the ST force acting on th e vortex in the free layer \ncan be written as \n                                          ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂× ⋅ =∂∂−= ∫ ∫\nα αα\nXd aL dVwXFs STmmρ P2,                                             (1) \nwhere J Masσ = , α=x, y, ()ϕρ, =ρ  is the in-plane radius vector, the derivative is taken with respect \nto the vortex core position X assuming an ansatz () ( ) [] t t Xρmρm , ,=  (m dependence on thickness \ncoordinate z is neglected). X has sense of the amplitude of the vortex gyrotropic eigenmode. \n     We use representation of m-components by the spherical angles ΦΘ, (Fig. 1) as  \n) cos, sin sin, cos (sin Θ Φ Θ Φ Θ =m  and find the expression for the ST force \n                                                          Xρ F∂Φ∂Θ =∫2 2sindaLST .                                                            (2) \n     In the main approximation we use the decompositions ( ) () () ( ) ρX Xρ ˆ cos ,0⋅ + =Θ = ρ ρ g m mz z , \n() ( ) [] ϕ ϕ ρ cos sin ,0 0 Y X m − +Φ= Φ Xρ , where ()ρ0\nzm , 0Φ are the static vortex core profile and phase, \n() () ( )22 2 2 2/ 1 4 ρ ρ ρ ρ + + = c pc g  is the excitation amplitude of the z-component of the vortex \nmagnetization ( RRcc/ = , cR is the vortex core radius, ρ, X are normalized to the dot radius R) and \n()() ρ ρ ρ / 12\n0 −= m  is the gyrotropic mode profile.12 One can conclude from Eq. (2) that only moving \nvortex core contribute to  the ST force because the contribu tion of the main dot area where 2/π=Θ  is \nequal to zero due to vanishing integrals on azimuthal angle φ from the gradient of the vortex phase Φ∂X 5 \n (it was checked accounting in ()Xρ,Φ  the terms up to cubic terms in X α-components). This is a reason \nwhy the ST contribution is relatively small bei ng comparable with the damping contribution.  \n     The integration in Eq. (2) yields the ST force ()Xz F × = ˆaLSTπ . This force contributes to the Thiele’s \nequation of motion ST D W FX XGX + + −∂=× \u0005 \u0005 ˆ , where γ π / 2ˆs pLMzG=  is the gyrovector, Dˆ is the \ndamping tensor. The vortex energy ()XW  and restoring force WR X F −∂=  can be calculated from an \nappropriate model14 (the force balance is shown in Fig. 2). For circular steady st ate vortex core motion \nthe XωX ×=\u0005  relation holds, which allows calculating Jc1. To calculate the vortex steady orbit radius \nX=sR  we need, however, to account non-linear on X α terms in the vortex damping and frequency (the \naccount only non-linear frequency as in Ref. 5 is not sufficient). The gyr ovector also depends on X, but \nthis dependence is essential only for the vortex core p reversal, where G changes its sign.  As we show \nbelow, the most important non-linearity co mes from the damping tensor defined as \n                                                   ()\nβ ααβγαX XdVMDs\nLLG∂∂⋅∂∂−=∫m mX ,                                                  (3)  \nor () [] Φ∂Φ∂Θ +Θ∂Θ∂ −= ∫ β α β α αβ γ α2 2sin /ρd LM Ds LLG  in ΦΘ, -representation Accounting \nαβ αβ δD D=  and introducing dimensionless damping parameter 0 /> −= GD d15 we can write the \nequation for a steady state vortex motion with the orbit radius X=sR : () ()ϕωST G F sRs Gsd =  from \nwhich RRss/ =  and the critical currents Jc1, Jc2 can be found. In the second order non-linear \napproximation ()2\n1 0 sd dsd + = , ()2\n1 0 s sG ω ω ω + =  and aLRs FSTπϕ= , whereGω is the vortex \nprecession frequency, 01>ω  is a function of the dot aspect ratio β=L/R  calculated from the vortex \nenergy decomposition ()sW  i n  s e r i e s  o f  R s /X= . It can be shown that () ( ) [] 3/41 9/200 β βγ βω − =sM  \nand () () βω βω0 1 4≈  for quite wide range of β= 0.01-0.2 of practi cal interest, whereas considerably larger 6 \n non-linearity () () 8.42 /0 1 =βωβω  was calculated in Ref. 5 due to in correct account of the magnetostatic \nenergy. We use the pole free model of the shifted vortex ()[] tXρm, , where the dynamic magnetization \nsatisfies the strong pinning boundary condition at the dot circumference16 R=ρ . The damping \nparameters are () () 2/ / ln8/50 c LLG RR d + =α , () 4/3/8 /2 2\n1 − =c LLG RR d α . We need also to account for \nthe Oersted field of the current, wh ich leads to contribution to the vo rtex frequency proportional to the \ncurrent density () ( ) J Jeω βω βω ω + = =0 0 0 , , where () ( ) CcRe ξ γ π ω / 15/8= , () R Rc8/ 2/12ln151 − +=ξ  \nis the correction for the finite core radius cR<<R. In the linear approximation we get the equality of the \ndamping force and the ST force (negative damp ing) as the condition to find the value of Jc1, with the \ncontribution of the Oersted field of the current accounted. The values of FST, Jc1 coincide with \ncalculation of Ref. 6 conducted by the method of the damping energy balance and differ by the \nmultiplayer 2 from Refs. 5, 8. The first critical current is ()e c d d J ω γσω0 00 1 2/ / − =  and the steady \nstate vortex orbit radius is \n \n                       () 1 /1− =cJJ Js λ ,     1cJJ> ,     ()[]10 1 011 2\n21\nω ωγσλd J dJ\ncc\n+=                      (4) \n     In this approximation the vortex trajectory radius ()Js  increases as square root of the current \novercriticality ()1 1/c cJ JJ−  (for the typical parameters and R=80-120 nm we get λ=0.25-0.30) and the \nvortex frequency () ( )1 12\n0 1 / ω λ ω βω ω − + + =c e G JJ J  increases linearly with J increasing. The vortex \nsteady orbit can exist until the m oving vortex crosses the dot border s=1 or its velocity X\u0005 reaches the \ncritical velocity cυ defined in Ref. 9. The later allows to write equation for the s econd critical current Jc2 \nas () ()c G RJsJ υ ω =. Substituting to this expression the equations for ()JGω  and ()Js  derived above 7 \n we get a cubic equation for Jc2 in the form () [ ] R xx J d Jc ce c λυ λω ω γσ / 2/2/1 2\n1 1 0 1 = + + , \n() 1 /1 2 − =c cJ J x . This equation has one positive root xc and the value of Jc2 can be easily calculated \n(Fig. 3). The former condition ( s=1) gives the second critical current ()12\n2 /11c c J J λ+=′ . More detailed \nanalysis shows that both the mechanisms of the hi gh current instability of the vortex motion are possible \ndepending on the dot sizes L, R, and the critical current is th e lower value of the currents Jc2, J’c2. The \nvortex core reversal inside the dot occurs for large enough R (> 100 nm) and L. For the typical sizes \nL=10 nm, R=120 nm and C=1, the critical currents are Jc1=6.3 106 A/cm2 (Ic1=2.9 mA), Jc2=1.13 108 \nA/cm2 (Ic2=51 mA), and for L= 5 nm, R=100 nm we get Jc1=1.8 106 A/cm2 (Ic1=0.56 mA), J’c2=2.7 107 \nA/cm2 (I’c2=8.4 mA). \n   In summary, we calculated the main physical para meters of the spin polar ized CPP current induced \nvortex oscillations in na nopillars, such as the cr itical current densities Jc1, Jc2, the vortex steady state \noscillations frequency and orbit radius. All the results are represented via the free layer sizes ( L, R), \nsaturation magnetization, Gilbert damp ing and the degree of the spin polarization of the fixed layer. \nThese parameters can be obtained from independent e xperiments. We demonstrated that the generalized \nThiele approach is applicable to the problem of the vortex STO excitations by the CPP spin polarized \ncurrent. The spin transfer torque force is related to the vortex core only. \n     The authors thank J. Grollier and A.K. Khvalkovsk iy for fruitful discussions. K.G. and G.R.A. \nacknowledge support by IKERBASQUE  (the Basque Science Foundation) and by the Program JAE-doc \nof the CSIC (Spain), respectively. The author s thank UPV/EHU (SGIker Arina) and DIPC for \ncomputation tools. The work was part ially supported by the SAIOTEK grant  S-PC09UN03. \n \n 8 \n References  \n1 G. Tatara, H. Kohno, and J. Shibata, Phys. Rep . 468, 213 (2008). \n2 A. Slavin and V. Tiberkevich, IEEE Trans. Magn.  45, 1875 (2009). \n3 V.S. Pribiag, I.N. Krivorotov, G.D. Fuchs et al., Nature Phys . 3, 498 (2007). \n4 A. Dussaux, B. Georges, J. Grollier et al. , submitted to Nature Phys.  (2009). \n5 B. A. Ivanov an d C. E. Zaspel, Phys. Rev. Lett.  99, 247208 (2007). \n6 A.V. Khvalkovskiy, J. Grollier, A. Dussaux, K.A. Zvezdin, and V. Cros, Phys. Rev . B 80, 140401 \n(2009).  \n7 J.-G. Caputo, Y. Gaididei, F.G. Mertens and D.D. Sheka, Phys. Rev. Lett.  98, 056604 (2007); D.D.  \n  Sheka, Y. Gaididei, and F.G. Mertens, Appl. Phys. Lett.  91, 082509 (2007). \n8 Y. Liu, H. He, and Z. Zhang, Appl. Phys. Lett.  91, 242501 (2007). \n9 K.Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett . 100, 027203 (2008); K.-S. Lee et al., Phys. \nRev. Lett.  101, 267206 (2008). \n10 S. Kasai, Y. Nakatani, K. Koba yashi, H. Kohno, and T. Ono, Phys. Rev. Lett.  97, 107204 (2006);  \n   K. Yamada, S. Kasai, Y. Naka tani, K. Kobayashi, and T. Ono, Appl. Phys. Lett.  93, 152502 (2008). \n11 A. A. Thiele, Phys. Rev. Lett . 30, 230 (1973).  \n12 K.Y. Guslienko, A.N. Slavin, V. Tiberkevich, S. Kim, Phys. Rev. Lett.  101, 247203 (2008).  \n13 J. Slonczewski, J. Magn. Magn. Mat . 159, L1 (1996); J. Magn. Magn. Mat . 247, 324 (2002).  \n14 K.Y. Guslienko, J. Nanosci. Nanotechn.  8, 2745 (2008). \n15 K.Y. Guslienko, Appl. Phys. Lett.  89, 022510 (2006).  \n16 K. Y. Guslienko et al., J. Appl. Phys . 91, 8037 (2002); V. Novosad et al. , Phys. Rev . B 72, 024455 \n(2005). 9 \n  \nCaptions to the Figures \n \nFig. 1. Sketch of the magnetic nan opillar with the coordinate system used. The upper (free) layer is in \nthe vortex state with non- uniform magnetization distribution. The polarizer layer (red color) is in \nuniform magnetization state w ith the magnetization along Oz axis. The positive current I (vertical arrow) \nflows from the polarizer to free layer.  \n Fig. 2. Top view of the free laye r with the moving vortex. The arrows  denote the force balance for the \nvortex core. The spin torque (\nFST), damping ( FD), restoring (RF) and gyro- ( FG) forces are defined in the \ntext. The vortex core steady trajectory Rs is marked by orange color. The vortex chirality is C=+1. \n \nFig. 3. Dependence of the critical currents Jc1 (solid red line), Jc2 (dashed green line) and J’c2 of the \nvortex motion instability on the radius R of the free layer. L= 10 nm, Ms =800 G, η =0.2, 01 .0=LLGα , \nγ/2 =2.95 MHz/Oe, Rc=12 nm.  The vortex STO motion is stable at Jc1 < J < min( Jc2, J’c2). \n \n \n     10 \n  \n  Fig. 1. \n \n  \n \n \n     \n \n     11 \n  \n  Fig. 2. \n \n \n \n \n   \n \n      \n 12 \n  \n  Fig. 3. \n \n \n60 80 100 120 14056789101112\nJc2\nJ'c2Current density, J (107 A/cm2)\nDot radius, R (nm)FeNi\nMs=800 G\nL=10 nm\nJc1x10\n "    },    {        "title": "1706.02296v2.Adiabatic_and_nonadiabatic_spin_torques_induced_by_spin_triplet_supercurrent.pdf",        "content": "Adiabatic and nonadiabatic spin torques induced by spin-triplet supercurrent\nRina Takashima,1,\u0003Satoshi Fujimoto,2and Takehito Yokoyama3\n1Department of Physics, Kyoto University, Kyoto 606-8502, Japan\n2Department of Materials Engineering Science, Osaka University, Toyonaka 560-8531, Japan\n3Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan\n(Dated: September 18, 2018)\nWe study spin transfer torques induced by a spin-triplet supercurrent in a magnet with the superconduct-\ning proximity e \u000bect. By a perturbative approach, we show that spin-triplet correlations realize new types of\ntorques, which are analogous to the adiabatic and nonadiabatic ( \f) torques, without extrinsic spin-flip scatter-\ning. Remarkable advantages compared to conventional spin-transfer torques are highlighted in domain wall\nmanipulation. Oscillatory motions of a domain wall do not occur for a small Gilbert damping, and the thresh-\nold current density to drive its motion becomes zero in the absence of extrinsic pinning potentials due to the\nnonadiabatic torque controlled by the triplet correlations.\nE\u000ecient manipulation of magnetization is of great techno-\nlogical importance. Spin-transfer torques (STT), which can\ncontrol magnetization with an electric current, have attracted\nattention[1–7], and STT can be applied to the so-called race-\ntrack memory using magnetic domain walls[8]. In a smooth\nmagnetic texture n, spin-polarized currents exert STT on n,\nwhich is given by [9–11]\n\u001cSTT=\u0000(js\u0001r)n+\fn\u0002(js\u0001r)n: (1)\nHerejs=\u0000(Pa3=2eS)j, and\fis a dimensionless parameter,\nwherejis a charge current density, Pis the spin polariza-\ntion of current, ais the lattice constant, Sis the spin size,\nand\u0000eis the electron charge. The first term in Eq. (1) arises\nwhen the electron spins follow the texture adiabatically. The\nsecond term, often referred to as the non-adiabatic torque, is\nknown to have two origins [10, 12, 13]. It appears from spin-\nflip impurity scatterings or the spin-orbit coupling. It also oc-\ncurs when electrons fail to follow magnetic textures because\nthe texture is not smooth enough. As demonstrated in several\nworks [10, 14], the nonadiabatic torque plays a crucial role\nin magnetization dynamics. For \f,0, the threshold current\ndensity for a steady motion of a domain wall becomes zero in\nthe absence of pinning potentials.\nRecently, superconductivity has opened up new possibil-\nities for spintronics with suppressed Joule heating[15, 16].\nIt has been pointed out that the Josephson current exerts\na spin torque on magnetization in ferromagnetic Joseph-\nson junctions[17–23]. Furthermore, with spin valves using\nsuperconductors[24–28], one can change the resistance dras-\ntically by a magnetic field, and the lifetime of spin density\nis enhanced in a superconducting state relative to a normal\nstate[29–31]. Such an interplay of superconductivity and\nmagnetic moments is important especially with spin-triplet\nCooper pairs due to the coupling between triplet order pa-\nrameters and localized moments[32–37]. Triplet pairs can\narise in the interface between a ferromagnet and singlet su-\nperconductor when there is magnetic inhomogeneity[38] or\nspin-orbit couplings [39, 40]. Experimentally, the proxim-\nity e\u000bect of triplet pairs has been observed in fully-spin\n\u0003Electronic address: takashima@scphys.kyoto-u.ac.jppolarized metals[41, 42] and multilayers with noncollinear\nmagnets[43, 44]. The spin-triplet proximity e \u000bect to a ferro-\nmagnet from Sr 2RuO 4, a candidate of a triplet superconductor,\nhas been also observed[45].\nGiven the experimental advances in the proximity-induced\ntriplet Cooper pairs in magnets, triplet supercurrent-induced\nSTT is a promising way to realize an e \u000ecient control of mag-\nnetization. Utilization of a supercurrent suppresses Joule heat-\ning and the tunablity of STT may be enhanced by pairing de-\ngrees of freedom. However, while several works showed that a\nsupercurrent exerts a spin-torque in ferromagnetic Josepshon\njunctions[17–23], it still remains unclear how a triplet super-\ncurrent acts on a localized moment, and how STT is changed\nby triplet-paring correlation.\nIn this work, we microscopically study STT induced by\ntriplet supercurrents considering the spin-triplet proximity ef-\nfect. We show that the derived STT have two parts, analogous\nto the adiabatic and non-adiabatic torques, which can be tuned\nby the triplet correlations. Remarkable advantages compared\nto conventional STT are highlighted in domain wall manipu-\nlation. In contrast to the non-adiabatic STT in normal metals,\nthe supercurrent-induced STT do not require extrinsic scatter-\ning processes, and hence, is more easy to control. Further-\nmore, a domain wall does not show oscillatory motions for\na small Gilbert damping, and hence an e \u000ecient manipulation\ncan be realized.\nLet us consider a thin-film magnet with the proximity e \u000bect\nofp-wave triplet superconductivity modeled by the Hamilto-\nnian :\nHel=\u0000tX\nhi;jicy\ni\u000bcj\u000b\u0000\u0016X\nicy\ni\u000bci\u000b\n\u0000JsdSX\nin(ri)\u0001\u001b\u000b\fcy\ni\u000bci\f\n+\u00010\n2X\ni;jeiQ\u0001(ri+rj)h\n(di j\u0001\u001b)i\u001byi\n\u000b\fcy\ni\u000bcy\nj\f+H:c:(2)\nHere cy\ni\u000b(ci\u000b) is the electron creation (annihilation) operator at\nsiteiwith spin\u000bon a square lattice, hi;jiis taken over the\nnearest neighbor pairs, tis the hopping amplitude, \u0016is the\nchemical potential, and \u001b=(\u001bx;\u001by;\u001bz) are the Pauli matri-\nces. Electrons couple to localized spins given by Sn(ri)=arXiv:1706.02296v2  [cond-mat.mes-hall]  16 Sep 20172\nTriplet SuperconductorDomain wall motion\nSupercurrentFM\nFIG. 1: (Color online) Proposed setup. A ferromagnet (FM) with a\ndomain wall is attached on a superconductor, which produces a spin-\ntriplet proximity e \u000bect. We apply a supercurrent in the xyplane and\ndrive the domain wall motion.\nS(sin\u0012cos\u001e;sin\u0012cos\u001e;cos\u0012) with the coupling constant Jsd.\nThe spin texture n(r) varies smoothly with the length scale\n`(\u001d\u0018SC) where\u0018SCdenotes the superconducting coherence\nlength. The last term in Eq. (2) is the proximity-induced triplet\np-wave pairing, given by di j=(dx\ni j;dy\ni j;dz\ni j), wheredi j=\u0000dji,\nand\u00010is the pairing amplitude.\nThe phase gradient of the paring function describes a super-\ncurrent. We consider a phase given by Q\u0001(ri+rj), which re-\nsults in the supercurrent density j'\u00002tenea2Q(forjQja\u001c1\nand at low temperature), where neis an electron density[56]\nthat participates in the supercurrent. Here, the supercurrent\ncan be supplied from an external dc current source. We also\nnote that we can restore the gauge invariance by redefining\nQin the current so as to include the vector potential. As\nshown in Fig. 1, a relevant experimental setup of the above\nmodel is a heterostructure composed of a metallic magnet and\na triplet superconductor (e.g., Sr 2RuO 4). A triplet supercon-\nductor can be replaced by a singlet superconductor (e.g., Nb)\nwith a conical magnetic layer such as Ho [44] or the spin-orbit\ncoupling[39, 40], which produces the triplet proximity e \u000bect.\nTo derive STT, we calculate the spin density induced by\na supercurrent to the linear order of j/Q. In the follow-\ning calculation, we perturbatively treat the spatial derivative\nofn(ri). This treatment can be simplified by rewriting the\nHamiltonian with electron operators ai\u000b, the spin quantization\naxis of which is parallel to n(ri)[10]. It is defined by ci\u000b=\n(U(ri))\u000b\fai\f, where U(ri)=m(ri)\u0001\u001bis a unitary matrix, and\nm(ri)=(sin(\u0012=2)cos\u001e;sin(\u0012=2)sin\u001e;cos(\u0012=2)) with\u0012and\n\u001ebeing the angles of n(ri). This satisfies Uy(n\u0001\u001b)U=\u001bz.\nThe Hamiltonian Eq. (2) is rewritten as\nHel'X\nk(\u0018kI2\u0000JsdS\u001bz)\u000b\u000bay\nk\u000bak\u000b\n+X\nk;q3\u0017\nk+q=2Aa\n\u0017(q)\u001ba\n\u000b\fay\nk+q\u000bak\f\n+1\n2X\nk\u0001\u000b\nkay\nk+Q\u000bay\n\u0000k+Q\u000b+H.c.; (3)\nwhere\u0018k=\u00002t(cos( kxa)+cos(kya))\u0000\u0016is the kinetic energy,\n3\u0017\nk=@\u0018k=@k\u0017is the velocity, I2=diag(1;1) , and ak\u000bis the\nFourier transform of ai\u000b. The second term arises from the\nhopping in the presence of a noncollinear texture. Here we\ndefine the spin gauge field Aa\n\u0017i\u001ba=\u0000iU(ri)@\u0017U(ri), where\nwe denote Ab\n\u0017(q)=N\u00001P\niAb\n\u0017ie\u0000iq\u0001riwith\u0017=fx;y;zgandb=fx;y;zg.Nis the total number of sites. Assuming a large\nexchange splitting \u00182JsdS\u001dj\u00010j, we focus on the equal spin\npairing given by\n\u0001\u000b\nk= \u0001 0(Uy(d(k)\u0001\u001b)i\u001byU\u0003)\u000b\u000b; (4)\n=\u0000\u00010\u0010\n(Rabdb(k)\u001ba)i\u001by\u0011\n\u000b\u000b; (5)\nwhered(k)=N\u00001P\nhi jidi je\u0000ik\u0001(ri\u0000rj). We neglect pairings be-\ntween spin-split bands, which correspond to the components\nofd(k) parallel ton.Rab=2mamb\u0000\u000eab(a;b=fx;y;zg) is a\nSO(3) rotation matrix corresponding to the unitary matrix U.\nIn the last term in Eq. (3), we neglect the spatial dependence\nof\u0012;\u001e, terms of the order of \u0018SC=`, assuming thatj\u00010(k)j=tis\nalso a small parameter.\nThe spin expectation value of electrons sa\ni=\n1\n2(\u001ba)\u000b\fhcy\ni\u000bci\fican be described by the operator ai\u000bas\nsa\ni=Rab(ri)˜sb\ni; (6)\nwhere ˜ sa\ni=1\n2(\u001ba)\u000b\fhay\ni\u000bai\fi. Noting this relation, we obtain\nthe spin density induced by a supercurrent as\n\u000e˜sa\nqB \nlim\nQ!0˜sa\nq\u0000˜sa\nqjQ=0\nQ\u0011!j\u0011\n(\u00002tenea2); (7)\n=\u0019ab\n\u0017\u0011Ab\n\u0017(q)j\u0011\n2ene; (8)\nwhere\u000e˜sq=N\u00001P\ni\u000e˜sie\u0000iq\u0001riand\n\u0019ab\n\u0017\u0011=lim\nq!0\u0000T\n4Nta2X\nn;k@2\u0018k\n@k\u0017@k\u0011Tr[SaGk+q(i\u000fn)SbGk(i\u000fn)]:(9)\nSee Supplemental Materials (SM) for detail [55]. Here,\nGk(i\u000fn) is the Green function in Nambu representation, the\nbasis of which is ( ak\";ak#;ay\n\u0000k\";ay\n\u0000k#)T, and Sais a spin ma-\ntrix in the Nambu representation, which is given by\nSa= \n\u001ba\n\u0000(\u001ba)T!\n: (10)\nIts inverse is defined by ( Gk(i\"n))\u00001=i\u000fnI4\u0000HBdG(k), where\nHBdG(k)= \n\u0018kI2\u0000M\u001bz\u0001(k)\n\u0001\u0003(k)\u0000\u0018kI2+M\u001bz!\n; (11)\nI4=diag(1;1;1;1),\u000fn=\u0019T(2n+1) is Matsubara frequency\nwith temperature T, and \u0001(k)=diag(\u0001\"\nk;\u0001#\nk).\nIn Eq. (9), we have neglected terms which are vanishingly\nsmall at low temperatures compared to the critical temper-\nature, in a system with a full gap or point nodes, see SM\n[55]. Also, we have taken the limit q!0assuming that the\nmomentum transfer from a smooth magnetic texture is small\ncompared to the Fermi momentum [10]. We note that siis\ninvariant by a unitary transformation U!Uei'\u001bzwith an\narbitrary spin rotational angle '(r) aroundn, while ˜siand\nRabchange their forms. In the following, we explicitly use\n@2\u0018k\n@k\u0017@k\u0011/\u000e\u0017\u0011and define\u0019ab\n\u0017\u0011=\u0019ab\n\u0017\u000e\u0017\u0011.3\nFrom Eqs. (6) and (8), we obtain the local STT, \u001cSTT=\n2Jsdn\u0002\u000esi, as\n\u001cSTT=X\n\u0017=x;y\u0000˜P\u0017a3\n2eSj\u0017\u0010\n\u0000@\u0017n+˜\f\u0017n\u0002@\u0017n\u0011\n: (12)Here ˜P\u0017and ˜\f\u0017are the analogs of the spin polarization of a\ncurrent Pand\fin Eq. (1), and they are given by\n˜P\u0017=JsdS\nnea3\"1\n2\u0010\n\u0019xx\n\u0017+\u0019yy\n\u0017\u0011\n+1\nj@\u0017nj2\u0010\n\u0000\u0019(1)\n\u0017\u0010\n(@\u0017\u0012)2\u0000sin2\u0012(@\u0017\u001e)2\u0011\n+2\u0019(2)\n\u0017sin\u0012@\u0017\u0012@\u0017\u001e\u0011#\n; (13)\n˜\f\u0017=\u0000JsdS\nnea31\n˜P\u00171\nj@\u0017nj2\u0010\n\u0019(2)\n\u0017\u0010\n(@\u0017\u0012)2\u0000sin2\u0012(@\u0017\u001e)2\u0011\n+2\u0019(1)\n\u0017sin\u0012@\u0017\u0012@\u0017\u001e\u0011\n; (14)\nwhere\n\u0019(1)\n\u0017=cos(2\u001e)1\n2\u0010\n\u0019xx\n\u0017\u0000\u0019yy\n\u0017\u0011\n+sin(2\u001e)\u0019xy\n\u0017; (15)\n\u0019(2)\n\u0017=sin(2\u001e)1\n2\u0010\n\u0019xx\n\u0017\u0000\u0019yy\n\u0017\u0011\n\u0000cos(2\u001e)\u0019xy\n\u0017: (16)\nThese are the central results of this paper, which are ap-\nplicable to any smooth magnetic textures. Notably, in the\nlow density limit and at low temperature, ˜P\u0017and ˜\f\u0017are\ngiven by the spin susceptibility perpendicular to nsince\n\u0019ab\n\u0017is equivalent to the bare spin susceptibility of the ak\u000b\nfield. To make ˜\f\u0017finite, anisotropy such as \u0019xx\n\u0017,\u0019yy\n\u0017or\n\u0019xy\n\u0017,0 is necessary. As is known in the spin susceptibil-\nity [46], such anisotropy naturally arises with a triplet pair-\ning. They depend on the relative phase between \u0001\"\nkand\n\u0001#\nkas1\n2\u0010\n\u0019xx\n\u0017\u0000\u0019yy\n\u0017\u0011\n=limq!0N\u00001P\nkRe(\u0001\u0003\"\nk\u0001#\nk)f\u0017(k;q) and\n\u0019xy\n\u0017=limq!0N\u00001P\nkIm(\u0001\u0003\"\nk\u0001#\nk)f\u0017(k;q);where f\u0017(k;q) is\npresented in the SM [55]. Therefore, a triplet pairing can\nmake ˜\f\u0017finite and cause the non-adiabatic torque without ex-\ntrinsic scattering processes. Furthermore, ˜P\u0017and˜\f\u0017depend on\nthe spatial position through the coupling between the dvector\nandn. This is important for a domain wall dynamics as we\nsee below.\nNow, we demonstrate domain wall dynamics[47] induced\nby the obtained STT. We consider the Hamiltonian Htot=\nHel+Hspin, where Helis given in Eq. (3) and\nHspin=S2\n2\n\u0002X\ni\u0010\n\u0000J(@\u0017n(ri;t))2\u0000Knz(ri;t)2+K?ny(ri;t)2\u0011\n:(17)\nHere, Jis the ferromagnetic exchange coupling, and K;K?\nare the onsite anisotropies that satisfy K?\u001c K\u001c\nJa\u00002. We consider a domain wall configuration given by\nn(r;t)=(cos\u001e0(t) sin\u0012(x;t);sin\u001e0(t) sin\u0012(x;t);cos\u0012(x;t)),\nwhere cos\u0012(x;t)=tanh\u0010x\u0000X(t)\n\u0015\u0011\n,\u0015=pJ=K, and X(t) is the\ndomain wall center. A schematic figure of a domain wall for\n\u001e0=\u0019is shown in Fig. 1. With this configuration, the STT\nin Eq. (12) is characterized by ˜Px=JsdS\nnea3h1\n2\u0010\n\u0019xx\nx+\u0019yy\nx\u0011\n\u0000\u0019(1)\nxiand˜\fx=\u0000JsdS\nnea3\u0019(2)\nx\n˜Px. Including the e \u000bects of damping, we ob-\ntain the equations of motion of \u001e(t) and X(t) as\n@tX=3c\n(1+\u000b2)(\u001c(\u001e0)jx+\u000bF(\u001e0)jx+sin 2\u001e0); (18)\n@t\u001e0=\u00001\n(1+\u000b2)t0(\u000b\u001c(\u001e0)jx\u0000F(\u001e0)jx+\u000bsin 2\u001e0);(19)\nwhere\u000bis the Gilbert damping constant, 3c=K?\u0015S=2, and\nt0=\u0015=3c.\u001c(\u001e0) and F(\u001e0) denote the coupling to the current\nvia STT. They read\n\u001c(\u001e0)=\u0000a\n23cX\ni˜Pxa3\n2eS@xnz; (20)\nF(\u001e0)=a\n23cX\ni˜Pxa3\n2eS˜\fx@xnz: (21)\nIn the above equations of motions, we have assumed that the\nelectron spin density induced by ˙ ndoes not change the dy-\nnamics qualitatively when the spin size Sis large. Also, we\ndid not consider pinning potentials for simplicity.\nIn the following, we consider triplet pairing given by\nd(k)=(\u0000sinkya;sinkxa;\u000esinkxa). Such a pairing can be sta-\nbilized by the spin-orbit coupling g(k)\u0001\u001b(g(k)=\u0000g(\u0000k))\nin a system without inversion symmetry; d(k)kg(k) is en-\nergetically favored[48]. The Rashba type spin-orbit coupling\ncan stabilized(k)=(\u0000sin(kya);sin(kxa);0), and in our case,\nit can originate from the boundary between the superconduc-\ntor and ferromagnet. Furthermore, we add dz(k)=\u000esin(kxa),\nwhich can be attributed to the additional spin-orbit coupling\ndue to the broken mirror symmetry about the xzplane.\nIn numerical calculations, we set \u0016=t=\u00001:8;\u00010=t=5\u0002\n10\u00002;JsdS=t=1, and T=t=5\u000210\u00003. Using the above d(k),\nwe first show ˜Pxand˜\fxin the domain wall configuration for\n\u001e0=\u0019=4 (Fig. 2 (a)-(c)). E \u000bective spin polarization ˜Pxis al-\nmost constant in space. On the other hand, ˜\fxhighly depends\non the spatial position. Importantly, ˜Px(˜\fx) is symmetric (an-\ntisymmetric) under x!\u0000 xfor\u000e=0, while ˜Pxand ˜\fxare\nslightly shifted for \u000e,0. Because of such symmetries, F(\u001e0)\n(Eq. (21)) vanishes for \u000e=0, where we note @xnzis an even\nfunction of x. On the other hand, for \u000e,0 such symmetries4\n×10-1\n×10-40.50.3(a)\n(b)\n(c)(d)\n(e)×10-4\n0.50.3\nFIG. 2: (Color online) (a) ˜Pxand (b) ˜\fxin a domain wall configu-\nration with \u001e0=\u0019=4 as functions of the position xfor di \u000berent\u000e,\nwhere dvector is d(k)=(\u0000sinkya;sinkxa;\u000esinkxa). (c) The profile\nof the domain wall. (d) \u001c(\u001e0) and (e) F(\u001e0) as functions of \u001e0for\ndi\u000berent\u000e, where we define j0=S e3ca\u00003.\n(a) (b)\n(c) (d)2\n5\n10\n152\n5\n10\n15\nFIG. 3: (Color online) Velocity of a domain wall center for \u000e=0\nin (a) and (c), and for \u000e=0:5 in (b) and (d). (a) shows the velocity\naveraged over the oscillation period after su \u000eciently long time, and\n(b) shows the velocity after su \u000eciently long time. In (c) and (d),\nthe time evolution of the velocity for di \u000berent jx=j0(indicated by\ndi\u000berent colors) is presented. We have set \u000b=10\u00004, and the initial\nconditions are \u001e0(t=0)=\u0019and ˙X(t=0).\nof˜Pxand ˜\fxare broken, and hence we obtain finite F(\u001e0).\nSimilar arguments apply to other \u001e0values. In the following,\nwe will show the resulting domain wall dynamics.\nWe next solve the equation of motion. For \u000e=0,\u001c(\u001e0)\nis well fitted by \u001c(\u001e0)'\u001c0+\u001c1cos 2\u001e0, and F(\u001e0)=0 as\nshown in Fig. 2 (d), (e). Note that such \u001e0dependence of \u001c(\u001e0)\narises from the triplet pairing. With this \u001c(\u001e0), we have solved\nEqs. (18) and (19). Fig. 3 (a) shows the averaged velocity\nafter su \u000eciently long time, and there is a threshold current\ndensity. It is given by jc=1p\n\u001c2\n0\u0000\u001c2\n1, which is obtained from\nEqs. (18) and (19) with \u001c(\u001e0)=\u001c0+\u001c1cos 2\u001e0andF(\u001e0)=\n0. As shown in Fig. 3 (c), ˙Xis zero after su \u000eciently longtime for jx<jcbecause of the intrinsic pinning due to the\nanisotropy K?. With a larger current density ( jx>jc), the\ndomain wall center oscillates with a finite drift velocity. The\nabove behavior is similar to a domain wall motion in normal\nferromagnetic metals without the non-adiabatic torque.\nWe now consider the case with \u000e,0. As shown in Fig. 2\n(d) and (e), \u001c(\u001e0) and F(\u001e0) are well fitted by\n\u001c(\u001e0)'\u001c0+\u001c1cos 2\u001e0; (22)\nF(\u001e0)'F0cos\u001e0: (23)\nImportantly, we have finite F(\u001e0) when\u000e,0. As shown\nin SM [55], finite dx(k)dz(k) and dy(k)dz(k) are necessary to\nhave finite F(\u001e0).\nThis F(\u001e0) changes the domain wall motion drastically. As\nshown in Fig. 3(b), there is no threshold current density for\ndriving the steady motion, which is similar to a situation in\nnormal metals with the non-adiabatic torque. An important\ndi\u000berence from conventional STT is that a domain wall shows\nno oscillatory motions ( ¨X=˙\u001e=0) even for a large current\ndensity depending on \u000b. According to Eqs. (19), oscillation\noccurs for j>jmax=max\u001e0f\u000bsin 2\u001e0=(\u000b\u001c(\u001e0)\u0000F(\u001e0))g.\nWith the numerically obtained parameters in Eqs. (22) and\n(23), jmaxis infinite when \u000bis small enough ( \u000b.10\u00003\u0018\nj\u00010j2=t2) because\u000b\u001c(\u001e0)\u0000F(\u001e0) can be zero. In contrast, for\nconventional STT, jmaxis always finite since \u001c(\u001e0) and F(\u001e0)\ndo not depend on \u001e0, and oscillatory motion always appears\nfor a current density larger than jmax. The absence of oscil-\nlatory motion is important for an e \u000ecient manipulation of a\ndomain wall. Let us estimate the required supercurrent den-\nsity. For example, in a ferromagnetic nanowire Ni 81Fe19, an\nexperimental value is S2K?\u0015a\u00003\u00180:05 J/m2[49], and hence\n3c'3\u0002102m/s and j0\u00184\u00021013A/m2. The required current\ndensity to achieve ˙X'0:4\u0016m=s isjx'105A/m2, which is\nlower than the critical current density in typical ferromagnetic\nJosephson junctions[50].\nTo summarize, we have microscopically derived STT in-\nduced by triplet supercurrents. We showed that spin-triplet\npairings give novel types of STT, which can be used for an ef-\nficient control of a domain wall. The results can be applied to\ndi\u000berent dvectors and magnetic textures such as a skyrmion,\nand we expect the possibilities for more interesting aspects of\ntriplet supercurrent-induced STT.\nThere are several comments and discussions. In nor-\nmal metals, a voltage drop occurs due to the domain wall\nmotion[51–54] in addition to the resistance of a sample. For\na superconducting system, while a supercurrent (dc current)\nis not accompanied by the voltage drop, the motion of a do-\nmain wall would also cause time evolution of the phase, and it\nmight result in a finite voltage drop. In this work, we have as-\nsumed such fluctuation is small compared to the overall phase\ngradient.\nIn this paper, we did not consider the Abrikosov vortices in\na superconductor, which might be induced by the stray field of\nthe ferromagnet. These vortices can cause voltage drop due to\ntheir dynamics and a non-uniform current pattern. To suppress\nthe vortices, we can use junctions with a ferromagnet with a\nsmall stray field, e.g., Sr 2RuO 4/permalloy junction with mag-\nnetization oriented in-plane. We can also use a singlet super-5\nconductor with a high critical field [38] such as niobium, and\nhence Nb /Ho/permalloy junction is another possible setup.\nWhen the superconducting pairing is proximity-induced, in\ngeneral, singlet pairing is expected to be mixed. However, the\ndecay length of the singlet proximity e \u000bect is much shorter\nthan that of the equal spin triplet pairing for a large exchange\ncoupling. Furthermore, the contribution from singlet pairing\nto STT is much smaller than that from triplet pairing in the\nadiabatic regime, which is justified in a smooth magnetic tex-ture.\nAcknowledgements. We would like to thank M. S. Anwar,\nH. Kohno, T. Nomoto, Y . Shiomi, and Y . Yanase for fruitful\ndiscussions. 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Lett. 98, 246601\n(2007).\n[54] R. A. Duine, Phys. Rev. B 79, 014407 (2009).\n[55] See Supplemental Material for the derivation of Eq. (9) and the\nexplicit form of F(\u001e0) (Eq. (21)).\n[56] For a high density, neshould be replaced with an e \u000bective elec-\ntron density modified by lattice e \u000bects.1\nSupplemental Material for\n“Adiabatic and non-adiabatic spin-torque induced by spin-triplet supercurrent”\nI. DERIVATION OF EQ. (9)\nWe start from the action:\nS=\u00001\n2X\nn;k;q\ty\nk+q(i\u000fn)(G\u00001\ntot)k+q;k(i\u000fn)\tk(i\u000fn); (S1)\nwhere\n\tk(i\u000fn)=0BBBBBBBBBBBBBB@ak+Q\"(i\u000fn)\nak+Q#(i\u000fn)\nay\n\u0000k+Q\"(\u0000i\u000fn)\nay\n\u0000k+Q#(\u0000i\u000fn)1CCCCCCCCCCCCCCA; (S2)\n(G\u00001\ntot)k+q;k(i\u000fn)\n=0BBBBB@\u0010\u0010\ni\u000fn\u0000\u0018k+Q\u0011\n1+JsdS\u001bz\u0011\n\u000eq;0\u00003\u0017\nk+q=2+QAa\n\u0017(q)\u001ba\u0000\u0001(k)\u000eq;0\n\u0000\u0001y(k)\u000eq;0\u0010\u0010\ni\u000fn+\u0018\u0000k+Q\u0011\n1\u0000JsdS\u001bz\u0011\n\u000eq;0+3\u0017\n\u0000k\u0000q=2+QAa\n\u0017(q)\u001baT1CCCCCA;(S3)\n'G\u00001\nk\u000eq;0+U(1)\nk+q;k+U(2)\nk+q;k: (S4)\nHere we define\nU(1)\nk+q;k=\u00003\u0017\nkQ\u0017\u000eq;0I4\u00003\u0017\nk+q=2Aa\n\u0017(q) \n\u001ba0\n0\u001bT\na!\n; (S5)\nU(2)\nk+q;k=\u0000@\u0018k+q=2\n@k\u0017@k\u0011Q\u0011Aa\n\u0017(q)Sa; (S6)\nandG\u00001\nkis defined in the main text. To restore the gauge invariance, we need to include the vector potential Aand redefine\n~Q=Q+e\ncA. Assuming the supercurrent, j/˜Q, is homogeneous, we can apply the perturbation with respect to ˜Qin the same\nway.\nThe spin density under a superconducting current is given by ˜ sa\nq=T\n2N1\n2P\nk;ntr[SaGtot;k+q;k(i\"n)]. We calculate it to the linear\norder of Q\u0011andAa\n\u0017(q), and obtain \u000e˜sa\nq=\u0019ab\n\u0017\u0011Ab\n\u0017(q)j\u0011\n2ene. Here\n\u0019ab\n\u0017\u0011=lim\nq!0\u0000T\n4Nta2X\nn;k@2\u0018k\n@k\u0017@k\u0011Tr[SaGk+q(i\u000fn)SbGk(i\u000fn)]+\u000eabLa\n\u0017\u0011; (S7)\nwhere\nLx\n\u0017\u0011=Ly\n\u0017\u0011=1\n2JsdS ta21\nNX\nk3\u0017\nk3\u0011\nk @nF(\")\n@\"\f\f\f\f\f\"=E\"\nk\u0000@nF(\")\n@\"\f\f\f\f\f\"=E#\nk!\n; (S8)\nLz\n\u0017\u0011=\u00001\n2ta21\nNX\nk3\u0017\nk3\u0011\nkX\n\u001b=\";#\u0018\u001b\nk\nE\u001b\nk@nF(\")\n@\"\f\f\f\f\f\"=E\u001b\nk\u0010\n2nF(E\u001b\nk)\u00001\u0011\n; (S9)\nwith E\"\nk=q\n(\u0018k\u0000JsdS)2+j\u0001\"\nkj2,E#\nk=q\n(\u0018k+JsdS)2+j\u0001#\nkj2, and nF(\")=(e\"=T+1)\u00001.La\n\u0017\u0011are the contributions from the\nFermi surface; they are proportional to the derivative of nF(\") and vanishingly small at low temperatures in systems with a full\ngap or point nodes. In the main text, we neglect La\n\u0017\u0017considering a low temperature compared to the superconducting critical\ntemperature. We note that for \u0001(k)=0, two terms in Eq. (S7) cancel each other and \u0019ab\n\u0017\u0011=0.\nAccording to Eq. (14) in the main text, finite1\n2(\u0019xx\n\u0017\u0000\u0019yy\n\u0017) or\u0019xy\n\u0017are necessary for ˜\f\u0017,0, and they depend on the relative2\nphase between \u0001\"\nkand\u0001#\nkas\n1\n2(\u0019xx\n\u0017\u0000\u0019yy\n\u0017)=lim\nq!0T\nNta2X\nk;n@2\u0018k+q=2\n@2k\u00170BBBBBB@Re(\u0001#\nk+q\u0001\"\u0003\nk)\n(\u000f2n+E#2\nk+q)(\u000f2n+E\"2\nk)1CCCCCCA; (S10)\n=lim\nq!01\nNX\nkRe(\u0001#\nk\u0001\"\u0003\nk)f\u0017(k;q); (S11)\n\u0019xy\n\u0017=lim\nq!0T\nNta2X\nk;n@2\u0018k+q=2\n@2k\u00170BBBBBB@Im(\u0001#\nk+q\u0001\"\u0003\nk)\n(\u000f2n+E#2\nk+q)(\u000f2n+E\"2\nk)1CCCCCCA: (S12)\n=lim\nq!01\nNX\nkIm(\u0001#\nk\u0001\"\u0003\nk)f\u0017(k;q); (S13)\nwhere\nf\u0017(k;q)=T\nta2X\nn@2\u0018k+q=2\n@2k\u00171\n(\u000f2n+E#2\nk+q)(\u000f2n+E\"2\nk): (S14)\nII. F(\u001e0)FOR A DOMAIN WALL\nLet us consider F(\u001e0) in a domain wall configuration when d\u0017(k)2R. We have\nF(\u001e0)=j\u00010j2a4\n2e3cSX\ni@xnzlim\nq!01\nNX\nkfx(k;q)\u0002dy(k)dz(k) sin\u001e0\u0000dx(k)dz(k) cos\u001e0\u0003sin\u0012\n+j\u00010j2a4\n2e3cSX\ni@xnzlim\nq!01\nNX\nkfx(k;q)\"\n\u00001\n2\u0010\ndx(k)2\u0000dy(k)2\u0011\nsin(2\u001e0)+dx(k)dy(k) cos(2\u001e0)#\ncos\u0012; (S15)\nwhere we have used Eq. (5) in the main text.\nIn the following, we show that F(\u001e0)=0 ford(k)=(\u0000sin(kya);sin(kxa);0). The first line in Eq. (S15) is zero since\ndz(k)=0. We note that in a domain wall configuration, \u0012(x)=\u0019\u0000\u0012(\u0000x) is satisfied. Since fx(k;q), which depends on n\nthroughj\u0001\"\nkj=j\u0001#\nkj=j\u00010j2jd(k)\u0002njinE\u001b\nk, and@xnzare invariant under the spatial reflection ( \u0012!\u0019\u0000\u0012), the second line in\nEq. (S15) vanishes after the spatial summationP\ni.\nForF(\u001e0),0, finite dx(k)dz(k) ordy(k)dz(k) is necessary. In this case, the first line in Eq. (S15) is nonzero, and jd(k)\u0002nj\nchanges its value under \u0012!\u0019\u0000\u0012so that the second term is also nonzero in general. As we show in the main text, d(k)=\n(\u0000sin(kya):sin(kxa);\u000esin(kxa)) is one way to satisfy this condition."    },    {        "title": "1003.3769v1.Dynamics_of_magnetization_on_the_topological_surface.pdf",        "content": "arXiv:1003.3769v1  [cond-mat.mes-hall]  19 Mar 2010Dynamics of magnetization on the topological surface\nTakehito Yokoyama1, Jiadong Zang2,3, and Naoto Nagaosa2,4\n1Department of Physics, Tokyo Institute of Technology, Toky o 152-8551, Japan\n2Department of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan\n3Department of Physics, Fudan University, Shanghai 200433, China\n4Cross Correlated Materials Research Group (CMRG), ASI, RIK EN, WAKO 351-0198, Japan\n(Dated: October 17, 2018)\nWe investigate theoretically the dynamics of magnetizatio n coupled to the surface Dirac fermions\nof athree dimensional topological insulator, byderiving t heLandau-Lifshitz-Gilbert (LLG) equation\nin the presence of charge current. Both the inverse spin-Gal vanic effect and the Gilbert damping\ncoefficient αare related to the two-dimensional diagonal conductivity σxxof the Dirac fermion,\nwhile the Berry phase of the ferromagnetic moment to the Hall conductivity σxy. The spin transfer\ntorque and the so-called β-terms are shown to be negligibly small. Anomalous behavior s in various\nphenomena including the ferromagnetic resonance are predi cted in terms of this LLG equation.\nPACS numbers: 73.43.Nq, 72.25.Dc, 85.75.-d\nTopologicalinsulator(TI) providesa new state of mat-\nter topologically distinct from the conventional band in-\nsulator[1]. In particular,the edge channelsorthe surface\nstates are described by Dirac fermions and protected by\nthe band gap in the bulk states, and backward scatter-\ning is forbidden by the time-reversal symmetry. From\nthe viewpoint of the spintronics, it offers a unique op-\nportunity to pursue novel functions since the relativistic\nspin-orbit interaction plays an essential role there. Actu-\nally, several proposals have been made such as the quan-\ntized magneto-electric effect [2], giant spin rotation [3],\nmagneto-transport phenomena [4], and superconducting\nproximity effect including Majorana fermions [5–7].\nAlso, a recent study focuses on the inverse spin-\nGalvanic effect in a TI/ferromagnet interface, predicting\nthe current-induced magnetization reversal due to the\nHall current on the TI [8]. In Ref. [8], the Fermi energy is\nassumed to be in the gap of the Dirac dispersion opened\nby the exchange coupling. In this case, the quantized\nHall liquid is realized, and there occurs no dissipation\ncoming from the surface Dirac fermions.\nHowever, in realistic systems, it is rather difficult to\ntune the Fermi energy in the gap since the proximity-\ninducedexchangefieldisexpectedtobearound5-50meV.\nTherefore, it is important to consider the generic case\nwhere the Fermi energy is at the finite density of states\nof Dirac fermions, where the diagonal conductivity is\nmuch larger than the transverse one, and the damping of\nthe magnetizationbecomes appreciable. Related systems\nare semiconductors and metals with Rashba spin-orbit\ninteraction, where the spin-Galvanic effect and current\ninduced magnetization reversal have been predicted [9]\nand experimentally observed [10, 11]. Compared with\nthese systems where the Rashba coupling constant is a\nkey parameter, the spin and momentum in TI is tightly\nrelated to each other corresponding to the strong cou-\npling limit of spin-orbit interaction, and hence the gigan-\ntic spin-Galvanic effect is expected.\n\u0001\u0000 \u0002 \u0003\u0004 \u0005 \u0006\n\u0007 \b \t\n\n\u000b \f \r \u000e\u000f\n\u0010\u0011 \u0012 \u0013 \u0014 \u0015\u0016 \u0017\u0018\u0019\u001a \u001b\u001c \u001d\n\u001e\u001f\nFIG. 1: (Color online) (a) Illustration of the Dirac dispers ion\non top of TI. The Fermi level εFis far above the surface\ngap opened by magnetization in the ferromagnetic layer. (b)\nSketch of FMR experiment in the soft magnetic layer. The\nsubstrate in the figure is TI, which is capped by a layer of\nsoft ferromagnet. The magnetization precesses around the\nexternal magnetic field Heff.\nIn this letter, we study the dynamics of the magnetiza-\ntion coupled to the surface Dirac fermion of TI. Landau-\nLifshitz-Gilbert (LLG) equationin the presenceofcharge\ncurrent is derived microscopically, and (i) inverse spin-\nGalvanic effect, (ii) Gilbert damping coefficient α, (iii)\ntheso-called β-terms, and(iv)thecorrectiontotheBerry\nphase, are derived in a unified fashion. It is found that\nthese are expressed by relatively small number of param-\neters, i.e., the velocity vF, Fermi wave number kF, ex-\nchange coupling M, and the transport lifetime τof the\nDirac fermions. It is also clarified that the terms re-\nlated to the spatial gradient are negligibly small when\nthe surface state is a good metal. With this LLG equa-\ntion, we propose a ferromagnetic resonance (FMR) ex-\nperiment, wheremodificationsoftheresonancefrequency\nand Gilbert damping are predicted. Combined with the\ntransport measurement of the Hall conductivity, FMR\nprovide several tests of our theory.2\nDerivation of LLG equation. — By attaching a ferro-\nmagnet on the TI as shown in Fig. 1, we can consider a\ntopological surface state where conducting electrons in-\nteract with localized spins, S, through the exchange field\nHex=−M/integraldisplay\ndrn(r)·ˆσ(r). (1)\nHere, we set S=Snwith a unit vector npointing in the\ndirection of spin, ˆσ(r) =c†(r)σc(r) represents (twice)\nthe electronspindensity, with c†= (c†\n↑,c†\n↓) beingelectron\ncreation operators, σthe Pauli spin-matrix vector, and\nMbeing the exchange coupling energy. The total Hamil-\ntonian of the system is given by Htot=HS+Hel+Hex,\nwhereHSandHelare those for localized spins and con-\nducting electrons, respectively.\nThe dynamics of magnetization can be described by\nthe LLG equation\n˙n=γ0Heff×n+α0˙n×n+t′\nel, (2)\nwhereγ0Heffandα0are an effective field and a Gilbert\ndamping constant, respectively, both coming from HS.\nEffects of conducting electrons are contained in the spin\ntorque\ntel(r)≡s0t′\nel(r) =Mn(r)×∝angbracketleftˆσ(r)∝angbracketrightne,(3)\nwhich arises from Hex. Here, s0≡S/a2is the local-\nized spin per area a2. In the following, we thus calculate\nspin polarization of conducting electrons perpendicular\nton,∝angbracketleftˆσ⊥(r)∝angbracketrightne, in such nonequilibrium states with cur-\nrent flow and spatially varying magnetization to derive\ntheβ-term, or with time-dependent magnetization for\nGilbert damping. Here and hereafter, ∝angbracketleft···∝angbracketrightnerepresents\nstatistical average in such nonequilibrium states.\nFollowing Refs. [12–14] we consider a small transverse\nfluctuation, u= (ux,uy,0),|u| ≪1, around a uniformly\nmagnetized state, n= ˆz, such that n= ˆz+u. In the\n‘unperturbed’ state, n= ˆz, the electrons are described\nby the Hamiltonian\nH0=/summationdisplay\nkvF(kyσx−kxσy)−Mσz−εF+Vimp(4)\nwhereVimpis the impurity potential given by Vimp=\nu/summationtext\niδ(r−Ri) in the first-quantization form. We take\na quenched average for the impurity positions Ri. The\nelectron damping rate is then given by γ= 1/(2τ) =\nπniu2νFin the first Born approximation. Here, niis the\nconcentration of impurities, and νF=εF/(2πv2\nF) is the\ndensity of states at εF. We assume that γ≪vFkF=/radicalbig\nε2\nF−M2,M, and calculate spin transfer torque in the\nlowest non-trivial order.\nIn the presence of u(r,t) =u(q,ω)ei(q·r−ωt), the con-\nducting electrons feel a perturbation (note that Hel+\nHex=H0+H1)\nH1=−M/summationdisplay\nkσc†\nk+qσck·u(q,ω)e−iωt,(5)and acquires a transverse component\n∝angbracketleftˆσ′α\n⊥(q,ω)∝angbracketrightne=Mχαβ\n⊥(q,ω+i0)uβ(q,ω) (6)\nin the first order in uin the momentum and frequency\nrepresentation. Here, χαβ\n⊥is the transverse spin suscep-\ntibility in a uniformly magnetized state with α,β=x,y,\nand summing over βis implied.\nNow, we study the ω-linear terms in the uniform ( q=\n0) part of the transverse spin susceptibility, χαβ\n⊥(q=\n0,ω+i0). We make the following transformation of the\noperator:\nc=U˜c=1/radicalbig\n2ε(ε+M)/parenleftbigg\nvF(ky+ikx)\nε+M/parenrightbigg\n˜c(7)\nwithε=/radicalbig\n(vFk)2+M2. Note U†U= 1,U†σxU=\nvFky/ε,andU†σyU=−vFkx/ε. This transformation\nmaps two component operator cinto one component op-\nerator on the upper Dirac cone ˜ c. With this new op-\nerator, we calculate the transverse spin susceptibility in\nMatsubara form\nχαβ\n⊥(0,iωλ) =/integraldisplayβ\n0dτeiωλτ/angbracketleftbig\nTτσα(0,τ)σβ(0,0)/angbracketrightbig\n=−T/summationdisplay\nk,nU†σαU˜G(k,iεn+iωλ)U†σβU˜G(k,iεn) (8)\nwith˜G(k,iεn) = (iεn−ε+εF+iγsgn(εn))−1. By sym-\nmetry consideration of the integrand in k-integral, we\nfindχαβ\n⊥(0,iωλ)∝δαβ. After some calculations, we ob-\ntain the torque stemming from the time evolution:\ntα\nel=M2iω\n2π1\n2v2\nF/parenleftbiggvFkF\nεF/parenrightbigg2\nεFτn×u (9)\n=1\n2/parenleftbiggMvFkF\nεF/parenrightbigg2\nνFτ˙ n×n.(10)\nThis result fits the conventional Gilbert damping with\nα=1\n2/parenleftbiggMvFkF\nεF/parenrightbigg2\nνFτa2\n¯hS. (11)\nWe next examine the case of finite current by applying\na d.c. electric field E, and calculate a linear response of\nσα\n⊥toE, i.e.,< σα\n⊥(q)>ne=Kα\ni(q)Ei. First, it is clear\nthatKα\ni(q=0) =−εiασxx/(evF) where εiαandσxx\nare the anti-symmetric tensor and diagonal conductivity,\nrespectively, because electron’s spin is ”attached” to its\nmomentum. This representsthe inversespin-Galvanicef-\nfect, i.e., chargecurrentinduces magneticmoment. Since\nwe assume that Fermi level is far away from the surface\ngap,σxx≫σxywhereσxyis the Hall conductivity. The\ndominant term in χis thusχxy∝σxx. This is quite\ndifferent from the case studied in Ref. [8], where Fermi\nlevel lies inside the surface gap and therefore σxxis van-\nishing. Hence, the only contribution to the inverse spin-\nGalvanic effect is χxx∝σxy, which is much smaller than3\nthe effect proposed in this letter. Compared with the in-\nverse spin-Galvanic effect in Rashba system [9–11], this\neffect is much stronger since the small Rashba coupling\nconstant, i.e., the small factor αRkF/EFin Eq. (16) of\nRef. [9], does not appear in the present case. Taking into\naccount the realistic numbers with α= 10−11eVmandvF= 3×105m/s, onefindsthat theinversespin-Galvanic\neffect in the present system is ∼50times largerthan that\nin Rashba systems.\nThe next leading order terms of the expansion in uβ\nandqjcan be obtained by considering the four-point ver-\ntices [12] as\n∝angbracketleftˆσα\n⊥(q)∝angbracketrightne=−eMπ\n45i\n8πε2\nFεikεjl[δαβδkl+δαkδβl+δαlδβk]qjuβEi (12)\n=−eM5i\n32ε2\nF[q·Euα−q·(u׈z)(E׈z)α+u·(E׈z)(q׈z)α]. (13)\nTherefore, the spin torque steming from the spatial gradient has the form:\ntβ\nel=−β1\n2e[n×(j·∇)n−(j−(j·n)ˆz)∇·(n׈z)+(∇−(n·∇)ˆz)n·(j׈z)] (14)\nwherej=σCEwith charge current jand conductivity\nσC=e2\n4π/parenleftBig\nvFkF\nεF/parenrightBig2\nεFτ. and\nβ=5π\n4εFτ/parenleftbiggM\nvFkF/parenrightbigg2\n. (15)\nFrom Eq.(14), one can find the followings: (i) The spin\ntransfer torque of the form ( j·∇)nis missing since\nwe consider the upper Dirac cone only. (ii) The β-\nterm has a form essentially different from that in the\nconventioal one.[12, 15, 16] In contrast to the conven-\ntionalferromagnet,[12] thisconstantcomesfromthe non-\nmagnetic impurity. Considering vFkF∼=εF, we get\nα/β∼(εFτ)2from Eqs. (11) and (15). Therefore,\ntheβ-terms are negligible for a good surface metal, i.e.,\nεFτ≫1.\nUp to now, we consider only one branch of the band\nwhere the Fermi energy is sitting. When we consider the\n2-band structure, i.e., the 2 ×2 matrix Hamiltonian H=\nvF[(ky+Mnx\nvF)σx−(kx−Mny\nvF)σy], we have the correctionto the Berry phase term. In analogy with the minimal\ncoupling of electromagnetic field, A=−M\nevF(−ny,nx)\nplays the same role as the U(1) gauge. By integrating\nthe fermions out, one can get a Chern-Simons term in\ntermsofthemagnetization LCS=σxyǫµνρAµ∂νAρwhere\nµ,ν,ρ=t,x,y. When the gradient of magnetization van-\nishes, it can be rewritten as\nLCS=σxy(M\nevF)2(nx˙ny−ny˙nx).(16)\nThis additional term can be interpreted as an additional\nBerryphase for the magnetization. In fact, as nzremains\nconstant in the present case, we have [ nx,ny] =inz.\nTherefore, nxandnybecome conjugate variables up\nto a factor, which naturally leads to a Berry phase:\nnx˙ny−ny˙nx. This term is exactly equivalent to the\nChern-Simons term.\nIncluding all the terms derived above, we finally arrive\nat a modified LLG equation:\n˙n−2σxy(M\nevF)2˙n/(s0N) =γ0Heff×n+/parenleftbiggM\nevFs0N/parenrightbigg\n(−j+(n·j)ˆz)+(α0+α/N)˙n×n+tβ\nel/(s0N) (17)\nwhereNisthenumberofferromagneticlayers. Notethat\nα-,β- andBerryphaseterms originatefromthe interplay\nbetween Dirac fermions and local magnetization which\npersists over a few layers of the ferromagnet. Therefore,\nthe overall coefficients are divided by the number of fer-\nromagnetic layers N.Ferromagnetic resonance. —Observingthe smallvalue\nofβ, the spatial gradient of magnetization can be ne-\nglected for the time being. Only one uniform domain in\nthe absence of current is taken into account for simplic-\nity. Without loss of generality, assume that an external\nmagnetic field is applied along zdirection, and consider4\nthe ferromagnet precession around that field. ˙ nz= 0\nis kept in the first order approximation, namely nzis a\nconstant in the time evolution. By inserting the ansatz\nnx(y)(t) =nx(y)e−iωtinto the modified LLG equation,\none obtains\nℜω=ξ\nξ2+η2ω0,ℑω=−η\nξ2+η2ω0(18)\nwhereη= (α0+α/N),ω0=γ0Heffandξ= 1−\n2σxy(M\nevF)2/(s0N). Expanding up to the first order in\nσxyandη, one gets ℜω=ω0+ 2σxy(M\nevF)2ω0/(s0N)\nandℑω=ηω0. Therefore, the precession frequency ac-\nquires a shift proportional to σxyin the presence of in-\nterplay between Dirac fermions and the ferromagnetic\nlayer. The relative shift of ℜωis 2σxy(M\nevF)2ω0/(s0N) =\n1\nπSNM\nεF(Ma\nvF)2∼1\nN(M\nεF)3[17]. By tuning the Fermi level,\nthis shift can be accessible experimentally.\nMeanwhile, the Gilbert damping constant αcan be\nmeasured directly without referring to the theoretical\nexpression in Eq. (11). One can investigate the fer-\nromagnetic layer thickness dependence of FMR line-\nwidth. While increasing the thickness Nof ferromagnet,\nthe Gilbert damping constant stemming from the Dirac\nfermions decreases inversely proportional to the thick-\nness. Taking into account the realistic estimation with\nεFτ∼100 and M/εF∼0.3, one has α/s0∼1, while\nα0∼0.001 usually. Therefore, even for a hundred of lay-\ners of ferromagnet, the contribution from the proximity\neffect is still significant compared to the one coming from\ntheferromagnetitself. Observingthattheimaginarypart\nof resonance frequency in Eq. (18) is proportional to η,\none may plot the relation between the FMR peak broad-\nening, namely ℑω, and 1/N. The broadening is a linear\nfunction of 1 /N, and approaches the value of the ferro-\nmagnet at large thickness limit. We can find the value of\nαfrom the slope of the plot.\nOn the other hand, the real part of FMR frequency\nprovides rich physics as well. Since in the presence of\nadditional Berry phase, the frequency shift is propor-\ntional to the Hall conductivity on the surface of TI, it\nleads to a new method to measure the Hall conductiv-\nity without four-terminal probe. In an ideal case when\nthe Fermi level lies inside the surface gap, this quantity\nis quantized as σ0\nxy=e2\n2h. However, in realistic case,\nFermi level is away from the surface gap, and therefore\nthe Hall conductivity is reduced to σxy=e2\n2hMnz\nεF[17]. As\na result, the shift of resonance frequency is proportional\nton2\nz∝cos2θ, and the FMR isotropy is broken. Here,\nθis the angle between effective magnetic field and the\nnormal to the surface of TI. One can perform an angle\nresolved FMR measurement. The signal proportional to\ncos2θcomes from additional Berry phase.\nSince parameters αandβdepend on Mandτ, it\nis quite important to measure these quantities directly.Molecular-beam epitaxy method can be applied to grow\nTI coated by a thin layer of soft ferromagnet. As is re-\nquired in the above calculation, Fermi level of TI should\nlie inside the bulk band gap. Also, the soft ferromag-\nnet should be an insulator or a metal with proper work\nfunction. One may employ angular resolved photoemis-\nsion spectroscopy(ARPES) or scanning tunneling micro-\nscope techniques to measure the surface gap ∆ opened\nby the ferromagnet, which is given by ∆ = Mnz. As the\neasy axis nzcan be found experimentally, Mcan be fixed\nas well. On the other hand, the lifetime τis indirectly\ndetermined by measuring the diagonal conductivity σxx\nviaσxx=e2\n4π/parenleftBig\nvFkF\nεF/parenrightBig2\nεFτ. Finally, Fermi surface can\nbe determined by ARPES, and all parameters in LLG\nequation Eq.(17) can be obtained.\nIn summary, we have investigated theoretically the\ndynamics of magnetization on the surface of a three\ndimensional topological insulator. We have derived\nthe Landau-Lifshitz-Gilbert equation in the presence of\ncharge current, and analyzed the inverse spin-Galvanic\neffect and ferromegnetic resonance predicting anomalous\nfeatures of these phenomena.\nThis work is supported by Grant-in-Aid for Scientific\nResearch (Grants No. 17071007, 17071005, 19048008\n19048015, and 21244053) from the Ministry of Educa-\ntion, Culture, Sports, Science and Technology of Japan.\n[1] M. Z. Hasan and C. L. Kane, arXiv:1002.3895; X. L.\nQi and S. C. Zhang, Physics Today, 63, 33 (2010) and\nreferences therein.\n[2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B\n78, 195424 (2008).\n[3] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev.\nLett.102, 166801 (2009).\n[4] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B\n81, 121401(R) (2010).\n[5] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407\n(2008).\n[6] Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev.\nLett.103, 107002 (2009).\n[7] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbø, and N.\nNagaosa, Phys. Rev. Lett. 104, 067001 (2010).\n[8] I. Garate and M. Franz, arXiv:0911.0106.\n[9] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[10] A. Chernyshov et al., Nature Phys. 5, 656 (2009).\n[11] I.M. Miron et al., Nature Materials 9, 230 (2010).\n[12] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[13] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W.\nBauer, Phys. Rev. B 74, 144405 (2006); Y. Tserkovnyak,\nA. Brataas, and G. E. Bauer, J. Magn. Magn. Mater.\n320, 1282 (2008).\n[14] Y. Tserkovnyak, G.A. Fiete and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[15] Clement H. Wong and Y. Tserkovnyak, Phys. Rev. B 80,5\n184411 (2009).\n[16] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).[17] J. Zang, and N. Nagaosa, arXiv:1001.1578"    },    {        "title": "1905.13042v1.Predicting_New_Iron_Garnet_Thin_Films_with_Perpendicular_Magnetic_Anisotropy.pdf",        "content": "1 \n Predicting New Iron Garnet Thin Film s with Perpendicular Magnetic Anisotropy  \nSaeedeh Mokarian Zanjani1, Mehmet C. Onbaşlı1,2,* \n \n1 Graduate School of Materials Science and Engineering, Koç University , Sarıyer, 34450 Istanbul, \nTurkey . \n2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul, \nTurkey.  \n* Corresponding Author: monbasli@ku.edu.tr  \n \nAbstract:  \nPerpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic \napplications like spin -orbit  torques  switching , logic and memory devices. An important class of \nmagnetic insulators with low Gilbert damping at room temperature are iron garnets, which only \nhave a few PMA types such as terbium  and samarium iron garnet. More and stable PMA garnet \noptions are necessary for researchers to be able to investigate new spintronic phenomena. In this \nstudy, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room \ntemperature. The effective anisotropy energies of 10 different garnet films that are lattice -matched \nto 5 different commercially available garnet substrates have been calculated using shape, \nmagnetoelastic and magnetocrystalline anisotropy  terms . Strain type, tensile o r compressive \ndepending on substrate choice, as well as the sign and the magnitude of the magnetostriction \nconstants of garnets determine if a garnet film may possess PMA. We show the conditions in which \nSamarium, Gadolinium, Terbium, Holmium, Dysprosium a nd Thulium garnets may possess PMA \non the investigated garnet substrate types . Guidelines for obtaining garnet films with low damping \nare presented. New PMA garnet film s with tunable saturation moment and field may improve spin -\norbit torque memory and compensated magnonic thin film devices.  \n \n \n \n \n 2 \n Introduction  \nWith the development of sputtering and pulsed laser deposition of high -quality iron garnet thin \nfilms with ultralow Gilbert damping, researchers have been able t o investigate a wide variety of \nmagnetization switching and spin wave phenomena1-3. The key enabler in many of these studies \nhas been Yttrium iron garnet (Y 3Fe5O12, YIG)4 which has a very low Gilbert damping allowing \nspin waves to propagate over multiple millimeters across chip. YIG thin films are useful for spin \nwave device applications, but since their easy axes lie along film plane, their utility cannot b e \nextended to different mechanisms such as spin -orbit torques, Rashba -Edelstein effect, logic \ndevices, forward volume magnetostatic spin waves1. At the same time, to have reliable and fast \nresponse using low current densities as in spin -orbit torque switching, magnetization orientation \nneeds to be perpendicular to the surface plane5. The possibility of having Dzyaloshinskii –Moriya \ninteraction (DMI) in TmIG/GGG may enable stabilizing skyrmions and help drive skyrmion \nmotion with pure spin currents6. \n \nThere is a number of studies on tuning anisotropy or obtaining perpendicular magnetic anisotropy \nin insulator thin films7-12. Among the materials studied, insulating magnetic garnet s whose \nmagnetic pr operties can be tuned have been a matter of interest over the past decades13-15 due to \ntheir low damping and high magnetooptical Faraday rotation. In order to obtain perpendicular \nmagnetic anisotropy in magnetic garnets, one needs to engineer the anisotropy terms that give rise \nto out -of-plane easy axis. Angular dependence of total m agnetization energy density is called \nmagnetic anisotropy energy and consists of contributions from shape anisotropy, strain -induced \n(magnetoelastic)  and magnetocrystalline anisotropy. A magnetic material preferentially relaxes its \nmagnetization vector tow ards its easy axis, which is the least energy axis , when there is no external \nfield bias . Such energy minimization process drives magnetic switching rates  as well as the stability \nof total magnetization v ector . Controlling magnetic anisotropy in thin film garnets not only offers \nresearchers different testbeds for experimenting new PMA -based switching phenomena, but also \nallows  the investigation of anisotropy -driven ultrafast dynamic magnetic response in thin film s and \nnanostructures.  \nThe most extensively studied garnet thin film is Yttrium Iron Garnet ( YIG). YIG films display in -\nplane easy axis because of their large shape anisotropy  and negligible magnetocrystalline \nanisotropy3. Although PMA  of ultrathin epitaxial YIG films has been reported16,17, the tolerance 3 \n for fabrication condition variations for PMA YIG is very limited and strain effects were found to \nchange magnetocrystalline anisotropy in YIG. Strain -controlled anisotr opy has been observed in \npolycrystalline ultrathin YIG films17,18. In case of YIG thin film grown on Gadolinium Gallium \nGarnet (GGG), only partial anisotropy control has been possible through significant change in \noxygen stoichiometry19, which increases damping. Since the fabrication of high -quality and highly \nPMA YIG films  is not easy for practical thicknesses on gadolinium gallium garnet substrates \n(GGG), researchers have explored tuning magnetic anisotropy by substituting Yttrium sites with \nother rare earth elements20,21. New garnet thin films that can exhibit PMA with different \ncoercivities, saturation fields, compensation points and tunable Gilbert damping values must be \ndeveloped  in order to evaluate the effect of these p arameters on optimized spintronic insulator \ndevices . \nSince the dominant anisotropy energy term is s hape anisotropy  in thin film YIG , some studies focus \non reducing the shape anisotropy contribution by micro and nanopatterning22-24. Continuous YIG \nfilms were etched  to form rectangular nanostrips with nanometer -scale thickness es, as \nschematic ally shown on Fig. 1(a) . Thus, least magnetic saturation field is needed along the longest \ndimension of YIG nanostrips . By growing ultrathin YIG, magnetoelastic strain contributions lead \nto a negative anisotropy field and thus PMA in YIG film s24. As the length -to-thickness ratio \ndecreases , the effect of shape anisotropy is reduced and in-plane easy axis is converted to PMA17.  \nWhile reducing the effect of shape anisotropy is necessary, one also needs to use m agnetoelastic \nanisotropy  contribut ion to reorient magnetic easy axis towards out of film plane , as schematically \nshown on Fig. 1 (b). Strain-induced perpendicular magnetic anisotropy in rare earth (RE) iron \ngarnets, especially in YIG, has been demonstrated to overcome shape anisotropy16,17,25,26. If \nmagnetoelastic anisotropy term induced by crystal lattice mismatch is large r than shape anisotropy \nand has opposite sign, then magnetoelastic anisotropy overcome s shape . Thus, the easy axis of the \nfilm becomes perpendicular to the film plane and the hyster esis loop becomes square -shaped with \nlow saturation field8. One can also achieve PMA in other RE magnetic iron garnets due to their \nlattice parameter  mismatch with their substrates . PMA has previously been achieved using \nSubstituted Gadolinium Gallium Garnet (SGGG) as substrate and a Samarium Gallium Garnet \n(SmGG) ultrathin film as buffer layer under (and on) YIG16. In case of thicker YIG films (40nm), \nthe magnetic easy axis becomes in -plane again. An important case shown by Kubota et.al19 \nindicates that increasing in-plane strain (ε ||) or anisotropy field (H a) helps achieve perpendicular 4 \n magnetic anisotropy. In ref.8,19, they reported that if magnetostriction coefficient (λ 111) is negative \nand large eno ugh to overcome shape anisotropy, and tensile strain is introduced to the thin film \nsample (ε ||>0), the easy axis becomes perpendicular to the sample plane  as in the case of Thul ium \niron garnet (Tm 3Fe5O12, TmIG ). \nA different form of magnetoelastic anisotropy effect can be induced by using porosity in garnet \nthin films. Mesoporous  Holmium Iron Garnet (Ho 3Fe5O12, HoIG) thin film on Si (001)27 exhibit s \nPMA due to reduced shape anisotropy, increased magnetostrictive and growth -induced anisotropy \neffects. Such combined effects lead to PMA in HoIG. In this porous thin film, the PMA was found \nto be independent of the substrate used, because the mechanical s tress does not result from a lattice \nor thermal expansion mismatch between the substrate and the film. Instead, the pore -solid \narchitecture itself imposes an intrinsic strain on the solution processed garnet film.  This example \nindicates that the film struc ture can be engineered in addition to the substrate choices in order to \novercome shape anisotropy in thin film iron garnets.  \nAnother key method for controlling anisotropy is strain doping through substitutional elements and \nusing their growth -induced aniso tropy effects , as schematically shown on Fig. 1(c) . Bi-doped \nyttrium iron garnet (Bi:YIG and Bi:GdIG) has been reported to possess perpendicular magnetic \nanisotropy due to the chemical composition change as the result of increased annealing \ntemperature21,28. Another reason for PMA in these thin films is strain from GGG substrate29. \nDoping of oxides by Helium implantation was shown to reversibly and locally tune magnetic \nanisotropy30. For TbxY3-xFe5O12 (x=2.5, 2.0, 1.0, 0.37 ) samples  grown by spontaneous nucleation \ntechnique31, magnetic easy axis was found to change from [111] to [100] direction as Tb \nconcentration was decreased. The first -order anisotropy constant K 1 undergoes a change of sign \nnear 190K . Another temperature -dependent lattice distortion effect that caused anisotropy chang e \nwas also reported for YIG films32. These results indicate  that temperature also plays an important \nrole in both magnetic compensation, lattice distortion and change in anisotropy.   \nIn this study, we systematically calculate the anisotropy  energ ies of  10 different types of lattice -\nmatched iron garnet compounds epitaxially -grown as thin films (X 3Fe5O12, X = Y, Tm, Dy, Ho, \nEr, Yb, Tb, Gd, Sm, Eu) on commercial ly available (111) -oriented garnet substrates (Gd 3Ga5O12-\nGGG , Y 3Al5O12-YAG , Gd 3Sc2Ga3O12 -SGGG, Tb 3Ga5O12 –TGG, Nd 3Ga5O12 -NGG).  Out of the \n50 different film/substrate pairs, we found that 20 cases are candidates for room temperature PMA. \nOut of these 20 cases, 7 film/substrate pairs were experimentally tested and shown to exhibit 5 \n characteristics originating from PMA. The remaining 13 pairs, to the best of our knowledge, have \nnot been tested for PMA experimentally.  We indicate through systematic anisotropy calculations \nthat large strain -induced magnetic anisotropy terms may overcome shape when the films are highly \nstrained . We use only the room temperature values of λ 11133 and only report predictions for room \ntemperature (300K) . Throughout the rest of this study, the films are labelled as XIG (X = Y, Tm, \nDy, Ho , Er, Yb, Tb, Gd, Sm, Eu), i.e. TbIG (Terbium iron garnet) or SmIG (Samarium iron garnet) \netc. to distinguish them based on the rare earth element.  Our model could accurately predict the \nmagnetic easy axis in almost all experimentally tested garnet film/substrate cases provided that the \nactual film properties are entered in the model and that the experimental film properties satisfy \ncubic lattice mat ching condition to the substrate.  6 \n  \nFigure 1.  Methods to achieve perpendicular magnetic anisotropy in iron garnet thin films. \n(a) Micro/nano -patterning reduces shape anisotropy and magnetoelastic anisotropies overcome \nshape.  (b) Large strain -induced anisotropy must over come  shape anisotropy to yield out -of-plane \neasy axis.  (c) Substitutional doping in garnets overcome shape anisotropy by enhancing \nmagnetocrystalline, growth -induced or magnetoelastic terms.   \n \nAnisotropy energy den sity calculation s \nTotal anisotropy energy density contains three main contributions; according the Equation 1, shape \nanisotropy ( Kshape ), first order cubic magnetocrystalline anisotropy ( K1), and strain -induced  \n7 \n (magnetoelastic)  anisotropy ( Kindu) parameters determine the total effective anisotropy energy \ndensity16. \nKeff=Kindu +Kshape +K1   (1) \nIn case of garnet film magnetized along [111] direction  (i.e. on a 111 substrate) , the magneto -elastic \nanisotropy energy density, resulting from magneto -elastic coupling is calculated by Equation 2: \nKindu =−3\n2λ111σ||      (2) \nwhere λ111 is magnetostriction constant along [111] direction  and it is usually  negative  at room \ntemperature34. In Eqn. 2,  σ|| is the in -plane  stress induced in the material  as a result of lattice \nmismatch between film and the substrate , and the in -plane stress is calculated from Equation 335: \nσ||=Y\n1−νε||      (3) \nwhere Y is elastic mo dulus, and ν is P oisson’s ratio36. \nFor calculation of in -plane strain, lattice parameter values obtaine d from the XRD characterization \nof the thin films are used . Equation 4  shows the strain relation as the lattice constant difference \nbetween the bulk form of the film and that of the substrate divided by the lattice constant  for the \nbulk form of the film16. \nε||=afilm −abulk\nafilm       (4) \nAssuming the lattice parameter of the thin film matches with that of the substrate, the lattice \nconstant of substrate can be used as the lattice constant of thin film for  calculation of strain in \nEquation 537: \nε||=asub−afilm\nafilm       (5) \nThe lattice constants used for the films and substrates examined for this study are presented on \nTable 1.  \nShape anisotropy energy density depends  on the geometry  and the intrinsic saturation magnetic \nmoment of the iron garnet material. Shape anisotropy has a demagnetizing effect on the total 8 \n anisotropy energy density. These significant anisotropy effects can be observed in magnetic \nhysteresis loop s and FMR measurements24.  \nThe most common anisotropies in magnetic materials are shape anisotropy and magneto -crystalline \nanisotropy38,39. Considering that the film is continuous, the shape anisotropy is calculated as16 \nKshape =2πMs2      (6) \nBy obtaining the values of M s for rare earth iron garnets as a function of temperature40,41, the value \nfor shape anisotropy energy density have been calculated using Equation 6.   \nIntrinsic magnetic anisotropy42, so called magnetocrystalline anisotropy, has the weakest \ncontribution to anisotropy energy densit y compared  to shape, and strain -induced \nanisotropies9,11,16,19. The values for the first order magnetocrystalline anisotropy is calculated and \nreported previously for rare earth iron garnets at different temperatures43. A key consideration in \nmagnetic thin films is saturation field. In anisotropic magnetic thin films, the anisotropy fiel ds have \nalso been calculated using equation 7 as a measure of how much field the films need for magnetic \nsaturation along the easy axis:  \nHA=2Keff\nMs       (7) \nTable 1. List of magnetic iron garnet thin films and garnet substrates available off -the-shelf used \nfor this study. The fourth column shows the lattice constants used for calculating the magnetoelastic \nanisotropy values of epitaxial garnets on the given substrates.  \nGarnet \nmaterial  Chemical \nformula  Purpose  Bulk \nlattice \nconstant  \n(Å) \nGGG  Gd3Ga5O12 Substrate  12.383  \nYAG  Y3Al5O12 Substrate  12.005  \nSGGG  Gd3Sc2Ga3O12 Substrate  12.480  \nTGG  Tb3Ga5O12 Substrate  12.355  \nNGG  Nd3Ga5O12 Substrate  12.520  \nYIG Y3Fe5O12 Film  12.376  \nTmIG  Tm 3Fe5O12 Film  12.324  \nDyIG  Dy3Fe5O12 Film  12.440  \nHoIG  Ho3Fe5O12 Film  12.400  \nErIG  Er3Fe5O12 Film  12.350  \nYbIG  Yb3Fe5O12 Film  12.300  9 \n TbIG  Tb3Fe5O12 Film  12.460  \nGdIG  Gd3Fe5O12 Film  12.480  \nSmIG  Sm 3Fe5O12 Film  12.530  \nEuIG  Eu3Fe5O12 Film  12.500  \n \nResults and Discussion  \nTable s 1 and 2 list in detail the parameters used and the calculated anisotropy energy density terms \nfor magnetic rare earth iron garnets  at 300K . These tables show only the cases predicted to be PMA  \nout of a total of 50 film/substrate pairs investigated . The extended version of Tables  1 and 2 for all \ncalculated anisotropy energy density terms for all combinations of the 50 film/subst rate pairs are \nprovided in the s upplementary tables. The tabulated values for saturation magnetization40,41 and \nlattice parameters44 have been  used for the calculations . In this study , we assumed the value of \nYoung’s modulus and Poisson ratio as 2.00×1012 dyne ·cm-2 and 0.29 for all garnet types , \nrespectively, based on ref.36. We also assume that the saturation magnetization, used for calculation \nof shape anisotropy, does not change with the film thickness. The saturation magnetization (Ms) \nvalues and  shape anisotropy  for iron garnet  films  are presented in the third  and fourth  columns , \nrespectively. The stress values for  fully lattice -matched film s σ calculated using equation 3 and \nmagnetostriction constants  of the films , λ111, are presented  on column s 6 and 7. Magnetoelastic \nanisotropy K indu, magnetocrystalline  anisotropy energy density K 1, and the total magnetic \nanisotropy energy density K eff are calculated  and listed on columns 8 , 9 and 10, respectively . H A \non column 11  is the anisotropy field ( the fields required to saturate the film s). \n \nTable 2 . Anisotropy energy density parameters calculation results. Rare earth iron garnets on GGG \n(as=12.3 83Å), YAG ( Y3Al5O12, as=12.005Å) , SGGG (a s=12.48Å) and TGG (Tb 3Ga5O12, \nas=12.355Å), and NGG ( Nd3Ga5O12, as=12.509Å)  substrates, with K eff < 0, are presented.   \nFilm  Substr ate Ms \n(emu·cm-\n3) Kshape \n(erg·cm-\n3) (× 103) ε \n(× \n10-3) σ \n(dyn·cm-\n2) \n(×1010) λ111 \n(×10-\n6) Kindu \n(erg·cm-\n3) (×104) K1 \n(300K) \n(erg·cm-\n3) \n(× 103) Keff \n(erg·cm-\n3) \n(× 103) HA \n(Oe) \n(× \n103) \nDyIG  GGG  31.85  6.37 -4.58 -1.29 -5.9 -11.4 -5.00 -113 -7.09 \nHoIG  GGG  55.73  19.5 -1.37 -0.386  -4 -2.3 -5.00 -8.66 -0.311  \nGdIG  GGG  7.962  0.398  -7.77 -2.19 -3.1 -10.2 -4.10 -106 -26.5 \nSmIG  GGG  140 123 -11.7 -3.30 -8.6 -42.6 -17.4 -321 -4.58 \nYIG YAG  141.7  126 -30.0 -8.44 -2.4 -30.4 -6.10 -184 -2.60 10 \n TmIG  YAG  110.9  77.2 -25.9 -7.29 -5.2 -56.9 -5.80 -497 -8.97 \nDyIG  YAG  31.85  6.37 -35.0 -9.85 -5.9 -87.2 -5.00 -870 -54.7 \nHoIG  YAG  55.73  19.5 -31.9 -8.97 -4 -53.8 -5.00 -524 -18.8 \nErIG  YAG  79.62  39.8 -27.9 -7.87 -4.9 -57.8 -6.00 -545 -13.7 \nYbIG  YAG  127.3  102 -24.0 -6.76 -4.5 -45.6 -6.10 -360 -5.66 \nGdIG  YAG  7.962  0.398  -38.1 -10.7 -3.1 -49.9 -4.10 -502 -126 \nSmIG  YAG  140 123 -41.9 -11.8 -8.6 -152.3  -17.4 -1420  -20.2 \nTbIG  SGGG  15.92  1.59 1.61 0.452  12 -8.14 -8.20 -88.0 -11.1 \nGdIG  SGGG  7.962  0.398  0.00 0.00 -3.1 0.00 -4.10 -3.70 -0.930  \nSmIG  SGGG  140 123 -3.99 -1.12 -8.6 -14.5 -17.4 -39.3 -0.562  \nDyIG  TGG  31.85  6.37 -6.83 -1.92 -5.9 -17.0 -5.00 -169 -10.6 \nHoIG  TGG  55.73  19.5 -3.63 -1.02 -4 -6.13 -5.00 -46.8 -1.68 \nGdIG  TGG  7.962  0.398  -10.0 -2.82 -3.1 -13.1 -4.10 -135 -33.9 \nSmIG  TGG  140 123 -14.0 -3.93 -8.6 -50.8 -17.4 -402 -5.74 \nTbIG  NGG  15.92  1.59 3.93 1.11 12 -19.9 -8.20 -206 -25.9 \n \nIn this study, we take the same sign convention as in ref. 16 and the film s exhibit PMA when Keff \n< 0. So for obtaining PMA, negative and large values for anisotropy energy density are desired. As \nall the garnets (except TbIG ) possess negative magnetostriction constant s at room temperature, the \nsign of the strain  (tensile or compressive)  determines whether the induced anisotropy is negative \nor positive . In the literature16,20,45,46 however , we observe that PMA was defined for either positive \nor ne gative effective anisotropy energy density (K eff). This inconsistency may cause confusion \namong researchers . Thermodynamically, a higher energy means an unstable state with respect to \nlower energy cases. Easy axis, by definition, is the axis along which the magnetic material can be \nsaturated with lowest external field or lowest total energy. A magnetic material would thus \nspontaneously minimize its energy and reorient it s magnetic moment along the easy axis. As a \nresult, we use here Keff < 0 for out-of-plane easy axis . Due to the thermodynamic arguments \nmentioned above, we suggest researchers to use K eff < 0 definition for PMA.  \n \nEffect of Substrate on Anisotropy Energy Density  \nChanging the substrate alters the strain in the film, which  also changes strain -induced anisotropy \nin the film. Figure 2 show s the calculated anisotropy energy density of rare earth iron garnet  thin \nfilms  grown on five commercially available differ ent substrates : Gadolinium Gallium Garnet \n(Gd3Ga5O12, GGG), Yttrium Aluminum Garnet ( Y3Al5O12, YAG), Substituted Gadolinium 11 \n Gallium Garnet ( Gd3Sc2Ga3O12, SGGG), Terbium Gallium Garnet (Tb 3Ga5O12, TGG ), and \nNeodymium Gallium Garnet (Nd 3Ga5O12, NGG) . As shown on Fig. 2(a), when gr own on GGG \nsubstrate ; Dysprosium Iron Garnet (DyIG), Holmium Iron Garnet (HoIG), Gadolinium Iron Garnet \n(GdIG), and Samarium Iron Garnet (SmIG) possess compressive strain  (afilm>asubstrate ). Considering \nthe large negative magnetostriction constant (λ 111) for each case, the strain -induced anisotropy \nenergy densit ies are estimated to cause negative total effective anisotropy energy density . As a \nresult, DyIG, HoIG, GdIG, and SmIG on GGG are predicted to be PMA cases.  \nBased o n the shape, magnetoelastic and magnetocrystalline anisotropy terms (room temperature \nK1), Thulium iron garnet (TmIG) on GGG  (111)  is estimated to be in -plane easy axis although \nunambiguous experimental evidence indicates that TmIG grows with PMA on GGG (111)  [1,19] . \nThe fact that only considering shape, magnetocrystalline and magnetoelastic anisotropy terms does \nnot verify this experimental result suggests that the PMA in TmIG/GGG (111) may originate from \na different anisotropy term such as surface anisotropy , growth -induced  or stoichiometry -driven \nanisotropy. Since the films used in the experiments are less than 10 nm or 5 -8 unit cells thick, \nsurface effects may become more significant and may require density functional theory -based \npredictions to account for surface anisotropy effects .  12 \n  \nFigure 2.  Calculated effective  anisotropy values for each rare earth iron garnet thin film when they \nare epitaxially grown on ( a) GGG, ( b) YAG, ( c) SGGG, ( d) TGG, (e) NGG  substrates . Note that \nthe scales of the axes are different in each part.  \n13 \n Yttrium Aluminum Garnet (YAG) is a substrate with smaller lattice parameter than all the rare \nearth iron garnet films  considered . With a substrate lattice parameter of as=12.005Å, YAG causes \nsignificant and varying amounts of strain on  YIG (af=12.376Å) , TmIG (af=12.324Å) , DyIG \n(af=12.440Å) , HoIG (af=12.400Å) , ErIG  (af=12.350Å) , YbIG (af=12.300Å) , TbIG (af=12.460Å) , \nGdIG (af=12.480Å) , SmIG (af=12.530Å)  and EuIG (a f=12.500Å) . Strain from YAG substrate \nyields negative strain -induced anisotropy energy density for these films . The strain -induced \nanisotropy term overcomes the shape anisotropy in these garnets when they are grown on YAG. \nConsequently , effective anisotropy energy densit ies become negative and these garnet  films are \nestimated to  possess perpendicular magnetic anisotropy . In the e xceptional case s of Terbium Iron \nGarnet (TbIG)  and Europium Iron Garnet (EuIG) , compressive strain is not enough to induce \nnegative strain anisotropy  because  the magnetostriction coefficient s of TbIG and EuIG are positive. \nSo the strain -induced anisotropy term s are also positive for both TbIG (Kindu(TbIG)  = 1.85×106 \nerg·cm-3) and EuIG (K indu(EuIG)  = 3.01×105 erg·cm-3) and do not yield PMA.  \nOther potential PMA garnets as a film on SGGG substrate are GdIG , TbIG , and SmIG. TbIG and \nGdIG cases are particularly interesting as growth conditions of these materials can be further \noptimized to achieve room temperature compensation and zero saturation magnetization. This \nproperty enables PMA garnet -based room temperature terahertz magnonics. The lattice parameters \nof GdIG (af=12.480Å) and SGGG (as=12.48Å) match exactly, so the in -plane strain value is zero \nand the effect of strain -induced anisotropy is eliminated completely. Consequently, due to small \nvalue for saturation magnetizat ion of GdIG, shape anisotropy (3.98 ×102 erg·cm-3) cannot compete \nwith magnetocrystalline anisotropy ( -4.1×103 erg·cm-3). In other words, in this case, the influence \nof magnetocrystalline anisotropy is not negligible compared to the other anisotropy terms.  \nConsequently , the anisotropy energy density is negative for GdIG when grown on SGGG due to \nthe influence of magnetocrystalline anisotropy energy density.  \nOne other candidate for a PMA rare earth iron garnet on SGGG substrate is SmIG. Since the film \nlattice parameter is greater than that of the substrate, compressive strain ( -3.99 ×10-3) is induced in \nthe film such that the resulting anisotropy energy density possesses a negative value of an order of \nmagnitude ( -1.45×105 erg·cm-3) comparable to the sh ape anisotropy energy density (1.23×105 \nerg·cm-3). With its relatively large magnetocrystalline anisotropy energy density ( -1.74×104 \nerg·cm-3), SmIG has a perpendicular magnetic anisotropy due to negative value for effective 14 \n anisotropy energy density (-4.80×105 erg·cm-3). TbIG film on SGGG substrate is a PMA candidate \nwith positive strain  and this film was also recently experimentally demonstrated to have PMA47.  \nSince TbIG’s lattice constant is smaller than that of the substrate, the film becomes subject to tensile \nstrain ( +1.61  ×10-3). Since TbIG also has a positive λ 111 (in contrast to that of SmIG), the film’s \nmagnetoelastic anisotropy term becomes large  and negative  and overcomes the shape anisotropy. \nIn case of TbIG, magnetocrystalline anisotropy alone overcomes shape and renders the film PMA \non SGGG. With the additio nal magnetoelastic anisotropy contribution ( -8.14×104 erg·cm-3), \nsignificant stability of PMA can be achieved.  \nTerbium Gallium  Garnet (TGG) is a substrate with lattice parameter (as=12.355Å ) such that it can \ninduce tensile strain on TmIG, ErIG and YbIG  and it induces compressive strain on the rest of the \nrare earth iron garnets (YIG, DyIG, HoIG, TbIG, GdIG, SmIG, EuIG) . In none of the tensile -\nstrained cases, PMA can be achieved since the sign of the magnetoelastic anisotropy is positive \nand has the same  sign as the shape anisotropy.  Among the compressively  strained cases, YIG, TbIG \nand EuIG are found to have weak magnetoelastic anisotropy terms which cannot overcome shape. \nAs a result, YIG, TbIG and EuIG on TGG substrate are expected to have in -plane easy axis. DyIG, \nHoIG, GdIG and SmIG films on TGG achieve large and negative effective  total anisotropy energy \ndensities  due to their negative λ 111 values . In addition, since the materials have compressive strain, \nthe signs cancel and lead to large magnetoelastic anisotropy energy terms that can overcome shape \nin these materials. So these cases are similar to the conditions explained for GGG substrate,  on \nwhich  only DyIG, HoIG, GdIG, and SmIG films with compressive strain can gain a large negative \nstrain -induced anisotropy energy density which can overcome shape a nisotropy . \nNeodymium gallium garnet  (NGG)  is a substrate 45 used for growing garnet thin films by pulsed \nlaser deposition method. NGG has large lattice constant compared with the rest of the bulk rare \nearth garnets and yield compressive strain in all rare earth garnets investigated except for Samarium \niron garnet (SmIG). For all cases other than SmIG, the sign of the magnetoelastic a nisotropy term \nis determined by the respective λ 111 for each rare earth iron garnet. YIG, TmIG, DyIG, HoIG, ErIG, \nYbIG cases have positive magnetoelastic anisotropy terms, which  lead to easy axes along the ir film \nplane s. For SmIG and EuIG on NGG, magnetoel astic strain and anisotropy terms are not large \nenough to overcome large shape anistropy. For GdIG, the weak tensile strain on NGG substrates \nactually causes in -plane easy axes as magnetoelastic strain offsets the negative magnetocrystalline 15 \n anisotropy. The only rare earth iron garnet that can achieve PMA on NGG is Terbium Iron Garnet \n(TbIG) due to its large negative strain -induced anisotropy energy density ( -2.44×105 erg·cm-3). Its \nlarge and negative magnetoelastic anisotropy can offset shape (1.59×103 erg·cm-3) and first order \nmagnetocrystalline anisotropy term ( -8.20×103 erg·cm-3), leading to a large negative effective \nmagnetic anisotropy energy density ( -2.51×105 erg·cm-3). Consequently, we predict that  growing \nTbIG on NGG substrate may yield PMA . \n \nFigure 3 show s the substrates on which one may expect PMA rare earth iron garnet films (or \nnegative Keff) due to strain only. Figures  3(a)-(d) compare the calculated effective energy densities \nas a function of strain type and sign for YIG, TmIG, YbIG, TbIG. For comparing the calculation \nresults presented here with the experimentally reported values for the anisotropy energy density of \nTmIG, we added the K eff directly from the experimental data in20 to Fig. 3(b). As shown on Fig. \n3(b), the experimental TmIG thin film shows positive K eff as the result of tensile in -plane strain \nand large negative magnetostriction constant.  \nFigure 4(a) -(f) shows the calculated effective energy densities as a function of strain type and sign \nfor GdIG, SmIG, EuIG, HoIG, DyIG, ErIG, respectively. K eff may get a positive or a negative value \nin both compressive and tensile strain cases due to vary ing signs of λ 111 constants of rare earth iron \ngarnets. In almost all cases that yield PMA on the given substrates, PMA iron garnets form under \ncompressive lattice strain. The only exception s in which tensile strain can yield PMA in garnet thin \nfilms is Tb IG on SGGG  and TbIG on NGG . In both of those cases, a small tensile strain enhances \nPMA but the magnetocrystalline anisotropy could already overcome shape and yield PMA without \nlattice strain. Therefore, experimental studies should target compressive latti ce strain.  \n 16 \n  \nFigure 3.  Effect of substrate strain on the effective anisotropy energy densities of ( a) YIG, ( b) \nTmIG, the data inserted on the graph, with red square symbol , GGG (Exp.) , is the experimental \nvalue of effective anisotropy energy of TmIG  on GGG based on ref.20 (c) YbIG, ( d) TbIG. Note \nthat the axes scales are different in each part.  \n17 \n  \nFigure 4. (a) GdIG, ( b) SmIG, ( c) EuIG, ( d) HoIG, ( e) DyIG, ( f) ErIG films on GGG, YAG, \nSGGG, TGG, and NGG substrates. Note that the axes scales are different in each part.  \n \n18 \n Based on Fig. 2 -4, the calculations  in this paper  numerically match with the reported values in the \nexperimental demonstrations in literature  both in sign and order of magnitude . However, since \nthere are inconsistent sign conventions for predictin g the magnetic anisotropy state  of the iron \ngarnet samples in the literature so far, some of the previous studies draw dif ferent conclusions on \nthe anisotropy despite the similar K eff. \nIn case of TmIG, as shown in Fig. 3(b), our model p redicts that there is a tensile strain -induced \nanisotropy resulting from the difference in film and substrate lattice parameters  and the film’ s \nnegative magnetostriction constant. The experimental results of magnetic anisotropy in \nTmIG/GGG8,20,48 are consistent with our model predictions . Previous studies identify PMA , if the \nfilm Keff is positive. A shortcoming of this approach is that such a definition would also identify \nYIG/GGG as PMA although its in-plane easy axis behavior has been experimentally de monstrated \nin numerous studies3,32,49. Keff < 0 for PMA definition  would be thermodynamically more \nappropriate and would also accurately explain almost  all cases including YIG/GGG.  Further \nexplanation about TmIG exceptional case is included in the Supplementary Information.   \nSensitivity of Anisotropy to Variations in Saturation Magnetic Moment and Film Relaxation  \nThe films with predicted effective anisotropy energy and field  may come out  differently when \nfabricated due to unintentional variability  in fabrication process conditions , film stoichiometry \n(rare earth ion to iron ratio and oxygen deficiency) as well as process -induced non -equilibrium \nphases in the garnet films.  These changes may partially or completely relax the films or increase \nstrain further due to secondary crystallin e phases. Practical minor changes in strain may \ndramatically alter both the sign and the magnitude of magnetoelastic ani sotropy energy and may \ncause a film predicted as PMA to come out with in -plane easy axis.  On the other hand, o ff-\nstoichiometry may cause reduction in saturation magnetic moment.  Reduction in saturation \nmagnetic moment decreases shape anisotropy term  quadratically (Kshape = 2πM s2), which implies \nthat a 1 0% reduction in Ms leads to a 19% decrease in K shape and anisotropy field may increase  (HA \n= 2K eff/Ms). Therefore, sample fabrication issues and the consequent changes in anisotropy terms \nmay weaken or completely eliminate the PMA of a film/substrate pair  and alter anisotropy field . \nWhile these effects may arise unintentionally, one can also use these effects deliberately for \nengineering garnet films for devices.  Therefore , the sensitivity of anisotropy properties of garnet 19 \n thin films such as anisotropy field and effective anisotropy energy density need s to be evaluated \nwith respect to changes in film strain and saturation magnetic moment.  \nFigure 5 shows the sensitivity of  the effective magnetic anisotropy energy density to deviation of \nboth strain and saturation magnetization, M s for five PMA film/substrate  combinations:  (a) \nHoIG/GGG, (b) YIG/YAG, (c) SmIG/ SGGG, (d) HoIG/TGG , and (e) SmIG/NGG ). The negative \nsign of effective magnetic anisotropy energy indicates PMA . The change of anisotropy energy from \nnegative to positive indicates a transition from PMA to in -plane easy axis . In these plots, calculated \nanisotropy energies are presented for saturation moments and strains scanned from 60% to 140% \nof tabulated bulk garnet Ms and of the strain s of fully lattice -matched films on the substrate s. The \ncolor scale indicates the anisotropy energy density in erg·cm-3. Although magnetocrystalline \nanisotropy energies are negative for all of the thin film rare earth garnets considered here, these \nterms are negligible with respect to shape anisotropy (K 1(300 K) ~ -5% of K shape). Therefore, \nmagnetoelastic anisotropy term must be large enough to overcome shape anisotropy. A derivation \nof anisotropy energy  as a function of Ms and strain in equations ( 8)-(10) shows that the negative \nλ111 values and negative strain states (compressive strain) for garnet films in Fig. 5(a)-(e) (HoIG, \nYIG, SmIG) enable these films to have PMA . To retain PMA state; λ 111 must be negative and large  \nassuming elastic moduli and the Poisson’s ratio are constant . The necessary condition for \nmaintai ning PMA is shown in equation 11 . \nKeff=Kindu +Kshape +K1    (8) \nKeff=−3\n2λ111Y\n1−vε||+2πMs2+K1   (9) \nKeff=3\n2λ111Y\n1−v|ε|||+2πMs2+K1   (10) \nKeff<0    if   |3\n2λ111Y\n1−v|ε|||+K1|>2πMs2  (11) \nRelaxing each fully strained and lattice -matched thin film towards unstrained state (ε → 0  or \nmoving from left  to right on each plot in Fig. 5  causes  the magnetoelastic anisotropy energy term \nto decrease in magnitude and gradually vanish . The total anisotropy energy decreases  in intensity \nfor decreasing strain and constant M s. When M s increases, shape anisotropy term also increases \nand overcomes magnetoelastic anisotropy term. As a result, higher M s for relaxed films (i.e. \nrelatively thick and iron -rich garnets) may lose PMA. Therefore, one needs to optimize the film 20 \n stoichiometry and deposition process conditions, especially growth temp erature, oxygen partial \npressure  and film thickness, to ensure that the films are strained and stoichiometric. Since strains \nare less than 1% in Fig. 5(a), 5(c)-(e), these samples are predicted to be exp erimentally more \nreproducible. For YIG on YAG, as shown in Fig. 5 (b), the strains are around 3%, which may be \nchallenging to reproduce.  The cases presented in Fig. 5 (a)-(e) are the only cases among 50 \nfilm/substrate pairs where reasonable changes in M s and strain may lead to complete loss of PMA. \nThe rest of the cases have not been found as sensitive to strain and M s variability and those \npredicted to be PMA are estimated to have stable anisotropy.  Effective a nisotropy energy plots \nsimilar to Fig. 5(a) -(e) are presented in the supplementary figures for all 50 film/substrate pairs.   \nWhile PMA is a useful metric for garnet films, the effective anisotropy energy of the films should \nalso not be too high ( < a few 105 erg·cm-3) otherwise the saturation fields for these films would \nreach or exceed 0.5 Tesla (5000 Oe). Supplementary figures present the calculated  effective \nanisotropy energy and anisotropy field values for all 50 film/substrate pairs for changing strain and \nMs values. These figures indicate that one can span anisotropy fields of about 300  Oersteds up to \n12.6 Tesla in PMA garnets. For practical integrated magnonic  devices, the effective anisotropy \nenergy should be large enough to have robust PMA although it shoul d not be too high such that \neffective anisotropy fields (i.e. saturation fields) would still be small and feasible.  Engineered strain \nand M s through controlled oxygen stoichiometry may help keep anisotropy field low while \nretaining PMA.  In addition, according to the recently published paper  on magnetic anisotropy of \nHoIG50, the lattice matching in case of the thick samples becomes  challenging to sustain, and the \nstrain relaxes inside the film. Thus, the decrease in the anisotropy field is one consequences of the \nlower strain state, which is an advantage for magnonics or spin -orbit torque devices. Below a \ncritical thickness, HoIG gr own on GGG has PMA. However, as the film reaches this critical \nthickness, the 40% or more strain relaxation is expected and the easy axis becomes in -plane. So \nthinner films are preferred to be grown in integrated device applications.  21 \n  \nFigure 5. Effect of partial film relaxation and saturation magnetic moment variability on the \neffective anisotropy  energy density of the films . Variation of effective magnetic anisotropy energy \ndensities for (a) HoIG on GGG, ( b) YIG on YAG, ( c) SmIG on SGGG, ( d) HoIG on TGG and ( e) \nSmIG on NGG are presented when strain relaxation and magnetic saturation moments change \nindependently. Film strain may vary from a completely lattice -matched state to the substrate to a \nrelaxed state or a highly strained state due to microparticle nucleation . Strain variability alter s \nmagnetoelastic  anisotropy and cause a PMA film become in -plane easy axis.  On the other hand, \n22 \n magnetic saturation moments may deviate from the tabulated values because of process -induced \noff-stoich iometry in the films (i.e. rare earth ion to iron ratio or iron deficiency or excess , oxygen \ndeficiency) . Relaxing the films reduces the magnetoelastic anisotropy term and diminishes PMA. \nIncreasing M s strengthens shape anisotropy and eliminates PMA for lo w enough strains for all five \ncases presented.  \nMinimizing Gilbert damping coefficient in garnet thin films is also an important goal for spintronic \ndevice applications. First principles predictions of physical origins of Gilbert damping  51 indicate \nthat magnetic materials with lower M s tend to have lower damping. Based on this prediction, DyIG, \nHoIG and GdIG films are predicted to have lower Gilbert damping with respect to the others. Since \nthe compen sation temperatures of these films could be engineered near room temperature, one may \noptimize their damping for wide bandwidths all the way up to terahertz (THz)52 spin waves  or \nmagnons . The first principles predictions  also indicate that higher magnetic susceptibility (χm) in \nthe films helps reduce damping (i.e. lower saturation field). Therefore, the PMA garnet films with \nlower anisotropy fields are estimated to have lower Gilbert dampi ng parameters with respect to \nPMA garnets with higher anisotropy fields . \n \nConclusion  \nShape, magnetoelastic and magnetocrystalline m agnetic anisotropy energy terms have been \ncalculated for ten different garnet thin films  epitaxially grown on five different garnet substrates. \nNegative K eff (effective magnetic anisotropy energy) corresponds to perpendicular magnetic \nanisotropy  in the convention used here .  By choosing a substrate with a lattice parameter smaller \nthan that of the film, one can  induce compressive strain in the films  to the extent that one can \nalways overcome shape anisotropy  and achieve PMA  for large and negative λ 111. Among the PMA \nfilms predicted, SmIG possesses  a high anisotropy energy density and this film is estimated to be \na robus t PMA when grown on all five different substrates.  \nIn order to obtain PMA, magnetoelastic anisotropy term must be large enough to overcome shape \nanisotropy. Magnetoelastic anisotropy overcomes shape anisotropy when the strain type \n(compressive or tensile) and magnetoelastic anisotropy constants λ 111 of the garnet film  have the \ncorrect signs (not necessarily opposite or same) and the magnetoelastic anisotropy term has a \nmagnitude larger than shape anisotropy. Both compressive and tensile -strained films can , in \nprinciple, become  PMA as long as shape anisotropy can be overcome with large magnetoelastic 23 \n strain effects.  Here, in almost all cases that yield PMA on the given substrates, PMA iron garnets \nform under compressive lattice strain, except TbIG on SGGG and  TbIG on NGG . These two cases \nhave tensile strain and relatively large magnetocrystalline anisotropy, which could already \novercome shape anisotropy without strain. Experiments are therefore suggested to target mainly \ncompressive lattice strain.  \n20 different garnet film/substrate pairs have been predicted to exhibit PMA and their properties are \nlisted on Table 2 . For 7 of these 20 potential PMA cases, we could find unambiguous experimental \ndemonstration of PMA. Among the  20 PMA cases, HoIG/GGG, YIG/Y AG, SmIG/SGGG, \nHoIG/ TGG and SmIG/ NGG  cases have been found to be sensitive to fabrication process or \nstoiochiometry -induced variations in M s and strain.  In order to control  effective anisotropy in rare \nearth iron garnets (RIGs), shape anisotropy could be tuned  by doping garnet film s with Ce 53, Tb \n31 and Bi 54 or by micro/nano -patterning . Saturation magnetization could also be increased \nsignificantly by doping, which results in increasing the shape anisotropy in the ma gnetic thin films.  \nAmong the cases predicted to possess PMA , anisotropy fields ranging from 310 Oe (0.31 T) to \n12.6 T have b een calculated . Such a wide anisotropy field range could be spanned and engineered \nthrough strain state, stoichiometry as we ll as substrate choice. For integrated magnonic devices and \ncircuits, garnets with low M s and lower anisotropy field s (HA < 0.5 T)  would require less energy \nfor switching and would be more appropriate due to their lower estimated Gilbert damping.  \nMethods  \nCalculation of anisotropy energy density. We used  Keff = K indu + K shape + K 1 equation to calculate \nthe total anisotropy energy density for each thin film rare earth iron garnet/substrate pair. Each \nanisotropy term consist of the following parameters:  Keff=−3\n2λ111Y\n1−vε||+2πMs2+K1. The \nenergy density is calculated based on the parameters reported in previous  references16,34,36,40,41,43. \nFirst order magnetocrystalline anisotropy, K 1, is an intrinsic, temperature -dependent constant \nreported for each REIG material. Young’s modulus  (Y), poison ratio  (ν) and magnetostriction \nconstant (λ111) parameters evolving in the magnetoelastic anisotropy energy density term  (first \nterm)  are considered to be constant according to the values previous ly reported . For shape \nanisotropy energy calculations (second term), bulk s aturation magnetization (Ms) for each film was \nused. Since each film may exhibit variability in M s with respect to bulk, the model presented here \nyields the most accurate predictions when the actual film Ms, λ 111, Y, ν and K 1, and in -plane strain 24 \n are enter ed for each term. 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AIP Advances  9, 035024 (2019).  \n51 Hickey , M. C. & Moodera, J. S. Origin of intrinsic Gilbert damping. Physical Review \nLetters  102, 137601 (2009).  \n52 Geprägs, S.  et al.  Origin of the spin Seebeck effect in compensated ferrimagnets. Nature \nCommunications  7, 10452 (2016).  \n53 Baños -López, E., Cortés -Escobedo, C., Sánchez -De Jesús, F., Barba -Pingarrón, A. & \nBolarín -Miró, A. Crystal structure and magnetic properties of cerium -doped YIG: Effect of \ndoping concentration and annealing temperature. Journal of Alloys and Compounds  730, \n127-134 (2018).  \n54 Lou, G., Kato, T., Iwata, S. & Ishibashi, T. Magneto -optical properties and magnetic  \nanisotropy of Nd 0.5Bi2.5Fe5-yGayO12 thin films on glass substrates. Optical Materials \nExpress  7, 2248 -2259 (2017).  \n Acknowledgments  \nM.C.O. acknowledges BAGEP 2017 Award and TUBITAK Grant No. 117F416.  \nCompeting interests  \nThere is no financial and non -financial competing interest among the authors.  \nAuthor contributions  \nM.C.O. designed the study. S.M.Z. performed the calculations and evaluated and analyzed the \nresults with M.C.O. Both authors discussed the results and wrote the manuscript together.  \n \n "    },    {        "title": "1809.01042v1.Separation_of_the_two_magnon_scattering_contribution_to_damping_for_the_determination_of_the_spin_mixing_conductance.pdf",        "content": "arXiv:1809.01042v1  [cond-mat.mtrl-sci]  4 Sep 2018Separation of the two-magnon scattering contribution to da mping for the\ndetermination of the spin mixing conductance\nA. Conca,1,∗S. Keller,1M. R. Schweizer,1E. Th. Papaioannou,1and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n(Dated: September 5, 2018)\nWe present angle dependent measurements of the damping prop erties of epitaxial Fe layers with\nMgO, Al and Pt capping layers. Based on the preferential dist ribution of lattice defects following the\ncrystal symmetry, we make use of a model of the defect density to separate the contribution of two-\nmagnon scattering to the damping from the isotropic contrib ution originating in the spin pumping\neffect, the viscous Gilbert damping and the magnetic proximi ty effect. The separation of the two-\nmagnon contribution, which depends strongly on the defect d ensity, allows for the measurement of\na value of the effective spin mixing conductance which is clos er to the value exclusively due to spin\npumping. The influence of the defect density for bilayers sys tems due to the different capping layers\nand to the unavoidable spread in defect density from sample t o sample is thus removed. This shows\nthe potential of studying spin pumping phenomena in fully or dered systems in which this separation\nis possible, contrary to polycrystalline or amorphous meta llic thin films.\nINTRODUCTION\nIn bilayers systems formed by a ferromagnetic (FM)\nlayer in contact with a metallic non-magnetic (NM) one,\na pure spin current can be generated and injected in the\nlatterwhen the ferromagneticresonanceisexcited. Typi-\ncally, a microwavemagnetic field is used for this purpose.\nThe whole processis commonly referredto as spin pump-\ning [1, 2]. If the non-magnetic layer is formed by a heavy\nmetal with large spin-orbit coupling (Pt, Ta or similar),\nthe spin current can be detected by using the inversespin\nHall effect (ISHE) for conversion into a charge current.\nSince the spin current leaving the magnetic layer car-\nries away angular momentum from the magnetization\nprecession,it representsanadditionallosschannelforthe\nmagnetic system and consequently causes an increase in\nthe measured Gilbert damping parameter α[1]:\n∆αsp=γ/planckover2pi1\n4πMsdFMg↑↓(1)\nwhereg↑↓is the real part of the spin mixing conductance\nwhich is controlling the magnitude of the generated spin\ncurrent and γis the gyromagnetic ratio.\nThis expression is only valid for sufficiently thick NM\nlayers where no reflection of the spin current takes place\nat the film surface or interface with other materials, i.e.\nno spin current is flowing back into the magnetic layer.\nIn principle, it allows the estimation of g↑↓by measur-\ning the increase in damping compared to the intrinsic\nvalue. However, to perform this measurement is not\nstraightforward. If the estimation of g↑↓for a FM/Pt\nsystem is needed, ideally one should measure the effec-\ntive Gilbert damping parameter α0for a single stand-\ning magnetic layer acting as a reference sample with no\nlosses due to spin pumping and repeat the same after de-\npositing a thick Pt layer. However, most of the commonferromagnetic materials, with exception of the magnetic\ninsulators like YIG, will change its properties due to ox-\nidation processes. Therefore, a capping layer is required\nand one has to find an appropriate one, in the sense that\nits introduction must not modify the damping properties\nof the magnetic layer. Examples in the literature show\nthat this is far to be a trivial task [3–5]. In addition to\nthis, the emergence of a finite magnetic polarization in\nPt in contact with a ferromagnetic layers has an impact\non damping which further hinders the estimation of g↑↓\n[5–12].\nFor the reference layers, the most convenient candi-\ndates as capping material are oxides like MgO, for which\nit has been proven that they are able to block the flow\nof spin current and therefore to deactivate spin pumping\n[13–15], or metals with weak spin-orbit interaction like\nAl or Ru. But even for these cases, it has been shown\nthat an increase of damping not related to spin pumping\nis possible. Ruiz et al.show for instance that a MgO\ncapping layer increases strongly the damping in permal-\nloy while this is not the case for Al capping layer [5].\nThe reason has nothing to do with the metallic char-\nacter of the capping layer since the increase for Ru is\neven larger than with MgO. The same work [5] already\nprovides a hint for a possible reason since the increase\nof damping roughly scales with the value of the inter-\nface perpendicular anisotropy constant K⊥\nS. Theoretical\nworks [16] show that the counterplay between the de-\nmagnetizing field responsible for the in-plane orientation\nof the magnetization and the perpendicular anisotropy\nfield can induce inhomogeneous magnetization states for\ncertain field strengths combinations which are responsi-\nble for an increased damping. In this sense, this effect\nhas been also adduced to explain the damping thickness\ndependence in Co 2FeAl/MgO systems [3].\nHere we present angle dependent measurements of the\ndamping properties of epitaxial Fe layers with MgO, Al2\nFIG. 1. (Color online) Dependence of the FMR linewidth on the frequency for different orientations φHof the external magnetic\nfield with respect to the [100] crystallographic axis of Fe fo r (a) Fe/Al and (b) Fe/Pt systems. The lines correspond to a li near\nfit to extract the effective damping parameter αeff. Forφ= 30◦a strong non-linearity due to magnetic dragging is observed .\nFor visibility reasons, each data set is shifted vertically by 1.25 mT with respect the previous one.\nand Pt capping layers. Fully epitaxial systems consti-\ntute a perfect ordered model with almost ideal and well\ndefined interfaces. Here, we will show that the angle de-\npendence of damping allows for a measurement of the\nstrength of the two-magnon scattering and of its contri-\nbution to the effective damping parameter. With the\nseparation of this contribution we access the increase\nof damping caused only by spin pumping and magnetic\nproximity effect and to an estimation of g↑↓without the\ncontamination of defects effects.\nEXPERIMENTAL DETAILS\nThe samples were deposited by e-beam evaporation\non MgO(100) substrates in a molecular beam epitaxy\n(MBE) chamber with a base pressure P b= 5×\n10−10mbar. A set of Fe/Pt bilayers with fixed Fe thick-\nness (12 nm) and varying Pt thickness were prepared.\nAdditional reference samples, where Pt is substituted by\nMgO or Al, have also been prepared. The Fe and Pt\nfilms were grown with a deposition rate of 0.05 ˚A/s. The\nsamples were deposited with a substrate temperature of\n300◦C and subsequently annealed at the same tempera-\nture.\nThe characterization by X-ray diffractometry (XRD)\n(presented elsewhere [17]) shows that the Fe/Pt bilayers\nare fully epitaxial with the Fe unit cell rotated by 45◦\nwith respect to the MgO substrate unit cell and with\nPt rotated again 45◦with respect to Fe. In the case of\nFe/Al, epitaxial growth of the upper layer could not be\nachieved.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). Forthis, the samples were placed facing the strip-line and\nthe˜S12transmission parameter was recorded.\nRESULTS AND DISCUSSION\nFigures 1 shows the dependence of the measured FMR\nline width ∆ Hon the frequency for the reference layer\nwith Al capping (a) and a Fe/Pt system (b). The data\nis shown for different orientation of the external static\nmagnetic field varying from φH= 0◦([100], easy axis) to\nφH= 45◦([110], hard axis). For visibility reasons, each\ndata set is shifted vertically by 1.25 mT with respect to\nthe previous one.\nAs commented before, the choice of capping layer can\nhave a large influence on the linewidth and effective\ndampingofthe magneticlayer,evenforlightmetals. The\nmagnetic proximity effect (MPE) in the case of Pt also\ncontributes to an increase on damping, [5, 9–12] which\nadditionally challenges the measurement of the contri-\nbution from the spin pumping. Taking into account all\nthese considerations, the effective increase on damping\nwhen comparing a reference system and a system with a\nheavy metal can be separated as follows:\nαeff=α0+αmpe+αsp+αi. (2)\nHereα0is the intrinsic damping parameter which can\nbe defined as characteristic of the material under in-\nvestigation (growth conditions however may influence it\nstrongly) and it is the sum of the losses by two-magnon\nscattering and by energy transfer to the phonon system.\nαmpeis the contribution due to the dynamic coupling be-\ntween the ordered spins in Pt due to the MPE and the3\nFIG. 2. (Color online) (a) Dependence of the FMR resonance\nfieldHFMRon the in-plane direction of the static magnetic\nfield for two values of the resonant frequency. (b) Dependenc e\nof the in-plane angle of the magnetization vector φMon the\nexternal field direction φH. Both angles are measured relative\ntothe[100] axis. Thedottedline represents thecase of perf ect\ncollinearity between magnetization and external field.\nmagnetization in the magnetic layer. αspis the result\nof the losses by the spin current generated in the fer-\nromagnetic layer by the precession of the magnetization\nand that flows into the Pt layer (spin pumping). The last\ntermαisummarizes the increase of damping due to other\ninterfacial effects such as interface PMA as commented\nabove, spin memory loss [18] or isotropic scattering at\ninterface defects [19].\nSeveral efforts have been made in order to separate\nsome of the contributions to αeff. In a recent work with\nCoFeB/Pt [9] we were able to separate αmpedue to the\ndependence on the Pt thickness. As already reported by\nCaminale et al.[11], a linear Pt thickness dependence of\nthe spin-current absorption in spin-sink layers exhibiting\nMPE and of αmpeis expected [12]. A detailed vector\nnetwork analyzer FMR study has also been recently re-\nported to separate the different contributions in NiFe/Pt\nsystems [20].\nTheterm α0isaresultoftwocontributions[22]. Oneis\nthe pure Gilbert damping, which is of viscous nature and\ngenerates a dissipation of energy and angular momentum\ntothe lattice. The secondoneisthe transfertospin-wavemodes with k/negationslash= 0 from the FMR mode via two-magnon\nscattering. For a pure Gilbert-like viscous damping the\nlinewidth dependence on the frequency is purely linear:\nµ0∆H=µ0∆H0+4παf\nγ. (3)\nHere, ∆H0is the inhomogeneous broadening and is re-\nlated to film quality.\nThe lines in Figs. 1 (a) and (b) are a fit to this ex-\npression. It has to be mentioned that although a viscous\ndamping generates a linear dependence, on the contrary\nit is not possible to assume that the observation of a lin-\nearbehavior provesthat only viscous damping is present.\nThe reason for that is that two-magnon scattering can\nmimic also a linear dependence [21–23]. For both sam-\nples, and for the MgO capped sample not shown here,\nforφ= 30◦a strongly non linear behavior with a large\nincrease in linewidth values for smaller frequencies is ob-\nserved. For this reason, the hollow points in Fig. 1 have\nbeen excluded from the fit. The non-linearity at low fre-\nquencies cannot be explained by viscous damping and it\nis caused by magnetic dragging. The magnetic dragging\neffect describesthe increaseofthe linewidth of precessing\nmagneticlayerswithlargemagneticcrystalineanisotropy\ndue to the non-collinearity of the magnetization and the\nexternal magnetic field. In Fig. 2 (a), the dependence\nof the resonance field HFMRon the in-plane direction of\nthe external magnetic field is shown for two fixed fre-\nquency values. As a result of the four-fold anisotropy ex-\npectedfromthecubiclatticeofFeandassumingaperfect\ncollinearity between magnetization vector and external\nfield,HFMRcan be modeled as: [10, 24]\nµ0HFMR=µ0˜HFMR+2K1\nMscos(4φ),(4)\nwhereK1is the cubic anisotropyconstant, φthe in-plane\nazimuthal angle and ˜HFMRis the averaged resonance\nfield value. The fraction2K1\nMsis directly the anisotropy\nfield H B. In Fig. 2(a) a deviation from this model is ob-\nserved for angles between the hard and easy axis and it\nis due to magnetic dragging, i.e., the magnetization is\nnot aligned to the external field due to the effect of the\nanisotropy field. The fact that the deviation from the\nmodel in Eq.4 is smallerfor largerfrequencies(i.e. larger\napplied field) alsosupportsthis interpretation. The same\nbehavior observed for φ= 30◦has been also been re-\nported for ultrathin Fe films [25] or for insulating LSMO\nfilms [28] and attributed to magnetic dragging. The de-\ngree of non-collinearity can be estimated by solving the\nequilibrium condition for the angle defining the orienta-\ntion of the magnetization φMfor each value of φH:\nHsin(φM−φH)+HB\n4sin(4φM) = 0,(5)4\nwhere the value for the cubic anisotropy field was taken\nfrom [10]. Fig. 2(b) shows the obtained value of φMfor\nthe data shown in Fig. 2(a). The angle between mag-\nnetization and magnetic field can be as large as 10◦for\n13 GHz and it is decreased to a maximum around 4.5◦\nfor 18 GHz. The magnetic dragging effect is largest for\nφHbetween the easy and hard axis and vanishes along\nthe main crystallographic axes.\nFigure 3 shows the value of the effective damping pa-\nrameter αeffas obtained from the fits in Fig. 1 for the\nthree capping layers. In all of them, an eight-fold sym-\nmetry on the in-plane angle φHis observed with maxima\nalong the easy and hard axis of the Fe layers and min-\nima in between. For the Fe/Al and Fe/MgO samples,\nwhere spin pumping has no influence, αeff=α0+αi\nwhile for the Fe/Pt sample, where both losses through\nspin pumping and due to the MPE are active, we obtain\nthe situation shown in Eq. 2. It is remarkable that the\ndifferent origins of the damping do not change the overall\nsymmetry of the angular dependence. It has though an\nimpact on the absolute values, which are larger for the\nFe/Pt sample.\nIn the literature concerning epitaxial layers, it is possi-\nble to find different symmetries for the dependence of the\nFMRlinewidthorthedampingparameteronthein-plane\nfield direction. For the Heusler alloy Co 2FeAl both four-\nand eight-fold symmetries for the linewidth have been\nreported. The situation differs depending on the thick-\nnessofthe film [23] andalsobetween different groups[30]\npointing out to a role of the growth conditions. For Fe 3Si\nfilms and Fe/V multilayer systems a four-fold symmetry\nis reported [22, 26] and for ultrathin Fe layers, where the\nrole of the interface is strong, a two-fold symmetry of\nαeffhas been measured [25]. Eight-fold symmetry has\nbeen also observed in epitaxial FeSi systems [26, 29]. In\na different work on Fe layers, a decrease on the obtained\nαvalue along the intermediate orientation between the\ntwomainaxisrelativetothe onemeasuredalongtheeasy\nand hard axis was reported [27], pointing to an angular\ndependence very similar to ours. Concerning insulating\nsystems, two- and four-fold symmetries have been ob-\nserved in LSMO films [28].\nTwo-magnon scattering can only occur if scattering\ncenters in form of defects are present. If, as expected,\nthese are present as point lattice defects or dislocation\nlinesalongthemaincrystallographicdirections, itisclear\nthat the scattering intensity should reflect the symmetry\nof the lattice. This fact would for certain explain a four-\nor eight-fold anisotropy in damping observed in some on\nthe reports mentioned above and the maxima in αefffor\nour samples for φ= 0◦,45◦,90◦,135◦.\nFollowingZakeri et al. andAria et al., the contribution\nto damping due to two-magnonscattering can be written\nas [21, 26]:α2M=/summationdisplay\n/angbracketleftxi/angbracketrightΓ/angbracketleftxi/angbracketrightf(φH−φ/angbracketleftxi/angbracketright), (6)\nwhere Γ /angbracketleftxi/angbracketrightrepresents the strength of the two-magnon\nscattering contribution along the in-plane crystallo-\ngraphic direction /angbracketleftxi/angbracketright. The function f(φH−φ/angbracketleftxi/angbracketright) al-\nlows for an angle dependent two-magnon contribution to\ndamping with respect to the orientation of the external\nfieldHrelative to the crystallographic directions /angbracketleftxi/angbracketright.\nThe physical interpretation of the function f(φH−φ/angbracketleftxi/angbracketright)\nlays in the Fourier transform of the defects in the film\n[26, 34]. By using the ansatz f(φH−φ/angbracketleftxi/angbracketright) = cos2(4φH−\nφ/angbracketleftxi/angbracketright) we can fit the damping dependence using a simpli-\nfied version:\nαeff=αiso+α2M=αiso+Γ2Mcos2(4φH−φ[100]) (7)\nwhereαisoincludes now all the isotropic contributions to\ndamping, i.e. αmpe,αsp, pure Gilbert damping and po-\ntentially isotropic interface contributions from the term\nαi, mainly spin memory loss and interface PMA related\neffects.\nThe red lines in Fig. 3 show the fit to this model. The\nobtained parameters are summarized in Table. I. A very\nlow value below 1 ×10−3is obtained for αisofor the Fe/Al\nsample. Since αsp,MPE= 0 is expected and due to the\nlow value we consider that the obtained αisomust be\nvery close to the value corresponding only to pure vis-\ncous Gilbert damping corresponding to high quality Fe.\nHowever, strictly speaking, the obtained value is only\nan upper limit since still other effects might contribute.\nConcerning 3d metals with no half-metallic character, a\nvery low damping value of 0.7 ×10−3has been reported\nby Leeet al.for CoFe [35]. This value is comparable\nto theαisomeasured here for Fe/Al. The fact that the\nCoFe samples in which the low value was obtained are\nalso fully epitaxial with an exceptionally high crystalline\nquality explains the similarity in values. The low defect\ndensity in CoFe almost suppresses two-magnon scatter-\ning in the CoFe samplesand thereforeis comparablewith\nourαisowhere that contribution is already separated.\nFor the Fe/MgO sample the value for αisoincreases by\na factor larger than 2 although also here αsp,MPE= 0.\nαiso Γ2M\n(10−3) (10−3)\nFe/Al 0.8 ±0.3 3.6 ±0.4\nFe/Pt 3.4 ±0.3 2.4 ±0.4\nFe/MgO 1.9 ±0.1 1.3 ±0.1\nTABLE I. Isotropic contribution αisoand two-magnon scat-\ntering contribution Γ 2Mto the total effective damping param-\neterαeff.5\nFIG. 3. (Color online) Angular dependence of the effective da mping parameter αeffin the in-plane direction of the static\nmagnetic field φHfor (a) Fe/Al, (b) Fe/Pt and (c) Fe/MgO. The red lines are a fit t o Eq. 7.\nThe main differences between Fe/Al and Fe/MgO are\nthat the MgO is single crystalline while Al is polycrys-\ntalline and the contrast between the metallic character\nof Al with the insulating oxide. The lattice mismatch\nbetween MgO and Fe is around 4% and introduces there-\nfore a certain degree of stress in the Fe layer which is\nnot present when the capping is polycrystalline Al and\nwhichcanhaveanimpactondamping. Atthesametime,\nsince the Gilbert damping is sensitive to the density of\nstates and this one is modified at the interface by the\nkind of bonds between the Fe atom and the atoms from\nthe cappinglayer, the simple materialdifferencemay also\nexplain the difference. In this sense it is remarkable that\nthe low damping value by Lee et al.commented before is\nonly observed for CoFe with a MgO capping layer and a\nlargervalue is measuredwhen MgAl 2O4is used [35]. Our\ndata confirms the important role of the capping layer on\ndamping observed in other works [5].\nA further increase in the value of αisois observed for\nthe Fe/Pt sample where additional losses through spin\npumping and MPE are present. Unfortunately the data\npresented in this paper does not allow to disentangle\nthese two contributions. For this reason, when using\nEq. 1 for the calculation of spin mixing conductance, it\nmakes sense to refer to an effective value g↑↓\neffwhich is at\nthe same time an upper limit for the corresponding value\nfor spin pumping alone. Using the Fe/Al sample as a\nreference we obtain a value for the spin mixing conduc-\ntance of 3 .7±0.9×1019m−2. This value is lower than\nthe one presented in our previous report [10] and shows\nthat the value of g↑↓\neffcan be easily overestimated if the\neffect of two-magnon scattering on damping is not sepa-\nrated, with the consequent overestimation of the injected\nspin current and underestimation of the spin Hall angle\nfrom the ISHE voltage [17]. The advantage of using epi-\ntaxial magnetic layers is that they allow the separation\nof the contribution of the two-magnon scattering due to\nthe strongangulardependenceand welldefinedcrystallo-\ngraphicdirections. Thesameisnotpossibleincommonly\nused material as CoFeB or NiFe where the amorphous or\npolycrystalline nature of the layers blends the scatteringdependence on the in-plane angle.\nThe parameter Γ 2Mprovides further insight into the\norigin of total damping in the samples. This parameter\nis larger for the Fe/Al sample in comparison to the fully\nepitaxial bilayers being almost three times larger than\nfor Fe/MgO. As a result, the total damping in the Fe/Al\nsample is dominated by the two-magnon scattering due\nalsoto the low αisowhile the sameis not true in the other\ntwo systems. It has to be taken into account that, since\nas scattering centers for magnon scattering the defects at\nthe interfaces play a role, they can be dominant in thin\nfilms. From TEM images (presented for instance in [10]),\nwe can prove the existence of a highly ordered interface\nin the fully epitaxial samples. Of course, the same is not\ntrue for the case with polycrystalline Al capping. We\nbelieve that the dominant role of the interface here is\npossible, also due to the overall low defect density in the\nbulk of the Fe layer.\nFor completeness we want to discuss two additional\neffects potentially affecting the linewidth and damping.\nDue to the spread of internal and anisotropy field due to\nmosaicity in the film, there is a contribution to the line\nbroadening which has the following form [26, 33]:\n∆Hmosaic=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂HFMR\n∂φH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆φH, (8)\nwhere ∆φHis the average spread of the direction of the\neasy axes in the film plane. From Fig. 1(c) it is clear\nthat this contribution should increase the linewidth in\ntheregion φ= 15−30◦andequivalentonesbutthis isnot\nobserved pointing to a weak impact of mosaicity. In any\ncase, the mosaicity term is frequency independent and\nwill be only visible in the inhomogeneous linebroadening\n∆H0and will not affect the determination of αeff.\nThe discussion followingthe introduction ofEqs.6 and\n7 was focused on crystalline lattice defects as the origin\nof two-magnon scattering. However any kind on inhomo-\ngeneity in the magnetic state of the sample may play the\nsame role. The presence of magnetic dragging, visible for\ninstance for φ= 30◦in Fig. 1 can create a slight inhomo-\ngeneity in the magnetization state for field orientations6\nclose to the hard axis direction and an increase of damp-\ning around the hard axis orientation. In any case, this\ncontribution follows also the symmetry of the lattice and\nit is accounted in the Γ 2Mparameter.\nAlthough certain theoretical works point to an\nanisotropic Gilbert damping in fully epitaxial systems\ndue to its dependence on the density of states at the\nFermi energy [31, 32], experimentally this has been only\nseen in ultrathin Fe films [22] due to the modification of\nthe electronic structure induced by the interfacial spin-\norbit coupling. The anisotropy in αeffpresented here can\nbe fully explained by two-magnon scattering, and there-\nfore an isotropic Gilbert damping can be assumed.\nCONCLUSIONS\nMaking use of the well defined dependence of the two-\nmagnon scattering mechanism on the in-plane field di-\nrection, we have been able to separate this contribution\nto damping from the isotropic contributions originating\nfrom the viscous Gilbert damping mechanism, from spin\npumping and from the magnetic proximity effect in Pt.\nThe method can be implemented thanks to the pref-\nerential ordering of crystalline defects with respect to\nthe crystallographic directions in epitaxial systems and\ntherefore cannot be extended to amorphous or polycrys-\ntalline magnetic films. This shows the potential of the\nstudy of spin pumping related phenomena in ordered\nsystems. Without the contribution of the two-magnon\nscattering, which depends strongly on the chosen cap-\nping layer and defect density, a value of the effective spin\nmixing conductance g↑↓\neffis obtained which is closer to\ntheg↑↓associated only to spin pumping. This approach\nallows for a better estimation of the spin Hall angle in\nmetals.\nACKNOWLEDGEMENTS\nFinancial support by M-era.Net through the\nHEUMEM project and by the Carl Zeiss Stiftung\nis gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and\nB. I. Halperin, Rev. Mod. Phys. 77, No. 4, 1375 (2005).\n[2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[3] A. Conca, A. Niesen, G. Reiss, and B. Hillebrands,\nJ. Phys. D: Appl. Phys. 51, 165303 (2018).\n[4] A. Natarajarathinam, Z. R. Tadisina, T. Mewes,\nS. Watts, E. Chen, and S. Gupta, J. Appl. Phys. 112,\n053909 (2012)[5] A. Ruiz-Calaforra, T. Br¨ acher, V. Lauer, P. Pirro,\nB. Heinz, M. 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P. White,\nW. T. Ruane, B. D. Esser, D. W. McComb, P. C. Ham-\nmel, andF. Yang, NatureCommunications 8, 234(2017)."    },    {        "title": "2401.09938v2.Real_space_nonlocal_Gilbert_damping_from_exchange_torque_correlation_applied_to_bulk_ferromagnets_and_their_surfaces.pdf",        "content": "Real-space nonlocal Gilbert damping from exchange torque correlation applied to\nbulk ferromagnets and their surfaces\nBalázs Nagyfalusi,1,2,∗László Szunyogh,2,3,†and Krisztián Palotás1,2,‡\n1Institute for Solid State Physics and Optics, HUN-REN Wigner Research Center for Physics,\nKonkoly-Thege M. út 29-33, H-1121 Budapest, Hungary\n2Department of Theoretical Physics, Institute of Physics,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n3HUN-REN-BME Condensed Matter Research Group,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n(Dated: February 29, 2024)\nIn this work we present an ab initio scheme based on linear response theory of exchange torque\ncorrelation, implemented into the real-space Korringa-Kohn-Rostoker (RS-KKR) framework to cal-\nculate diagonal elements of the atomic-site-dependent intrinsic Gilbert damping tensor. The method\nis first applied to bcc iron and fcc cobalt bulk systems. Beside reproducing earlier results from the\nliterature for those bulk magnets, the effect of the lattice compression is also studied for Fe bulk,\nand significant changes for the Gilbert damping are found. Furthermore, (001)-oriented surfaces\nof Fe and Co are also investigated. It is found that the on-site Gilbert damping increases in the\nsurface atomic layer and decreases in the subsurface layer, and approaches the bulk value moving\nfurther inside the magnets. Realistic atomic relaxation of the surface layers enhances the identified\neffects. Thefirst-neighbordampingparametersareextremelysensitivetothesurfacerelaxation. De-\nspite their inhomogeneity caused by the surface, the transverse Gilbert damping tensor components\nremain largely insensitive to the magnetization direction.\nI. INTRODUCTION\nIt is highly demanded to understand and control\nthe dynamical processes governing the manipulation\nof various magnetic textures, such as atomic chains1,2,\nmagnetic skyrmions3,4or domain walls5, which can\nbe potentially used in future magnetic recording and\nlogic devices. These processes are often described by\nthe phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation6,7,\n∂ ⃗ mi\n∂t=−γ ⃗ mi×⃗Beff\ni+α\nmi⃗ mi×∂ ⃗ mi\n∂t,(1)\nwhere ⃗ miis the magnetic moment at site i,mi=|⃗ mi|\nis its length, and γis the gyromagnetic ratio. The\nfirsttermonthe rhsofEq.(1)describestheprecession\nof⃗ miaround the effective magnetic field ⃗Beff\ni, while\nthe second term is the Gilbert damping due to the\nenergy dissipation to the lattice. Clearly, this latter\nterm causes the relaxation of the magnetization to its\nequilibrium value, which is controlled by the damping\nconstant αand plays a crucial role in the realization\nof high-speed spintronic devices.\nThe Gilbert damping constant αcan be deter-\nmined experimentally from the ferromagnetic reso-\nnance (FMR) spectroscopy where the damping pa-\nrameter is related to the line-width in the measured\nspectra8. FMR spectroscopy is a well-established\nmethod for bulk materials9,10, but especially in the\nlow temperature measurement it is controversial be-\ncause the intrinsic Gilbert damping needs to be sepa-\nrated from various extrinsic sources of the line-width,\ne.g., two-magnon scattering, eddy-current damping,\nradiative damping, spin-pumping, or the slow relaxer\nmechanism11–16. The comparison of experimental\nmeasurement to theoretical calculations is also made\ndifficult bythe sampleproperties likethe exactatomic\nstructure.From a theoretical perspective the ultimate goal is\nto develop a method to calculate the Gilbert damp-\ning parameters from the electronic structure of the\nmaterial. In the last decades there have been sev-\neral efforts to understand the damping process. The\nfirst successful method was developed by Kamberský\nwhorelatedthedampingprocesstothespin-orbitcou-\npling (SOC) in terms of the breathing Fermi surface\nmodel17, while he also proposed the spin-orbit torque\ncorrelation model18,19. Later on several other meth-\nods were introduced such as the spin-pumping20and\nlinear-response approaches11,21,22. A recent summary\nof these methods was published by Guimarães et al.23\nDue to the increased interest in noncollinear mag-\nnetism Fähnle et al.24suggested an inhomogeneous\ntensorial damping. The replacement of a scalar αby\na damping matrix αmeans that the damping field in\nEq.(1)isnolongerproportionaltothetimederivative\nof⃗ mi, it becomes a linear function of ∂ ⃗ mi/∂t. More-\nover, nonlocality of the damping process implies that\nthe damping field at site iexperiences ∂ ⃗ mj/∂tfor any\nsitej. The LLG equation (1) is then replaced by the\nset of equations25,\n∂ ⃗ mi\n∂t=⃗ mi×\n−γ⃗Beff\ni+X\njαij1\nmj∂ ⃗ mj\n∂t\n,(2)\nwhere the damping term is unfolded to pairwise con-\ntributions of strength αij. The appearance of non-\nlocal damping terms was evidenced for magnetic do-\nmain walls26,27by linking the Gilbert damping to the\ngradients of the magnetization. In NiFe, Co, and\nCoFeB thin films Li et al.28measured wave-number-\ndependent dissipation using perpendicular spin wave\nresonance, validating thus the idea of nonlocal damp-\ning terms. Different analytical expressions for αijare\nalready proposed22,25,29,30, and the nonlocal damp-\ning is found for bulk materials25,31as well as its ef-arXiv:2401.09938v2  [cond-mat.mtrl-sci]  28 Feb 20242\nfect on magnon properties of ferromagnets have been\ndiscussed32. Recent studies went further and, anal-\nogously to the higher order spin-spin interactions in\nspin models, introduced multi-body contributions to\nthe Gilbert damping33.\nThe calculation of the Gilbert damping prop-\nerties of materials has so far been mostly fo-\ncused on 3D bulk magnets, either in chemically\nhomogeneous11,19,23,25,34–36or heterogeneous (e.g.\nalloyed)11,22,31forms. There are a few studies avail-\nable reporting on the calculation of the Gilbert damp-\ning in 2D magnetic thin films12,23,37,38, or at surfaces\nand interfaces of 3D magnets31,35,37. The calculation\nof the Gilbert damping in 1D or 0D magnets is, due\nto our knowledge, not reported in the literature. Fol-\nlowing the trend of approaching the atomic scale for\nfunctional magnetic elements in future spintronic de-\nvices, the microscopic understanding of energy dissi-\npation through spin dynamics in magnets of reduced\ndimensions is inevitable and proper theoretical meth-\nods have to be developed.\nOur present work proposes a calculation tool for\nthediagonalelementsofthenon-localintrinsicGilbert\ndamping tensor covering the 3D to 0D range of mag-\nnetic materials on an equal footing, employing a real-\nspace embedding Green’s function technique39. For\nthis purpose, the linear response theory of the Gilbert\ndamping obtained by the exchange torque correlation\nis implemented in the real-space KKR method. As a\ndemonstration of the new method, elemental Fe and\nCo magnets in their 3D bulk form and their (001)-\noriented surfaces are studied in the present work. Go-\ning beyond comparisons with the available literature,\nnew aspects of the Gilbert damping in these materials\nare also reported.\nThe paper is organized as follows. In Sec. II the\ncalculation of the Gilbert damping parameters within\nthe linear response theory of exchange torque corre-\nlation using the real-space KKR formalism is given.\nSec. III reports our results on bulk bcc Fe and fcc Co\nmaterials and their (001)-oriented surfaces. We draw\nour conclusions in Sec. IV.\nII. METHOD\nA. Linear response theory within real-space\nKKR\nThe multiple-scattering of electrons in a finite clus-\nter consisting of NCatoms embedded into a 3D or 2D\ntranslation-invariant host medium is fully accounted\nfor by the equation39\nτC=τH\u0002\nI−(t−1\nH−t−1\nC)τH\u0003−1,(3)\nwhere τCand τHare the scattering path operator\nmatrices of the embedded atomic cluster and the host,\nrespectively, tCandtHare the corresponding single-\nsite scattering matrices, all in a combined atomic site\n(j, k∈ {1, ..., N C}) and angular momentum ( Λ,Λ′∈\n{1, ...,2(ℓmax+ 1)2}) representation: τ={τjk}=\n{τjk\nΛΛ′}andt={tj\nΛΛ′δjk}, where ℓmaxis the angularmomentum cutoff in describing the scattering events,\nand for simplicity we dropped the energy-dependence\nof the above matrices.\nFor calculating the diagonal Cartesian elements\nof the nonlocal Gilbert damping tensor connecting\natomic sites jandkwithin the finite magnetic atomic\ncluster, we use the formula derived by Ebert et al.22,\nαµµ\njk=2\nπmj\nsTr\u0010\nTj\nµ˜τjk\nCTk\nµ˜τkj\nC\u0011\n, (4)\nwhere µ∈ {x, y, z}, the trace is taken in the\nangular-momentum space and the formula has to\nbe evaluated at the Fermi energy ( EF). Here,\nmj\nsis the spin moment at the atomic site j,\n˜τjk\nC,ΛΛ′= (τjk\nC,ΛΛ′−(τkj\nC,Λ′Λ)∗)/2i, and Tj\nµis the\ntorque operator matrix which has to be calculated\nwithin the volume of atomic cell j,Ωj:Tj\nµ;ΛΛ′=R\nΩjd3rZj\nΛ(⃗ r)×βσµBxc(⃗ r)Zj\nΛ′(⃗ r),wherethenotationof\nthe energy-dependence is omitted again for simplicity.\nHere, βis a standard Dirac matrix entering the Dirac\nHamiltonian, σµare Pauli matrices, and Bxc(⃗ r)is the\nexchange-correlation field in the local spin density ap-\nproximation (LSDA), while Zj\nΛ(⃗ r)are right-hand side\nregular solutions of the single-site Dirac equation and\nthe superscript ×denotes complex conjugation re-\nstricted to the spinor spherical harmonics only22. We\nshould emphasize that Eq. (4) applies to the diagonal\n(µµ) elements of the Gilbert tensor only. To calcu-\nlate the off-diagonal tensor elements one needs to use,\ne.g., the more demanding Kubo-Bastin formula40,41.\nNote also that in noncollinear magnets the exchange\nfield Bxc(⃗ r)is sensitive to the spin noncollinearity42\nwhich influences the calculated torque operator ma-\ntrix elements, however, this aspect does not concern\nour present study including collinear magnetic states\nonly.\nNote that the nonlocal Gilbert damping is, in gen-\neral, not symmetric in the atomic site indices, αµµ\njk̸=\nαµµ\nkj, instead\nαµµ\nkj=mj\ns\nmksαµµ\njk(5)\nholds true. This is relevant in the present work for\nthe ferromagnetic surfaces. On the other hand, in\nferromagnetic bulk systems αµµ\njk=αµµ\nkjsince mj\ns=\nmk\ns=msfor any pair of atomic sites.\nIn practice, the Gilbert damping formula in Eq. (4)\nis not directly evaluated at the Fermi energy, but a\nsmall imaginary part ( η) of the complex energy is ap-\nplied, which is called broadening in the following, and\nits physical effect is related to the scattering rate in\nother damping theories19,25,37,43. Taking into account\nthe broadening η, the Gilbert damping reads\nαµµ\njk(η) =−1\n4h\n˜αµµ\njk(E+, E+) + ˜αµµ\njk(E−, E−)\n−˜αµµ\njk(E+, E−)−˜αµµ\njk(E−, E+)i\n,(6)\nwhere E+=EF+iηandE−=EF−iη, and the3\nindividual terms are\n˜αµµ\njk(E1, E2) =\n2\nπmj\nsTr\u0010\nTj\nµ(E1, E2)τjk\nC(E2)Tk\nµ(E2, E1)τkj\nC(E1)\u0011\n(7)\nwith E1,2∈ { E+, E−}, and the ex-\nplicitly energy-dependent torque opera-\ntor matrix elements are: Tj\nµ;ΛΛ′(E1, E2) =R\nΩjd3rZj×\nΛ(⃗ r, E1)βσµBxc(⃗ r)Zj\nΛ′(⃗ r, E2).\nB. Effective damping and computational\nparameters\nEq.(6)givesthebroadening-dependentspatiallydi-\nagonal elements of the site-nonlocal Gilbert damping\ntensor: αxx\njk(η),αyy\njk(η), and αzz\njk(η). Since no longitu-\ndinal variation of the spin moments is considered, the\ntwo transversal components perpendicular to the as-\nsumed uniform magnetization direction are physically\nmeaningful. Given the bulk bcc Fe and fcc Co sys-\ntems and their (001)-oriented surfaces with C4vsym-\nmetry under study in the present work, in the follow-\ning the scalar αrefers to the average of the xxand\nyyGilbert damping tensor components assuming a\nparallel magnetization with the surface normal z[001]-\ndirection: αjk= (αxx\njk+αyy\njk)/2 =αxx\njk=αyy\njk. From\nthe site-nonlocal spatial point of view in this work we\npresent results on the on-site (\" 00\"), first neighbor\n(denoted by \" 01\") and second neighbor (denoted by\n\"02\") Gilbert damping parameters, and an effective,\nso-called total Gilbert damping ( αtot), which can be\ndefined as the Fourier transform of αjkat⃗ q= 0. The\nFourier transform of the Gilbert damping reads\nα⃗ q=∞X\nj=0α0jexp(−i⃗ q(⃗ r0−⃗ rj))\n≈X\nr0j≤rmaxα0jexp(−i⃗ q(⃗ r0−⃗ rj)),(8)\nwhere r0j=|⃗ r0−⃗ rj|and the effective damping is\ndefined as\nαtot=α⃗ q=⃗0=∞X\nj=0α0j≈X\nr0j≤rmaxα0j.(9)\nSince we have a real-space implementation of the\nGilbert damping, the infinite summation for both\nquantities is replaced by an approximative summation\nfor neighboring atoms upto an rmaxcutoff distance\nmeasured from site \"0\". Moreover, note that for bulk\nsystems the effective damping αtotis directly related\nto the ⃗ q= 0mode of FMR experiments.\nThe accuracy of the calculations depends on many\nnumerical parameters such as the number of ⃗kpoints\nused in the Brillouin zone integration, the choice of\nthe angular momentum cutoff ℓmax, and the spatial\ncutoff rmaxused for calculating α⃗ qandαtot. Previ-\nous research25showed that the Gilbert damping heav-\nily depends on the broadening η, so we extended ourstudies to a wider range of η= 1meV to 1 eV. The\nsufficient k-point sampling was tested at the distance\nofrmax= 7a0(where a0is the corresponding 2D lat-\ntice constant) from the reference site with the broad-\nening set to 1mRy, and the number of ⃗kpoints was\nincreased up to the point, where the 5th digit of the\ndamping became stable. Maximally, 320400 ⃗kpoints\nwere used for the 2D layered calculation but the re-\nquested accuracy was reached with 45150 and 80600\n⃗kpoints for bulk bcc Fe and fcc Co systems, respec-\ntively.\nThe choice of ℓmaxwas tested through the whole η\nrange for bcc Fe, and it was based on the comparison\nof damping calculations with ℓmax= 2andℓmax= 3.\nThe maximal deviation for the on-site Gilbert damp-\ning was found at around η= 5mRy, but it was still\nless than 10%. The first and second neighbor Gilbert\ndamping parameters changed in a more significant\nway (by ≈50%) in the whole ηrange upon changing\nℓmax, yet the effective total damping was practically\nunchanged, suggesting that farther nonlocal damping\ncontributions compensate this effect. Since αtotis the\nmeasurable physical quantity we concluded that the\nlower angular momentum cutoff of ℓmax= 2is suffi-\ncient to be used further on.\nThe above choice of ℓmax= 2for the angular mo-\nmentumcutoff, themathematicalcriterionofpositive-\ndefinite αjk(which implies α⃗ q>0for all ⃗ qvectors),\nand the prescribed accuracy for the effective Gilbert\ndamping in the full considered η= 1meV to 1 eV\nrange set rmaxto 20 a0for both bcc Fe and fcc Co. It\nis worth mentioning that the consideration of lattice\nsymmetries made possible to decrease the number of\natomic sites in the summations for calculating α⃗ qand\nαtotby an order of magnitude.\nIII. RESULTS AND DISCUSSION\nOur newly implemented method was employed to\nstudy the Gilbert damping properties of Fe and Co\nferromagnetsintheirbulkand(001)-orientedsurfaces.\nIn these cases only unperturbed host atoms form\nthe atomic cluster, and the so-called self-embedding\nprocedure44is employed, where Eq. (3) reduces to\nτC=τHforthe3Dbulkmetalsand2Dlayeredmetal-\nvacuum interfaces.\nA. Bulk Fe and Co ferromagnets\nFirst we calculate and analyze the nonlocal and ef-\nfective dampings for bulk bcc Fe by choosing a 2D\nlattice constant of a0= 2.863Å. The magnitude of\nthe magnetic moments are obtained from the self-\nconsistent calculation. The spin and orbital moments\narems= 2.168µBandmo= 0.046µB, respectively.\nThe broadening is set to η= 68meV. The inset of Fig.\n1a) shows the typical function of the nonlocal Gilbert\ndamping α0jdepending on the normalized distance\nr0j/a0between atomic sites \"0\" and \" j\". In accor-\ndance with Ref. 25 the nonlocal Gilbert damping\nquickly decays to zero with the distance, and can be4\na)\n5 10 15 2005\n5 10 15 20−505\nr0j/a0α0j[×10−4]\nr0j/a0α0j·(r0j/a0)2[×10−4]\nb)\n5 10 15 20−202468\nrmax/a0αtot[×10−3]\nFIG. 1. a) Nonlocal Gilbert damping in bulk bcc Fe as a\nfunction of distance r0jbetween atomic sites \"0\" and \" j\"\nshown upto a distance of 20 a0(the 2D lattice constant is\na0= 2.863Å): the black squares are calculated α0jval-\nues times the normalized squared-distance along the [110]\ncrystallographic direction, and the red line is the corre-\nsponding fitted curve based on Eq. (10). The inset shows\nthe nonlocal Gilbert damping α0jvalues in the given dis-\ntance range. b) Convergence of the effective damping pa-\nrameter αtot, partial sums of α0jupto rmaxbased on Eq.\n(9), where rmaxis varied. The broadening is chosen to be\nη=68 meV.\nwell approximated with the following function:\nα(r)≈Asin (kr+ϕ0)\nr2exp(−βr).(10)\nTo test this assumption we assorted the atomic sites\nlying in the [110] crystallographic direction and fit-\nted Eq. (10) to the calculated data. In practice, the\nfit is made on the data set of α0j(r0j/a0)2, and is\nplotted in Fig. 1a). Although there are obvious out-\nliers in the beginning, the magnitude of the Gilbert\ndamping asymptotically follows the ∝exp(−βr)/r2\ndistance dependence. The physical reason for this de-\ncay is the appearance of two scattering path operators\n(Green’s functions) in the exchange torque correlation\nformula in Eq. (4) being broadened due to the finite\nimaginary part of the energy argument.\nIn our real-space implementation of the Gilbert\ndamping, an important parameter for the effective\ndamping calculation is the real-space cutoff rmaxin\nEq. (9). Fig. 1b) shows the evolution of the ef-\nfective (total) damping depending on the rmaxdis-\ntance, within which all nonlocal damping terms α0jare summed up according to Eq. (9). An oscillation\ncan similarly be detected as for the nonlocal damping\nitself in Fig. 1a), and this behavior was fitted with\na similar exponentially decaying oscillating function\nas reported in Eq. (10) in order to determine the ex-\npected total Gilbert damping αtotvalue in the asymp-\ntotic r→ ∞limit. In the total damping case it is\nfound that the spatial decay of the oscillation is much\nslower compared to the nonlocal damping case, which\nmakes the evaluation of αtotmore cumbersome. Our\ndetailed studies evidence that for different broaden-\ningηvalues the wavelength of the oscillation stays\nthe same but the spatial decay becomes slower as\nthe broadening is decreased (not shown). This slower\ndecay together with the fact that the effective (to-\ntal) damping value itself is also decreasing with the\ndecreasing broadening results that below the 10meV\nrange of ηthe amplitude of the oscillation at the dis-\ntance of 20 a0is much larger than its asymptotic limit.\nIn practice, since the total damping is calculated as\nther→ ∞limit of such a curve as shown in Fig. 1b),\nthis procedure brings an increased error for αtotbelow\nη= 10meV, and this error could only be reduced by\nincreasing the required number of atomic sites in the\nreal-space summation in Eq. (10).\nFig. 2 shows the dependence of the calculated on-\nsite, first- and second-neighbor and effective total\nGilbert damping parameters on the broadening η.\nThe left column shows on-site ( α00) and total ( αtot)\nwhile the right one the first ( α01) and second ( α02)\nneighbor Gilbert dampings. We find very good agree-\nment with the earlier reported results of Thonig et\nal.25, particularly that the on-site damping has the\nlargest contribution to the total damping being in the\nsame order of magnitude, while the first and second\nneighbors are smaller by an order of magnitude. The\nobtained dependence on ηis also similar to the one\npublished by Thonig et al.25:α00andαtotare in-\ncreasing with η, and α01andα02do not follow a\ncommon trend, and they are material-dependent, see,\ne.g., the opposite trend of α02with respect to ηfor\nFe and Co. The observed negative values of some of\nthesite-nonlocaldampingsarestillconsistentwiththe\npositive-definiteness of the full (infinite) αjkmatrix,\nwhich has also been discussed in Ref. 25.\nThe robustness of the results was tested against a\nsmall change of the lattice constant simulating the ef-\nfect of an external pressure for the Fe bulk. These re-\nsults are presented in the second row of Fig. 2, where\nthe lattice constant of Fe is set to a0= 2.789Å. In this\ncase the magnetic moments decrease to ms= 2.066µB\nandmo= 0.041µB. It can clearly be seen that the on-\nsite, first and second neighbor Gilbert dampings be-\ncome smaller upon the assumed 2.5% decrease of the\nlattice constant, but the total damping remains prac-\ntically unchanged in the studied ηrange. This sug-\ngests that the magnitudes of more distant non-local\ndamping contributions are increased.\nThe third row of Fig. 2 shows the selected damp-\ning results for fcc Co with a 2D lattice constant of\na0= 2.507Å. The spin and orbital moments are\nms= 1.654µBandmo= 0.078µB, respectively. The\nincreaseofthetotal, theon-site, andthefirst-neighbor5\n10−310−210−11000246810Fe -a0= 2.863˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.863˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Fe -a0= 2.789˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.789˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Co -a0= 2.507˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Co -a0= 2.507˚A\nη(eV)α[×10−4]α01\nα02\nFIG. 2. Left column: Local on-site ( α00, black square)\nand total ( αtot, red triangle) Gilbert damping as a func-\ntionofthebroadening ηforbccFe(001)with a0= 2.863Å,\nbcc Fe(001) with a0= 2.789Å, and fcc Co(001) with\na0= 2.507Å. Right column: Nonlocal first nearest neigh-\nbor (α01, black square) and second nearest neighbor ( α02,\nred triangle) Gilbert damping for the same systems.\ndampings with increasing ηis similar to the Fe case,\nand the on-site term dominates αtot. An obvious\ndifference is found for the second-neighbor damping,\nwhich behaves as an increasing function of ηfor Co\nunlike it is found for Fe.\nConcerning the calculated damping values, there is\na large variety of theoretical methods and calculation\nparameters, as well as experimental setups used in\nthe literature, which makes ambiguous to compare\nour results with others. Recently, Miranda et al.31\nreported a comparison of total and on-site damping\nvalues with the available theoretical and experimen-\ntal literature in their Table S1. For bcc Fe bulk they\nreported total damping values in the range of 1.3–\n4.2×10−3and for fcc Co bulk within the range of 3.2–\n11×10−3, and our results fit very well within theseranges around η≈100meV for Fe and for η >100\nmeV for Co. Moreover, we find that our calculated on-\nsite damping values for bcc Fe are larger ( >5×10−3)\nthan the reported values of Miranda et al.(1.6×10−3\nand 3.6 ×10−3), but for fcc Co the agreement with\ntheir reported total (3.2 ×10−3) and on-site damping\n(5.3×10−3) values is very good at our η= 136meV\nbroadening value.\n10−310−210−110010−510−410−310−2Fe\nη(eV)αtot\nαSOC=1\nαSOC=0\n10−310−210−110010−510−410−310−2Co\nη(eV)αtot\nαSOC=1\nαSOC=0\nFIG. 3. Effective (total) Gilbert damping for bcc Fe\n(left) and fcc Co (right) as a function of broadening ηon\na log-log scale. The error bars are estimated from the\nfitting procedure of Eq. (10). The red triangles show the\ncase with normal SOC ( αSOC=1), and the blue diamonds\nwhere SOC is switched off ( αSOC=0).\nNext, weinvestigatethespin-orbit-coupling-(SOC)-\noriginated contribution to the Gilbert damping. Our\nmethodmakesitinherentlypossibletoincludeaSOC-\nscaling factor in the calculations45. Fig. 3 shows the\nobtained total Gilbert damping as a function of the\nbroadening ηwith SOC switched on/off for bcc Fe\nand fcc Co. It can be seen that the effect of SOC\nis not dominant at larger ηvalues, but the SOC\nhas an important contribution at small broadening\nvalues ( η < 10−2eV), where the calculated total\nGilbert damping values begin to deviate from each\nother with/without SOC. As discussed in Ref. 23,\nwithout SOC the damping should go toward zero for\nzero broadening, which is supported by our results\nshown in Fig. 3.\nB. (001)-oriented surfaces of Fe and Co\nferromagnets\nIn the following, we turn to the investigation of the\nGilbertdampingparametersatthe(001)-orientedsur-\nfaces of bcc Fe and fcc Co. Both systems are treated\nas a semi-infinite ferromagnet interfaced with a semi-\ninfinite vacuum within the layered SKKR method46.\nIn the interface region 9 atomic layers of the ferromag-\nnet and 3 atomic layers of vacuum are taken, which is\nsandwiched between the two semi-infinite (ferromag-\nnet and vacuum) regions. Two types of surface atomic\ngeometries were calculated: (i) all atomic layers hav-\ning the bulk interlayer distance, and (ii) the surface\nand subsurface atomic layers of the ferromagnets have6\nTABLE I. Geometry relaxation at the surfaces of the fer-\nromagnets: change of interlayer distances relative to the\nbulk interlayer distance at the surfaces of bcc Fe(001) and\nfcc Co(001), obtained from VASP calculations. \"L1\" de-\nnotes the surface atomic layer, \"L2\" the subsurface atomic\nlayer, and \"L3\" the sub-subsurface atomic layer. All other\ninterlayer distances are unchanged in the geometry opti-\nmizations.\nL1-L2 L2-L3\nbcc Fe(001) -13.7% -7.7%\nfcc Co(001) -12.4% -6.4%\nbeenrelaxedintheout-of-planedirectionusingtheVi-\nenna Ab-initio Simulation Package (VASP)47within\nLSDA48. For the latter case the obtained relaxed\natomic geometries are given in Table I.\nFigure 4 shows the calculated layer-resolved on-\nsite and first-neighbor Gilbert damping values (with\nη= 0.68eV broadening) for the bcc Fe(001) and fcc\nCo(001) surfaces. It can generally be stated that the\nsurface effects are significant in the first 4 atomic lay-\ners of Fe and in the first 3 atomic layers of Co. We\nfind that the on-site damping ( α00) increases above\nthe bulk value in the surface atomic layer (layer 1:\nL1), and decreases below the bulk value in the sub-\nsurface atomic layer (L2) for both Fe and Co. This\nfinding is interesting since the spin magnetic moments\n(ms, shown in the insets of Fig. 4) are also consider-\nably increased compared to their bulk values in the\nsurface atomic layer (L1), and the spin moment enters\nthe denominator when calculating the damping in Eq.\n(4).α00increases again in L3 compared to its value in\nL2, thus it exhibits a nonmonotonic layer-dependence\nin the vicinity of the surface. The damping results ob-\ntained with the ideal bulk interlayer distances and the\nrelaxed surface geometry (\"R\") are also compared in\nFig. 4. It can be seen that the on-site damping is in-\ncreasedinthesurfaceatomiclayer(L1), anddecreased\nin the subsurface (L2) and sub-subsurface (L3) atomic\nlayers upon atomic relaxation (\"R\") for both Fe and\nCo. The first-neighbor dampings ( α01) are of two\ntypes for the bcc Fe(001) and three types for the fcc\nCo(001), see caption of Fig. 4. All damping values\nare approaching their corresponding bulk value mov-\ning closer to the semi-infinite bulk (toward L9). In\nabsolute terms, for both Fe and Co the maximal sur-\nface effect is about 10−3for the on-site damping, and\n2×10−4for the first-neighbor dampings. Given the\ndamping values, the maximal relative change is about\n15% for the on-site damping, and the first-neighbor\ndampings can vary by more than 100% (and can even\nchangesign)inthevicinityofthesurfaceatomiclayer.\nNote that Thonig and Henk35studied layer-resolved\n(effective) damping at the surface of fcc Co within the\nbreathing Fermi surface model combined with a tight-\nbinding electronic structure approach. Although they\nstudied a different quantity compared to us, they also\nreported an increased damping value in the surface\natomic layer, followed by an oscillatory decay toward\nbulk Co.\nSo far the presented Gilbert damping results cor-\nrespond to spin moments pointing to the crystallo-\n1 3 5 7 90.81Fe\n1 92.43\nlayermsms\nmR\ns\nlayerα00[×10−2]\nα00\nαR\n00\n1 3 5 7 90.81\n1 91.71.8\nlayermsms\nmR\nsCo\nlayerα00[×10−2]α00\nαR\n00\n1 3 5 7 9−2−101Fe\nlayerα01[×10−4]\nα01+αR\n01+\nα01−αR\n01−\n1 3 5 7 91234\nCo\nlayerα01[×10−4]α01+αR\n01+\nα01−αR\n01−\nα01 αR\n01FIG. 4. Evolution of the layer-resolved Gilbert damping\nfrom the surface atomic layer (L1) of bcc Fe(001) and fcc\nCo(001) toward the bulk (L9), depending also on the out-\nof-plane atomic relaxation \"R\". On-site ( α00) and first\nneighbor ( α01) Gilbert damping values are shown in the\ntop two and bottom two panels, respectively. The broad-\nening is η= 0.68eV. The empty symbols belong to the\ncalculations with the ideal bulk interlayer distances, and\nthe full symbols to the relaxed surface geometry, denoted\nwith index \"R\". Note that α01is calculated for nearest\nneighbors of atomic sites in the neighboring upper, lower,\nand the same atomic layer (for fcc Co only), and they are\nrespectively denoted by \" +\" (L-(L+1)), \" −\" (L-(L −1)),\nand no extra index (L-L). The insets in the top two panels\nshow the evolution of the magnitudes of the layer-resolved\nspin magnetic moments ms. The horizontal dashed line in\nall cases denotes the corresponding bulk value.\ngraphic [001] ( z) direction, and the transverse compo-\nnents of the damping αxxandαyyare equivalent due\ntothe C4vsymmetryofthe(001)-orientedsurfaces. In\norder to study the effect of a different orientation of\nall spin moments on the transverse components of the\ndamping, we also performed calculations with an ef-\nfective field pointing along the in-plane ( x) direction:\n[100] for bcc Fe and [110] for fcc Co. In this case, due\nto symmetry breaking of the surface one expects an\nanisotropy in the damping, i.e., that the transverse\ncomponents of the damping tensor, αyyandαzz, are\nnot equivalent any more. According to our calcula-7\ntions, however, the two transverse components of the\non-site ( αyy\n00andαzz\n00) and nearest-neighbor ( αyy\n01and\nαzz\n01) damping tensor, at the Fe surface differed by less\nthan 0.1 % and at the Co surface by less than 0.2 %,\ni.e., despite the presence of the surface the damping\ntensor remained highly isotropic. The change of the\ndamping with respect to the orientation of the spin\nmoments in zorxdirection (damping anisotropy)\nturned out to be very small as well: the relative dif-\nference in αyy\n00was 0.1 % and 0.3 %, while 0.5 % and\n0.1 % in αyy\n01for the Fe and the Co surfaces, respec-\ntively. For the farther neighbors, this difference was\nless by at least two orders of magnitude.\nIV. CONCLUSIONS\nWe implemented an ab initio scheme of calculat-\ning diagonal elements of the atomic-site-dependent\nGilbert damping tensor based on linear response the-\nory of exchange torque correlation into the real-space\nKorringa-Kohn-Rostoker (KKR) framework. To val-\nidate the method, damping properties of bcc Fe and\nfcc Co bulk ferromagnets are reproduced in good com-\nparison with the available literature. The lattice com-\npression is also studied for Fe bulk, and important\nchanges for the Gilbert damping are found, most pro-\nnounced for the site-nonlocal dampings. By investi-\ngating (001)-oriented surfaces of ferromagnetic Fe andCo, we point out substantial variations of the layer-\nresolved Gilbert damping in the vicinity of the sur-\nfaces depending on various investigated parameters.\nThe effect of such inhomogeneous dampings should be\nincluded into future spin dynamics simulations aim-\ning at an improved accuracy, e.g., for 2D surfaces and\ninterfaces. We anticipate that site-nonlocal damping\neffects become increasingly important when moving\ntoward physical systems with even more reduced di-\nmensions (1D).\nACKNOWLEDGMENTS\nThe authors acknowledge discussions with Danny\nThonig. Financial support of the National Research,\nDevelopment, and Innovation (NRDI) Office of Hun-\ngary under Project Nos. 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In materials with  large spin-orbit coupling, the spin Hall effect \nmay convert charge current to pure spin current and a large conversion efficiency, which is quantified by spin Hall angle (SHA), is desirable for the realization of miniaturized and energy efficient spintronic devices. Here, we report a giant SHA in beta-tungsten ( β-W) thin \nfilms in Sub/W( t)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures wi th variable W thickness. \nWe employed an all-optical time-resolved magneto-optical Kerr effect microscope for an unambiguous determination of SHA using the principle of modulation of Gilbert damping of the adjacent ferromagnetic layer by the spin-orbit torque from the W layer. A non-monotonic variation of SHA with W layer thickness ( t) is observed with a maximum of about 0.4 at \nabout t = 3 nm, followed by a sudden reduction to a very low value at t = 6 nm. This variation \nof SHA with W-thickness correlates well with the thickness dependent structural phase transition and resistivity variation of W above the spin diffusion length of W, while below this length the interfacial electronic effect at W/CoFeB influences the estimation of SHA.    \n \n     \n2 \n                                                           I. INTRODUCTION      \n     The exciting new frontier of spintronics [1] and magnonics [2, 3] research is driven by the \nneed of utilizing spin-orbit (SO) effect for obtaining pure spin current [4]. An important aspect of improving the performance of the device is to minimize Joule heating, which \nrequires fundamentally pure spin cu rrent [4, 5]. It is quite non-tr ivial to generate and transport \nthe spin current. Some of the earlier studies have used non-local spin-valve based techniques \n[6, 7], spin pumping [8, 9, 10], and Rashba effect  [11] for generating sp in currents. Moreover, \nutilization of pure spin current for magnetiz ation manipulation poses additional challenges. \nRecent finding of spin Hall effect (SHE) [12] has opened up the possibility of utilizing pure \nspin current for manipulation of magnetic moments [4, 13, 14]. To quantify the SHE, an \nimportant parameter, namely, spin Hall angle (S HA), has been proposed and it is related to \nthe conversion efficiency of charge-to-spin current [15, 16]. Considerable efforts have been devoted for estimating and understanding the valu e of SHA for various heavy metals (HMs). \nParticularly, the SHE in HM layers can generate sufficiently large spin current to manipulate \nmagnetic moments of a ferromagnetic layer adjacent to the HM layer as it exerts significant spin torque [13, 17]. Furthermore, the SHE i nduced spin-orbit torques (SOTs) have been \nshown to induce the large domain wall velo city [14], excite precessional magnetization \ndynamics [18, 19] as well as result in magneti zation switching [20]. Remarkably, it has been \nrecently demonstrated that by using sophi sticated device structuring, SOT induced \nmagnetization switching can be triggered in the absence of any magnetic field [21, 22, 23]. Some key requirements for technological implementation of the above mentioned interesting applications are to search for HMs with reas onably large SHA, investigation of various \nfactors affecting the SHA of HM thin films, and to understand the variation of SHA in such \nHM thin films by controlling those factors. An important issue in this research is to establish \nan accurate and unambiguous measurement techni que of SHA. The precise quantification of \nSHA and its origin in a conventional metal-based system is of technological interest for    \n3 \n spintronics based device applications. In gene ral, the techniques used for determining the \nSHA are the spin torque ferromagnetic resonance (ST-FMR) technique [13, 24, 25] spin torque switching of perpendicularly magneti zed films [22] and measurement based on non-\nlocal spin valves [5]. All these techniques primar ily rely on electrical excitation, detection and \nextremely delicate micro-fabrication [26]. Recently, it has been demonstrated that by using \ntime resolved magneto-optical Kerr effect (TRMOKE) [27] technique SHA can be measured more conveniently in a non-invasive manner without the requirement of advanced micro-fabrication and electrical detection, and more precise estimate of SHA may be obtained [28]. \n   The highly resistive β-tungsten (distorted tetragonal phase commonly referred to as A15 \nstructure) is known to be one of the efficient materials for exhibiting large SHA due to strong \nSO coupling [29]. Also, in ferromagnetic thin film heterostructures, use of tungsten (W) leads to highly stable perpendicular magnetic anisotropy [30] and interfacial Dzyaloshinskii Moriya interaction [31]. Another important characteristics associated with W is the thickness \ndependent phase transition exhibited by it, usua lly observed in the thickness range of sub-10 \nnm [32, 33]. In general, sputte r deposited W films with thickness below 5 nm is found to have \nβ phase with resistivity larger than 150 µ Ω-cm, whereas the films with thickness above 5 nm \npossess predominantly α phase (BCC structure) with resistivity of about 40 µ Ω-cm [20, 34, \n35]. Till date, the SHA for W has been reported in few studies for mostly in β phase. However, \na systematic study of SHA in W/ferromagnet (FM)/oxide heterostructure with W layer \nthickness varying across the structural phase transition  is missing. Depending on the deposition condition, SHA values of up to about  0.3 have been reported specifically for β \nphase of W [20, 29]. Few reports ha ve mentioned relatively small SHA for α phase of W [29]. \nTherefore, it calls for investigating the systemat ic dependence of SHA on the structural phase \nof W which is intricately related to its thickness. Furthermore, few important recent studies \nhave suggested that the transparency of HM/F M interface plays a crucial role in evaluating \nthe SHA of HM layer [36, 37, 38]. Additionally, a theoretical study has classified the bulk and    \n4 \n interface SHE and claimed that the interface spin Hall effect may be as large as 25 times than \nthe bulk SHE [39]. Recently, by alloying different  HM layers, attempts have been made to \nachieve large SHA [40]. All these studies relate to the intricate role of spin orbit coupling \ninduced SHE in generating pure spin current th at is aimed towards utilization for device \napplications.    \n    Here, we present the correlation between thickness dependent phase transition in W thin \nfilms and large SHE induced modulation of damping (MOD) in technologically important Sub/W( t)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures. All- optical detection technique \nTRMOKE is used for investigating the magnetization dynamics [28]. Utilizing the sensitive variation of MOD, we estimate the SHA. We  observe a clear variation in the value of \nestimated SHA with the phase of W underlayer. However, even within the β phase, when the \nthickness of W is smaller than its spin diffusion length, the value of SHA is found to be significantly low. We correlate this variation of SHA with the bulk of the HM layer as well as \nthe interfacial electronic effect at the HM/FM interface. \n \n                      II. SAMPLE PREPARATION AND CHARACTERIZATION                                              \nThe thin film heterostructures Sub/W( t)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm) with t = 2 to 7 \nnm in the step of 1 nm were deposited by dc/rf magnetron sputtering on Si (100) wafers \ncoated with 100 nm SiO 2. The purpose of varying W underlayer thickness was to choose W \nthickness across the phase transition regime. The base pressure of the deposition chamber was better than 2 × 10\n-7 Torr. CoFeB and W were grown using dc power of 20 Watt whereas the \nSiO 2 was grown using rf power of 60 Watt at 13.56 MHz. All thin films were grown in Ar gas \natmosphere of 1 mTorr pressure and deposition conditions were carefully optimized [31]. \nUsing a shadow mask, 5 nm thick chromium/25 nm thick gold contact electrodes were first \nprepared, followed by deposition of the sample stack of 3 mm × 1 mm dimension between the contact electrodes using another shadow mask. The dc charge current was applied along the    \n5 \n length of the sample using a standard sour ce-meter [U3606A, Agilent Technologies] and \nexperimental arrangement allowed us to suitably  choose the applied bias magnetic field angle \nwith respect to the current flow direction. Tw o-colour optical pump probe technique was used \nin the TRMOKE experiment. The second harmonic (wavelength: 400 nm, pulse width: 100 fs) \nof a mode locked Ti-sapphire oscillator (Tsunami, Spectra Physics) was used as the pump \nbeam to excite the magnetization dynamics in the samples, whereas the fundamental laser beam (wavelength: 800 nm, pulse width: 80 fs) was used as the probe beam to detect the pump induced magneto-optical Kerr rotation from the sample as a function of the time delay \nbetween the pump and the probe beam [28]. All the experiments were performed under ambient condition. Atomic force microscope wa s used to investigate the surface topography, \nwhereas vibrating sample magnetometer was used to characterize the static magnetic properties of these heterostructures. Using standard four probe technique the resistivity of W film was determined and the grazing incidence X-ray diffraction was used for investigating \nthe phase of W.  \n                                              III. RESULTS AND DISCUSSION \nFigure 1 (a) shows the grazing incidence X-ray diffraction (XRD) patterns for \nSub/W( t)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm). In these XRD plots, the peaks corresponding to α \nand β phase of W are marked. The high intensity XRD peak at ~40.5° corresponds primarily \nto the α phase (BCC structure) of W (~40.5°) (110) orientation. Interestingly, we find that the \npeak in the vicinity of 40.5° is present for all thicknesses of W but when the W thickness is less than 6 nm, then the peaks (~34.8° and ~42.1°) corresponding to β-W (A15 structure) with \n(200) and (211) crystal orientations appear [34, 37].  One may note that in close proximity of \n40°, β-W peak for (210) crystal orientation could also be present, which is quite difficult to \nidentify. We wish to clarify here that for W thickness below 5 nm the β-rich phase along with \nsmall amount of α phase exists while with increasing thickness of the W layer the fraction of \nα phase increases and starts to dominate for W thickness above 5 nm. For the sake of simple    \n6 \n notation we refer the phase below 5 nm W as β-phase and above this thickness as α-phase. \nThese findings are consistent with some of the existing literature reports, where it is described that W exhibits a transition from β phase (A15 structure) to α phase (BCC structure) with \nincreasing film thickness in the range of about 5 to 6 nm [32, 33]. It has also been shown in \nsome other studies that this transition thickness may be increased or decreased by carefully \ntuning the deposition conditions of the W thin films [34, 41]. \nIn Fig. 1 (b), the atomic force microsco pe images for all the heterostructures \ninvestigated in the present study are shown. Fr om these images we observe that the average \ntopographical roughness for the samples with t = 2, 3, 4, 5, 6 and 7nmis 0.21, 0.21, 0.16, 0.19, \n0.14 and 0.23 nm, respectively. The roughness values vary by about 10% when measured at \nvarious regions of space of the same sample. Overall, the topographical roughness in all film stacks is found to be significantly small irre spective of whether the W thickness corresponds \nto its β or α phase. Due to the small thicknesses of the thin film heterostructures, presumably, \nthe interfacial roughness will clearly show its imprint on the topographical roughness. We \nthus infer that the interfacial roughness, if any, present in these heterostructures is very small and is similar in all samples. To determine the variation of resistivity of W with its thickness across the two different phases, we performed four probe measurements on all the samples. The inverse of sheet resistance of the film stack as a function of W thickness is plotted in Fig. \n1(c). A change of the slope is observed beyo nd 5 nm, which indicates a change in the W \nresistivity. We estimate the average resistivity of W in β and α phase to be about 260 µ Ω-cm \nand 105 µ Ω-cm, respectively.  \n \n                          A. Principle behind the determination of spin Hall angle   \n         We next focus on the mechanism involved in the tuning of magnetization dynamics \nunder the influence of spin current. Figure 2 (a) shows the schematic of experimental arrangement in which the flow of charge current through the W layer and consequent spin    \n7 \n current generation due to SHE are shown. Under the influence of spin current, the CoFeB \nlayer experiences an anti-damping like SOT and the magnetization dynamics is governed by modified Landau-Lifshitz-Gilbert equation  [28] as given below: \n)ˆˆˆ(2)ˆˆ() ˆ(ˆ\n0m mJdMe dtmdm Hmdtmd\ns\nseff ×× + × + × −= σμα γh\n                        (1). \nHere, γ is the gyromagnetic ratio, σˆ is the spin polarization vector, mˆis the magnetization \nvector, Ms is saturation magnetization, Js is spin current density, Heff is the effective magnetic \nfield, d is the ferromagnetic layer thickness and α is the Gilbert damping constant [5, 42]. \nDepending on the polarity of σˆ, the spin torque [24] acts collinearly against or towards the \nintrinsic Gilbert damping of the precessing ma gnetization. Effective damping, in turn, gets \nmodulated depending on the injected spin current  density and relative orientation between the \nmagnetic moment (which lies along the direction of magnetic field) and charge current density [24]. The modulation of damping (MOD) under the influence of spin current [28, 43] \ncan be expressed as: \nfdeM Js s π γ αα α 2 2/ ) (0h= − =Δ                                                                (2), \nwhere α0 is the damping in the absence of applied charge current, e is electronic charge, f is \nthe precessional frequency and other symbols have  the same meaning as described before in \nthe text. Thus, the SHA (charge current to spin current conversion efficiency) is given by: \nθ γ απ θ sin / 2 2c s\ncs\nSH J fd eMJJh Δ = =                                                      (3), \nwhere, θ is the angle between Jc and bias magnetic field direction. Experimentally, we find \nthe value of ∆α/Jc and estimate the SHA using the above expression. \n                           B. All-optical investigation of magnetization dynamics \n         In Fig. 2(b), the as measured data using TRMOKE set-up is shown for the Sub/W(4 \nnm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample at H = 1.46 kOe without any applied charge \ncurrent. The femtosecond laser excites the sample thereby triggering the magnetization    \n8 \n dynamics. Overall, the dynamics can be divided into three different temporal regimes as \nshown in the plot. Regime I (sharp drop immediately after negative delay, ~700 fs) corresponds to ultrafast demagnetization and regime II corresponds to the fast relaxation (1.7 ps) due to the spin lattice re laxation. Subsequently, in regime III, we observe a slower \nrelaxation (~ 40 ps) along with magnetization precession, which gets damped in few ns. We \nmainly concentrate in regime III to estimate the damping and its modulation due to the action \nof spin torque. The blue line in Fig. 2 (b) corresponds to the bi-exponential background present in the preces sional data in regime III. We subtr act this background from the raw data \nand fit the resulting data using standard damped harmonic function. From the fit we estimate the damping α using the expression α = 1/2πfτ, where f is the precessional frequency and τ is \nthe relaxation time corresponding to magnetization oscillation.  \nWe further studied the bias field dependent magnetization dynamics and Fig. 3(a) \nshows a representative experimental data of precessional magnetization dynamics along with \nthe theoretical fit using a damped sinusoidal function for Sub/W(4 nm)/Co\n20Fe60B20(3 \nnm)/SiO 2(2 nm) sample. From the fitting, we extract the relaxation time ( τ) as 0.52, 0.59 and \n0.70 ns for bias field values of 1.46, 1.05 and 0.65 kOe, respectively. Figure 3 (b) shows the corresponding fast Fourier transform (FFT, power vs. frequency), from which the precessional mode frequency is extracted. The frequency ( f) versus bias magnetic field ( H) is \nplotted in Fig. 3 (c) for the same film stack. Standard Kittel expression mentioned below is \nused to fit the f vs. H data: \n                                                                                                                    (4), \nwhere γ = gµ\nB/ħ, g is the Lande g factor, H is the applied bias magnetic field and Meff is the \neffective magnetization. From the fit Meff and g are determined as fitting parameters. For these \nfilm stacks we obtain Meff ~ 1000 ± 30 emu/cc and g = 2.0 ± 0.5, except for the W thickness \nof 5 nm, where Meff is found to be lower (770 ± 24 emu/cc). In Fig. 3(d), the Meff obtained \nfrom the dynamic measurement is plotted as a function of W thickness, t. Interestingly, for all (() )2142effM HH f ππγ+ =   \n9 \n the film stacks investigated in this study, Meff is found to be close to the saturation \nmagnetization Ms obtained using vibrating sample magnetometer. From this we infer that \ninterface anisotropy is negligibly small in these heterostructures [31]. \n                             C. Spin current induced modulation of damping \n    Figure 4 shows some typical time-resolved Kerr rotation data after the application of dc \ncharge current through the film stack with diffe rent polarities and the MOD as a function of \nthe dc charge current density. The applied charge current through the heterostructure gets distributed into W and CoFeB layers according to the resistivity of each metallic layer. Here, \nJ\nc represents the current density through the W underlayer. Figure 4 (a) shows the \nmagnetization precession data at a bias field H = 1.46 kOe and for positive and negative Jc (θ \n= 90°) along with the fit using damped sine function to extract the damping. The magnitude and sign of J\nc and the corresponding extracted value of effective damping α are mentioned in \neach panel. A direct comparison of MOD with applied Jc is shown for three representative \nthickness of W in Fig. 4 (b) i.e., t = 3 nm, 4 nm (corresponding to β-rich phase) and 7 nm \n(corresponding to α phase). From these plots, it is evident that the spin current induced spin \ntorque generated by SHE almost linearly modulat es the damping consistent with the existing \nliterature [24, 28]. We use a linear fit and extract the slope of MOD ( Δα/Jc) in order to \nestimate the SHA using Eq. 3 for a given thickness of W. The values of the slope extracted \nfrom the fit for 2 nm, 3 nm, 4 nm, 5 nm, 6 nm and 7 nm thick W are (0.47 ± 0.04) × 10-12 \nm2/A, (1.29 ± 0.10) × 10-12 m2/A, (1.20 ± 0.06) × 10-12 m2/A, (1.02 ± 0.04) × 10-12 m2/A, (0.39 \n± 0.06) × 10-12 m2/A and (0.11 ± 0.01) × 10-12 m2/A, respectively. Note that the slope of MOD \ndue to the spin current generated by SHE is generally larger for the β phase W (3, 4, 5 nm) in \ncomparison to the α phase W (6, 7 nm). The sample with 2 nm W thickness is an exception \nand it will be discussed later in this article. The limited Jc value for the high resistive β phase \nW (in Fig. 4 (b), for t = 3 and 4 nm) is to avoid Joule heating in these films. Nevertheless,    \n10 \n damping variation in α up to ±15% is observed for a reasonably small current density of 0.3 × \n1010 A/ m2 for the sample with t = 3 nm. \n                         D. Tungsten layer thickness dependence of spin Hall angle \n Figure 5(a) shows the plot of S HA as a function of W thickness 2 nm ≤ t ≤ 7 nm in \nSub/W( t)/Co 20Fe60B20(3 nm)/SiO 2(2 nm). Interestingly, in this plot we notice that the SHA is \nquite small when the W layer thickness is 2 nm, subsequently, SHA increases to a large value for W layer thickness of 3 nm and 4 nm. For t>4 nm, the SHA decreases monotonically up to \nt = 7 nm. It is important to emphasize here that  we observe a giant value of SHA as large as \n0.4 ± 0.04 for t = 3 nm, which is about 30% larger than the value of SHA reported using \nelectrical detection technique (~0.3)  [29]. Within the β phase of W, the SHA decreases as the \nthickness of W becomes comparable to spin diffusion length of W ( λ\nsf) [41, 44].  The \nobserved dependence of SHA on W thickness from 2 nm to 4 nm can be explained by considering drift diffusion analysis of the spin fl ow that incorporates spin Hall effect. Earlier, \ntheoretical and experimental studies have propo sed that within the spin diffusion length, the \ncounter flowing spin current generated due to ve rtical gradient in the spin dependent electron \nchemical potential adjacent to the HM surface (under the assumption that no spin current penetrate out of HM) cancel the spin Hall generated spin current [20, 41, 43, 45, 46]. Due to this, the magnitude of the spin Hall spin current reduces significantly (resulting in smaller \nMOD and underestimation of SHA) as the thickn ess of the HM layer becomes comparable to \nλ\nsf.  From our experimental data we understand that the λsf of our W thin film is less than 3 nm \nas it is difficult to extract this parameter precisely using standard fit with few numbers of data points available in β-W phase. In order to understand whether the variation of SHA is directly \ncorrelated with the variation of resistivity, we plot the W resistivity with its thickness in Fig. 5 \n(b). The significant drop in the resistivity value for t > 5 nm indicates a transition from β-W to \nα-W phase in the W film used in our experiment. It is important to notice here that the \nvariation in SHA above spin diffusion length is primarily correlated with the thickness    \n11 \n dependent β to α phase transition (structural change which is also related to the resistivity \nchange c.f. Fig. 5(b)) of W [20, 29, 32, 34, 44]. Though from Fig. 5 (b), the resistivity of 5 nm \nthick W film is found to be primarily in β phase, but from the trend of SHA values, it appears \nthat there is probably a mixed β and α phase of W at this thickness. Hence, apart from W \nthickness of 2 and 5 nm, SHA is found to be in direct correspondence with the resistivity, \nwhich is similar to the variation of SHA with conductivity as observed for other HMs [40, 47].            In the phase transition regime, the change in crystal structure indicates a change in SO coupling strength of W which may play an important role in modifying the SHA [32]. There \nis a possibility that in the low resistive regime (for t > 5 nm), the SHA is influenced by the \nchange in SO coupling strength. As the phase transition in W with its thickness and the \nassociated resistivity change predominantly originates from the bulk portion of W, thus, it indicates that the bulk part of W plays dominant role in defining the SHA and thereby the spin current above the spin diffusion length of W. Th e bulk SHA in the NM layers is theoretically \npredicted and experimentally found to consist of  contributions from the intrinsic, the side-\njump, and the skew scattering mechanisms [4, 48, 49]. The estimated large value of SHA in β-\nW films indicates that in these heterostructures all the mechanisms efficiently contribute for the large SHA as well as the W/CoFeB interface is highly transparent. The trend found in the variation of SHA with W thickness ( β-W has larger SHA in comparison to α-W) is mostly \nconsistent with earlier reported results [29] while the  values of SHA obtained using all-optical \ndetection technique is a new addition in this field. The method employed here is non-invasive and more unambiguous as it helps to eliminat e any experimental artifacts involved in the \nelectrical detection schemes  [50]. Being a local technique, the all-optical method does not \nsuffer from the large area averaging, which could have produced spurious effects due to \ninhomogeneities and defects present in the sample. Further, in the time-domain measurement \ntechnique, magnetization damping can be directly extracted from time-resolved precession data, which is more advantageous than other te chniques such as FMR line width measurement,    \n12 \n where excitation of multiple modes may lead to inhomogeneous line broadening, which could \nartificially increase the damping.  \n \n                                                  IV. CONCLUSION \nIn summary, we have used all-optical time-resolved magneto-optical Kerr effect \nmicroscopy to investigate the ma gnetization dynamics in Sub/W( t)/Co 20Fe60B20(3 nm)/SiO 2(2 \nnm) with varying W layer thickness under the influence of spin current generated by SHE. The W layer thickness is so chosen  that it undergoes a transition from β-rich to α-rich phase at \na thickness above 5 nm. For highly resistive β phase W, large modulation of damping of upto \n±15% at a modest current density of 0.3 × 10\n10 A/m2 and corresponding SHA as large as 0.4 is \nachieved. The SHA above the spin diffusion length of W follows the thickness dependent phase transition of W. On the other hand, a smaller modulation of damping and underestimation of SHA is observed for W laye r thickness smaller than its spin diffusion \nlength. In order to realize the full value of spin  current due to SHE, it is thus important to use \nthe HM thickness above its spin diffusion lengt h. The variation of SHA with W thickness ( β-\nW has larger SHA in comparison to α-W) is mostly consistent with the change in resistivity. \nThough in some of the recent theoretical and experimental studies, direct correspondence of SHA with conductivity has been discussed, how ever, we believe our investigations will \ntrigger more studies to get further deep insight  into the relationship between conductivity and \nspin Hall angle specifically for materials with mixed phase. Our results of detailed variation \nof SHA for different values of W thickness, w ill be beneficial for in-depth understanding of \ncorrelation between the thickness dependent phase transition in W and SHA. Furthermore, these studies will be significantly important from the application perspective as the future \nspintronic devices are expected to use larg e SHA material and spin current induced \nmagnetization switching.    \n    \n13 \n  \nAcknowledgements: We gratefully acknowledge the support from S. 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Lett.  109, 042403 (2016). \n[42] L. Liu, T. Moriyama, D. C. Ralph a nd R. A. Buhrman, Spin-Torque Ferromagnetic \nResonance Induced by the Spin Hall Effect, Phys. Rev. Lett.  106, 036601 (2011). \n[43] L. Liu, R. A. Buhrman and D. C. Ralph, Review and Analysis of Measurements of the \nSpin Hall Effect in Platinum, arxiv 1111.3702 , (2011). \n[44] S. Cho, S. H. C. Baek, K. D. Lee, Y. Jo and B. G. Park, Large spin Hall magnetoresistance and its correlation to the spin-orbit torque in W/CoFeB/MgO structures, \nSci. Rep.  \n5, 14668 (2015). \n[45] P. C. Vanson, H. Vankempen and P. Wyde r, Boundary resistance of the ferromagnetic-\nnonferromagnetic metal interface, Phys. Rev. Lett.  58, 2271 (1987). \n[46] M. H. Nguyen, D. C. Ralph and R. A. Buhrman, Spin Torque Study of the Spin Hall \nConductivity and Spin Diffusion Length in Platinum Thin Films with Varying Resistivity, \nPhys. Rev. Lett.  116, 126601 (2016). \n[47] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani and F. Casanova, Tuning the spin Hall effect of Pt from the moderately dirty to the superclean \nregime, Phys. Rev. B  \n94, 060412 (2016). \n[48] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T. Jungwirth, Spin Hall \neffects, Rev. Mod. Phys.  87, 1213 (2015). \n[49] X. Zhou, M. Tang, X. L. Fan, X. P. Qiu and S. M. Zhou, Disentanglement of bulk and interfacial spin Hall effect in ferroma gnet/normal metal interface, Phys. Rev. B  \n94, 144427 \n(2016). \n[50] I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti and C. H. Back, \nComparison of frequency, field, and time domain ferromagnetic resonance methods, J. Mag. \nMagn. Mater.  307, 148 (2006). \n \n \n \n \n    \n \n     \n17 \n \nFigure 1. (a) X-ray diffraction patterns measured  at grazing angle incidence for W films with \nthickness of 3 nm, 4 nm, 5 nm and 6 nm. Peaks corresponding to β and α phase of W are \nmarked in the plots. (b)  Atomic force microscope images showing the surface topography of the Sub/W( t)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm) samples with t = 2 nm to 7 nm. (c) Variation of \ninverse of sheet resistance of Sub/W( t)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) as a function of W \nthickness ( t) measured using linear four probe technique.  \n  \n    \n18 \n  \n \n Figure 2. (a) Schematic of sample geometry and illustration of pump probe experimental \ngeometry. Co-ordinate convention as followed is also shown.  (b) Time-resolved Kerr rotation \ndata for Sub/W(4 nm)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm) sample at H = 1.46 kOe is shown. The \nthree different temporal regimes are indicated in the graph.   \n  \n \n   \n \n \n \n  \n \n \n   \n19 \n  \nFigure 3. (a) Time-resolved precessional  magnetization dynamics for Sub/W(4 \nnm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample at different bias  magnetic field values. (b) The \ncorresponding FFT power spectra to extract the precession frequency. (c) Plot of variation of \nfrequency as a function of bias magnetic field. The solid line is the fit with Kittel formula. (d) Saturation magnetization of the Sub/W( t)/Co\n20Fe60B20(3 nm)/SiO 2(2 nm) samples as a \nfunction of W layer thickness.   \n  \n   \n \n   \n   \n20 \n  \nFigure 4. (a) Representative TRMOKE traces for extraction of damping under the influence of \npositive and negative current densities. Here, W thickness is mentioned in the left panel. The \nestimated damping values at mentioned current densities (in A/m2) are also shown. \nComparison of left and right panel indicates that the damping value changes with the polarity of charge current. (b) Modulation of damp ing plot for W thickness of 3 and 4 nm \ncorresponding to the β phase and 7 nm corresponding to the α phase. Solid line is the linear fit \nto the modulation of damping with current density. Error bars correspond to the fitting error \nobtained during the estimation of damping.   \n  \n \n \n \n   \n   \n21 \n  \nFigure 5. (a) Variation of spin Hall angle with W thickness. Error bars are estimated by \nconsidering errors in damping, saturation ma gnetization and resistivity measurements. The \ncolor contrast shows the transition from β to α phase of W. (b) Variation of resistivity of W \nwith thickness ( t). \n"    },    {        "title": "1604.02998v1.All_Optical_Study_of_Tunable_Ultrafast_Spin_Dynamics_in__Co_Pd__NiFe_Systems__The_Role_of_Spin_Twist_Structure_on_Gilbert_Damping.pdf",        "content": "All-Optical Study of Tunable Ultrafast Spin Dynamics in [Co/Pd]-NiFe Systems: The\nRole of Spin-Twist Structure on Gilbert Damping\nChandrima Banerjee,1Semanti Pal,1Martina Ahlberg,2T. N.\nAnh Nguyen,3, 4Johan \u0017Akerman,2, 4and Anjan Barman1,\u0003\n1Department of Condensed Matter Physics and Material Sciences,\nS. N. Bose National Centre for Basic Sciences, Block JD, Sec. III, Salt Lake, Kolkata 700 098, India\n2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden\n3Laboratory of Magnetism and Superconductivity,\nInstitute of Materials Science, Vietnam Academy of Science and Technology,\n18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam.\n4Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden\n(Dated: April 12, 2016)\nWe investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time-\nresolved magneto-optical Kerr e\u000bect magnetometer. The competition between the out-of-plane\nanisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias \feld reor-\nganizes the spins in the soft layer, which are modi\fed further with the variation in t. The spin-wave\nspectrum, the ultrafast demagnetization time, and the extracted damping coe\u000ecient all depend on\nthe spin distribution in the soft layer, while the latter two also depend on the spin-orbit coupling\nbetween the Co and Pd layers. The spin-wave spectra change from multimode to single-mode as t\nincreases. At the maximum \feld reached in this study, H=2.5 kOe, the damping shows a nonmono-\ntonic dependence on twith a minimum at t= 7.5 nm. For t<7.5 nm, intrinsic e\u000bects dominate,\nwhereas for t>7.5 nm, extrinsic e\u000bects govern the damping mechanisms.\nI. INTRODUCTION\nNonuniform magnetic structures, including exchange\nbias (ferromagnet/antiferromagnet)3,24and exchange-\nspring (ferromagnet/ferromagnet)5{8systems, have\nrecently been explored extensively on account of their\nintrinsic advantages for applications in both permanent\nmagnets and recording media. Exchange-spring (ES)\nmagnets are systems of exchanged-coupled hard and soft\nmagnetic layers that behave as a single magnet. Here,\nthe high saturation magnetization ( Ms) of the soft phase\nand the high anisotropy ( Hk) of the hard phase result in\na large increase in the maximum energy product. This\nmakes them useful as permanent magnets in energy ap-\nplications such as engines or generators in miniaturized\ndevices. On the other hand, for spintronic applications,\nthe soft phase is used to improve the writability of\nthe magnetic media, which in turn is stabilized by the\nmagnetic con\fguration of the hard layer. Consequently,\na wealth of research has been devoted to investigating\nthe static and dynamic magnetic properties, including\nthe switching behavior and exchange coupling strength,\nin ES systems.\nIn case of ES systems with tilted anisotropy, the hard\nand soft phases consist of materials with out-of-plane\n(OOP) and in-plane (IP) anisotropies, respectively. This\ncombination results in a canting of the magnetization\nof the soft layer with a wide and tunable range of tilt\nangles. The advantage of such a hybrid anisotropy sys-\ntem is that it is neither plagued by the poor writability\nand thermal instability of systems with IP anisotropy,\nnor does it lead to very high switching \felds, as in OOPsystems. As a result, these materials provide additional\ndegrees of freedom to control the magnetization dynam-\nics in magnetic nanostructures, and hint at potential\napplications in novel spintronic devices utilizing the\nspin-transfer torque (STT) e\u000bect|such as spin-torque\noscillators (STOs)25,26and STT-MRAMs.\nSo far, numerous studies have been performed on\nsuch systems where the exchange coupling between\nthe hard and soft layers has been tailored by varying\nthe layer thickness,12,13layer composition,19number\nof repeats,15and interfacial anisotropy.13The litera-\nture describes investigations of domain structure and\nother static magnetic properties for [Co/Pd]/Co,14\n[Co/Pd]/NiFe,12,14,19,21[Co/Pd]/CoFeB,14,15,20\n[Co/Pd]-Co-Pd-NiFe,13[Co/Ni]/NiFe,4and CoCrPt-\nNi11|these systems being studied with static mag-\nnetometry, magnetic force microscopy (MFM), and\nmicromagnetic simulations. The magnetization dy-\nnamics in such systems have also been measured using\nBrillouin light scattering (BLS)19,20and ferromagnetic\nresonance (FMR)21experiments, where the spin-wave\n(SW) modes have been investigated by varying the thick-\nness of the soft layer and changing the con\fguration of\nthe hard layer. In any process involving magnetization\ndynamics, the Gilbert damping constant ( \u000b) plays a key\nrole in optimizing writing speeds and controlling power\nconsumption. For example, in case of STT-MRAM\nand magnonic devices, low \u000bfacilitates a lower writing\ncurrent and the longer propagation of SWs, whereas a\nhigher\u000bis desirable for increasing the reversal rates and\nthe coherent reversal of magnetic elements, which are\nrequired for data storage devices.arXiv:1604.02998v1  [cond-mat.mtrl-sci]  11 Apr 20162\n46810121416350400450500\n  )sf( emit noitazitengameDt (nm)(d)(a) \n-202 600 1200 1800-2-10\n  Kerr rotation (a rb. unit)\nTime (ps)(b)0 10 20 30\n  Power (arb. unit)\nFrequency (GHz)(c)\n(b)\n-202 60012001800-2-10Kerrrotation(arb.unit)\nTime(ps)\nFigure 1. (color online) (a) Schematic of the two-color pump-\nprobe measurement of the time-resolved magnetization dy-\nnamics of exchange-spring systems. The bias \feld is applied\nwith a small angle to the normal of the sample plane. (b)\nTypical time-resolved Kerr rotation data revealing ultrafast\ndemagnetization, fast and slow relaxations, and precession\nof magnetization for the exchange-spring system with t=\n7.5 nm at H= 2.5 kOe. (c) FFT spectrum of the background-\nsubtracted time-resolved Kerr rotation. (d) Variation of de-\nmagnetization time with t.\nIn this paper, we present all-optical excitation and de-\ntection of magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) tilted anisotropy ES systems, with varying\nsoft layer thickness ( t), using a time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer. The dy-\nnamical magnetic behavior of similar systems has previ-\nously been studied using BLS19and FMR21measure-\nments. However, a detailed study of the precessional\nmagnetization dynamics and relaxation processes in such\ncomposite hard/soft systems is yet to be carried out.\nThe advantage of implementing TR-MOKE is that here\nthe magnetization dynamics can be measured on di\u000ber-\nent time scales and the damping is measured directly\nin the time domain, and is therefore more reliable. We\ninvestigate the ultrafast magnetization dynamics over pi-\ncosecond and picosecond time scales. The ultrafast de-\nmagnetization is examined and found to change due to\nthe modi\fed spin structure in the soft layer for di\u000berent\ntvalues. The extracted SW spectra are strongly depen-\ndent on t. An extensive study of the damping coe\u000ecient\nreveals that the extrinsic contribution to the damping\nis more dominant in the higher thickness regime, while\nintrinsic mechanisms govern the behavior at lower thick-\nnesses.II. EXPERIMENTAL DETAILS\nA. Sample fabrication\nThe samples were fabricated using dc mag-\nnetron sputtering and have the following structure:\nTa(5nm)/Pd(3nm)/[Co(0.5nm)/Pd(1nm)] \u00025=Ni80Fe20(t)\n/Ta(5nm), where t= 4{20 nm. The chamber base pres-\nsure was below 3 \u000210\u00008Torr, while the Ar work\npressure was 2 and 5 mTorr for the Ta, NiFe and Co,\nPd layers, respectively. The samples were deposited\nat room temperature on naturally oxidized Si(100)\nsubstrates. The 5 nm Ta seed layer was used to induce\nfcc-(111) orientation in the Pd layer, which improves\nthe perpendicular magnetic anisotropy of the Co/Pd\nmultilayers; a Ta cap layer was used to avoid oxidation,\nwhich has been reported in previous studies.12{14The\nlayer thicknesses are determined from the deposition\ntime and calibrated deposition rates.\nB. Measurement technique\nTo investigate the precessional frequency and damp-\ning of these samples, the magnetization dynamics were\nmeasured by using an all-optical time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer2based on\na two-color optical pump-probe experiment. The mea-\nsurement geometry is shown in Fig. 1(a). The magne-\ntization dynamics were excited by laser pulses of wave-\nlength (\u0015) 400 nm (pulse width = 100 fs, repetition rate\n= 80 MHz) of about 16 mJ/cm2\ruence and probed by\nlaser pulses with \u0015= 800 nm (pulse width = 88 fs, rep-\netition rate = 80 MHz) of about 2 mJ/cm2\ruence. The\npump and probe beams are focused using the same micro-\nscope objective with N.A. of 0.65 in a collinear geometry.\nThe probe beam is tightly focused to a spot of about\n800 nm on the sample surface and, as a result, the pump\nbecomes slightly defocused in the same plane to a spot\nof about 1 \u0016m. The probe beam is carefully aligned at\nthe centre of the pump beam with slightly larger spot\nsize. Hence, the dynamic response is probed from a ho-\nmogeneously excited volume. The bias \feld was tilted\nat around 15\u000eto the sample normal (and its projection\nalong the sample normal is referred to as Hin this ar-\nticle) in order to have a \fnite demagnetizing \feld along\nthe direction of the pump beam. This \feld is eventually\nmodi\fed by the pump pulse which induces precessional\nmagnetization dynamics in the samples. The Kerr rota-\ntion of the probe beam, back-re\rected from the sample\nsurface, is measured by an optical bridge detector us-\ning phase sensitive detection techniques, as a function of\nthe time-delay between the pump and probe beams. Fig-\nure 1(b) presents typical time-resolved Kerr rotation data\nfrom the ES sample with t= 7.5 nm at a bias \feld H=\n2.5 kOe. The data shows a fast demagnetization within\n500 fs and a fast remagnetization within 8 ps, followed by\na slow remagnetization within 1800 ps. The precessional3\n(b) \n010 20 30 0 1 2\nPower (arb. unit)  Kerr Rotation(arb. unit)\n  \n  \n    \n  \n    \n  \nFrequency (GHz)4.5 nm\n5.5 nm\n7.5 nm\n8 nm\n15 nm\n  \nTime (ns)20 nm \n  \n  B\nA \n   \nNiFe  (t = 20 nm)  \nCo/Pd \n1 -1 Normalized Mz Co/Pd NiFe  (t = 6 nm)  \nCo/Pd NiFe ( t = 10 nm) (a) \nFigure 2. (color online) (a) Background-subtracted time-\nresolved Kerr rotation and the corresponding FFT spectra\nfor samples with di\u000berent tvalues at H= 2.5 kOe. The\nblack lines show the \ft according to Eq. 1. (b) Simulated\nstatic magnetic con\fgurations for samples with t= 20, 10,\nand 6 nm with a bias \feld H= 2.5 kOe in the experimental\ncon\fguration. The simulated samples are not to scale. The\ncolor map is shown at the bottom of the \fgure.\ndynamics appear as an oscillatory signal above the slowly\ndecaying part of the time-resolved Kerr rotation data.\nThis part was further analyzed and a fast Fourier trans-\nform (FFT) was performed to extract the corresponding\nSW modes, as presented in Fig. 1(c).III. RESULTS AND DISCUSSIONS\nIn order to closely observe the ultrafast demagnetiza-\ntion and fast remagnetization, we recorded the transient\nMOKE signals for delay times up to 30 ps at a resolution\nof 50 fs. In Fig. 1(d), the demagnetization times are plot-\nted as a function of t. We observe that the demagnetiza-\ntion is fastest in the thinnest NiFe layer ( t= 4 nm) and\nincreases sharply with the increase in t, becoming con-\nstant at 500 fs at t= 5 nm. At t= 10 nm, it decreases\ndrastically to 400 fs and remains constant for further in-\ncreases in t. For t<5 nm, the laser beam penetrates\nto the Co/Pd layer. In this regime, the large spin-orbit\ncoupling of Pd enhances the spin-\rip rate, resulting in a\nfaster demagnetization process. As tincreases, the top\nNiFe layer is primarily probed. Here, the spin con\fgura-\ntion across the NiFe layer, which is further a\u000bected by the\ncompetition between the in-plane and the out-of-plane\nanisotropies of the NiFe and [Co/Pd] layers, governs the\ndemagnetization process. Qualitatively, ultrafast demag-\nnetization can be understood by direct transfer of spin\nangular momentum between neighboring domains10,23.\nwhich may be explained as follows: For t>8 nm, the\nmagnetization orientation in the NiFe layer varies over a\nwide range of angles across the \flm thickness, where the\nmagnetization gradually rotates from nearly perpendicu-\nlar at the Co/Pd and NiFe interface to nearly parallel to\nthe surface plane in the topmost NiFe layer. Such a spin\nstructure across the NiFe layer thickness can be seen as a\nnetwork of several magnetic sublayers, where the spin ori-\nentation in each sublayer deviates from that of the neigh-\nboring sublayer. This canted spin structure accelerates\nthe spin-\rip scattering between the neighboring sublay-\ners and thus results in a shorter demagnetization time,\nsimilar to the work reported by Vodungbo et al.23On the\nother hand, for 5 nm <t<8 nm, the strong out-of-plane\nanisotropy of the Co/Pd layer forces the magnetization\nin the NiFe \flm towards the direction perpendicular to\nthe surface plane, giving rise to a uniform spin structure.\nThe strong coupling reduces the transfer rate of spin an-\ngular momentum and causes the demagnetization time\nto increase.\nTo investigate the variation of the precessional dynamics\nwith t, we further recorded the time-resolved data for a\nmaximum duration of 2 ns at a resolution of 10 ps. Fig-\nure 2(a) shows the background-subtracted time-resolved\nKerr rotation data for di\u000berent values of tatH=\n2.5 kOe and the corresponding fast Fourier transform\n(FFT) power spectra. Four distinct peaks are observed\nin the power spectrum of t= 20 nm, which reduces to\ntwo for t= 15 nm. This is probably due to the rela-\ntive decrease in the nonuniformity of the magnetization\nacross the NiFe thickness, which agrees with the varia-\ntion in the demagnetization time, as described earlier. To\ncon\frm this, we simulated the static magnetic con\fgura-\ntions of these samples in a \feld of H= 2:5 kOe using the\nLLG micromagnetic simulator.18Simulations were per-\nformed by discretizing the samples in arrays of cuboidal4\ncells with two-dimensional periodic boundary conditions\napplied within the sample plane. The simulations assume\nthe Co/Pd multilayer as an e\u000bective medium16with sat-\nuration magnetization Ms= 690 emu/cc, exchange sti\u000b-\nness constant A= 1.3\u0016erg/cm, and anisotropy constant\nKu1= 5.8 Merg/cc along the (001) direction, while the\nmaterial parameters used for the NiFe layer were Ms=\n800 emu/cc, A= 1.3\u0016erg/cm, and Ku1= 0.17The in-\nterlayer exchange between Co/Pd and NiFe is set to 1.3\n\u0016erg/cm and the gyromagnetic ratio \r= 18.1 MHz Oe\u00001\nis used for both layers. The Co/Pd layer was discretized\ninto cells of dimension 5 \u00025\u00022 nm3and the NiFe layer\nwas discretized into cells of dimension 5 \u00025\u00021 nm3.\nThe results are presented in Fig. 2(b) for t= 20, 10, and\n6 nm samples. The nonuniform spin structure is promi-\nnent in the NiFe layer of the t= 20 nm sample, which\nmodi\fes the SW spectrum of this sample, giving rise to\nthe new modes.17With the reduction of t, the spin struc-\nture in the NiFe layer gradually becomes more uniform,\nwhile at t= 7.5 nm it is completely uniform over the\nwhole thickness pro\fle. Hence, for low values of t, the\npower spectra shows a single peak due to the collective\nprecession of the whole stack. The variation in precession\nfrequency with tis plotted in Fig. 3(a). The frequency\nof the most intense mode shows a slow decrease down to\nt= 7.5 nm, below which it increases sharply down to the\nlowest thickness t= 4.5 nm, exhibiting precessional dy-\nnamics. This mode is basically the uniform mode of the\nsystem and follows Kittel's equation.9The variation in\nfrequency depicts the evolution of the e\u000bective anisotropy\nfrom OOP to IP with increasing t, which is in agreement\nwith previously reported results.21For lower t, the sys-\ntem displays an OOP easy axis, owing to the strong OOP\nanisotropy of the Co/Pd multilayer. This is manifested\nas a sharp increase in the frequency with decreasing t\nbelow 7.5 nm. For greater thicknesses, the e\u000bect of the\nperpendicular anisotropy of the Co/Pd multilayer gradu-\nally decreases and the e\u000bect of the in-plane NiFe becomes\nmore prominent. These two anisotropies cancel near t=\n7.5 nm, resulting in a minimum frequency, as shown in\nFig. 3(a).\nTo extract the damping coe\u000ecient, the time domain\ndata was \ftted with an exponentially damped harmonic\nfunction given by Eq. 1.\nM(t) =M(0)e\u0000t\n\u001csin(2\u0019ft\u0000\u001e) (1)\nwhere the relaxation time \u001cis related to the Gilbert\ndamping coe\u000ecient \u000bby the relation \u001c= 1/(2\u0019f\u000b).\nHere, fis the experimentally obtained precession fre-\nquency and \u001eis the initial phase of the oscillation. The\n\ftted data for various values of tis shown by the solid\nblack lines in Fig. 2. We did not extract a value of \u000bfort\n= 15 and 20 nm due to the occurrence of multimode os-\ncillations, which may lead to an erroneous estimate of the\ndamping. The extracted \u000bvalues are plotted against tin\nFig. 3(b) for two di\u000berent \feld values of 2.5 and 1.3 kOe.\nThe evolution of \u000bas a function of tdepends signi\fcantly\n5 6 7 8 9 100.0160.0200.0240.0280.032\n5 10 15 2041216\nt(nm) t(nm)(a) (b)Frequency(GHz)Figure 3. (color online) Evolution of (a) spin-wave frequency\nand (b) Gilbert damping constant as a function of tat 1.3 kOe\n(green circles) and 2.5 kOe (violet circles).\nonH, as can be seen from the \fgure. This is because of\nthe di\u000berent mechanisms responsible for determining the\ndamping in di\u000berent samples, as will be discussed later.\nAn interesting trend in the \u000bvs.tplot is observed\nforH= 2.5 kOe. For 10 nm \u0014t\u00147.5 nm,\u000bdecreases\nwith decreasing tand reaches a minimum value of about\n0.014 for t= 7.5 nm. Below this thickness, \u000bincreases\nmonotonically and reaches a value of about 0.022 for the\nlowest thickness. This variation of \u000bis somewhat cor-\nrelated with the variation of precession frequency with\nthickness. In the thinner regime, we probe both the\nNiFe layer and a fraction of the Co/Pd multilayer and\nthe relative contribution from the latter increases as t\ndecreases. The occurrence of a single mode oscillation\npoints towards a collective precession of the stack, which\nmay be considered a medium with e\u000bective magnetic pa-\nrameters consisting of both NiFe and Co/Pd layers. The\nvariation in damping may be related to the variation in\nthe anisotropy of the material. The competing IP and\nOOP anisotropies of the NiFe and Co/Pd layers lead\nto the appearance of a minimum in the damping. The\ndamping in this system may have multiple contributions,\nnamely (a) dephasing of the uniform mode in the spin-\ntwist structure1(b) interfacial d-dhybridization at the\nCo/Pd interface16, and (c) spin pumping into the Pd\nlayer.22The \frst is an extrinsic mechanism and is dom-\ninant in samples with higher NiFe thicknesses, while the\nother two mechanisms are intrinsic damping mechanisms.\nFort>7.5 nm, due to the nonuniformity of the spin\ndistribution, the dominant mode undergoes dynamic de-\nphasing and the damping thus increasescompared to the\nmagnetically uniform samples. With the increase in NiFe\nthickness, the nonuniformity of spin distribution and the\nconsequent mode dephasing across its thickness increases,\nleading to an increase in the damping value. Hence, in\nsamples with higher tvalues, dephasing is the dominant\nmechanism, while at lower tvalues|i.e., when the con-\ntribution from the Co/Pd multilayer is dominant|the\nspin-orbit coupling and spin pumping e\u000bects dominate.\nAt intermediate tvalues, the extrinsic and intrinsic ef-\nfects compete with each other, leading to a minimum\nin the damping. However, the damping increases mono-\ntonically with tin a lower \feld of H=1.3 kOe. For a\ndeeper understanding of this e\u000bect, we have measured \u000b5\n24681012140.0120.0160.0200.0240.0280.0324\n56789100.0140.0210.0280.0350.042(b) \n  \n5nm \n5.5nm \n6.5nm \n7nm/s61537F\nrequency (GHz)(a) \n  \n10nm \n8.5nm \n8nm \n7.5nm \n7nm/s61537F\nrequency (GHz)\nFigure 4. (color online) Dependence of Gilbert damping co-\ne\u000ecient on soft layer thickness ( t) for (a) 7{10 nm and (b)\n5{7 nm, respectively.\nas a function of precession frequency f. Figures 4(a){(b)\nshow the variation of \u000bwith f. Two di\u000berent regimes in\nthe thickness are presented in (a) and (b) to show the\nrate of variation more clearly. For 10 nm \u0014t\u00147 nm,\u000b\ndecreases strongly with the decrease in fand the rate of\nvariation remains nearly constant with t. This is the sig-\nnature of extrinsic damping generated by the nonuniform\nspin distribution. However, for t= 6.5 nm, the rate falls\ndrastically and for t\u00145.5 nm,\u000bbecomes nearly indepen-\ndent of t, which indicates that purely intrinsic damping is\noperating in this regime. This con\frms the competition\nbetween two di\u000berent types of damping mechanisms in\nthese samples.\nThe study demonstrates that various aspects of ul-\ntrafast magnetization dynamics|namely demagnetiza-\ntion time, precession frequency, number of modes, and\ndamping|are in\ruenced by the spin distribution in the\nsoft magnetic layer, as well as by the properties of the\nhard layer. By changing the thickness of the soft layer,\nthe relative contributions of these factors can be tuned\ne\u000bectively. This enables e\u000ecient control of the damp-\ning and other magnetic properties over a broad range,\nand will hence be very useful for potential applications\nin spintronic and magnonic devices.IV. CONCLUSION\nIn summary, we have employed the time-resolved\nMOKE technique to measure the evolution of ul-\ntrafast magnetization dynamics in exchange-coupled\n[Co/Pd] 5/NiFe( t) multilayers, with varying NiFe layer\nthicknesses, by applying an out-of-plane bias magnetic\n\feld. The coupling of a high-anisotropy multilayer with\na soft layer allows broad control over the spin struc-\nture, and consequently other dynamic magnetic prop-\nerties which are strongly dependent on t. The ultra-\nfast demagnetization displayed a strong variation with\nt. The reason for this was ascribed to the chiral-spin-\nstructure-dependent spin-\rip scattering in the top NiFe\nlayer, as well as to interfacial 3 d-4dhybridization of\nCo/Pd layer. The precessional dynamics showed mul-\ntiple spin-wave modes for t= 20 nm and 15 nm, whereas\na single spin-wave mode is observed for thinner NiFe lay-\ners following the change in the magnetization pro\fle with\ndecreasing t. The precession frequency and the damp-\ning show strong variation with the thickness of the NiFe\nlayer. The changes in frequency are understood in terms\nof the modi\fcation of the anisotropy of the system, while\nthe variation in damping originates from the competition\nbetween intrinsic and extrinsic mechanisms, which are\nsomewhat related to the anisotropy. The observed dy-\nnamics will be important for understanding the utiliza-\ntion of tilted anisotropy materials in devices such as spin-\ntransfer torque MRAM and spin-torque nano-oscillators.\nV. ACKNOWLEDGEMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Coun-\ncil (VR), the Knut and Alice Wallenberg Foundation\n(KAW), and the Swedish Foundation for Strategic Re-\nsearch (SSF). This work was also supported by the Euro-\npean Research Council (ERC) under the European Com-\nmunity's Seventh Framework Programme (FP/2007{\n2013)/ERC Grant 307144 \"MUSTANG\". AB acknowl-\nedges the \fnancial support from the Department of Sci-\nence and Technology, Government of India (Grant no.\nSR/NM/NS-09/2011(G)) and S. N. Bose National Centre\nfor Basic Sciences, India (Grant no. SNB/AB/12-13/96).\nC.B. thanks CSIR for the senior research fellowship.\n\u0003abarman@bose.res.in\n1A. Barman and S. Barman. Dynamic dephasing of mag-\nnetization precession in arrays of thin magnetic elements.\nPhys. Rev. B , 79:144415, 2009.\n2A. Barman and A. Haldar. 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Phys. , 105:07D116, 2009."    },    {        "title": "1911.00728v1.Tuning_Non_Gilbert_type_damping_in_FeGa_films_on_MgO_001__via_oblique_deposition.pdf",        "content": "Tuning Non -Gilbert -type damping in  FeGa film s on MgO(001)  via oblique \ndeposition  \nYang Li1,2, Yan Li1,2, Qian Liu3, Zhe Yuan3, Qing -Feng Zhan4, Wei He1, Hao-Liang \nLiu1, Ke Xia3, We i Yu1, Xiang-Qun Zhang1, Zhao -Hua Cheng1,2,5 a) \n1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina  \n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China  \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China  \n4State Key Laboratory of Precision Spectroscopy, School of Physics and Materials \nScience, East Ch ina Normal University, Shanghai 200241, China  \n5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China   \na) Corresponding author , e-mail: zhcheng@iphy.ac.cn   \nAbstract  \nThe ability to tailor the damping factor is essential for spintronic and spin- torque \napplication s. Here, we report an approach to manipulate the damping factor of \nFeGa/MgO(001) films by oblique deposition. Owing to  the defects at the surface or \ninterface in thin films , two -magnon scatterin g (TMS)  acts as a non -Gilbert damping \nmechanism in magnetization relaxation.  In this work, the contribution of TMS was \ncharacterized  by in-plane angul ar dependent  ferromagnetic resonance (FMR) . It is \ndemonstrated that the intrinsic Gilbert damping is isotropic and invariant , while the \nextrinsic mechanism  related to TMS is anisotropic  and can be tuned by oblique \ndeposition. Furthermore, the two and fourfold TMS related to  the uniaxial magnetic \nanisotropy  (UMA)  and m agnetocrystalline anisotropy were discussed. Our result s open \nan avenue to manipulate  magnetization relaxation in spintronic devices.  \n1 \n Keywords : Gilbert damping , two -magnon scattering, FMR, oblique deposition, \nmagnetic anisotropy  \n2 \n 1. Introduction  \nIn the past decades, controlling magnetization dynamics in magnetic \nnanostructures has been extensively studied due to its great importance for spintronic \nand spin- torque applications  [1,2] . The magnetic relaxation is described within the \nframework of the Landa u-Lifshitz  Gilbert (LLG)  phenomenology  using the Gilbert \ndamping factor  α [3]. The intrinsic  Gilbert damping depends primarily on the spin- orbit \ncoupling (SOC) [4,5] . It has been demonstrated that alloying or doping with non-\nmagnetic transition metals  provides an opportunity to tune the intrinsic damping  [6,7] . \nUnfortunately, in this way the soft magnetic properties will reduce . In addition to the \nintrinsic damping, the two-magnon scatterin g (TMS) process se rves as a n important  \nextrinsic  mechanism i n magnetization relaxation  in ultrathin films  due to the defects at \nsurface or interface [8,9] . This process  describes the scattering between the uniform \nmagnons and degener ate final -state spin wave modes  [10]. The existence of TMS has \nbeen demonstrated in many systems of ferrites  [11-13]. Since the anisotropic scattering \ncenters , the angular dependence of the extrinsic TMS process  exhibi ts a strong in -plane \nanisotropy  [14], which allows  us to adjust the overall magnetic relaxation , including \nboth the int ensity of relaxation rate and  the anisotropic behavior.  \nHere, we report an approach to engineer  the damping factor of Fe81Ga19 (FeGa ) \nfilms by oblique deposition. The FeGa alloy exhibits large  magnetostriction and narrow \nmicrowave resonance linewidth  [15] , which  could assure  it as a  promising material for \nspintronic devices. For the geometry of off -normal deposition, it has been demonstrated \nto provoke shadow effects and create a periodic stripe defect matrix. This can introduce \na strong uniaxial magnetization anisotropy (UMA) pe rpendicular to the projection of \nthe atom flux  [16-19]. Even though some reports have shown oblique deposition \nprovokes a  twofold TMS channel  [20-22], the oblique angle dependence of the intrinsic \n3 \n Gilbert damping and the TMS  still remain in doubt. For our case, on the basis of the \nfirst-principles calculation and the in -plane angular -dependent FMR measurements, we \nfound that the intrinsic Gilbert damping is isotropic and invariant with varying oblique \ndeposition angles, while the extrinsic mechanism related to the two -magnon -scattering \n(TMS ) is anisotropic and can be tuned by oblique deposition. In addition, importantly \nwe firstly observe a phenomenon that the cubic magnetocrystalline anisotropy \ndetermines the area including degenerate magnon modes, as well as the intensity of fourfold TMS. In general , the strong connection between the extrinsic TMS and the \nmagnetic anisotropy , as well their direct impact on the damping constants , are \nsystem ically investigated, which offer us a useful approach to tailor the damping factor.\n \n2. Experimental details  \nFeGa thin films with a thickness of 20 nm were grown on MgO(001) substrates in \na magnetron sputtering system with a base pressure below 3 × 10−7 Torr. Prior to \ndeposition, t he substrates were annealed at 700 °C for 1 h in a vacuum chamber to \nremove surface contaminations and then held at 250 °C during deposition. The incident \nFeGa beam was at different obl ique angles of ψ =0°, 15°, 30°, and 45°, with respect to \nthe surface normal , and named S1, S2, S3, and S4 in this paper , respectively.  The \nprojection of FeGa beam  on the plane of the substrates  was set perpendicular to the \nMgO[110] direction, which induces a UMA perpendicular to the projection of FeGa \nbeam , i.e., parallel to the MgO[110] direction, due to the we ll-known  self-shadowing \neffect.  Finally , all the samples  were covered with a 5 nm  Ta capping layer to avoid \nsurface oxidation [see figure  1(a)]. The epitaxial relation  of \nFeGa(001)[ 110]||MgO (001)[ 100] was characterized by using t he X -ray in -plane Φ-\nscans , as described elsewhere [23]. Magnetic hy steresis loops were measured  at various \nin-plane magnetic field orientations φ H with respect to the FeGa [100] axis using \n4 \n magneto -optical Kerr effect (MOKE) technique at room temperature . The d ynamic \nmagnetic properties were investigated by broadband FMR measurements  based on a \nbroadband vector network analyzer  (VNA) with a  transmission geometry coplanar \nwaveguide (VNA- FMR ) [24]. This setup allows both frequency and field- sweeps \nmeasurements with external field applied parallel to the sample plane.  During \nmeasurements,  the sampl es were placed face down on the coplanar wavegu ide and the \ntransmission coefficient S 21 was recorded.  \n3. Results and discussion \nFigure  1(b) displays the Kerr  hysteresis loops  of sample  S1 and S4 recorded along \nwith the main crystallographic directions of FeGa [100], [110],  and [010] . The sample \nS1 exhibit s rectangular hysteresis curves with sm all coercivities for the magnetic  field \nalong  [100] and [010]  easy axes. In contrast, the S4 displays a  hysteresis curve with \ntwo step s for the magnetic field along the [010]  axis, which indicates  a UMA along the \nFeGa[100 ] axis superimposed on the four fold magnetocrystalline anisotropy . As a \nresult,  with increasing the oblique angle, the angular dependence of normalized remnant \nmagnetization ( Mr/Ms) gradually reveals a four fold symmetry combined with a uniaxial \nsymmetry, as shown i n the inset of f igure 1(b).  \nSubsequently,  the magnetic anisotropic properties can be further precisely \ncharacterized  by the in -plane angular -dependent FMR measurements.  Figure 1(c) and \n1(d) show  typical  FMR spectra for the real and imaginary part s of coefficient S 21 for \nthe sample S2 . Recorded  FMR spectra contain a symmetric and an antisymmetric \nLorentzian peak , from which the resonant field H r with linewidth ∆𝐻𝐻 can be obtained  \n[24,25] . \nFigure 2(a) shows the in -plane angular dependence of H r measured at 13 .0 GHz  \nand can be fitted  by the following expression [26,27] : \n5 \n 𝑓𝑓=𝛾𝛾𝜇𝜇0\n2𝜋𝜋�𝐻𝐻𝑎𝑎𝐻𝐻𝑏𝑏                                                     (1 ) \nHere,𝐻𝐻𝑎𝑎=𝐻𝐻4(3+𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M)/4+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H)+𝑀𝑀eff and 𝐻𝐻𝑏𝑏=\n𝐻𝐻4𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H), H4 and Hu represent the fourfold \nanisotropy field and the UMA field caused by the self -shadowing effect , respectively. \n𝜑𝜑H(𝜑𝜑M) is the  azimuthal angles of the applied field ( the tipped magnetization ) with \nrespect to the [100] direction , as depicted  in figure 1(a). 𝜇𝜇0𝑀𝑀eff=𝜇𝜇0𝑀𝑀𝑠𝑠−2𝐾𝐾out\n𝑀𝑀𝑠𝑠, Ms is \nthe saturation magnetization  and Kout is the out -of-plane uniaxial anisotropy constant . \n𝑓𝑓 is the resonance frequency , 𝛾𝛾 is the gyromagnetic ratio and here used as the accepted \nvalue for Fe films, 𝛾𝛾=185 rad GHz/T  [28] . \nThe angular dependent  Hr reveals  only a  fourfold symmetry for the none -\nobliquely deposited sample , which indicates  the cubic lattice texture of FeGa on MgO . \nWith increasing the oblique angle , a uniaxial symmetry is found  to be superimposed \non the four fold symmetry, clearly confirming a UMA is produced by the oblique \ngrowth , which agrees with the MOKE’ results.  The fitted  parameter  𝜇𝜇0𝑀𝑀eff=1.90±\n 0.05T is found to be independent on the oblique deposition and close to 𝜇𝜇0𝑀𝑀𝑠𝑠=\n1.89 ± 0.02 T  estimated using VSM, which is almost same as the value  of the  \nliterature  [29] . This indicates negligible out- of-plane ma gnetic anisotropy in the thick \nFeGa films . As shown in figure 2(b), it is observed  that the UMA (Ku=HuMs/2) exhibits \na general increasing  trend with oblique angle , which coincides with the fact  the \nshadowing effect is stronger  at larger  angles of incidence [16-19]. Interestingly the \noblique deposition also affects the cubic anisotropy  K4 (K4=H4Ms/2). Different from  \nthe K4 increases slightly with deposition angle  in Co/Cu system  [16], here the value of \nK4 is the lowest at a n oblique angle of 15°. It is well known th at film stress significantly \ninfluences  the crystallization tendency  [30,31] . FeGa alloy is highly stress ed sensitive \n6 \n due to its larger magnetostriction . Thus, t he change in  K4 of FeGa films  may be \nattributed to the anisotropy dispersion created due to the stress variations  during grain \ngrowth. It should be mentioned that the best way to determine magnetic parameters is \nto measure the out -of-plane  FMR . But the effective saturation magnetization 𝜇𝜇0𝑀𝑀eff=\n1.90T of FeGa alloy leads to the perpendicular applied field beyond our instrument \nlimit. Meanwhile, t he results obtained above are also in accord with those extracted \nby fitting field dependence of the resonance frequency with H//FeGa[100]  shown in \nfigure 2(c).  \nThe effective Gilbert damping 𝛼𝛼eff is extracted by linearly fitting the dependence  \nof linewidth  on frequency : 𝜇𝜇0∆H=𝜇𝜇0∆H0+2𝜋𝜋𝜋𝜋𝛼𝛼𝑒𝑒ff\n𝛾𝛾, where ∆𝐻𝐻0 is the inhomogeneous \nbroadening. For the sake of clarity , figure 3(a) only shows the frequency  dependence \nof linewidth for the  samples S1 and S2 along  [110]  and [100]  axes. It is evident that, \nfor the sample  S1, both linear slopes of two direction s are almost  same. While w ith \nregard to the sample S2 , the slope of the ∆H-f curve along  the easy axis is approximately \na factor of 2 greater than that of the hard axis. The obtained values of  𝛼𝛼eff are shown in \nfigure 3(b). Firstly, the results clearly indicate that the effective damping exhibits \nanisotropy , with  higher value along the easy axis . Secondly, f or the easy  axis, the \noblique angle dependence on the damping parameter indicates an extraordinary trend \nand has a peak at  deposition angle 15°. However, the damping shows an increasing \ntrend  with the oblique angle for the field along the hard axis. In the following part, we \nwill explore the effect of oblique deposition on the mechanism of the anisotropic \ndamping and the magnetic relaxation pr ocess.     \nSo far, convincing experimental evidence is still lacking to prove the existence of \nanisotropic damping in bulk magnets. Chen et al. have shown the emergence of \nanisotropic Gilbert damping in ultrathin Fe (1.3nm)/GaAs and its anisotropy disappears \n7 \n rapidly when the Fe thickness  increases  [32]. We perform  the first-principles \ncalculation of the Gilber t damping of Fe Ga alloy considering  the effect induced by the \nlattice distortion. W e artificially make a tetragonal lattice with varying the lattice \nconstant of the c -axis. The electronic structure of Fe -Ga alloy is calculated self -\nconsistently using the coherent potential approx imation implemented with the tight-\nbinding linear muffin- tin orbitals. Then the atomic potentials of Fe and Ga are randomly \ndistributed in a 5× 5 lateral supercell, which is connected to two semi -infinite Pd leads . \nA thermal lattice disorder is included via  displacing atoms randomly from the perfect \nlattice sites following a Gaussian type of distribution [ 33]. The root -mean -square \ndisplacement at room temperature is determined by the Debye model with the Debye \ntemperature 470 K. The length of the supercell is variable and the calculated total \ndamping is scaled linearly with this length. Thus, a linear least- squares fitting can be \nperformed to extract the bulk damping of the Fe -Ga alloy  [34].  The calculated Gilber t \ndamping is plotted in f igure 3(c) as a funct ion of the lattice distortion (𝑐𝑐−𝑎𝑎)𝑎𝑎⁄.  The \nGilbert damping is nearly independent of the lattice distortion and there is no evidence of anisotropy in t he intrinsic bulk damping of Fe Ga alloy.  \nSo the extrinsic contributions  are responsible for  the anisotropic behavior of \ndamping , which can be separated from the in -plane angular dependent linewidth. The \nrecorded FMR  linewidth  have the following different cont ributions  [11] :  \n     𝜇𝜇0∆𝐻𝐻=𝜇𝜇0∆𝐻𝐻inh+2𝜋𝜋𝛼𝛼𝐺𝐺𝑓𝑓\n𝛾𝛾𝛾𝛾+�𝜕𝜕𝐻𝐻r\n𝜕𝜕𝜑𝜑H∆𝜑𝜑H�+�Γ<𝑥𝑥𝑖𝑖>𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>�\n<𝑥𝑥𝑖𝑖>𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎 \n                �(�𝜔𝜔2+(𝜔𝜔0\n2)2−𝜔𝜔0\n2)/(�𝜔𝜔2+(𝜔𝜔0\n2)2+𝜔𝜔0\n2)+Γtwofoldmaxcos4(φM- φtwofold)   (2) \n∆Hinh is both frequency and angle independent  term due to the  sample  \ninhomogeneity . The second term is the intrinsic Gilbert damping  (𝛼𝛼𝐺𝐺) contribution. 𝛾𝛾  \n8 \n is a correction factor owing to the field dragging effect caused by magnetic anisotropy  \n[12], 𝛾𝛾 =cos (φM-φH). The 𝜑𝜑M as a function of φH  for the sample S2 at fixed 13 GHz \nis calculated  and show n in figure 4(a).  Note that the draggi ng effect  vanishes  (𝜑𝜑M=\nφH) when the field is along the hard or  easy axes . The third term  describes the mosaicity \ncontribution originating from the angular dispersion of the crystallographic cubic axes \nand yield s a broader linewidth  [35]. The four th term is the TMS contribution. The \nΓ<𝑥𝑥𝑖𝑖> signifies  the intensity  of the TMS along the principal in -plane crystallographic \ndirection  <𝑥𝑥𝑖𝑖>. The 𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� term indica tes the TMS contribution depending \non the in- plane direction of the field rel ative to <𝑥𝑥𝑖𝑖> and commonly expressed as \ncos2[2(φM-φ<xi>)] [14].  In addition, 𝜔𝜔 is the angular resonant frequency and  𝜔𝜔0=\n𝛾𝛾𝜇𝜇0𝑀𝑀eff. In our case, besides the fourfold TMS caused by expected lattice geometric \ndefects, the other twofold TMS channel is induced by the dipolar fields emerging from \nperiodic stripelike defects  [20,21] . This term is parameterized by its strength Γtwofoldmax \nand the axis of maximal scattering  rate φtwofold.  \nAs an example, t he angle- dependent linewidth measured  at 13 .0 GHz for  the \nsample S2 is  shown in f igure 4(b).  It clearly exhibits a strong in -plane anisotropy, and  \nthe linewidth along the [100] direction is significantly  larger than that along the [110] \ndirection . Taking only  isotropic Gilbert damping  into account , the dragging effect \nvanishes  with field applied  along the hard and easy axes . Meanwhile, the mosaicity \nterm gives an angular variation of the linewidth proportional to |𝜕𝜕𝐻𝐻𝑟𝑟𝜕𝜕𝜑𝜑𝐻𝐻|⁄ , which is \nalso zero along with the principal <100> and < 110>  directions. This gives direct \nevidence that the rel axation is not exclusively governed only considering the intrinsic \nGilbert mechanism and mosaicity term. Because the probability of defect formation \nalong with <100> directions is higher than that  along the <110> directions  [12], the \n9 \n TMS contribution is stronger along the easy axes , which is in accordance with t he fact \nthat the linewidth s along the [100] and [110]  direction s are non -equivalent. Moreover , \nthe linewidth of [010] direction is slightly  larger  than that along the [100] dire ction, \nsuggesting that  another twofold TMS channel is induce d by oblique deposition. As \nindicated by the red solid line  in figure 4(b), the linewidth can be well fitted. D ifferent \nparts making sense to the  linewidth can therefore be sepa rated  and summarized in Tab le \nI. As we know, the TMS predicts the curved non- linear frequency dependence of \nlinewidth, which not appear in a small frequency range for our case (as shown in f igure \n3(a)).  The linewidth as function of frequency was also well fitted including the TMS -\ndamping using the parameters in Table I  (not shown here) . \nThe larger strength of TMS along the easy axis can clearly explain the anisotropic \nbehavior of da mping , with higher value along the easy axis shown in f igure 3(b). The \nobtained Gilbert damping  factor  of ~ 7×10-3 is isotropic and invariant  with different \noblique angle s. The value of damping is slightly larger than the bulk value of 5.5×10-3 \n[29], which may be attributed to spin pumping of the Ta capping layer.   \n The obtained maxi ma of twofold TMS exhibits  an increasing  trend  with the \noblique angle  [shown in f igure 4(c)]. According  to previous works  on the shadowing \neffect [16-19], the larger  deposition angle  makes  the shadow ing effect  stronger , and the \ndipolar fields within stripe like defects  increase just like the UMA. This  can clearly \nexplain that the intensity of two fold TMS  follows exactly the same trend with the \ndeposition angle as the UMA . The axis of the maximal intensity of two fold TMS is \nparal lel to the projection of the  FeGa atom flux  from the fitting data.   As shown in Table  \nI, amazingly the modified growth conditions also influence  the fourfold TMS, \nespecially the strength of TMS along the <100>  axis. Figure 4(c) also presents the \nchanges of the fourfold TMS intensity as the  deposition angle  and shows a peak at 15° , \n10 \n which follows a similar trend as that of 𝛼𝛼eff along [100] axis as shown in f igure 3(b). \nThis indeed confirm s TMS -damping plays an important role in FeGa thin films.    \nFor the dispersion relation ω(k∥) in thin magnetic films , the propagation angle \n𝜑𝜑𝑘𝑘∥����⃗ defined as the angle between k∥���⃗ and the projection of the saturation magnetization \nMs into the sample plane is less than the critical value : 𝜑𝜑max =\n𝑐𝑐𝑎𝑎𝑎𝑎−1�𝜇𝜇0𝐻𝐻r(𝜇𝜇0𝐻𝐻r+𝜇𝜇0𝑀𝑀eff) ⁄  [9,36,37] . This implies  no degenerate modes are \navailable for the angle 𝜑𝜑𝑘𝑘∥����⃗ larger than φmax. Based on this theory, we propose a \nhypothesis that the crystallographic anisotropy  determine s the area including \ndegenerate magnon modes , as well as  the intensity of the fourfold TMS.  The resonance \nfield along <100>  axis change s due to the various crystallographic anisotropy , which \nhas a great effect on the φmax. The values of φmax of samples are shown in f igure 4(d). \nThe data follow the  same trend with the oblique angle as Γ<100>. During the grain \ngrowth,  the cubic anisotropy is  influenced possibly  since the anisotropy dispersion due \nto the stress. For the lower anisotropy of  sample S2 , a relatively larger  amount of stress \nand defects present in the sample and lead to a larger  four fold TMS.  \n4. Conclusions  \nIn conclusion, the effects of oblique deposition on the dynamic properties of FeGa \nthin films have been investigated  systematically . The pronounced TMS  as non-Gilbert \ndamping  results in an anisotropic magnetic relaxation . As the oblique angle increases, \nthe magnitude of the twofold TMS  increases due to the larger shadowing effect . \nFurthermore, the cubic anisotropy dominates  the area including degenerate magnon \nmodes, as well as the intensity of fourfold TMS. The reported results  confirm  that the \nmodified anisotropy can influence the extrinsic relaxation pr ocess  and open a n avenue \nto tailor magnetic  relaxation in spintronic devices. \n11 \n Acknowledgments  \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212)  and \nthe Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Norma l University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03).  \n12 \n References  \n[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159  L1 \n[2] Žutić  I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 \n[3] Gilbert T L2004 IEEE Trans. Magn. 40 3443  \n[4] He P, Ma  X, Zhang  J W, Zhao  H B, Lüpke  G, Shi  Z and Zhou S M 2013 Phys. \nRev. Lett.  110 077203 \n[5] Heinrich B, Meredith D J and Cochran J F 1979 J. Appl. Phys. 50 7726 \n[6] Lee A J, Brangham  J T, Cheng  Y, White  S P, Ruane  W T, Esser  B D, \nMcComb  D W, Hammel  P C and Yang  F Y 2017 Nat. Commun. 8 234 \n[7] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey W E 2007 Phys. Rev. Lett.  \n98 117601 \n[8] Azzawi  S, Hindmarch A and Atkinson D 2017 J. Phys. D: Appl. 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B  80 224421  \n \n \n  \n   \n            \n15 \n Figure Captions  \nFigure  1 (color online) (a) Schematic illustration of the film deposition geometry and \ncoordinate system (b) I n-plane hysteresis loops of samples S1 and S4 with the field \nalong [100], [110], and [010]. The inset shows the polar plot of the normalized \nremanence (M r/Ms) as a functi on of the in- plane angle.  FMR spectrum for the sample \nS2 with H along [100] and [110] axes showing the real (c ) and imaginary (d ) part s of \nthe S 21.  \nFigure  2 (color online) (a) H r vs. φH for FeGa films. (b) The  anisotropy  constants  K4 \nand Ku vs. deposition angle. (c) f  vs. Hr plots measured at H //[100], Symbols are \nexperimental data and the solid lines are the fitted results.  \nFigure  3 (color online) (a) ∆H as a function of f  for samples S1 and S2 with  field along \neasy and hard axis. (b) The dependence of the damping parameter on  the oblique angle \nwith field along [100] and [110] directions. (c) The calculated damping of FeGa alloy \nas a function of lattice distortion.  Figure  4 (color online)  (a) φ\nM and (b) ∆H as a function of φH for the  sample S2  \nmeasured at 13.0 GHz. (c) Oblique angle dependences of Γ<100> and Γtwofoldmax. (d) The \nlargest angle including degenerate magnon modes as a function of the oblique angle \nwith the applied field along <100> direction. \nTable Caption  \nTable I. The magnetic relaxation parameters of the FeGa films prepared via oblique \ndeposition (with experimental errors in parentheses).  \n \n  \n16 \n Figure 1  \n \n \n  \n \n  \n \n  \n \n  \n \n17 \n Figure  2 \n \n \n \n \n \n  \n \n  \n \n  \n \n  \n \n18 \n Figur e 3 \n \n \n  \nFigure  4 \n \n \n \n \n \n \n \n19 \n TableⅠ \nSample  𝜇𝜇0ΔHinh\n(mT) 𝛼𝛼G Δ𝜑𝜑H \n(deg.) Γ<100> \n(107Hz) Γ<110> \n(107Hz) Γtwofoldmax\n(107Hz) 𝜑𝜑twofold  \n(deg. ) \nS1 0 0.007  0.62 17(3)  5.8(1.8)  0(2) 90 \nS2 0.7 0.007  1.2 81.4(3.7)  9.3(1.9)  7.4(3) 90 \nS3 0 0.007  1.0 59.2(4.5)  11.1(2) 13(3.7)  90 \nS4 0 0.007  1.1 33.3(6)  14.8(3.7)  26(4)  90 \n \n20 \n "    },    {        "title": "1604.01389v1.Homodyne_detected_ferromagnetic_resonance_of_in_plane_magnetized_nano_contacts__composite_spin_wave_resonances_and_their_excitation_mechanism.pdf",        "content": "Homodyne-detected ferromagnetic resonance of in-plane magnetized nano-contacts:\ncomposite spin wave resonances and their excitation mechanism\nMasoumeh Fazlali, Mykola Dvornik, Ezio Iacocca,\u0003Philipp D urrenfeld, and Mohammad Haidar\nDepartment of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden\nJohan \u0017Akerman\nDepartment of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden\nMaterials Physics, School of ICT, KTH Royal Institute of Technology, 164 40 Kista, Sweden and\nNanOsc AB, Electrum 205, 164 40. Kista, Sweden\nRandy K. Dumasy\nDepartment of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden and\nNanOsc AB, Electrum 205, 164 40. Kista, Sweden\nThis work provides a detailed investigation of the measured in-plane \feld-swept homodyne-\ndetected ferromagnetic resonance (FMR) spectra of an extended Co/Cu/NiFe pseudo spin valve\nstack using a nanocontact (NC) geometry. The magnetodynamics are generated by a pulse-\nmodulated microwave current and the resulting recti\fed dcmixing voltage, which appears across\nthe NC at resonance, is detected using a lock-in ampli\fer. Most notably, we \fnd that the mea-\nsured spectra of the NiFe layer are composite in nature and highly asymmetric, consistent with the\nbroadband excitation of multiple modes. Additionally, the data must be \ft with two Lorentzian\nfunctions in order to extract a reasonable value for the Gilbert damping of the NiFe. Aided by mi-\ncromagnetic simulations, we conclude that (i)for in-plane \felds the rfOersted \feld in the vicinity\nof the NC plays the dominant role in generating the observed spectra, (ii)in addition to the FMR\nmode, exchange dominated spin waves are also generated, and (iii) the NC diameter sets the mean\nwavevector of the exchange dominated spin wave, in good agreement with the dispersion relation.\nPACS numbers: 75.78.-n, 76.50.+g, 85.75.-d\nI. INTRODUCTION\nSpin torque ferromagnetic resonance (ST-FMR)1{9is a\npowerful and versatile tool that enables the characteriza-\ntion of magnetodynamics on the nanoscale. Unlike more\nconventional FMR measurement techniques, where a res-\nonant cavity or waveguide is used to generate rfmagnetic\nexcitation \felds, the resonant precession in an ST-FMR\nmeasurement is assumed to be primarily a result of the\nST from a spin polarized current. However, ST-FMR\nrepresents a speci\fc type of a more general homodyne\ndetection scheme where the excitation mechanism itself\ncan originate from a variety of physical mechanisms apart\nfrom, or in combination with ST, including, e.g.,rfOer-\nsted \felds10and electric \felds11.\nWhile homodyne-detected FMR studies on magnetic\ntunnel junction (MTJ) and all-metallic spin valve nano-\npillar devices have dominated the literature2{6, there\nhave been an increasing number of studies utilizing\npoint- and nanocontacts (NCs) on extended multilayer\n\flm stacks7,12{15. The NC geometry is particularly\npromising for high-frequency spin torque oscillators (NC-\nSTOs)16{22and for the emerging \feld of ST-based\nmagnonics,23{27where highly non-linear auto-oscillatory\nmodes are utilized for operation.\nIn the NC geometry literature7,12{15,28, the observed\nST-FMR spectra of the NiFe-based free layers have been\nanalyzed as a single resonance, despite a signi\fcant peak\nasymmetry hinting at additional contributions. Thelinewidth of this asymmetric peak has not been under-\nstood so far7. The same studies also note that the typical\n\feld condition of an in-plane \feld aligning both magnetic\nlayers in parallel should not result in any ST, calling\ninto question the fundamental excitation mechanism of\nthe observed spectra. This signi\fcant discrepancy has\nbeen tentatively explained as being caused by local mis-\nalignments due to sample imperfections. However, given\nhow robust ST-FMR measurements are over sets of dif-\nferent devices, it is rather unsatisfactory to have to refer\nto unknown extrinsic factors for the ST-FMR technique\nto function. It appears that the rfOersted \feld gener-\nated by the injected microwave current into NC could be\nat play15. Therefore, both a better fundamental under-\nstanding of the linear spin wave (SW) modes in the NC\ngeometry and of their excitation mechanism are therefore\nhighly desirable.\nIn this work we show that the observed resonance\nspectrum in a NiFe NC-STO free layer is composite in\nnature and can be described as a sum of two distinct\nresonances with very di\u000berent behavior and origin. Ex-\nperimentally, the data must be \ft with two Lorentzian\nfunctions in order to extract a reasonable value for the\nGilbert damping of the FMR mode. From these \fts, it\nis also clear that only the width of one of the resonances\nshows a damping-like linear dependence on the frequency,\nwhereas the width of the other is mostly frequency inde-\npendent. Aided by micromagnetic simulations, we con-\nclude that (i)the composite resonance is a sum of a FMRarXiv:1604.01389v1  [cond-mat.mes-hall]  5 Apr 20162\nmode and an exchange dominated spin wave mode, (ii)\nthe NC diameter sets the mean wavevector of the ex-\nchange dominated spin waves, in good agreement with\nthe dispersion relation, and (iii) for in-plane \felds the\nrfOersted \feld, not ST, in the vicinity of the NC plays\nthe dominant role in exciting the observed spectra. We\nargue that homodyne-detected FMR studies in the NC\ngeometry must account for such additional excitations to\naccurately extract the fundamental magnetodynamical\nproperties.\nII. EXPERIMENT\nNC-STO fabrication starts with a blanket Pd(8 nm)\n/ Cu(15 nm) / Co(8 nm) / Cu(8 nm) / NiFe(4.5 nm)\n/ Cu(3 nm) / Pd(3 nm) \flm stack deposited by mag-\nnetron sputtering on a thermally oxidized Si substrate,\nwhere the NiFe (Ni 80Fe20) and Co play the role of the\nfree and \fxed layers, respectively, as shown in Fig. 1(a).\nThe blanket \flm is then patterned into 16 \u0016m x 8\u0016m\nspin valve mesas and a 30 nm SiO 2layer is deposited by\nrfmagnetron sputtering. Circular NCs of nominal di-\nameters,D, of 90 nm, 160 nm and 240 nm are de\fned\nthrough the SiO 2insulating layer using e-beam lithogra-\nFIG. 1. (a) Schematic of the NC-STO and the measurement\nset-up. (b) Inset: ST-FMR spectra at four di\u000berent frequen-\ncies for the D=160 nm sample. Main \fgure: Plot of the\n\feld position of the dominant resonance peak. The resonance\n\felds can be well \ft by the Kittel equation using \u00160Ms=0.85\n\u00060.02 T for the NiFe layer.phy at the center of the mesa. A \fnal photolithographic\nprocess then de\fnes a coplanar waveguide for electrical\nconnections and e\u000ecient microwave signal pick-up. All\nmeasurements were performed at room temperature in a\ncustom built probe station utilizing a uniform in-plane\nmagnetic \feld. Our homodyne-detected FMR measure-\nments utilized both a microwave generator and lock-in\nampli\fer, which were connected to the device using a\nbias-tee, as schematically shown in Fig. 1. The rfpower\ninjected into the NC is -14 dBm, which ensures that\nthe excited magnetodynamics are in the linear regime.\nThe resulting dcmixing voltage3,Vmix, is measured as\na function of the magnetic \feld and at a \fxed excita-\ntion frequency. The microwave current was amplitude\nmodulated at a low (98.76 Hz) modulation frequency for\nlock-in detection of Vmix.\nIII. EXPERIMENTAL RESULTS\nThe \feld-swept spectra measured for di\u000berent frequen-\ncies, which are vertically o\u000bset for clarity, are shown in\nFig. 1(b, inset) for the D=160 nm sample. As shown in\nthe main panel of Fig. 1(b), the dominant resonance peak\n(data points) can be well \ft (solid line) with the Kittel\nequation, which results in \u00160Ms= 0:85\u00060:02 T and a\nnegligible magnetocrystalline anisotropy.\nInterestingly, upon closer inspection it becomes clear\nthat the measured spectra are highly asymmetric, ex-\nhibiting a signi\fcant shoulder on the low-\feld side of the\ndominant resonance peak. In Fig. 2 we show a single rep-\nresentative resonance at f= 18 GHz for the D=160 nm\nsample. While it is well known that the mixing voltage\ncan be intrinsically asymmetric2,29,30, it is important to\npoint out that we cannot \ft our data with a single reso-\nFIG. 2. Zoom-in of a representative ST-FMR spectrum of the\nD=160 nm sample taken at f= 18 GHz and Irf= 1.3 mA,\ntogether with a \ft (red line) based on two Lorentzians as\ndescribed in the text. The inset shows the two individual\ncontributions of the quasi-uniform FMR mode (black), and\nthe spin wave resonance (blue).3\nnance having both symmetric and antisymmetric contri-\nbutions. Most importantly, the prior theoretical results\nare virtually independent of the NC diameter, in direct\ncontrast to our experimental observations. In order to\nproperly \ft (red solid line) the entire spectrum we must\ninstead use twoLorentzian functions, each with its own\nresonance \feld and linewidth, as shown in Fig. 2(inset).\nThe \ft shows vanishing antisymmetric contribution to\nthe lineshape for both of the resonances\nf= o\u000bset +X\ni=FMR;SWRslope\u0002B\n+1\n\u0001Bi\u0014\nSi \u0001Bi2\n4 (B\u0000Bires)2+ \u0001Bi2\n+Ai \u0001Bi(B\u0000Bi\nres)\n4 (B\u0000Bires)2+ \u0001Bi2\u0015\n,(1)\nwhereBandBi\nresare applied and resonance \felds, re-\nspectively, and \u0001 Biis the linewidth of the corresponding\npeak.SiandAiare amplitudes of its symmetric and anti-\nsymmetric components, respectively. As the frequency\nvs. \feld behavior of the main resonance mode can be\nwell \ft with the Kittel equation, Fig. 1(b), we ascribe\nthis peak to the FMR mode of the NiFe layer and the\nsecond low \feld mode with a higher order spin wave res-\nonance (SWR), which will be discussed in detail later.\nThe linewidth vs. frequency of both the FMR and\nSWR modes are plotted in Fig. 3 for three di\u000berent NC\ndiameters of 90, 160, and 240 nm. Three di\u000berent signif-\nicant observations can be made. First, the FMR mode\nshows a clear linear increase of linewidth with the fre-\nquency, from which the Gilbert damping, \u000b, can be ex-\ntracted using the following relation:\n\u0001Bi=4\u0019\u000b\n\rf+ \u0001Bi\n0 (2)\nFIG. 3. The measured (dots) and \ftted (solid lines)\nlinewidths of the FMR and SWR modes are shown for the\ndi\u000berent NC diameters.\nFIG. 4. Measured (dots) and calculated (solid lines) reso-\nnance \felds of the FMR and SWR modes for the di\u000berent\nNC diameters. The black solid line is a \ft to an average of\nthe the FMR mode for all three devices. Inset: A plot of\nthe \ftted NC diameter ( D0) vs. the nominal diameter ( D),\ntogether with a line indicating D0=D.\nwhere \u0001Bi\n0is an inhomogeneous broadening of the cor-\nresponding resonance. Our measured values of \u000b, which\nare all on the order of 0.01, are also consistent with those\nmeasured in Ref. 31. This provides further evidence that\nthe dominant resonance mode can indeed be correlated\nwith the usual FMR mode of NiFe. Second, the linewidth\nof the SWR mode is mostly independent of frequency, in-\ndicating that the primary origin of the linewidth is not\ndamping. Third, the inhomogeneous broadening is ap-\nproximately inversely proportional to the NC diameter,\nwhich at \frst seems counter intuitive as one would ex-\npect a larger NC to sample a larger sample volume and\ntherefore include more inhomogeneities. The origin of\nthis interesting e\u000bect will be explained later.\nThe frequency versus \feld dependence of the measured\nFMR and SWR modes are summarized in Fig. 4. The\nblack solid line shows the average behavior of the FMR\nmode for NC diameters of 90, 160, and 240 nm and es-\nsentially reproduces what is shown in Fig. 1. For a \fxed\nfrequency, we \fnd that the SWR mode shifts to lower\n\felds as the NC diameter decreases. Assuming that the\norigin of the SWR mode is the exchange interaction, the\ndiameter of the NC, D0, can be estimated using the fol-\nlowing dispersion relation:\nf=\r\n2\u0019\u0014\u0010\nBSWR\nres +\u00160MS(\u0015exk)2\u0011\n\u0002\u0010\nBSWR\nres +\u00160MS+\u00160MS(\u0015exk)2\u0011\u00151=2\n;(3)\nwhere\u0015ex=p\n2A=\u0016 0M2sandk=\u0019=D0are the ex-\nchange length and the SWR wave vector, respectively.\nThe room temperature value of the exchange sti\u000bness is\nset toA= 11 pJ/m.32The estimated sizes of the NCs are\nin reasonable agreement with the corresponding nominal\nvalues as shown in the inset of Fig. 4.4\nIV. MICROMAGNETIC SIMULATIONS\nThe micromagnetic simulations were performed using\nthe mumax3 solver33. Since the actual spin valve mesa is\ntoo large to be simulated in its entirety in a reasonable\ntime frame, we limited our calculations to a 5.120 \u0016m\u0002\n2.560\u0016m\u00024 nm volume with periodic boundary con-\nditions tailored to mimic the lateral aspect ratio of the\nexperimental spin valve mesa. To break the symmetry of\nthe system, which might otherwise fully forbid any STT-\nrelated e\u000bects and non-conservative SW scattering, we\nassume that the applied \feld points 5 °out-of-plane, com-\nparable to the possible error in the experimental \feld\nalignment. As a \frst step, the evolution of the ground\nstate of the entire Co/Cu/NiFe stack is calculated, con-\n\frming that (i)the Co and NiFe layers remain virtually\ncollinear in the given range of the applied magnetic \felds\nand (ii)there are no mutual stray \felds produced be-\ntween the layers in the vicinity of the NC. Since there\nis a signi\fcant spin wave dispersion mismatch between\nCo and NiFe, we do not expect any resonant dynamic\nmagnetic coupling between the layers. Under these three\nconsiderations we can con\fdently simulate the dynamics\nof the NiFe free layer alone.\nIn the simulations we replicate the experimental data\nacquisition routine by performing the \feld sweeps with\na harmonic excitation of f= 18 GHz. The in\fnite wire\napproximation is used to calculate the Oersted \feld pro-\nduced by the NC.34,35For every value of the applied\n\feld we \frst let the system reach the steady state and\nthen sample the spatial map of the magnetization for\nthe following 5 ns at 3.5 ps time intervals with a sub-\nsequent point-wise FFT applied and the amplitude and\nphase of the magnetization precession extracted at the\nexcitation frequency. Where applicable, the direction of\nthe spin-current polarization is assumed to be collinear\nwith the magnetization in the nominally \fxed Co layer.\nThe implemented saturation magnetization, gyromag-\nnetic ratio, and damping constant are estimated by \ft-\nting a Kittel equation to the experimental data. The\nroom temperature value of the exchange sti\u000bness is set\ntoA= 11 pJ/m.32\nThe simulated magnetic response shown in Fig. 5(b)\nagrees well with the experimentally measured data shown\nin Fig. 5(a). To identify the origin of the observed peak\nasymmetry we investigate the spatial pro\fles of the mag-\nnetization precession amplitude in the vicinity of the res-\nonance, see Fig. 5(c). The snapshots clearly show propa-\ngating SWs on the low-\feld side of the main peak, while\nno SWs are resolved on the high-\feld side. Looking closer\nat the phase pro\fles of the corresponding modes, that es-\nsentially depict the wavelength of the excited magnons,\nthe following conclusions can be made: (a) The propaga-\ntion of SWs perpendicular to the saturation direction is\nsuppressed and (b) the lowest excited mode is not uni-\nform, but anti-symmetric with respect to the NC center.\nFIG. 5. (a) The normalized measured mixing voltage ( Vmix)\nand (b) the normalized simulated magnetization precesssion\namplitude for the three NC diameters as a function of the\napplied in-plane magnetic \feld. (c) Spatial maps of the mag-\nnetization precession amplitude (top row) and phase (bottom\nrow) simulated for a D= 160 nm NC diameter taken at the\n\felds corresponding to the main peak and its1⁄2and1⁄4heights\n(as shown by the corresponding black symbols in (b)). Prop-\nagating spin waves are clearly seen for the two lowest \felds.\nV. DISCUSSION\nIf the free layer is magnetized in-plane, then both back-\nward volume magnetostatic SWs (BVMSSW) and sur-\nface magnetostatic-exchange SWs (SMSSW) can be ex-\ncited:\nfBVMSSW =\u0014\u0010\nfB+fM(\u0015exk)2\u0011\n\u0002\u0012\nfB+fM(\u0015exk)2+fM\u00121\u0000e\u0000kd\nkd\u0013\u0013\u0015 1=2\n;\nfSMSSW =\u0014\u0010\nfB+fM(\u0015exk)2\u0011\n\u0002\u0010\nfB+fM(\u0015exk)2+fM\u0011\n+f2\nM\n4\u0000\n1\u0000e\u00002kd\u0001\u00151=2\n;\nwherefB=\r\n2\u0019B,fM=\r\n2\u0019\u00160MS\nThey are calculated using Eq. (5.97b) and Eq. (5.111b)\nfrom Ref. 36 for propagation along and perpendicu-5\nlar to the saturation direction, respectively. The ex-\nchange contribution is included by substituting B!\nB+\u00160Ms(\u0015exk)2.\nThe corresponding dispersion relations are shown in\nFig. 6 for Py thicknesses, d, of 100 nm (green lines)\nand 4.5 nm (red lines). There is always a region of\nresonance \felds, where magnetostatic and exchange-\ndominated SWs co-exist, as highlighted by the shaded\narea in Fig. 6 for the Py thickness of 100 nm. Al-\nthough the band is broad for relatively thick layers, it\nonly amounts to 1.16 mT for the 4.5 nm Py, i.e., an\norder of magnitude smaller than the intrinsic broaden-\ning of the FMR peak. We therefore conclude that SWs\ncontributing to the low-\feld tail of the FMR peak are\nexchange-dominated. Note that the calculated dispersion\nrelations di\u000ber from what is found using Eq. (3) (thick\nsolid line in Fig. 6). This di\u000berence arises as the disper-\nsion relations also strongly depend on the exact boundary\nconditions at the free layer surfaces. For instance if the\nPy \flm is pinned on both surfaces, e.g., if placed in be-\ntween su\u000eciently thick metallic layers, the dispersion of\nthe exchange-dominated backward volume SWs is given\nby the following equation (shown by the dotted line in\nFig. 6)37:\nfBVMSSW\u0000pinned =\u0014\u0010\nfB+fM(\u0015exk)2\u0011\n\u0002\u0012\nfB+fM(\u0015exk)2+fM\n1 + (kd=\u0019 )2\u0013\u00151=2\n;\nNote that in this case the spectrum of the exchange-\ndominated surface SWs will be dispersionless and not\naccessible experimentally\nSince Eq. (3) \fts the NC diameter reasonably well, we\nconclude that (a) the detected mixing voltage is gener-\nated by the exchange-dominated backward volume SWs\nFIG. 6. The dispersion relations for the SWs propagating par-\nallel and perpendicular to the saturation direction are shown\nfor the di\u000berent thicknesses of the NiFe layer. The points\ncorrespond to the minimum of the SW group velocity.and (b) there is undoubtedly some surface pinning of the\nPy layer. The exact origin of the pinning and its strength\nis beyond the scope of the present study.\nDue to the collinear free and \fxed layers we do not\nexpect any signi\fcant contribution from the ST to the\nobserved magnetization dynamics, which is con\frmed\nby comparing micromagnetic simulations performed with\nand without ST included. Correspondingly, for the in-\nplane applied magnetic \felds in the NC geometry with\nnodcbias currents applied, the rfOersted \feld is the pri-\nmary excitation mechanism responsible for the observed\ndynamics.\nIn a linear approximation the response of the system\nis essentially determined by the spectrum of the excita-\ntion, which in our case is provided by the rfOersted\n\feld. If the excitation has a \fnite amplitude at some\npoint of reciprocal space ( i.e.at the given frequency and\nwavevector), then, if allowed, the corresponding magnon\nwill be excited. The spatial pro\fle of the Oersted \feld\nand its spectrum for the D= 160 nm NC are shown in\nFig. 7(a) and (b), respectively. We can identify both lo-\ncal and global anti-symmetries with respect to the NC\ncenter with corresponding periods determined by twice\nFIG. 7. (a) The spatial distribution and (b) the correspond-\ning 2D FFT of the out-of-plane component of the Oersted\n\feld by the D = 160 nm diameter NC. (c) The Oersted \feld\nas the function of the wavevector component along the satu-\nration direction is shown for the D = 160 nm diameter NC.\nThe inset shows a zoom-in of the small wavevector part of\nthe spectrum. (d) The dispersion relation of the exchange\ndominated SWs. (e) The experimentally acquired magneti-\nzation dynamics spectrum using nanocontact of D = 160 nm\ndiameter.6\nthe NC diameter and mesa width, L, respectively. Since\nboth spatial components are naturally con\fned to their\nunit periods, the linewidth of the corresponding excita-\ntion peak is \fnite. Therefore the Oersted \feld most e\u000e-\nciently couples to the SW bands having widths of 2 \u0019=L\nand\u0019=D corresponding to the wavevectors of 2 \u0019=L and\nn\u0019=D , respectively, where n= 1;3;:::(see Fig. 7(c) and\nits inset). As the NC diameter decreases, the position\nand width of the former band stays constant, while the\nlatter one shifts towards lower resonance \felds and in-\ncreases its width, leading to the observed extension of\nthe tail in the excitation spectrum.\nIt is important to mention that the circular NC can-\nnot e\u000bectively couple to the uniform FMR. Instead, the\nmain peak observed in the experiments and simulations\ncorresponds to the anti-symmetric mode with k= 2\u0019=L.\nHowever, due to the vanishing magnetostatic dispersion,\nits resonant \feld is virtually indistinguishable from the\nuniformk= 0 FMR mode.\nConsidering a typical FMR experiment where the ex-\ncitation frequency is \fxed, according to Eq. (3) the\nwavevector of the generated propagating SW is ulti-\nmately determined by the value of the applied magnetic\n\feld. As the \feld is swept towards zero past the dominant\nFMR resonance, the NC continuously excites propagat-\ning SWs of increasing wavevectors. Since the excitation\namplitude drops rapidly for the low values of the ap-\nplied \feld ( i.e.for short wavelength SWs), the detected\nmagnetic signal vanishes accordingly, leading to the ap-\npearance of the low-\feld tail, see Fig. 7(c) and (e).\nBy assuming that the extent of the tail is estimated at\n1/10 of the its peak amplitude, we can project the corre-\nsponding experimentally observed applied magnetic \feld\nto the cuto\u000b wavevector of the excitation spectrum as\nschematically demonstrated by the shaded rectangles in\nFig. 7(c), (d) and (e). This gives us the cut-o\u000b wavevec-\ntors (in units of \u0019=D0) of 1.93, 1.95 and 2.34 for the NCs\nof 90, 160 and 240 nm nominal diameters, respectively.\nSince these values fall roughly inside the \frst two funda-\nmental SW bands attributed to k= 2\u0019=L andk=\u0019=D,\nthe two-peak scheme used to \ft the experimental data is\nfully justi\fed.\nIt should be noted that the micromagnetic simulations\ndo not reproduce the shoulder as it is observed exper-\nimentally for all the NC diameters. According to our\nmodel, the shoulder should be inherited from the exci-\ntation spectrum. Perhaps the approximation we used to\ncalculate the Oersted \feld, an in\fnite wire, is not su\u000e-\ncient to bring out this feature. Nevertheless, this does not\nchange the interpretation of the results and conclusions\nof the present study.\nFinally, the NC size dependence of the FMR and SWR\ninhomogeneous broadenings shown in Fig. 3 can be well\nunderstood by assuming that it is inherited from the\nlinewidth of the corresponding excitation peaks. For the\nSWR mode, the expected extrinsic contribution to themagnonic linewidth is 96 mT, 43 mT and 15 mT for the\nNC diameters of 82 nm, 122 nm and 205 nm, respectively,\nin excellent agreement with the \ftted values. In contrast,\nfor the FMR the contribution is vanishing and should be\nvirtually independent of the NC size. However, if the NC\nhad shape imperfections, the corresponding irregularities\nin the Oersted \feld pro\fle should broaden the excitation\npeaks and, eventually, the FMR and SWR. As we typ-\nically observe a less perfect NC for smaller diameters,\nthe inhomogeneous broadening of FMR should increase\naccordingly, consistent with the experimental data.\nVI. CONCLUSIONS\nIn conclusion, using homodyne based measurement\ntechniques we provide an in-depth study of the magneto-\ndynamics in a quasi-con\fned system, namely a NC pat-\nterned on an extended pseudo spin valve \flm stack. The\nobserved spectra are highly asymmetric and cannot be\nexplained by a single resonance mode, as has been done\nin the past7,12. Instead, each spectra is \ft by a com-\nbination of two Lorentzians from which we can extract\nthe FMR mode resonance \feld and linewidth. The sec-\nondary mode corresponds to the generation of exchange-\ndominated spin waves with a wavevector inversely pro-\nportional to the NC diameter. The results are reproduced\nby the micromagnetic simulations that show the rfOer-\nsted \feld generated by the injected rfcurrent is the dom-\ninant excitation mechanism of the observed magnetiza-\ntion dynamics. We thereby demonstrate experimentally\na highly tunable point source of the propagating SW with\nthe wavevectors limited only by the resolution of the fab-\nrication process used. This is of the paramount impor-\ntance for the applications of sub-THz and THz magnonics\nand spintronics.\nVII. ACKNOWLEDGMENTS\nWe would like to thank Pranaba Muduli and Mo-\njtaba Ranjbar for useful discussions. This work was\nsupported by the European Commission FP7-ICT-2011-\ncontract No. 317950 \\MOSAIC\". It was also supported\nby the European Research Council (ERC) under the\nEuropean Community \rs Seventh Framework Programme\n(FP/2007-2013)/ERC Grant 307144 \\MUSTANG\". 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Melkov, Magnetization Oscilla-\ntions and Waves (CRC Press, Boca Raton, FL, 1996)."    },    {        "title": "1911.03408v1.Giant_anisotropy_of_Gilbert_damping_in_a_Rashba_honeycomb_antiferromagnet.pdf",        "content": "Giant anisotropy of Gilbert damping in a Rashba honeycomb antiferromagnet\nM. Baglai,1R. J. Sokolewicz,2A. Pervishko,1, 3M. I. Katsnelson,2O. Eriksson,1, 4D. Yudin,5, 1and M. Titov2, 3\n1Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden\n2Institute for Molecules and Materials, Radboud University Nijmegen, NL-6525 AJ Nijmegen, the Netherlands\n3ITMO University, Saint Petersburg 197101, Russia\n4School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n5Skolkovo Institute of Science and Technology, Moscow 121205, Russia\n(Dated: November 11, 2019)\nGiant Gilbert damping anisotropy is identi\fed as a signature of strong Rashba spin-orbit coupling\nin a two-dimensional antiferromagnet on a honeycomb lattice. The phenomenon originates in spin-\norbit induced splitting of conduction electron subbands that strongly suppresses certain spin-\rip\nprocesses. As a result, the spin-orbit interaction is shown to support an undamped non-equilibrium\ndynamical mode that corresponds to an ultrafast in-plane N\u0013 eel vector precession and a constant\nperpendicular-to-the-plane magnetization. The phenomenon is illustrated on the basis of a two\ndimensional s-dlike model. Spin-orbit torques and conductivity are also computed microscopically\nfor this model. Unlike Gilbert damping these quantities are shown to reveal only a weak anisotropy\nthat is limited to the semiconductor regime corresponding to the Fermi energy staying in a close\nvicinity of antiferromagnetic gap.\nI. INTRODUCTION\nA gapless character of the spin-wave spectrum in\nisotropic Heisenberg magnets in two dimensions re-\nsults in the homogeneity of magnetic ordering being\ndestroyed by thermal \ructuations at any \fnite tem-\nperatures. In contrast, in van der Waals magnets,\ncharacterized by intrinsic magnetocrystalline anisotropy\nthat stems from spin-orbit coupling1, an ordered mag-\nnetic state can be retained down to a monolayer\nlimit. Two-dimensional (2D) van der Waals magnets\nare currently experiencing a revived attention2{8driven\nby the prospects of gateable magnetism9{12, a con-\ntinuing search for Kitaev materials13,14and Majorana\nfermions15, topologically driven phenomena16as well as\nvarious applications3,4,7. The trade-o\u000b between quan-\ntum con\fnement, nontrivial topology and long-range\nmagnetic correlations determines their unique magneto-\nelectronic properties, in particular a tunable tunneling\nconductance17and magnetoresistance18{20depending on\nthe number of layers in the sample, as well as long-\ndistance magnon transport21.\nFerromagnetic thin \flms have already entered commer-\ncial use in hard drives, magnetic \feld and rotation angle\nsensors and in similar devices7,22,23, while keeping high\npromises for technologically competitive ultrafast mem-\nory elements24and neuromorphic chips25. Moreover, it\nhas recently been suggested that current technology may\nhave a lot to gain from antiferromagnet (AFM) materi-\nals. Indeed, manipulating AFM domains does not induce\nstray \felds and has no fundamental speed limitations\nup to THz frequencies26. Despite their ubiquitousness,\nAFM materials have, however, avoided much attention\nfrom technology due to an apparent lack of control over\nthe AFM order parameter { the N\u0013 eel vector. Switching\nthe N\u0013 eel vector orientation by short electric pulses has\nbeen put forward only recently as the basis for AFM\nspintronics27{29. The proposed phenomenon has beensoon observed in non-centrosymmetric crystals such as\nCuMnAs30{33and Mn 2Au34{36. It should be noted that\nin most cases AFMs are characterized by insulating type\nbehavior37, limiting the range of their potential appli-\ncations, e.g., for spin injection38. Interestingly, antifer-\nromagnetic Mn 2Au possesses a typical metal properties,\ninheriting strong spin-orbit coupling and high conductiv-\nity, and is characterized by collective modes excitations\nin THz range36.\nDespite a lack of clarity concerning the microscopic\nmechanisms of the N\u0013 eel vector switching, these experi-\nments have been widely regarded as a breakthrough in\nthe emerging \feld of THz spintronics26,30,36,39{43. It has\nbeen suggested that current-induced N\u0013 eel vector dynam-\nics in an AFM is driven primarily by the so-called N\u0013 eel\nspin-orbit torques29,32,44{56. The N\u0013 eel spin-orbit torque\noriginates in a non-equilibrium staggered polarization\nof conduction electrons on AFM sublattices29,32,48,50.\nCharacteristic magnitude of the non-equilibrium stag-\ngered polarization and its relevance for the experiments\nwith CuMnAs and Mn 2Au remain, however, debated.\nThe N\u0013 eel vector dynamics in an AFM is also strongly\na\u000bected by an interplay between di\u000berent types of Gilbert\ndampings. Unlike in a simple single-domain ferromagnet\nwith a single sublattice, the Gilbert damping in an AFM\nis generally di\u000berent on di\u000berent sublattices and includes\nspin pumping from one sublattice to another. A proper\nunderstanding of Gilbert damping is of key importance\nfor addressing not only the mechanism of spin pumping\nbut also domain wall motion, magnon lifetime, AFM res-\nonance width and many other related phenomena57{61.\nIt is also worth noting that spin pumping between two\nthin ferromagnetic layers with antiparallel magnetic ori-\nentations share many similarities with Gilbert damping\nin a bipartite AFM62,63.\nA conduction electron mechanism for Gilbert damp-\ning in collinear ferromagnet requires some spin-orbit in-\nteraction to be present. It is, therefore, commonly as-arXiv:1911.03408v1  [cond-mat.str-el]  8 Nov 20192\nFIG. 1. A model of Rashba honeycomb antiferromagnet with\ntwo sublattices, AandB, and on-site exchange interaction\nbetween localized momenta and conduction electrons (see\nEq. 1). The large blue arrow represents the N\u0013 eel vector vector,\nn, that is in general, characterized by non-vanishing in-plane,\nnk, and perpendicular-to-the-plane, n?, components. We re-\nfer to a speci\fc coordinate system with ^xaxis chosen to be\nin the direction of nk.\nsumed that spin-orbit interaction of electrons naturally\nenhances the Gilbert damping. Contrary to this in-\ntuition, we show that Rashba spin-orbit coupling does\ngenerally suppress one of the Gilbert damping coe\u000e-\ncients and leads to the appearance of undamped non-\nequilibrium N\u0013 eel vector precession modes in the AFM.\nSpin dynamics in a bipartite AFM is described in terms\nof two mutually orthogonal vector \felds, namely the vec-\ntorn(t) that is proportional to the N\u0013 eel vector (di\u000berence\nbetween sublattice moments) and the vector m(t) that is\nproportional to the net magnetization (sum of sublattice\nmoments) of a sample. Even though the AFM ground\nstate corresponds to m= 0, it is widely understood that\nno N\u0013 eel dynamics is possible without formation of a small\nbut \fnite nonequilibrium magnetization m. It appears,\nhowever, that Gilbert damping terms associated with the\ntime dynamics of m(t) andn(t) are essentially di\u000berent\nfrom a microscopic point of view.\nIndeed, the Gilbert damping that is proportional to\n@tnis characterized by a coe\u000ecient \u000bn, which is van-\nishing in the absence of spin-orbit interaction, much like\nit is the case in the ferromagnets. This behavior can be\ntraced back to a spin-rotational symmetry of the collinear\nAFM. Indeed, the absolute value of nis conserved up to\nthe orderm2. Thus, the dynamics of the N\u0013 eel vector is\nessentially a rotation that does not change the conduction\nelectron spectrum as far as the spin-rotation invariance\nis present. Breaking the spin-rotation symmetry by spin-\norbit interaction induces, therefore, a \fnite \u000bn, which is\nquadratic with respect to spin-orbit interaction strength.\nIn contrast, the Gilbert damping that is proportional\nto@tmoriginates directly in the conduction electron\nscattering even in the absence of any spin-orbit inter-action. The strength of the damping in a simple sym-\nmetric AFM is characterized by a coe\u000ecient \u000bm, which\nis typically much larger than \u000bn. As a rule, the spin-\norbit interaction tends to suppress the coe\u000ecient \u000bmby\nrestricting the ways in which electrons can damp their\nmagnetic moments. The condition \u000bm\u001d\u000bnhas been\nindeed well documented in a metallic AFM57,59.\nIn this paper, we uncover the microscopic mechanism\nof strong and anisotropic Gilbert damping suppression\ndue to the in\ruence of spin-orbit interaction in a 2D AFM\nmodel on a honeycomb lattice.\nBelow we focus mainly on the AFM in the regime of\ngood metallic behavior, such that the Fermi energy of\nelectrons exceeds by order of magnitude that of an ef-\nfectives-dexchange coupling between electron spins and\nlocalized AFM magnetic momenta. In this case, the tran-\nsition to the highly anisotropic regime takes place pro-\nvided the characteristic spin-orbit energy \u0015exceeds the\nscale ~=\u001c, where\u001cis the electron scattering time. Alter-\nnatively, one may think of characteristic spin-orbit length\nbecoming smaller than the mean free path of conduction\nelectrons. We show here that the splitting of 2D Fermi\nsurfaces by spin-orbit interaction leads to a dramatic sup-\npression of electron spin \rips in certain directions. This\nresults in a strong anisotropy of both Gilbert damping\ntensors ^\u000bnand ^\u000bm, that get some of their principal com-\nponents vanishing. This extreme anisotropy in the damp-\ning leads to essentially undamped N\u0013 eel vector dynamics\nfor certain nonequilibrium modes.\nIn particular, we identify a speci\fc undamped mode\nthat corresponds to perpendicular-to-the-plane magneti-\nzationm/^zand in-plane N\u0013 eel vector n(t)?^z. The\nN\u0013 eel vector corresponding to the mode has a precission\naroundmwith the frequency Jexm=~, whereJexis the\nvalue of the isotropic AFM exchange.\nThe presence of the undamped mode identi\fed here,\nillustrates how lowering the symmetry of the electronic\nbath (by spin-orbit interaction) may induce a conserva-\ntion law in the localized spin subsystem. Based on this\nmicroscopic mechanism we provide qualitative arguments\nin favor of a generality of the giant Gilbert damping\nanisotropy in a 2D metalic AFM with spin-orbit cou-\npling. Even though the undamped mode cannot be asso-\nciated with a single spin-wave or a magnon, its presence\nhas a strong impact on the nonequilibrium N\u0013 eel vector\ndynamics in 2D Rashba AFMs.\nApart from the Gilbert damping our results extend to\ncover conductivity and spin-orbit torques in the Rashba\nhoneycomb AFM model. We also demonstrate how weak\nanisotropy of all these quantities emerge with Fermi en-\nergies approaching the AFM band gap.\nII. PHENOMENOLOGY OF AFM DYNAMICS\nIn this paper, we choose to describe the AFM with a\nclassical Heisenberg model for localized spins SX=SnX\non two sublattices X=A;B. The spins have the same3\nmodulusSand antiparallel directions nA=\u0000nBin the\nground state. The AFM Heisenberg model is coupled to\nan e\u000bective tight-binding model of conduction electrons\n(see Appendix A) by means of exchange interaction,\nHsd=\u0000JX\niX\n\u001b\u001b0Si\u0001\u001b\u001b\u001b0cy\ni\u001bci\u001b0; (1)\nwhereJstands for an s-d-like exchange energy that is the\nsame onAandBsublattices, the operators cy\ni\u001b(ci\u001b) are\nthe standard creation (annihilation) operators for an elec-\ntron on the lattice site iwith the spin index \u001b, and the no-\ntation\u001b= (\u001bx;\u001by;\u001bz) represents the three-dimensional\nvector of Pauli matrices.\nThe real-time dynamics of AFM is, then, de\fned\nby two coupled di\u000berential equations (Landau-Lifshitz-\nGilbert equations) on the unit vectors nAandnB,\n_nA=HA\u0002nA+ (JA=~)nA\u0002sA; (2a)\n_nB=HB\u0002nB+ (JA=~)nB\u0002sB; (2b)\nwhere dot stands for the time derivative, sXis the spin\ndensity of conduction electrons on the sublattice X,\nsA,B(r) =1\n2X\ni\u001b\u001b0D\ncy\ni\u001b\u001b\u001b\u001b0ci\u001b0E2\nA; (3)\nandAis the area of the unit cell in the AFM. The no-\ntationsHA,Brefer to e\u000bective \felds on the sublattices A\nandBthat are de\fned by the Heisenberg model.\nFor an isotropic antiferromagnet, one \fnds an e\u000bec-\ntive \feld28HA+HB=Jexm=~+ 2H, whereHis an\nexternal magnetic \feld in frequency units and Jexis a\ndirect antiferromagnetic exchange energy that is one of\nthe largest energies in the problem. In turn, the combi-\nnationHA\u0000HBis proportional to magnetic anisotropy\nthat we do not specify in this paper.\nMagnetization dynamics in AFM is conveniently for-\nmulated in terms of the N\u0013 eel and magnetization vectors,\nn=\u0000\nnA\u0000nB\u0001\n=2;m=\u0000\nnA+nB\u0001\n=2;(4)\nthat remain mutually perpendicular n\u0001m= 0 and yield\nthe constraint n2+m2= 1. The dynamics necessarily\ninduces a \fnite nonequilibrium magnetization vector m,\nwhile the condition m\u001c1 remains to be ful\flled.\nFrom Eqs. (2) we obtain\n_n=\u0000\nn\u0002m+H\u0002n+n\u0002s++m\u0002s\u0000;(5a)\n_m=H\u0002m+m\u0002s++n\u0002s\u0000; (5b)\nwhere \n = 2 JexS=~ands\u0006=JA(sA\u0006sB)=2~. In\nEqs. (5) we have deliberately skipped terms that are in-\nduced by anisotropy of AFM exchange since the latter\ndepend on particularities of the AFM Heisenberg model\nthat we do not discuss here.\nThe vectors+is proportional to average polariza-\ntion of conduction electrons, while the vector s\u0000is pro-\nportional to the staggered polarization. The quantities\n−2.50.02.5\nvp/∆−2024ε/∆θ=π/2\n−2.50.02.5\nvp/∆θ=π/4K-valley K/prime-valley\n−2.50.02.5\nvp/∆θ= 0FIG. 2. Electronic band structure of the honeycomb AFM\nmodel of Eq. (9) for di\u000berent orientations of the N\u0013 eel vec-\ntor (nz= cos\u0012). Two-dimensional momenta pare measured\nwith respect to the wave-vectors KandK0that specify two\nnonequivalent valleys. Deviation of the N\u0013 eel vector from the\nperpendicular-to-the plane con\fguration ( \u0012= 0) lifts the val-\nley degeneracy. We restrict our analysis to the metallic regime\nwith Fermi energies corresponding to two Fermi surfaces per\nvalley (an example is shown by a black dotted line). The\nenergy scale \u0001 characterizes the strength of s-dexchange in-\nteraction.\ns\u0006=s\u0006\n0+\u000es\u0006contain equilibrium contributions s\u0006\n0that\ncharacterize various interactions induced by conduction\nelectrons. These contributions do renormalize the pa-\nrameters of the AFM Heisenberg model and are not the\nsubject of the present paper.\nThe nonequilibrium contributions \u000es\u0006originate from\nvarious forces applied to conduction electrons. One nat-\nural example is the electric \feld that not only induces an\nelectric current in the sample but also contributes to \u000es\u0006.\nThe electric \feld can be further related to electric current\nby the resistivity tensor. The response of spin densities\nto electric current de\fnes the so-called spin-orbit torques\nin Eqs. (5) that we also compute.\nSimilarly, the response of \u000es\u0006to the time derivatives _n\nand _mdescribe various types of Gilbert damping induced\nby conduction electrons. Quite generally, such a response\ncan be written in the form of a tensor\n\u0012\n\u000es+\n\u000es\u0000\u0013\n=\u0012\n^\u000bm^\u000bmn\n^\u000bnm ^\u000bn\u0013\u0012\n_m\n_n\u0013\n; (6)\nwhere all tensor components may themselves depend on\nthe vectorsnandm.\nGilbert dampings, in their original meaning, corre-\nspond to the contributions to \u000es\u0006that are symmet-\nric under the time reversion. The terms that change\nsign should, more appropriately, be referred to as ef-\nfective spin renormalizations. Both types of terms are,\nhowever, obtained from the microscopic analysis of the\nGilbert damping tensors in Eq. (6) similarly to the case\nof ferromagnets64.4\nTime reversion, mentioned above, applies exclusively\nto the Heisenberg model, while keeping the tight-binding\nmodel (a bath) non-reversed. In other words we do not\nreverse the electron scattering time \u001c. Such a de\fnition\nhelps to identify the dissipative (even with respect to\nthe time reversion) contributions to \u000es\u0006that describe\nGilbert dampings. These contributions must, however,\nchange sign under the transformation \u001c!\u0000\u001c, because\nspin densities s\u0006are always odd with respect to complete\ntime reversion (the one which also includes that of the\nelectron bath). We will see below, indeed, that all Gilbert\ndampings are proportional to the scattering time \u001cin the\nsame way as the longitudinal conductivity does.\nBefore we proceed with the microscopic analysis of \u000es\u0006\nfor a particular model, it is instructive to draw some gen-\neral consequences for Eqs. (5) based on symmetry argu-\nments in the case of collinear AFM with sublattice sym-\nmetry and spin-rotational invariance (i. e. for vanishing\nspin-orbit interaction).\nAssuming that deviations from the AFM ground state\nremain small we shall limit ourselves to the linear order\ninmin Eq. (7a) and to the quadratic order in min\nEq. (7b). Thus, we shall retain terms up to linear order\ninmin the tensors ^ \u000bm, ^\u000bnm, and ^\u000bmnand terms up to\nquadratic order in min ^\u000bn.\nMixing tensors ^ \u000bmnand ^\u000bnmmust be odd in m, which\nimplies, for our precision, a linear in mapproximation.\nAs a result, the sublattice symmetry (the symmetry with\nrespect to renaming AandB) prescribes that the mix-\ning tensors must also be linear in n. In the absence of\nspin-orbit coupling we are also restricted by spin-rotation\ninvariance that (together with the sublattice and time-\nreversion symmetries) dictates the following form of the\nGilbert damping contributions to the non-equilibrium\nspin densities\n\u000es+=\u000bm_m+\u000b0\nmn\u0002(n\u0002_m)+\u000bmnm\u0002(n\u0002_n);(7a)\n\u000es\u0000=\u000bn_n+\u000b0\nnm\u0002(m\u0002_n) +\u000bnmn\u0002(m\u0002_m);(7b)\nwhere all coe\u000ecients are assumed to be constants.\nIt is easy to see that the vector forms n\u0002(m\u0002_n) and\nm\u0002(n\u0002_m), which could have respectively entered the\nspin densities \u000es+and\u000es\u0000, do not contribute to Eqs. (5)\nin the precision explained above. Substitution of Eqs. (7)\ninto Eqs. (5) gives\n_n=\u0000\nn\u0002m+H\u0002n+ \u0016\u000bmn\u0002_m+\u000bnm\u0002_n;(8a)\n_m=H\u0002m+\u000bnn\u0002_n\n+ \u0016\u000bmm\u0002_m+\r(n\u0002m)(n\u0001_m)\u0000\u000b0\nnm2n\u0002_n;(8b)\nwhere \u0016\u000bm=\u000bm\u0000\u000b0\nmand\r=\u000bmn+\u000bnm+\u000b0\nm\u0000\u000b0\nn.\nDiscarding the three last terms in Eq. (8b), which are all\nof the second order in m, we indeed arrive at a set of\nGilbert damping terms that is widely used in the AFM\nliterature57,58,60.\nThe symmetry consideration behind Eqs. (8) has es-\nsentially relied upon the spin-rotation invariance. This\nalso implies \u000bn= 0 as has been pointed out in the in-\ntroductory section. The coe\u000ecient \u000bmcan, in turn, be\fnite and large, even in the absence of spin-orbit inter-\naction. As we will show below, the presence of spin-\norbit interaction does not only provide us with a \fnite\n\u000bnbut also drastically change the symmetry structure of\nEqs. (8). We will demonstrate that the onset of spin-orbit\ninteraction strongly a\u000bects the coupling of the localized\nspin subsystem to the electron bath (described by the\ntight-binding model) resulting in a strong reduction in\nthe ability of conduction electrons to \rip spins in certain\ndirections and, therefore, to impose a friction on magne-\ntization dynamics.\nIn the following, we turn to the microscopic analysis\nof the conductivity (Sec. IV), spin-orbit torques (Sec. V)\nand Gilbert dampings (Sec. VI) in a particular model\nof Rashba honeycomb AFM that has been put forward\nrecently by some of the authors65. Rashba spin-orbit in-\nteraction breaks spin-rotational invariance of the model\nby singling out the direction ^zperpendicular to the 2D\nplane. We, therefore, investigate how such spin-rotation\nbreaking manifests itself in the anisotropy of the above-\nmentioned quantities.\nIII. MICROSCOPIC MODEL\nFor the sake of a microscopic analysis we adopt a sub-\nlattice symmetric s-d-like model of a 2D honeycomb an-\ntiferromagnet with Rashba spin-orbit coupling, that was\nintroduced in Ref. 65. The energy dispersion of this\nmodel is illustrated schematically in Fig. 2. The low en-\nergy model for conduction electrons responsible for the\ndispersion in Fig. 2, is described by an e\u000bective Hamilto-\nnian (see Appendix A) that in a valley-symmetric rep-\nresentation reads\nHe\u000b=vp\u0001\u0006+1\n2\u0015[\u001b\u0002\u0006]^z\u0000\u0001n\u0001\u001b\u0006z\u0003z+V(r):(9)\nHere\u0006,\u0003, and\u001bare the vectors of Pauli matrices in\nsublattice, valley and spin space, respectively, vis the\ncharacteristic Fermi velocity, while \u0015and \u0001 =JSare\nthe energy scales characterizing the strength of Rashba\nspin-orbit coupling and s-d-like exchange energy, corre-\nspondingly.\nThe termV(r) stands for a scalar Gaussian white-noise\ndisorder potential, which is proportional to the unit ma-\ntrix in sublattice, valley and spin space. The potential\nhas a zero mean value hV(r)i= 0 and is fully character-\nized by the pair correlator,\nhV(r)V(r0)i= 2\u0019(~v)2\u000bd\u000e(r\u0000r0); (10)\nwhere the angular brackets denote the averaging over dis-\norder realizations. The dimensionless parameter \u000bd\u001c1\nquanti\fes the disorder strength.\nThe disorder potential is responsible for a momentum\nrelaxation of conduction electrons. Exchange interaction\nand spin-orbit scattering (or the scattering on a non-\ncollinear con\fgurations with m6= 0) enable coupling be-\ntween localized angular momenta and kinetic momenta5\nof electrons. Together these mechanisms form a channel\nto dissipate angular momentum of localized spins into\nthe lattice. Thus, our model provides us with a micro-\nscopic framework to study dissipative quantities such as\nGilbert dampings, anti-damping spin-orbit torques and\nconductivity that we compute below. We also note that\nthe computation of spin-relaxation time can be directly\nrelated to our analysis of Gilbert damping66,67.\nThe spectrum of the model (9) with V(r) = 0 consists\nof two electron and two hole branches for each of the\nvalleys as illustrated in Fig. 2,\n\u000fe\n\u0006;&(p) =p\nv2p2+ \u00012\u0006&\u0015\u0001nz+\u00152=4\u0007\u0015=2;(11a)\n\u000fh\n\u0006;&(p) =\u0000p\nv2p2+ \u00012\u0007&\u0015\u0001nz+\u00152=4\u0006\u0015=2;(11b)\nwhere&=\u0006is the valley index. All spectral branches\nare manifestly isotropic with respect to the direction of\nthe electron momentum pirrespective of the N\u0013 eel vector\norientation (as far as m= 0).\nIn order to limit the complexity of our microscopic\nanalysis we restrict ourselves to the metallic regime that\ncorresponds to the Fermi energy \"F>\u0001 +\u0015above\nthe minimum of the top electron branches, \u000fe\n+;&(p), as\nshown schematically in Fig. 2. Note that the Fermi en-\nergy\"Fis counted in the model from the center of the\nAFM gap. We also focus on the limit of weak disorder\n\"F\u001c=~\u001d1 where\u001c=~=(\u0019\u000bd\"F) stands for the electron\nscattering time. Also, in order to describe spin-orbit in-\nduced anisotropy we \fnd it convenient to decompose the\nN\u0013 eel vector (as well as the magnetization vector) to the\nin-plane and perpendicular-to-the-plane components as\nn=nk+n?, wheren?=nz^z.\nIV. CONDUCTIVITY\nThe electric conductivity in the metallic regime is dom-\ninated by electron di\u000busion. Despite the fact that the\nFermi surface (line) is isotropic and does not depend on\nthe direction of nk, the conductivity appears to be weakly\nanisotropic with respect to in-plane rotations of the N\u0013 eel\nvector due to the onset of spin-orbit interaction. In par-\nticular, fornz= 0 we \fnd the diagonal conductivity com-\nponents\n\u001bxx=4e2\nh\"F\u001c\n~\"2\nF\u0000\u00012\n\"2\nF+ 3\u00012; (12a)\n\u001byy=\u001bxx+4e2\nh\"F\u001c\n~\"2\nF\n\"2\nF+ \u00012\u00152\u00012\n\"4\nF+\"2\nF\u00012+ 2\u00014;\n(12b)\nwhere the principal axes correspond to choosing ^xdirec-\ntion alongnk(see Fig. 1). In the deep metal regime, and\nfor a general direction of n, this anisotropy is evidently\nsmall\n\u001axx\u0000\u001ayy\n\u001axx+\u001ayy=\u00152\u00012\n\"4\nF(1\u0000n2\nz); \"F\u001d\u0015+ \u0001; (13)where\u001aaa= 1=\u001baais the corresponding resistivity tensor\ncomponent. We note that the anomalous Hall conduc-\ntivity is identically vanishing in the model of Eq. (9).\nThe results of Eq. (12) and all subsequent results of our\npaper are technically obtained from linear response Kubo\nformulas evaluated in the di\u000busive approximation (ladder\ndiagram summation). The details of these calculations\ncan be found in Appendixes B, C, and D.\nV. SPIN-ORBIT TORQUE\nBefore proceeding with the microscopic analysis of\nGilbert damping we shall discuss the e\u000bects of spin-orbit\ninduced anisotropy for spin-orbit torques in the model of\nEq. (9). Since this anisotropy appears to be weak in the\nmetal regime, we shall touch on it only brie\ry.\nAs was already mentioned, the spin-orbit torques origi-\nnate in the response of nonequilibrium spin polarizations\n\u000es\u0006to electric current. Technically, we compute \frst\nthe response of \u000es\u0006to electric \feld and, then, express\nthe electric \feld in terms of 2D electric current density j\nusing the conductivity tensor of Eq. (12).\nA straightforward computation of such a response gives\n\u000es\u0000= 0 (see Appendixes B and C for more detail) and\n\u000es+=a(n2\nz)^z\u0002j+b(n2\nz)nk\u0002(nk\u0002(^z\u0002j))\n+c(n2\nz)nk\u0002(n?\u0002(^z\u0002j)); (14)\nwhere the coe\u000ecients a,bandcdo generally depend on\nn2\nz= 1\u0000n2\nx\u0000n2\nyand are shown in Fig. 3. It is appropriate\nto recall here that the computation of the responses from\nthe model of Eq. (9) refers to the case when m= 0. The\nsymmetry form of Eq. (14) in this case has been also\nestablished recently from numerical simulations65.\nImportantly, the \frst term in the right-hand side of\nEq. (14) represents the well-known Rashba-Edelstein\ne\u000bect68, while the other two terms represent higher har-\nmonics of the same \feld-like e\u000bect that arise due to spin-\nrotation symmetry breaking. Anti-damping like torques\n(that are even under time-reversal) are vanishing in the\nmodel due to the valley symmetry constraint. This sym-\nmetry reads \u0003 xH[n]\u0003x=H[\u0000n], from which it follows\nthat the response of \u000es+to charge current must be an\neven function of n.\nThe behavior of the coe\u000ecients a,bandcas a function\nofnzis illustrated in Fig. 3 for two di\u000berent choices of\nthe Fermi energy. For in-plane N\u0013 eel vector orientations\n(nz= 0) we \fnd\na=a01 + 3\u000e2\n1 + 2 \u0016\u00152\u000e2+\u000e4\u00002\u000e6; (15a)\nb= 2a\u000e21\u00002\u0016\u00152\u00004\u000e2+\u000e4\n1 + 2\u000e2\u00003\u000e4; (15b)\nc=\u00002a\u000e21 + 2 \u0016\u00152\u000e2\u00002\u000e2\u00003\u000e4\u00004\u000e6\n1 + 4\u000e2+ 5\u000e4+ 6\u000e6; (15c)6\nFIG. 3. The coe\u000ecients a,b, andcin Eq. (14) as a function of\nthe direction of the N\u0013 eel vector, nz= cos\u0012, for two di\u000berent\nFermi energies: \"F= 4\u0001 (left panel) and \"F= 16\u0001 (right\npanel). We use \u0015= 1:5\u0001. Fornz= 0 the results correspond\nto Eqs. (15).\nwhere\na0=AJ\ne~v\u0015\n\"F; \u0016\u0015=\u0015\n\"F; \u000e =\u0001\n\"F: (16)\nIn the metal regime, \"F\u001d\u0015+\u0001, the results of Eqs. (15)\nare reduced to\na=AJ\ne~v\u0015\n\"F; b =\u0000c= 2AJ\ne~v\u0015\n\"F\u0012\u0001\n\"F\u00132\n:(17)\nOne can, therefore, see that the high harmonics terms\n(proportional to bandc) become irrelevant in the metal\nregime.\nVanishing response of the staggered polarization,\n\u000es\u0000= 0, for the model of Eq. (9) is a simple consequence\nof the sublattice symmetry. As shown below the presence\nof a \fnite, though small, mbreaks such a symmetry and\nleads to a \fnite \u000es\u0000. Taking into account a linear in m\nterm in the Hamiltonian is also necessary to obtain \fnite\nmixed Gilbert damping tensors ^ \u000bnmand ^\u000bmnin Eq. (6).\nA low-energy model that takes into account \fnite mag-\nnetization vector reads (see also Appendix D)\nH=He\u000b\u0000\u0001m\u0001\u001b; (18)\nwhereHe\u000bis given by Eq. (9). The conductivity tensor\ndoes not acquire a linear in mterms in the leading order\nwith respect to the large metal parameter \"F\u001c=~, because\nthe anomalous Hall e\u000bect remains subleading with respet\nto the metal parameter. Similarly, the result of Eq. (14)\nis not a\u000bected by the linear in mcorrections.\nHowever, the direct computation of the staggered po-\nlarization response (in the linear order with respect to m)\ngives rise to a \fnite result. In the limit of large Fermi\nenergy\"F\u001d\u0015+ \u0001, we \fnd\n\u000es\u0000=AJ\ne~v\u0015\n\"F\u0012\u0001\n\"F\u00132h\n2n?\u0002(m?\u0002(^z\u0002j)) (19)\n+ 2mk\u0002(n?\u0002(^z\u0002j))\u00003nk\u0002(m?\u0002(^z\u0002j))i\n;where the overall strength of the e\u000bect is again of the\norder of the coe\u000ecients bandc. This makes the e\u000bects\nof nonequilibrium staggered polarization (including the\ncelebrated N\u0013 eel spin-orbit torque) irrelevant in the metal\nregime. Indeed, staggered polarization can hardly be in-\nduced by electrons with wavelengths that strongly exceed\nthe distance between AandBsublattices.\nThe results of Eqs. (14), (15) clearly suggest that the\nonly torques surviving in the large energy limit are those\nrelated to non-equilibrium polarization \u000es+=a0^z\u0002j,\nwhich is nothing but the standard Rashba-Edelstein\ne\u000bect68. These torques have a form Tn=a0n\u0002(^z\u0002j) in\nthe right-hand side of Eq. (5a) and Tm=a0m\u0002(^z\u0002j)\nin the right-hand side of Eq. (5b). The anisotropy of\ntorques is, however, irrelevant in this limit.\nVI. GILBERT DAMPING\nSurprisingly, the situation is di\u000berent when we consider\nGilbert damping terms. In this case we \fnd that the gi-\nant anisotropy of Gilbert damping persists to arbitrarily\nlarge Fermi energy as soon as spin orbit energy \u0015exceeds\n~=\u001c. The latter condition ensures that the scattering be-\ntween spin-split subbands is suppressed.\nThe direct computation of the Gilbert damping tensors\nfor\u0015\u001d~=\u001cgives\n\u000es+=\u000bk\nm_mk+\r\u0000mm+\r\u0000mn; (20a)\n\u000es\u0000=\u000b?\nn_n?+\r\u0000nm+\r\u0000nn; (20b)\nwhere the terms \u0000 abcontain various vector forms.\nFar in the metal regime, \"F\u001d\u0015+ \u0001, we \fnd\n\u000bk\nm= 2\"F\u001c\n~AJ2S\n\u0019~2v2\u0014\n1\u0000\u00012\n\"2\nF(2 +n2\nz) +:::\u0015\n;(21a)\n\u000b?\nn=\"F\u001c\n~AJ2S\n\u0019~2v2\"\u0012\u0015\n\"F\u00132\n+:::#\n; (21b)\n\r= 2\"F\u001c\n~AJ2S\n\u0019~2v2\"\u0012\u0001\n\"F\u00132\n+:::#\n; (21c)\nwhile the vectors forms \u0000 abcan be written as\n\u0000mm=n\u0002(n\u0002_m) +nk\u0002(nk\u0002_mk)\n\u00002nk\u0002(nk\u0002_m?); (22a)\n\u0000mn=n\u0002(mk\u0002_n?)\u0000m?\u0002(nk\u0002_n?)\n+n?\u0002(m?\u0002_nk)\u0000nk\u0002(mk\u0002_nk)\n\u00003m?\u0002(n?\u0002_nk); (22b)\n\u0000nm= 2nk\u0002(mk\u0002_m?) + 2mk\u0002(n?\u0002_mk)\n\u0000n?\u0002(m?\u0002_m) + 2m?\u0002(n\u0002_m)\n+mk\u0002(n\u0002_m?)\u0000m?\u0002(n?\u0002_mk);(22c)\n\u0000nn=\u0000m\u0002(n\u0002_nk): (22d)\nThus, we see from Eqs. (21) that the coe\u000ecients \u000b?\nnand\n\rare vanishingly small in the metal regime. Moreover,7\nin the limit \"F\u001d\u0001 the only non-vanishing contribu-\ntions to Gilbert dampings are given by the \frst terms\non the right-hand sides of Eqs. (20) that are manifestly\nanisotropic.\nThe onset of spin-orbit interactions therefore makes\nGilbert dampings ultimately anisotropic, also in the deep\nmetal regime. This is in contrast to conductivity and\nspin-orbit torques that are quickly becoming isotropic in\nthe metal limit. For \"F\u001d\u0015+ \u0001, we \fnd the well known\nLandau-Lifshitz-Gilbert equations\n_n=\u0000\nn\u0002m+H\u0002n+ \u0016\u000bk\nmn\u0002_mk+\u000b?\nnm\u0002_n?;\n_m=H\u0002m+\u000b?\nnn\u0002_n?+ \u0016\u000bk\nmm\u0002_mk; (23)\nwhere we again omit terms that originate e. g. from mag-\nnetic anisotropy of the AFM. Eqs. (23) are clearly dif-\nferent from Eqs. (8) derived on the basis of symmetry\nanalysis in the absence of spin-orbit interaction.\nThe very pronounced, highly anisotropic Gilbert\ndamping terms in the Landau-Lifshitz-Gilbert equations\nof Eqs. (23) represent the main result of our paper. The\nphenomenon of the giant Gilbert damping anisotropy in\nthe 2D AFM clearly calls for a qualitative understanding\nthat we provide in Sec. VII.\nVII. QUALITATIVE CONSIDERATION\nThe results of Eqs. (20), (21) suggest that the\nanisotropy of Gilbert damping is most pronounced in the\nmetal limit, \"F\u001d\u0001 +\u0015as far as\u0015\u001c=~\u001d1. In partic-\nular, certain spin density responses are vanishing in this\nlimit. One of them is the response of the average spin\ndensity\u000es+\nzto _mzthat is de\fned by the tensor com-\nponent\u000bzz\nmin Eq. (6). The other four vanishing tensor\ncomponents \u000bxx\nn,\u000bxy\nn,\u000byx\nnand\u000byy\nncorrespond to the re-\nsponses of the in-plane staggered spin densities \u000es\u0000\nxand\n\u000es\u0000\nyto _nxand _ny.\nIt is important to stress that the component \u000bzz\nmis not\nonly \fnite but also quite large in the absence of spin-orbit\ninteraction, i. e. for \u0015= 0. It is, therefore, instructive to\nunderstand how the onset of spin-orbit interaction may\ncancel\u000bzz\nmresponse and lead to the conservation of z\nprojection of magnetization vector.\nSuch a qualitative understanding can be achieved by\nconsidering the Kubo-Greenwood formula for \u000bzz\nmfor the\nmodel of Eq. (18) in the limit \u0001 !0 and\u001c!1 ,\n\u000bzz\nm/X\npX\ns;s0=\u0006jh\tp;sj\u001bzj\tp;s0ij2\u000e(\"F\u0000\u000fe\np;s)\u000e(\"F\u0000\u000fe\np;s0);\n(24)\nwhere\u000fe\np;\u0006=p\nv2p2+\u00152=4\u0007\u0015=2 correspond to the two\nelectronic branches of Eq. (11a) that are evidently valley\ndegenerate in the limit \u0001 !0.\nThe states \t p;sare simply the eigenstates of theHamiltonian H0=vp\u0001\u0006+ (\u0015=2) [\u001b\u0002\u0006]z,\nH0=0\nB@0 0 v(px\u0000ipy) 0\n0 0 \u0000i\u0015 v (px\u0000ipy)\nv(px+ipy)i\u0015 0 0\n0v(px+ipy) 0 01\nCA;\n(25)\nthat can be explicitly written as\n\tp;\u0006=1\n2p\nv2p2\u0007\u0015\u000fe\n\u0006=20\nBB@vpe\u0000i\u001e\n\u0006i\u000fe\n\u0006\n\u000fe\n\u0006\n\u0006ivpei\u001e1\nCCA; (26)\nwhere we have used px=pcos\u001e,py=psin\u001e.\nOne may notice that h\tp;sj\u001bzj\tp;si= 0 for any value\nof\u0015suggesting that the response function \u000bzz\nmin Eq. (24)\nis vanishing. This is, however, not the case for \u0015= 0. In-\ndeed, in the absence of spin-orbit interaction the electron\nbranches become degenerate \u000fe\np;\u0006=vpsuch that the in-\nplane spin-\rip processes contribute to the Kubo formula,\nh\tp;+j\u001bzj\tp;\u0000ij\u0015=0=h\tp;\u0000j\u001bzj\tp;+ij\u0015=0= 1:(27)\nThese processes are exactly the ones responsible for a\n\fnite Gilbert damping component \u000bzz\nmin the absence of\nspin-orbit interaction. The spin-orbit induced splitting\nof the subbands forbids these spin-\rip processes as soon\nas\u0015\u001c=~\u001d1 and leads to a giant anisotropy of Gilbert\ndamping in the metal limit. Indeed, the other elements\nof the Gilbert damping tensor \u000bxx\nmand\u000byy\nmremain \fnite\nirrespective of the subband splitting,\nh\tp;\u0006j(\u001bx+i\u001by)j\tp;\u0006i=\u0006ivpei\u001e\np\nv2p2+\u00152=4: (28)\nOne can further show that for \u0015= 0 the entire Gilbert\ndamping tensor ^ \u000bmbecomes isotropic ^ \u000bxx\nm= ^\u000byy\nm= ^\u000bzz\nm\nas it have been expected on the basis of the symmetry\nanalysis.\nVery similar physics is also responsible for the\nanisotropy of the tensor ^ \u000bn. It is worth noting that\nthe same type of anisotropy is known to take place in\nthe limit of large spin-orbit interaction in 2D Rashba\nferromagnets64. Spin-orbit induced anisotropy of Gilbert\ndamping plays, however, a lesser role in 2D ferromagnets\ndue to the much stricter constraint on the single mag-\nnetization vector. A less restricted dynamics of mand\nnvectors make the Gilbert damping anisotropy play a\nbigger role in 2D AFMs.\nIndeed, it can be directly seen from Eqs. (23) that a\nnonequilibrium state with m=m^zandn=nkbecomes\nundamped in the absence of external \feld H= 0. Such\na state corresponds to the undamped N\u0013 eel vector pre-\ncession around ^zaxis with a frequency given by Jexm.\nThe state clearly survives in the presence of easy plane\nmagnetic anisotropy in the AFM. We believe that such\na phenomenon remains to be rather generic for a vari-\nety of 2D or layered AFM systems with strong spin-orbit\ncoupling of Rashba type.8\nVIII. CONCLUSIONS\nIn this paper, we demonstrate that the presence of suf-\n\fciently strong spin-orbit coupling \u0015\u001c=~\u001d1 results in\nthe ultimate anisotropy of the Gilbert damping tensor\nin the metal regime, \"F\u001d\u0001 +\u0015. One can trace the\nphenomenon to the spin-orbit induced splitting of Fermi\nsurfaces that forbids scattering processes that are respon-\nsible for the relaxation of certain magnetization and N\u0013 eel\nvector components.\nWe also demonstrate that a \fnite in-plane projection\nnkof the N\u0013 eel vector is responsible for a weak anisotropy\nof conductivity and spin-orbit torques for Fermi energies\napproaching the band edge, \"F\u0018\u0001+\u0015. This anisotropy\nis, however, absent in the metallic regime.\nGilbert damping is, however, in the absence of spin-\norbit interaction as it is required by symmetry consider-\nations. Thus, we demonstrate that the onset of Rashba\nspin-orbit interaction in 2D or layered AFM systems\nleads to a giant anisotropy of Gilbert damping in the\nmetallic regime. The physics of this phenomenon origi-\nnates in spin-orbit induced splitting of the electron sub-\nbands that destroys a particular decay channel for mag-\nnetization and leads to undamped precession of the N\u0013 eel\nvector. The phenomenon is based on the assumption that\nother Gilbert damping channels (e. g. due to phonons)\nremain suppressed in the long magnon wavelength limit\nthat we consider. The predicted giant Gilbert damp-\ning anisotropy may have a profound impact on the N\u0013 eel\nvector dynamics in a variety of 2D and layered AFM ma-\nterials.\nACKNOWLEDGMENTS\nWe are thankful to I. Ado, H. Gomonay and J. Sinova\nfor fruitful discussions. This research was supported by\nthe JTC-FLAGERA Project GRANSPORT. D.Y. and\nM.T. acknowledge the support from the Russian Science\nFoundation Project No. 17-12-01359. A.P. acknowledges\nsupport from the Russian Science Foundation Project\n18-72-00058. The work of D.Y. was also supported by\nthe Swedish Research Council (Vetenskapsr\u0017 adet, 2018-\n04383). M.T. is especially thankful to the KITP visitor\nprogram \\Spintronics Meets Topology in Quantum Ma-\nterials\". O.E. acknowledges support from the Swedish\nResearch Council (Vetenskapsr\u0017 adet) and the Knut and\nAlice Wallenberg foundation.\nAppendix A: Model system\nIn this Appendix, we shall brie\ry justify Eqs. (9) and\n(18) of the main text. We start from an s-d-like model\nfor two-dimensional antiferromagnet on a honeycomb\nlattice65. The model includes a local exchange interac-\ntion between localized magnetic moments and conduction\nelectron spins as given by Eq. (1). Itinerant electrons inthe model are, therefore, governed by the tight-binding\nHamiltonian\nH0=\u0000tX\niX\n\u001b\u001b0cy\ni\u001bci\u001b0\u0000JX\niX\n\u001b\u001b0Si\u0001\u001b\u001b\u001b0cy\ni\u001bci\u001b0\n+i\u0015\n3aX\nhi;jiX\n\u001b\u001b0^z\u0001(\u001b\u0002dij)\u001b\u001b0cy\ni\u001bcj\u001b0; (A1)\nwhere we do ignore disorder for a moment. The model\nis characterized by the nearest neighbor hopping energy\ntand the Rashba spin-orbit coupling energy \u0015,z-axis is\naligned perpendicular to the two-dimensional plane, the\nin-plane vectors dijconnect the neighboring sites iandj\non a honeycomb lattice. For any site ion the sublattice\nAwe choose\nd1=a\u0012\n0\n1\u0013\n;d2=a\n2\u0012p\n3\n\u00001\u0013\n;d3=\u0000a\n2\u0012p\n3\n1\u0013\n;\n(A2)\nwhereais the length of the bond between AandB.\nBy projecting the tight-binding model of Eq. (A1) on\nstates in a vicinity of the valley wave-vectors,\nK=4\u0019\n3p\n3a\u0012\n1\n0\u0013\n;andK0=\u0000K; (A3)\nwe \fnd, in the valley symmetric approximation, the ef-\nfective Hamiltonian of Eq. (9) with the assumption that\nSA=\u0000SB, wherev= 3ta=2~. By relaxing the assump-\ntion we obtain the model of Eq. (18).\nAppendix B: Linear Response tensors\nIn order to keep technical expressions compact we let\n~= 1 and\"F=\"below. Our technical analysis is based\non linear response of electron spin density to various per-\nturbations at zero frequency ( dc) limit. In particular, we\nconsider three types of responses: the one with respect\nto electric current (via electric \feld and inverse conduc-\ntivity tensor), the one with respect to the time derivative\nof the N\u0013 eel vector and the other one with respect to the\ntime derivative of magnetization vector. These responses\nare summed up as\n\u000es+=^SSOT\n+j+^SGD\nmn_n+^SGD\nm_m; (B1a)\n\u000es\u0000=^SSOT\n\u0000j+^SGD\nnm_m+^SGD\nn_n; (B1b)\nwhere we de\fne the response tensors ^SSOT\n\u0006 that are\ndescribing spin-orbit torques (both \feld-like and anti-\ndamping) and various ^SGDtensors that are describing\nvarious contributions to Gilbert dampings (and to e\u000bec-\ntive spin renormalizations)64.\nIn order to compute the linear response tensors in\nEqs. (B1) we apply the standard Kubo formula\n\u000es\u0006\n\u000b=J2Sv2A\n2VX\n\fcTrD\n^GR^s\u0006\n\u000b^GA^F\fE@X\f\n@t; (B2)9\nwhereVis the system area, cTr is an operator trace,\n^GR(A)= (\"\u0000H\u0006i0) are retarded (advanced) Green func-\ntion operators, ^ s+\n\u000b=\u001b\u000b, ^s\u0000\n\u000b= \u0003z\u0006z\u001b\u000bare the operators\ncorresponding to the average spin-polarization s+and\nstaggered spin-polarization s\u0000, the product ^F\u0001X(t) rep-\nresents the time-dependent perturbation in the Hamil-\ntonian, while the angular brackets denote the disorder\naveraging that we consider in di\u000busive (ladder) approxi-\nmation.\nThe linear-response formula Eq. (B2) assumes zero\ntemperature and zero frequency limit that corresponds to\ntaking both Green's functions at the same energy \"=\"F.\nWe also neglect the Fermi-sea contribution (also known\nas St\u0014 reda contribution) since such a contribution appears\nto be either zero or subleading in the metal parameter\n\"\u001c\u001d1 with respect to our results.\nThus, in order to compute Gilbert dampings and spin-\norbit torque tensors we consider linear response of \u000es\u0006\nto the three perturbations mentioned above. Each per-\nturbation is parameterized by the term \u000eH=^F\u0001X(t)\nwith\n_X=_n; ^F=\u0000\u0001 \u0003z\u0006z\u001b; (B3a)\n_X=_m; ^F=\u0000\u0001\u001b; (B3b)\n_X= (\u0019v=e )^\u001b\u00001j;^F=\u0006; (B3c)\nwhere ^\u001bis the conductivity tensor (this is computed from\nthe standard Kubo formula which is analogous to the one\nin Eq. (B2) but for the response of current density to\nelectric \feld). The disorder averaging amounts to replac-\ning Green's functions in Eq. (B2) with the corresponding\ndisorder-averaged Green's functions and to replacing one\nof the operators, ^ s\u000bor^F, with the corresponding vertex-\ncorrected operator.\nDisorder-averaged Green's functions become diagonal\nin the momentum space due to restored translational in-\nvariance and take the form GR(A)\np = [\"\u0000H\u0000\u0006R(A)]\u00001,\nwhere the Hamiltonian His de\fned in Eq. (9) of the\nmain text, while the self-energy \u0006R(A)is evaluated in the\nBorn-approximation depicted schematically in Fig. 4a.\nWe \fnd that the real part of the self-energy does renor-\nmalize the Fermi energy \"and thes-dexchange coupling\nstrength \u0001, while the imaginary part reads\nIm \u0006R(A)=\u0007\u0019\u000bd\n2(\"\u0000\u0001 \u0003z\u0006zn\u0001\u001b): (B4)\nIn order to evaluate linear response tensors in the lead-\ning order with respect to the metal parameter \"\u001c\u001d1\none also needs to sum up the ladder diagrams as shown\nin Fig. 4b-c.\nTo do that one de\fnes the vertex corrected operator\n^Fvc=^F+^F(1)+^F(2)+^F(3)+\u0001\u0001\u0001; (B5)\nwhere we denote by ^F(i)the operator ^Fthat is dressed\nby the number of idisorder lines,\n^F(i)= 2\u0019\u000bdZd2p\n(2\u0019)2GR\np^F(i\u00001)GA\np: (B6)\nFIG. 4. Diagrammatic illustration. a) Born-approximation.\nb) Ladder-approximation. c) Disorder-averaged polarization\nbubble. d) Perturbative expansion of the disorder-averaged\npolarization bubble.\nIt appears that the summation in Eq. (B5) can be re-\nduced to geometric series in a \fnite operator space. In-\ndeed, let us de\fne the operator space that is spanned by\n16 operators in each of the valleys\nBi=1\n2\u0003\u0010\u0006\u000b\u001b\f; i=f\u0010;\u000b;\fg; (B7)\nwhereiis a cumulative index with \u0010= 0;za valley parity\nindex and\u000b;\ftaking on the four values f0;x;y;zgeach.\nForB= (B1;B2;:::;B 32) we de\fne the vertex cor-\nrected operator vector as\nBvc=B+FB+F2B+F3B+\u0001\u0001\u0001=1\n1\u0000FB;(B8)\nwhereFstands for a matrix of vertex corrections. Using\nthe normalization condition Tr BiBj= 2\u000eijwe \fnd\nFij=\u0019\u000bdZd2p\n(2\u0019)2Tr\u0002\nGA\npBiGR\npBj\u0003\n; (B9)\nwhere Tr stands for the usual matrix trace in the valley,\nspin and sublattice spaces.\nIt easy to imagine that the matrix inversion in Eq. (B8)\nmight be a daunting analytical task. We note, however,\nthat the matrix Fis evidently diagonal in the valley\nspace, and it can also become block-diagonal in sublattice\nand spin spaces by choosing a more convenient basis.\nA particularly useful choice of basis corresponds to in-\nplane rotation of both spin and sublattice Pauli matrices\nto the frame associated with the in-plane projection nk\nof the N\u0013 eel vector. For spin Pauli matrices this transfor-\nmation is given by\n\u001bx!\u0003znx\u001bx+ny\u001byq\nn2x+n2y; \u001by!\u0003zny\u001bx\u0000nx\u001byq\nn2x+n2y;(B10)\nwhere we took advantage of the fact that the direction of\nnis opposite in the two valleys. The same transformation\n(B10) has to be applied to \u0006 xand \u0006y.\nThe matrixFis instrumental for the analysis of all\nlinear response tensors in Eq. (B1). Indeed, using the10\nde\fnition of Eq. (B9) in Eq. (B2) and summing up the\ndi\u000busion ladders we \fnd\n\u000es\u0006\n\u000b=J2Sv2A\n2\u0019\u000bdX\n\fX\nijTr[^s\u0006\n\u000bBi]RijTr[^F\fBj]@X\f\n@t;\n(B11)\nwhereR=F(1\u0000F)\u00001. Thus, the computation of all\nresponse tensors is reduced in the di\u000busive approximation\nto the computation of the vertex correction matrix Fand\nsubsequent matrix inversion.\nAppendix C: Vertex correction\nStill, \fnding an inverse matrix (1 \u0000F)\u00001is not that\nstraightforward due to a pair of eigenvalues (one per val-\nley) that equal exactly 1. The presence of such eigenval-\nues roots in the particle conservation and is, therefore,\nnot an arti\fcial problem. The unit eigenvalues do evi-\ndently prevent the matrix inversion in Eq. (B8). Nev-\nertheless, it can be shown that the corresponding eigen-\nvectors do not enter the \fnal equations of motion for\nlocalized spins. In the next section, we brie\ry illustrate\nhow one can formally avoid the particle conservation di-\nvergence in the computation of vertex corrections.\nLet us de\fne by a\u0010the eigenvectors of Fthat corre-\nspond to two unit eigenvalues, Fa\u0010=a\u0010, with\u0010= 0;z.\nFor the normalized vector a\u0010we de\fne special operators\n\u0016B\u0010=a\u0010\u0001B=\"\u0000\u0001\u0003\u0010\u0006zn\u0001\u001b\n2p\n\"2+ \u00012; (C1)\nwhich are conserved with respect to impurity dressing\n\u0016B\u0010=\u0016B(i)\n\u0010for any order i. This means that the vertex\ncorrected operator \u0016Bvc\n\u0010is formally diverging in the dc\nlimit. In what follows, we formally write \u0016Bvc\n\u0010=R1\u0016B\u0010,\nwhere the limit R1! 1 is taken at the end of the\ncalculation.\nThe response tensors de\fned by Eqs. (6) consist of\ndi\u000berent correlators of the operators \u0006 \u000b,s+\n\u000b=\u001ba, and\ns\u0000\n\u000b= \u0003z\u0006z\u001b\u000b. It is evident that most of these operators\nare already orthogonal to \u0016B\u0010,\nTr\u0002\n\u0006\u000b\u0016B\u0010\u0003\n= Tr\u0002\ns+\n\u000b\u0016B\u0010\u0003\n= Tr\u0002\ns\u0000\n\u000b\u0016B0\u0003\n= 0; (C2)\nwhile the only dangerous sector is related to the projec-\ntion\nTr\u0002\ns\u0000\n\u000b\u0016Bz\u0003\n=\u00004\u0001n\u000bp\n\"2+ \u00012; (C3)\nwhich is evidently \fnite. The result of Eq. (C3) leads\nto formally diverging contribution \u000es\u0000\ndivthat is generally\npresent in all components of \u000es\u0000,\n\u000es\u0000\ndiv;\u000b/R1X\n\fTr[^s\u0000\n\u000b\u0016Bz] Tr[^F\f\u0016Bz]@n\f\n@t: (C4)One can immediately see, however, that such a diverging\ncontribution corresponds to a particular vector form,\n\u000es\u0000\ndiv;\u000b/R1n\u000bn\u0001@n\n@t= 0; (C5)\nthat manifestly vanishes due to the constraint jnj= 1\nwhich is exact in the limit m= 0. Thus, the diver-\ngency inBvc\ndiv(which originates in the di\u000busion pole of\nthe density-density response) is, in fact, harmless for the\nresponse tensors we are discussing.\nIt is interesting to note that the irrelevance of the di-\nvergency in Bvc\ndivoperator extends to higher orders in m,\neven though it becomes much harder to see. We touch\non this problem in Appendix D.\nAppendix D: Finite magnetization\nThe deviation from a collinear antiferromagnetic order\ncan be accounted by considering a \fnite net magnetiza-\ntion term in the Hamiltonain perturbatively,\nH=He\u000b+U; U =\u0000\u0001m\u0001\u001b: (D1)\nIn the paper, we build the \frst order perturbation theory\nwith respect to U.\nFirst of all, it can be shown that the self-energy ac-\nquires the linear in mcontribution as\nIm \u0006R(A)=\u0007\u0019\u000bd\n2(\"\u0000\u0001 \u0003z\u0006zn\u0001\u001b+ \u0001m\u0001\u001b):(D2)\nSecond, the Dyson expansion of the disorder-averaged\nGreen's functions GR(A)with respect to mreads\nGR(A)!GR(A)+GR(A)UR(A)GR(A); (D3)\nwhereUR(A)=U(1\u0006i\u0019\u000bd=2) and we disregarded terms\nstarting from quadratic order in m. Note, that we\nhave kept the notations GR(A)for the disorder averaged\nGreen's functions of the unperturbed system.\nThe computation of linear response tensors amounts\nto considering an additional contribution to the response\ntensor represented by a complex class of diagrams de-\npicted schematically in Fig. 4d. Before ladder summa-\ntion is applied the diagrams of Fig. 4d correspond to a\ncontribution to the correlator of two operators BiandBj\nof the type\nUij= 2\u0019\u000bdZd2p\n(2\u0019)2Tr\u0002\nGAUAGABiGRBj\n+GABiGRURGRBj\u0003\n; (D4)\nwhich has yet be dressed. The dressing amounts to re-\nplacing both BiandBjoperators with the corresponding\nvertex corrected operators Bvc\niandBvc\nj, respectively.\nThe \fnal result for the response of spin density is still\ngiven by Eq. (B11), where the matrix R=F(1\u0000F)\u00001\nis, however, replaced with\nR=F\n1\u0000F+1\n1\u0000FU1\n1\u0000F; (D5)11\nwhich corresponds to diagrams Fig. 4c-d. It is again con-\nvenient to consider a particular basis for the matrix Fas\nde\fned in Eq. (B10) to simplify analytical computation.\nThe problem of divergence in the operators \u0016B\u0010does\nnow become less trivial. Careful analysis shows that the\nlinear terms in mincluded in Eq. (D5) lead to additional\ndiverging contributions to \u000es\u0000of the form\n\u000es\u0000;(1)\ndiv;\u000b/\u0000R1n\u000bm\u0001@m\n@t; (D6)\nthat is analogous to the one in Eq. (C5) for a \fnite m.\n(We remind that the constraint n2+m2= 1 provides\na relation between these terms). The contribution in\nEq. (D6) is, however, of too high order in min Eq. (5a)\nand cancels out completely in Eq. 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Although, the damping constant is believed to be anisotropic by theories.\nIt is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate\nangle dependent broadband ferromagnetic resonance study of high quality epitaxial La 0:7Sr0:3MnO 3films.\nExtrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold\nsymmetry at room temperature. The observed anisotropic relaxation is attributed to the magnetization\norientation dependence of the band structure. In addition, we demonstrated that such anisotropy can\nbe tailored by manipulating the stain. This work provides new insights to understand the mechanism of\nmagnetization relaxation.\nA. INTRODUCTION\nThe magnetization relaxation process determines the speed of magnetization relaxation and the\nenergy required for current-induced magnetization reversal [1–6]. Understanding the mechanism\nand controlling of magnetization relaxation [7–12], including intrinsic Gilbert damping and extrinsic\neffects, pave the way for ultra-low power and high performance spintronic devices based on spin\ntransferandspinorbittorques[13–15]. IthasbeendemonstratedthatGilbertdampingconstant( \u000b)\ncanbetunedeffectivelybyengineeringthedensityofstatesandspinorbitcoupling(SOC)[9,16–18].\nIn addition, magnetization relaxations subjected to finite size and interfacial effects have also been\nextensively investigated [8, 19, 20]. However, it is still an open question that if magnetic damping\nis anisotropic. In principle, \u000bis magnetization orientation dependent and should be a 3 \u00023 tensor\nin the phenomenological Gilbert equation [21, 22], yet it is often treated as a scalar (isotropic). In\nthe case of polycrystalline thin films prepared by sputtering, such treatment is reasonable due to\nthe smearing of long range structural order. Whereas for single crystal thin films, it is still difficult\nto draw a conclusion due to the lack of convincing experimental evidence. From the view of theo-\nries, the Gilbert damping is determined by two scattering processes, the interband resistivity-like\nscattering and the intraband conductivity-like scattering [12]. Both terms vary with temperature\nthrough their dependence on electron relaxation time. The interband scattering which dominates\ndamping in most ferromagnets becomes isotropic at room temperature [23]. Therefore, anisotropic\nlinewidth in 3d magnetic metals was only observed at low temperature[24]. From the aspect of\nexperimental technique, Seib et al. have predicted that the precession trajectory of magnetization\nin a ferromagnetic resonance (FMR) measurement (standard technique for measuring damping)\nmay partially average out the anisotropy [25]. Hence, detecting the anisotropy in Gilbert damping\nis extremely difficult. Furthermore, the existence of several angle dependent extrinsic contributions\nto damping in most materials further hinders the determination of a possible weak anisotropic\ndamping [11, 26–28]. We note that in a ferromagnet with nearly half-metallic band structure, the\nisotropic interband term is suppressed [29] and the damping can be dominated by the anisotropic\nintraband contribution[23]. Recent reports have claimed the observation of anisotropic damping in\nhalf-metallic Heusler alloy[30, 31]. However, unavoidable chemical disorder [32, 33]of Heusler alloy\nintroduces extrinsic effects such as spin wave scattering hence complicates the verification procedure\nof such anisotropy.\n\u0003heshikun@gmail.com msecj@nus.edu.sg\n2La0:7Sr0:3MnO 3(LSMO) is an oxide perovskite material exhibited half-metallic band structure\nand ultra-low damping at room temperature [34, 35]. In this work, we studied the magnetiza-\ntion relaxation of LSMO films deposited on NdGaO 3(NGO) (110) substrates using angle-resolved\nbroadband ferromagnetic resonance. The purpose of choosing NGO (110) substrates is to utilize its\nnon-equalaandbaxis value. Such asymmetry will potentially lead to non-spherical Fermi surface.\nTwo types of high quality samples with different static magnetic anisotropies were investigated. The\nnormal LSMO film (hereafter denoted as S-LSMO) exhibited weak uniaxial magnetic anisotropy\nwhereas the other with modulated strain relaxation mode (hereafter denoted as W-LSMO) have\nboth uniaxial and cubic anisotropy fields. The angle dependence of the in-plane intrinsic Gilbert\ndampingshowedtwo-foldsymmetryinbothtypeofsamples. Strikingly, theorientationofminimum\ndamping differs 90 degree. This work provided strong evidence of anisotropic nature of magneti-\nzation relaxation and demonstrated the tuning of anisotropy in damping through stress relaxation\nengineering.\nB. RESULTS\nEpitaxial growth of LSMO\nPulsed laser deposition (PLD) was used to deposit LSMO thin films with a thickness of 25nm\non (110) NGO substrates. The energy and repetition frequency of KrF laser (248nm) were 225mJ\nand 2Hz, respectively. During deposition, the substrate temperature was fixed at 950\u000eC. The\noxygen pressure was 225mTorr for S-LSMO and 200mTorr for W-LSMOAfter deposition, S-LSMO\nwas cooled down to room temperature at 10K/min under the oxygen pressure of 1 Torr, whereas\nW-LSMO at 5K/min under the oxygen pressure of 100 Torr in order to promote the modification\nof strain hence micro-structurestructure.\nCrystalline quality analysis\nThe crystallographic structures of the films were characterized by synchrotron high resolution\nX-ray diffraction. Reciprocal space maps (RSMs) taken at room temperature around {013} pc(here\nthe subscript pc stands for pseudocubic) reflections confirm the epitaxial growth of LSMO layers\non the NGO substrate as shown in Fig. 1 (a). The vertical alignment of LSMO and NGO reciprocal\nlatticepointclearlyshowsthattheLSMOfilmiscompletelystrainedontheNGOsubstrate. Lattice\nmismatch along [100] pcand [010] pcare 1.03% and 0.8%, respectively. Considering the position of\nthe LSMO reciprocal lattice point in the {013} pcmappings, equal Lvalues of (103) pcand (-103) pc\nindicates the perpendicular relation between vector aandcin the lattice, whereas different L values\nfor (013) pcand (0-13) pcshows that the angle between bandcis not equal to 90Âř. Thus, the LSMO\nis monoclinic phase which is consistent with previous reports [36]. The good crystalline quality was\nfurther verified by aberration-corrected scanning transmission electron microscopy (AC-STEM).\nFig. 1 (b, c) are the simultaneously acquired high angle annular dark field (HAADF) and annular\nbright field (ABF) images of S-LSMO along [100]pc direction, while Fig. 1 (d, e) are for [010] pc\ndirection. The measurement directions can be differentiated from the diffraction of NGO substrate:\n31/2[010] superlattices for [100] pcdirection (inset of Fig. 1(c)) and 1/2[101] superlattices for [100] pc\ndirection (inset of Fig. 1(e)). High quality single crystalline films are essential for the present\npurposes because high density of defects will result in spin wave scattering [26].\nMagnetic anisotropy fields\nThe magnetic dynamic properties were investigated by a home-built angle-resolved broadband\nFMR with magnetic field up to 1.5T. All measurements were performed at room temperature.\nShown in Fig. 2(a) is the color-coded plot of the transmission coefficient S21 of the S-LSMO sample\nmeasured at 10GHz. 'His the in-plane azimuth angle of the external magnetic field counted\nfrom [010] pcdirection (Fig. 2(b)). This relative orientation was controlled by a sample mounting\nmanipulator with a precision of less than 0.1\u000e. The olive shape of the color region indicates the\nexistence of anisotropy field, whereas the very narrow field region of resonances is an evidence of low\ndamping. Three line cuts at 'H=0, 45 and 90 degrees are plotted in Fig. 2(c), showing the variation\nof both FMR resonance field ( Hres) and line shape with 'H. All curves are well fitted hence both\ntheHresand resonance linewidth \u0001H are determined. The 'Hdependence of H resat two selected\nfrequencies (20 and 40 GHz) are shown in Fig. 2(d) for S-LSMO. The angle dependencies of the\nresonance field Hres('H)is calculated starting from the total energy [37]:\nE=\u0000MH [cos\u0012Hcos\u0012M+ sin\u0012Hsin\u0012Mcos('M\u0000'H)] + 2\u0019M2cos2\u0012M\u00001\n2MH 2?cos2\u0012M\n\u00001\n4MH 4?cos4\u0012M\u00001\n2MH 2ksin2\u0012Mcos2('M\u0000\u001e2IP)\u00001\n4MH 4k3+cos 4('M\u0000\u001e4IP)\n4sin4\u0012M(1)\nwhere\u0012Mand'Mare the polar angle and the azimuth angle of the magnetization ( M),H2?,\nH4?,H2k,H4kare the uniaxial and cubic out-fo-plane and in-plane anisotropy fields. The easy axes\nof in-plane anisotropies are along \u001e2IPand\u001e4IP, respectively. According to Smit-Beljers equation\nthe resonance condition for \u0012M=\u0019/2 is [38]:\n2\u0019f=\r\nMsin\u0012p\nE\u0012\u0012E'' (2)\nHere,E\u0012\u0012=Hrescos('M\u0000'H) + 4\u0019Me\u000b\u0000H2kcos2('M\u0000\u001e2IP) +H4k(3 + cos 4('M\u0000\u001e4IP)=4)and\nE''=Hrescos('M\u0000'H)+H2kcos 2('M\u0000\u001e2IP)+H4kcos 4('M\u0000\u001e4IP)aresecondpartialderivatives\nof the total energy with respect to the polar and azimuth angles. \r=1.76\u0002107s\u00001G\u00001denotes the\ngyromagnetic ratio, 4\u0019Me\u000b= 4\u0019M\u0000H2?is the effective magnetization. The resonance field of\nS-LSMO shows pronounced minimum at 'H=n\u0001\u0019, indicating the existence of uniaxial magnetic\nanisotropy with easy axis along \u001e2IP= 0or [010] pcdirection. Cubic anisotropy is negligible hence\nH4k=0. Such uniaxial anisotropy observed in S-LSMO is consistent with previous reports [39],\nwhich is attributed to anisotropic strain produced by the NGO(110) substrate [40–42]. Compared\nto the resonance fields in our measurement, the magnetic anisotropy fields are orders of magnitude\nsmaller. Therefore, the calculated difference between 'Hand'Mare always smaller than 1\u000eand\n'='H='Mis assumed in the following discussion.\n4Magnetization orientation dependence of Gilbert damping\nIn order to study the symmetry of magnetization relaxation of the sample. The FMR linewidth\n\u0001Hfor a matrix of parameter list (72 field orientations and 36 frequency values) are extracted.\nThe results are shown by 3-D plots in Fig. 3(a) . Here, zaxis is \u0001Handx,yaxes aref\u0001cos'\nandf\u0001sin', respectively. The figure clearly shows that the linewidth depends on magnetization\norientation. At a given frequency, the linewidth is maximum (minimum) at '= 0('=\u0019=2) for\nS-LSMO. Fig. 3(c) shows the \u0001Hversus frequency for three field orientations. The FMR linewidth\ndue to intrinsic magnetic damping scales linearly with frequency \u0001HGL= 4\u0019\u000bf=\r cos ('M\u0000'H)\naccording to Laudau-Lifshitz-Gilbert phenomenological theory [43, 44]. However, a weak non-\nlinearity in the low frequency range can be identified. In general, extrinsic linewidth contributions\nsuch as inhomogeneity and magnon scattering will broaden the FMR spectrum hence result in\nadditionallinewidthcontributionsscalesnon-linearlywithfrequency[9,11]. Theinterfacialmagnon\nscattering is suppressed due to relative large film thickness (25 nm) and the bulk magnon scattering\ncontribution to the linewidth is negligible in our samples with very good atomic order. However, the\nstatic magnetic properties of the thin film may vary slightly in the millimeter scale. Since the FMR\nsignal is an averaged response detected by the coplanar waveguide (5mm long), a superposition of\nlocation resonance modes broadens the FMR spectrum. Such well-known contribution to linewidth,\ndefined as \u0001Hinhom, are generally treated as a constant [9, 44, 45]. However, it is frequency\ndependent for in-plane configuration and need to be treated carefully for samples with ultra-low\ndamping. Here, we fit the data with \u0001H= \u0001HGL+ \u0001Hinhom, taking into account the frequency\nand orientation dependence of \u0001Hinhom. As can be seen from Fig. 3(c), the data are well reproduced\nfor every field orientations. Hence, the magnetization orientation dependence of intrinsic damping\nconstant is determined and plotted in Fig. 3(e). Remarkably, the damping constant shows two-fold\nsymmetry. The lowest damping of S-LSMO with in-plane magnetization, observed at '= 0and\n'=\u0019, is(8:4\u00060:3)\u000210\u00004and comparable to the value measured under a perpendicular field\n(Tbl. I). The maximum damping at '=\u0019=2and'= 3\u0019=2is about 25% higher.\nSince the magnetization damping and resonance field of the S-LSMO sample exhibited identical\nsymmetry (Fig. 2 (d) and Fig. 2(e)), it seems that the observed anisotropic damping is directly\nrelatedtocrystallineanisotropy. Therefore, wepreparedtheW-LSMOsamplewithslightlydifferent\nstructureandhencemodifiedstaticmagneticanisotropyproperties. TheW-LSMOsampleexhibited\n1D long range atomic wave-like modulation [36] (twining domain motif) along [100] pcaxis near the\ninterface between substrate and film. Due to different strain relaxation mechanism as compared to\nS-LSMO, the 'Hdependence of Hresfor the W-LSMO have additional features and can only be\nreproduced by including both H2k(13:9\u00060:9Oe) andH4k(11:8\u00061:2Oe) terms. The easy axis of\nthe uniaxial anisotropy ( \u001e2IP=0 ) is the same as S-LSMO whereas the additional cubic anisotropy\nis minimum at \u001e4IP=45Âř. The magnetization orientation dependence of the FMR linewidth for\nW-LSMO is significantly different (Fig. 3(b)) as compared to S-LSMO. Such change in trend can be\nclearlyidentifiedfromthefrequencydependenceoflinewidthforselectedmagnetizationorientations\nshown in Fig. 3(d). Magnetization damping values are extracted using the same procedure as S-\nLSMObecausethespinwavecontributionisexcluded. Thedampingconstantagainshowedtwo-fold\nin-plane symmetry. However, in contrast to S-LSMO, the maximum damping value of W-LSMO is\nobserved at '= 0and'=\u0019.\n54\u0019Meff(T)H2k(Oe)H4k(Oe) \u000b? \u000b('= 0)\u000b('=\u0019=2)\nS-LSMO 0.3280 \u00060.0011 37\u000640 (8:6\u00060:5)\u000210\u00004(8:4\u00060:3)\u000210\u00004(11\u00060:6)\u000210\u00004\nW-LSMO 0.3620 \u00060.002513.9\u00060:911.8\u00061:2(4:7\u00060:7)\u000210\u00004(6:5\u00060:3)\u000210\u00004(5:3\u00060:3)\u000210\u00004\nTable I. Summary of the parameters for S-LSMO and W-LSMO samples.\nC. DISCUSSION\nAnisotropy in linewidth at low temperatures have been reported decades ago, however, data in\nmost early publications were taken at a fixed frequency in a cavity-based FMR [24, 46]. Due to\nlack of frequency dependence information, it is not clear if the anisotropy in linewidth is due to\nintrinsic damping or extrinsic effects [47–49]. In this study, besides wide range of frequencies, we\nalso adopted samples with effective anisotropy orders of magnitude smaller than the external field.\nTherefore, the field dragging effect and mosaicity broadening, both of which are anisotropic in natur\ne[50], are negligibly small and the Gilbert damping constant is determined reliably. Furthermore,\nthemechanisminthissimplesystemisdifferentfrompreviousreportsrelatedtointerfacialexchange\ncoupling and spin pumping[51, 52]. Since both S-LSMO and W-LSMO exhibited in-plane uniaxial\nmagnetic anisotropy, the opposite trends observed in these two samples exclude the existence of a\ndirect link between anisotropic damping and effective field. Both magnetic anisotropy and damping\nare related to the band structure but in quite different ways. According to perturbation theory,\nthe magnetic anisotropy energy is determined by the matrix elements of the spin-orbit interaction\nbetween occupied states. Hence, the contributions from all the filled bands must be considered to\ncalculate the absolute value of magnetic anisotropy. On the other hand, the magnetic damping is\nrelated to the density of states at the Fermi level.\nThe damping term in the Landau-Lifshitz-Gilbert equation of motion is\u000b\njMj\u0000\nM\u0002dM\ndt\u0001\n, there-\nfore, anisotropy in damping can have two origins, one related to the equilibrium orientation of\nmagnetization M(orientation anisotropy) and the other depends on the instantaneous change in\nmagnetization dM=dt(rotational anisotropy). In FMR experiments the magnetization vector ro-\ntates around its equilibrium position, therefore, the rotational anisotropy may be smeared out [25].\nThe orientation anisotropy is described by both interband and intraband scattering process. Ac-\ncording to Gilmore et al.[23], the latter is isotropic at sufficiently high scattering rates at room\ntemperature. We suspect that the anisotropic damping in LSMO is due to its half-metallic band\nstructure. As a result of high spin polarization, interband scattering is suppressed and the room\ntemperature damping is dominated by intraband scattering. The intraband contribution to damp-\ning exhibit anisotropy for all scattering rates [23] which agree well with our experiments. The\nsuppression of interband scattering is evidenced by the ultra-low damping in the order of 10\u00004.\nNotably, the absolute value of the observed anisotropy, 2.6 \u000210\u00004for S-LSMO and 1.2 \u000210\u00004for\nW-LSMO, is so small that could not be identified reliably for a material with typical damping\nvalues between 5 \u000210\u00003to 2\u000210\u00002.\nIn a microscopic picture, the Gilbert damping is proportional to the square of SOC constant ( \u0018)\nand density of states at the Fermi level, \u000b\u0018\u00182D(EF). The shape of the Fermi surface depends on\nthe orientation of the magnetization due to SOC. 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Back, Nature Physics (2018).\n9Figure 1. Structure characterization of S-LSMO sample. (a) and (b) XRD profiles around S-LSMO\n(00L) reflections (L=1,2,3,4) with the incident beam aligned along the [100] pcand [010] pc, respectively.\n(b) and (c) STEM-HAADF/ABF lattice images of S-LSMO along [100] pcdirection. (d) and (e) STEM-\nHAADF/ABF images of S-LSMO along [010] pcdirection. the insets are the intensity profile and FFT\nimage; The red dashed line indicates the interface.\n10(b)\n(c)\n10GHz(a)\n30\n21060\n24090\n270120\n300150\n330180 010GHz\n242022002420Hres(Oe)\n5.625.70\n12.6512.712.75\nHres(kOe)(d)\n40GHz\n20GHzS-LSMO\n0 90 180 270 360\n0\n0 fit\n45\n45 fit\n90\n90 fitS21 (a.u.)\n2200 2300 2400\nHres(Oe)Figure 2. Magnetic anisotropy characterization. (a) The 2D polar color plot of the FMR spectra of\nS-LSMO. The frequency is 10GHz. (b) Schematics of the FMR setup and the definition field orientation.\n(c) FMR spectra for 'H=0, 45 and 90 degrees for S-LSMO. (d) Field orientation ( 'H) dependence of the\nresonance fields ( Hres) of the S-LSMO sample at f=20 and 40GHz. The solid lines in (c) and (d) are\ncalculated values.\n11404040\n0 060\n-40\n40 4020\n0 030\n-40\nf (GHz) f (GHz)0 10 20 30 40204060ΔH (Oe)\n102030ΔH (Oe)\n0 10 20 30 40\n810120\nfit\n45\n90fit\nfit0\nfit\n45\n90fit\nfit(a) (b)\n(c) (d)\n(e) (f)ΔH (Oe)\nΔH (Oe)\nf (GHz) f (GHz)α (10-4)\n0 360 270 180 90\nφ567\n0 90 180 270 360α (10-4)\nφf (GHz) f (GHz)Figure 3. Anisotropic linewidth and damping: (a)-(b) 3-D plot of frequency and in-plane field ori-\nentation dependence of FMR linewidth. (c)-(d) frequency dependence of FMR linewidth for seleted field\norientations. Solid symbols are experimental data and the lines are calculated value. (e)-(f) Damping\nconstant as a function of '. (a),(c), (e) are for S-LSMO and (b),(d), (f) are for W-LSMO.\n12"    },    {        "title": "0811.0425v1.Amplitude_Phase_Coupling_in_a_Spin_Torque_Nano_Oscillator.pdf",        "content": "arXiv:0811.0425v1  [cond-mat.mtrl-sci]  4 Nov 2008Amplitude-Phase Coupling in a Spin-Torque Nano-Oscillato r\nKiwamu Kudo,∗Tazumi Nagasawa, Rie Sato, and Koichi Mizushima\nCorporate Research and Development Center, Toshiba Corpor ation, Kawasaki, 212-8582, Japan\n(Dated: October 31, 2018)\nThe spin-torque nano-oscillator in the presence of thermal fluctuation is described by the normal\nform of the Hopf bifurcation with an additive white noise. By the application of the reduction\nmethod, the amplitude-phase coupling factor, which has a si gnificant effect on the power spectrum of\nthe spin-torque nano-oscillator, is calculated from the La ndau-Lifshitz-Gilbert-Slonczewski equation\nwith the nonlinear Gilbert damping. The amplitude-phase co upling factor exhibits a large variation\ndepending on in-plane anisotropy under the practical exter nal fields.\nWhen a direct current Iflows into a magnetoresistive\n(MR) device, a stationary magnetic state becomes un-\nstable and a steady magnetic oscillation is excited by the\nspin-transfer torque. The oscillation is expected to be\napplicableto ananoscalemicrowavesource, i.e., the spin-\ntorque nano-oscillator (STNO).1,2According to the the-\nory based on the spin-wave Hamiltonian formalism,3,4,5,6\nthe frequency nonlinearity plays a key role in determin-\ning the behavior of the oscillator. It has been shown that\nthe strong frequency nonlinearity leads to significant ef-\nfects on the power spectrum of STNO in the presence\nof thermal fluctuation: a linewidth enhancement5and\nnon-Lorentzian lineshapes6. In this paper, the impor-\ntantnonlinearityisexamined. FromtheLandau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation as the model of\nSTNO,wecalculateexplicitlythemagnitudeofthequan-\ntity corresponding to the normalized frequency nonlin-\nearityN/Γeff(see, e.g., Eq. (4) in Ref. 6) of the spin-\nwaveapproach. Inparticular,wetakeaccountofin-plane\nanisotropy of a magnetic film which has been neglected\nin the early studies3,4,5,6, finding the large effect of the\nanisotropy on the nonlinearity.\nWe describe STNO by a generic oscillator model. It\nis known that small-amplitude oscillations near the Hopf\nbifurcation point are generally governed by the simple\nevolution equation for a complex variable W(t) known\nas the Stuart-Landau (SL) equation.7The SL equation\nis derived as a normal form of the supercritical Hopf bi-\nfurcation from the general system of ordinary differen-\ntial equations. Accordingly, the LLGS equation similarly\nreduces to the SL equation in the case where the Hopf\nbifurcation,whichrepresentsagenerationofmagneticos-\ncillations in STNO, occurs. The reduction of the LLGS\nequation can be executed by the reduction method based\non the center-manifold theorem. At finite temperature,\nthere exists inevitable thermal magnetization fluctuation\nin STNO.8,9We include the thermal effect into the mag-\nnetization dynamics by just adding white noise term to\nthe SL equation, i.e., STNO in the presence of thermal\nfluctuation is described by the ‘noisy’ Hopf normal form:\nd˜W\nd˜t=i˜Ω˜W+(1+iδ)(p−|˜W|2)˜W+η(˜t),(1)\nwhere˜Wis the normalized complex variable represent-\ning the amplitude and phase of a magnetization vec-torM(see Eq. (7) below). In Eq. (1), ˜Ω represents a\nfundamental frequency, ˜tis a normalized dimensionless\ntime, andη(˜t) is the zero-mean, white Gaussian noise\nwith the only non-vanishing second moment given by\n∝an}bracketle{tη(˜t)¯η(˜t′)∝an}bracketri}ht= 4δ(˜t−˜t′).pis the bifurcation parame-\nter. An oscillation is generated when pbecomes positive.\nIn the context of STNO, p∝(I−Ic) whereIcis the\nthreshold current. The parameter δquantifies the cou-\npling between the amplitude and phase fluctuations and\nis called the amplitude-phase coupling factor . It isδthat\nwe calculate numerically in this paper and that corre-\nsponds to the normalized frequency nonlinearity N/Γeff\nof the spin-wave approach. The amplitude-phase cou-\npling factor δaffects the power spectrum of an oscillator\nand leads to linewidth enhancement and non-Lorentzian\nlineshapes.10,11Due toits effect, the factor δisalsocalled\nthelinewidth enhancement factor .12Eq. (1) is often used\nas the simplest model of a noisy auto-oscillator in many\nfields, for example, electrical engineering, chemical reac-\ntions, optics, biology, and so on.10,13Therefore, we can\neasily compare STNO with conventional oscillators and\nclarify its features.\nThe amplitude-phase coupling factor δis obtained in\nthe procedure of the reduction of the LLGS equation. In\nthe following, we first explain the LLGS equation. Then,\nfollowing Kuramoto’s monograph7, we consider an insta-\nbility of a steady solution and execute the reduction of\nthe LLGS equation.\nThe magnetic energy density of the free layer of STNO\nis assumed to have the form\nE=−M·Hext−Ku\nM2s(M·ˆx)2+1\n24πM·N ·M,(2)\nwhereMsisthesaturationmagnetization, Hext=Hxˆx+\nHyˆy+Hzˆzis an external field, Kuis uniaxial anisotropy\nalong thexdirection, and Nis the demagnetizing ten-\nsor;N= diag(Nx,Ny,Nz). Using the spherical coordi-\nnate system (see Fig. 1), we describe the magnetization\ndynamics of STNO by the LLGS equation\n/braceleftBigg\ncosψ˙φ=−α(ξ)˙ψ−F1(φ,ψ,ω J)\n˙ψ=α(ξ)cosψ˙φ+F2(φ,ψ,ω J),(3)\nwhereF1(φ,ψ,ω J)≡(γ/Ms)∂E/∂ψ−a(φ,ωJ) and\nF2(φ,ψ,ω J)≡(γ/(Mscosψ))∂E/∂φ+b(φ,ψ,ω J).γis2\nφHH\nM\nxHxyyz\nψz\nuK\npI\nFIG. 1: The spherical coordinate system ( φ,ψ) for the direc-\ntion of the free layer magnetization m=M/Msof STNO.\npdenotes the direction of the pinned layer magnetization;\np= (cosψpcosφp,cosψpsinφp,sinψp).\nthe gyromagnetic ratio. The second terms of Fire-\nsult from the Slonczewski term TJ= (γaJ/Ms)M×\n(M×p) in which aJis proportional to the cur-\nrent density Jthrough the free layer14. Therefore,\na(φ,ωJ)≡ωJcosψpsin(φ−φp) andb(φ,ψ,ω J)≡\nωJ[cosψpsinψcos(φ−φp)−sinψpcosψ], whereωJ=\nγaJ.α(ξ)-terms of Eqs. (3) are the generalized Gilbert\ndamping terms proposed by Tiberkevich and Slavin.15\nWe take into account only the first non-trivial term of\nthe Taylor series expansion for α(ξ) by the magneti-\nzation change rate ξ≡(∂m/∂t)2/(γ4πMs)2;α(ξ) =\nαG(1+q1ξ). According to Ref. 15, the nonlinear LLGS\nmodel with q1= 3 gives a good agreement with the ex-\nperimental results of Ref. 1 and Ref. 16.\nAn instability of a steady solution of Eq. (3) is consid-\nered. A steady solution ( φ0(ωJ),ψ0(ωJ)) is derived from\nFi(φ0,ψ0,ωJ) = 0. Shifting the variables as u1≡φ−φ0\nandu2≡ψ−ψ0, we have the Taylor series of Eq. (3) as\nfollows,\n˙u=Lu+N2uu+N3uuu+··· (4)\nwhereu= (u1,u2)T. Here, the diadic and triadic nota-\ntions7have been used. The stability of a steady solution\nis determined by the eigenvalues of the linear coefficient\nmatrixL:λ±= Γ±(Γ2−detL)1/2. Γ is defined as\nΓ = Γ(ωJ)≡(1/2)trLand plays the role as a control pa-\nrameter since it depends on ωJ. We confine ourselves to\nthecasewheretheHopfbifurcationoccurs. Then, λ±isa\npair of complex-conjugate eigenvalues. The point, Γ = 0,\nis the Hopf bifurcation point; while a steady solution re-\nmains stable for Γ <0, it becomes unstable for Γ >0.\nThe bifurcation point corresponds to the threshold ωc\nJ\nwhich is determined by tr L= 0 andFi(φ0,ψ0,ωc\nJ) = 0.\nNear the bifurcation point, we divide Linto the two\nparts;L=L0+ ΓL1, whereL0is the critical part and\nΓL1istheremainingpart. Correspondingto L,λ+isalso\ndivided into the two parts; λ+=λ0+Γλ1. Although L1\nandλ1generally depend on Γ further, we neglect their\ndependence and evaluate them by the values at Γ = 0.\nAccordingly, λ0=iω0and\nλ1= 1−1\n2iω0d\ndΓdetL/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΓ=0, (5)\nwhereω0≡√detL0. The right and left eigenvector ofFIG.2: (Color online)(a)Power PdividedbyR0I2withR0=\n13.6 Ω and (b) linewidth (FWHM) of the signal of STNO as\na function of applied current I. Dots are experimental data\natT= 150 K taken from Ref. 16. Red lines are theoretical\nfitting curves based on the model of Eq. (1).\nL0corresponding to the eigenvalue λ0are denoted as U\nandU∗, respectively. These are normalized as U∗U=\n¯U∗¯U= 1 where ¯Umeans a complex conjugate of U.\nLet us apply the reduction method to Eq. (4). The SL\nequation for a complex amplitude W(t),\n˙W= Γλ1W−g|W|2W (6)\nand the neutral solution for the magnetization dynamics,\n/parenleftbigg\nφ\nψ/parenrightbigg\n=/parenleftbigg\nφ0\nψ0/parenrightbigg\n+W(t)eiω0tU+¯W(t)e−iω0t¯U(7)\nareobtainedwithinthelowestorderapproximation.7Un-\nder the approximation, only the Taylor expansion coef-\nficients up to the third order are needed. The complex\nconstantgin Eq. (6) is given by\ng≡ν1+iν2=−3(U∗,N3¯UUU)\n+4(U∗,N2UV0)+2(U∗,N2¯UV+),(8)\nwhereV0=L−1\n0N2U¯UandV+= (L0−2iω0)−1N2UU.\nThe amplitude-phase coupling factor δis obtained from\nthe complex constant gand is given by\nδ=ν2/ν1. (9)\nIn this way, the factor δfor STNO can be calculated\nnumerically from the parameters of the LLGS equation.\nThe noisy Hopf normal form given by Eq. (1) is de-\nrived when we add the noise term f(t) with∝an}bracketle{tf(t)¯f(t′)∝an}bracketri}ht=\n4Dγ2δ(t−t′) to the SL Eq. (6). f(t) has the di-\nmension of frequency. The components in Eq. (1) are\ndefined as ˜W(t) = (Dγ2/ν1)−1/4W(t)ei(ω0+Γδ−ΓImλ1)t,\n˜t≡/radicalbig\nDγ2ν1t,p≡Γ//radicalbig\nDγ2ν1, and˜Ω≡ω0//radicalbig\nDγ2ν1.\nTherefore, we can make the most of many well-known\nproperties of Eq. (1)10,11to examine the behavior of\nSTNO. It is known, for example, that the spectrum\nlinewidth ∆ ωFWHMfar abovethe threshold ( p≫0) is in-\ncreased by a factor of (1+ δ2).10In the context of STNO,\nwhen Γ≫0, the linewidth can be expressed as\n∆ωFWHM= ∆ωres×kBT\nEosci×1\n2(1+δ2),(10)3\nFIG.3: (Color online)(a)Dependenceof δonthenonlinearity\nof the damping q1for various values of an external magnetic\nfieldH. An uniaxial anisotropy field is taken as Hk/4πMs=\n0.04. (b) Dependence of δon an external magnetic field H\nfor various values of an uniaxial anisotropy field Hk.\nwhich corresponds to Eq. (11) in Ref. 5. Here, kBTis\nthe thermal energy. ∆ ωresis the linewidth at thermal\nequilibrium ( ωJ= 0) given by ∆ ωres= 2Γeq, where\nΓeq≡ −Γ(ωJ= 0). Moreover, Eosciis the magneti-\nzation oscillating energy and can be written as Eosci≃\n2U†[∂(∂u1E,∂u2E)\n∂(u1,u2)]u=0UPWVfree=1\n2ΓeqkBT\nDγ2PWwhen it is\nassumed that Eosci≃kBTnear thermal equilibrium ( en-\nergy equipartition ). Here,Vfreeis the volume of the free\nlayer andPWis the total power of W(t) given byPW=/radicalbig\nDγ2/ν1{p+2/F(p)}withF(p)≡√πep2/4[1+ erf(p/\n2)]. From the expression of Eq. (10), it is found that\nthe MR device in STNO itself is nothing but a resonator\non the analogy of electrical circuits. The other one of\nwell-known properties of Eq. (1) is that the amplitude-\nphasecouplingfactordistortsthepowerspectrumtonon-\nLorentzian lineshapes especially near the threshold (see,\ne.g., FIG. 5 of Ref. 11). The degree of the lineshape\ndistortion is determined by the magnitude of δandp,\ncorresponding to the calculation in Ref. 6. We com-\nment on the validity of Eq. (1) for large-amplitude os-\ncillations. In Fig. 2, the theoretical fitting curves based\non the model Eq. (1) are compared with the experimen-\ntal data of Ref. 16 and give a good agreement with them\nup toI≃5.6 mA (p≃8.2) beyond the threshold cur-\nrentIc= 4.8 mA (p= 0) estimated by the fitting.17\nTherefore, although the derivation of Eq. (1) is based on\na perturbation expansion around the bifurcation point,\nit is considered to be valid for rather large-amplitude os-\ncillations with p∼10.\nWe briefly mention the oscillating frequency ωosci.From Eqs. (1) and (7), the oscillating frequency of a\nfree layer magnetization far above threshold is written as\nωosci=ω0−Γδ+ΓImλ1. Although the calculationresults\nfor Imλ1of Eq. (5) are not shown here, we have found\nthat this quantity has a small value with Im λ1∼αG\nfor wide range of parameters of the LLGS equation. Ac-\ncordingly,ωosciisapproximatelygivenby ωosci≃ω0−Γδ.\nSince Γ∝(I−Ic), while the frequency ωoscidecreases as\nthe current I(>Ic) increaseswhen δ>0 (red shift), ωosci\nincreases when δ<0 (blue shift) in accordance with the\nspin-wave models3,4,5,6.\nAsillustratedabove,theamplitude-phasecouplingfac-\ntorδplays a key role to determine the behavior of an\noscillator. Therefore, the features of STNO can be found\nout by the calculation of δ.\nSome calculation examples of δare shown in Fig. 3.\nIt is considered the case where a free layer is an in-\nplane magnetic film with an in-plane external field ap-\nplied along the xdirection, Hext=Hˆx. It is assumed\nthatN= diag(0,0,1),αG= 0.02, and (φp,ψp) = (0,0).\nIn Fig. 3(a), the dependence of δon the nonlinearity\nof the damping q1is shown. It is found that δmono-\ntonically decreases for q1and the variation of δis very\nlarge. This result suggests that a nonlinear damping sig-\nnificantly changes the LLG dynamics.15In Fig. 3(b), the\ndependence of δon an external magnetic field Hfor var-\nious values of an uniaxial anisotropy field Hk(= 2Ku/\nMs) is shown. The nonlinearity of the damping is taken\nasq1= 3.15In the practical external field region, δis\nvery sensitive to an uniaxial anisotropy field and varies\nlargely. Therefore, when the dynamics of STNO is con-\nsidered, it is necessary to take the effect of an uniaxial\nanisotropy field into account seriously. This is the main\nresult of the present paper.\nIn summary, we have considered the dynamics of\nSTNO by reducing the LLGS equation to a generic oscil-\nlatormodelandcalculatedexplicitlytheamplitude-phase\ncoupling factor which is the key factor for the power\nspectrum. The amplitude-phase coupling factor δis very\nsensitive to magnetic fields, in-plane anisotropy, and the\nnonlinearity of damping. The large variation of δis the\nremarkable feature of STNO in comparison with conven-\ntional oscillators. The calculation way for δshown is ap-\nplicable for an arbitrarymagnetization configurationand\nis useful for finding a stable STNO with small ∆ ωFWHM\n(Eq. (10)), which is preferable for applications.\n∗Electronic address: kiwamu.kudo@toshiba.co.jp\n1S. I. Kiselev et al, Nature 425, 380 (2003).\n2W. H. Rippard et al, Phys. Rev. Lett. 92, 027201 (2004).\n3A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264\n(2005).\n4V. Tiberkevich, A. N. Slavin, and J.-V. Kim, Appl. Phys.\nLett.91, 192506 (2007).\n5J.-V. Kim, V. Tiberkevich, and A. N. Slavin, Phys. Rev.\nLett.100, 017207 (2008).6J.-V. Kim et al., Phys. Rev. Lett. 100, 167201 (2008).\n7Y. Kuramoto, Chap. 2 of Chemical Oscillations, Waves,\nand Turbulence (Springer-Verlag, Berlin, 1984).\n8J.-V. Kim, Phys. Rev. B 73, 174412 (2006).\n9K. Mizushima, K. Kudo, and R. Sato, J. Appl. Phys. 101,\n113903 (2007).\n10H. Risken, Chap. 12 of Fokker-Planck Equation (2nd Ed.\nSpringer-Verlag, Berlin, 1989).\n11J. P. Gleeson and F. O’Doherty, SIAM J. Appl. Math. 66,4\n1669 (2006).\n12C. H. Henry, IEEE Journal of Quantum Electronics, QE-\n18, 259 (1982).\n13H. Haken, Advanced Synergetics (Springer-Verlag, New\nYork, 1993).\n14J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n15V. Tiberkevich and A. Slavin, Phys. Rev. B 75, 014440\n(2007).\n16Q. Mistral et al., Appl. Phys. Lett. 88, 192507 (2006).17The dimensionless power in Fig. 2(a) is given by P/R0I2=\na{p+2/F(p)}witha≃2.3063×10−9andp≃10.202(I−\nIc). To obtain the linewidth in Fig. 2(b), we have used the\nparameters ofp\nDγ2ν1/2π= 11.24 MHz and δ= 0.5, and\nhave solved the eigenvalue problem of the Fokker-Planck\nequation corresponding to Eq. (1) as done in Ref. 10 or\nRef. 6."    },    {        "title": "2305.09231v1.Non_Hermitian_Casimir_Effect_of_Magnons.pdf",        "content": "Non-Hermitian Casimir Effect of Magnons\nKouki Nakata1,\u0003and Kei Suzuki1,y\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n(Dated: May 17, 2023)\nThere has been a growing interest in non-Hermitian quantum mechanics. The key concepts of\nquantum mechanics are quantum fluctuations. Quantum fluctuations of quantum fields confined in a\nfinite-size system induce the zero-point energy shift. This quantum phenomenon, the Casimir effect,\nis one of the most striking phenomena of quantum mechanics in the sense that there are no classical\nanalogs and has been attracting much attention beyond the hierarchy of energy scales, ranging from\nelementary particle physics to condensed matter physics, together with photonics. However, the\nnon-Hermitian extension of the Casimir effect and the application to spintronics have not yet been\ninvestigated enough, although exploring energy sources and developing energy-efficient nanodevices\nare its central issues. Here we fill this gap. By developing a magnonic analog of the Casimir effect\ninto non-Hermitian systems, we show that this non-Hermitian Casimir effect of magnons is enhanced\nas the Gilbert damping constant (i.e., the energy dissipation rate) increases. When the damping\nconstant exceeds a critical value, the non-Hermitian Casimir effect of magnons exhibits an oscillating\nbehavior, including a beating one, as a function of the film thickness and is characterized by the\nexceptional point. Our result suggests that energy dissipation serves as a key ingredient of Casimir\nengineering.\nIntroduction. —Recently, non-Hermitian physics has\nbeen drawing considerable attention not only for funda-\nmental science but also for applications such as energy-\nefficient nanodevices [1]. Thanks to the complete absence\nof any metallic elements from conduction electrons, insu-\nlating magnets are free from drawbacks of conventional\nelectronics,e.g.,substantialenergylossduetoJouleheat-\ning. However, the effect of energy dissipation on spins is\nunavoidable even in insulating magnets. Hence, to ex-\nplore energy sources and develop the potential for appli-\ncations is a crucial issue.\nWe shed light on this problem by using a quantum ef-\nfect, the Casimir effect [2–4], which arises from the zero-\npoint energy. Quantum fluctuations of quantum fields\nrealize a zero-point energy shift under spatial boundary\nconditions. This Casimir effect is one of the most strik-\ning phenomena of quantum mechanics in the sense that\nthere areno classical analogs. Although the original plat-\nform for the Casimir effect [2–4] is the photon field [5],\nthe concept can be extended to various fields such as\nscalar, tensor, and spinor fields. Thanks to this universal\nproperty, the Casimir effects have been investigated in\nvarious research areas [6] beyond the hierarchy of energy\nscales [7–12], ranging from elementary particle physics\nto condensed matter physics, together with photonics.\nHowever, the non-Hermitian extension of the Casimir ef-\nfect and the application to spintronics remain missing in-\ngredients, although exploring energy sources and devel-\noping the potential for energy-efficient nanodevices are\nthe central issues of spintronics [13–16].\nHere we fill this gap. The Casimir effects are charac-\nterized by the energy dispersion relation. We therefore\n\u0003(Equal contribution) nakata.koki@jaea.go.jp\ny(Equal contribution) k.suzuki.2010@th.phys.titech.ac.jp\nFIG. 1. Schematic of the thin film of the AFMs.\nincorporate the effect of energy dissipation into the en-\nergy dispersion relation of magnons through the Gilbert\ndamping constant [17] and thus develop a magnonic ana-\nlog of the Casimir effect, called the magnonic Casimir ef-\nfect[18], intonon-Hermitiansystems. Wethenshowthat\nthisnon-HermitianextensionofthemagnonicCasimiref-\nfect, which we call the magnonic non-Hermitian Casimir\neffect, is enhanced as the Gilbert damping constant (i.e.,\nthe energy dissipation rate) increases. When the damp-\ning constant exceeds a critical value, the magnonic non-\nHermitian Casimir effect exhibits an oscillating behavior\nas a function of the film thickness and is characterized by\nthe exceptional point [19] (EP). We refer to this behav-\nior as the magnonic EP-induced Casimir oscillation. We\nemphasize that this magnonic EP-induced Casimir oscil-\nlation is absent in the dissipationless system of magnons.\nThe magnonic EP-induced Casimir oscillation exhibits a\nbeating behavior in the antiferromagnets (AFMs) where\nthe degeneracy between two kinds of magnons is lifted.\nOur result suggests that energy dissipation serves as a\nnew handle on Casimir engineering [20] to control and\nmanipulate the Casimir effect of magnons. Thus, we\npave a way for magnonic Casimir engineering through\nthe utilization of energy dissipation.\nMagnonic non-Hermitian system. —We consider the\ninsulating AFMs of two-sublattice systems in three di-\nmensions described by the Heisenberg model (see Fig. 1),arXiv:2305.09231v1  [quant-ph]  16 May 20232\nwhere the AFMs have the Néel magnetic order and there\nexists the zero-point energy [21, 22]. Throughout this\nstudy, we work under the assumption that the Néel phase\nremains stable in the presence of energy dissipation. El-\nementary magnetic excitations are two kinds of magnons\nhaving the spin angular momentum \u001b~with the index\n\u001b=\u0006and the reduced Planck constant ~. By incor-\nporating the effect of energy dissipation into the energy\ndispersion relation of magnons through the two-coupled\nLandau-Lifshitz-Gilbert equation where the value of the\nGilbert damping constant \u000b > 0for each sublattice is\nidentical to each other, we study the low-energy magnon\ndynamics [23] described by the energy dispersion relation\n\u000f\u001b;k;\u000b2Cand the wavenumber k= (kx;ky;kz)2R[24]\nin the long wavelength limit as [25]\n\u000f\u001b;k;\u000b=2S\n1 +\u000b2\u0010\n\u0000i\u000bC+q\n(E\u001b;k;\u000b)2\u0011\n(1)\nand\n(E\u001b;k;\u000b)2:=A\u001b;\u000b2(ak)2+\u000e\u001b2\u0000D\u001b2\u000b2;(2)\nwherek:=jkj, the length of a magnetic unit cell is a, the\nspin moment in a magnetic unit cell is S, and the others\nare material-dependent parameters which are indepen-\ndent of the wavenumber, 0< A\u001b;\u000b2R,0< \u000e\u001b2R,\n0<D\u001b2R, and 0< C2R: In the presence of spin\nanisotropies, the parameters are given as [25]\nA\u001b;\u000b=r\n(1 +\u000b2)\u0010\nJ2+\u001bKh\n2J\u0011\n; (3a)\n\u000e\u001b=p\nKe(2J+Ke) +Kh(J\u0000\u001bJ+Ke);(3b)\nD\u001b=s\nJ2+\u001bKhJ+Kh2\n4; (3c)\nC=J+Ke+Kh\n2; (3d)\nwhereJ > 0parametrizes the antiferromagnetic ex-\nchange interaction between the nearest-neighbor spins,\nKh>0is the hard-axis anisotropy, and Ke>0is the\neasy-axis anisotropy along the ydirection (see Fig. 1).\nThese are generally Kh=J\u001c1andKe=J\u001c1. In the\nabsence of the hard-axis anisotropy Kh= 0, two kinds\nof magnons \u001b=\u0006are in degenerate states, whereas the\ndegeneracy is lifted by Kh>0. Note that, in general, the\neffect of dipolar interactions is negligibly small in AFMs,\nand we neglect it throughout this study.\nThe Gilbert damping constant \u000bis a dimensionless\nconstant, and the energy dissipation rate increases as the\nGilbert damping constant grows. In the dissipationless\nsystem [18], the Gilbert damping constant (i.e., the en-\nergy dissipation rate) is zero \u000b= 0. The dissipative sys-\ntem of\u000b>0described by Eq. (1) can be regarded as a\nnon-Hermitian system for magnons in the sense that the\nenergy dispersion takes a complex value. Note that the\nconstant term in Eq. (1), \u0000i\u000bC, is independent of thewavenumber and just shifts the purely imaginary part of\nthe magnon energy dispersion \u000f\u001b;k;\u000b. For this reason [see\nalso Eq. (9a)], the constant term, \u0000i\u000bC, is not relevant\nto the magnonic Casimir effect [26]. We then define the\nmagnon energy gap of Eq. (1) as \u0001\u001b;\u000b:=Re(\u000f\u001b;k=0;\u000b),\ni.e.,\n\u0001\u001b;\u000b=2S\n1 +\u000b2Re\u0010q\n(E\u001b;k=0;\u000b)2\u0011\n; (4)\nand investigate the \u000b-dependence of the magnon energy\ndispersion.\nMagnonic EP. —Whenthedampingconstant \u000bissmall\nand (E\u001b;k=0;\u000b)2>0,E\u001b;k=0;\u000btakes a real value and\ndecreases as \u000bincreases. This results in\nd\u0001\u001b;\u000b\nd\u000b<0: (5)\nThus, the magnon energy gap decreases as the damping\nconstant increases [27] [compare the solid line with the\ndashed one in the left panel of Fig. 2 (i)]. When the\ndamping constant is large enough, the magnon energy\ngap vanishes \u0001\u001b;\u000b= 0at\u000b=\u000bcri\n\u001b,\n\u000bcri\n\u001b:=\u000e\u001b\nD\u001b; (6)\nwhere there exists the gapless magnon mode which be-\nhaves like a relativistic particle with the linear energy\ndispersion. From the property of Eq. (5), we call (i)\n\u000b\u0014\u000bcri\n\u001bthe gap-melting regime.\nWhen the damping constant exceeds the critical value\n\u000bcri\n\u001b, i.e.,\u000b > \u000bcri\n\u001b,E\u001b;k=0;\u000btakes a purely imaginary\nvalue as (E\u001b;k=0;\u000b)2<0. In this regime, the real part of\nthe magnon energy dispersion remains zero Re (\u000f\u001b;k;\u000b) =\n0for the region 0\u0014k\u0014kcri\n\u001b;\u000b,\nkcri\n\u001b;\u000b:=1\nas\nD\u001b2\u000b2\u0000\u000e\u001b2\nA\u001b;\u000b2; (7)\nwhereas Re (\u000f\u001b;k;\u000b)>0fork >kcri\n\u001b;\u000b[see the highlighted\nin yellow in the left panel of Figs. 2 (ii) and (iii)]. The\ncritical point kcri\n\u001b;\u000bcan be regarded as the EP [16, 27] for\nthe wavenumber k, and we refer to it as the magnonic\nEP.Asthevalueofthedampingconstantbecomeslarger,\nthat of the EP increases\ndkcri\n\u001b;\u000b\nd\u000b>0: (8)\nAt the EP k=kcri\n\u001b;\u000b, the group velocity v\u001b;k;\u000b:=\nRe[@\u000f\u001b;k;\u000b=(@~k)]becomes discontinuous [see the solid\nlines in the left panel of Figs. 2 (ii) and (iii)]. We em-\nphasize that this behavior is absent in the dissipationless\nsystem\u000b= 0[18]. In the presence of the EP, the group\nvelocity becomes much larger than the usual such as in\nthe gap-melting regime (i) [compare the solid lines in the\nleftpanelofFigs.2(ii)and(iii)withtheoneofFig.2(i)].3\nAssuming\u000bcri\n\u001b=+< \u000bcri\n\u001b=\u0000, the non-Hermitian system\nfor magnons described by Eq. (1) with \u000b>0can be di-\nvided into three regimes (i)-(iii) in terms of the magnonic\nEPs as follows [see the left panel of Figs. 2 (i), (ii),\nand (iii)]:\n(i)\u000b\u0014\u000bcri\n\u001b=+<\u000bcri\n\u001b=\u0000. No magnonic EPs.\n(ii)\u000bcri\n\u001b=+<\u000b<\u000bcri\n\u001b=\u0000. One EP,kcri\n\u001b=+;\u000b.\n(iii)\u000bcri\n\u001b=+<\u000bcri\n\u001b=\u0000\u0014\u000b. Two EPs, kcri\n\u001b=+;\u000bandkcri\n\u001b=\u0000;\u000b.\nMagnonic Casimir energy. —The magnonic analog of\nthe Casimir energy, called the magnonic Casimir en-\nergy [18], is characterized by the energy dispersion re-\nlation of magnons (see Ref. [18] for dissipationless sys-\ntems). Therefore, by incorporating the effect of energy\ndissipationintotheenergydispersionrelationofmagnons\nthrough the Gilbert damping constant [see Eq. (1)], a\nnon-Hermitian extension of the magnonic Casimir effect\ncan be developed. We remark that the Casimir energy\ninduced by quantum fields on the lattice, such as the\nmagnonic Casimir energy [18], can be defined by using\nthe lattice regularization [28–34]. In this study, we focus\non thin films confined in the zdirection (see Fig. 1). In\nthe two-sublattice systems, the wavenumber on the lat-\ntice is replaced as (akj)2!2[1\u0000cos(akj)]along thej\naxis forj=x;y;z. Here by taking into account the Bril-\nlouin zone (BZ), we set the boundary condition for the\nzdirection in wavenumber space so that it is discretized\naskz!\u0019n=Lz, i.e.,akz!\u0019n=Nz, whereLz:=aNz\nis the film thickness, Nj2Nis the number of magnetic\nunit cells along the jaxis, andn= 1;2;:::;2Nz. Thus,\nthe magnonic Casimir energy ECas[18] per the number\nof magnetic unit cells on the surface for Nzis defined as\nthe difference between the zero-point energy Esum\n0for the\ndiscrete energy \u000f\u001b;k;\u000b;ndue to discrete kz[see Eq. (9b)]\nand the one Eint\n0for the continuous energy \u000f\u001b;k;\u000b[see\nEqs. (9c) and (1)] as follows [28–34]:\nECas(Nz) :=Esum\n0(Nz)\u0000Eint\n0(Nz); (9a)\nEsum\n0(Nz) :=X\n\u001b=\u0006Z\nBZd2(ak?)\n(2\u0019)2\"\n1\n2\u00101\n22NzX\nn=1\u000f\u001b;k;\u000b;n\u0011#\n;\n(9b)\nEint\n0(Nz) :=X\n\u001b=\u0006Z\nBZd2(ak?)\n(2\u0019)2\"\n1\n2NzZ\nBZd(akz)\n2\u0019\u000f\u001b;k;\u000b#\n;\n(9c)\nwherek?:=q\nkx2+ky2,d2(ak?) =d(akx)d(aky),\nthe integral is over the first BZ, and the factor 1=2in\nEqs. (9b) and (9c) arises from the zero-point energy.\nWe remark that [23] assuming thin films of Nz\u001c\nNx;Ny(see Fig. 1), the zero-point energy in the thin film\nof the thickness NzisEsum\n0(Nz)NxNyand consists of two\nparts asEsum\n0(Nz) =ECas(Nz)+Eint\n0(Nz)[see Eq. (9a)],\nwhereEint\n0(Nz)exhibits the behavior of Eint\n0(Nz)/Nz\n[see Eq. (9c)]. Then, to see the film thickness dependence\nofECas(Nz), we introduce the rescaled Casimir energyC[b]\nCasin terms of Nzbforb2Ras\nC[b]\nCas(Nz) :=ECas\u0002Nzb(10)\nand callC[b]\nCasthe magnonic Casimir coefficient in the\nsense thatECas=C[b]\nCasNz\u0000b.\nNote that the zero-point energy arises from quan-\ntum fluctuations and does exist even at zero tempera-\nture. The zero-point energy defined at zero temperature\ndoes not depend on the Bose-distribution function [see\nEqs. (9b) and (9c)]. Throughout this work, we focus on\nzero temperature [23].\nMagnonic non-Hermitian Casimir effect. —Finally, we\ninvestigate the magnonic Casimir effect in the non-\nHermitian system \u000b > 0, which we call the magnonic\nnon-HermitianCasimireffect, foreachregime(i)-(iii). As\nan example, we consider NiO, an insulating AFM. From\nRefs. [25, 35, 36], we roughly estimate the model param-\neter values for NiO as follows [see Eq. (1)]: J= 47:0859\nmeV,Kh= 0:039 5212 meV,Ke= 0:001 71829 meV,\nS= 1:206 83, anda= 0:417nm. NiO is a biaxial\nAFM ofKh>0andKe>0. Due to the hard-axis\nanisotropy Kh>0, the degeneracy between two kinds\nof magnons \u001b=\u0006is lifted in NiO. These parameters\nprovide\u000bcri\n\u001b=+\u00180:008 5414< \u000bcri\n\u001b=\u0000\u00180:041 8709. Fig-\nure 2 shows the magnon energy dispersion [Eq. (1)] and\nthe magnonic Casimir energy [Eq. (9a)] with its Casimir\ncoefficient [Eq. (10)] for each regime (i)-(iii).\n(i) Gap-melting regime \u000b\u0014\u000bcri\n\u001b=+< \u000bcri\n\u001b=\u0000. The\nmagnonic Casimir energy takes a real value as shown\nin the middle and right panels of Fig. 2 (i), see also\nEq. (10), and there are no magnonic EPs [see the left\npanel of Fig. 2 (i)]. This property is the same as its\nCasimir effect in the dissipationless system \u000b= 0[18].\nWhen\u000b < \u000bcri\n\u001b=+, the magnon energy gap for both\n\u001b=\u0006is nonzero \u0001\u001b=\u0006;\u000b>0and both magnons \u001b=\u0006\nare the gapped modes. For each gapped mode, the abso-\nlute value of the magnonic Casimir coefficient C[3]\nCasde-\ncreases and approaches asymptotically to zero as the film\nthickness increases. We emphasize that the magnon en-\nergy gap decreases as the damping constant \u000bincreases\n[see Eq. (5)]. Then, the magnitude of the magnonic\nCasimir energy and its coefficient increase as the value\nof the damping constant becomes larger and approaches\nto the critical value \u000b!\u000bcri\n\u001b=+[see the middle panel of\nFig. 2 (i)].\nWhen\u000b=\u000bcri\n\u001b=+, the magnon \u001b=\u0000remains the\ngapped mode, whereas the magnon energy gap for \u001b= +\nvanishes \u0001\u001b=+;\u000b= 0and the magnon \u001b= +becomes\nthe gapless mode which behaves like a relativistic par-\nticle with the linear energy dispersion. In the gapless\nmode, the magnonic Casimir coefficient C[3]\nCasapproaches\nasymptotically to a nonzero constant as the film thick-\nness increases. This means that although the magnonic\nCasimir effect is realized on the lattice, the behavior of\nthe gapless magnon mode is analogous to the conven-\ntional Casimir effect of a massless scalar field in con-4\nNiOMagnon energy dispersion \u000f\u001b;k;\u000b Re(ECas) and Re(C[b]\nCas) Im(ECas) and Im(C[b]\nCas)\n(i)\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.00854Re(εσ=−)\nRe(εσ=+)\nIm(εσ=±)Energy εσ,k,α [meV]\nWavenumber akεσ=± at α=0\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20Nzα=0.00854Casimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of aα=0\n         \n−1.5−1−0.5\n 0  20  40  60  80Re(CCas[3]) [meV]\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20Nzα=0.00854Casimir energy Im( ECas) [meV]\nNz: Thickness of magnet in units of aα=0\n         \n−0.1 0 0.1\n 0  20  40  60  80Im(CCas[3]) [meV]\n(ii)\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.04Re(εσ=−)\nRe(εσ=+)\nIm(εσ=+)\nIm(εσ=−)EPEnergy εσ,k,α [meV]\nWavenumber ak\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20α=0.04\nNzCasimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of a−20−15−10−5 0 5 10\n 0 50 100  150  200  250  300Re(CCas[1.5]) [meV]\n 0 0.05 0.1 0.15 0.2\n 0  50  100  150  200  250  300α=0.04NzCasimir energy Im( ECas) [µeV]\nNz: Thickness of magnet in units of a−10−5 0 5 10\n 0 50 100  150  200  250  300Im(CCas[1.5]) [µeV]\n(iii)\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.05Re(εσ=−)\nRe(εσ=+)\nIm(εσ=+)\nIm(εσ=−)EP EPEnergy εσ,k,α [meV]\nWavenumber ak−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20α=0.05\nNzCasimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of a−20−10 0 10 20\n 0 50 100  150  200  250  300Re(CCas[1.5]) [meV]\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35\n 0  50  100  150  200  250  300α=0.05NzCasimir energy Im( ECas) [µeV]\nNz: Thickness of magnet in units of aα=0.05\n−20−10 0 10 20\n 0 50 100  150  200  250  300Im(CCas[1.5]) [µeV]\nFIG. 2. Plots of the magnon energy dispersion \u000f\u001b;k;\u000b, the real part of the magnonic Casimir energy Re (ECas), and the\nimaginary part Im (ECas)for NiO in (i) the gap-melting regime, (ii) the oscillating regime, and (iii) the beating regime. Inset:\nEach magnonic Casimir coefficient C[b]\nCas[see Eq. (10)].\ntinuous space [37] except for a-dependent lattice effects,\nwhereas that of the gapped magnon modes is similar\nto the Casimir effect known for massive degrees of free-\ndom [37, 38].\n(ii) Oscillating regime \u000bcri\n\u001b=+< \u000b < \u000bcri\n\u001b=\u0000. The\nmagnonic Casimir energy takes a complex value as\nshown in the middle and right panels of Fig. 2 (ii), see\nalso Eq. (10). There is one EP, e.g., akcri\n\u001b=+;\u000b=0:04\u0018\n0:039 05437773 for\u000b= 0:04[see the left panel of\nFig. 2 (ii)]. Then, the magnonic non-Hermitian Casimir\neffect exhibits an oscillating behavior as a function of Nz\nfor the film thickness Lz:=aNz.\nAn intuitive explanation for the oscillation of the\nmagnonic non-Hermitian Casimir effect and its relation\nto the EP is given as follows: Through the lattice regu-\nlarization, the magnonic Casimir energy is defined as the\ndifference [see Eq. (9a)] between the zero-point energy\nwith the discrete wavenumber kz[see Eq. (9b)] and the\none with the continuous wavenumber [see Eq. (9c)]. On\nthe lattice, the wavenumber kzunder the boundary con-\ndition is discretized in units of \u0019=aNzaskz!(\u0019=aNz)n.\nAs the film thickness Nzincreases, the unit becomessmaller, and finally, it matches the EP as \u0019=aNz=kcri\n\u001b;\u000b,\ni.e.,Nz=\u0019=akcri\n\u001b;\u000b, where the magnonic non-Hermitian\nCasimir effect is enhanced because the group velocity be-\ncomes much larger than the usual such as in the gap-\nmelting regime (i) due to the EP [compare the solid lines\nin the left panel of Figs. 2 (ii) and (iii) with the one of\nFig. 2 (i)]. Then, the magnonic non-Hermitian Casimir\neffect is enhanced periodically where the film thickness\nNzis multiples of \u0019=akcri\n\u001b;\u000b. Thus, the oscillating behavior\nofthemagnonicnon-HermitianCasimireffectstemsfrom\nthe EP,kcri\n\u001b;\u000b, and the oscillation is characterized in units\nof\u0019=akcri\n\u001b;\u000b. We refer to this oscillating behavior as the\nmagnonic EP-induced Casimir oscillation. The period of\nthis Casimir oscillation is\n\u0003Cas\n\u001b;\u000b:=\u0019\nakcri\u001b;\u000b: (11)\nAs an example, the period is \u0003Cas\n\u001b=+;\u000b=0:04\u001880:441 49815\nfor\u000b= 0:04. This agrees with the numerical result in the\nmiddle and right panels of Fig. 2 (ii), see the highlighted\nin red. We call (ii) \u000bcri\n\u001b=+< \u000b < \u000bcri\n\u001b=\u0000the oscillat-\ning regime. The middle and right panels of Fig. 2 (ii)5\nshow that the magnonic EP-induced Casimir oscillation\nis characterized by its Casimir coefficient C[b]\nCasofb= 1:5.\n(iii) Beating regime \u000bcri\n\u001b=+< \u000bcri\n\u001b=\u0000\u0014\u000b. The\nmagnonic Casimir energy takes a complex value as shown\nin the middle and right panels of Fig. 2 (iii), see also\nEq. (10). There are two EPs, kcri\n\u001b=+;\u000bandkcri\n\u001b=\u0000;\u000b, which\ninduce two types of the Casimir oscillations character-\nized by \u0003Cas\n\u001b=+;\u000band\u0003Cas\n\u001b=\u0000;\u000b, respectively. As an exam-\nple,akcri\n\u001b=+;\u000b=0:05\u00180:049 21389535 andakcri\n\u001b=\u0000;\u000b=0:05\u0018\n0:027 28830018 provide \u0003Cas\n\u001b=+;\u000b=0:05\u001863:835 48044 and\n\u0003Cas\n\u001b=\u0000;\u000b=0:05\u0018115:125 9929, respectively, for \u000b= 0:05\n[see the left panel of Fig. 2 (iii)]. Due to the interference\nbetween the two Casimir oscillations, the magnonic non-\nHermitian Casimir effect exhibits a beating behavior as\na function of Nzfor the film thickness Lz:=aNzwith a\nperiod of\n1\nj1=\u0003Cas\n\u001b=+;\u000b\u00001=\u0003Cas\n\u001b=\u0000;\u000bj: (12)\nAs an example, the period is j1=\u0003Cas\n\u001b=+;\u000b=0:05\u0000\n1=\u0003Cas\n\u001b=\u0000;\u000b=0:05j\u00001\u0018143:284 2588 for\u000b= 0:05. This\nagrees with the numerical result in the middle and right\npanels of Fig. 2 (iii), see the highlighted in blue. We call\n(iii)\u000bcri\n\u001b=+<\u000bcri\n\u001b=\u0000\u0014\u000bthe beating regime. The middle\nand right panels of Fig. 2 (iii) show that the beating be-\nhavior of the magnonic EP-induced Casimir oscillation is\ncharacterized by its Casimir coefficient C[b]\nCasofb= 1:5.\nWe remark that in the absence of the hard-axis\nanisotropy Kh= 0, two kinds of magnons \u001b=\u0006\nare in degenerate states [see Eq. (1)]. This results in\n\u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000[see Eq. (6)] and \u0003Cas\n\u001b=+;\u000b= \u0003Cas\n\u001b=\u0000;\u000b[see\nEq. (11)]. Hence, the Casimir oscillation is one type with\naperiodof \u0003Cas\n\u001b;\u000b. Thismeansthatthebeatingbehavioris\nabsent in the uniaxial AFMs of Kh= 0andKe>0[23].\nMagnonic Casimir engineering. —In each regime (i)-\n(iii), the Gilbert damping (i.e., energy dissipation) serves\nas a key ingredient of Casimir engineering [20] to con-trol and manipulate the Casimir effect of magnons. The\nGilbert damping can be enhanced and controlled by the\nestablished experimental techniques of spintronics such\nas spin pumping [23]. In addition, microfabrication tech-\nnology can control the film thickness and manipulate the\nmagnonic non-Hermitian Casimir effect. The Casimir\npressure of magnons, which stems from the real part\nof its Casimir energy, contributes to the internal pres-\nsure of thin films. We find from the middle panel of\nFigs. 2 (ii) and (iii) that depending on the film thick-\nness, the sign of the real part of the magnonic Casimir\ncoefficient changes. This means that by tuning the film\nthickness, we can control and manipulate the direction\nof the magnonic Casimir pressure as well as the magni-\ntudethankstotheEP-inducedCasimiroscillation. Thus,\nour study utilizing energy dissipation, the magnonic non-\nHermitian Casimir effect, provides the new principles of\nnanoscale devices, such as highly sensitive pressure sen-\nsors and magnon transistors [39], and paves a way for\nmagnonic Casimir engineering.\nConclusion. —We have shown that as the Gilbert\ndamping constant (i.e., the energy dissipation rate) in-\ncreases, the non-Hermitian Casimir effect of magnons in\nantiferromagnets is enhanced and exhibits the oscillat-\ning behavior which stems from the exceptional point.\nThis exceptional point-induced Casimir oscillation also\nexhibits the beating behavior when the degeneracy be-\ntween two kinds of magnons is lifted. These magnonic\nCasimir oscillations are absent in the dissipationless sys-\ntem of magnons. 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Princep, R. A. Ewings, S. Ward, S. Tóth, C. Dubs,\nD. Prabhakaran, and A. T. Boothroyd, npj Quantum\nMater. 2, 1 (2017).\n[67] Y. Nambu and S.-i. Shamoto, J. Phys. Soc. Jpn. 90,\n081002 (2021), arXiv:2106.15752.\n[68] O. I. Gorbatov, G. Johansson, A. Jakobsson,\nS. Mankovsky, H. Ebert, I. Di Marco, J. Minár,\nand C. Etz, Phys. Rev. B 104, 174401 (2021).7\nSupplemental Material\nIn this Supplemental Material, we add an explanation\nabout the Casimir energy induced by quantum fields on\nthe lattice and provide some details about the magnonic\nCasimir effect in the absence of the hard-axis anisotropy.\nWe also add remarks on, in order, observation of the\nmagnonic Casimir effect in the AFMs, Casimir effects\nfrom other origins, thermal effects, higher energy bands,\nedge or surface magnon modes, and the effect of the edge\ncondition.\nI. THE CASIMIR ENERGY ON THE LATTICE\nIn the main text, following the Casimir energy\nfor photon fields (i.e., quantum fields in continuous\nspace) [2], the magnonic Casimir energy is defined as in\nEqs. (9a), (9b), and (9c) through the lattice regulariza-\ntion. In contrast to the Casimir effect for photon fields\n(i.e., quantum fields in continuous space), the magnonic\nCasimir energy is induced byits quantum field on the lat-\ntice, and there is no ultraviolet divergence in each com-\nponent [see Eqs. (9b) and (9c)]. Here we remark that the\nCasimir energy induced by quantum fields on the lat-\ntice, such as the magnonic Casimir energy ECas(Nz)[see\nEq. (9a)], plays a key role in finding the film thickness\ndependence of the zero-point energy in the thin film (see\nFig. 1). The zero-point energy in the thin film of the\nthicknessNzisEsum\n0(Nz)NxNy[see Eq. (9b)] and con-\nsists of two parts as Esum\n0(Nz) =ECas(Nz) +Eint\n0(Nz)\n[see Eq. (9a)], where Eint\n0(Nz)exhibits the behavior of\nEint\n0(Nz)/Nz[see Eq. (9c)].\nII. THE HARD-AXIS ANISOTROPY\nA. In the absence of the hard-axis anisotropy\nIn the main text, we have considered NiO. NiO is a bi-\naxial AFM of Kh>0andKe>0: There exist not only\nthe easy-axis anisotropy Ke= 0:001 71829 meV but also\nthe hard-axis anisotropy Kh= 0:039 5212 meV, see the\nmain text for other parameter values. Here, by chang-\ning only the value of KhtoKh= 0with leaving other\nparameter values unchanged, we estimate the magnonic\nCasimir effect and provide some details about its behav-\nior in the absence of the hard-axis anisotropy.\nFigure S1 shows the magnon energy dispersion \u000f\u001b;k;\u000b\nfor the gap-melting regime (i) in the absence of the\nhard-axis anisotropy Kh= 0. Figure S2 shows the real\npart of the magnonic Casimir energy Re (ECas)for the\ngap-melting regime (i) in the absence of the hard-axis\nanisotropy Kh= 0and that in the presence of hard-axis\nanisotropy Kh= 0:039 5212 meV. The latter is the same\nas the middle panel of Fig. 2 (i).\nIn the absence of the hard-axis anisotropy Kh= 0, two\nkinds of magnons \u001b=\u0006are in degenerate states [see\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.00854, Kh=0\nRe(εσ=±)\nIm(εσ=±)Energy εσ,k,α [meV]\nWavenumber akεσ=± at α=0, Kh=0FIG. S1. Plots of the magnon energy dispersion \u000f\u001b;k;\u000bfor\nthe gap-melting regime (i) in the absence of the hard-axis\nanisotropy Kh= 0, where\u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000= 0:008 54.\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20Nzα=0.00854 α=0.00854, Kh=0Casimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of aα=0\n             α=0,Kh=0\n            \n−1.5−1−0.5\n 0  20  40  60  80Re(CCas[3]) [meV]\nFIG.S2. PlotsoftherealpartofthemagnonicCasimirenergy\nRe(ECas)for the gap-melting regime (i) in the absence of the\nhard-axis anisotropy Kh= 0and those in the presence of\nhard-axis anisotropy Kh= 0:039 5212 meV. The latter is the\nsame as the middle panel of Fig. 2 (i). Inset: Its Casimir\ncoefficientC[b]\nCas=ECas\u0002Nzb.\nEq. (1)]. This results in \u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000= 0:008 54[see\nEq. (6)]. When the damping constant reaches the critical\nvalue\u000b=\u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000= 0:008 54, the magnon energy\ngaps for both \u001b=\u0006vanish, \u0001\u001b=\u0006;\u000b= 0, and both\nmagnons\u001b=\u0006become the gapless modes which behave\nlike relativistic particles with the linear energy disper-\nsion (see the solid line in Fig. S1). Then, the magnonic\nCasimir coefficient C[3]\nCasasymptotically approaches to\na nonzero constant as the film thickness increases (see\nFig. S2), which means that its Casimir energy exhibits\nthe behavior of ECas/1=Nz3. Figure S2 also shows\nthat the magnitude of the magnonic Casimir energy and\nits coefficient for Kh= 0become larger than that for\nKh= 0:039 5212 meV.8\nTABLE I. Magnonic non-Hermitian Casimir effects in the\nAFMs ofKe>0.\nKh>0Kh= 0\nThe degeneracy of magnons ( \u001b=\u0006) No Yes\nThe number of the magnonic EPs 2 1\nThe EP-induced Casimir oscillation Yes Yes\nThe beating behavior of the oscillation Yes No\nB. An example of the uniaxial AFM\nAs an example, Cr 2O3can be regarded as a uniaxial\nAFMofKh= 0andKe>0, wheretwokindsofmagnons\n\u001b=\u0006areindegeneratestates. Hence, themagnonicEP-\ninduced Casimir oscillation is one type, and its beating\nbehavior is absent in Cr 2O3. Magnonic non-Hermitian\nCasimir effects in the AFMs of Ke>0are summarized\nin Table I. Note that Ref. [56] reported the experimental\nrealization of sub-terahertz spin pumping in Cr 2O3, and\nRef. [57] reported that in NiO.\nIII. REMARKS ON OBSERVATION\nIn the main text, we have explained that the Gilbert\ndampingcanbeenhancedandcontrolledbytheuseofthe\nestablished experimental techniques of spintronics such\nas spin pumping. Here we add remarks on observation of\nour theoretical prediction. We expect that the magnonic\nCasimir effect in the AFMs can be experimentally ob-\nserved in principle through measurement of magnetiza-\ntion. The reason is as follows.\nExternal magnetic fields induce magnetostriction,\nwhich can be regarded as a kind of lattice deformation,\nand its correction for the length of a magnetic unit cell\nais characterized by the magnetostriction constant [58–\n65]. The magnonic Casimir energy of the AFMs does not\ndepend on external magnetic fields usually, whereas the\nmagnonic Casimir effect is influenced by magnetostric-\ntion, and its correction for the magnonic Casimir energy\ndepends on magnetic fields and contributes to magne-\ntization. Thus, although the correction is small, the\nmagnonic Casimir effect in the AFMs can be experimen-\ntally observed in principle through measurement of mag-\nnetization and its film thickness dependence by using ex-\nternal magnetic fields (i.e., magnetostriction).\nWe remark that the magnetic-field derivative of the\nreal part of the Helmholtz free energy is magnetiza-\ntion. At zero temperature, assuming thin films of Nz\u001c\nNx;Ny(see Fig. 1), the Helmholtz free energy of quan-\ntum fields for magnons in the thin film of the thickness\nNzisEsum\n0(Nz)NxNy[see Eq. (9b)] and consists of two\nparts asEsum\n0(Nz) =ECas(Nz)+Eint\n0(Nz)[see Eq. (9a)],\nwhereEint\n0(Nz)exhibits the linear-in- Nzbehavior as\nEint\n0(Nz)/Nz[see Eq. (9c)]. Since ECas(Nz)exhibits an\noscillatingandabeatingbehaviorasafunctionofthefilmthickness in the regimes (ii) and (iii), respectively [see\nEq. (10) and the middle panels of Figs. 2 (ii) and (iii)],\nthe Helmholtz free energy of the thin film shows a dif-\nferentNz-dependence from the linear-in- Nzbehavior. In\nother words, magnetization of the thin film exhibits an\noscillating or a beating behavior as a function of the film\nthickness due to the magnonic non-Hermitian Casimir ef-\nfect. Hence, our prediction, the non-Hermitian Casimir\neffect of magnons, can be observed in principle through\nmeasurement of magnetization, its oscillating or beating\nbehavior as a function of the film thickness.\nIV. CASIMIR EFFECTS FROM OTHER\nORIGINS\nIn the main text, we have focused on the magnonic\nCasimir effect. Here we add a remark on Casimir ef-\nfects from other origins such as phonons and photons.\nEven excluding magnetostriction, the energy dispersion\nof magnons depends strongly on magnetic fields through\nZeeman coupling, whereas those of phonons and photons\ndo not. Therefore, we expect that the magnonic Casimir\neffect can be distinguished experimentally from the oth-\ners by manipulating external magnetic fields.\nV. THERMAL EFFECTS\nIn the main text, we have focused on zero tempera-\nture. Here we remark on thermal effects. At nonzero\ntemperature, a thermal contribution to the Helmholtz\nfree energy, called the thermal Casimir energy, arises ad-\nditionally and is characterized by the Boltzmann factor.\nIt should be emphasized that, although it is called the\nthermal Casimir energy, there is a significant distinction\nbetween the thermal Casimir effect and the Casimir ef-\nfect: The thermal Casimir effect is independent of the\nzero-pointenergy. ThethermalCasimireffectarisesfrom\nthermal fluctuations and is affected by temperatures,\nwhereas the Casimir effect arises from the zero-point en-\nergy due to quantum fluctuations and is not affected by\ntemperatures. Hence, we expect that the Casimir effect\nof magnons can be distinguished experimentally from its\nthermalCasimireffectbymanipulatingtemperature. For\ndetailsofamagnonicanalogofthethermalCasimireffect\nin a Hermitian system, see Ref. [48] as an example.\nVI. HIGHER ENERGY BANDS\nIn the main text, we have assumed that the magnonic\nCasimir energy of the AFM, NiO, is dominated by the\ntwo bands of Eq. (1). Here we remark on the contri-\nbution from higher energy bands than those of Eq. (1).\nThe magnonic Casimir energy or the zero-point energy\n[see Eq. (9a)] arises from quantum fluctuations and does9\nexistevenatzerotemperature. Thezero-pointenergyde-\nfined at zero temperature does not depend on the Bose-\ndistribution function [see Eqs. (9b) and (9c)]. Hence,\nhigher energy bands than those of Eq. (1) also can con-\ntribute to the magnonic Casimir energy. However, the\ncontribution becomes smaller as the shape of the bands\nis flatter. Numerical calculations of Refs. [66–68] show\nthat higher energy bands of a ferrimagnet tend to be flat.\nThe ferrimagnet has an alternating structure of up and\ndown spins like the Néel magnetic order of the AFM, and\nin this sense, the ferrimagnet is similar to the AFM. We\ntherefore assume that higher energy bands of the AFM\nalso tend to be flat. Thus, throughout this study, we\nwork under the assumption that the magnonic Casimir\nenergy of the AFM, NiO, is dominated by the two bands\nof Eq. (1). For a more accurate estimation, inelastic neu-\ntron scattering measurement of its higher energy bands\nis essential.\nVII. EDGE OR SURFACE MAGNON MODES\nWe add an explanation about the effect of edge or\nsurface magnon modes on the magnonic Casimir energy.\nThe magnonic Casimir effect in our setup (see the thin\nfilm of Fig. 1) is induced by quantum fields for magnons\nof wavenumbers kzdiscretized by small Nz: Its necessary\ncondition is a kz-dependent dispersion relation through\nthe discretization of kz. In this study, we consider thin\nfilms ofNz\u001cNx;Ny. Even if there are edge or surface\nmagnon modes, they are confined only on the xyplane,\nand their wavenumber in the zdirection is always zero,\ni.e.,kz= 0, where its energy dispersion relation is in-\ndependent of kz. Therefore, such edge or surface modes\ncannot contribute to the magnonic Casimir effect. In\nthis sense, the magnonic Casimir effect in our setup (see\nFig. 1) is not affected by the presence or absence of edge\nor surface magnon modes.\nVIII. THE EFFECT OF EDGE CONDITIONS\nWe add a remark on the edge condition. Details of\nthe edge condition, such as the presence or absence of\ndisorder, may affect the boundary condition for the wave\nfunction of magnons, but the magnonic Casimir effect\nis little influenced as long as one does not assume an\nultrathin film such as Nz= 1;2;3. Even if there is a\nchange in the magnon band structure near the edge due\nto some reasons, such as changed spin anisotropies, the\nexistence of the magnonic Casimir effect remains valid\nas long as its necessary condition (see Sec. VII of this\nSupplemental Material) is satisfied."    },    {        "title": "2202.06154v1.Generalization_of_the_Landau_Lifshitz_Gilbert_equation_by_multi_body_contributions_to_Gilbert_damping_for_non_collinear_magnets.pdf",        "content": "Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to\nGilbert damping for non-collinear magnets\nSascha Brinker,1Manuel dos Santos Dias,2, 1,\u0003and Samir Lounis1, 2,y\n1Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n2Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg, Germany\n(Dated: February 15, 2022)\nWe propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing\nthe dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises\nfrom multi-body scattering processes. The tensor consists of a damping-like term and a correction\nto the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend\non e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: ei\u0001ej,\n(nij\u0001ei)(nij\u0001ej),nij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where some terms are subjected to the\nspin-orbit \feld nijin \frst and second order. We explore the magnitude of the di\u000berent contributions\nusing both the Alexander-Anderson model and time-dependent density functional theory in magnetic\nadatoms and dimers deposited on Au(111) surface.arXiv:2202.06154v1  [cond-mat.mtrl-sci]  12 Feb 20222\nI. INTRODUCTION\nIn the last decades non-collinear magnetic textures have been at the forefront in the \feld of spintronics due to the\npromising applications and perspectives tied to them1,2. Highly non-collinear particle-like topological swirls, like\nskyrmions3,4and hop\fons5, but also domain walls6can potentially be utilized in data storage and processing devices\nwith superior properties compared to conventional devices. Any manipulation, writing and nucleation of these various\nmagnetic states involve magnetization dynamical processes, which are crucial to understand for the design of future\nspintronic devices.\nIn this context, the Landau-Lifshitz-Gilbert (LLG) model7,8is widely used to describe spin dynamics of materials\nranging from 3-dimensional bulk magnets down to the 0-dimensional case of single atoms, see e.g. Refs.9{12. The\nLLG model has two important ingredients: (i) the Gilbert damping being in general a tensorial quantity13, which can\noriginate from the presence of spin-orbit coupling (SOC)14and/or from spin currents pumped into a reservoir15,16;\n(ii) the e\u000bective magnetic \feld acting on a given magnetic moment and rising from internal and external interactions.\nOften a generalized Heisenberg model, including magnetic anisotropies and magnetic exchange interactions, is utilized\nto explore the ground state and magnetization dynamics characterizing a material of interest. Instead of the con-\nventional bilinear form, the magnetic interactions can eventually be of higher-order type, see e.g.17{23. Similarly to\nmagnetic interactions, the Gilbert damping, as we demonstrate in this paper, can host higher-order non-local contri-\nbutions. Previously, signatures of giant anisotropic damping were found24, while chiral damping and renormalization\nof the gyromagnetic ratio were revealed through measurements executed on chiral domain wall creep motion24{28.\nMost \frst-principles studies of the Gilbert damping were either focusing on collinear systems or were case-by-case\nstudies on speci\fc non-collinear structures lacking a general understanding of the fundamental behaviour of the Gilbert\ndamping as function of the non-collinear state of the system. In this paper, we discuss the Gilbert damping tensor\nand its dependencies on the alignment of spin moments as they occur in arbitrary non-collinear state. Utilizing linear\nresponse theory, we extract the dynamical magnetic susceptibility and identify the Gilbert damping tensor pertaining\nto the generalized LLG equation that we map to that obtained from electronic structure models such as the single\norbital Alexander-Anderson model29or time-dependent density functional theory applied to realistic systems10,30,31.\nApplying systematic perturbative expansions, we \fnd the allowed dependencies of the Gilbert damping tensor on the\ndirection of the magnetic moments. We identify terms that are a\u000bected by SOC in \frst and second order. We generalize\nthe LLG equation by a simple form where the Gilbert damping tensor is amended with terms proportional to scalar,\nanisotropic, vector-chiral and scalar-chiral products of magnetic moments, i.e. terms like ei\u0001ej, (nij\u0001ei)(nij\u0001ej),\nnij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where we use unit vectors, ei=mi=jmij, to describe the directional dependence\nof the damping parameters and nijrepresents the spin-orbit \feld.\nThe knowledge gained from the Alexander-Anderson model is applied to realistic systems obtained from \frst-principles\ncalculations. As prototypical test system we use 3 dtransition metal adatoms and dimers deposited on the Au(111)\nsurface. Besides the intra-site contribution to the Gilbert damping, we also shed light on the inter-site contribution,\nusually referred to as the non-local contribution.\nII. MAPPING THE GILBERT DAMPING FROM THE DYNAMICAL MAGNETIC SUSCEPTIBILITY\nHere we extract the dynamical transverse magnetic response of a magnetic moment from both the Landau-Lifshitz-\nGilbert model and electronic structure theory in order to identify the Gilbert damping tensor Gij10,11,32,33. In linear\nresponse theory, the response of the magnetization mat siteito a transverse magnetic \feld bapplied at sites jand\noscillating at frequency !reads\nm\u000b\ni(!) =X\nj\f\u001f\u000b\f\nij(!)b\f\nj(!); (1)\nwith the magnetic susceptibility \u001f\u000b\f\nij(!) and\u000b;\fare thex;ycoordinates de\fned in the local spin frame of reference\npertaining to sites iandj.\nIn a general form13the LLG equation is given by\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njGij\u0001dmj\ndt1\nA; (2)3\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)°2°101234Energy [U]°1.00°0.75°0.50°0.250.000.250.500.751.00DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8\n°2°101234Energy [U]°1.0°0.50.00.51.0DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8dI/dVVexcitationa)b)\nabc\nFIG. 1. Illustration of the Landau-Lifshitz-Gilbert model and local density of states within the Alexander-Anderson model.\n(a) A magnetic moment (red arrow) precesses in the the presence of an external \feld. The blue arrow indicates the direction\nof a damping term, while the green arrow shows the direction of the precession term. (b) Density of states for di\u000berent\nmagnetizations in the range from 0 :2 to 0:8. Density of states of dimers described within the Alexander-Anderson model for\ndi\u000berent magnetizations in the range from 0 :2 to 0:8. Shown is the ferromagnetic reference state. The magnetizations are\nself-consistently constrained using a longitudinal magnetic \feld, which is shown in the inset. Model parameters: U= 1:0 eV,\nEd= 1:0 eV; t= 0:2 eV;\u0000 = 0:2 eV; 'R= 0 °.\nwhere\r= 2 is the gyromagnetic ratio, Be\u000b\ni=\u0000dHspin=dmiis the e\u000bective magnetic \feld containing the contributions\nfrom an external magnetic \feld Bext\ni, as well as internal magnetic \felds originating from the interaction of the\nmoment with its surrounding. In an atomistic spin model described by e.g. the generalized Heisenberg hamiltonian,\nHspin=P\nimiKimi+1\n2P\nijmiJijmj, containing the on-site magnetic anisotropy Kiand the exchange tensor Jij,\nthe e\u000bective \feld is given by Be\u000b\ni=Bext\ni\u0000Kimi\u0000P\njJijmj(green arrow in Fig. 1a). The Gilbert damping tensor\ncan be separated into two contributions { a damping-like term, which is the symmetric part of the tensor, S, (blue\narrow in Fig. 1a), and a precession-like term A, which is the anti-symmetric part of the tensor. In Appendix A we\nshow how the antisymmetric intra-site part of the tensor contributes to a renormalization of the gyromagnetic ratio.\nTo extract the magnetic susceptibility, we express the magnetic moments in their respective local spin frame of\nreferences and use Rotation matrices that ensure rotation from local to globa spin frame of reference (see Appendix B).\nThe magnetic moment is assumed to be perturbed around its equilibrium value Mi,mloc\ni=Miez\ni+mx\niex\ni+my\niey\ni,\nwhere e\u000b\niis the unit vector in direction \u000bin the local frame of site i. Using the ground-state condition of vanishing\nmagnetic torques, Miez\ni\u0002\u0000\nBext\ni+Bint\ni\u0001\n= 0 and the inverse of the transverse magnetic susceptibility can be identi\fed\nas\n\u001f\u00001\ni\u000bj\f(!) =\u000eij\u0012\n\u000e\u000b\fBe\u000b\niz\nMi+i!\n\rMi\u000f\u000b\f\u0016\u0013\n+1\nMiMj(RiJijRT\nj)\u000b\f+ i!(RiGijRT\nj)\u000b\f; (3)\nfrom which it follows that the Gilbert damping is directly related to the linear in frequency imaginary part of the\ninverse susceptibility\nd\nd!=[\u001f\u00001]\u000b\f\nij=\u000eij\u00121\n\rMi\u000f\u000b\f\u0016\u0013\n+ (RiGijRT\nj)\u000b\f: (4)\nNote thatRiandRjare rotation matrices rotating to the local frames of site iandj, respectively, which de\fne the\ncoordinates \u000b;\f=fx;yg(see Appendix B).\nBased on electronic structure theory, the transverse dynamical susceptibility can be extracted from a Dyson-like\nequation:\u001f\u00001(!) =\u001f\u00001\n0(!)\u0000U, where\u001f0is the susceptibility of non-interacting electron while Uis a many-body\ninteraction Kernel, called exchange-correlation Kernel in the context of time-dependent density functional theory30.\nThe Kernel is generally assumed to be adiabatic, which enables the evaluation of the Gilbert damping directly from4\nthe non-interacting susceptibility. Obviously:d\nd!\u001f\u00001(!) =d\nd!\u001f\u00001\n0(!). For small frequencies !,\u001f0has a simple\n!-dependence11:\n\u001f0(!)\u0019<\u001f0(0) +i!=d\nd!\u001f0j!=0 (5)\nand as shown in Ref.33\nd\nd!\u001f\u00001\n0(!)\u0019[<\u001f0(0)]\u00002=d\nd!\u001f0j!=0: (6)\nStarting from the electronic Hamiltonian Hand the corresponding Green functions G(E\u0006i\u0011) = (E\u0000H\u0006i\u0011)\u00001, one\ncan show that the non-interacting magnetic susceptibility can be de\fned via\n\u001f\u000b\f\n0;ij(!+ i\u0011) =\u00001\n\u0019TrZEF\ndE\u0002\n\u001b\u000bGij(E+!+ i\u0011)\u001b\fImGji(E) +\u001b\u000bImGij(E)\u001b\fGji(E\u0000!\u0000i\u0011)\u0003\n;(7)\nwith\u001bbeing the vector of Pauli matrices. Obviously to identify the Gilbert damping and how it reacts to magnetic\nnon-collinearity, we have to inspect the dependence of the susceptibility, and therefore the Green function, on the\nmisalignment of the magnetic moments.\nIII. MULTI-SITE EXPANSION OF THE GILBERT DAMPING\nAssuming the hamiltonian Hconsisting of an on-site contribution H0and an inter-site term encoded in a hopping\ntermt, which can be spin-dependent, one can proceed with a perturbative expansion of the corresponding Green\nfunction utilizing the Dyson equation\nGij=G0\ni\u000eij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+::: : (8)\nWithin the Alexander-Anderson single-orbital impurity model29,H0\ni=Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b, whereEd\nis the energy of the localized orbitals, \u0000 is the hybridization in the wide band limit, Uiis the local interaction\nresponsible for the formation of a magnetic moment and Biis an constraining or external magnetic \feld. SOC can be\nincorportated as tsoc\nij=i\u0015ijnij\u0001\u001b, where\u0015ijandnij=\u0000njirepresent respectively the strength and direction of the\nanisotropy \feld. It can be parameterized as a spin-dependent hopping using the Rashba-like spin-momentum locking\ntij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b)34.\nDepending on whether the considered Green function is an on-site Green function Giior an inter-site Green function\nGijdi\u000berent orders in the hopping are relevant. On-site Green functions require an even number of hopping processes,\nwhile inter-site Green functions require at least one hopping process.\nThe on-site Green function G0\nican be separated into a spin-less part Niand a spin dependent part Mi,\nG0\ni=Ni\u001b0+Mi\u0001\u001b ; (9)\nwhere the spin dependent part is parallel to the magnetic moment of site i,Mikmi(note that SOC is added later on\nto the hoppings). Using the perturbative expansion, eq. (8), and the separated Green function, eq. (9), to calculate\nthe magnetic susceptibility, eq. (7), one can systematically classify the allowed dependencies of the susceptibility with\nrespect to the directions of the magnetic moments, e.g. by using diagrammatic techniques as shown in Ref.18for a\nrelated model in the context of higher-order magnetic exchange interactions.\nSince our interest is in the form of the Gilbert damping, and therefore also in the form of the magnetic susceptibility, the\nperturbative expansion can be applied to the magnetic susceptibility. The general form of the magnetic susceptibility\nin terms of the Green function, eq. (7), depends on a combination of two Green functions with di\u000berent energy\narguments, which are labeled as !and 0 in the following. The relevant structure is then identi\fed as33,\n\u001f\u000b\f\nij(!)\u0018Tr\u001b\u000b\niGij(!)\u001b\f\njGji(0): (10)\nThe sake of the perturbative expansion is to gather insights in the possible forms and dependencies on the magnetic\nmoments of the Gilbert damping, and not to calculate explicitly the strength of the Gilbert damping from this5\nexpansion. Therefore, we focus on the structure of eq. (10), even though the susceptibility has more ingredients,\nwhich are of a similar form.\nInstead of writing all the perturbations explicitly, we set up a diagrammatic approach, which has the following\ningredients and rules:\n1. Each diagram contains the operators NandM, which are \u001b\u000band\u001b\ffor the magnetic susceptibility. The\noperators are represented by a white circle with the site and spin index: i\u000b\n2. Hoppings are represented by grey circles indicating the hopping from site itoj:ij. The vertex corresponds\ntotij.\n3. SOC is described as a spin-dependent hopping from site itojand represented by: ij;\u000b. The vertex\ncorresponds to tsoc\nij=i\u0015ij^n\u000b\nij\u001b\u000b.\n4. The bare spin-independent (on-site) Green functions are represented by directional lines with an energy at-\ntributed to it: !. The Green function connects operators and hoppings. The line corresponds to Ni(!).\n5. The spin-dependent part of the bare Green function is represented by: !;\u000b .\u000bindicates the spin direction.\nThe direction ensures the right order within the trace (due to the Pauli matrices, the di\u000berent objects in the\ndiagram do not commute). The line corresponds to Mi(!)m\u000b\ni\u001b\u000b.\nNote that the diagrammatic rules might be counter-intuitive, since local quantities (the Green function) are represented\nby lines, while non-local quantities (the hopping from itoj) are represented by vertices. However, these diagrammatic\nrules allow a much simpli\fed description and identi\fcation of all the possible forms of the Gilbert damping, without\nhaving to write lengthy perturbative expansions.\nSpin-orbit coupling independent contributions.\nTo get a feeling for the diagrammatic approach, we start with the simplest example: the on-site susceptibility without\nany hoppings to a di\u000berent site, which describes both the single atom and the lowest order term for interacting atoms.\nThe possible forms are,\n\u001fii\n\u000b\f(!)/\n!0\ni\u000b i\f+\n!;\r0\ni\u000b i\f\n+\n!0;\r\ni\u000b i\f+\n!;\u000e0;\r\ni\u000b i\f; (11)6\nwhich evaluate to,\n!0\ni\u000b i\f= Tr\u001b\u000b\u001b\fNi!)Ni(0) =\u000e\u000b\fNi(!)Ni(0) (12)\n!;\r0\ni\u000b i\f= Tr\u001b\u000b\u001b\r\u001b\fMi(!)Ni(0)m\r\ni= i\u000f\u000b\r\fMi(!)Ni(0)m\r\ni (13)\n!0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\f\u001b\rNi(!)Mi(0)m\r\ni= i\u000f\u000b\f\rMi(!)Mi(0)m\r\ni (14)\n!;\u000e0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\u000e\u001b\f\u001b\rMi(!)Mi(0)m\u000e\nim\r\ni\n= (\u000e\u000b\u000e\u000e\f\r+\u000e\u000b\r\u000e\f\u000e\u0000\u000e\u000b\f\u000e\r\u000e)Mi(!)Mi(0)m\u000e\nim\r\ni: (15)\nThe \frst diagram yields an isotropic contribution, the second and third diagrams yield an anti-symmetric contribution,\nwhich is linear in the magnetic moment, and the last diagram yields a symmetric contribution being quadratic in the\nmagnetic moment. Note that the energy dependence of the Green functions is crucial, since otherwise the sum of\neqs. (13) and (14) vanishes. In particular this means that the static susceptibility has no dependence linear in the\nmagnetic moment, while the the slope of the susceptibility with respect to energy can have a dependence linear in\nthe magnetic moment. The static part of the susceptibility maps to the magnetic exchange interactions, which are\nknown to be even in the magnetic moment due to time reversal symmetry.\nCombining all the functional forms of the diagrams, we \fnd the following possible dependencies of the on-site Gilbert\ndamping on the magnetic moments,\nG\u000b\f\nii(fmg)/f\u000e\u000b\f;\u000f\u000b\f\rm\r\ni;m\u000b\nim\f\nig: (16)\nSince we work in the local frames, mi= (0;0;mz\ni), the last dependence is a purely longitudinal term, which is not\nrelevant for the transversal dynamics discussed in this work.\nIf we still focus on the on-site term, but allow for two hoppings to another atom and back, we \fnd the following new7\ndiagrams,\n!00\n0\ni\u000b i\fij ji\n+\n!;\r00\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0\n0\ni\u000b i\fij ji\n+:::\n+\n!;\r0;\u000e0;\u0011\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0;\u0011\n0;\u0017\ni\u000b i\fij ji\n: (17)\nThe dashed line in the second diagram can be inserted in any of the four sides of the square, with the other possibilities\nomitted. Likewise for the diagrams with two or three dashed lines, the di\u000berent possible assignments have to be\nconsidered. The additional hopping to the site jyields a dependence of the on-site magnetic susceptibility and\ntherefore also the on-site Gilbert damping tensor on the magnetic moment of site j.\nAnother contribution to the Gilbert damping originates from the inter-site part, thus encoding the dependence of the\nmoment site ion the dynamics of the moment of site jviaGij. This contribution is often neglected in the literature,\nsince for many systems it is believed to have no signi\fcant impact. Using the microscopic model, a di\u000berent class\nof diagrams is responsible for the inter-site damping. In the lowest order in t=Um the diagrams contain already two\nhopping events,\n! !0 0\ni\u000b j\f\nijij\n+\n!;\r !0 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0 0\ni\u000b j\f\nijij\n+:::\n+\n!;\r !;\u000e0;\u0011 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0;\u0011 0;\u0010\ni\u000b j\f\nijij\n: (18)\nIn total, we \fnd that the spin-orbit independent intra-site and inter-site Gilbert damping tensors can be respectively\nwritten as\nGii=\u0010\nSi+Sij;(1)\ni (ei\u0001ej) +Sij;(2)\ni (ei\u0001ej)2\u0011\nI\n+\u0010\nAi+Aij\ni(ei\u0001ej)\u0011\nE(ei);(19)8\nand\nG\u000b\f\nij=\u0000\nSij+Sdot\nij(ei\u0001ej)\u0001\n\u000e\u000b\f\n+\u0000\nAij+Adot\nij(ei\u0001ej)\u0001\n(E(ei) +E(ej))\u000b\f\n+Scross\nij(ei\u0002ej)\u000b(ei\u0002ej)\f+Sba\nije\f\nie\u000b\nj; (20)\nwhere as mentioned earlier SandArepresent symmetric and asymmetric contributions, Iis the 3\u00023 identity while\nE(ei) =0\n@0ez\ni\u0000ey\ni\n\u0000ez\ni0ex\ni\ney\ni\u0000ex\ni01\nA.\nRemarkably, we \fnd that both the symmetric and anti-symmetric parts of the Gilbert damping tensor have a rich\ndependence with the opening angle of the magnetic moments. We identify, for example, the dot and the square\nof the dot products of the magnetic moments to possibly play a crucial role in modifying the damping, similarly to\nbilinear and biquadratic magnetic interactions. It is worth noting that even though the intra-site Gilbert damping can\nexplicitly depend on other magnetic moments, its meaning remains unchanged. The anti-symmetric precession-like\nterm describes a precession of the moment around its own e\u000bective magnetic \feld, while the diagonal damping-like\nterm describes a damping towards its own e\u000bective magnetic \feld. The dependence on other magnetic moments\nrenormalizes the intensity of those two processes. The inter-site Gilbert damping describes similar processes, but with\nrespect to the e\u000bective \feld of the other involved magnetic moment. On the basis of the LLG equation, eq. (2), it can be\nshown that the term related to Sba\nijwith a functional form of e\f\nie\u000b\njdescribes a precession of the i-th moment around\nthej-th moment with a time- and directional-dependent amplitude, @tmi/(mi\u0002mj) (mi\u0001@tmj). The double\ncross product term yields a time dependence of @tmi/(mi\u0002(mi\u0002mj)) ((mi\u0002mj)\u0001@tmj). Both contributions\nare neither pure precession-like nor pure damping-like, but show complex time- and directional-dependent dynamics.\nSpin-orbit coupling contributions. The spin-orbit interaction gives rise to new possible dependencies of the\ndamping on the magnetic structure. In particular, the so-called chiral damping, which in general is the di\u000berence\nof the damping between a right-handed and a left-handed opening, rises from SOC and broken inversion symmetry.\nUsing our perturbative model, we can identify all possible dependencies up to second order in SOC and third order\nin the magnetic moments.\nIn the diagramms SOC is added by replacing one spin-independent hopping vertex by a spin-dependent one,\n!00\n0\ni\u000b i\fij ji\n!\n!00\n0\ni\u000b i\fij\r ij\n: (21)\nUp to \frst-order in SOC, we \fnd the the following dependencies were found for the on-site Gilbert damping\nGii(fmg)/f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\nji;^n\f\nijm\u000b\ni;^n\u000b\nijm\f\ni;\u000e\u000b\f(^nij\u0001mi);\u000e\u000b\f(^nij\u0001mj);\n^n\f\nijm\u000b\nj;^n\u000b\nijm\f\nj;m\u000b\ni(^nij\u0002mi)\f;m\f\ni(^nij\u0002mi)\u000b;\n\u000e\u000b\f^nij\u0001(mi\u0002mj);m\u000b\ni(^nij\u0002mj)\f;m\f\ni(^nij\u0002mj)\u000b;(^nij\u0001mj)\u000f\u000b\f\rm\r\ni;\nm\u000b\nim\f\ni(^nij\u0001mj);(m\u000b\nim\f\nj\u0000m\f\nim\u000b\nj)(^nij\u0001mj);^n\f\nijm\u000b\ni(mi\u0001mj);^n\u000b\nijm\f\ni(mi\u0001mj)g: (22)\nWe identi\fed the following contributions for the on-site and intersite damping to be the most relevant one after the\nnumerical evaluation discussed in the next sections:\nGsoc\nii=Ssoc;ij\ni nij\u0001(ei\u0002ej)I\n+Ssoc;ij;(2)\ni (nij\u0001ei)(nij\u0001ej)I\n+Asoc;ij\ni nij\u0001(ei\u0002ej)E(ei)\n+Asoc;ij;(2)\ni (nij\u0001ej)E(nij); (23)9\nand\nGsoc;\u000b\f\nij =Ssoc\nijnij\u0001(ei\u0002ej)\u000e\u000b\f+Ssoc;ba\nijn\f\nij(ei\u0002ej)\u000b\n+Asoc\nijE\u000b\f(nij): (24)\nThe contributions being \frst-order in SOC are obviously chiral since they depend on the cross product, ei\u0002ej. Thus,\nsimilar to the magnetic Dzyaloshinskii-Moriya interaction, SOC gives rise to a dependence of the Gilbert damping\non the vector chirality, ei\u0002ej. The term chiral damping used in literature refers to the dependence of the Gilbert\ndamping on the chirality, but to our knowledge it was not shown so far how this dependence evolves from a microscopic\nmodel, and how it looks like in an atomistic model.\nExtension to three sites. Including three di\u000berent sites i,j, andkin the expansions allows for a ring exchange\ni!j!k!iinvolving three hopping processes, which gives rise to new dependencies of the Gilbert damping on\nthe directions of the moments.\nAn example of a diagram showing up for the on-site Gilbert damping is given below for the on-site Gilbert damping\nthe diagram,\n!00 0;\r\n0\ni\u000b i\fijjk\nki(25)\nApart from the natural extensions of the previously discussed 2-site quantities, the intra-site Gilbert damping of site i\ncan depend on the angle between the sites jandk,ej\u0001ek, or in higher-order on the product of the angles between site\niandjwithiandk, (ei\u0001ej)(ei\u0001ek). In sixth-order in the magnetic moments the term ( ei\u0001ej)(ej\u0001ek)(ek\u0001ei) yields\nto a dependence on the square of the scalar spin chirality of the three sites, [ ei\u0001(ej\u0002ek)]2. Including SOC, there are\ntwo interesting dependencies on the scalar spin chirality. In \frst-order one \fnds similarly to the recently discovered\nchiral biquadratic interaction18and its 3-site generalization19, e.g. ( nij\u0001ei) (ei\u0001(ej\u0002ek)), while in second order a\ndirect dependence on the scalar spin chirality is allowed, e.g. n\u000b\nijn\f\nki(ei\u0001(ej\u0002ek)). The scalar spin chirality directly\nrelates to the topological orbital moment35{37and therefore the physical origin of those dependencies lies in the\ntopological orbital moment. Even though these terms might not be the most important ones in our model, for speci\fc\nnon-collinear con\fgurations or for some realistic elements with a large topological orbital moment, e.g. MnGe20, they\nmight be important and even dominant yielding interesting new physics.\nIV. APPLICATION TO THE ALEXANDER-ANDERSON MODEL\nMagnetic dimers. Based on a 2-site Alexander-Anderson model, we investigated the dependence of the Gilbert\ndamping on the directions of the magnetic moments using the previously discussed possible terms (see more details\non the method in Appendix C). The spin splitting Ude\fnes the energy scale and all other parameters. The energy of\norbitals is set to Ed= 1:0. The magnetization is self-consistently constrained in a range of m= 0:2 tom= 0:8 using\nmagnetic constraining \felds. The corresponding spin-resolved local density of states is illustrated in Fig. 1b, where\nthe inter-site hopping is set to t= 0:2 and the hybridization to \u0000 = 0 :2. We performed two sets of calculations: one\nwithout spin-dependent hopping, 'R= 0 °, and one with a spin-dependent hopping, 'R= 20 °.\nThe di\u000berent damping parameters are shown in Fig. 2 as function of the magnetization. They are obtained from a\nleast-squares \ft to several non-collinear con\fgurations based on a Lebedev mesh for `= 238. The damping, which is\nindependent of the relative orientation of the two sites, is shown in Fig. 2a. The symmetric damping-like intra-site\ncontributionSidominates the damping tensor for most magnetizations and has a maximum at m= 0:3. The anti-\nsymmetric intra-site contribtuion Ai, which renormalizes the gyromagnetic ratio, approximately changes sign when\nthe Fermi level passes the peak of the minority spin channel at m\u00190:5 and has a signi\fcantly larger amplitude\nfor small magnetizations. Both contributions depend mainly on the broadening \u0000, which mimics the coupling to an10\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)abc\nab\nFIG. 2. Damping parameters as function of the magnetization for the dimers described within the Alexander-Anderson\nmodel including spin orbit coupling. A longitudinal magnetic \feld is used to self-consistently constrain the magnetization. The\nparameters are extracted from \ftting to the inverse of the transversal susceptibility for several non-collinear con\fgurations\nbased on a Lebedev mesh. Model parameters in units of U:Ed= 1:0; t= 0:2;\u0000 = 0:2; 'R= 20 °.\nelectron bath and is responsible for the absorption of spin currents, which in turn are responsible for the damping of\nthe magnetization dynamics15,16.\nThe directional dependencies of the intra-site damping are shown in Fig. 2b. With our choice of parameters, the\ncorrection to the damping-like symmetric Gilbert damping can reach half of the direction-independent term. This\nmeans that the damping can vary between \u00190:4\u00001:0 for a ferromagnetic and an antiferromagnetic state at m= 0:4.\nAlso for the renormalization of the gyromagnetic ratio a signi\fcant correction is found, which in the ferromagnetic case\nalways lowers and in the antiferromagnetic case enhances the amplitude. The most dominant contribution induced\nby SOC is the chiral one, which depends on the cross product of the moments iandj, which in terms of amplitude is\ncomparable to the isotropic dot product terms. Interestingly, while the inter-site damping term is in general known\nto be less relevant than the intra-site damping, we \fnd that this does not hold for the directional dependence of the\ndamping. The inter-site damping is shown in Fig. 2c. Even though the directional-independent term, Sij, is nearly\none order of magnitude smaller than the equivalent intra-site contribution, this is not necessarily the case for the\ndirectional-dependent terms, which are comparable to the intra-site equivalents.\nV. APPLICATION TO FIRST-PRINCIPLES SIMULATIONS\nTo investigate the importance of non-collinear e\u000bects for the Gilbert damping in realistic systems, we use DFT and\ntime-dependent DFT to explore the prototypical example of monoatomic 3 dtransition metal adatoms and dimers\ndeposited on a heavy metal surface hosting large SOC (see Fig. 3a for an illustration of the con\fguration). We\nconsider a Cr, Mn, Fe and Co atoms deposited on the fcc-Au(111) surface (details of the simulations are described in\nAppendix D). The parameters and the corresponding functional forms are \ftted to our \frst-principles data using 196\nnon-collinear states based on a Lebedev mesh for `= 238.\nAdatoms on Au(111). To illustrate the di\u000berent e\u000bects on the Gilbert damping, we start by exploring magnetic\nadatoms in the uniaxial symmetry of the Au(111) surface. For the adatoms no non-local e\u000bects can contribute to the\nGilbert damping.\nThe Gilbert damping tensor of a single adatom without SOC has the form shown in relation to eq. (16),\nG0\ni=SiI+AiE(ei): (26)\nNote that SOC can induce additional anisotropies, as shown in eq. (22). The most important ones for the case of a\nsingle adatom are f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\njig, which in the C3vsymmetry result in\nGi=G0\ni+Ssoc\ni0\n@0 0 0\n0 0 0\n0 0 11\nA+Asoc\ni0\n@0 1 0\n\u00001 0 0\n0 0 01\nA; (27)11\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:083 0 :014 0 :242 0 :472\nAi 0:204 0 :100 0 :200 0 :024\nSsoc\ni 0:000 0 :000 0 :116 0 :010\nAsoc\ni 0:000 0 :000\u00000:022 0 :012\n\rrenorm\nx=y 1:42 1 :67 1 :43 1 :91\n\rrenorm\nz 1:42 1 :67 1 :48 1 :87\nTABLE I. Gilbert damping parameters of Cr, Mn, Fe and Co adatoms deposited on the Au(111) surface as parametrized in\neqs. (26) and (27). The SOC \feld points in the z-direction due to the C3vsymmetry. The renormalized gyromagnetic ratio\n\rrenormis calculated according to eqs. (28) for an in-plane magnetic moment and an out-of-plane magnetic moment.\nsince the sum of all SOC vectors points in the out-of-plane direction with ^nij!ez. Thus, the Gilbert damping tensor\nof adatoms deposited on the Au(111) surface can be described by the four parameters shown in eqs. (26) and (27),\nwhich are reported in Table I for Cr, Mn, Fe and Co adatoms. Cr and Mn, being nearly half-\flled, are characterized\nby a small damping-like contribution Si, while Fe and Co having states at the Fermi level show a signi\fcant damping\nof up to 0:47 in the case of Co. The antisymmetric part Aiof the Gilbert damping tensor results in an e\u000bective\nrenormalization of the gyromagnetic ratio \r, as shown in relation to eq. (A5), which using the full LLG equation,\neq. (2), and approximating mi\u0001dmi\ndt= 0 is given by,\n\rrenorm=\r1\n1 +\r(ei\u0001Ai); (28)\nwhere Aidescribes the vector Ai=\u0000\nAi;Ai;Ai+Asoc\ni\u0001\n. For Cr and Fe there is a signi\fcant renormalization of the\ngyromagnetic ratio resulting in approximately 1 :4. In contrast, Co shows only a weak renormalization with 1 :9 being\nclose to the gyromagnetic ratio of 2. The SOC e\u000bects are negigible for most adatoms except for Fe, which shows a\nsmall anisotropy in the renormalized gyromagentic ratio ( \u001910 %) and a large anisotropy in the damping-like term of\nnearly 50 %.\nDimers on Au(111). In contrast to single adatoms, dimers can show non-local contributions and dependencies on\nthe relative orientation of the magnetic moments carried by the atoms. All quantities depending on the SOC vector\nare assumed to lie in the y-z-plane due to the mirror symmetry of the system. A sketch of the dimer and its nearest\nneighboring substrate atoms together with adatoms' local density of states are presented in Fig. 3.\nThe density of states originates mainly from the d-states of the dimer atoms. It can be seen that the dimers exhibit\na much more complicated hybridization pattern than the Alexander-Anderson model. In addition the crystal \feld\nsplits the di\u000berent d-states resulting in a rich and high complexity than assumed in the model. However, the main\nfeatures are comparable: For all dimers there is either a fully occupied majority channel (Mn, Fe, and Co) or a fully\nunoccupied minority channel (Cr). The other spin channel determines the magnetic moments of the dimer atoms\nf4:04;4:48;3:42;2:20g\u0016Bfor respectively Cr, Mn, Fe and Co. Using the maximal spin moment, which is according\nto Hund's rule 5 \u0016B, the \frst-principles results can be converted to the single-orbital Alexander-Anderson model\ncorresponding to approximately m=f0:81;0:90;0:68;0:44g\u0016Bfor the aforementioned sequence of atoms. Thus by\nthis comparison, we expect large non-collinear contributions for Fe and Co, while Cr and Mn should show only weak\nnon-local dependencies.\nThe obtained parametrization is given in Table II. The Cr and Mn dimers show a weak or nearly no directional\ndependence. While the overall damping for both nanostructures is rather small, there is a signi\fcant correction to\nthe gyromagnetic ratio.\nIn contrast, the Fe and Co dimers are characterized by a very strong directional dependence. Originating from the\nisotropic dependencies of the damping-like contributions, the damping of the Fe dimer can vary between 0 :21 in\nthe ferromagnetic state and 0 :99 in the antiferromagnetic state. For the Co dimer the inter-site damping is even\ndominated by the bilinear and biquadratic term, while the constant damping is negligible. In total, there is a very\ngood qualitative agreement between the expectations derived from studying the Alexander-Anderson model and the\n\frst-principles results.12\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:0911 0 :0210 0 :2307 0 :5235\nSij;(1)\ni 0:0376 0 :0006\u00000:3924\u00000:2662\nSij;(2)\ni 0:0133\u00000:0006 0 :3707 0 :3119\nAi 0:2135 0 :1158 0 :1472 0 :0915\nAij\ni 0:0521 0 :0028\u00000:0710\u00000:0305\nSij\u00000:0356 0 :0028 0 :2932 0 :0929\nSdot\nij\u00000:0344\u00000:0018\u00000:3396\u00000:4056\nSdot;(2)\nij 0:0100 0 :0001 0 :1579 0 :2468\nAij\u00000:0281\u00000:0044 0 :0103 0 :0011\nAdot\nij\u00000:0175 0 :0000\u00000:0234\u00000:0402\nScross\nij 0:0288 0 :0002\u00000:2857\u00000:0895\nSba\nij 0:0331 0 :0036 0 :2181 0 :2651\nSsoc;ij;y\ni 0:0034 0 :0000 0 :0143\u00000:0225\nSsoc;ij;z\ni 0:0011 0 :0000\u00000:0104 0 :0156\nAsoc;ij;y\ni 0:0024\u00000:0001\u00000:0036 0 :0022\nAsoc;ij;z\ni 0:0018\u00000:0005 0 :0039\u00000:0144\nSsoc;y\nij 0:0004 0 :0001 0 :0307 0 :0159\nSsoc;z\nij\u00000:0011 0 :0000\u00000:0233 0 :0206\nSba,soc ;y\nij\u00000:0027 0 :0000\u00000:0184\u00000:0270\nSba,soc ;z\nij 0:0005\u00000:0001 0 :0116\u00000:0411\nTABLE II. Damping parameters of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The possible forms of the\ndamping are taken from the analytic model. The SOC \feld is assumed to lie in the y-zplane and inverts under permutation\nof the two dimer atoms.\n\u00003\u00002\u000010123E\u0000EF[eV]\u00006\u0000303DOS [#states/eV]Cr dimerMn dimerFe dimerCo dimersurface\nab\nFIG. 3. aIllustration of a non-collinear magnetic dimer (red spheres) deposited on the (111) facets of Au (grey spheres).\nFrom the initial C3vspatial symmetry of the surface the dimers preserve the mirror plane (indicated grey) in the y-zplane. b\nLocal density of states of the Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The grey background indicates the\nsurface density of states. The dimers are collinear in the z-direction.\nVI. CONCLUSIONS\nIn this article, we presented a comprehensive analysis of magnetization dynamics in non-collinear system with a special\nfocus on the Gilbert damping tensor and its dependencies on the non-collinearity. Using a perturbative expansion\nof the two-site Alexander-Anderson model, we could identify that both, the intra-site and the inter-site part of the\nGilbert damping, depend isotropically on the environment via the e\u000bective angle between the two magnetic moments,\nei\u0001ej. SOC was identi\fed as the source of a chiral contribution to the Gilbert damping, which similarly to the\nDzyaloshinskii-Moriya and chiral biquadratic interactions depends linearly on the vector spin chirality, ei\u0002ej. We\nunveiled dependencies that are proportional to the three-spin scalar chirality ei\u0001(ej\u0002ek), i.e. to the chiral or\ntopological moment, and to its square. Using the Alexander-Anderson model, we investigated the importance of the13\ndi\u000berent contributions in terms of their magnitude as function of the magnetization. Using the prototypical test\nsystem of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface, we extracted the e\u000bects of the non-collinearity\non the Gilbert damping using time-dependent DFT. Overall, the \frst-principles results agree qualitatively well with\nthe Alexander-Anderson model, showing no dependence for the nearly half-\flled systems Cr and Mn and a strong\ndependence on the non-collinearity for Fe and Co having a half-\flled minority spin-channel. The realistic systems\nindicate an even stronger dependence on the magnetic texture than the model with the used parameters. The Fe and\nthe Co dimer show signi\fcant isotropic terms up to the biquadratic term, while the chiral contributions originating\nfrom SOC have only a weak impact on the total Gilbert damping. However, the chiral contributions can play the\ndeciding role for systems which are degenerate in the isotropic terms, like e.g. spin spirals of opposite chirality.\nWe expect the dependencies of the Gilbert damping on the magnetic texture to have a signi\fcant and non-trivial\nimpact on the spin dynamics of complex magnetic structures. Our \fndings are readily implementable in the LLG\nmodel, which can trivially be amended with the angular dependencies provided in the manuscript. Utilizing multiscale\nmapping approaches, it is rather straightforward to generalize the presented forms for an implementation of the\nmicromagnetic LLG and Thiele equations. The impact of the di\u000berent contributions to the Gilbert damping, e.g. the\nvector (and/or scalar) chiral and the isotropic contributions, can be analyzed on the basis of either free parameters\nor sophisticated parametrizations obtained from \frst principles as discussed in this manuscript. It remains to be\nexplored how the newly found dependencies of the Gilbert damping a\u000bect the excitations and motion of a plethora of\nhighly non-collinear magnetic quasi-particles such as magnetic skyrmions, bobbers, hop\fons, domain walls and spin\nspirals. Future studies using atomistic spin dynamics simulations could shed some light on this aspect and help for\nthe design of future devices based on spintronics.\nACKNOWLEDGMENTS\nThis work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research\nand innovation program (ERC-consolidator grant 681405 { DYNASORE) and from Deutsche Forschungsgemeinschaft\n(DFG) through SPP 2137 \\Skyrmionics\" (Project LO 1659/8-1). The authors gratefully acknowledge the computing\ntime granted through JARA-HPC on the supercomputer JURECA at the Forschungszentrum J ulich39.\nVII. METHODS\nAppendix A: Analysis of the Gilbert damping tensor\nThe Gilbert damping tensor Gcan be decomposed into a symmetric part Sand an anti-symmetric part A,\nA=G\u0000GT\n2andS=G+GT\n2: (A1)\nWhile the symmetric contribution can be referred to as the damping-like contribution including potential anisotropies,\nthe anti-symmetric Atypically renormalizes the gyromagnetic ratio as can be seen as follows: The three independent\ncomponents of an anti-symmetric tensor can be encoded in a vector Ayielding\nA\u000b\f=\u000f\u000b\f\rA\r; (A2)\nwhere\u000f\u000b\f\ris the Levi-Cevita symbol. Inserting this into the LLG equation yields\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njAijdmj\ndt1\nA (A3)\n\u0019\u0000\rmi\u00020\n@Be\u000b\ni\u0000\rX\njAij\u0000\nmj\u0002Be\u000b\nj\u00011\nA: (A4)\nThe last term can be rewritten as\n(Aij\u0001mj)Be\u000b\nj\u0000\u0000\nAij\u0001Be\u000b\nj\u0001\nmj: (A5)14\nFor the local contribution, Aii, the correction is kmiandkBe\u000b\niyielding a renormalization of \rmi\u0002Be\u000b\ni. However,\nthe non-local parts of the anti-symmetric Gilbert damping tensor can be damping-like.\nAppendix B: Relation between the LLG and the magnetic susceptibility\nThe Fourier transform of the LLG equation is given by\n\u0000i!mi=\u0000\rmi\u00020\n@Bext\ni\u0000X\njJijmj\u0000i!X\njGijmj1\nA: (B1)\nTransforming this equation to the local frames of site iandjusing the rotation matrices RiandRjyields\ni!\n\rMimloc\ni=mloc\ni\nMi\u00020\n@RiBext\ni\u0000X\njRiJijRT\njmloc\nj\u0000i!X\njRiGijRT\njmloc\nj1\nA; (B2)\nwhere mloc\ni=Rimiandmloc\nj=Rjmj. The rotation matrices are written as R(#i;'i) = cos(#i=2)\u001b0+\ni sin(#i=2)\u0000\nsin('i)\u001bx\u0000cos('i)\u001by\u0001\n, with (#i;'i) being the polar and azimuthal angle pertaining to the moment\nmi. In the ground state the magnetic torque vanishes. Thus, denoting mloc\ni= (mx\ni; my\ni; Mi), wheremx=y\niare\nperturbations to the ground states, yields for the ground state\n0\n@(RiBext\ni)x\u0000P\nj(RiJijRT\njMjez)x\n(RiBext\ni)y\u0000P\nj(RiJijRT\njMjez)y\n(RiBext\ni)z\u0000P\nj(RiJijRT\njMjez)z1\nA=0\n@0\n0\n(RiBe\u000b\ni)z1\nA: (B3)\nLinearizing the LLG and using the previous result and limiting our expansion to transveral excitations yield\ni!\n\rMimx\ni=my\ni(RiBe\u000b\ni)z\nMi\u0000(RiBext\ni)y+X\nj(RiJijRT\njmloc\nj)y+ i!X\nj(RiGijRT\njmloc\nj)y(B4)\ni!\n\rMimy\ni=\u0000mx\ni(RiBe\u000b\ni)z\nMi+ (RiBext\ni)x\u0000X\nj(RiJijRT\njmloc\nj)x\u0000i!X\nj(RiGijRT\njmloc\nj)x; (B5)\nwhich in a compact form gives\nX\nj\n\f=x;y0\n@\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f1\nAm\f\nj= (RiBext\ni)\u000b; (B6)\nand can be related to the inverse of the magnetic susceptibility\nX\nj\n\f=x;y\u001f\u00001\ni\u000b;j\f(!)m\f\nj= (RiBext\ni)\u000b: (B7)\nThus, the magnetic susceptibility in the local frames of site iandjis given by\n\u001f\u00001\ni\u000b;j\f(!) =\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f(B8)\nAppendix C: Alexander-Anderson model{more details\nWe use a single orbital Alexander-Anderson model,\nH=X\nij[\u000eij(Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b)\u0000(1\u0000\u000eij)tij]; (C1)15\nwhereiandjsum over all n-sites,Edis the energy of the localized orbitals, \u0000 is the hybridization in the wide band\nlimit,Uiis the local interaction responsible for the formation of a magnetic moment, miis the magnetic moment of site\ni,Biis an constraining or external magnetic \feld, \u001bare the Pauli matrices, and tijis the hopping parameter between\nsiteiandj, which can be in general spin-dependent. SOC is added as spin-dependent hopping using a Rashba-like\nspin-momentum locking tij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b), where the spin-dependent hopping is characterized by its\nstrength de\fned by 'Rand its direction nij=\u0000nji34. The eigenenergies and eigenstates of the model are given by,\nHjni= (En\u0000i \u0000)jni: (C2)\nThe single particle Green function can be de\fned using the eigensystem,\nG(E+ i\u0011) =X\nnjnihnj\nE\u0000En+ i\u0011; (C3)\nwhere\u0011is an in\fnitesimal parameter de\fning the retarded ( \u0011!0+) and advanced ( \u0011!0\u0000) Green function. The\nmagnitude of the magnetic moment is determined self-consistently using\nmi=\u00001\n\u0019Im TrZ\ndE\u001bGii(E); (C4)\nwhereGii(E) is the local Green function of site idepending on the magnetic moment. Using the magnetic torque\nexerted on the moment of site i,\ndH\nd^ei=\u0000miBe\u000b\ni; (C5)\nmagnetic constraining \felds can be de\fned ensuring the stability of an arbitrary non-collinear con\fguration,\nBconstr=\u0000Pm\n?mi\njmijBe\u000b\ni) Hconstr=\u0000Bconstr\u0001\u001b ; (C6)\nwherePm\n?is the projection on the plane perpendicular to the moment m. The constraining \felds are added to the\nhamiltonian, eq. (C1), and determined self-consistently.\nAppendix D: Density functional theory{details\nThe density functional theory calculations were performed with the Korringa-Kohn-Rostoker (KKR) Green function\nmethod. We assume the atomic sphere approximation for the the potential and include full charge density in the\nself-consistent scheme40. Exchange and correlation e\u000bects are treated in the local spin density approximation (LSDA)\nas parametrized by Vosko, Wilk and Nusair41, and SOC is added to the scalar-relativistic approximation in a self-\nconsistent fashion42. We model the pristine surfaces utilizing a slab of 40 layers with the experimental lattice constant\nof Au assuming open boundary conditions in the stacking direction, and surrounded by two vacuum regions. No\nrelaxation of the surface layer is considered, as it was shown to be negligible43. We use 450\u0002450k-points in the\ntwo-dimensional Brillouin zone, and the angular momentum expansions for the scattering problem are carried out up\nto`max= 3. Each adatom is placed in the fcc-stacking position on the surface, using the embedding KKR method.\nPreviously reported relaxations towards the surface of 3 dadatoms deposited on the Au(111) surface44indicate a\nweak dependence of the relaxation on the chemical nature of the element. Therefore, we use a relaxation towards the\nsurface of 20 % of the inter-layer distance for all the considered dimers. The embedding region consists of a spherical\ncluster around each magnetic adatom, including the nearest-neighbor surface atoms. 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B 98(9) 094428 URL https://link.aps.org/doi/10.1103/\nPhysRevB.98.094428"    },    {        "title": "0802.1740v1.Temperature_dependent_magnetization_dynamics_of_magnetic_nanoparticles.pdf",        "content": "arXiv:0802.1740v1  [cond-mat.other]  12 Feb 2008Temperature dependent magnetization dynamics of\nmagnetic nanoparticles\nA. Sukhov1,2and J. Berakdar2\n1Max-Planck-Institut f¨ ur Mikrostrukturphysik, Weinberg 2, D-0 6120 Halle/Saale,\nGermany\n2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle-Wittenbe rg,\nHeinrich-Damerow-Str. 4, 06120 Halle, Germany\nAbstract. Recent experimental and theoretical studies show that the switc hing\nbehavior of magnetic nanoparticles can be well controlled by extern al time-dependent\nmagnetic fields. In this work, we inspect theoretically the influence o f the temperature\nand the magnetic anisotropy on the spin-dynamics and the switching properties of\nsingle domain magnetic nanoparticles (Stoner-particles). Our theo retical tools are\nthe Landau-Lifshitz-Gilbert equation extended as to deal with finit e temperatures\nwithin a Langevine framework. Physical quantities of interest are t he minimum\nfield amplitudes required for switching and the corresponding rever sal times of the\nnanoparticle’s magnetic moment. In particular, we contrast the ca ses of static and\ntime-dependent external fields and analyze the influence of dampin g for a uniaxial and\na cubic anisotropy.\nPACS numbers: 75.40.Mg, 75.50.Bb, 75.40.Gb, 75.60.Jk, 75.75.+aTemperature dependent magnetization dynamics of magnetic nanoparticles 2\n1. Introduction\nInrecent years, therehasbeenasurgeofresearchactivities fo cusedonthespindynamics\nand the switching behavior of magnetic nanoparticles [1]. These stud ies are driven\nby potential applications in mass-storage media and fast magneto- electronic devices.\nIn principle, various techniques are currently available for controllin g or reversing the\nmagnetization of a nanoparticle. To name but a few, the magnetizat ion can be reversed\nby a short laser pulse [2], a spin-polarized electric current [3, 4] or an alternating\nmagnetic field [5, 6, 7, 8, 9, 10, 11, 12, 13]. Recently [6], it has been sh own for a\nuniaxial anisotropythattheutilizationofaweak time-dependent ma gneticfieldachieves\na magnetization reversal faster than in the case of a static magne tic field. For this case\n[6], however, the influence of the temperature and the different ty pes of anisotropy\non the various dependencies of the reversal process have not be en addressed. These\nissues, which are the topic of this present work, are of great impor tance since, e.g.\nthermal activation affects decisively the stability of the magnetizat ion, in particular\nwhen approaching the superparamagnetic limit, which restricts the density of data\nstorage [14]. Here we study the possibility of fast switching at finite t emperature with\nweak external fields. We consider magnetic nanoparticles with an ap propriate size as to\ndisplayalong-rangemagneticorderandtobeinasingledomainremane ntstate(Stoner-\nparticles). Uniaxial and cubic anisotropies are considered and show n to decisively\ninfluence the switching dynamics. Numerical results are presented and analyzed for\niron-platinum nanoparticles. In principle, the inclusion of finite tempe ratures in spin-\ndynamics studies is well-established (cf. [19, 20, 23, 15, 16, 1] and references therein)\nand will be followed here by treating finite temperatures on the level of Langevine\ndynamics. For the analysis of switching behaviour the Stoner and Wo hlfarth model\n(SW) [17] is often employed. SW investigated the energetically metas table and stable\nposition of the magnetization of a single domain particle with uniaxial an isotropy in\nthe presence of an external magnetic field. They showed that the minimum static\nmagnetic field (generally referred to as the Stoner-Wohlfarth (SW ) field or limit) needed\nto coherently reverse the magnetization is dependent on the direc tion of the applied\nfield with respect to the easy axis. This dependence is described by t he so-called\nStoner-Wohlfarth astroid. The SW findings rely, however, on a sta tic model at zero\ntemperature. Application of a time-dependent magnetic field reduc es the required\nminimum switching field amplitude below the SW limit [6]. It was, however, no t yet\nclear how finite temperatures will affect these findings. To clarify th is point, we utilize\nan extension of the Landau-Lifshitz-Gilbert equation [18] including fi nite temperatures\non the level of Langevine dynamics [19, 20, 23]. Our analysis shows t he reversal time\nto be strongly dependent on the damping, the temperature and th e type of anisotropy.\nThese dependencies are also exhibited to a lesser extent by the crit ical reversal fields.\nThepaperisorganizedasfollows: nextsection2presents detailsof thenumerical scheme\nand the notations whereas section 3 shows numerical results and a nalysis for Fe 50Pt50\nand Fe 70Pt30nanoparticles. We then conclude with a brief summary.Temperature dependent magnetization dynamics of magnetic nanoparticles 3\n2. Theoretical model\nIn what follows we focus on systems with large spins such that their m agnetic dynamics\ncan be described by the classical motion of a unit vector Sdirected along the particle’s\nmagnetization µ, i.e.S=µ/µSandµSisthe particle’s magnetic moment at saturation.\nThe energetics of the system is given by\nH=HA+HF. (1)\nwhereHA(HF) stands for the anisotropy (Zeeman energy) contribution. Furt hermore,\nthe anisotropy contribution is expressed as HA=−Df(S) withDbeing the anisotropy\nconstant. Explicit formof f(S)isprovidedbelow. Themagnetizationdynamics, i.e. the\nequation of motion for S, is governed by the Landau-Lifshitz-Gilbert (LLG) equation\n[18]\n∂S\n∂t=−γ\n(1+α2)S×/bracketleftBig\nBe(t) +α(S×Be(t))/bracketrightBig\n. (2)\nHere we introduced the effective field Be(t) =−1/(µS)∂H/∂Swhich contains the\nexternal magnetic field and the maximum anisotropy field for the unia xial anisotropy\nBA= 2D/µS.γis the gyromagnetic ratio and αis the Gilbert damping parameter. The\ntemperature fluctuations will be described on the level of the Lang evine dynamics [19].\nThis means, a time-dependent thermal noise ζ(t) adds to the effective field Be(t) [19].\nζ(t) is a Gaussian distributed white noise with zero mean and vanishing time correlator\n/angbracketleftζi(t′)ζj(t)/angbracketright=2αkBT\nµsγδi,jδ(t−t′). (3)\ni,jare Cartesian components, Tis the temperature and kBis the Boltzmann constant.\nIt is convenient to express the LLG in the reduced units\nb=Be\nBA, τ=ωat, ωa=γBA. (4)\nThe LLG equation reads then\n∂S\n∂τ=−1\n(1+α2)S×/bracketleftBig\nb(τ) +α(S×b(τ))/bracketrightBig\n, (5)\nwhere the effective field is now given explicitly by\nb(τ) =−1\nµSBA∂H\n∂S+Θ(τ) (6)\nwith\n/angbracketleftΘi(τ′)Θj(τ)/angbracketright=ǫδi,jδ(τ−τ′);ǫ=2αkBT\nµsBA. (7)\nThereducedunitsareindependent ofthedampingparameter α. Inthefollowingsections\nwe use extensively the parameter\nq=kBT\nD. (8)\nqis a measure for the thermal energy in terms of the anisotropy ene rgy. And\nd=D/(µSBA) expresses the anisotropy constant in units of a maximum anisotro pyTemperature dependent magnetization dynamics of magnetic nanoparticles 4\nenergy for the uniaxial anisotropy and is always 1 /2. The stochastic LLG equation (5)\nin reduced units (4) is solved numerically using the Heun method which c onverges in\nquadratic mean to the solution of the LLG equation when interprete d in the sense of\nStratonovich [20]. For each type of anisotropy we choose the time s tep ∆τto be one\nthousandth part of the corresponding period of oscillations. The v alues of the time\ninterval in not reduced units for uniaxial and cubic anisotropies are ∆tua= 4.61·10−15s\nand ∆tca= 64.90·10−15s, respectively, providing us thus with correlation times on the\nfemtosecond time scale. The reason for the choice of such small tim e intervals is given in\n[19], where it is argued that the spectrum of thermal-agitation forc es may be considered\nas white up to a frequency of order kBT/hwithhbeing the Planck constant. This value\ncorresponds to 10−13sfor room temperature. The total scale of time is limited by a\nthousand of such periods. Hence, we deal with around one million iter ation steps for\na switching process. Details of realization of this numerical scheme c ould be found in\nreferences [21, 22, 20]. We note by passing, that attempts have b een made to obtain,\nunder certain limitations, analytical results for finite-temperatur e spin dynamics using\nthe Fokker-Planck equation (cf. [15, 16] and references therein ). For the general case\ndiscussed here one has however to resort to fully numerical appro aches.\n3. Results and interpretations\nWeconsider a magneticnanoparticleina singledomainremanent state (Stoner-particle)\nwith aneffective anisotropy whose origin can be magnetocrystalline, magnetoelastic and\nsurface anisotropy. We assume the nanoparticle to have a spheric al form, neglecting\nthus the shape anisotropy contributions. In the absence of exte rnal fields, thermal\nfluctuations may still drive the system out of equilibrium. Hence, the stability of\nthe system as the temperature increases becomes an important is sue. The time tat\nwhich the magnetization of the system overcomes the energy barr ier due to the thermal\nactivation, also called the escape time , is given by the Arrhenius law\nt=t0eD\nkBT, (9)\nwhere the exponent is the ratio of the anisotropy to the thermal e nergy. The coefficient\nt0may be inferred when D≫kBTand for high damping [19] (see [25] for a critical\ndiscussion)\nt0=1+α\nαγπµS\n2D/radicalBigg\nkBT\nD. (10)\nHere we focus on two different types of iron-platinum-nanoparticle s: The compound\nFe50Pt50which has a uniaxial anisotropy [26, 27], whereas the system Fe 70Pt30possesses\na cubic anisotropy [24]. Furthermore, the temperature dependen ce will be studied by\nvaryingq(cf. eq.(8)).\nFor Fe 50Pt50the important parameters for simulations are the diameter of the\nnanoparticles 6 .3nm, the strength of the anisotropy Ku= 6·106J/m3, the magnetic\nmoment per particle µp= 21518 ·µBand the Curie-temperature Tc= 710K[26, 27].Temperature dependent magnetization dynamics of magnetic nanoparticles 5\nThe relation between KuandDuisDu=KuVu, whereVuis the volume of Fe 50Pt50\nnanoparticles. In the calculations for Fe 50Pt50nanoparticles the following qvalues\nwere chosen: q1= 0.001,q2= 0.005 orq3= 0.01 which correspond to the real\ntemperatures 56 K, 280Kor 560K, respectively (these temperatures are below the\nblocking temperature). The corresponding escape times are tq1≈2·10217s,tq2≈1075s\nandtq3≈7·1031s, respectively. In some cases we also show the results for an additio nal\ntemperature q01= 0.0001 with the corresponding real temperature to be equal to 5 K.\nThecorrespondingescapetimeforthisis tq01≈104300s. Thesetimesshouldbecompared\nwith the measurement period which is about tm≈5ns, endorsing thus the stability of\nthe system during the measurements.\nFor Fe 70Pt30the parameters are as follows: The diameter of the nanoparticles 2 .3nm,\nthe strength of the anisotropy Kc= 8·105J/m3, the magnetic moment per particle\nµp= 2000·µB, the Curie-temperature is Tc= 420K[24], and Dc=KcVc(Vcis the\nvolume.) For Fe 70Pt30nanoparticles the values of qwe choose in the simulations are\nq4= 0.01,q5= 0.03 orq6= 0.06 which means that the temperature is respectively\n0.3K, 0.9Kor 1.9K. The escape times are tq4≈1034s,tq5≈2·105sandtq6≈2·10−2s,\nrespectively. Here we also choose an intermediate value q04= 0.001 and the real\ntemperature 0 .03Kwith the corresponding escape time to be equal to tq04≈10430s.\nThe measurement period is the same, namely about 5 ns. All values of the escape times\nwere given for α= 0.1.\nCentral to this study are two issues: The critical magnetic field and the corresponding\nreversal time . The critical magnetic field we define as the minimum field amplitude\nneeded to completely reverse the magnetization. The reversal tim e is the corresponding\ntime for this process. In contrast, in other studies [6] the rever sal time is defined as the\ntime needed for the magnetization to switch from the initial position t o the position\nSz= 0, our reversal time is the time at which the magnetization reaches the very\nproximity of the antiparallel state (Fig. 1). The difference in the defi nition is in so far\nimportant as the magnetization position Sz= 0 at finite temperatures is not stabile so\nit may switch back to the initial state due to thermal fluctuations an d hence the target\nstate is never reached.\n3.1. Nanoparticles having uniaxial anisotropy: Fe 50Pt50\nA Fe50Pt50magnetic nanoparticle has a uniaxial anisotropy whose direction defi nes the\nzdirection. The magnetization direction Sis specified by the azimuthal angle φand\nthe polar angle θwith respect to z. In the presence of an external field bapplied at\nan arbitrarily chosen direction, the energy of the system in dimensio nless units derives\nfrom\n˜H=−dcos2θ−S·b. (11)\nThe initial state of the magnetization is chosen to be close to Sz= +1 and we aim at\nthe target state Sz=−1.Temperature dependent magnetization dynamics of magnetic nanoparticles 6\n00.511.522.533.544.5 5\nTime, [ns]-1-0.500.51Magnetization SzT0=0 K T3=560 K\nFigure 1. (Color online) Magnetization reversal of a nanoparticle when a stat ic field\nis applied at zero Kelvin ( q0= 0, black) and at reduced temperature q3= 0.01≡560K\n(blue). The strengths of the fields in the dimensionless units (4) and (8) areb= 1.01\nandb= 0.74, respectively. The damping parameter is α= 0.1. The start position of\nthe magnetization is given by the initial angle θ0=π/360 between the easy axis and\nthe magnetization vector.\n3.1.1. Static field For an external static magnetic field applied antiparallel to the z\ndirection ( b=−bez) eq.(11) becomes\n˜H=−dcos2θ+bcosθ. (12)\nTo determine the critical field magnitude needed for the magnetizat ion reversal we\nproceed as follows (cf. Fig. 1): At first, the external field is increa sed in small steps.\nWhen the magnetization reversal is achieved the corresponding va lues of the critical\nfield versus the damping parameter αare plotted as shown in the inset of Fig. 3. The\nreversal times corresponding to the critical static field amplitudes of Fig. 3 are plotted\nversus damping in Fig. 4.\nIn the Stoner-Wohlfarth (static) model the mechanism of magnet ization reversal is not\ndue to damping. It is rather caused by a change of the energy profi le in the presence of\nthe field. The curves displayed on the energy surface in Fig. 2 mark t he magnetization\nmotion in the E(θ,φ) landscape. The magnetization initiates from φ0= 0 and θ0and\nends up at θ=π. As clearly can be seen from the figure, reversal is only possible if th e\ninitial state is energetically higher than the target state. This ”low d amping” reversal\nis, however, quite slow, which will be quantified more below. For the re versal at T= 0,\nthe SW-model predicts a minimum static field strength, namely bcr=B/BA= 1 (the\ndashed line in Fig. 3 ).\nThis minimum field measured with respect to the anisotropy field stren gth does not\ndepend on the damping parameter α, provided the measuring time is infinite. For T >0\nthe simulations were averaged over 500 cycles with the result shown in Fig. 3. The one-\ncycle data are shown in the inset. Fig. 3 evidences that with increasin g temperature\nthermal fluctuations assist a weak magnetic field as to reverse the magnetization.\nFurthermore, the required critical field is increased slightly at very large and strongly\nat very small damping with the minimum critical field being at α≈1.0. The reason forTemperature dependent magnetization dynamics of magnetic nanoparticles 7\nFigure 2. (Color online) The trajectories of the magnetization unit vector\nparameterized by the angles θandφat zero temperature. Other parameters are as in\nFig. 1 for q0.\n00.511.522.533.544.5 5 5.5 6\nDamping α00.20.40.60.81Critical DC field T0=0 K (SW)\nT01=5 K\nT1=56 K\nT2=280 K\nT3=560 K0 1 2 3 4 5 6\nDamping α00.20.40.60.81Critical DC field T3=560 K\nFigure 3. (Color online) Critical static field amplitudes vs. the damping paramet ers\nfor different temperatures averaged over 500 times. Inset show s not averaged data for\nq3= 0.01≡560K.\nthis behavior is that for low damping the second term of equation (2) is much smaller\nthan the first one, meaning that the system exhibits a weak relaxat ion. In the absence\nof damping, higher fields are necessary to switch the magnetization . For high α, both\nterms in equation (2) become small (compared to a low-damping case ) leading to a\nstiff magnetization and hence higher fields are needed to drive the ma gnetization. For\nmoderate damping, we observe a minimum of switching fields which is due to an optimal\ninterplay between precessional and damping terms. Obviously, finit e temperatures do\nnot influence this general trend.\nFor the case of q0= 0, the Landau-Lifshitz-Gilbert equation of motion can be solved\nanalytically in spherical coordinates. The details of the solution can b e found in Ref.\n[20] (eq. (A1)-(A8)). The final result of the solution in this refere nce differs, however,\nfrom the one given here due to to different geometries in these syst ems. In contrast to\nour alignment of the magnetization and the external field, the stat ic field in Ref. [20] is\napplied parallel to the initial position of the magnetization. For the so lution, we assume\nthat the magnetization starts at θ=θ0=π/360 and arrives at θ=π. Note, that the\nexpression θ/negationslash= 0 is important only for zero Kelvin since the switching is not possible ifTemperature dependent magnetization dynamics of magnetic nanoparticles 8\nthe magnetization starts at θ0= 0 (the vector product in equation (2) vanishes). The\nreversal time in the SW-limit is then given by\ntrev=g(θ0,b)1+α2\nα, (13)\nwheregis defined as\ng(θ0,b) =µS\n2γD1\nb2−1ln/parenleftBiggtg(θ/2)bsinθ\nb−cosθ/parenrightBigg/vextendsingle/vextendsingle/vextendsingleπ\nθ0. (14)\nFromthisrelationweinferthatswitchingispossibleonlyiftheappliedfie ldislargerthan\nthe anisotropy field and the reversal time decreases with increasin gb. This conclusion is\nindependent of the Stoner-Wohlfarth model and follows directly fr om the solution of the\nLLG equation. An illustration is shown by the dashed curve in Fig. 4, wh ich was a test\nto compare the appropriate numerical results with the analytical o ne. As our aim is the\nstudy of the reversal-time dependence on the magnetic moment an d on the anisotropy\nconstant, we deem the logarithmic dependence in Eq.(14) to be weak and write\ng(b,µS,D)≈µS\nγ2D\nB2µ2\nS−4D2. (15)\nThis relation indicates that an increase in the magnetic moment result s in a decrease\nof the reversal time. The magnetic moment enters in the Zeeman en ergy and therefore\nthe increase in magnetic moment is very similar to an increase in the mag netic field.\nAn increase of the reversal time with the increasing anisotropy orig inates from the fact\nthat the anisotropy constant determines the height of the poten tial barrier. Hence, the\nhigher the barrier, the longer it takes for the magnetization to ove rcome it.\nFor the other temperatures the corresponding reversal times ( also averaged over 500\ncycles) are shown in Fig. 4. In contrast to the case T= 0, where an appreciable\ndependence on damping is observed, the reversal times for finite t emperatures show\na weaker dependence on damping. If α→0 only the precessional motion of the\nmagnetization is possible and therefore trev→ ∞. At high damping the system relaxes\non a time scale that is much shorter than the precession time, giving t hus rise to an\nincrease in switching times. Additionally, one can clearly observe the in crease of the\nreversal times with increasing temperatures, even though these time remain on the\nnanoseconds time scale.\n3.1.2. Alternating field As was shown in Ref. [6, 7, 15] theoretically and in Ref. [5]\nexperimentally, a rotating alternating field with no static field being ap plied can also\nbe used for the magnetization reversal. A circular polarized microwa ve field is applied\nperpendicularly to the anisotropy axis. Thus, the Hamiltonian might b e written in form\nof equation (11) and the applied field is\nb(t) =b0cosωtex+b0sinωtey, (16)\nwhereb0is the alternating field amplitude and ωis its frequency. For a switching of\nthe magnetization the appropriate frequency of the applied altern ating field should beTemperature dependent magnetization dynamics of magnetic nanoparticles 9\n00.511.522.533.544.5 5 5.5 6\nDamping α012345Reversal time, [ns]    Theory T0=0 K\nT01=5 K\nT1=56 K\nT2=280 K\nT3=560 K 0 1 2 3 4 5 6\nDamping α012345Reversal time, [ns]    T3=560 K\nFigure 4. (Color online) Reversal times corresponding to the critical static fi elds in\nFig. 3 vs. damping averaged over 500 cycles. Inset shows the as-c alculated numerical\nresults for q3= 0.01≡560K(one cycle).\n00.050.10.150.20.25\nTime, [ns]-1-0.500.51Magnetization SzT0=0 K\n0 1 2 3 4 5\nTime, [ns]-1-0.500.51Magnetization Sz T3=560 K\nFigure 5. (Color online) Magnetization reversal in a nanoparticle using a time\ndependentfieldfor α= 0.1andatazerotemperature. Thefieldstrengthandfrequency\nin the units (4) are respectively b0= 0.18 andω=ωa/1.93. Inset shows for this case\nthe magnetization reversal for the temperature q3= 0.01≡560Kwithb0= 0.17 and\nthe same frequency.\nchosen. In Ref. [15] analytically and in [6] numerically a detailed analysis of the optimal\nfrequency is given which is close to the precessional frequency of t he system. The role\nof temperature and different types of anisotropy have not yet be en addressed, to our\nknowledge.\nFig. 5 shows our calculations for the reversal process at two differ ent temperatures.\nIn contrast to the static case, the reversal proceeds through many oscillations on a\ntime scale of approximately ten picoseconds. Increasing the tempe rature results in an\nincrease of the reversal time.\nFig. 6 shows the trajectory of the magnetization in the E( θ,φ) space related to the\ncase of the alternating field application. Compared with the situation depicted in Fig.\n2, the trajectory reveals a quite delicate motion of the magnetizat ion. It is furthermore,\nnoteworthy that the alternating field amplitudes needed for the re versal (cf. Fig. 7) are\nsubstantially lower than their static counterpart, meaning that th e energy profile of theTemperature dependent magnetization dynamics of magnetic nanoparticles 10\nFigure 6. (Color online) Trajectories followed by magnetization as specified by θand\nφforq0= 0. Other parametersare b0= 0.18,α= 0.1 andω=ωa/1.93. Energy-profile\nvariations due to the oscillating external field are not visible on this sc ale.\nsystem is not completely altered by the external field.\nFig. 7 inspects the dependence of the minimum switching field amplitude on\ndamping. The critical fields are obtained upon averaging over 500 cy cles. The SW-\nlimit lies by 1 on this scale. In contrast to the static case, the critical fields increase\nwith increasing α. In the low damping regime the critical field is smaller than in the\ncase of a static field. This behavior can be explained qualitatively by a r esonant energy-\nabsorptionmechanism when thefrequencies oftheappliedfieldmatc hes thefrequency of\nthe system. Obviously, at very low frequencies (compared to the p recessional frequency)\nthe dynamics resembles the static case.\nThe influence of the temperature on the minimum alternating field amp litudes is\ndepicted in Fig. 7. With increasing temperatures, the minimum amplitud es become\nsmaller due to an additional thermal energy pumped from the enviro nment. The curves\nin this figure can be approached with two linear dependencies with diffe rent slopes for\napproximately α <1 and for α >1; for high damping it is linearly dependent on α,\nmore specifically it can be shown that for high damping the critical field s behave as\nbcr≈1+α2\nα. (17)\nThe proportionality coefficient contains the frequency of the alter nating field and the\ncritical angle θ. The solution (17) follows from the LLG equation solved for the case\nwhen the phase of the external field follows temporally that of the m agnetization, which\nwe checked numerically to be valid.\nThe reversal times associated with the critical switching fields are s hown in (Fig.\n8). Qualitatively, we observe the same behavior as for the case of a static field. The\nvalues of the reversal times for T= 0 are, however, significantly smaller than for the\nstatic case. For the same reason as in the static field case, an incre ased temperature\nresults in an increase of the switching times.Temperature dependent magnetization dynamics of magnetic nanoparticles 11\n00.511.522.533.544.5 5 5.5 6\nDamping α00.511.522.5 Critical AC fieldT0=0 K\nT1=56 K\nT2=280 K\nT3=560 K0 1 2 3 4 5 6\nDamping α00.511.522.5 Critical AC fieldT3=560 K\nFigure 7. (Color online) Critical alternating field amplitudes vs. damping for\ndifferent temperatures averaged over 500 times. Inset shows no t averaged data for\nq3= 0.01≡560K.\n00.511.522.533.544.5 5 5.5 6\nDamping α012345Reversal time, [ns]T1=56 K\nT2=280 K\nT3=560 K\n0 1 2 3 4 5 6\nDamping α00.10.20.30.40.5Reversal time, [ns]T0=0 K\nFigure 8. (Coloronline) The damping dependence ofthe reversaltimes corre sponding\nto the critical field amplitudes of Fig. 7 for different temperatures. Inset shows the\ncase of zero Kelvin.\n3.2. Nanoparticles with cubic anisotropy: Fe 70Pt30\nNow we focus on another type of the anisotropy, namely a cubic anis otropy which is\nsupposed to be present for Fe 70Pt30nanoparticles [24]. The energetics of the system is\nthen described by the functional form\n˜H=−d(S2\nxS2\ny+S2\nyS2\nz+S2\nxS2\nz)−S·b, (18)\nor in spherical coordinates\n˜H=−d(cos2φsin2φsin4θ+cos2θsin2θ)−S·b. (19)\nIn contrast to the previous section, there are more local minima or in other words more\nstable states of the magnetization in the energy profile for the Fe 70Pt30nanoparticles.\nIt can be shown that the minimum barrier that has to be overcome is d/12 which is\ntwelve times smaller than that in the case of a uniaxial anisotropy. Th e maximal one is\nonlyd/3.\nThe magnetization of these nanoparticles is first relaxed to the initia l state close toTemperature dependent magnetization dynamics of magnetic nanoparticles 12\nFigure 9. (Color online) Trajectories of the magnetization in the θ(φ) space (q0= 0).\nIn the units (4) we choose b= 0.82 andα= 0.1.\nφ0=π/4 andθ0= arccos(1 /√\n3), whereas in the target state it is aligned antiparallel to\nthe initial one, i. e. φe= 3π/4 andθe=π−arccos(1/√\n3). In order to be close to the\nstarting state for the uniaxial anisotropy case we choose φ0= 0.2499·π,θ0= 0.3042·π.\n3.2.1. Static driving field A static field is applied antiparallel to the initial state of the\nmagnetization, i.e.\nb=−b/√\n3(ex+ey+ez). (20)\nIn Fig. 9 the trajectory of the magnetization in case of an applied st atic field is shown.\nSimilar to the previous section the energy of the initial state lies highe r than that of\nthe target state. The magnetization rolls down the energy landsca pe to eventually\nend up by the target state. The trajectory the magnetization fo llows is completely\ndifferent from the one for the uniaxial anisotropy. Fig. 10 suppleme nts this scenario\nof the magnetization reversal by showing the time evolution of the Szvector. Because\nof the different anisotropy type, the trajectory is markedly differ ent from the case of\nthe uniaxial anisotropy and a static field. Here we show only the Szmagnetization\ncomponent even though the other components also have to be tak en into account in\norder to avoid a wrong target state.\nThe procedure to determine the critical field amplitudes is similar to th at described in\nthe previous section. In Fig. 11 the critical fields versus the dampin g parameter for\ndifferent temperatures are shown. For q0, the critical field strength is smaller than 1.\nThis is consistent insofar as the maximum effective field for a cubic anis otropy is2\n3BA.\nIn principle, the critical field turns out to be constant for all αbut for an infinitely large\nmeasuring time. Since we set this time to be about 5 nanoseconds, th e critical fields\nincrease for small and high damping. On the other hand, at lower tem peratures smaller\ncritical fields are sufficient for the (thermal activation-assisted) reversal process.\nThe behaviour of the corresponding switching times presented in Fig . 12 only\nsupplements the fact of too low measuring time, which is chosen as 5 nsfor a better\ncomparison of these results with ones for uniaxial anisotropy. Ind eed, constant jumps\nin the reversal times for T= 0Kas a function of damping can be observed. The reasonTemperature dependent magnetization dynamics of magnetic nanoparticles 13\n00.511.522.533.544.5 5\nTime, [ns]-1-0.500.51Magnetization Sz\nT0=0 K\nT6=1.9 K\nFigure 10. (Color online) Magnetization reversal of a nanoparticle when a stat ic field\nb= 0.82 is applied and for α= 0.1 at zero temperature (black). The magnetization\nreversal for α= 0.1,b= 0.22 andq6= 0.06≡1.9Kis shown with blue color.\n0 1 2 3 4 56\nDamping α00.20.40.60.811.2 Critical DC field T0=0 K\nT04=0.03 K\nT4=0.3 K\nT5=0.9 K\nT6=1.9 K0 1 2 3 4 5 6\nDamping α00.51Critical DC field T6=1.9 K\nFigure 11. (Color online) Critical static field amplitudes vs. the damping paramet ers\nfor different temperatures averaged over 500 times. Inset show s not averaged data for\nq6= 0.06≡1.9K.\n0 1 2 3 4 56\nDamping α012345Reversal time, [ns]   T0=0 K\nT04=0.03 K\nT4=0.3 K\nT5=0.9 K\nT6=1.9 K\nFigure 12. (Color online) Reversal times corresponding to the critical static fi elds of\nFig. 11 vs. damping averaged over 500 times.\nwhy the reversal times for finite temperatures are lower is as follow s: The initial state\nforT= 0Kis chosen to be very close to equilibrium. This does not happen for finit eTemperature dependent magnetization dynamics of magnetic nanoparticles 14\nFigure 13. (Color online) Trajectories of the magnetization vector specified b y the angles θandφat\nzero temperature. The chosen parameters are b0= 0.055 and ω= ˜ωa/1.93, where ˜ ωa= 2/3ωa.\ntemperatures, where the system due to thermal activation jump s out of equilibrium (cf.\nsee Fig. 10).\n3.2.2. Time-dependent external field Here we consider the case of an alternating field\nthat rotates in the plain perpendicularly to the initial state of the ma gnetization. It is\npossible to switch the magnetization with a field rotating in the xy−plane but the field\namplitudes turn out to be larger than those when the field rotates p erpendicular to the\ninitial state. For the energy this means that the field entering equa tion (19) reads\nb(t) = (b0cosω1tcosφ0+b0sinω1tsinφ0cosθ0)ex\n+(−b0cosω1tsinφ0+b0sinω1tcosφ0cosθ0)ey+(−b0sinθ0sinω1t)ez,(21)\nwhereb0is the alternating field amplitude and ω1is the frequency associated with the\nfield. This expression is derived upona rotationof thefield planeby th e anglesφ0=π/4\nandθ0= arccos(1 /√\n3).\nThe magnetization trajectories depicted in Fig. 13 reveal two inter esting features:\nFirstly, particularly for small damping, the energy profile changes v ery slightly (due\nto the smallness of b0) while energy is pumped into the system during many cycles.\nSecondly, thesystemswitchesmostlyinthevicinityoflocalminimatoa cquireeventually\nthe target state. Fig. 14 hints on the complex character of the ma gnetization dynamics\nin this case. As in the static field case with a cubic anisotropy the critic al field\namplitudes shown in Fig. 15 are smaller than those for a uniaxial anisot ropy. Obviously,\nthe reason is that the potential barrier associated with this anisot ropy is smaller in this\ncase, giving rise to smaller amplitudes. As before an increase in tempe rature leads to a\ndecrease in the critical fields.\nThe reversal times shown inFig. 16 exhibit the same feature asin the cases for uniaxial\nanisotropy: With increasing temperatures the corresponding rev ersal times increase. A\nphysically convincing explanation of the (numerically stable) oscillation s for the reversal\ntimes is still outstanding.Temperature dependent magnetization dynamics of magnetic nanoparticles 15\n00.511.522.533.544.5 5\nTime, [ns]-1-0.500.51Magnetization Sz\nT0=0 K\nT6=1.9 K\nFigure 14. (Color online) Magnetization reversal in a nanoparticle using a time de pendent field for\nα= 0.1 andq0(black) and for q6= 0.06≡1.9K(blue). Other parameters are as in Fig. 13.\n0 1 2 3 4 5 6\nDamping α00.51Critical AC fieldT6=1.9 K\n00.511.522.533.544.5 5 5.5 6\nDamping α00.511.52Critical AC fieldT0=0 K\nT4=0.3 K\nT5=0.9 K\nT6=1.9 K\nFigure 15. (Color online) Critical alternating field amplitudes vs. damping for diffe rent temperatures\naveraged over 500 cycles. Inset shows the single cycle data at q6= 0.06≡1.9K.\n00.511.522.533.544.5 5 5.5 6\nDamping α012345Reversal time, [ns]    T4=0.3 K\nT5=0.9 K\nT6=1.9 K\n0 1 2 3 4 5 6\nDamping α012345Reversal time, [ns]T0=0 K\nFigure 16. (Color online) The damping dependence of the reversal times corre sponding to the critical\nfields of the Fig. 15 for different temperatures averaged over 500 runs. Inset shows the T= 0 case.\n4. Summary\nIn this work we studied the critical field amplitudes required for the m agnetization\nswitching of Stoner nanoparticles and derived the corresponding r eversal times forTemperature dependent magnetization dynamics of magnetic nanoparticles 16\nstatic and alternating fields for two different types of anisotropies . The general trends\nfor all examples discussed here can be summarized as follows: Firstly , increasing the\ntemperature results in a decrease of all critical fields regardless o f the anisotropy type.\nAnisotropy effects decline with increasing temperatures making it ea sier to switch the\nmagnetization. Secondly, elevating thetemperature increases th e corresponding reversal\ntimes. Thirdly, thesametrendsareobservedfordifferenttemper atures: Thecriticalfield\namplitudes for a static field depend only slightly on α, whereas the critical alternating\nfield amplitudes exhibit a pronounced dependence on damping. In the case of a uniaxial\nanisotropy we find the critical alternating field amplitudes to be smalle r than those for a\nstatic field, especially in the low damping regime and for finite temperat ures. Compared\nwithastaticfield, alternating fieldsleadtosmaller switching times( T= 0K). However,\nthis is not the case for the cubic anisotropy. The markedly different trajectories for the\ntwo kinds of anisotropies endorse the qualitatively different magnet ization dynamics.\nIn particular, one may see that for a cubic anisotropy and for an alt ernating field\nthe magnetization reversal takes place through the local minima lea ding to smaller\namplitudesoftheappliedfield. Generally, acubicanisotropyissmallert hantheuniaxial\none giving rise to smaller slope of critical fields, i.e. smaller alternating fi eld amplitudes.\nIt is useful to contrast our results with those of Ref. [15]. Our re versal times for\nAC-fields increase with increasing temperatures. This is not in contr adiction with the\nfindings of [15] insofar as we calculate the switching fields at first, an d then deduce the\ncorresponding reversal times. If the switching fields are kept con stant while increasing\nthe temperature [15] the corresponding reversal times decreas e. We note here that\nexperimentally known values of the damping parameter are, to our k nowledge, not\nlarger than 0 .2. The reason why we go beyond this value is twofold. Firstly, the valu es\nof damping are only well known for thin ferromagnetic films and it is not clear how to\nextend them to magnetic nanoparticles. For instance, in FMR exper iments damping\nvalues are obtained from the widths of the corresponding curves o f absorption. The\ncurves for nanoparticles can be broader due to randomly oriented easy anisotropy axes\nand, hence, the values of damping could be larger than they actually are. Secondly, due\ntoaverystrongdependenceofthecriticalAC-fields(Fig. 7, e.g.) t heycanevenbelarger\nthan static field amplitudes. This makes the time-dependent field disa dvantageous for\nswitching in an extreme high damping regime.\nFinally, as can be seen from all simulations, the corresponding rever sal times are much\nmore sensitive a quantity thantheir critical fields. This follows from t he expression (13),\nwhere a slight change in the magnetic field bleads to a sizable difference in the reversal\ntime. This circumstance is the basis for our choice to average all the reversal times and\nfields over many times. This is also desirable in view of an experimental r ealization, for\nexample, in FMR experiments or using a SQUID technique quantities like critical fields\nand their reversal times are averaged over thousands of times. T he results presented\nin this paper are of relevance to the heat-assisted magnetic recor ding, e.g. using a\nlaser source. Our calculations do not specify the source of therma l excitations but\nthey capture the spin dynamics and switching behaviour of the syst em upon thermalTemperature dependent magnetization dynamics of magnetic nanoparticles 17\nexcitations.\nAcknowledgments\nThis work is supported by the International Max-Planck Research School for Science\nand Technology of Nanostructures.\nReferences\n[1]Spindynamics in confined magnetic structures III B. Hillebrands, A. Thiaville (Eds.) (Springer,\nBerlin, 2006); Spin Dynamics in Confined Magnetic Structures II B. Hillebrands, K. Ounadjela\n(Eds.) (Springer, Berlin, 2003); Spin dynamics in confined magnetic structures B. Hillebrands,\nK. Ounadjela (Eds.) (Springer, Berlin, 2001); Magnetic Nanostructures B. Aktas, L. Tagirov, F.\nMikailov (Eds.), (Springer Series in Materials Science, Vol. 94) (Spring er, 2007) and references\ntherein.\n[2] M. Vomir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, Phys. Rev. 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Asti (Eds.), Applied Magnetism (Kluwer, Academic Pub., Dordrecht, 1994).\n[15] S. I. Denisov, T. V. Lyutyy, P. H¨ anggi, and K. N. Trohidou, Ph ys. Rev. B 74, 104406 (2006).\n[16] S. I. Denisov, T. V. Lyutyy, and P. H¨ anggi, Phys. Rev. Lett. 97, 227202 (2006).\n[17] E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. Londo n, Ser A 240, 599 (1948).\n[18] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935).\n[19] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[20] J. L. Garcia-Palacios and F. Lazaro, Phys. Rev. B 58, 14937 (1998).\n[21]Algorithmen in der Quantentheorie und Statistischen Physi kJ. Schnakenberg (Zimmermann-\nNeufang, 1995).\n[22] U. Nowak, Ann. Rev. Comp. Phys. 9, 105 (2001).\n[23] K. D. Usadel, Phys. Rev. B 73, 212405 (2006).\n[24] C. Antoniak, J. Lindner, and M. Farle, Europhys. Lett. 70, 250 (2005).\n[25] I. Klik and L. Gunther, J. Stat. Phys. 60, 473 (1990).\n[26] C. Antoniak, J. Lindner, M. Spasova, D. Sudfeld, M. Acet, and M. Farle, Phys. Rev. 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Phys. 93, 453\n(2003)."    },    {        "title": "2211.02490v1.Derivation_of_Interacting_Two_Qubit_Dynamics_from_Spin_Boson_Model.pdf",        "content": "arXiv:2211.02490v1  [quant-ph]  4 Nov 2022Derivation of Interacting Two-Qubit Dynamics from\nSpin-Boson Model\nHiroaki Matsueda1,2, Yukiya Ide1, and Sadamichi M aekawa3,4\n1Department of Applied Physics, Graduate School of Engineee ring, Tohoku University, Sendai\n980-8579, Japan\n2Center for Science and Innovation in Spintronics, Tohoku Un iversity, Sendai 980-8577, Japan\n3RIKEN Center for Emergent Matter Science, Wako, Saitama 351 -0198, Japan\n4Kavli Institute for Theoretical Physics, University of Chi nese Sciences, Beijing 100190, China\nE-mail: hiroaki.matsueda.c8@tohoku.ac.jp\n(Received July 31, 2022)\nWe derive damping equations of motion for interacting two-s pin states from a spin-boson model\nin order to examine qubit dynamics in quantum computers. On t he basis of the composite operator\nmethod, we develop the Caldeira-Leggett approach for open q uantum systems so that the entangle-\nment dynamics originated from the two-spin correlation can be taken. We demonstrate numerical\nresults for time dependence on the two-spin dynamics. We find that the relaxation of the total spin is\ndescribed by a quantum version of the Landau-Lifshitz-Gilb ert equation for magnetic materials. We\nalso find that a two-spin composite mode keeps oscillation ev en after the total spin has been fully\nrelaxed. We thus conclude that the two-spin correlation due to the presence of the composite mode\nis stable against dissipation. We consider the mechanism of why the correlation is maintained.\nKEYWORDS: qubit, spin-boson model, relaxation, composite operator, equation of motion,\nCaldeira-Leggett approach\n1. Introduction\nNowadays, exploring quantum technologies such as quantum c omputation, quantum cryptogra-\nphy, and quantum sensing is turning into a realistic goal for current engineering. Ten years have\npassed already after the D-wave machine, a kind of quantum an nealers, was commercially provided,\nand the machine is getting used for various optimization pro blems. Recent flagship research projects\nassociated with quantum technologies aim to construct larg e-scale fault-tolerant universal quantum\ncomputers in the middle of this century. For this purpose, de velopment of noisy intermediate-scale\nquantum (NISQ) computers is an important milestone at the pr esent stage. We believe that future\nsophisticated society will be highly supported from these t echnologies, and thus it is necessary to\npromote basic science behind the technologies. There are ma inly two directions for the advanced re-\nsearch. One is direct treatment of fault tolerance algorith m and implementation as long-term research,\nand the other is deep examination of NISQ itself as short or me dium-term plan. The latter is closely\nrelated to non-equilibrium physics in which the dynamics of our qubit system is highly disturbed\nby the environmental noise and interaction among qubits the mselves when the qubits are massively\nintegrated on the substrate. Since a qubit can be identified w ith a quantum spin, our target model is\nthe so-called spin-boson model in which the interacting qua ntum spins couple with bosonic degrees\nof freedom. Thus, our interest is to understand operational stability of single spin and entanglement\namong multiple spins in this model. This is because quantum c omputation is realized by sequential\nchange of qubit states with external perturbation as unitar y gates.\nMotivated by the abovementioned consideration, we theoret ically examine the spin dynamics in\n1the spin-boson model. Here, the entanglement control of qub its is a key for various quantum tech-\nnologies, and thus we particularly focus on whether dynamic al behavior of non-local correlation or\nentanglement is stably controlled against dissipation due to the presence of the environment. This\ntype of works was recently done as a toy model for the D-wave ma chine [1]. However, this is a very\nspecial case in which we can successfully integrate out boso nic degrees of freedom in terms of the\nSuzuki-Trotter decomposition. We would like to get versati le techniques for more general cases.\nFor this purpose, we first derive damping equations of motion for the total spin from our spin-\nboson model in which two spins interact with each other. We de veloped old approaches such as Feyn-\nman’s influential functional and Caldeira-Leggett model. W e assume Ohmic spectral distribution of\nenvironmental degrees of freedom and Markovian approximat ion to get a closed equation of motion.\nWe then find that the result is equivalent to a quantum version of the Landau-Lifshitz-Gilbert (LLG)\nequation for macroscopic spin precession in magnetic mater ials [2, 3]. Here, the Gilbert constant is\nproportional to the coe fficient of the distribution function. On the basis of the total -spin dynamics, we\nnext focus on the internal dynamics of two spins in order to un derstand the stability of their nonlocal\ncorrelation. The abovementioned theoretical method is als o applied to a composite operator associ-\nated with the correlation. We find that the two-spin correlat ion is maintained even after the total spin\nhas been relaxed and the stability of two-spin correlation i s different from that of the total-spin dy-\nnamics [3]. In the usual spectroscopy in quantum many-body s ystems, low-lying states are dominated\nby composite spins [4]. We expect that the stability of the dy namics strongly depends on the spatial\nsize of correlated spin cloud. We demonstrate numerical res ults and mention why the stability of the\nnonlocal correlation appears.\n2. Model and Equations of Motion for Interacting Two-Spin Dy namics\nWe consider interacting two qubits (quantum spins) coupled to the bosonic environment. We start\nwith the following Hamiltonian\nH=/summationdisplay\nk,αωkbα†\nkbα\nk+/summationdisplay\nαBα\n0Sα+J/summationdisplay\nαSα\n1Sα\n2+/summationdisplay\nk,α/parenleftBig\nνkbα\nk+ν∗\nkbα†\nk/parenrightBig\nSα, (1)\nwhere we consider two S=1/2 spins, Sα=Sα\n1+Sα\n2(α=1,2,3),bα†\nkandbα\nkare boson operators\nwith mode kand index of angular momentum α,Bα\n0is associated with the energy di fference of qubit\nstates,ǫ, and transverse field, h,/vectorB0=(h,0,ǫ), and Jis antiferromagnetic coupling between spins.\nHere we assume a special form of boson operators with index αso that we can find simple damping\nequations for spins. Furthermore, the coupling between qub its is assumed to be of Heisenberg type\nfor simplicity, but this is an ideal situation. We should not e that the coupling actually depends on the\ntype of qubit design. For simplicity we introduce\nAα=Bα\n0+Lα,Lα=/summationdisplay\nk/parenleftBig\nνkbα\nk+ν∗\nkbα†\nk/parenrightBig\n, (2)\nand then the Hamiltonian is simply represented as H=/summationtext\nkωk/vectorb†\nk·/vectorbk+J/vectorS1·/vectorS2+/vectorA·/vectorS.\nThe equations of motion for single spin operators, /vectorS1and/vectorS2, are represented as\n∂\n∂t/vectorS1=J/vectorm−/vectorS1×/vectorA,∂\n∂t/vectorS2=−J/vectorm−/vectorS2×/vectorA, (3)\nwhere the composite spin operator /vectormis defined by\nmα=ǫαβγSγ\n1Sβ\n2, /vectorm=/vectorS2×/vectorS1, (4)\n2and this operator characterizes entanglement between two s pins. To understand the relation between\n/vectormand entanglement, it is useful to remember the definition of t he single spin /vectorS1. The states,|↑∝angbracketright\nand|↓∝angbracketright, are the eigenstates of Sz\n1in the single-spin case. The transition between these state s are\nrepresented by Sx\n1andSy\n1. Here, the eigenstates of mzare the product and entangled states ( |↑↑∝angbracketright,\n|↓↓∝angbracketright, and|↑↓∝angbracketright± i|↓↑∝angbracketright). Then, mxandmycorrespond to the transition from the product state ( |↑↑∝angbracketrightor\n|↓↓∝angbracketright) to the singlet state, and these transition operators play a crucial role on generating or keeping\nentanglement. Thus, we particularly focus on the relaxatio n dynamics of /vectorm. Note that the equation of\nmotion for the total spin /vectorSdoes not include /vectormbecause/vectorScommutes with the Heisenberg coupling:\n∂\n∂t/vectorS=−/vectorS×/vectorA=−1\n2/parenleftBig/vectorS×/vectorA−/vectorA×/vectorS/parenrightBig\n. (5)\nThus, the alternative treatment of /vectormis important for the examination of internal dynamics betwe en\nspins. The equation of motion for /vectormis given by\n∂\n∂t/vectorm=1\n2J/parenleftBig/vectorS2−/vectorS1/parenrightBig\n−/vectorm×/vectorA=1\n2J/parenleftBig/vectorS2−/vectorS1/parenrightBig\n−1\n2/parenleftBig\n/vectorm×/vectorA−/vectorA×/vectorm/parenrightBig\n. (6)\nNote that we have taken symmetrized procedure in Eqs. (5) and (6) in order to avoid technical di ffi-\nculty associated with noncommutativity of quantum operato rs.\nThese equations still contain bosonic operators through /vectorA. Let us remove the bosonic degrees of\nfreedom. For this purpose, the Heisenberg equation of motio n for environmental boson is given by\ni∂\n∂tbα\nk=ωkbα\nk+ν∗\nkSα, (7)\nand the formal solution can be obtained as\nbα\nk(t)=e−iωktbα\nk(0)−iν∗\nk/integraldisplayt\n0dt′e−iωk(t−t′)Sα/parenleftbigt′/parenrightbig. (8)\nWe would like to obtain a closed form of equations of motion fo r/vectorSand/vectorm, and for this purpose we\nsubstitute the bosonic solution into the equations. We assu me the bosonic spectrum as\nJ(ω)=/summationdisplay\nk|νk|2δ(ω−ωk)=ηω, (9)\nwhere this assumption represents the Ohmic process and the c oefficientηplays a central role on the\nrelaxation of spin dynamics.\nBy combining these equations with use of Markovian approxim ation (ωcis the cut-offfrequency\nforJ(ω), and we take it as a large constant), the final form of total sp in dynamics is given by\n∂\n∂t/vectorS(t)=/vectorB(t)×/vectorS(t)−ηsinωct\nt/parenleftBig/vectorS(t)×/vectorS(0)−/vectorS(0)×/vectorS(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorS(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorS(t)/parenrightBigg\n, (10)\nwhere the coefficientα(t) is defined by\nα(t)=η/integraldisplayωct\n0dτsinτ\nτ,lim\nωct→∞α(t)=πη\n2, (11)\nand/vectorB(t) is defined by\nBα(t)=Bα\n0+/summationdisplay\nk/parenleftBig\nνke−iωktbα\nk(0)+ν∗\nkeiωktbα†\nk(0)/parenrightBig\n. (12)\n3The result is essentially a quantum version of the LLG equati on with the damping coe fficientα(t). In\nthe right hand side of Eq. (10), the second term represents co rrelation between the initial state and the\nstate at time t. For largeωcvalues, the second term becomes negligible with time. The la st term in\nEq. (10) shows damping, and also produces quantum e ffects that do not contain in the classical LLG\ndynamics. The quantum e ffects are originated in non-commutativity between /vectorSand∂/vectorS/∂t, and then\nthe magnitude of the expectation value of /vectorSmay not be a conserved quantity. We will briefly discuss\nthis point later.\nOn the other hand, /vectormshows the following dynamics\n∂\n∂t/vectorm(t)=/vectorB(t)×/vectorm(t)+1\n2(J−2ηωc)/parenleftBig/vectorS2(t)−/vectorS1(t)/parenrightBig\n−ηsinωct\nt/parenleftBig\n/vectorm(t)×/vectorS(0)−/vectorS(0)×/vectorm(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorm(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorm(t)/parenrightBigg\n. (13)\nBy substituting /vectorm×/vectorS=/parenleftBig/vectorS1−/vectorS2/parenrightBig\n/2+i/vectormand/vectorS×/vectorm=/parenleftBig/vectorS2−/vectorS1/parenrightBig\n/2+i/vectorminto Eq. (13), we obtain\n∂\n∂t/vectorm(t)=/vectorB(t)×/vectorm(t)−1\n2(J−2ηωc)/parenleftBig\n/vectorm(t)×/vectorS(t)−/vectorS(t)×/vectorm(t)/parenrightBig\n−ηsinωct\nt/parenleftBig\n/vectorm(t)×/vectorS(0)−/vectorS(0)×/vectorm(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorm(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorm(t)/parenrightBigg\n. (14)\nA striking feature of this equation is that the relaxation te rm (the last term in the right hand side)\nturns offwhen/vectorSis in a stationary condition ∂/vectorS/∂t=0. In this case, the operator /vectorSin the second\nterm behaves as a static field. The third term represents quan tum correlation between initial state and\nthe state at time t, but this term is negligible as we have already pointed out. F urthermore, decoupling\nof the nature of /vectorSand/vectormis facilitated by taking J=2ηωc. Therefore, the dynamics of /vectormis very\nstable against the relaxation of /vectorS. More precisely, the stationary condition must be represen ted by\n∝angbracketleftψ|∂/vectorS/∂t|ψ∝angbracketright=∂∝angbracketleft/vectorS∝angbracketright/∂t=0 with the initial quantum state |ψ∝angbracketright. Thus, the abovementioed statement\nwould be too strong. We expect slow damping of the composite s pin/vectorm, and the damping behavior of\n/vectormwould be represented by sophisticated treatment of higher- order equation of motion. We thus think\nthat the energy relaxation time T1is determined by the time scale of relaxation of the total spi n∝angbracketleft/vectorS∝angbracketright,\nand the decoherence time T2between two spins is determined by the time scale of relaxati on of the\ncomposite spin∝angbracketleft/vectorm∝angbracketright. In the next section, we analyze these coupled equations of m otion to examine\nthe feature of the dynamics of ∝angbracketleft/vectorS∝angbracketrightand∝angbracketleft/vectorm∝angbracketright.\nIn the previos works associated with inertial spin dynamics in ferromagnets, higher-order terms\nof the LLG equation have been considered [5, 6]. It is an inter esting future work to examine their\nrelationship with the present result.\n3. Numerical Results\nBefore going into numerical details, we briefly examine the q ubit system without bosons in order\nto find a guideline for determining the magnitudes of Jandǫ(h=0). In this case, the Hamiltonian\nwithout bosons, H0, is transformed into\nH0=J/vectorS1·/vectorS2+ǫSz=J\n2S(S+1)−3J\n4+ǫSz, (15)\nwhere/vectorS·/vectorS=S(S+1)=2/vectorS1·/vectorS2+3/2 and thus H0can be represented by using the total spin\nS. The singlet is characterized by S=0 and Sz=0, while the triplet is characterized by S=1\nandSz=−1,0,+1. Here, we compare the energy of singlet E(S=0,Sz=0)=−3J/4 with\nthe energy of one of triplets E(S=1,Sz=−1)=J/4−ǫ. Then, E(S=1,Sz=−1) becomes\n4lower than E(S=0,Sz=0) forǫ > J. When we take a parameter range in which the triplet\n(disentangled product state) is stabilized, a viewpoint of classical LLG dynamics would be reasonable\nfor describing the dynamics of /vectorS. In this proceeding, we would like to start with such a simple case,\nand then consider the coupling with bosonic environment. We are interested in a parameter region in\nwhich singlet and triplet states are strongly competing wit h each other, but this is a future work. For\ncomparison, we show that the operator /vectormsatisfies the following relation\n/vectorm·/vectorm=3\n8−1\n2/vectorS1·/vectorS2=3\n4−1\n4S(S+1). (16)\nThus the magnitude of this quantity is also characterized by the total spin S. This value for the singlet\nstate with finite amont of entanglement is larger than that fo r the triplet state. This result also supports\nthat/vectormcharacterizes entanglement between two spins.\nFor numerical simulation, we take J=1,ωc=200, and/vectorB0=(h,0,ǫ)=(0,0,2). We intro-\nduce the initial quantum state as a product (disentangled) s tate, and then consider how the two-spin\ncorrelation or entanglement is generated by the time evolut ion:\n|ψ∝angbracketright∝(a|↑∝angbracketright1+(1−a)|↓∝angbracketright1)⊗(b|↑∝angbracketright2+(1−b)|↓∝angbracketright2)⊗|ϕ∝angbracketright, (17)\nwhere we take a=0.7 and b=0.3, and|ϕ∝angbracketrightis a bosonic part. For this initial state, the expectation\nvalues of/vectorSand/vectormare, respectively, given by\n∝angbracketleft/vectorS∝angbracketright=∝angbracketleftψ|/vectorS|ψ∝angbracketright=(0.72,0,0),∝angbracketleft/vectorm∝angbracketright=∝angbracketleftψ|/vectorm|ψ∝angbracketright=(0,−0.25,0). (18)\nNote that the magnitudes of these vectors depend on the selec tion of the initial quantum state |ψ∝angbracketright.\nWe do not take a full polarized state |↓↓∝angbracketright(or the maximally-entangled singlet state) at t=0, since\n∝angbracketleft/vectorm∝angbracketright(or∝angbracketleft/vectorS∝angbracketright) is zero in this case. Our equations of motion, Eqs. (10) and ( 14), are operator relations,\nnot classical vector equations. Thus, we must take expectat ion values by the state |ψ∝angbracketrightin order to\nintroduce graphical representation. In this process, /vectorS×∂/vectorS/∂tin Eq. (10) and /vectorm×∂/vectorS/∂tin Eq. (14)\nare respectively decomposed into two independent terms:\n∝angbracketleft/vectorS×∂\n∂t/vectorS∝angbracketright∼∝angbracketleft/vectorS∝angbracketright×∂\n∂t∝angbracketleft/vectorS∝angbracketright,∝angbracketleft/vectorm×∂\n∂t/vectorS∝angbracketright∼∝angbracketleft/vectorm∝angbracketright×∂\n∂t∝angbracketleft/vectorS∝angbracketright. (19)\nThe dynamics of∝angbracketleft/vectorS∝angbracketrightafter this approximation becomes equivalent to the classic al LLG equation,\nexcept that∝angbracketleft/vectorB∝angbracketrightstill contains information of bosons. Here we neglect time d ependence on /vectorB(t) that\nappears as a result of the second term in Eq. (12). In this case , we can solve the equation of motion,\nand we find that the relaxation time scale T1is proportional to (1 +π2η2|∝angbracketleft/vectorS∝angbracketright|2)/2πηǫ|∝angbracketleft/vectorS∝angbracketright|. Unfor-\ntunately, quantum e ffects originated from non-commutativity between /vectorSand∂/vectorS/∂tand polaronic\neffects are lost in this approximation, and then |∝angbracketleft/vectorS∝angbracketright|is kept. Thus, we suppose that the realistic relax-\nation time scale may change. In the present approximation, t he decoherence time T2becomes infinity\ndue to the stability of the dynamics of /vectorm. The precise estimation of T2is an important future work,\nbut we can say T2>T1even within the present simple analysis.\nWe demonstrate time evolution of ∝angbracketleft/vectorS∝angbracketrightand∝angbracketleft/vectorm∝angbracketrightforη=0.008 (2ηωc=3.2>J=1) in Fig. 1.\nWe find that∝angbracketleft/vectorS∝angbracketrightdecays into the direction of −∝angbracketleft/vectorB∝angbracketright. This feature is consistent with the classical LLG\ndynamics. As we have already discussed, the length of ∝angbracketleft/vectorS∝angbracketright, 0.72, is conserved in the present approx-\nimation, and∝angbracketleft/vectorS∝angbracketrightdoes not become (0 ,0,−1) even after the long time. We particularly focus on the\nentanglement dynamics represented by ∝angbracketleft/vectorm∝angbracketright. In contrast to∝angbracketleft/vectorS∝angbracketright, the coherent oscillation of ∝angbracketleft/vectorm∝angbracketrightis main-\ntained even after∝angbracketleft/vectorS∝angbracketrighthas been relaxed to the stationary point. The coherent oscil lation corresponds to\ncontinuous spin flip ( |↑↓∝angbracketright↔|↓↑∝angbracketright ) between two spins. We find that the phase di fference between∝angbracketleftmx∝angbracketright\nand∝angbracketleftmy∝angbracketrightisπ/2. As we have already mentioned, the equation of motion for /vectormdoes not contain the\ndamping term if ∂S/∂tbecomes zero. This is the origin of the stable oscillation of ∝angbracketleft/vectorm∝angbracketright. Therefore, the\ntwo-spin dynamics is essentially di fferent from total-spin dynamics.\n5Fig. 1. Spin dynamics for η=0.008. (a)∝angbracketleft/vectorS∝angbracketright, (b)/angbracketleftbig/vectorm/angbracketrightbig, and (c) Graphical representation of (a) and (b). In\nfigure (c), the blue curve represents ∝angbracketleft/vectorS∝angbracketright, and the red curve represents/angbracketleftbig/vectorm/angbracketrightbig.\n4. Concluding Remarks\nWe derived the spin dynamics in the spin-boson model in order to examine the entanglement\ncontrol of qubits against dissipation due to the presence of the environment. For the total spin, we\nfound that the result is consistent with the LLG equation for macroscopic spin precession in magnetic\nmaterials although the result also contains some quantum e ffects. Here, the Gilbert constant is propor-\ntional to the coefficient of the distribution function of the bosonic degrees of freedom. However, the\nentanglement dynamics originated from a composite spin sho ws different behavior. We numerically\nshowed the relaxation dynamics of the total spin and the stab ility of the entanglement dynamics. In\nthe present approximation, the decoherence time T2is infinity, and more precise treatment based on\nthe higher-order equation of motion is an interesting futur e work.\nH.M. is supported by JPSJ KAKENHI (Nos. 21K03380, 21H04446, 21H03455) from MEXT\nJapan and CSIS, Tohoku University, Japan. S.M is supported b y JST CREST Grant (Nos. JPMJCR19J4,\nJPMJCR1874, and JPMJCR20C1) and JSPJ KAKENHI (Nos. 17H0292 7 and 20H01865) from MEXT,\nJapan.\nReferences\n[1] Y . Bando and H. Nishimori, Phys. Rev. A 104, 022607 (2021).\n[2] J. Anders, C. R. J. Sait, and S. A. R. Horsley, New. J. Phys. 24, 033020 (2022).\n[3] H. Matsueda, Y . Ide, and S. Maekawa, in preparation.\n[4] T. Otaki, Y . Yahagi, and H. Matsueda, J. Phys. Soc. Jpn. 86, 084709 (2017).\n[5] K. Neeraj et al., Nat. Phys. 17, 245 (2021).\n[6] M.-C. Ciornei, J. M. Rubi, and J.-E. Wegrowe, Phys. Rev. B 83, 020410(R) (2011).\n6"    },    {        "title": "1309.7483v1.High_efficiency_GHz_frequency_doubling_without_power_threshold_in_thin_film_Ni81Fe19.pdf",        "content": "arXiv:1309.7483v1  [cond-mat.mtrl-sci]  28 Sep 2013High-efficiency GHz frequency doubling without power thresh old in thin-film Ni 81Fe19\nCheng Cheng1and William E. Bailey1\nMaterials Science and Engineering Program, Department of A pplied\nPhysics and Applied Mathematics, Columbia University, New York,\nNY 10027\nWe demonstrate efficient second-harmonic generation at moderat e input power for\nthin film Ni 81Fe19undergoing ferromagnetic resonance (FMR). Powers of the gene r-\natedsecond-harmonicareshowntobequadraticininputpower, wit hanupconversion\nratio three orders of magnitude higher than that demonstrated in ferrites1, defined\nas ∆P2ω/∆Pω∼4×10−5/W·Pω, where ∆ Pis the change in the transmitted rf\npower and Pis the input rf power. The second harmonic signal generated exhibit s\na significantly lower linewidth than that predicted by low-power Gilbert damping,\nand is excited without threshold. Results are in good agreement with an analytic,\napproximate expansion of the Landau-Lifshitz-Gilbert (LLG) equa tion.\n1Nonlinear effects in magnetizationdynamics, apart frombeing offun damental interest1–4,\nhave provided important tools for microwave signal processing, es pecially in terms of fre-\nquency doubling and mixing5,6. Extensive experimental work exists on ferrites1,4,6, tradi-\ntionally used in low-loss devices due to their insulating nature and narr ow ferromagnetic\nresonance (FMR) linewidth. Metallic thin-film ferromagnets are of int erest for use in these\nand related devices due to their high moments, integrability with CMOS processes, and\npotential for enhanced functionality from spin transport; low FMR linewidth has been\ndemonstrated recently in metals through compensation by the spin Hall effect7. While some\nrecent work has addressed nonlinear effects8–10and harmonic generation11–13in metallic\nferromagnets and related devices14–16, these studies have generally used very high power\nor rf fields, and have not distinguished between effects above and b elow the Suhl instabil-\nity threshold. In this manuscript, we demonstrate frequency dou bling below threshold in\na metallic system (Ni 81Fe19) which is three orders of magnitude more efficient than that\ndemonstrated previously in ferrite materials1. The results are in good quantitative agree-\nment with an analytical expansion of the Landau-Lifshitz-Gilbert (L LG) equation.\nFor all measurements shown, we used a metallic ferromagnetic thin fi lm structure, Ta(5\nnm)/Cu(5 nm)/Ni 81Fe19(30 nm)/Cu(3 nm)/Al(3 nm). The film was deposited on an oxi-\ndized silicon substrate using magnetron sputtering at a base press ure of 2.0 ×10−7Torr. The\nbottom Ta(5 nm)/Cu(5 nm) layer is a seed layer to improve adhesion a nd homogeneity of\nthe film and the top Cu(3 nm)/Al(3 nm) layer protects the Ni 81Fe19layer from oxidation.\nA diagram of the measurement configuration, adapted from a basic broadband FMR setup,\nis shown in Fig.1. The microwave signal is conveyed to and from the sam ple through a\ncoplanar waveguide (CPW) with a 400 µm wide center conductor and 50 Ω characteristic\nimpedance, which gives an estimated rf field of 2.25 Oe rms with the inpu t power of +30\ndBm. We examined the second harmonic generation with fundamenta l frequencies at 6.1\nGHz and 2.0 GHz. The cw signal from the rf source is first amplified by a solid state am-\nplifier, then the signal power is tuned to the desirable level by an adj ustable attenuator.\nHarmonics of the designated input frequency are attenuated by t he bandpass filter to less\nthan the noise floor of the spectrum analyzer (SA). The isolator limit s back-reflection of\nthe filtered signal from the sample into the rf source. From our ana lysis detailed in a later\nsection of this manuscript, we found the second harmonic magnitud e to be proportional to\n2FIG. 1. Experimental setup and the coordinate system, θ= 45◦; see text for details. EM:\nelectromagnet; SA: spectrum analyzer. Arrows indicate the transmission of rf signal.\nthe product of the longitudinal and transverse rf field strengths , and thus place the center\nconductor of CPW at 45◦from H Bto maximize the Hrf\nyHrf\nzproduct. The rf signal finally\nreaches the SA for measurements of the power of both the funda mental frequency and its\nsecond harmonic.\nFig.2(a) demonstrates representative field-swept FMR absorptio n and the second har-\nmonic emission spectra measured by the SA as 6.1 GHz and 12.2 GHz pea k intensities as\na function of the bias field H B. We vary the input rf power over a moderate range of +4\n- +18 dBm, and fit the peaks with a Lorentzian function to extract t he amplitude and\nthe linewidth of the absorbed (∆ Pω) and generated (∆ P2ω) power. Noticeably, the second\nharmonic emission peaks have a much smaller linewidth, ∆ H1/2∼10 Oe over the whole\npower range, than those of the FMR peaks, with ∆ H1/2∼21 Oe. Plots of the absorption\nand emission peak amplitudes as a function of the input 6.1 GHz power, shown in Fig.2(b),\nclearly indicate a linear dependence of the FMR absorption and a quad ratic dependence\nof the second harmonic generation on the input rf power. Taking th e ratio of the radiated\nsecond harmonic power to the absorbed power, we have a convers ion rate of 3.7 ×10−5/W,\nas shown in Fig.2(c).\nSince the phenomenon summarized in Fig.2 is clearly not a threshold effe ct, we look into\nthe second-harmonic analysis of the LLG equation with small rf fields , which is readily de-\nscribed in Gurevich and Melkov’s text for circular precession relevan t in the past for low-M s\n3FIG. 2. Second harmonic generation with ω/2π= 6.1 GHz. a) left panel : 6.1 GHz input power\n+17.3 dBm; right panel : 6.1 GHz input power +8.35 dBm. b) amplitudes of the ω(FMR) and\ngenerated 2 ωpeaks as a function of input power Pω; right and top axes represent the data set\nin log-log plot (green), extracting the power index; c) rati o of the peak amplitudes of FMR and\nsecond harmonic generation as a function of the input 6.1 GHz power; green: log scale.\n4ferrites18. For metallic thin films, we treat the elliptical case as follows. As illustra ted in\nFig.1, the thin film is magnetized in the yzplane along /hatwidezby the bias field H B, with film-\nnormal direction along /hatwidex. The CPW exerts both a longitudinal rf field hrf\nzand a transverse\nrf field hrf\nyof equal strength. First consider only the transverse field hrf\ny. In this well es-\ntablished case, the LLG equation ˙m=−γm×Heff+αm×˙mis linearized and takes the\nform \n˙/tildewidermx\n˙/tildewidermy\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidermx\n/tildewidermy\n+\nγ/tildewiderhrf\ny\n0\n (1)\n, whereγis the gyromagnetic ratio, αis the Gilbert damping parameter, ωM≡γ4πMs, and\nωH≡γHz. Introducing first order perturbation to mx,yunder additional longitudinal hrf\nz\nand neglecting the second order terms, we have\n\n˙/tildewidermx+˙/tildewidest∆mx\n˙/tildewidermy+˙/tildewidest∆my\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidermx+/tildewidest∆mx\n/tildewidermy+/tildewidest∆my\n+\nγ/tildewiderhrf\nz/tildewidermy\n−γ/tildewiderhrf\nz/tildewidermx\n+\nγ/tildewiderhrf\ny\n0\n(2)\nSubtracting (1) from (2) and taking/tildewiderhrfy,z=Hrf\ny,ze−iωt,/tildewidemx,y= (Hrf\ny/Ms)e−iωt/tildewideχ⊥,/bardbl(ω), the\nequation for the perturbation terms is\n\n˙/tildewidest∆mx\n˙/tildewidest∆my\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidest∆mx\n/tildewidest∆my\n+Hrf\nzHrf\ny\nMse−i2ωt\nγ/tildewiderχ/bardbl(ω)\n−γ/tildewiderχ⊥(ω)\n(3)\nSinceχ⊥is one order of magnitude smaller than χ/bardbl, we neglect the term −γ/tildewiderχ⊥(ω).\nIn complete analogy to equation (1), the driving term could be viewed as an effective\ntransverse field of Hrf\nz(Hrf\ny/Ms)/tildewiderχ/bardbl(ω)e−i2ωt, and the solutions to equation (3) would be\n/tildewidest∆mx= (Hrf\nzHrf\ny/M2\ns)/tildewiderχ/bardbl(ω)/tildewiderχ⊥(2ω)e−i2ωt,/tildewidest∆my= (Hrf\nzHrf\ny/M2\ns)/tildewiderχ/bardbl(ω)/tildewiderχ/bardbl(2ω)e−i2ωt. We\ncan compare the power at frequency fand 2fnow that we have the expressions for\nboth the fundamental and second harmonic components of the pr ecessing M. The time-\naveraged power per unit volume could be calculated as /angbracketleftP/angbracketright= [/integraltext2π\nω\n0P(t)dt]/(2π/ω), P(t) =\n−∂U/∂t= 2M∂H/∂twhere only the transverse components of MandHcontribute\nto P(t). Using the expression for /angbracketleftP/angbracketright,MandH, we have Pω=ωH2\ny,rfχ(ω)′′\n/bardbland\nP2ω= 2ωH2\nz,rf(Hrf\ny/Ms)2|˜χ(ω)/bardbl|2χ(2ω)′′\n/bardbl, from which we conclude that under H Bfor FMR\nat frequency f=ω/(2π), we should see a power ratio\nP2ω/Pω= 2(Hrf\nz/Ms)2χ(ω)′′\n/bardblχ(2ω)′′\n/bardbl (4)\nWithMs= 844 Oe, α= 0.007 as measured by FMR for our Ni 81Fe1930 nm sample and\n2.25 Oe rf field amplitude at input power of 1 W for the CPW, we have a ca lculated 2 f/f\n5power ratio of 1.72 ×10−5/W, which is in reasonable agreement with the experimental data\n3.70×10−5/W as shown in Fig.2(c). To compare this result with the ferrite exper iment in\nref.[1], we further add the factor representing the ratio of FMR ab sorption to the input rf\npower, which is 3 .9×10−2in our setup. This leads to an experimental upconversion ratio of\n1.44×10−6/W in ref.[1]’s definition (∆ P2ω/Pω\nin2), compared with 7 .1×10−10/W observed\nin Mg 70Mn8Fe22O (Ferramic R-1 ferrite).\nExamining Eq.(4), we noticethatthereshouldbetwo peaksinthefield -swept 2femission\nspectrum: the first coincides with the FMR but with a narrower linewid th due to the term\n|˜χ(ω)/bardbl|2, and the second positioned at the H Bfor the FMR with a 2 finput signal due to\nthe term χ(2ω)′′\n/bardbl. The second peak should have a much smaller amplitude. Due to the fie ld\nlimit of our electromagnet, we could not reach the bias field required f or FMR at 12.2 GHz\nunder this particular configuration and continued to verify Eq.(4) a t a lower frequency of 2.0\nGHz. We carried out an identical experiment and analysis and observ ed an upconversion\nefficiency of 0.39 ×10−3/W for the 4.0 GHz signal generation at 2.0 GHz input, again in\nreasonable agreement with the theoretical prediction 1.17 ×10−3/W. Fig.3 demonstrates the\ntypical line shape of the4 GHzspectrum, in which the input 2 GHzpowe r being +18.9 dBm.\nA second peak at the H Bfor 4 GHz FMR is clearly visible with a much smaller amplitude\nand larger linewidth than the first peak, qualitatively consistent with Eq.(4). A theoretical\nline (dashed green) from equation (4) with fixed damping parameter α= 0.007 is drawn to\ncompare with the experimental data. The observed second peak a t the 2fresonance H B\nshows a much lower amplitude than expected. We contribute this diffe rence to the possible\n2fcomponent in the rf source which causes the 2 fFMR absorption. The blue line shows\nthe adjusted theoretical line with consideration of this input signal impurity.\nSummary : We have demonstrated a highly efficient frequency doubling effect in thin-\nfilm Ni 81Fe19for input powers well below the Suhl instability threshold. An analysis of\nthe intrinsically nonlinear LLG equation interprets the observed phe nomena quantitatively.\nThe results explore new opportunities in the field of rf signal manipula tion with CMOS\ncompatible thin film structures.\nWe acknowledge Stephane Auffret for the Ni 81Fe19sample. We acknowledge support\nfrom the US Department of Energy grant DE-EE0002892 and Natio nal Science Foundation\n6FIG. 3. 4 GHz generation with input signal at 2 GHz, +18.9 dBm. A second peak at the bias field\nfor 4 GHz FMR is clearly present; red dots: experimental data ; dashed green: theoretical; blue:\nadjusted theoretical with input rf impurity. See text for de tails.\nECCS-0925829.\nREFERENCES\n1W. P. Ayres, P. H. Vartanian, and J. L. Melchor, J. Appl. Phys. 27, 188 (1956)\n2N. Bloembergen and S. Wang, Phys. Rev. 93, 72 (1954)\n3H. Suhl, J. Phys. Chem. Solids. 1, 209 (1957)\n4J. D. Bierlein and P. M. Richards, Phys. Rev. B 1, 4342 (1970)\n5G. P. Ridrigue, J. Appl. Phys. 40, 929 (1969)\n6V. G. Harris, IEEE Trans. Magn. 48, 1075 (2012)\n7V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. B aither, G. Schmitz\nand S. O. Demokritov, Nature Mat. 11, 1028 (2012)\n8A. Berteaud and H. Pascard, J. Appl. Phys. 37, 2035 (1966)\n9T. Gerrits, P. Krivosik, M. L. Schneider, C. E. Patton, and T. J. Silv a, Phys. Rev. Lett.\n98, 207602 (2007)\n10H. M. Olson, P. Krivosik, K. Srinivasan, and C. E. Patton, J. Appl. Ph ys.102, 023904\n(2007)\n711M. Bao, A. Khitun, Y. Wu, J. Lee, K. L. Wang, and A. P. Jacob, Appl. Phys. Lett. 93,\n072509 (2008)\n12Y. Khivintsev, J. Marsh, V. Zagorodnii, I. Harward, J. Lovejoy, P . Krivosik, R. E. Camley,\nand Z. Celinski, Appl. Phys. Lett. 98, 042505 (2011)\n13J. Marsh, V. Zagorodnii, Z. Celinski, and R. E. Camley, Appl. Phys. Le tt.100, 102404\n(2012)\n14M. Yana, P. Vavassori, G. Leaf, F.Y. Fradin, and M. Grimsditch, J. M agn. Magn. Mater\n320, 1909 (2008)\n15V. E. Demidov, H. Ulrichs, S. Urazhdin, S. O. Demokritov, V. Besson ov, R. Gieniusz, and\nA. Maziewski, Appl. Phys. Lett. 99, 012505 (2011)\n16C. Bi, X. Fan, L. Pan, X. Kou, J. Wu, Q. Yang, H. Zhang, and J. Q. Xia o, Appl. Phys.\nLett.99, 232506 (2011)\n17S. E. Bushnell, W. B. Nowak, S. A. Oliver, and C. Vittoria, Rev. Sci. In strum.63, 2021\n(1992)\n18A. G. Gurevich and G. A. Melkov, Magnetization Oscillation and Waves ( CRC, Boca\nRaton, 1996)\n8"    },    {        "title": "2008.06253v1.Large_enhancement_of_spin_pumping_due_to_the_surface_bound_states_in_normal_metal_superconductor_structures.pdf",        "content": "Large enhancement of spin pumping due to the surface bound states in normal\nmetal/superconductor structures\nM.A. Silaev1, 2, 3\n1Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia\n3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\nWe show that the spin pumping from ferromagnetic insulator into the adjacent metallic spin\nsink can be strongly stimulated by the superconducting correlations. The key physical mechanism\nresponsible for this e\u000bect is the presence of quasiparticle surface states at the ferromagnetic insu-\nlator/superconductor interface. We consider the minimal model when these states appear because\nof the suppressed pairing constant within the interfacial normal layer. For thin normal layers we\nobtain a strongly peaked temperature dependence of the Gilbert damping coe\u000ecient which has been\nrecently observed in such systems. For thicker normal layers the Gilbert damping monotonically\nincreases down to the temperatures much smaller than the critical one. The suggested model paves\nthe way to controlling the temperature dependence of the spin pumping by fabricating hybrid normal\nmetal/superconductor spin sinks.\nPACS numbers:\nIntroduction Spin transport and spin dynamics\nin superconductors have attracted signi\fcant attention\nrecently1{7. Quite interesting experimental results have\nbeen obtained for the spin pumping e\u000bects8{18which\nin general play the central role in spintronics19{21. It\nwas found that superconducting correlations can lead ei-\nther to the signi\fcant suppression8or to the signi\fcant\nenhancement9{13,17of Gilbert damping (GD) coe\u000ecient\nFIG. 1: Schematic setup of the ferromagnetic insulator (FI)\n\flm with the adjacent metallic spin sink consisting of of nor-\nmal (N) and superconducting (S) layers. The constant ex-\nternal magnetic \feld is H0x. The magnetization precession\nm(t) is driven by the external magnetic \feld H\nei\nty. It\ngenerates spin current i\npumped from F to the spin sink.\nUpper panel shows the coordinate dependencies of the order\nparameter \u0001( x) and local density of states N(x) at the energy\n\"= 0:5\u00010fordN= 0:2\u00180,dS= 3\u00180,T= 0:7Tc.in systems consisting of superconducting and ferromag-\nnetic layers, such as in the generic example shown in\nFig.1. The basic mechanism for changing GD in such\nsystems is the spin pumping e\u000bect. This mechanism\nis based on the spin angular momentum transfer from\nthe ferromagnet into the the adjacent metallic \flm via\nthe pumped spin current i(t) generated by the time-\ndependent magnetization m(t). The spin relaxation in\nthe metallic spin sink leads to the damping-like spin\ntorque and modi\fes the e\u000bective GD coe\u000ecient of the\nsystem.\nIn this way the suppression of GD with decreasing\ntemperature T < Tcin systems with superconducting\nspin sink8can be qualitatively understood as result-\ning from the the freezing out of quasiparticles in the\nsuperconductor22. However, the strong increase of GD\nwith lowering temperature9{13,17seems to be counter-\nintuitive and its understanding requires further theoret-\nical e\u000borts.\nIn ferromagnetic insulator (FI) /superconductor (S) bi-\nlayers GdN/NbN the peaked behaviour of GD as a func-\ntion of temperature has been observed13. The maximal\nGD reached at about T\u00190:7Tcis several times larger\nthan in the normal state \u000e\u000b=\u000e\u000bN\u00182\u00003, where\u000e\u000bis\nthe spin-pumping related change of GD. Because of the\nseveral reasons such behaviour cannot be explained23by\nthe coherence peak of spin susceptibility in homogeneous\nsuperconductors24. First, this peak occurs at T\u00190:9Tc\nand for the realistic values of the Dynes parameter25\n\u0000\u00190:1Tcin NbN its magnitude is23\u000e\u000b=\u000e\u000bN\u00180:2\u00000:3.\nSuch behaviour is typical for the line widths of nu-\nclear magnetic resonance26,27and electronic paramag-\nnetic resonance28in superconductors. It is clearly di\u000ber-\nent from the observed behaviour of GD in FI/S systems13\nwhich has an order of magnitude larger peak \u000e\u000b=\u000e\u000bN\u0018\n2\u00003 at signi\fcantly lower temperatures T\u00190:7Tc.\nIn this Letter we suggest a minimal theoretical modelarXiv:2008.06253v1  [cond-mat.supr-con]  14 Aug 20202\nwhich explains the large enhancement of GD in FI/S\nstructures. The key physical mechanism responsible for\nthis e\u000bect is the existence of quasiparticle states localized\nat the FI/S interface. Such states appear due to the sup-\npressed pairing within the interfacial normal layer29{32\n(N) as illustrated in Fig.1. Shown on top of the Fig.1\nare the spatial pro\fles of the order parameter \u0001( x) and\nthe local density of states (DOS) N(x) at the subgap\nenergy\"= 0:5\u00010, where \u0001 0is the bulk energy gap at\nT= 0. The overall N/S \flm thickness is dS= 3\u00180, where\n\u00180=p\nDS=Tc0is the coherence length, DSis the di\u000bu-\nsion constant in S, Tc0is the bulk critical temperature.\nNear the interface at x= 0 the DOS is enhanced due to\nthe subgap quasiparticle states which are formed in the\nN/S structure33{36and occupy the certain energy interval\nbetween the bulk gap and Thouless energy DN=d2\nNwhere\nDNis the di\u000busion coe\u000ecient and dNis the thickness of\nN. The existence of surface bound states in N/S struc-\ntures is demonstrated37in Fig.2a,c where the N(x;\") pro-\n\fles are shown to have a maximum at x= 0 and energies\nwhich depend on dN. The order parameter and DOS\nin Figs.1,2 are calculated within the Usadel theory38as\nexplained below. In Fig.1 we choose identical di\u000busion\ncoe\u000ecient in N and S layers DN=DS=Dwhile in\nFig.2DN= 0:05DS.\nAt low frequencies \n \u001c\u00010the DOS enhancement\nleads to the increased probability of the magnon ab-\nsorption by conductivity electrons in the N/S layer.\nQualitatively, at a given energy level this probability\nis determined by number of available states for transi-\ntionN(\")N(\"+ \n)\u0019N2(\") and the di\u000berence of oc-\ncupation numbers n0(\"+ \n)\u0000n0(\")\u0019\n@\"n0where\nn0(\") = tanh(\"=2T) is the equilibrium distribution func-\ntion. The product of these factors leads to the energy-\nresolved magnon absorption probability Pm= \nN2@\"n0.\nIn Fig.2b,d one can that of Pm(\") atT= 0:7Tc0is en-\nhanced at the boundary of N layer x= 0 (red curves)\nas compared to x=dS(blue curves). Besides that, the\nlocalization of surface states is qualitatively equivalent\nto the decrease of the spin sink volume which and the\ncorresponding increase of the non-equilibrium spin po-\nlarization. As we show by an exact calculation below\nthese mechanisms lead to the large enhancement of spin\npumping in the N/S \flms.\nInterestingly, besides explaining the large peak of the\nspin pumping for dN\u001c\u00180the model described above\nyields also the qualitatively di\u000berent regime with almost\nmonotonic increase of GD down to the temperatures\nT\u001cTc. This behaviour is obtained for dN\u0018\u00180when\nthe bound states are pushed down to lower energies as\nshown in Fig.2c and the absorption probability us en-\nhanced for quasiparticles with \"\u001c\u00010which are not\nfrozen out down to the signi\fcantly low temperatures\ndetermined by the Thouless energy Tth\u0019DN=d2\nN. Sim-\nilar behaviour of GD has been observed experimentally\nin Py/Nb/Pt superconducting heterostructures12,17, al-\nthough its physical origin can be di\u000berent.\nModel of spin pumping To quantify the spin\n(a)\n (b)\n(c)\n (d)\nFIG. 2: (a,c) Density of states pro\fle N(\";x) in the N/S\nstructure. The position of N/S boundary shown by the dashed\nline is at (a) dN= 0:2\u00180and (c)dN= 0:8\u00180.T= 0:7Tc0,\n\u0000 = 0:1Tc0,dS= 5\u00180,DN= 0:05DS. Plots for other dSare\nshown in Appendix37. (b,d) Magnon absorption probability\nPm(\") = \n@\"n0N2for the frequency \n = 0 :02Tc0, Red and\nblue curves are taken at x= 0 andx=dS, respectively.\nParameters are the same as in (a,b).\npumping e\u000bect we consider the microscopic model of\nthe spin-dependent scattering of electrons at the FI\ninterface37,39,40. As we show below, it formally yields\nthe spin current identical to the one given by the inter-\nfacial exchange interaction between the localized spins\nin FI and conduction elections in the adjacent metal41.\nWithin this model the local spin polarization close to the\ninterfaceS(t) acts as e\u000bective \feld for the localized mag-\nnetic moments. This process can be taken into account\nby introducing the additional term i(t) into the Landau-\nLifshitz-Gilber equation\n(1 +\u000bm\u0002)@tm+\rm\u0002Heff=i=SF0dF (1)\ni(t) =JsdS(t)\u0002m(t) (2)\nHereSF0is the equilibrium spin density in F, dFis the\nF \flm thickness, Heffis the e\u000bective \feld and \u000bis the\nintrinsic Gilbert damping coe\u000ecient. The term i(t) can\nbe interpreted as the spin current between FI and metal.\nTo calculateS(t) we need to \fnd the spin response of\nthe superconductor to the interfacial exchange \feld. In\nthe linear regime it is given by\nS\n=\u0017heff\u001fmm\n (3)\nwhere we introduce the e\u000bective exchange \feld heff=\nJsd=dS, normal metal DOS at the Fermi level \u0017and the\nlocal spin susceptibility \u001fm.3\nThe spin-pumping related change of the GD is deter-\nmined by the dissipative part of the susceptibility\n\u000e\u000b=CTc0Im\u001fm=\n (4)\nwhere the dimensionless coe\u000ecient determining the cou-\npling strength between the FI and metallic \flms is23\nC=heff\nTc0\u0017heff\nSF0dS\ndF(5)\nFrom there one can see that since heff/1=d2\nSthe cou-\npling coe\u000ecient is C/1=dS. Localization of surface\nstates provides the e\u000bective decrease of dSwhich leads\nto the increase of Cand the spin response.\nCalculation of the time-dependent spin re-\nsponse. What is left is to calculate the local spin\nsusceptibility \u001fmin the Eq.4 for the FI/N/S structure\nin Fig.1. We do so by developing the microscopic ki-\nnetic theory of spin pumping generalizing the quasiclas-\nsical approach2,40,42,43to the time-dependent situation.\nThe magnetization of conduction electrons is deter-\nmined by spin accumulation and can be written in terms\nof the Keldysh quasiclassical Green's function (GF) as\nS(t) =\u0000\u0017Tr [^\u001c3^\u001bgK(t;t)]=8 (6)\ngKis the (2\u00022 matrix) Keldysh component of the qua-\nsiclassical GF matrix \u0014 g=\u0012\n^gR^gK\n0 ^gA\u0013\nwhich depends\non two times and a single spatial coordinate variable\n\u0014g= \u0014g(t1;t2;r). GF \u0014gobeys the Usadel equation\nf^\u001c3@t;\u0014ggt+r(D\u0014g\u000er\u0014g) = \u0001[^\u001c1;\u0014g]+[\u0014\u0000;\u0014g]\u0000[\u0014\u0006so;\u0014g]t:(7)\nwhere ^\u001bk;^\u001ck,k= 0;1;2;3 are Pauli matrices, Dis the\ndi\u000busion coe\u000ecient. The commutator operator is de\fned\nas [X;g]t=X(t1)g(t1;t2)\u0000g(t1;t2)X(t2), similarly for\nanticommutatorf;gt. The symbolic product operator is\ngiven by (A\u000eB)(t1;t2) =R\ndtA(t1;t)B(t;t2).\nSpin relaxation is determined by the spin-orbital scat-\ntering self energy\n^\u0006so=\u001b\u0001^g\u001b=(6\u001cso) (8)\nThe self-consistency equation for the gap function is\n\u0001 =\u0015Tr[^\u001c1^gK]=4 (9)\nwhere\u0015is the pairing coe\u000ecient. In our model we assume\nthe pairing constant to be suppressed in the N region\n\u0015(x<dN) = 0:05\u0015(x>dN) as compared to its value in\nS. We scan over the values of the di\u000busion coe\u000ecient in\nthe N layer DNwhile keeping it \fxed in S layer DS. The\ninelastic scattering is described by the Dynes44param-\neter which enters to the Eq.7 as the matrix in Nambu-\nKeldysh space with ^\u0000R;A=\u0006\u0000^\u001c3which described both\nthe DOS singularity broadening and the relaxation of\nnon-equilibrium distribution functions as described be-\nlow. Note that this terms conserves the total spin in ac-\ncordance with the general property of spin-independent\nelectron-phonon scattering.Eq.7 is supplemented by the dynamical boundary con-\nditions atx= 0 describing the spin splitting and pump-\ning induced by the electron scattering at the FI inter-\nface with time-dependent magnetization. These bound-\nary conditions are derived37from the spin-dependent\nscattering matrix ^Sconnecting the incident ^ iand re-\n\rected ^ relectronic waves ^ r=^S(t)^ i. For frequen-\ncies small compared to the exchange \feld in FI we use\nthe adiabatic approximation which yields the expression\n^S=ei(m^\u001b)^\u001c3\u0002=2, where \u0002 is the time-independent spin-\nmixing angle. Then, assuming that j\u0002j\u001c1 and\nD\u0014g\u000e@x\u0014g(x= 0) =iJsd[\u001bm^\u001c3;^g]t (10)\nwherem=m(t) is the time-dependent magnetiza-\ntion. Within the minimal band model of the FI39,40\nthe interfacial exchange constant is expressed through\nthe spin-mixing angle as Jsd=\u0017vF\n4R1\n\u00001d^pxj^pxj\u0002(^px),\nwhere ^pxis the electron momentum projection on the\ninterface normal. Eq.10 generalizes the static bound-\nary condition at the spin-active interface39,40,43,45to the\ncase of time-pendent magnetization. The induced spin\ncurrent is obtained using the general expression i(t) =\n\u0019\u0017DTr[^\u001b\u0014g\u000e@x\u0014g](t;t). With the help of Eqs.(10,6) it\nyields the phenomenological Eq.(2).\nIntroducing the usual parametrization of quasiclassical\nKeldysh function in terms of the distribution function\n^gK= ^gR\u000e^f\u0000^f\u000e^gAwe can identify the terms which are\nessential to calculate linear response in the low-frequency\nlimit. Expanding the energy representation of ^ gKto the\n\frst order in \n we obtain the non-equilibrium correction\n\u000e^gK= (^\u001bm \n)\u0014\n(^gR\n0\u0000^gA\n0)fh+\n@\"n0\n2(gR\nh+gA\nh)\u0015\n(11)\nwhere we parametrise the spin-dependent corrections\nas follows ^f= (^\u001bm \n)fhand\u000egR;A= (^\u001bm \n)\u000egR;A\nh.\nIn contrast to stationary non-equilibrium situations42\nwhen only the \frst term in (11) is important the time-\ndependent case requires taking into account also the sec-\nond term with the corrections of spectral functions23. In\nthe low-frequency limit the calculation is simpli\fes by ne-\nglecting the frequency dependence of the perturbed spec-\ntral GF in (11). Using (11) we write the time-dependent\nspin polarization in the metallic \flm as follows\nS\n=i\nm\nZ1\n\u00001d\"[2Nfh+ (gR\n3h+gA\n3h)@\"n0] (12)\nwhereN= Tr(^\u001c3^gR)=2 is the local DOS and gR;A\n3h=\nTr(^\u001c3^gR;A\nh)=2 . Equations for zero-order spectral func-\ntion ^gR;A\n0(\";x), corrections ^ gR;A\nh(\";x) and the distribu-\ntion function fh(\";\n;x) are obtained straightforwardly37\nfrom Eqs.(7, 10). The zero-order GF ^ gR;A\n0(\";x) are calcu-\nlated in the N/S structure self-consistently together with\nthe order parameter 9. This gives in particular the \u0001( x)\nandN(\";x) pro\fles shown in Fig.1,2. The corrections fh\nand ^gR;A\nhare determined by the linear equation37.4\nDN=DS, \u0000 = 0:1Tc0\ndN\n\u00180=\n(a)\nDN= 0:05DS, \u0000 = 0:1Tc0\n(b)\nDN=DS, \u0000 = 0:01Tc0\n(c)\nDN= 0:05DS, \u0000 = 0:01Tc0\n(d)\n(e)\n (f)\n (g)\n (h)\nFIG. 3: Upper row: temperature dependencies of the GD \u000e\u000b(T) in FI/N/S systems. The three curves in each plot correspond\ntodN=\u00180= 0:8; 0:2; 0. Lower row: color plots of the functions \u000e\u000b(dN;T)=\u000e\u000bN. Horizontal lines in each panel are positioned\nas guide for eyes at dN=\u00180= 0:8; 0:2; 0 corresponding to the curves in the upper plot. The four columns correspond to various\nDynes parameters \u0000 =Tc0= 0:1; 0:01 and ratios of di\u000busion coe\u000ecients in N and S layers DN=DS= 1; 0:05 speci\fed on top of\nthe panels. Common parameters are dS= 3\u00180,\u001csnTc0= 1, \n = 0:02Tc0.\nResults and discussion Using the described formal-\nism we calculate the non-equilibrium spin polarization\n(12) in the N/S structure shown in Fig.1. This gives us\nthe local susceptibility (3) and the excess GD (4 ). The\nresulting temperature dependencies of \u000e\u000b(T) are shown\nin Fig. 3 for various parameters. The \frst column in\nFig.3 corresponds to \u0000 = 0 :1Tc0and identical di\u000busion\ncoe\u000ecients in N ans S layers. In the absence of N layer\ndN= 0 there is a usual coherence peak at T\u00190:9Tc\nwith the small amplitude \u000e\u000b=\u000e\u000bN\u00191:4. Adding the\nthin N layer with dN>0:1\u00180leads to the increase of the\npeak amplitude to \u000e\u000b=\u000e\u000bN\u00191:9 and shifting to lower\ntemperatures.\nThe peak is enhanced by decreasing the di\u000busion coef-\n\fcientDNin the normal layer. Qualitatively, this leads\nto better localization of surface bound states and hence\nto the increase of surface DOS. As shown in the second\ncolumn of Fig.3 for DN= 0:05DSand \u0000 = 0:1Tc0the\npeak is enhanced to \u000e\u000b=\u000e\u000bN\u00192:5 reached at T\u00190:7Tc\nwithdN= 0:2\u00180. This behaviour is quite similar to the\nexperimental observation13. For larger dN>0:5\u00180the\ntemperature dependence is qualitatively changed to the\nmonotonic increase down to the low temperatures. As\nshown by the yellow curve with dN= 0:8\u00180the increase\ncontinues to T\u00190:1Tc.\nEven larger increase is obtained for smaller Dynes pa-\nrameters \u0000 = 0 :01Tc0as shown in the third and fourth\ncolumns of the Fig. 3. For DN=DSwe obtain the max-imal value\u000e\u000b=\u000e\u000bN= 3. ForDN= 0:05DSwe obtain the\nmaximal value \u000e\u000b=\u000e\u000bN= 4:8. For all values of \u0000 we note\nthat forDN\u001cDSthe monotonically increasing \u000e\u000b(T) is\nobtained down to the threshold temperature of the order\nof Thouless ennergy Tth\u0019DN=d2\nN. As one can see in\nthe color plots Fig.3f,h for increasing dNit can be rather\nsmallTth\u001cTc.\nThe introduced model can explain the observed spin-\npumping enhancement in GdN/NbN system13assuming\nthat there is a naturally formed thin normal layer at\nthe FI/S interface. The pairing suppression at the inter-\nface can result from various reasons, including magnetic\ndisorder46,47, strong usual disorder48or the band struc-\nture modi\fcation49. It is straightforward to check our\nprediction of the enhanced GD by fabricating arti\fcial\nFI/N/S structures with various parameters.\nThe behaviour of \u000e\u000b(T) obtained in Figs.3b,d with\ndN= 0:8\u00180is qualitatively similar to the one observed ex-\nperimentally in Py/Nb/Pt heterostructures12,17. In the\nequilibrium state of our model the spin-triplet supercon-\nductivity is absent. Therefore the monotonic increase\nof GD due to the supercondducting correlations is not\nin principle an exclusive feature of the system with spin\nsuper-currents. However, the spin-triplet correlations are\ngenerated in the non-equilibrium case (11) providing23\nsigni\fcant contribution to the spin response (12).\nThe developed quasiclassical theory of spin pumping\ncan be generalized to the case of metallic ferromagnets5\nby introducing the \fnite spin-dependent tunnelling prob-\nability through the F/S interface43,50,51to the boundary\ncondition (10). This provides the way to study charge\nand heat transport induced by the magnetic precession\nas well as spin torques induced by voltage and tempera-\nture biases52{56.\nConclusions We have developed the general\nformalism to calculate spin-pumping in spatially-\ninhomogeneous metallic \flms with spin-active interfaces.\nAs an example we have considered the FI/N/S structure\nand found that the the presence of quasiparticle bound\nstates localized near the spin-active interface provides\nstrong enhancement of spin pumping which shows up in\nthe strong increase of the GD coe\u000ecient with decreasing\ntemperature below Tc. The model explains large peak\nof GD in Gd/NbN structures and shows the way to\ncontrolling spin pumping properties in superconducting\nsystems.\nAcknowledgements This work was supported by the\nAcademy of Finland (Project No. 297439) and Russian\nScience Foundation, Grant No. 19-19-00594. I thank\nYakov Fominov for comments.\nAppendix A: Stationary spin-mixing scattering\nmatrix\nNear the \rat FI/M surface we write wave functions in\nthe form kkeikkrwherekk=kzz+kyyis the conserved\nmomentum parallel to the interface. Along zcoordinate\nwe have 1D Shrodinger equations\ni@t = (^H\u0000\"F?) (A1)\n^H=\u0000@2\nx=2m+ [\"F+V+ (m^\u001b)Vs]\u0012(\u0000x) (A2)\nwherem=m(t).\nLet us \frst \fnd the frozen scattering matrix which de-\npends adiabatically on time. In this case the energy of\nincoming and scattered electrons coincide so that writing\n /ei\"twe get stationary 1D Shrodinger equation\n^H = (\"+\"F?) (A3)\n^H=\u0000@2\nx=2m+ [\"F+V0+ (m^\u001b)Vs]\u0012(\u0000x) (A4)\nwhere\"F?=\"F\u0000k2\nk=2m. For the energy we have \"=\nk2=2m\u0000\"Fwherek2=k2\nx+k2\nk. First, we \fnd the\nscattering matrix writing solutions\n kk=A+eikxx+A\u0000e\u0000ikxx(A5)\n kk=Bex=\u0015\u001b(A6)\nwhere\u0015\u00002\n\u001b= 2mV\u001b\u0000k2\nxandV\"(#)=V0+ (\u0000)Vsare\nthe spin-up (down) band energies in FI. The re\rection\ncoe\u000ecientS\u001b=A+=A\u0000is then\nS\u001b=ei'ei\u001b\u0002=2=1 +ikx\u0015\u001b\n1\u0000ikx\u0015\u001b(A7)Since we are interested in spin-dependent re\rection phase\nwe get the spin-mixing angle\nei\u0002=1 +k2\nx\u0015+\u0015\u0000+ikx(\u0015+\u0000\u0015\u0000)\n1 +k2x\u0015+\u0015\u0000\u0000ikx(\u0015+\u0000\u0015\u0000)(A8)\nwhich yields\n\u0002=2 = arcsin \nkx(\u0015+\u0000\u0015\u0000)p\n(1 +k2x\u0015+\u0015\u0000)2+k2x(\u0015+\u0000\u0015\u0000)2!\n(A9)\nFinally, the spin-dependent part of the scattering ma-\ntrix connecting the incident ^ iand re\rected ^ relectronic\nwaves written in the basis-independent form\n^ r=^S^ i (A10)\n^S=ei(m^\u001b)^\u001c3\u0002=2(A11)\nAppendix B: Time-dependent boundary conditions\nat the FI/metal interface\nHere we derive boundary conditions () starting from\nthe scattering theory of the interface between FI and\nmetal, either normal or superconducting one. The main\ndi\u000berence from the previous works deriving boundary\nconditions at FI/M interface is that the magnetization\nof FI depends on time m=m(t).\nWe consider matrix GF de\fned in a Keldysh-Nambu-\nspin space\n\u0014G(r1;r2;t1;t2) =\u0012^GR^GK\n0^GA\u0013\n(B1)\nwhere retarded, advanced and Keldysh parts are de\fned\nin a standard way as follows\n^GR(r1;r2;t1;t2) =\u0012(t1\u0000t2)\u0002 (B2)h\nh^\t(r1;t1)^\t+(r2;t2)i+h^\t(r1;t1)^\t+(r2;t2)ii\n^GA(r1;r2;t1;t2) =\u0012(t2\u0000t1)\u0002 (B3)h\nh^\t(r1;t1)^\t+(r2;t2)i+h^\t(r1;t1)^\t+(r2;t2)ii\n^GK(r1;r2;t1;t2) = (B4)\nh^\t(r1;t1)^\t+(r2;t2)i+h^\t(r1;t1)^\t+(r2;t2)i\nwhere the \feld operators ^\t = ( ^ \";^ #;\u0000^ +\n#;^ +\n\") satisfy\nthe equations of motion\ni@t^\t = ^H(t)^\t (B5)\nand the Hamiltonian has time-dependent order parame-\nter \u0001 = \u0001( t), boundary potential V=V(t)\n^H(t) = ^\u001c3(k2=2m\u0000\"F) + ^\u001c2\u0001(t) +^V(t) (B6)6\nThe GF satis\fes Gor'kov equations\n[i@t1\u0000^H(t1;r1)]^G=\u000e(t12)\u000e(r12) (B7)\n^G[\u0000i@t2\u0000^H(t2;r2)] =\u000e(t12)\u000e(r12) (B8)\nwherer12=r1\u0000r2andt12=t1\u0000t2. Assuming the \rat\nFI/M interface we consider transverse momentum com-\nponentskz;yas conserved quantities. The perpendicular\ncomponent kxchanges to the opposite one upon elec-\ntron re\rection. We are interested in the components of\nGF which are slowly varying as function of the center of\nmass coordinate r= (r1+r2)=2 and thus can be written\nas follows\n\u0014Gkk(x1;x2;t1;t2) =Z\ndr12e\u0000ikkr12\u0014G(r1;r2;t1;t2)\nThe GF satis\fes Gor'kov equations\n[i@t1\u0000^H(t1;z1)]\u0014Gkk=\u000e(t12)\u000e(x12) (B9)\n\u0014Gkk[\u0000i@t2\u0000^H(t2;x2)] =\u000e(t12)\u000e(x12) (B10)\n^H(t;x) =\u0000(@2\nx=2m+\"?)^\u001c3+ ^\u001c2\u0001(t) +^V(t) (B11)\nwhere\"?=\"F\u0000k2\nk=2m.\nLet's consider the Fourier expansion\n\u0014Gkk(x1;x2) =X\nk1;2ei(k1x1\u0000k2x2)\u0014Gkk(k1;k2) (B12)\nNear the M/FI interface z= 0 we can establish the con-\nnection between amplitudes\n\u0014Gkk(\u0000k1;k2) =^S(t1)\u0014Gkk(k1<0;k2) (B13)\n\u0014Gkk(k1;\u0000k2) =\u0014Gkk(k1;k2<0)^S+(t2) (B14)\nFrom these two relations we get\n\u0014Gkk(\u0000k1;\u0000k2) =^S(t1)\u0014Gkk(k1<0;k2<0)^S+(t2)\n(B15)\nRelations (B13,B14) can be obtained as follows. First,\nconsider the vicinity of interface jx1;2j\u001c\u0018where\u0018=\nvx=\u0001. In this case we can use the simpli\fed equation for\nGF neglecting the time derivative and order parameter\n[(@2\nx1=2m+\"?)^\u001c3\u0000^V(t1;x1)]\u0014G(x1<x2) = 0 (B16)\n(@2\nx2=2m+\"?)\u0014G(x2<x1)^\u001c3\u0000\u0014G(x2<x1)^V(t2;x2)] = 0\n(B17)\nThese are two independent equations identical to the\nShrodinger equation (A3) at \"= 0. Thus we can write\nthe solution\n\u0014Gkk(x1<x2) =X\nk1>0[e\u0000ik1x1+eik1x1^S(t1)]\u0014F2(x2)\n(B18)\n\u0014Gkk(x2<x1) =X\nk2>0\u0014F1(x1)[eik2x2+e\u0000ik2x2^S+(t2)]\n(B19)where ^F1;2(x) in principle can be arbitrary functions.\nComparing these relations with the general Fourier ex-\npansion (B12) we get Eqs. (B13,B14).\nThe quasiclassical GF in general is introduced accord-\ning to the following general procedure\n\u0014gp(r) =1\n\u0019Z1\n\u00001d\u0018pZ\ndr12e\u0000ipr12^\u001c3\u0014G(r1;r2)\nNear the \rat surface we have only the z-dependence\n^gp(x) =1\n\u0019Z1\n\u00001dqe\u0000iqxZ1\n\u00001d\u0018p^\u001c3^Gkk(kx+q;kx\u0000q)\nwhere we denote r12=r1\u0000r2,\u0018p= (k2\nz+k2\nk)=2m\u0000\"F.\nThen atx= 0 we have\n\u0014gp(x= 0) =1\n\u0019ZZ1\n\u00001dqd\u0018p^\u001c3\u0014Gkk(\u0000kx\u0000q;\u0000kx+q)\n\u0014gp(x= 0) =1\n\u0019ZZ1\n\u00001dqd\u0018p^\u001c3\u0014Gkk(kx+q;kx\u0000q)\n(B20)\nThen using relations B15\n\u0014gp(x= 0) =1\n\u0019ZZ1\n\u00001dqd\u0018p^\u001c3\u0014Gkk(kx+q;kx\u0000q) =\n1\n\u0019ZZ1\n\u00001dqd\u0018p^S(t1)^\u001c3\u0014Gkk(\u0000kx\u0000q;\u0000kx+q)^S+(t2)\u0019\n^S(t1)^gp(z= 0) ^S+(t2)\nwhere in the last relation we assume that ^Sdoes not de-\npend onq. Finally we get the time-dependent boundary\ncondition for quasiclassical functions\n\u0014gp(x= 0) = ^S(t1)\u0014gp(x= 0) ^S+(t2) (B21)\nExpanding ^S(t)\u00191 +i\u0002^\u001bm(t)=2 in Eq.B21 we get\nthe matrix current at the M/FI boundary\n^I(t1;t2) =\u0000Zd\np\n4\u0019(n\u0001vF)\u0014gp(t1;t2) = (B22)\n\u0000vFZ1\n0d^px^pz[\u0014gp(t1;t2)\u0000\u0014gp(t1;t2)]\u0019\n\u0000ivF\n2Z1\n0d^px^px\u0002(^px)[^\u001bm^\u001c3;\u0014g]t\nwhere we denote [ ^X;\u0014g]t=^X(t1)\u0014g(t1;t2)\u0000\u0014g(t1;t2)^X(t2),\nn=zis the normal to FI interface and denote the inci-\ndent ^p\u0001n<0 and re\rected ^p\u0001n>0 momenta.\nThis expression can be simpli\fed even more if we as-\nsume that due to the impurity scattering the anisotropic\nparts of GF are small. Then we can use two lowest order\nterms in the spherical harmonics expansion\n\u0014gp= \u0014g+p\u0001\u0014ga=p (B23)7\nKeeping only the s-wave term we get for the matrix cur-\nrent (B22)\n\u0014I(t1;t2) =i\u0017\u00001Jsd[^\u001c3^\u001bm;\u0014g]t (B24)\nwhere the conductance is given by\nJsd=\u0017vF\n4Z1\n\u00001d^pxj^pxj\u0002(^px) (B25)\nWe can \fnd the spin current using the general expres-\nsion\ni(t) =\u0019\u0017Tr4[^\u001b^IK(t;t)]: (B26)\nTaking into account the de\fnition of the spin density\nS(t) =\u0000\u0017Tr [^\u001c3^\u001bgK(t;t)]=8 (B27)\nthe spin current B26 \rowing from FI to the spin sink can\nbe written as\ni(t) =JsdS(t)\u0002m(t) (B28)\nAppendix C: Equation for the spectral and\ndistribution functions\nKinetic equation From the Keldysh-Usadel equation\nin the main text we obtain the \fnite-frequency kinetic\nequation\nr(Drfh) = [\u001c\u00001\nso+ 2(2\u0000 +i\n)N]fh (C1)\nD@xfh(x= 0) =\u00002iheffN@\"n0 (C2)\n@xfh(x=dS) = 0 (C3)\nwhereD=DTr(1\u0000^gR^gA)=2 and\u001c\u00001\nso= 4D=3D\u001csn. The\nsystem (C1, C2, C3) is linear with the coe\u000ecients de-\ntermined by the zero-order spectral function. Solving it\nwe \fnd the spin-dependent non-equilibrium distribution\nfunction generated by the dynamical spin-active inter-\nface.Spectral functions\nIn the adiabatic approximation we \fnd the spectral\nfunctions from the stationary Usadel equation\ni[(\"+i\u0000)^\u001c3;^g] +@x(D^g@x^g) = \u0001[^\u001c1;^g]\u0000[^\u0006so;^g] (C4)\nwith the boundary conditions\nD^g@x^g=iJsd[^\u001c3^\u001bm(t);\u0014g] (C5)\nUsing the normalization condition (^ gR)2= 1 we use the\nfollowing parametrization for equilibrium GF and correc-\ntions in the low-frequency adiabatic approximation\n^gR\n0= cos\u00120^\u001c3+ sin\u00120^\u001c1 (C6)\n^gR\nh= (\u0000sin\u00120^\u001c3+ cos\u00120^\u001c1)\u0012h (C7)\nThen we get the following equations for the parameters\n\u00120,\u0012h\ni(\"+i\u0000) sin\u00120+ \u0001 cos\u00120+@x\u0012D\n2@x\u00120\u0013\n= 0 (C8)\n@x\u00120(x= 0;dS) = 0 (C9)\n\u0012h\u0014\ni(\"+i\u0000) cos\u00120\u00002\n3\u001cso\u0000\u0001 sin\u00120\u0015\n+@x\u0012D\n2@x\u0012h\u0013\n= 0\n(C10)\nDN@x\u0012h(x= 0) = 2iheffsin\u00120; (C11)\n@x\u0012h(x=dS) = 0 (C12)\nSolving the nonlinear Eq.(C8,C9) together with the self-\nconsistency equation for \u0001 we obtain the zero-order spec-\ntral functions in the N/S structure. 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Virtanen,\nPhysical Review B 101, 115406 (2020)."    },    {        "title": "2307.00903v1.Magnetic_lump_motion_in_saturated_ferromagnetic_films.pdf",        "content": "Magnetic lump motion in saturated ferromagnetic films\nXin-Wei Jin,1, 2Shi-Jie Shen,2Zhan-Ying Yang,1, 3and Ji Lin2,∗\n1School of Physics, Northwest University, Xi’an 710127, China\n2Department of Physics, Zhejiang Normal University, Jinhua 321004, China\n3Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n(Dated: July 4, 2023)\nIn this paper, we study in detail the nonlinear propagation of magnetic soliton in a ferromagnetic film. The\nsample is magnetized to saturation by an external field perpendicular to film plane. A new generalized (2+1)-\ndimensional short-wave asymptotic model is derived. The bilinear-like forms of this equation are constructed\nand exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be\ngenerated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable\nand can maintain their shapes and velocities during evolution or collision. The interaction between lump and\nmagnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss\nthe nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The\nresults show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay\nexponentially during propagation. And the shock waves are generated from a lump when quenching the strength\nof inhomogeneous exchange.\nI. INTRODUCTION\nThe propagation of electromagnetic wave in ordered\nmagnetic materials, especially in a ferromagnetic medium,\nplays a vital role in faster and higher density storage fields [1–\n3]. In particular, magnetic soliton(MS), which exists in both\nferro- and antiferro-magnets, is becoming a very promising\ninformation carrier because of its particle-like behavior and\nmaneuverability [4–9]. In the past few decades, a wide range\nof soliton-type propagation phenomena has been theoretically\npredicted [10–13], and some of them have been confirmed\nexperimentally [14, 15].\nIndeed, wave propagation in ferromagnetic media is well-\nknown as a highly nonlinear problem. A complete description\nof all types of nonlinear excitations is governed by the\nMaxwell equations coupled with Landau-Lifschitz equation.\nFor this moment, let us notice that a fully nonlinear theory has\nnot been developed. But the linear theory for sufficiently small\namplitudes was established and validated experimentally [16].\nIn order to obtain results valid in nonlinear regimes, or at\nleast weakly nonlinear, one has to resort to intermediate\nmodels (by introducing a small perturbative parameter related\nto the soliton wavelength) [17]. These models include long-\nwave model [18–20], modulational asymptotic model [21],\nand short-wave model [22–25]. Both long-wave model and\nmodulational asymptotic model are mainly used to explain\nand predict the behavior of large-scale phenomena owing to\ntheir long-wave-type approximate condition [26]. However,\nthis condition is not always applicable because the scale of\nmagnetic materials and devices are getting more refined and\nmore sophisticated. Moreover, the main practical interest of\nferrites is that they propagate microwaves [27, 28]. On the\ncontrary, from the viewpoint of applied physics, the short-\nwave-type approximation is much more relevant to available\nexperiments than the former one.\nSince Kraenkel et al. first proposed the short-wave model\n[29], quite a few related nonlinear evolution equations have\nbeen derived, which belong to the Kraenkel-Manna-Merle\n(KMM) system [22, 23, 30–32]. Some significant works\n∗Corresponding author: linji@zjnu.edu.cnhave been devoted to searching and explaining different\nexcitation patterns of ferromagnetic insulators. As for (1+1)-\ndimensional KMM system, the existence of multi-valued\nwaveguide channel solutions has been verified, and the\nnonlinear interaction properties were investigated between\nthe localized waves alongside the depiction of their energy\ndensities [22]. By applying the Hirota bilinear transformation\nmethod, the one- and two-soliton solutions were constructed\nwhile studying in details the solitons scattering properties\n[23]. This system is also solvable using the inverse scattering\nmethod [25]. It is noteworthy that this system possesses\nthe loop-soliton and spike-like soliton [33, 34], and the\nmagnetic loop-soliton dynamics have been extensively studied\n[35–37]. The propagation of electromagnetic waves in\nhigher-dimensional ideal ferromagnets has also been studied,\ncorresponding to the (2+1)-dimensional KMM system [26, 31,\n38, 39]. The analytical one-line-soliton solution as well as its\ntransverse stability have been reported [26]. It has been shown\nthat these structures were stable under certain conditions.\nOn the other hand, most previous studies have only focused\non the propagation of MS in ideal ferrites, which means\nsome important properties of the magnetic material were\nneglected. The main reason is that the nonlinear wave\nequation describing the propagation of electromagnetic waves\nin non-ideal ferromagnetic materials is no longer integrable.\nHowever, the Gilbert-damping and inhomogeneous exchange\neffects are essential features in a real ferromagnetic film, and\ntheir connection with MS motion is an important issue that has\nnot been explored so far. In this paper, we aim to investigate\ntheoretically and numerically the dynamics of the MS in a\nferromagnetic film including damping and the inhomogeneous\nexchange effect. The rest of this paper is organized as follows.\nIn Section 2, we review the physical background and derive\na new (2+1)-dimensional short-wave asymptotic model in\nferromagnetic media. In Section 3, the bilinear-like form\nof the reduced system is constructed and the analytical MS\nsolutions are acquired. In Section 4, the transmission stability\nof the magnetic soliton is numerically explored. The results\nshow that an unstable MS will split to some magnetic lumps\nby a small perturbation. The motions of these lumps under\nthe influence of damping and inhomogeneous exchange are\nanalysed in detail. We end this work in Section 5 with a brief\nconclusion and perspectives.arXiv:2307.00903v1  [nlin.PS]  3 Jul 20232\nII. PHYSICAL BACKGROUND\nA. Basic equations\nThe physical system under consideration is a saturated\nmagnetized ferrite film lying in the x−yplane, as shown in\nFig. 1. Different from Ref. [32], we consider the external\nfield H∞\n0perpendicular to the film, i.e., M0= (0,0,m). So the\ntransverse drift is avoided. The typical thickness of the film\nis about 0.5mm, and the width is approximately 10mm. We\nassume the propagation distance is large enough with regard\nto the wavelength, say more than 50cm. The evolution of the\nmagnetic field Hand the magnetization density Mis governed\nby the Maxwell equations coupled with Landau-Lifschitz-\nGilbert equation, which read as\n−∇(∇·H)+∆H=1\nc2∂2\n∂t2(H+M), (1a)\n∂\n∂tM=−γµ0M×Heff+σ\nMsM×∂\n∂tM, (1b)\nwhere c=1/p\nµ0˜εis the speed of light with the scalar\npermittivity ˜εof the medium, γis the gyromagnetic ratio,\nµ0being the magnetic permeability of the vacuum, σis the\ndamping constant, and Msis the saturation magnetization. The\neffective field Heffis given by [30]\nHeff=H−βn(n·M)+α∆M. (2)\nHere αandβare the constants of the inhomogeneous\nFigure 1. Ferrite film under consideration. The sample is magnetized\nto saturation by long strong magnetic field H∞\n0applied in the\nz-direction. The x-direction of the short wave propagation is\nperpendicular to the direction of static magnetization.\nexchange and the magnet anisotropy ( β>0 corresponds to\nthe easy-plane case), respectively. For a simple tractability, the\nunit vector nof the anisotropy axis is assumed to be along the\nzaxis (i.e., n≡ez). In order to transform the above systems to\ndimensionless equation, we rescale the quantities M,H, and t\nintoµ0γM/c,µ0γH/c, and ct. Thus, the constants µ0γ/cand\ncin Eqs.(2) and (3) are replaced by 1, Msbym=µ0γMs/c,\nandσby˜σ=σ/µ0γ[30].\nB. Linear analysis\nTo study the linear regime we look at a small perturbation of\na given solution. Equations (1) are linearized about the steady\nstate:\nM0= (0,0,m),H0=µM0. (3)where µis the strength of the internal magnetic field. Before\nproceeding further we assume that the ferromagnetic materials\nhave weak damping ¯σ∼ε˜σ. The exchange interaction\nparameter αand anisotropy parameter βare of order ε2and\nε3, respectively (i.e. ¯α=ε2α,¯β=ε3β). Let us seek for\nthe plane wave perturbation solution propagating along the x-\ndirection such as\nM=M0+εmexp[i(kx+ly−ωt)],\nH=H0+εhexp[i(kx+ly−ωt)],(4)\nwhere kandlare the wave numbers in the xandydirections, ω\nis the frequency. Vectors m= (mx,my,mz)andh= (hx,hy,hz)\nare arbitrary real scalar quantities.\nSubstituting Eq. (4) into (1) and (2) in the linear limit, it is\nreduced to\n\nω20 0 ω2−l2kl 0\n0 ω20 kl ω2−k20\n0 0 ω20 0 ω2−k2−l2\n−iωmµ 0 0 −m 0\n−mµ−iω 0 m 0 0\n0 0 −iω 0 0 0\n·\nmx\nmy\nmz\nhx\nhy\nhz\n=0\nThen we obtain the following dispersion relation\nm2(µ+1)\u0002\nµ(k2+l2−ω2)−ω2\u0003\n−ω2(k2+l2−ω2) =0\n(5)\nNote that we focus on studying the short-wave approximation\nk→∞[2]. It comes k0∼ε−1through a small parameter ε≪1\nlinked to the magnitude of the wavelength. Consequently, the\nfrequency expands accordingly as\nω=ω−1ε−1+ω1ε+ω3ε3+.... (6)\nThis assumption guarantees the phase velocity ω(k)/kand\nthe group velocity ∂ω/∂kare always bounded [3]. Now,\nreplacing Eq. (6) into the dispersion relation above, we obtain\na set of equations:\n•At order of ε−4:ω−1=±k0\n•At order of ε−2:ω1=\u0002\n(µ+1)m2+l2\u0003\n/2k0\n•higher order equations which determines ω3,ω5,...\nThe direction of the wave propagation is assumed to be\nclose to the xaxis, thus yvariable gives only account of a\nslow transverse deviation[40, 41]. Therefore lis assumed\nto be very small with respect to kand we write l=l0of\norder 0 with respect to ε. The phase up to order εis thus\n(x−t)/ε+l0y−εω1t,which motivates the introduction of\nnew variables:\nζ=1\nε(x−Vt),y=y,τ=εt. (7)\nThe variable ζdescribes the shape of the wave propagating at\nspeed V; it assumes a short wavelength about 1 /ε. The slow\ntime variable τaccounts for the propagation during very long\ntime on very large distances with regard to the wavelength.\nThe transverse variable yhas an intermediate scale, as in KP-\ntype expansions [26, 41]\nC. Multiple scale approach\nIn order to derive the nonlinear model, fields MandHare\nexpanded in power series of εas\nM=M0+εM1+ε2M2+ε3M3+...,\nH=H0+εH1+ε2H2+ε3H3+....(8)3\nwhere M0,H0,M1,H1,...are functions of (ζ,y,τ).\nWe consider the boundary conditions: lim\nζ→−∞M0=\n(0,0,m),lim\nζ→−∞Mj=lim\nζ→−∞Hj=0,(j̸=0). We derive\nthe following expressions by substituting Expansions (8) into\nequation (1):\n•At order ε−2:\nM0is a constant vector M0=(0,0,m),\n•At order ε−1:\nHx\n0=0,My\n1=0,Mz\n1=0,\n•At order ε0:\nMx\n1ζ=mHy\n0,\nMx\n2ζζ=−Hx\n2ζζ−Hy\n1ζτ\nMy\n2ζζ=−Hx\n1ζy+Hx\n0ζy\nMz\n2ζζ=Hz\n2ζτ+Hz\nyy\n•At order ε1:\nMx\n2ζ=−mHy\n1\nMy\n2ζ=m¯αMx\n1ζζ+¯σM1ζx−Mx\n1Hz\n0+mHx\n1\nMz\n2ζ=Mx\n1Hy\n0\nlet us introduce some independent variables XandTdefined\nasX=−mζ/2,Y=my,T=mτ.\nAfter eliminating H2andM2, we finally obtain the (2+1)-\ndimensional KMM equation:\nCXT=−BBX+CYY,\nBXT=BCX+BYY−sBX+ρBXX,(9)\nwhere observables B,Cand constants s,ρare defined by\nC=−X−ZX\n(Hz\n0/m)dX,B=Mx\n1/2m,\ns=−¯σ/2,ρ=¯αm2/4.(10)\nThis equation is new, which describes the evolution of\nmagnetization field Mand magnetic field Hwithin a ferrite\nfilm in presence of Gilbert-damping and inhomogeneous\nexchange. The quantities H0andM1refer to the zeroth and\nfirst-order expansion coefficients of the external magnetic field\nand the magnetization, respectively. For some simplicity,\nin the next, the independent variables X,YandTwill be\nrewritten as their lower cases x,yandt, respectively.\nIII. HIROTA’S BILINEARIZATION AND SOLITON\nSOLUTIONS OF THE (2+1)-DIMENSIONAL KMM\nEQUATION\nTo explore soliton solutions for the (2+1)-dimensional\nKMM equation (9), we consider a specific dependent variable\ntransformation\nB=G\nF,C=δx−2(lnF)t−2(lnF)y, (11)\nwhere δis an arbitrary constant. Consequently, the bilinear-\nlike forms of the (2+1)-dimensional KMM equation can be\nderived as follow\nF·(DxDt+sDx−D2\ny)G·F+G·(DxDy+D2\ny)F·F=δF2G\n(12a)\n∂x\u0014G2\n2F2−(DyDt+D2\nt)F·F\nF2\u0015\n+∂y\u0014(DyDt+D2\nt)F·F\nF2\u0015\n=0\n(12b)where G,Fare all differential functions of (x,y,t)to be\ndetermined. The symbols Dx,Dtrefer to the Hirota’s operators\nwith respect to the variable x,t, respectively. In order to\nconstruct the solitary wave solutions of Eq.(6), we expand\nGandFwith respect to a formal expansion parameter as\nG=εG1+ε3G3+ε5G5+...,F=1+ε2F2+ε4F4+ε6F6+...,\nin which εis a perturbation parameter and functions Gi,Fi,(i=\n1,2,3,...)are expansion coefficients of the above series. The\none-soliton solution could be constructed by truncating the\nperturbation expansion of GandFas follow\nG=eη1,F=1+k2A2\n16δ2e2η1. (13)\nSubstituting these expressions into Eq.(9) and solving the\nbilinear system recursively, in the absence of damping,\nthe analytical one-soliton solution of the (2+1)-dimensional\nKMM equation can be transformed into\nB=2δ\nksech(η1+η0),C=δx−2δ\nk[tanh(η1+η0)+1],\n(14)\nwhere η1=kx+ly+ [(l2−kl)/2k]t,η0=ln(k/4δ),kandl\nare arbitrary real constants. It should be noted that this soliton\nsolution exists only when the damping is neglected (s=0).\nSimilar to the procedure for constructing one-soliton solution,\nthe two-soliton solution can be given by treating the truncated\nperturbation expansions of GandFas\nG=A1eξ1+A2eξ2+C12eξ1+2ξ2+C21e2ξ1+ξ2, (15a)\nF=1+B11e2ξ1+B22e2ξ2+B12eξ1+ξ2+E12e2ξ1+2ξ2,(15b)\nwhere A1,A2,k1,k2are real constants, ξi=kix+liy+\u0002\n(l2\ni+δ)/ki\u0003\nt,(i=1,2), and the remaining parameters have\nthe following forms:\nBii=A2\nik2\ni\n16δ2,B12=A1A2\n2δ2k2\n1k2\n2\nk2+,k1l2=k2l1,\nCi j=AiA2\nj\n16δ2k2\njk2\n−\nk2+,E12=A2\n1A2\n2\n256δ4k2\n1k2\n2k4\n−\nk4+,(16)\nwhere k+=k1+k2,k−=k1−k2. Parameters Ai,Aj,ki,kj\nandli,(i=1,2,j=3−i)are arbitrary real constants.\nIV . NUMERICAL INVESTIGATION OF LINE-SOLITON\nAND MAGNETIC LUMPS\nA. Unstable MS splits into lumps\nWe now turn to the stability and interactions between MSs\nin a ferromagnetic film. The initial data is a MS perturbed\nby some position-dependent Gaussian wave packets with the\nfollowing expression:\nf=bexp\"\n−\u0012x−x0\nxr\u00132\n−\u0012y\nyr\u00132#\n, (17)\nwhere b,xrandyrcorrespond to the shape of the wave packet\nandx0is related to the perturbation position.\nThe time evolution results clearly show the instability of\nthe MS. For small bi, the MS will break up and eventually4\n(a)\n (b)\n(c)\n (d)\nFigure 2. Propagation of MS perturbed by a Gaussian disturbance.\n(a) Component Hz, (b) Component Hy, (c) and (d) are enlarged views\nof the indicated areas circled in red and black, respectively. The\nparameters are chosen as A1=A2=1,δ=−1,l1=l2=0,k1=\n1,k2=2,x0=−29,b=0.1,xr=1.5,yr=2.5 in (16) and (17).\nevolve into some stable two-dimensionally localized lumps , as\ndisplayed in Figs. 2(a) and 2(b). We observe that most of the\nenergy is always propagated as a lump, even if its speed may\ndiffer from the input. Such a magnetic lump is a solitary wave\npacket that maintains its shape and speed during propagation\nor collision.\nA complete single lump of magnetic field component Hz\n(component Hy) is circled in red (black) in Fig.2. The enlarged\nviews (see Figs.2(c) and 2(d)) provide a clear picture of the\nshape and contour map of the lump. It can be found that\ncomponent Hzis a dipole-mode lump, whereas component\nHyis a standard KP-lump. We also show the vector field\nof the magnetic lump in Fig.3(a). Note that magnetic field\ncomponent Hxis zero, the magnetic field is always in the y−z\nplane, hence the lump can be regarded as a 360◦domain wall\nlocalized in xandydirections. Fig.3(b) presents the magnetic\nfield along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of component Hz,Hy, respectively.\nThe rest of this work is concerned with the propagation and\ninteraction behavior of these lumps in ferrite medium.\n(a)\n (b)\nFigure 3. (a) The vector field of the magnetic lump. (b) The magnetic\nlump along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of components Hz,Hy, respectively.B. Lump motion in ferromagnets with damping or\ninhomogeneous exchange effects\nFigure 4. Three dimensional projections of lump at t=0,HandW\nrepresent the definitions of lump height and width, respectively.\nThe evolution behavior of the magnetic lump in the ideal\nferrite is quite simple and imaginable. Each lump maintains\nits shape while it travels at a constant speed. However, in most\nof real ferromagnetic materials, we have to take the Gilbert-\ndamping into account . For instance, the dimensionless\ndamping constant sranges from 0.048 to about 0.385 in\ngarnet ferrite films. Here we are going to study the dynamics\nof magnetic lump in a damped ferrite film. The typical\nferromagnetic film under consideration is a garnet ferrite film\nwith the dimensionless damping constant s=0.1. For a\nclearer view of the change in shape of the lump, we define\nHandWas the height and width of the lump, which\nare the vertical distance between the highest point and the\nlowest point and the horizontal distance along the propagation\ndirection, respectively. All of these are summarized in Fig.4.\nThe propagation of a lump on the garnet ferrite film\nis presented in Fig.5. As shown in Fig.5(a), the lump\ntravels forward a visible distance in the damped ferrite.\nBeyond that, comparing the profiles of lump between t=0\nand t=10, we evidently observe that the lump becomes\nsmaller and narrower. Fig.5(b) shows the lump height\nand width exhibit a tendency of exponential decay. The\nsolid blue line is the exponential fitting curve to H(t),\nwith the function expression being H(t) = A0e−st. We\nconfirm the above-mentioned amplitude attenuation law is\nuniversal by simulating the motion of lump in ferrites with\nvirous damping factors. Moreover, a definite relationship\nbetween the amplitude and the localization region of solitons\nis important for the soliton excitations. We analyze different\nsizes of numerical lumps and mark the width and height of\nlumps in the phase diagram (see Fig. 5(c)). The results show\nthat for a magnetic lump excitation, its width and height meet\na linear relationship within the error range ( W/H∼0.305).\nSo the lump excitation, upon decay, retains a soliton form.\nTherefore, in this system, the Gilbert-damping plays a role of\ndissipating energy during the motion of magnetic lumps and\nit is characterized by decreasing the amplitude and width of\nlump.\nThe inhomogeneities otherwise referred to as deformities is\ninevitable in real magnetic materials, and it can be caused by\neither external fields or the presence of defects, voids and gaps\nin the material. It has already been reported that the MS may\nbe deformed by the presence of inhomogeneities, in particular5\n(a)\n (b)\n (c)\nFigure 5. Evolution of a magnetic lump in a damped ferrite film with dimensionless damping constant s=0.1. (a) Comparison picture of lump\nwave at t=0 and t=10. (b) The variation of lump height H, lump width Wand velocity V . (c) Numerical relationship between the width and\nheight of magnetic lump.\nits structure and speed [35, 42]. In this present system,\nthe inhomogeneous exchange process is unignorable when\nthe wavelength of lump is comparable to the characteristic\nexchange length.\n(a)\n (b)\n(c)\n (d)\nFigure 6. Propagation of lump with and without the inhomogeneous\ninteraction, respectively.\nWe now move to study the lump motion in the presence of\ninhomogeneous exchange effect. The initial data is the stable\nmagnetic lump shown in Fig.5. As can be observed from Fig.\n6(a) and 6(b), in ferrite without exchange interaction, the lump\nsolution propagates at a constant speed and along the previous\npath. We then consider the non-equilibrium dynamics of lump\nby performing a sudden interaction quench. The pictures\nof component Hyat dimensionless times t=2 and t=4.5\nare shown in Fig. 6(c) and 6(d). As we see, for a quench\nfrom the non-interacting to strong inhomogeneous exchange\nferrite film, the lump oscillates rapidly and diffracts alongthe propagation direction. A two-dimensional shock wave\nis generated and propagates forward. The shock wave front\ncontinues to propagate in the negative direction along x-axis.\nFinally, the energy of lump will be dissipated into numberless\ntiny waves. Accordingly, considering that the lump would be\ndestroyed by the inhomogeneous exchange process, one has to\nconsider keeping its wavelength away from the characteristic\nexchange length in the lump-based microwave applications.\nC. Some examples of excitations and interactions\nThe evolution pattern given in Fig.2 reveals that the lump\nmoves at a larger velocity than the broken MS in the\npropagation. The reason is that the velocity of soliton solution\nis proportional to the soliton amplitude. During the formation\nof the lump, the original MS will be destroyed, and most of the\nenergy is concentrated in some certain centers, which causes\nthe amplitude (and velocity) of the lump to be greater than that\nof MS. These lumps with various speeds enable us to explore\nthe interaction between lump and soliton, as well as between\ntwo lumps.\nA typical example of lump-MS collision is shown in\nFig.7(a). The MS begins to break up around at t=4.\nSubsequently, the splitting lump is going to catch up and\ncollide with the front-MS. After the collision, the front-\nMS is destroyed and broken into several lumps with various\nsizes. It is remarkable that the lump keep its localized\nform before and after the collision almost unchanged. This\nphenomenon implies such two-component lumps are natural\nresults from this nonlinear propagation equations. Further\nsimulation shows these lump structures could be generated\nby a MS with random disturbance. Fig.7(b) depicts a\ncharacteristic inelastic collision between two lumps. We\ninitially generate two adjoining lumps. They are emitted by\nMS at dimensionless time t=6.5. The merging process can\nbe performed as follows. From t=7.5 tot=9.5, two lumps\nmerge simultaneously together and give birth to a new lump\nwhose amplitude is significantly greater than the amplitude of\nprevious lumps. Obviously there is a weak attraction between\ntwo lumps which results in their fusion. In addition to the\nfusion of the two lumps, we also observed an extraordinary\npeak at a specific moment (about t=9.5), which looks like a6\n(a)\n(b)\nFigure 7. (a) Collision between lump and MS. (b) Mergence of two lumps and the formation of a second-order rogue wave-like structure.\nsecond-order rogue wave. It appears to be the result of the\ninteraction between the ripples surrounding the two lumps.\nAfter the fusion, the rouge wave-like structure disappears and\nthe dynamics of the output is determined mainly by a single\nhigh-amplitude lump.\nV . CONCLUSION\nAs a conclusion, the nonlinear propagation of MS in a\nsaturation magnetized ferromagnetic thick film is studied in\ndetail. In the starting point, we derive the (2+1)-dimensional\nKMM system that governs the evolution of short MS waves\nin a saturated ferromagnetic film. The bilinear form of the\nKMM system is constructed and the MS solutions are obtained\nanalytically.\nAfter that, numerical simulations are performed to analyse\nthe evolution behaviours of MS. A significant observation\nis that the unstable MS can be destroyed by Gaussian\nperturbation and broken into some stable magnetic lumps.\nThese lumps exhibit high stability during the propagation.\nFurthermore, some examples are given to analyse the collision\nbehaviours between lump and MS, and the interaction between\ntwo lumps. It is found the lump keeps its shape and speed in\nthe collision with MS. The results confirm that the lump is astable propagation mode in this system and, more to the point,\nthe velocity of lump can be adjusted by its amplitude. Their\nrobustness and controllability provide the possibility for future\ninformation memory and logic devices. We also study the\npropagation of such a lump in ferrites subjected to influence\nof damping and inhomogeneous exchange effects. When the\nGilbert-damping of ferrite is considered, the lumps undergo\nthe following changes: the amplitude and the speed of lump\nare decreased, and the width of lump along the propagation\ndirection is getting narrow. It would cause a strong diffraction\nof the lump if we quench the interaction strength.\nWe hope our work will invoke follow-up experimental\nstudies of lump-based microwave applications. Addition-\nally, since only one- and two-line-soliton are obtained,\nthe integrability of the (2+1)-dimensional system Kraenkel-\nManna-Merle (KMM) remains an open issue. The existence\nof the higher-dimensional evolution system as well as the\nbulk polariton solution is an intriguing avenue for future\nexploration.\nACKNOWLEDGMENT\nThis work was supported by the National Natural Science\nFoundation of China under Great Nos. 11835011; 11675146;\n11875220;.\n[1] M Daniel, V Veerakumar, and R Amuda. 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Wu1 \n1 Department of Physics and Astronomy, University of California, Irvine, California \n92697 -4575, USA  \n \nAbstract  \nEngineering Gilbert damping of ferromagnetic metal films is of great importance to \nexploit and design spintronic devices that are operated with an ultrahigh speed. Based on \nscattering theory of Gilbert damping, we extend the torque method originally  used in \nstudies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of \nferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3 d \ntransition metal ferromagnet iron, cobalt and nickel monolayers that are co ntacted by the \nprototypical topological insulator Bi 2Se3. Amazingly, we find that their Gilbert dampings \nare strongly enhanced by about one order in magnitude, compared with dampings of their \nbulks and free -standing monolayers, owing to the strong spin -orbit coupling of Bi 2Se3. \nOur work provides an attractive route to tailoring Gilbert damping of ferromagnetic \nmetallic films by putting them in contact with topological insulators.  \n  \n \n \n \n \nEmail: wur@uci.edu   \n \n \n \n \n \n 2 \n I. INTRODUCTION  \nIn ferromagnets, the time -evolution of their magnetization M can be described by the \nLandau -Lifshitz -Gilbert (LLG) equation [1-3]  \n1Meff\nSdd\ndt dt   MMM H M\n, \nwhere \n0B g  \n  is the gyromagnetic ratio, and \nMSM  is the saturation \nmagnetization. The first term describes the precession motion of magnetization M about \nthe effective magnetic field, Heff, which  includes contributions from external field, \nmagnetic anisotropy, exchange, dipole -dipole and Dzyaloshinskii -Moriya  interactions [3]. \nThe second term represents the decay of magnetization prece ssion with a dimensionless \nparameter \n , known as the Gilbert damping [4-8]. Gilbert  damping is known to be \nimportant for the performance of various spintronic devices such as hard drives, magnetic \nrandom access memories, spin filters, and magnetic sensors [3, 9, 10]. For example, \nGilbert damping in the free layer of reader head in a magnetic hard drive determines its \nresponse speed and signal -to-noise ratio [11, 12]. The bandwidth, insertion loss , and \nresponse time of a magnetic thin film microwave device also critically depend on the \nvalue of \n  in the film [13]. \n \nThe rapid developm ent of spintronic technologies calls for the ability of tuning Gilbert  \ndamping in a wide range. Several approaches have been proposed for the engineering of \nGilbert damping in ferromagnetic (FM) thin films, by using non -magnetic or rare earth \ndopants, addi ng differ ent seed layers for growth, or adjusting composition ratios in the \ncase of alloy films [9, 14-16]. In par ticular, tuning \n  via contact with other materials \nsuch as heavy metals, topological insulators (TIs), van der Waals monolayers or magnetic \ninsulators is promising as the selection of material combinations is essentially unlimited.  \nSome of these materials may have fundamentally different damping mechanism and offer \nopportunity for studies of new phenomena such as spin -orbit torque, spin -charge \nconversion, and thermal -spin-behavior [17, 18].  \n \nIn this work, we systematically investigate the effect of Bi 2Se3 (BS), a prototypical TI, on \nthe Gilbert damping of 3d transitio n metal (TM) Fe, Co and Ni monolayers (MLs) as they 3 \n are in contacted with each other. We find that the Gilbert dampings in the TM/TI \ncombinations are enhanced by about an order of magnitude than their counterparts in \nbulk Fe, Co and Ni as well as in the fr ee-standing TM MLs.  This drastic enhancement \ncan be attributed to the strong spin -orbit coupling (SOC) of the TI substrate and might \nalso be related to its topological nature . Our work introduces an appealing way to \nengineer Gilbert dampings of FM metal fi lms by using the peculiar physical properties of \nTIs.  \n \nII. COMPUTATIONAL DETAILS  \nOur density functional theory (DFT) calculations are carried out using the Vienna Ab-\ninitio  Simulation Package (VASP) at the level of the generalized gradien t approximation \n[19-22]. We treat Bi -6s6p, Se -4s4p, Fe -3d4s, Co -3d4s and Ni -3d4s as valence electrons \nand employ the projector -augmented wave pseudopotentials to d escribe core -valence \ninteractions [23, 24].  The energy cutoff of plane -wave expansion is 450 eV [22]. The BS \nsubstrate is simulated by five quintuple layers ( QLs), with an in -plane lattice constant of \naBS = 4.164 Å and a vacuum space of 13 Å between slabs along the normal axis. For the \ncomputational convenience, we put Fe, Co and Ni MLs on both sides of the BS slab. For \nthe structural optimization of the BS/TM slabs, a 6× 6× 1 Gamma -centered k -point grid is \nused, and the positions of all atoms except those of the three central BS QLs are fully \nrelaxed with a criterion that the force on each atom is less than 0.01 eV/Å. The van der \nWaals (vdW) correction in the form of the nonlocal vdW functional (optB86b -vdW) [25, \n26] is included in all calculations.  \n \nThe Gilbert dampings are determined by extending the torque method that we developed \nfor the study of magnetocrystalline anisotropy [27, 28]. To ensure the numerical \nconvergence, we use very dense Gamma -centered k -point grids and, furthermore, large \nnumbers of unoccupied bands. For example, the first Bri llouin zone of BS/Fe is sampled \nby a 37× 37× 1 Gamma -centered k -point grid, and the number of bands for the second -\nvariation step is set to 396, twice of the number (188)  of the total valence electrons. More \ncomputational details are given in Appendix A. Mag netocrystalline anisotropy energies \nare determined by computing total energies with different magnetic orientations [29].  4 \n  \nIII. TORQUE METHOD OF DETERMINING GILBERT DAMPING  \nAccording to the scattering theory of Gilbert damping [30, 31], the energy dissipation \nrate of the electroni c system with a Hamiltonian, H(t), is determined by  \n dis 2i j j i F i F j\nijE E E E Euu\n         \nHHuu\n. \nHere, EF is the Fermi level and \nu  is the deviation of  normalized magnetic moment away \nfrom its equilibrium, i.e., \n0m m u  with \n00 MsM m . On the other hand, the time \nderivative of the magnetic energy in the LLG equation is [32] \n mag 3S\neffM dEdt\n MH\n mm\n. \nBy taking \ndis magEE\n  , one obtains the Gilbert damping as:  \n4i j j i F i F j\nij SE E E EM u u\n        \nHH\n. \nNote that, to obtain Eq. (4), we use \n mu  since the eq uilibrium normalized \nmagnetization m0 is a constant. In practical numerical calculations, \nFEE  is \ntypically substituted by the Lorentzian function \n 22\n0 0.5 0.5 L          . \nThe half maximum parameter, \n1 , is adjusted to reflect different scattering rates of \nelectron -hole pairs created by the precession of magnetization M [10]. This procedure  \nhas been already used in several ab initio  calculations for Gilbert dampings of metallic \nsystems [8, 9, 32-35], where the electronic responses play the major role for energy \ndissipation .  \n \nIn this work, we focus on the primary Gilbert damping in FM metals that arises from \nSOC [10, 36-38]. There are two important effects in a uniform precession of \nmagnetization M, when SOC is taken into consideration. The first is the F ermi surface \nbreathing as M rotates, i.e., some occupied states shift to above the Fermi level and some \nunoccupied states shift to below the Fermi level. The second is the transition between \ndifferent states across the Fermi level as the precession can be viewed as a perturbation to 5 \n the system. These two effects generate electron -hole pairs near the Fermi level and their \nrelaxation through lattice scattering leads to the Gilbert damping.  \n \nNow we demonstrate how to obtain the Gilbert damping due to SOC thro ugh extending \nour previous torque method [27]. The general Hamiltonian in Eq. (4) can be replaced by \n SOC j j jr\n H l s\n [4, 27] where  the index j refers to atoms, and \njji lr and s \nare orbital and spin operators, respectively. This is in the same spirit for the determination \nof the magnetocrystal line anisotropy [27], for which  our torque meth od is recognized as a \npowerful tool in the framework of spin -density theory [27]. When m points at the \ndirection of \n , , ,x y zm m m n , the term \nls  in HSOC is written as follows:   \n22\n2211cos sin sin22\n1sin sin cos 52 2 2\n1sin cos sin2 2 2ii\nz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n\n\n\n\n  \n\n     \n   \n  \nn ls\n \nTo obtain the derivatives of H in Eq. (4), we assume that the magnitude of M is constant \nas its direction changes [36]. The processes of getting angular derivatives of H are \nstraightforward and the results are given by Eq. (A1) -(A5) in Appendix B.  \n \nIV. RESULTS AND DISCUSSION  \nIn this section, we first show that our approach of determining  Gilbert damping works \nwell for  FM metals such as 3d TM Fe, Co and Ni bulks. Following that, we demonstrate \nthe strongly enhanced Gilbert dampings of Fe, Co and Ni MLs due to the contact with BS \nand then discuss the underlying physical mechanism of these enhancements.  \n \nA. Gilbe rt dampings of 3d TM Fe, Co and Ni bulks  \nGilbert dampings of 3d TM bcc Fe, hcp Co and fcc Ni bulks calculated by means of our \nextended torque method are consistent with previous theoretical results [10]. As shown in \nFig. 1, the intraband contributions decrease whereas the interband contributions increase \nas the scattering rate \n  increases. The minimum values of  \n have the same magnitude 6 \n as those in Ref. [10] for all three metals, showing the applicability of our approach for the \ndetermination of Gilbert dampings of FM metals.  \n \n \nFigure 1 (color online) Gilbert dampings of (a) bcc Fe, (b) hcp Co and (c) fcc Ni bulks. Black \ncurves give the total Gilbert damping. Red and blue curves give the intraband and interband \ncontributi ons to the total Gilbert damping, respectively.  \n \nB. Strongly enhanced Gilbert dampings of Fe, Co and Ni MLs in contact with BS  \nWe now investigate the magnetic properties of heterostructures of BS and Fe, Co and Ni \nMLs. BS/Fe is taken as an example and its atom arrangement is shown in Fig. 2a. From \nthe spatial distribution of charge density difference \nBS+Fe-ML BS Fe-ML        in Fig. \n2b, we see that there is fairly obvious charge transfer between Fe and the topmost Se \natoms. By taking  the average of \n  in the xy plane, we find that charge transfer mainly \ntakes place near the interface (Fig.2c). Furthermore, the charge transfer induces non -\nnegligible magnetization  in the topmost QL of BS (Fig. 2b). Similar charge  transfers and \ninduced magnetization are also found in BS/Co and BS/Ni (Fig. A1 and Fig. A2 in \n7 \n Appendix C). These suggest that interfacial interactions between BS and 3 d TMs are very \nstrong.  Note that BS/Fe and BS/Co have in -plane easy axes whereas the BS/ Ni has an \nout-of-plane one.     \n \n \nFigure 2 (color online) (a) Top view of atom arrangement  in BS/Fe. (b) Charge density difference \n\n near the interface in BS/Fe. Numbers give the induced magnetic moments (in units of \nB ) in \nthe top most  QL BS. Color bar indicates the weight of negative (blue) and positive (red) charge \ndensity differences. (c) Planer -averaged charge density difference \n in BS/Fe. In (a), (b), (c), \ndark green, light gra y and red balls represent Fe, Se and Bi atoms, respectively.  \n \nFig. 3a and 3b show the \n  dependent Gilbert dampings of BS/Fe, BS/Co and BS/Ni. It is \nstriking that Gilbert dampings of BS/Fe, BS/Co and BS/Ni are enhanced by about one  or \ntwo order s in magnitude  from the counterparts of Fe, Co and Ni bulks as well as their \nfree-standing MLs, depending on the choice of scattering rate in the range from 0. 001 to \n1.0 eV.  Similar to Fe, Co and Ni bulks, the intraband contributions monotonically  \ndecrease while the interband contributions increase as the scattering rate \n  gets larger \n(Fig. A3 in Appendix D).  Note that our calculations indicate that there is no obvious \ndifference between the Gilbert dampings of BS/Fe when f ive- and six -QL BS slabs are \nused (Fig. A4 in Appendix E). This is consistent with the experimental observation that \nthe interaction  between the top and bottom topological surface states is negligible  in BS \nthicker than five QLs [39].  \n \n8 \n  \nFigure 3 (color online) Scattering rate \n  dependent Gilbert dampings of (a) Fe ML, bcc Fe bulk, \nBS/Fe and PbSe/Fe, (b) Co ML, hcp Co bulk, BS/Co, Ni ML, fcc Ni bulk and BS/Ni. (c) \nDependence of the Gilbert dampin g of BS/Fe on the scaled SOC \nBS  of BS in the range from \nzero (\nBS0 ) to full strength (\nBS1 ). Solid lines show the fitting of Gilbert damping \nBS/Fe  \nto Eq. (6). The inse t shows Gilbert damping comparisons between BS/Fe at \nBS0 , bcc Fe bulk \nand Fe ML.  \n \nAs is well -known, TIs are characterized by their strong SOC and topologically nontrivial \nsurface states. An important issue is how they affect the Gilbert damping s in BS/TM \nsystems. Using BS/Fe as an example, we artificially tune the SOC parameter \nBS of BS \nfrom zero to full strength and fit the Gilbert damping \nBS/Fe  in powers of \nBS  as \n2\nBS/Fe 2 BS BS/Fe BS 0 (6)       \n. \nAs shown in Fig. 3c, we obtain two interesting results: (I) when \nBS  is zero, the \ncalculated residual Gilbert damping \nBS/Fe BS 0    is comparable to Gilbert dampings of \nbcc Fe bulk and Fe free -standin g ML (see the inset in Fig. 3c) ; (II) Gilbert damping \nBS/Fe\n increases almost linearly with \n2\nBS  , simi lar to previous results [36]. These reveal \nthat the strong SOC of BS is crucial for the enhancement of Gilbert damping.    \n  \nTo gain insight i nto how the strong SOC of BS affects the damping of BS/Fe, we explore \nthe k-dependent contributions to Gilbert damping, \nBS/Fe . As shown in Fig. 4a , many \nbands near the Fermi  level show strong intermixing between Fe and BS orbitals (mar ked \nby black arrows ). Accordingly, these k-points have large contributions to the Gilbert \n9 \n damping  (marked by red arrows in Fig. 4b). However, if the hybridized states are far \naway from the Fermi level, they make almost zero contribution to the Gilbert damp ing. \nTherefore, we conclude that only hybridizations at or close to Fermi level have dominant \ninfluence on the Gilbert damping. This is understandable, since energy differences EF-Ei \nand EF-Ej are important in the Lorentzian functions in Eq. (4).   \n \n \nFigu re 4 (color online) (a) DFT+SOC calculated band structure of BS/Fe. Color bar indicates \nthe weights of BS (red) and Fe ML (blue). Black dashed line indicates the Fermi level. (b) k -\ndependent contributions to Gilbert damping \nBS/Fe at sc attering rate \n26meV . Inset shows \nthe first Brillouin zone and high -symmetry k -points \n , \nK and \n .  \n \nIt appears that there is no direct link between the topologic al nature of BS and the strong \nenhancement of Gilbert damping. The main contributions to Gilbert damping are not \nfrom the vicinity around the \n -point,  where the topological nature of BS manifests.  \nBesides,  BS should undergo a topol ogical phase transition from trivial to topological as \nits SOC \nBS  increases [40]. If the topological nature of BS dictates the e nhancement of \nGilbert damping, one should expect a kink in the \nBS  curve at this phase transition \npoint but this is obviously absent i n Fig. 3c.  \n \n10 \n To dig deeper into this interesting issue, we replace the topologically nontrivial BS with a \ntopologically trivial insulator PbSe, because the latter has a nearly the same SOC as the \nformer. As shown in Fig. 3a, the Gilbert damping of PbSe/Fe is noticeably smaller than \nthat of BS/Fe, although both are significantly enhanced from the values of \n  of Fe bulk \nand Fe free -standing ML. Taking the similar SOC and surface geometry between BS and \nPbSe  (Fig. A5 in Appendix F) , the large difference between the Gilbert dampings of \nBS/Fe and PbSe/Fe suggests that the topological nature of BS still has an influence on \nGilbert damping. One possibility is that the BS surface is metallic with the presence of \nthe time -reversal protected t opological surface states and hence the interfacial \nhybridization is stronger.      \n \n \nFigure 5 (color online) Comparisons between Gilbert damping \n  of BS/Fe at \n26meV  and \n(a) total DOS, (b) Fe projected DOS and (c) BS projected DOS. Red arrows and light cyan \nrectangles highlight the energy windows where Gilbert damping \n  and the total DOS and Fe \nPDOS have a strong correlation . In (a), (b) and (c), all DOS are in units of state per eV and \nFermi level EF indicated by the vertical green lines is set to be zero.  \n \nA previous study of Fe, Co and Ni bulks suggested a strong correlation between Gilbert \ndamping and total density of states (DOS) around the Fermi level [36]. To attest if this is \n11 \n applicable here, we show the total DOS and Gilbert damping \nBS/Fe  of BS/Fe as a \nfunction of the Fermi level based on the rigid band a pproximation. As shown in Fig. 5a, \nGilbert damping \nBS/Fe and the total DOS behave rather differently in most energy \nregions. From the Fe projected DOS (PDOS) and BS projected PDOS (Fig. 5b and 5c), \nwe see a better  correlation between G ilbert damping \nBS/Fe and Fe -projected DOS, \nespecially in regions highlighted by the cyan rectangles . We perceive that although the \n\n-DOS correlation might work for simple systems, it doesn’t hold when hybridiza tion and \nSOC are complicated as the effective SOC strength may vary from band to band.  \n \nV. SUMMARY  \nIn summary, we extend our previous torque method from determining magnetocrystalline \nanisotropy energies [27, 28] to calculating Gilbert da mping of FM metals and apply this  \nnew approach to Fe, Co and Ni MLs in contact with TI  BS. Remarkably, the presence of \nthe TI BS substrate causes order of magnitude enhancements in their Gilbert dampings. \nOur studies demonstrate such strong enhancement is mainly due to the strong SOC of TI \nBS substrate . The topological nature of BS may also play a role by facilitati ng the \ninterfacial hybridiz ation and leaving more states around the Fermi level . Although \nalloying with heavy elements also enhances Gilbert dampings  [32], the use of  TIs pushes \nthe enhancement into a much wide r range. Our work thus establishes an attractive way \nfor tuning the Gilbert damping of FM metallic films, especially in the ultrathin reg ime.  \n \nACKNOWLEDGMENTS  \nWe thank Prof. A. H. MacDonald and Q. Niu at University Texas, Austin, for insightful \ndiscussions. We also thank Prof. M. Z. Wu at Colorado State University and Prof. J. Shi \nat University of California, Riverside for sharing their ex perimental data before \npublication. Work was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). \nDensity functional theory calculations were performed on parallel computers at NERSC \nsupercomputer centers.   \n \n 12 \n Appendix A: Details of Gilbert damping calcula tions  \nTo compare Gilbert dampings of Fe, Co and Ni free -standing MLs with BS/Fe, BS/Co, \nand BS/Ni, we use \n33  supercells containing three atoms and set their lattice \nconstants to 4.164 Å, same as that of the BS substrate. This means that the lattice \nconstant of their primitive unit cell (containing one atom) is fixed at 2.40 Å. The relaxed \nlattice constants of Fe (2.42 Å), Co (2.35 Å) and Ni (2.36 Å) free -standing MLs are close \nto this value.   \nSystems  a (Å)  b (Å) c (Å) k-point grid            \nFe bulk  2.931  2.931  2.931  35× 35× 35  16 36 2.25 \n Co bulk  2.491  2.491  4.044  37× 37× 23  18 40 2.22 \nNi bulk  3.520  3.520  3.520  31× 31× 31  40 80 2.00 \nFe ML  4.164  4.164  -- 38× 38× 1  24 56 2.33 \nCo ML  4.164  4.164  -- 37× 37× 1  27 64 2.37    \nNi ML  4.164  4.164  -- 39× 39× 1  30 72 2.40 \nBS/Fe  4.164  4.164  -- 37× 37× 1  188 396 2.11 \nBS/Co  4.164  4.164  -- 37× 37× 1  194 408 2.10 \nBS/Ni  4.164  4.164  -- 37× 37× 1  200 432 2.16 \nPbSe/Fe  4.265  4.265  -- 37× 37× 1  174 376 2.16 \n \nTable A1. Here are details of Gilbert damping calculations of all systems that are studied \nin this work.     is abbreviated for the number of  valence electrons and     stands for the \nnumber of total bands.   is the ratio between     and    , namely,          . Note that \nfive QLs of BS are used in calculations for BS/Fe, B S/Co and BS/Ni.   \n \n \n \n \n \nAppendix B: Derivatives of SOC Hamiltonian HSOC with respect to the \nsmall deviation \nu  of magnetic moments  \nBased on the SOC Hamiltonian HSOC in Eq. (5) in the main text, derivatives of the term \nls\n against the polar angle \n  and azimuth angle \n  are    13 \n \n11sin cos cos22\n1 1 1cos sin sin A1 ,2 2 2\n1 1 1cos sin sin2 2 2ii\nnz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n  \n  \n\n\n  \n\n       \n   \n  \nls \nand \n \n  \n 22\n22110 sin sin22\n10 sin cos A2 .2 2 2\n10 cos sin2 2 2ii\nn\nii\niis i l e i l e\ns i l e i l e\ns i l e i l e\n\n\n\n\n\n\n  \n\n         \n     \n    \nls\n \nNote that magnetization M is assumed to have a constant magnitude when it precesses , so \nwe have \n0SOC SOC    H M H m . When the normalized magnetization  m points \nalong the direction of \n , , ,x y zm m m n , we have: \nsin cosxm , \nsin sinym  \nand \ncoszm . Taking \n0m m u  and the chain rule together, we obtain derivatives of \nSOC Hamiltonian HSOC with respect to the small deviation  of magnetic moments as \nfollows:   \nsincos cos A3 ,sinSOC SOC SOC SOC SOC\nx x x x x\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \ncoscos sin A4 ,sinSOC SOC SOC SOC SOC\ny y y y y\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \nand  \nu14 \n \n sin A5 .SOC SOC SOC SOC SOC\nz z z z z\nSOCu m m m m\n\n                 \nH H H H H m\nm\nH \nCombining Eq. (5) and Eq. (A1 -A6), we can easily obtain the final formulas of \nderivatives of SOC Hamiltonian HSOC of magnetization m. \n \n \n \n \nAppendix C: Charge transfers and induced magnetic moments in BS/Fe, \nBS/Co and BS/Ni  \n \nFigure A1 (color online) Planar -averaged char ge difference \nBS TM ML BS TM-ML         \n(TM = Fe, Co and Ni) of  (a) BS/Fe, (b) BS/Co and (c) BS/Ni . The atoms positions are given along \nthe z axis.  \n \n15 \n  \nFigure A2 (Color online) Charge density difference \nBS TM ML BS TM ML          (TM = Fe, \nCo and Ni) nea r the interface betwee n the TM monolayer and the top most  QL BS of (a) BS/Fe, (b) \nBS/Co and (c) BS/Ni. The color bar shows the weights of the negative (blue) and positive (red) \ncharge density differences. Numbers give the induced magnetic moments (in units  of \nB ) in the \ntopmost  QL BS. Bi and Se atoms are shown by the purple and light green balls, respectively.  \n \n \nAppendix D: Contributions of intraband and interband to the Gilbert \ndampings of BS/Fe, BS/Co and BS/Ni  \n \nFigure A3 (color  online) Calculated Gilbert dampings of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. \nBlack curves give the total damping. Red and blue curves give the intraband and interband \ncontributions, respectively.  \n16 \n Appendix E: Gilbert dampings of BS/Fe with five - and six -QLs of BS slabs  \n8  \nFigure A4 (color online). Gilbert dampings of BS/Fe with five (red) and six (black) QLs of BS \nslabs. In the calculations of the Gilbert damping of BS/Fe with six QLs of BS, we use a 39 ×39×1 \nGamma -centered k -point grid, and the number of the  total bands is 448 which is twice \nlarger than the number of the total valence electrons (216).    \n \nAppendix F: Structural c omparisons between BS/Fe and PbSe/Fe  \n \n17 \n Figure A5 (color online) (a) Top view and (c) side view of atom arrangement  in BS/Fe. (b) Top  \nview and (d) side view of atom arrangement  in PbSe/Fe. In (a) and (c), the xyz -coordinates are \nshown by the red arrows. In (b) and (d), the rectangles with blue dashed lines highlight the most \ntop QL BS in BS/Fe which is similar to the Pb and Se atom laye rs in PbSe/Fe. 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Saslow2,3\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan,\n2Department of Physics, Texas A&M University, College Stati on, Texas, 77843-4242, U.S.A.,\n3Center for Nanoscale Science and Technology, National Inst itute of\nStandards and Technology, Gaithersburg, Maryland, 20899- 6202, U.S.A.\n(Dated: June 16, 2021)\nBased on spin-dependent transport theory and thermodynami cs, we develop a generalized theory\nof the Joule heating in the presence of a spin current. Along w ith the conventional Joule heating\nconsisting of an electric current and electrochemical pote ntial, it is found that the spin current and\nspin accumulation give an additional dissipation because t he spin-dependent scatterings inside bulk\nand ferromagnetic/nonmagnetic interface lead to a change o f entropy. The theory is applied to\ninvestigate the dissipation due to pure spin-current gener ated by spin pumping across a ferromag-\nnetic/nonmagnetic/ferromagnetic multilayer. The dissip ation arises from an interface because the\nspin pumping is a transfer of both the spin angular momentum a nd the energy from the ferromagnet\nto conduction electrons near the interface. It is found that the dissipation is proportional to the\nenhancement of the Gilbert damping constant by spin pumping .\nPACS numbers: 72.25.Ba, 72.10.Bg, 85.75.-d, 72.25.Mk\nI. INTRODUCTION\nDissipation due to electron transport in a conductor\nis an important issue for both fundamental and applied\nphysics [1–5]. According to electron transport theory [6],\nthe conductivity of the electron becomes finite because\nof impurity scattering inside the conductor, which leads\nto Joule heating JeE, whereJeandEare the electric\ncurrent density and electric field, respectively. Motivated\nto reduce power consumption due to Joule heating, as\nwell as because of a fundamental interest in its quantum\nmechanical nature, the generation of a pure spin-current\nby spin pumping, spin-Seebeck effect, or spin-Hall effect\nhas been extensively investigated [7–15].\nDissipation is associated with the production of en-\ntropy. Spin-flip processes and spin-dependent scatterings\nwithin a bulk ferromagnet (F) or nonmagnet (N) and at\nan F/N interface mix the spin-up and spin-down states,\nleading to a change of the entropy. Therefore a physi-\ncal system, such as a F/N metallic multilayer, carrying a\npure spin-current, still dissipates energy even in the ab-\nsence of an electric current. A quantitative evaluation\nof the dissipation due to pure spin-current therefore is a\nfundamentally important problem.\nIn 1987, Johnson and Silsbee [1] studied the surface\nand bulk transport coefficients for spin conduction, and\nthe associated entropy production rates, without consid-\nering the rate of interface heating. More recently, Sears\nandSaslow[4]used irreversiblethermodynamicstostudy\ninterface heating due to electric current in a magnetic\nsystem, and Tulapurkar and Suzuki [5] used the Boltz-\nmann equation to investigate bulk and interface heating\nfor spin conduction. Reference [5] shows that, roughly\nspeaking, the dissipation due to spin current is propor-\ntional to the square of the spin polarization of the con-\nduction electrons, indicating that the heating associatedwith the spin current is much smaller than that due to\nthe electric current. However, these works consider only\nacollinearalignmentofthemagnetizationsinaF/Nmul-\ntilayer, so only the longitudinal components of the spin\ncurrent and spin accumulation (i.e., spin chemical poten-\ntial, proportional to the nonequilibrium spin density) ap-\npear. (Longitudinal andtransverse will be used to mean\nthat the direction of the spin polarization is collinear or\nnormal to the local magnetization.) On the other hand,\nin many physical phenomena, such as spin torque switch-\ning[16]andspinpumping[7,8], anon-collinearalignment\nof the magnetizations generally appears, in which trans-\nverse spin current and spin accumulation exist. For ex-\nample, spin pumping is a generation of the transverse\nspin current by the transfer of spin angular momentum\nfrom the ferromagnetic layer to the conduction electrons\n[7,8,17–22]. Bulk heating due to spin pumping in a mag-\nnetic wire within a domain wall (driven by m×H) has\nalso been studied [3], but was not extended to include\ninterface heating. In these works, the main contribu-\ntion to the dissipation arises from the electric current.\nThe present work develops a unified theory of dissipation\nwhich enables the simultaneous evaluation of both bulk\nand interface heating in a ferromagneticsystem, with the\nspin current having arbitrary alignment of the magneti-\nzations. Also, an evaluation of the dissipation due to a\npure spin-current is indispensable for comparison with\nexperiments that determine the rate of heating.\nThis paper develops a general theory of dissipation in\nthe presence of spin current based on the spin-dependent\ntransport theory and thermodynamics. It is found that,\nalong with the conventional Joule heating, the spin cur-\nrentIs(or its density Js) and spin accumulation µcon-\ntribute to the bulk and interfacedissipations, asshownin\nEqs. (17) and (18). We apply the theory to evaluate the\ndissipation due to a pure spin-current generated by spin2\nm1m2\nF1F2Nm1.m1×m1.\nxd2d1\n0Ispump\nIsF1→N \nIsF2→N θ\nFIG. 1: Schematic view of the F 1/N/F2ferromagnetic multi-\nlayer system. The directions of ˙m1andm1×˙m1are indicated\nby arrows.\npumping in the ferromagnetic (F 1) / nonmagnetic (N)\n/ ferromagnetic (F 2) multilayer. Spin pumping provides\nan interesting example to study the dissipation problem\nof pure spin-current. In spin pumping, electric current is\nabsent throughout the system. The electron transport is\ndescribedbyaone-dimensionalequation, andanexternal\ntemperature gradient is absent, which makes evaluation\nof the dissipation simple comparedwith the spin-Seebeck\neffectorspin-Halleffect. Itisfoundthatthedissipationis\nproportional to the enhancement of the Gilbert damping\nby spin pumping. The amount of the dissipation due to\nthe spin pumping is maximized for an orthogonal align-\nment of the two magnetizations. For the conditions we\nstudy, the maximum dissipation is estimated to be two\nto three orders of magnitude smaller than the dissipa-\ntion due to the electric current when there is spin torque\nswitching.\nThe paper is organized as follows. In Sec. II, the sys-\ntem we consider is illustrated. Section III formulates a\ntheory of dissipation of spin-polarized conduction elec-\ntrons, using diffusive spin transport theory and thermo-\ndynamics. Section IV studies the relationship between\nthe dissipation due to spin pumping and the equation de-\nveloped in the previous section. Section V quantitatively\nevaluates the dissipation due to spin pumping. Section\nVI, compares the spin pumping dissipation with the dis-\nsipation in the case of spin torque switching. Section VII\nprovides our conclusions.\nII. SPIN PUMPING IN F/N/F SYSTEM\nFigure 1 shows a schematic view of the F 1/N/F2ferro-\nmagnetic multilayer system, where m1andm2are unit\nvectors pointing along the magnetizations of the F 1and\nF2layers,respectively. Whereneeded, subscripts k= 1,2\ndenote the F klayer. The thickness of the F klayer is de-\nnoted by dk. The F 1and F 2layers lie in the regions\n−d1≤x≤0 and 0≤x≤d2, respectively. We assume\nthat the spincurrentisconservedinthe Nlayer,andthus\nconsider its zero-thickness limit because a typical valuefor the spin diffusion length of an N layer is much greater\nthan its thickness: for example, the spin diffusion length\nfor Cu is on the order of 100 nm, whereas experimental\nthicknesses are less than 5 nm [7,8,23].\nSteady precession of m1with the cone angle θcan be\nexcited by microwave radiation of the angular velocity\nωfor ferromagnetic resonance (FMR) in the F 1layer.\nThen, the F 1layer pumps the pure spin-current\nIpump\ns=/planckover2pi1\n4π/parenleftbigg\ng↑↓\nr(F1)m1×dm1\ndt+g↑↓\ni(F1)dm1\ndt/parenrightbigg\n,(1)\nwherethe realandimaginarypartsofthe mixing conduc-\ntance are denoted by g↑↓\nrandg↑↓\ni, respectively [24,25].\nThe pumped spin current creates spin accumulations in\nthe ferromagnetic ( µF) and nonmagnetic ( µN) layers,\nwhich induce backflow spin current (into N) [20,24–26],\ngiven by\nIF→N\ns=1\n4π/bracketleftbigg(1−γ2)g\n2m·(µF−µN)m\n−g↑↓\nrm×(µN×m)−g↑↓\niµN×m\n+t↑↓\nrm×(µF×m)+t↑↓\niµF×m/bracketrightBig\n.(2)\nThe total interface conductance g=g↑↑+g↓↓and\nthe spin polarization of the interface conductance γ=\n(g↑↑−g↓↓)/(g↑↑+g↓↓)aredefined fromthe interfaceresis-\ntanceofthespin- ν(ν=↑,↓)electrons rνν= (h/e2)S/gνν,\nwhereSis the crosssection area. The real and imaginary\nparts of the transmission mixing conductance at the F/N\ninterfacearedenoted by t↑↓\nr(i). The conditionthat the spin\ncurrent is conserved in the N layer can be expressed as\nIpump\ns+IF1→N\ns+IF2→N\ns=0. (3)\nIII. DISSIPATION FORMULAS\nTo obtain the dissipation due to spin pumping, it\nis necessary to investigate how the spin accumulation\nrelaxes inside the F layers and at the F/N interfaces.\nFor generality we include the terms related to the elec-\ntric current and field, although these are absent in the\nspin-pumped system. The spin accumulation in the\nferromagnetic layer relates to the distribution function\nˆF= (f0+f·σ)/2, which is a 2 ×2 matrix in spin\nspace and satisfies the Boltzmann equation [5,26–33],\nvia [34] µ=/integraltext\nεFTr[σˆF]dε,σbeing the Pauli matri-\nces. The charge and spin distributions are denoted by\nf0andf, respectively. The distributions for spin paral-\nlel,f↑= (f0+m·f)/2, or antiparallel, f↓= (f0−m·f)/2,\nto the local spin, give the longitudinal spin. On the other\nhand, the components of forthogonal to mcorrespond\nto the transverse spin. Below, we introduce the follow-\ning notations to distinguish the longitudinal (”L”) and\ntransverse (”T”) components of the spin current Isand3\nspin accumulation µ:\nIL\ns= (m·Is)m, (4)\nIT\ns=m×(Is×m), (5)\nµL= (m·µ)m, (6)\nµT=m×(µ×m), (7)\nwhereIsequals to Ipump\ns+IF1→N\nsat the F 1/N interface\nand−IF2→N\nsat the F 2/N interface, respectively. The\nspin current density is denoted as Js=Is/S.\nWe first consider the diffusive transport for the longi-\ntudinal spin [27–33]. The longitudinal spin accumulation\nrelates to the electrochemical potential ¯ µν=µ0+δµν−\neV(ν=↑,↓) viaµL= (¯µ↑−¯µ↓)m, whereµ0,δµν, and\n−eVare the chemical potential in equilibrium, its devi-\nation in nonequilibrium, and the electric potential. The\nlongitudinal electron density nν=/integraltext\nd3k/(2π)3fνand its\ncurrent density jν=/integraltext\nd3k/(2π)3vxfνsatisfy [27]\n∂nν\n∂t+∂jν\n∂x=−nν\n2τν\nsf+n−ν\n2τ−ν\nsf, (8)\nwhere the spin-flip scattering time from spin state νto\n−ν(up to down or down to up) is denoted by τν\nsf. The\ncharge density ne=−e(n↑+n↓) and electric current\ndensityJe=−e(j↑+j↓) satisfy the conservation law,\n∂ne/∂t+∂Je/∂x= 0. The electron density nνis related\ntoδµνvianν≃ Nνδµν, whereNνis the density of states\nof the spin- νelectron at the Fermi level. In the diffusive\nmetal,jνcan be expressed as\njν=−σν\ne2∂¯µν\n∂x, (9)\nwheretheconductivityofthespin- νelectronσνrelatesto\nthe diffusion constant Dνand the density of state Nνvia\nthe Einstein law σν=e2NνDν. Detailed balance [35],\nN↑/τ↑\nsf=N↓/τ↓\nsf, is satisfied in the steady state. The\nspin polarizations of the conductivity and the diffusion\nconstantare denoted by β= (σ↑−σ↓)/(σ↑+σ↓) andβ′=\n(D↑−D↓)/(D↑+D↓). FromEq. (8), thelongitudinalspin\naccumulation in the steady state satisfies the diffusion\nequation [27]\n∂2\n∂x2µL=1\nλ2\nsd(L)µL, (10)\nwhereλsd(L)is the longitudinal spin diffusion length de-\nfined as 1 /λ2\nsd(L)= [1/(D↑τ↑\nsf) + 1/(D↓τ↓\nsf)]/2. The lon-\ngitudinal spin current density can be expressed as\nJL\ns=−/planckover2pi1\n2e2∂\n∂x(σ↑¯µ↑−σ↓¯µ↓)m. (11)\nThe issue of whether transport of the transverse spin\nin the ferromagnet is ballistic or diffusive has been dis-\ncussed in [16,25,36] and [29–32]. These two theories are\nsupported by different experiments [26,37–39], and thevalidity of each theory is still controversial. The present\nwork considers the case of diffusive transport for gen-\nerality. Ballistic transport corresponds to the limit of\nλJ,t↑↓\nr(i)→0, where λJis the spin coherence length in-\ntroduced below. In the steady state, the transverse spin\naccumulation µT=µ−µLobeys [26,29]\n∂2\n∂x2µT=1\nλ2\nJµT×m+1\nλ2\nsd(T)µT,(12)\nwhere the first term on the right-hand-side describes the\nprecession of the spin accumulation around the magne-\ntization due to the exchange coupling. The exchange\ncoupling constant Jsdis in relation to the spin coher-\nence length λJviaλJ=/radicalbig\n/planckover2pi1(D↑+D↓)/(2Jsd) [28–33].\nThe spin diffusion length of the transverse spin is λsd(T)\n[29]. The transverse spin current density is related to the\ntransverse spin accumulation via [26,29]\nJT\ns=−/planckover2pi1σ↑↓\n2e2∂\n∂xµT, (13)\nwhereσ↑↓=e2[(N↑+N↓)/2][(D↑+D↓)/2]. The so-\nlutions of the transverse spin accumulation and cur-\nrent are linear combinations of e±x/ℓande±x/ℓ∗with\n1/ℓ=/radicalBig\n(1/λ2\nsd(T))−(i/λ2\nJ).\nIn the nonmagnetic layer, the distinction between the\nlongitudinal and transverse spin is unnecessary. In fact,\nin the limit of zero-spin polarization ( β=β′= 0) and in\nthe absenceofthe exchangecouplingbetweenthe magne-\ntization and electrons’ spin ( Jsd= 0), as for the nonmag-\nnet, Eqs. (10) and (12), or Eqs. (11) and (13), become\nidentical.\nThe relation between the spin accumulation and dis-\nsipation is as follows. The heat density of the longitu-\ndinal spin- νelectrons dqνrelates to the energy density\nuν=/integraltext\nd3k/(2π)3εfν, chemical potential µν=µ0+δµν,\nand the electron density nνvia [40,41]\ndqν=duν−µνdnν. (14)\nThe energydensity uL=u↑+u↓for the longitudinal spin\nsatisfies [6]\n∂uL\n∂t+∂jL\nu\n∂x=JeE, (15)\nwherejL\nu=ju,↑+ju,↓, andju,ν=/integraltext\nd3k/(2π)3εvxfνis\nthe energy current density [6]. Here, the term JeEis\nthe Joule heating due to the electric current. On the\nother hand, the energy current of the transverse spin jT\nu\nsatisfies∂jT\nu/∂x= 0 in the steady state, where the right-\nhand-side is zero because there is no source of the trans-\nverse spin inside the F and N layers. We introduce the\nheat current density by [34]\njq=jL\nu−/summationdisplay\nν=↑,↓µνjν+jT\nu−µT·JT\ns\n/planckover2pi1.(16)4\nIn steady state, the heat current is related to the dis-\nsipation via [42] ∂QV/∂t=T[∂(jq/T)/∂x], where the\ntemperature Tis assumed to be spatially uniform in the\nfollowing calculations. The subscript ” V” is used to em-\nphasize that this is the dissipation per unit volume per\nunit time. Then, ∂QV/∂tis\n∂QV\n∂t=Je\ne∂¯µ\n∂x−∂\n∂xJs\n/planckover2pi1·µ, (17)\nwhere ¯µ= (¯µ↑+ ¯µ↓)/2 is the electrochemical potential.\nThe interface resistance also gives the dissipation, where\nthe dissipation per unit area per unit time is\n∂QA\n∂t=Je\neδ¯µ−Js\n/planckover2pi1·δµ, (18)\nwhereδ¯µandδµarethedifferencesof ¯ µandµattheF/N\ninterface. The subscript ” A” is used to emphasize that\nthis is the dissipation per unit area per unit time. Equa-\ntions(17)and(18)aregeneralizedJouleheatingformulas\nin the presence of spin current, and the main results in\nthis section. The total spin current Jsand spin accu-\nmulation µinclude both the longitudinal and transverse\ncomponents, whereas only the longitudinal components\nappeared in the previous work [5]. The amount of the\ndissipation can be evaluated by substituting the solution\nof the diffusion equation of the spin accumulation into\nEqs. (17) and (18) with accurate boundary conditions\nprovided by Eqs. (1) and (2). We call Eqs. (17) and (18)\nthe bulk and interface dissipations, respectively.\nIV. DISSIPATION DUE TO SPIN PUMPING\nInspin pumping, transversespinangularmomentum is\nsteadily transferred from the magnetic system (F 1layer)\nto the conduction electrons near the F 1/N interface. The\nnetspinangularmomentum, ds= [Ipump\ns+m1×(IF1→N\ns×\nm1)]dt, transferred from the ferromagnet should over-\ncome the potential difference µN−µF1to be pumped\nsteadily from the F 1/N interface to the N layer during\nthe time dt. This means that not only the spin angu-\nlar momentum but also the energy is transferred from\nthe F1layer to the conduction electrons. The trans-\nferred energy per unit area per unit time is given by\n(µN−µF1)·(ds/dt)/(/planckover2pi1S). In terms of the spin cur-\nrent and spin accumulation, this transferred energy is\nexpressed as\n∂QSP\nA\n∂t=1\n/planckover2pi1S/bracketleftbig\nIpump\ns+m1×/parenleftbig\nIF1→N\ns×m1/parenrightbig/bracketrightbig\n·[µN(x= 0)−µF1(x= 0)].(19)\nComparing Eq. (19) with Eq. (18), we find the relation\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF1/N=−∂QSP\nA\n∂t, (20)where (∂QA/∂t)T\nF1/Nis defined by\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF1/N=/parenleftbigg∂QA\n∂t/parenrightbigg\nF1/N−/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N.(21)\nHere, (∂QA/∂t)F1/Nis the F 1/N interface dissipation de-\nfined by Eq. (18), whereas\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N=−1\n/planckover2pi1S/parenleftbig\nm1·IF1→N\ns/parenrightbig\nm1\n·[µN(x= 0)−µF1(x= 0)].(22)\nBecause Eq. (22) is defined by the longitudinal compo-\nnents of the spin current and spin accumulation in Eq.\n(18), we call this quantity the longitudinal part of the\nF1/N interface dissipation. On the other hand, Eq. (21)\nis defined by the transverse components of the spin cur-\nrent and spin accumulation at the F 1/N interface. More-\nover, using Eqs. (17), (18) and (21), Eq. (19) can be\nrewritten as\n∂QSP\nA\n∂t=/parenleftbigg∂QA\n∂t/parenrightbigg\nF2/N+/integraldisplayd2\n0dx/parenleftbigg∂QV\n∂t/parenrightbigg\nF2\n+/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N+/integraldisplay0\n−d1dx/parenleftbigg∂QV\n∂t/parenrightbigg\nF1,(23)\nwhere the F 2/N interface dissipation, ( ∂QA/∂t)F2/N\nin Eq. (23), and the F 1and F 2bulk dissipations,\n(∂QV/∂t)F1and (∂QV/∂t)F2, are defined from Eqs. (17)\nand (18). As discussed below, Eq. (23) describes the\nenergy dissipation process carried by the spin current.\nTherefore, we define Eq. (23), or equivalently, Eq. (19),\nthe dissipation due to spin pumping.\nWith the help of Figs. 2 (a) and 2 (b) we now discuss\nthe physical interpretation of Eq. (23), which schemati-\ncallyshowtheflowsofspinangularmomentumandofen-\nergy. In spin pumping one usually focuses attention only\non the flow of spin angular momentum, i.e., spin current,\nbut because we are also interested in energy dissipation\nwealsoshowenergyflow. Whenthepumped angularmo-\nmentum reachesthe F 2/Ninterface, partofit isabsorbed\nin the F 2layer, and is depolarized by scattering at the\nF2/N interface and by spin flip and spin diffusion within\nthe F2layer. The remaining part returns to the F 1/N in-\nterface, which we call back flow. The back flow to the F 1\nlayer is relaxed by scattering at the F 1/N interface and\nby spin flip and spin diffusion within the F 1layer, where\nthe transverse component of the back flow at the F 1/N\ninterfacerenormalizesthe pumped spin current. In terms\nof the energy flow shown in Fig. 2 (b), spin absorption\nat the F 2/N interface leads to the interface dissipation\n(∂QA/∂t)F2/Nand bulk dissipation ( ∂QV/∂t)F2due to\nspin depolarization. The back flow at the F 1layer also\ngives the interface dissipation ( ∂QA/∂t)L\nF1/Nand bulk\ndissipation ( ∂QV/∂t)F1. The total dissipation is the sum\nof these dissipations, as indicated by Eq. (23). In other\nwords, the transferred energy from the F 1layer to the5\nT\nLpump\nabsorbed\nback(a)\nF1 F2m1 m2\npump\nback(b)\nF1 F2(∂ ̦V/∂t)F2\n(∂ ̦V/∂t)F1(∂ ̦A/∂t)F2/N \n(∂ ̦A/∂t)F1/N L∂̦ASP /∂t \nFIG. 2: Schematic views of the flows of (a) angular momen-\ntum and (b) energy from the microwave to the ferromagnetic\nmultilayer, in which ”L” and ”T” define the longitudinal and\ntransverse components with respect to m1.\nconduction electrons at the F 1/N interface is not local-\nized, and is dissipated throughout the system. Then,\nEq. (23), or equivalently, Eq. (19), can be regarded as\nthe dissipation due to spin pumping. Also, Eq. (21) is\nregarded as the energy transfer from the F 1layer to the\nconduction electrons near the F 1/N interface. Appendix\nA shows that all terms on the right-hand side of Eq. (23)\nare positive, thus guaranteeingthe second law of thermo-\ndynamics.\nTo conclude this section, it is of interest to compare\nEq. (19) with the dissipation due to electric current.\nLet us assume that an electric current is flowing through\na multilayer, driven by a voltage difference across two\nelectrodes. The total dissipation per unit area per unit\ntime is obtained from Eqs. (17) and (18) as [5]\n∂QEC\nA\n∂t=Je\ne[¯µ(∞)−¯µ(−∞)], (24)\nwhere[¯µ(∞)−¯µ(−∞)]/eisthevoltagedifferencebetween\nthe electrodes. Comparing Eq. (19) with (24), we notice\nthat the net transverse spin current and the difference in\nthe spin accumulation at the F 1/N interface correspond\nto the electric current and applied voltage, respectively,\nand that in spin pumping the F 1/N interface plays the\nrole of the electrode, This is because the angular momen-\ntum and the energy transferred from the magnetization\nof the F 1layer to the conduction electron are pumped\nfrom this interface to the multilayer.V. EVALUATION OF DISSIPATION\nIn this section, we quantitatively evaluate the dissipa-\ntion due to spin pumping, Eq. (19). Substituting the\nsolutions of Eqs. (10) and (12) into Eq. (2), the to-\ntal spin currents at the F 1/N and F 2/N interfaces are,\nrespectively, expressed as\nIpump\ns+IF1→N\ns=/planckover2pi1\n4π/parenleftbigg\n˜g↑↓\nr(F1)m1×dm1\ndt+ ˜g↑↓\ni(F1)dm1\ndt/parenrightbigg\n−1\n4π/bracketleftBig\n˜g∗\nF1(m1·µN)m1+ ˜g↑↓\nr(F1)m1×(µN×m1)\n+˜g↑↓\ni(F1)µN×m1/bracketrightBig\n,\n(25)\nIF2→N\ns=−1\n4π/bracketleftBig\n˜g∗\nF2(m2·µN)m2+ ˜g↑↓\nr(F2)m2×(µN×m2)\n+˜g↑↓\ni(F2)µN×m2/bracketrightBig\n.\n(26)\nThe renormalized conductances, ˜ g∗and ˜g↑↓\nr,i, are defined\nby the following ways:\n1\n˜g∗=2\n(1−γ2)g+1\ngsdtanh(d/λsd(L)),(27)\n/parenleftbigg˜g↑↓\nr\n˜g↑↓\ni/parenrightbigg\n=1\nK2\n1+K2\n2/parenleftbigg\nK1K2\n−K2K1/parenrightbigg/parenleftbiggg↑↓\nr\ng↑↓\ni/parenrightbigg\n,(28)\nwheregsd=h(1−β2)S/(2e2ρλsd(L)), andρ= 1/(σ↑+σ↓)\nis the resistivity. The terms K1andK2are defined as\nK1= 1+t↑↓\nrRe/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n+t↑↓\niIm/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n,\n(29)\nK2=t↑↓\niRe/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n−t↑↓\nrIm/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n,\n(30)\nwheregt=hSσ↑↓/(e2ℓ). In the ballistic transport limit\nfor the transverse spin, ˜ g↑↓equals to g↑↓. Then, we ex-\npandµNasµN=/planckover2pi1(ωasinθm1+b˙m1+cm1×˙m1), where\ny=δy/∆ (y=a,b,c) are dimensionless coefficients de-\ntermined by Eq. (3) with Eqs. (25) and (26). In the\nlimit ofg↑↓\nr≫g↑↓\ni[25],δb= 0, and ∆, δa, andδcare\ngiven by\n∆ =/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)/parenrightBig/bracketleftBig/parenleftBig\n˜g∗\nF1+ ˜g∗\nF2cos2θ+˜g↑↓\nr(F2)sin2θ/parenrightBig\n×/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)cos2θ+ ˜g∗\nF2sin2θ/parenrightBig\n−/parenleftBig\n˜g↑↓\nr(F2)−˜g∗\nF2/parenrightBig2\nsin2θcos2θ/bracketrightbigg\n,\n(31)6\nδa= ˜g↑↓\nr(F1)/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)/parenrightBig/parenleftBig\n˜g↑↓\nr(F2)−˜g∗\nF2/parenrightBig\nsinθcosθ,\n(32)\nδc=˜g↑↓\nr(F1)/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)/parenrightBig\n×/parenleftBig\n˜g∗\nF1+ ˜g∗\nF2cos2θ+ ˜g↑↓\nr(F2)sin2θ/parenrightBig\n.(33)\nEquation (19) in the limit of g↑↓\nr≫g↑↓\niis then given\nby\n∂QSP\nA\n∂t=/planckover2pi1ω2sin2θ˜g↑↓\nr(F1)(1−c)\n4πS\n×/braceleftbigg\nc+ ˜g↑↓\nr(F1)(1−c)Re/bracketleftbigg1\ngttanh(d1/ℓ)/bracketrightbigg/bracerightbigg\n.\n(34)\nIn the ballistic transport limit of the transverse spin, Eq.\n(34) is simplified to /planckover2pi1ω2g↑↓\nr(F1)(1−c)c/(4πS). We empha-\nsize that Eq. (34) is proportional to the enhancement of\nthe Gilbert damping by spin pumping [20,26]:\nα′=γ0/planckover2pi1˜g↑↓\nr(F1)(1−c)\n4πMSd 1, (35)\nwhereγ0is the gyromagnetic ratio. Here, α′is derived\nin the following way. According to the conservation law\nof the total angular momentum, the pumped spin from\nthe F1/N interface per unit time, ds/dt, should equal to\nthe time change of the magnetization in the F 1layer,\ni.e., a torque dm1/dt= [(gµB)/(/planckover2pi1MSd)]ds/dtacts on\nm1, whereM/(gµB) is the number of the magnetic mo-\nments in the F 1layer, and the Land´ e g-factor satisfies\ngµB=γ0/planckover2pi1. This torque, [( gµB)/(/planckover2pi1MSd)]ds/dt, with\nds/dt=Ipump\ns+m1×(IF1→N\ns×m1), can be expressed as\nα′m1×(dm1/dt). Then, α′is identified as the enhance-\nment of the Gilbert damping constant due to the spin\npumping. The present result indicating that the dissi-\npation is proportional to α′represents that the pumped\nspin current at the F 1/N interface carries not only the\nangular momentum but also the energy from the F 1to\nN layer.\nWe quantitatively evaluate Eq. (34) by using parame-\nters taken from experiments for the NiFe/Cu multilayer\nwith the assumption β=β′[23,26,29,43]; ( h/e2)S/[(1−\nγ2)g] = 0.54kΩnm2,γ= 0.7,g↑↓\nr/S= 15nm−2,g↑↓\ni/S=\n1 nm−2,t↑↓\nr/S=t↑↓\ni/S= 4 nm−2,ρ= 241 Ωnm,\nβ= 0.73,λsd(L)= 5.5 nm,λsd(T)=λsd(L)//radicalbig\n1−β2,\nλJ= 2.8 nm,d= 5 nm, γ0= 1.8467×1011rad/(T s),\nM= 605×103A/m, and ω= 2π×9.4×109rad/s, re-\nspectively, where the parameters of the F 1and F2layers\nare assumed to be identical, for simplicity. In Fig. 3 (a),\nwe show the dissipation due to spin pumping, Eq. (34),\nfor an arbitrary cone angle θ. The damping α′, Eq. (35),\nis also shown in Fig. 3 (b). The cone angle θin typi-\ncal FMR experiments [7,8] is small. However, the spin\npumping affects not only the FMR experiment but alsocone angle of magnetization m 1, θ (deg)0 30 60 90 120 150 180 dissipation (fJ/nm 2s) \n080 \n20 \ncone angle of magnetization m 1, θ (deg)0 30 60 90 120 150 18000.0010.0020.004\n0.003damping,  α \n‘40 60 (a)\n(b)\nFIG. 3: Dependencies of (a) the dissipation due to pure spin-\ncurrent, Eq. (19), and (b) the damping, α′, Eq. (35), on the\ncone angle θ.\nspin torque switching [37], in which θvaries from 0◦to\n180◦. Therefore, we show the dissipation and damping\nfor the whole range of θin Fig. 3.\nThe dissipation is zero for θ= 0◦and 180◦because\ndm1/dt=0at these angles. The maximum dissipation\nis about 60 fJ/(nm2s). To understand how large this\ndissipation is, we compare this value with the dissipation\ndue to spin torque switching current in the same system;\nwe discuss this in the next section.\nTo conclude this section, we briefly mention that the\ndissipation due to spin pumping can be evaluated not\nonly from Eq. (19) but also from Eq. (23). Appendix B\ngives explicit forms for each term on the right-hand side\nofEq. (23), from whichthe dissipationcan be calculated.\nVI. COMPARISON WITH SPIN TORQUE\nSWITCHING\nSpinpumpingoccursnotonlyinFMRexperimentsbut\nalso in spin torque switching experiments. An important\nissue in the spin torque switching problem is the reduc-\ntion of power consumption due to heating [44]. Whereas\nheating has usually meant the dissipation due to electric\ncurrent, the results of the previous section indicate that\nspin pumping also contributes to the dissipation. Thus\nit is of interest to quantitatively evaluate the dissipation\ndue to the electric current, and compare it with that due7\nto spin pumping studied in the previous section, which\nwill clarify the ratio of the contribution of spin pumping\nto heating in the spin torque switching experiment.\nWe assume that an electric current Iis injected from\nthe F2layer to the F 1layer. Then, a term\nIFk→N\ns(e)=/planckover2pi1γ\n2eIFk→Nmk, (36)\nshould be added to Eq. (2), which represents a spin cur-\nrent due to the electric current [25]. The current IFk→N\nis the electric current which flows from the F klayer to\nthe N layer, meaning that IF1→N=−IF2→N=−I. As\nin the system studied in the previous section, we assume\nthat the spin current is zero at both ends of the ferro-\nmagnet. Taking into account Eq. (36), Eqs. (25) and\n(26) are replaced by\nIpump\ns+IF1→N\ns=/planckover2pi1\n4π˜g↑↓\nrm1×dm1\ndt\n−1\n4π/bracketleftbigg\n˜g∗(m1·µN)m1+h˜g∗I\n˜geem1+˜g↑↓\nrm1×(µN×m1)/bracketrightbigg\n,\n(37)\nIF2→N\ns=−1\n4π/bracketleftbigg\n˜g∗(m2·µN)m2−h˜g∗I\n˜geem2\n+˜g↑↓\nrm2×(µN×m2)/bracketrightbig\n,(38)\nwhere, as done in the previous section, we assume that\nthe material parameters of two ferromagnets are iden-\ntical, and thus, omit subscripts ”F k” from the conduc-\ntances, for simplicity. We also assume that g↑↓\nr≫g↑↓\ni. A\nnew conductance ˜ geis defined as\n1\n˜ge=2γ\n(1−γ2)g+β\ngsdtanh/parenleftbiggd\n2λsd(L)/parenrightbigg\n.(39)\nA characteristic current of the spin torque switching\nis the critical current of the magnetization dynamics Ic,\nwhich can be defined as the current canceling the Gilbert\ndamping torque of the F 1layer at the equilibrium state\n[38]. The equilibrium state in the present study corre-\nsponds to θ= 0◦. In this limit ( θ→0), Eq. (35) is\nreplaced by\nα′=γ0/planckover2pi1˜g↑↓\nr\n4πMSd 1/parenleftbigg1\n2−π˜g∗I\neω˜g↑↓\nr˜ge/parenrightbigg\n. (40)\nWe assume that the Gilbert damping purely comes from\nthe spin pumping. Then, the critical current is defined\nas the current satisfying α′= 0; i.e.,\nIc=eω˜g↑↓\nr˜ge\n2π˜g∗. (41)\nUsing the same parameter values as in the previous sec-\ntion, the critical current density Jc=Ic/Sis estimated\nas 6.3×106A/cm2. This value is about the same or-\nder of an experimentally observed value [45] ( ∼6×106A/cm2on average) of the critical current having a mag-\nnetic anisotropy field HK, whose magnitude (1-3 kOe) is\nabout the same order of the parameter value, ω/γ0≃3.2\nkOe, used here. The dissipation due to this electric\ncurrent based on the conventional Joule heating for-\nmula,∂QEC\nA/∂t=/summationtext\nk[ρJ2\ncdk+rFk/NJ2\nc], is evaluated\nas 11.8×103fJ/(nm2s), where rF/N= (h/e2)S/gis the\nF/N interface resistance. This value of the dissipation is\ntwo to three orders of magnitude larger than the dissi-\npation due to the spin pumping studied in the previous\nsection.\nWe briefly investigate the origins of a large differ-\nence between the dissipations due to the spin and elec-\ntric currents. Let us assume that the bulk and inter-\nface spin polarizations ( βandγ) are identical, and that\nthe thickness of the ferromagnetic layer is much larger\nthan the spin diffusion length ( d≫λsd(L)), for sim-\nplicity, from which the critical current is simplified as\nIc=eω˜g↑↓\nr/(2πβ). Then the ratio between the dissi-\npations due to spin pumping and electric current be-\ncomes (∂QSP\nA/∂t)/(∂QEC\nA/∂t)∼β2/planckover2pi1/[e2(˜g↑↓\nr/S)(ρd+r)].\nThe square of the spin polarization, β2, is on the or-\nder of 10−1. Also, the orders of [( h/e2)S/(˜g↑↓\nrr)] and\nr/ρdare 1 and 0 .1, respectively. Then, the ratio\n(∂QSP\nA/∂t)/(∂QEC\nA/∂t) is roughly 10−2, which is roughly\nconsistent with the above evaluation. This consideration\nimplies that a largedissipation due to the electric current\ncomes from the smallness of the spin polarization. Also,\na large bulk resistivity ( ρ), in addition to the interface\nresistance ( r), also contributes to the large dissipation\ndue to the electric current, whereas only the interface\nresistance contributes to the spin pumping dissipation\nbecause spin pumping is an interface effect.\nToconclude this section, wemention that the total dis-\nsipation in the FMR consists of that due to spin pump-\ning, Eq. (34), and that due to the intrinsic damping in\nthe F1layer. One can consider the possibility that the\ntotal dissipation in the FMR might become comparable\nto or exceed the dissipation due to the electric current\n(calculated above) when the dissipation due to intrinsic\nmagnetic damping is included, despite the fact the dissi-\npation due to spin pumping is small. However, we found\nthat the intrinsic damping constant α0should be at least\nontheorderof0 .1−1tomakethedissipationinthe FMR\ncomparable with that due to the electric current; see Ap-\npendix C. On the other hand, the experimental value of\nthe intrinsic Gilbert damping constant is on the order of\n0.001−0.01 [46]. Therefore, the dissipation in the FMR\nis still much smaller than that due to the electric current\neven after the dissipation due to the intrinsic damping\nis taken into account. The energy supplied by the mi-\ncrowave to the F 1layer is divided into the power to sus-\ntain the magnetization precessionand that transferredto\nthe conduction electrons near the F 1/N interface, where\ntheir ratio is roughly α0:α′. The former ( ∝α0) is dissi-\npatedbythebulkmagneticdissipationwhereasthe latter\n(∝α′) is dissipated by the spin-flip processes and spin-\ndependent scatterings within bulk and at the interface,8\nas shown by Eq. (23).\nVII. CONCLUSION\nThe dissipation and heating due to a pure spin-current\ngenerated by spin pumping in a ferromagnetic/ nonmag-\nnetic /ferromagneticmultilayerwasquantitativelyinves-\ntigated. Usingspin-dependenttransporttheoryandther-\nmodynamics we generalized the Joule heating formula in\nthe presence of spin current flowing in a ferromagnetic\nmultilayer. The bulk and interface dissipation formulas\nare given by Eqs. (17) and (18), respectively. For spin\npumping, the transferred energy from the ferromagnet to\nthe conduction electrons is not localized at the interface,\nand is dissipated throughout the system by the flow of a\npure spin-current, as shown by Eq. (23). The dissipation\ndue to the spin pumping, Eq. (34), is proportional to the\nenhancement of the Gilbert damping by spin pumping,\nEq. (35). Usingtypicalvaluesofparametersinametallic\nmultilayer system, the amount of the dissipation at max-\nimum is estimated to be twoto three ordersof magnitude\nsmaller than the dissipation due to the electric current\nfor spin torque switching.\nAcknowledgement\nThe authors would like to acknowledge M. D. Stiles, P.\nM. Haney, G. Khalsa, R. Jansen, T. Yorozu, H. Maehara,\nA. Emura, T. Nozaki, H. Imamura, S. Tsunegi, H. Kub-\nota, S. Yuasa, and Y. Utsumi. This work was supported\nby JSPS KAKENHI Grant-in-Aid for Young Scientists\n(B) 25790044.\nAppendix A: Non-negativity of bulk and interface\ndissipations\nIn this Appendix, we prove that all terms on the right-\nhand side of Eq. (23) are positive, which guarantees the\nsecond law of thermodynamics; i.e., the dissipation, or\nrate of the entropy production, is positive [41]. Here, we\nomit the subscript “F k” (k= 1,2) from conductances,\nfor simplicity.\nFirst, we prove the non-negativity of the longitudinal\nand transverse parts of the bulk dissipation. The longi-\ntudinal part of Eq. (17) can be rewritten as\n/parenleftbigg∂QV\n∂t/parenrightbiggL\n=Je\ne∂¯µ\n∂x−∂\n∂xJL\ns\n/planckover2pi1·µL\n=−/summationdisplay\nν=↑,↓jν∂¯µν\n∂x−(¯µ↑−¯µ↓)\n2∂\n∂x(j↑−j↓)\n=/summationdisplay\nν=↑,↓e2\nσν(jν)2+(1−β2)\n4e2ρλ2\nsd(L)(¯µ↑−¯µ↓)2,\n(A1)which is clearly positive. Here, we use the relation ∂(j↑−\nj↓)/∂x=−(1−β2)(¯µ↑−¯µ↓)/(2e2ρλ2\nsd(L)). Also, we can\nconfirm from Eqs. (12) and (13) that the transversepart,\n/parenleftbigg∂QV\n∂t/parenrightbiggT\n=−∂\n∂xJT\ns\n/planckover2pi1·µT\n=2e2\n/planckover2pi12σ↑↓/parenleftbig\nJT\ns/parenrightbig2+σ↑↓\n2e2λ2\nsd(T)/parenleftbig\nµT/parenrightbig2,(A2)\nis positive. Therefore, the bulk dissipation is positive at\nanyx.\nNext, let us prove the non-negativity of the interface\ndissipation by using the solutions of the spin current and\nspin accumulation (see also Appendix B). The longitudi-\nnal part of the F 1/N interface dissipation can be written\nas\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N=˜g∗\n4π/planckover2pi1S/bracketleftbigg\n1−˜g∗\ngsdtanh(d1/λsd(L))/bracketrightbigg\n(m1·µN)2.\n(A3)\nAccording to Eq. (27), 1 −˜g∗/[gsdtanh(d1/λsd(L))] is\nlarger than zero. Therefore, the longitudinal part of the\nF1/N interface dissipation is positive. The longitudinal\npart of the F 2/N interface dissipation,\n/parenleftbigg∂QA\n∂t/parenrightbigg\nF2/N=JF2→N\ns\n/planckover2pi1·(µF2−µN),(A4)\nis positive because of the same reason. The transverse\npart of the F 2/N interface dissipation,\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF2/N=˜g↑↓\nr\n4π/planckover2pi1S/braceleftbigg\n1−˜g↑↓\nrRe/bracketleftbigg1\ngttanh(d2/ℓ)/bracketrightbigg/bracerightbigg\n×/bracketleftBig\nµ2\nN−(m2·µN)2/bracketrightBig\n,\n(A5)\nis also positive due to similar reasons, where we use ap-\nproximation ˜ g↑↓\nr≫˜g↑↓\niused in Sec. V for simplicity.\nAppendix B: Theoretical formulas for bulk and\ninterface dissipation\nIn this Appendix, we discuss how to calculate the dis-\nsipation due to spin pumping from Eq. (23). To this\nend, we first show the solutions for the spin current and\nspin accumulation in the F 1and F2layers because each\nterm on the right-hand-side of Eq. (23) consists of spin\ncurrent and spin accumulation, as shown in Eqs. (17)\nand (18). The general solution for the spin current and\nspin accumulation are summarized in our previous work\n[47]. Here, we use these solutions, and express the spin\ncurrentand spin accumulation in terms ofthe coefficients\naandcofµNdefined in Sec. V with the assumptions\n˜g↑↓\nr≫˜g↑↓\ni.\nFirst, we present the theoretical formulas for the spin\ncurrent and spin accumulation within the F 1layer. We9\nintroduce two unit vectors t1=m1×˙m1/|m1×˙m1|and\nt2=−˙m1/|˙m1|, which are orthogonal to the magnetiza-\ntionm1and satisfy t1×t2=m1, because the transverse\ncomponents of the spin current and spin accumulation,\nEqs. (5) and (7), can be projected to these two direc-\ntions. Then, the longitudinal and transverse components\nof the spin current in the F 1layer are given by\nm1·Is(F1)=−/planckover2pi1ω˜g∗asinθ\n4πsinh[(x+d1)/λsd(L)]\nsinh(d1/λsd(L)),(B1)\nt1·Is(F1)=/planckover2pi1ω˜g↑↓\nr(1−c)sinθ\n4πRe/bracketleftbiggsinh[(x+d1)/ℓ]\nsinh(d1/ℓ)/bracketrightbigg\n,\n(B2)\nt2·Is(F1)=/planckover2pi1ω˜g↑↓\nr(1−c)sinθ\n4πIm/bracketleftbiggsinh[(x+d1)/ℓ]\nsinh(d1/ℓ)/bracketrightbigg\n.\n(B3)\nWe can confirm that the sum of these components,\n(m1·I)m1+(t1·Is)t1+(t2·Is)t2, atx= 0 is identical\nto the spin current at the F 1/N interface, Ipump\ns+IF1→N\ns.\nSimilarly, the longitudinal and transverse spin accumu-\nlation in the F 1layer are given by\nm1·µF1=/planckover2pi1ω˜g∗asinθ\ngsdcosh[(x+d1)/λsd(L)]\nsinh(d1/λsd(L)),(B4)\nt1·µF1=−/planckover2pi1ω˜g↑↓\nr(1−c)sinθRe/bracketleftbiggcosh[(x+d1)/ℓ]\ngtsinh(d1/ℓ)/bracketrightbigg\n,\n(B5)\nt2·µF1=−/planckover2pi1ω˜g↑↓\nr(1−c)sinθIm/bracketleftbiggcosh[(x+d1)/ℓ]\ngtsinh(d1/ℓ)/bracketrightbigg\n.\n(B6)\nNext, we present the explicit forms of the spin current\nand spin accumulation in the F 2layer. The magneti-\nzationm2can be expressed in terms of ( t1,t2,m1) as\nm2= cosθm1+ sinθt1. We introduce two unit vec-\ntors,u1=−sinθm1+ cosθt1andu2=t2satisfying\nu1×u2=m2, to decompose the transverse compo-\nnent. In terms of ( u1,u2,m2),µNcan be expressed as\nµN=/planckover2pi1ωsinθ[(acosθ+csinθ)m2+(−asinθ+ccosθ)u1].\nThen, the longitudinal and transverse spin currents are\ngiven by\nm2·Is(F2)=−/planckover2pi1ω˜g∗(asinθcosθ+csin2θ)\n4π\n×sinh[(x−d2)/λsd(L)]\nsinh(d2/λsd(L)),(B7)\nu1·Is(F2)=−/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n4π\n×Re/bracketleftbiggsinh[(x−d2)/ℓ]\nsinh(d2/ℓ)/bracketrightbigg\n,(B8)u2·Is(F2)=−/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n4π\n×Im/bracketleftbiggsinh[(x−d2)/ℓ]\nsinh(d2/ℓ)/bracketrightbigg\n.(B9)\nWe can confirm that the sum of these components, ( m2·\nI)m2+ (u1·Is)u1+ (u2·Is)u2, atx= 0 is identical\nto the spin current at the F 2/N interface, −IF2→N\ns. The\nlongitudinal and transverse spin accumulations are given\nby\nm2·µF2=/planckover2pi1ω˜g∗(asinθcosθ+csin2θ)\ngsd\n×cosh[(x−d2)/λsd(L)]\nsinh(d2/λsd(L)),(B10)\nu1·µF2=/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n×Re/bracketleftbiggcosh[(x−d2)/ℓ]\ngtsinh(d2/ℓ)/bracketrightbigg\n,(B11)\nu2·µF2=/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n×Im/bracketleftbiggcosh[(x−d2)/ℓ]\ngtsinh(d2/ℓ)/bracketrightbigg\n.(B12)\nFigures 4 (a) and (b) show the spatial distributions of\nthe spin current density and spin accumulation, respec-\ntively. The spin current density and spin accumulation\nare decomposed into the longitudinal and transverse di-\nrections, where the solid lines correspond to the longitu-\ndinal components whereas the dotted ( ∝bardblt1oru1) and\ndashed (∝bardblt2oru2) correspond to the transverse compo-\nnents. The valuesoftheparametersareidenticalto those\nused in Sec. V with θ= 45◦. Because spin pumping oc-\ncurs at the F 1/N interface, the spin current density and\nspin accumulation are concentrated near this interface.\nWe emphasize that the spatial directions of the longitu-\ndinal and transverse spin are different between the F 1\nand F2layers when the magnetizations, m1andm2, are\nnoncollinear; as a result the spin current in Fig. 4 (a)\nlooks discontinuous at the interface, although Eq. (3) is\nsatisfied.\nWe now consider the dissipation formulas. The lon-\ngitudinal and transverse parts of the bulk dissipation in\nthe F1layer can be expressed as\n/parenleftbigg∂QV\n∂t/parenrightbiggL\nF1=/planckover2pi1ω2\n4πS˜g∗2a2sin2θ\ngsdλsd(L)sinh2(d1/λsd(L))\n×cosh/bracketleftbigg2(x+d1)\nλsd(L)/bracketrightbigg\n,(B13)\n/parenleftbigg∂QV\n∂t/parenrightbiggT\nF1=/planckover2pi1ω2˜g↑↓2\nr(1−c)2sin2θ\n4πS2e2\nhσ↑↓\n×/braceleftBigg\n1\nλ2\nsd(T)/vextendsingle/vextendsingle/vextendsingle/vextendsingleℓcosh[(x+d1)/ℓ]\nsinh(d1/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinh[(x+d1)/ℓ]\nsinh(d1/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracerightBigg\n.(B14)10\nposition, x (nm)-5 0 5-1010 \n1 2 3 4 -1 -2 -3 -4 30 \n0F1 F2(b)position, x (nm)-5 0 5 spin current density (10 -25  J/nm 2)\n010 \n1 2 3 4 -1 -2 -3 -4 15 \n5F1 F2(a)\nposition, x (nm)-5 0 5 bulk dissipation (fJ/nm 3s) \n010 \n1 2 3 4 -1 -2 -3 -4 15 \n5F1 F2(c)-5  spin accumulation (10 -25  J) \n20 \nFIG. 4: Examples of the distributions of (a) longitudi-\nnal (solid) and transverse (dotted and dashed) spin current\ndensities, (b) longitudinal (solid) and transverse (dotte d and\ndashed) spin accumulations, and (c) bulk dissipations for\nθ= 45◦.\nSimilarly, the longitudinal and transverse parts of the\nbulk dissipation in the F 2layer can be expressed as\n/parenleftbigg∂QV\n∂t/parenrightbiggL\nF2=/planckover2pi1ω2\n4πS˜g∗2(asinθcosθ+csin2θ)2\ngsdλsd(L)sinh2(d2/λsd(L))\n×cosh/bracketleftbigg2(x−d2)\nλsd(L)/bracketrightbigg\n.(B15)\n/parenleftbigg∂QV\n∂t/parenrightbiggT\nF2=/planckover2pi1ω2˜g↑↓2\nr(−asin2θ+csinθcosθ)2\n4πS2e2\nhσ↑↓\n×/braceleftBigg\n1\nλ2\nsd(T)/vextendsingle/vextendsingle/vextendsingle/vextendsingleℓcosh[(x−d2)/ℓ]\nsinh(d2/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinh[(x−d2)/ℓ]\nsinh(d2/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracerightBigg\n.\n(B16)\nFigure 4 (c) shows the spatial distribution of the bulk\ndissipation, which is also concentrated near the interface.\nThe longitudinal part of the F 1/N interface dissipation\nand the longitudinal and transverse parts of the F 2/Ninterface dissipations are given by\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N=/planckover2pi1ω2˜g∗a2sin2θ\n4πS/bracketleftbigg\n1−˜g∗\ngsdtanh(d1/λsd(L))/bracketrightbigg\n,\n(B17)\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF2/N=/planckover2pi1ω2˜g∗(asinθcosθ+csin2θ)2\n4πS\n×/bracketleftbigg\n1−˜g∗\ngsdtanh(d2/λsd(L))/bracketrightbigg\n,(B18)\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF2/N=/planckover2pi1ω2˜g↑↓\nr(−asin2θ+csinθcosθ)2\n4πS\n×/braceleftbigg\n1−˜g↑↓\nrRe/bracketleftbigg1\ngttanh(d2/ℓ)/bracketrightbigg/bracerightbigg\n.\n(B19)\nForθ= 45◦, we quantitatively evaluate\nthat/integraltext0\n−d1dx(∂QV/∂t)L\nF1= 3.34 fJ/(nm2s),/integraltext0\n−d1dx(∂QV/∂t)T\nF1= 6 .51 fJ/(nm2s),/integraltextd2\n0dx(∂QV/∂t)L\nF2= 18 .15 fJ/(nm2s), and/integraltextd2\n0dx(∂QV/∂t)T\nF2= 4.95 fJ/(nm2s), respectively. Also,\nthe interface dissipations are quantitatively evaluated as\n(∂QA/∂t)L\nF1/N= 0.44 fJ/(nm2s), (∂QA/∂t)L\nF2/N= 2.39\nfJ/(nm2s), and ( ∂QA/∂t)T\nF2/N= 8.03 fJ/(nm2s) for\nθ= 45◦, respectively. We can confirm that the value of\nthe dissipation evaluated from these values as Eq. (23)\nis the same with that evaluated from Eq. (19) with Fig.\n3.\nAppendix C: Dissipation due to intrinsic damping\nIn this Appendix, we briefly evaluate the dissipation\ndue to the magnetization precession in the FMR experi-\nment, whicharisesfromtheintrinsicGilbertdamping. In\nthe FMR, the energysupplied by the microwavebalances\nwith the dissipation due to the damping, and the mag-\nnetization precesses practically on the constant energy\ncurve. The magnetization dynamics with the macrospin\nassumption is described by the Landau-Lifshitz-Gilbert\n(LLG) equation\ndm1\ndt=−γ0m1×H−α0γ0m1×(m1×H),(C1)\nwherethemagneticfield Hrelatestothemagneticenergy\ndensityEviaH=−∂E/∂(Mm1). From Eq. (C1), the\nchange of the energy density averaged on the constant\nenergy curve is given by\ndE\ndt≡1\nτ/contintegraldisplay\ndtdE\ndt\n=−αγ0M\nτ/contintegraldisplay\ndt/bracketleftBig\nH2−(m1·H)2/bracketrightBig\n,(C2)11\nwhere,τ=/contintegraltext\ndtis the precession period on a constant\nenergy curve. Assuming that the ferromagnet has uniax-\nial anisotropy H= (0,0,HKmz) as done in Sec. VI, Eq.\n(C2) is given by\ndE\ndt=−α0γ0MH2\nKsin2θcos2θ. (C3)\nThe microwaveshould supply the energydensity −dE/dt\nto sustain the precession. Then, the energy sup-\nplied by the microwave per unit area per unit time is\nα0γ0MH2\nKd1sin2θcos2θ, whered1is the thickness of the\nferromagnet. Comparing this energywith the dissipation\ndue to the spin pumping carried by the spin current, Eq.\n(34), the ratio of the dissipation between the intrinsicdamping and spin pumping is\n|dE/dt|d1\n∂QSP\nA/∂t∼α0\nα′, (C4)\nwhereα′is given by Eq. (35). 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Tang,2and Florian Marquardt1, 3\n1Institute for Theoretical Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany\n2Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA\n3Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1, 91058 Erlangen, Germany\nExperiments during the past two years have shown strong resonant photon-magnon coupling in\nmicrowave cavities, while coupling in the optical regime was demonstrated very recently for the first\ntime. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical\nsystems. This line of research promises to evolve into a new field of optomagnonics, aimed at\nthe coherent manipulation of elementary magnetic excitations by optical means. In this work we\nderive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the\nwell-known optomechanical case, with remarkably large coupling. Going beyond that, we study the\noptically induced nonlinear classical dynamics of a macrospin. In the fast cavity regime we obtain\nan effective equation of motion for the spin and show that the light field induces a dissipative term\nreminiscent of Gilbert damping. The induced dissipation coefficient however can change sign on\nthe Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is\nconsidered, the system can enter a chaotic regime by successive period doubling of the oscillations.\nI. INTRODUCTION\nThe ability to manipulate magnetism has played his-\ntorically an important role in the development of infor-\nmation technologies, using the magnetization of materi-\nals to encode information. Today’s research focuses on\ncontrolling individual spins and spin currents, as well as\nspin ensembles, with the aim of incorporating these sys-\ntems as part of quantum information processing devices.\n[1–4]. In particular the control of elementary excitations\nof magnetically ordered systems –denominated magnons\nor spin waves, is highly desirable since their frequency is\nbroadly tunable (ranging from MHz to THz) [2, 5] while\ntheycanhaveverylonglifetimes, especiallyforinsulating\nmaterials like the ferrimagnet yttrium iron garnet (YIG)\n[6]. The collective character of the magnetic excitations\nmoreoverrendertheserobustagainstlocalperturbations.\nWhereas the good magnetic properties of YIG have\nbeenknownsincethe60s, itisonlyrecentlythatcoupling\nandcontrollingspinwaveswithelectromagneticradiation\nin solid-state systems has started to be explored. Pump-\nprobe experiments have shown ultrafast magnetization\nswitching with light [7–9], and strong photon-magnon\ncoupling has been demonstrated in microwave cavity ex-\nperiments [10–18] –including the photon-mediated cou-\npling between a superconducting qubit and a magnon\nmode [19]. Going beyond microwaves, this points to the\ntantalizing possibility of realizing optomagnonics : the\ncoupled dynamics of magnons and photons in the op-\ntical regime, which can lead to coherent manipulation\nof magnons with light. The coupling between magnons\nand photons in the optical regime differs from that of\nthe microwave regime, where resonant matching of fre-\nquencies allows for a linear coupling: one magnon can be\nconvertedintoaphoton, andviceversa[20–22]. Intheop-\ntical case instead, the coupling is a three-particle process.\nThis accounts for the frequency mismatch and is gener-\nz\nyzx\noptical mode\noptical shift\u0000GSx0magnonmodeopticalmodeGˆ~SˆaabcFigure 1. (Color online) Schematic configuration of the model\nconsidered. (a)Optomagnoniccavitywithhomogeneousmag-\nnetization along the z-axis and a localized optical mode with\ncircular polarization in the y-z-plane. (b) The homogeneous\nmagnon mode couples to the optical mode with strength G.\n(c) Representation of the magnon mode as a macroscopic spin\non the Bloch sphere, whose dynamics is controlled by the cou-\npling to the driven optical mode.\nally called parametric coupling. The mechanism behind\nthe optomagnonic coupling is the Faraday effect, where\nthe angle of polarization of the light changes as it prop-\nagates through a magnetic material. Very recent first\nexperiments in this regime show that this is a promising\nroute, by demonstrating coupling between optical modes\nand magnons, and advances in this field are expected to\ndevelop rapidly [23–27].\nIn this work we derive and analyze the basic op-\ntomagnonic Hamiltonian that allows for the study of\nsolid-state cavity optomagnonics. The parametric op-\ntomagnonic coupling is reminiscent of optomechanicalarXiv:1604.07053v3  [cond-mat.mes-hall]  19 Sep 20162\nmodels. In the magnetic case however, the relevant oper-\natorthatcouplestotheopticalfieldisthespin, insteadof\nthe usual bosonic field representing a mechanical degree\nof freedom. Whereas at small magnon numbers the spin\ncan be replaced by a harmonic oscillator and the ideas of\noptomechanics [28] carry over directly, for general trajec-\ntories of the spin this is not possible. This gives rise to\nrich non-linear dynamics which is the focus of the present\nwork. Parametric spin-photon coupling has been studied\npreviously in atomic ensembles [29, 30]. In this work we\nfocus on solid-state systems with magnetic order and de-\nrive the corresponding optomagnonic Hamiltonian. After\nobtaining the general Hamiltonian, we consider a simple\nmodel which consists of one optical mode coupled to a\nhomogeneous Kittel magnon mode [31]. We study the\nclassical dynamics of the magnetic degrees of freedom\nand find magnetization switching, self-sustained oscilla-\ntions, and chaos, tunable by the light field intensity.\nThe manuscript is ordered as follows. In Sec. (II) we\npresent the model and the optomagnonic Hamiltonian\nwhich is the basis of our work. In Sec. (IIA) we discuss\nbriefly the connection of the optomagnonic Hamiltonian\nderived in this work and the one used in optomechanic\nsystems. In Sec. (IIB) we derive the optomagnonic\nHamiltonian from microscopics, and give an expression\nfor the optomagnonic coupling constant in term of ma-\nterial constants. In Sec. (III) we derive the classical\ncoupled equations of motion of spin and light for a ho-\nmogeneous magnon mode, in which the spin degrees of\nfreedomcanbetreatedasamacrospin. InSec. (IIIA)we\nobtain the effective equation of motion for the macrospin\nin the fast-cavity limit, and show the system presents\nmagnetization switching and self oscillations. We treat\nthe full (beyond the fast-cavity limit) optically induced\nnonlinear dynamics of the macrospin in Sec. (IIIB), and\nfollow the route to chaotic dynamics. In Sec. (IV) we\nsketch a qualitative phase diagram of the system as a\nfunction of coupling and light intensity, and discuss the\nexperimental feasibility of the different regimes. An out-\nlook and conclusions are found in Sec. (V). In the Ap-\npendix we give details of some of the calculations in the\nmain text, present more examples of nonlinear dynamics\nasafunctionofdifferenttuningparameters, andcompare\noptomagnonic vs.optomechanic attractors.\nII. MODEL\nFurther below, we derive the optomagnonic Hamilto-\nnian which forms the basis of our work:\nH=\u0000~\u0001^ay^a\u0000~\n^Sz+~G^Sx^ay^a; (1)\nwhere ^ay(^a) is the creation (annihilation) operator for a\ncavity mode photon. We work in a frame rotating at the\nlaser frequency !las, and \u0001 =!las\u0000!cavis the detuning\nwith respect to the optical cavity frequency !cav. Eq. (1)assumes a magnetically ordered system with (dimension-\nless) macrospin S= (Sx;Sy;Sz)with magnetization axis\nalong ^ z, and a precession frequency \nwhich can be con-\ntrolled by an external magnetic field [32]. The coupling\nbetween the optical field and the spin is given by the\nlast term in Eq. (1), where we assumed (see below) that\nlight couples only to the x\u0000component of the spin as\nshown in Fig. (1). The coefficient Gdenotes the para-\nmetric optomagnonic coupling. We will derive it in terms\nof the Faraday rotation, which is a material-dependent\nconstant.\nA. Relation to optomechanics\nClose to the ground state, for deviations such that\n\u000eS\u001cS(withS=jSj), we can treat the spin in the\nusual way as a harmonic oscillator, ^Sx\u0019p\nS=2(^b+^by),\nwithh\n^b;^byi\n= 1. Then the optomagnonic interaction\n~G^Sx^ay^a\u0019~Gp\nS=2^ay^a(^b+^by)becomes formally equiv-\nalent to the well-known opto mechanical interaction [28],\nwith bare coupling constant g0=Gp\nS=2. All the phe-\nnomena of optomechanics apply, including the “optical\nspring” (here: light-induced changes of the magnon pre-\ncession frequency) and optomagnonic cooling at a rate\n\u0000opt, and the formulas (as reviewed in Ref. [28]) can be\ntaken over directly. All these effects depend on the light-\nenhanced coupling g=g0\u000b, where\u000b=pnphotis the\ncavity light amplitude. For example, in the sideband-\nresolved regime ( \u0014\u001c\n, where\u0014is the optical cavity\ndecay rate) one would have \u0000opt= 4g2=\u0014. Ifg > \u0014,\none enters the strong-coupling regime, where the magnon\nmode and the optical mode hybridize and where coher-\nent state transfer is possible. A Hamiltonian of the form\nof Eq. (1) is also encountered for light-matter interaction\nin atomic ensembles [29], and its explicit connection to\noptomechanics in this case was discussed previously in\nRef. [30]. In contrast to such non-interacting spin en-\nsembles, the confined magnon mode assumed here can\nbe frequency-separated from other magnon modes.\nB. Microscopic magneto-optical coupling G\nIn this section we derive the Hamiltonian presented in\nEq. (1) starting from the microscopic magneto-optical\neffect in Faraday-active materials. The Faraday effect is\ncaptured by an effective permittivity tensor that depends\non the magnetization Min the sample. We restrict our\nanalysis to non-dispersive isotropic media and linear re-\nsponseinthemagnetization, andrelegatemagneticlinear\nbirefringence effects which are quadratic in M(denomi-\nnated the Cotton-Mouton or Voigt effect) for future work\n[5, 33]. In this case, the permittivity tensor acquires an\nantisymmetric imaginary component and can be written3\nas\"ij(M)=\"0(\"\u000eij\u0000ifP\nk\u000fijkMk), where\"0(\") is the\nvacuum (relative) permittivity, \u000fijkthe Levi-Civita ten-\nsorandfamaterial-dependentconstant[33](hereandin\nwhat follows, Latin indices indicate spatial components).\nThe Faraday rotation per unit length\n\u0012F=!fMs\n2cp\"; (2)\ndepends on the frequency !, the vacuum speed of light\nc, and the saturation magnetization Ms. The magneto-\noptical coupling is derived from the time-averaged energy\n\u0016U=1\n4\u0001\ndrP\nijE\u0003\ni(r;t)\"ijEj(r;t), using the complex\nrepresentation of the electric field, (E+E\u0003)=2. Note\nthat \u0016Uis real since \"ijis hermitean [5, 33]. The magneto-\noptical contribution is\n\u0016UMO=\u0000i\n4\"0f\u0002\ndr M(r)\u0001[E\u0003(r)\u0002E(r)]:(3)\nThis couples the magnetization to the spin angular mo-\nmentum density of the light field. Quantization of this\nexpression leads to the optomagnonic coupling Hamilto-\nnian. A similar Hamiltonian is obtained in atomic en-\nsemble systems when considering the electric dipolar in-\nteraction between the light field and multilevel atoms,\nwhere the spin degree of freedom (associated with M(r)\nin our case) is represented by the atomic hyperfine struc-\nture [29]. The exact form of the optomagnonic Hamil-\ntonian will depend on the magnon and optical modes.\nIn photonic crystals, it has been demonstrated that opti-\ncal modes can be engineered by nanostructure patterning\n[34], and magnonic-crystals design is a matter of intense\ncurrent research [3]. The electric field is easily quantized,\n^E(+)(r;t) =P\n\fE\f(r)^a\f(t), where E\f(r)indicates the\n\ftheigenmode of the electric field (eigenmodes are indi-\ncated with Greek letters in what follows). The magne-\ntization requires more careful consideration, since M(r)\ndependsonthelocalspinoperatorwhich, ingeneral, can-\nnot be written as a linear combination of bosonic modes.\nThere are however two simple cases: (i) small deviations\nof the spins, for which the Holstein-Primakoff representa-\ntion is linear in the bosonic magnon operators, and (ii) a\nhomogeneous Kittel mode M(r) =Mwith macrospin S.\nIn the following we treat the homogeneous case, to cap-\nture nonlinear dynamics. From Eq. (3) we then obtain\nthe coupling Hamiltonian ^HMO =~P\nj\f\r^SjGj\n\f\r^ay\n\f^a\r\nwith\nGj\n\f\r=\u0000i\"0fMs\n4~SX\nmn\u000fjmn\u0002\ndrE\u0003\n\fm(r)E\rn(r);(4)\nwhere we replaced Mj=Ms=^Sj=S, withSthe extensive\ntotal spin (scaling like the mode volume). One can diago-\nnalize the hermitean matrices Gj, though generically not\nsimultaneously. In the present work, we treat the con-\nceptually simplest case of a strictly diagonal coupling tosome optical eigenmodes ( Gj\n\f\f6= 0butGj\n\u000b\f= 0). This is\nprecludedonlyiftheopticalmodesarebothtime-reversal\ninvariant ( E\freal-valued) and non-degenerate. In all the\nother cases, a basis can be found in which this is valid.\nFor example, a strong static Faraday effect will turn op-\ntical circular polarization modes into eigenmodes. Al-\nternatively, degeneracy between linearly polarized modes\nimplies we can choose a circular basis.\nConsider circular polarization (R/L) in the y\u0000z-plane,\nsuch thatGxis diagonal while Gy=Gz= 0. Then we\nfind\nGx\nLL=\u0000Gx\nRR=G=1\nSc\u0012F\n4p\"\u0018; (5)\nwhere we used Eq. (2) to express the coupling via the\nFaraday rotation \u0012F, and where \u0018is a dimensionless over-\nlap factor that reduces to 1if we are dealing with plane\nwaves (see App. A). Thus, we obtain the coupling Hamil-\ntonianHMO=~G^Sx(^ay\nL^aL\u0000^ay\nR^aR). This reduces to\nEq. (1) if the incoming laser drives only one of the two\ncircular polarizations.\nThe coupling Ggives the magnon precession frequency\nshift perphoton. It decreases for larger magnon mode\nvolume, in contrast to GS, which describes the overall\nopticalshift for saturated spin ( Sx=S). For YIG,\nwith\"\u00195and\u0012F\u0019200ocm\u00001[5, 35], we obtain\nGS\u00191010Hz(taking\u0018= 1), which can easily become\ncomparable to the precession frequency \n. The ultimate\nlimit for the magnon mode volume is set by the optical\nwavelength,\u0018(1\u0016m)3, which yields S\u00181010. There-\nforeG\u00191Hz, whereas the coupling to a single magnon\nwould be remarkably large: g0=Gp\nS=2\u00190:1MHz.\nThis provides a strong incentive for designing small mag-\nnetic structures, by analogy to the scaling of piezoelectri-\ncal resonators [36]. Conversely, for a macroscopic volume\nof(1mm)3, withS\u00181019, this reduces to G\u001910\u00009Hz\nandg0\u001910Hz.\nIII. SPIN DYNAMICS\nThe coupled Heisenberg equations of motion are ob-\ntainedfromtheHamiltonianinEq. (1)byusing\u0002\n^a;^ay\u0003\n=\n1,h\n^Si;^Sji\n=i\u000fijk^Sk. Wenextfocusontheclassicallimit,\nwhere we replace the operators by their expectation val-\nues:\n_a=\u0000i(GSx\u0000\u0001)a\u0000\u0014\n2(a\u0000\u000bmax)\n_S= (Ga\u0003aex\u0000\nez)\u0002S+\u0011G\nS(_S\u0002S):(6)\nHere we introduced the laser amplitude \u000bmaxand the in-\ntrinsic spin Gilbert-damping [37], characterized by \u0011G,\ndue to phonons and defects ( \u0011G\u001910\u00004for YIG [38]).\nAfter rescaling the fields (see App.. B), we see that the4\nclassical dynamics is controlled by only five dimension-\nless parameters:GS\n\n;G\u000b2\nmax\n\n;\u0001\n\n;\u0014\n\n; \u0011G. These are inde-\npendent of ~as expected for classical dynamics.\nIn the following we study the nonlinear classical dy-\nnamics of the spin, and in particular we treat cases where\nthe spin can take values on the whole Bloch sphere and\ntherefore differs significantly from a harmonic oscilla-\ntor, deviating from the optomechanics paradigm valid\nfor\u000eS\u001cS. The optically induced tilt of the spin\ncan be estimated from Eq. (6) as \u000eS=S =Gjaj2=\n\u0018\nG\u000b2\nmax=\n =B\u000bmax=\n, whereB\u000bmax=G\u000b2\nmaxis an op-\ntically induced effective magnetic field. We would ex-\npect therefore unique optomagnonic behavior (beyond\noptomechanics) for large enough light intensities, such\nthatB\u000bmaxis of the order of or larger than the preces-\nsion frequency \n. We will show however that, in the case\nof blue detuning, even small light intensity can destabi-\nlize the original magnetic equilibrium of the uncoupled\nsystem, provided the intrinsic Gilbert damping is small.\nA. Fast cavity regime\nAs a first step we study a spin which is slow compared\nto the cavity, where G_Sx\u001c\u00142. In that case we can\nabyzx-0.10-0.0500.050.10\n-0.2-0.100.10.2\nFigure 2. (Color online) Spin dynamics (fast cavity limit)\nat blue detuning \u0001 = \n and fixedGS=\n = 2,\u0014=\n = 5,\n\u0011G= 0. The left column depicts the trajectory (green full\nline) of a spin (initially pointing near the north pole) on the\nBloch sphere. The color scale indicates the optical damping\n\u0011opt. The right column shows a stereographic projection of\nthe spin’s trajectory (red full line). The black dotted line\nindicates the equator (invariant under the mapping), while\nthe north pole is mapped to infinity. The stream lines of the\nspin flow are also depicted (blue arrows). (a) Magnetization\nswitching behavior for light intensity G\u000b2\nmax=\n = 0:36. (b)\nLimit cycle attractor for larger light intensity G\u000b2\nmax=\n =\n0:64.expand the field a(t)in powers of _Sxand obtain an ef-\nfective equation of motion for the spin by integrating out\nthe light field. We write a(t) =a0(t) +a1(t) +:::, where\nthe subscript indicates the order in _Sx. From the equa-\ntion fora(t), we find that a0fulfills the instantaneous\nequilibrium condition\na0(t) =\u0014\n2\u000bmax1\n\u0014\n2\u0000i(\u0001\u0000GSx(t));(7)\nfrom which we obtain the correction a1:\na1(t) =\u00001\n\u0014\n2\u0000i(\u0001\u0000GSx)@a0\n@Sx_Sx:(8)\nTo derive the effective equation of motion for the spin,\nwe replacejaj2\u0019ja0j2+a\u0003\n1a0+a\u0003\n0a1in Eq. (6) which\nleads to\n_S=Be\u000b\u0002S+\u0011opt\nS(_Sxex\u0002S) +\u0011G\nS(_S\u0002S):(9)\nHereBe\u000b=\u0000\nez+Bopt, where Bopt(Sx) =Gja0j2ex\nacts as an optically induced magnetic field. The second\nterm is reminiscent of Gilbert damping, but with spin-\nvelocity component only along ex. Both the induced field\nBoptand dissipation coefficient \u0011optdepend explicitly on\nthe instantaneous value of Sx(t):\nBopt=G\n[(\u0014\n2)2+ (\u0001\u0000GSx)2]\u0010\u0014\n2\u000bmax\u00112\nex(10)\n\u0011opt=\u00002G\u0014SjBoptj(\u0001\u0000GSx)\n[(\u0014\n2)2+ (\u0001\u0000GSx)2]2:(11)\nThis completes the microscopic derivation of the optical\nLandau-Lifshitz-Gilbert equation for the spin, an impor-\ntant tool to analyze effective spin dynamics in different\ncontexts [39]. We consider the nonlinear adiabatic dy-\nnamics of the spin governed by Eq. (9) below. Two\ndistinct solutions can be found: generation of new sta-\nble fixed points (magnetic switching) and optomagnonic\nlimit cycles (self oscillations).\nGiven our Hamiltonian (Eq. (1)), the north pole is sta-\nble in the absence of optomagnonic coupling – the se-\nlection of this state is ensured by the intrinsic damping\n\u0011G>0. By driving the system this can change due to\nthe energy pumped to (or absorbed from) the spin, and\nthe new equilibrium is determined by Be\u000band\u0011opt, when\n\u0011optdominates over \u0011G. Magnetic switching refers to the\nrotation of the macroscopic magnetization by \u0018\u0019, to a\nnew fixed point near the south pole in our model. This\ncan be obtained for blue detuning \u0001>0, in which case\n\u0011optis negative either on the whole Bloch sphere (when\n\u0001> GS) or on a certain region, as shown in Fig. (2)a.\nSimilar results were obtained in the case of spin opto-\ndynamics for cold atoms systems [30]. The possibility of\nswitching the magnetization direction in a controlled way\nis of great interest for information processing with mag-\nnetic memory devices, in which magnetic domains serve5\nas information bits [7–9]. Remarkably, we find that for\nblue detuning, magnetic switching can be achieved for\narbitrary small light intensities in the case of \u0011G= 0.\nThis is due to runaway solutions near the north pole for\n\u0001>0, as discussed in detail in App. C. In physical sys-\ntems, the threshold of light intensity for magnetization\nswitching will be determined by the extrinsic dissipation\nchannels.\nFor higher intensities of the light field, limit cycle at-\ntractors can be found for j\u0001j<GS, where the optically\ninduced dissipation \u0011optcan change sign on the Bloch\nsphere (Fig. (2)b). The combination of strong nonlinear-\nity and a dissipative term which changes sign, leads the\nsystem into self sustained oscillations. The crossover be-\ntween fixed point solutions and limit cycle attractors is\ndetermined by a balance between the detuning and the\nlight intensity, as discussed in App. C. Limit cycle at-\ntractors require B\u000bmax=\n>j\u0001j=GS(note that from (11)\nBopt\u0018B\u000bmaxif\u0014\u001d(\u0001\u0000GS)).\nWe note that for both examples shown in Fig. (2), for\nthe chosen parameters we have \u0011opt\u001d\u0011Gin the case of\nYIG, and hence taking \u0011G= 0is a very good approx-\nimation. More generally, from Eqs. (10) we estimate\n\u0011opt\u0018GSB opt=\u00143and therefore we can safely neglect\n\u0011Gfor(\u000bmaxG)2S\u001d\u0011G\u00143. The qualitative results (limit\ncycle, switching) survive up to \u0011opt&\u0011G, although quan-\ntitatively modified as \u0011Gis increased: for example, the\nsize of the limit cycle would change, and there would be\na threshold intensity for switching.\nB. Full nonlinear dynamics\nThe nonlinear system of Eq. (6) presents even richer\nbehavior when we leave the fast cavity regime. For limit\ncycles near the north pole, when \u000eS\u001cS, the spin is\nwell approximated by a harmonic oscillator, and the dy-\nnamics is governed by the attractor diagram established\nfor optomechanics [40]. In contrast, larger limit cycles\nwill display novel features unique to optomagnonics, on\nwhich we focus here.\nBeyond the fast cavity limit, we can no longer give\nanalytical expressions for the optically induced magnetic\nfield and dissipation. Moreover, we can not define a coef-\nficient\u0011optsince an expansion in _Sxis not justified. We\ntherefore resort to numerical analysis of the dynamics.\nFig. (3) shows a route to chaos by successive period dou-\nbling, upon decreasingthe cavity decay \u0014. This route can\nbe followed in detail as a function of any selected param-\neter by plotting the respective bifurcation diagram. This\nis depicted in Fig. (4). The plot shows the evolution\nof the attractors of the system as the light intensity is\nincreased. The figure shows the creation and expansion\nof a limit cycle from a fixed point near the south pole,\nfollowed by successive period doubling events and finally\nentering into a chaotic region. At high intensities, a limit\nt⌦\nGSz⌦\nyzxabc\ndeIncreasing period of the limit cycle\nChaos2⇡\u00003\u00000.5\n\u000013\u00002.52\n\u00003\u00002\u00001Figure 3. (Color Online) Full non-linear spin dynamics and\nroute to chaos for GS=\n = 3andG\u000b2\nmax=\n = 1(\u0011G= 0).\nThe system is blue detuned by \u0001 = \n and the dynamics,\nafter a transient, takes place in the southern hemisphere. The\nsolid red curves represent the spin trajectory after the initial\ntransient, on the Bloch sphere for (a) \u0014=\n = 3, (b)\u0014=\n = 2,\n(c)\u0014=\n = 0:9, (d)\u0014=\n = 0:5. (f)Szprojection as a function\nof time for the chaotic case \u0014=\n = 0:5.\nspin projectionGSz/⌦chaoslimit cycleperioddoublingcoexistence\n1.0laser amplitudepG|↵max|2/⌦210-1-21.5\nFigure 4. Bifurcation density plot for GS=\n = 3and\u0014=\n = 1\nat\u0001 = \n(\u0011G= 0), as a function of light intensity. We plot\ntheSzvalues attained at the turning points ( _Sz= 0). For\nother possible choices ( eg. _Sx= 0) the overall shape of\nthe bifurcation diagram is changed, but the bifurcations and\nchaotic regimes remain at the same light intensities. For the\nplot, 30 different random initial conditions were taken.\ncycle can coexist with a chaotic attractor. For even big-\nger light intensities, the chaotic attractor disappears and\nthesystemprecessesaroundthe exaxis, asaconsequence\nof the strong optically induced magnetic field. Similar bi-\nfurcation diagrams are obtained by varying either GS=\nor the detuning \u0001=\n(see App. D).6\n11\n2\nxy-plane limit cycles\"optomechanics\"chaosoptomagnonic limit cyclesswitching\nchaos yz plane limit cycles⌦GS\nxy-plane limit cyclesB↵max⌦\nFigure 5. Phase diagram for blue detuning with \u0001 = \n, as a\nfunction of the inverse coupling strength \n=GSand the op-\ntically induced field B\u000bmax=\n =G\u000b2\nmax=\n. Boundaries are\nqualitative. Switching, in white, refers to a fixed point solu-\ntion with the spin pointing near the south pole. Limit cycles\nin thexyplane are shaded in blue, and they follow the op-\ntomechanical attractor diagram discussed in Ref. [40]. For\nhigherB\u000bmax, chaos can ensue. Orange denotes the param-\neter space in which limit cycles deviate markedly from op-\ntomechanical predictions. These are not in the xyplane and\nalso undergo period doubling leading to chaos. In red is de-\npicted the area where pockets of chaos can be found. For\nlargeB\u000bmax=\n, the limit cycles are in the yzplane. In the\ncase of red detuning \u0001 =\u0000\n, the phase diagram remains as\nis, except that instead of switching there is a fixed point near\nthe north pole.\nIV. DISCUSSION\nWe can now construct a qualitative phase diagram for\nour system. Specifically, we have explored the qualitative\nbehavior (fixed points, limit cycles, chaos etc.) as a func-\ntion of optomagnonic coupling and light intensity. These\nparameters can be conveniently rescaled to make them\ndimensionless. We chose to consider the ratio of magnon\nprecessionfrequencytocoupling, intheform \n=GS. Fur-\nthermore, we express the light intensity via the maxi-\nmal optically induced magnetic field B\u000bmax=G\u000b2\nmax.\nThe dimensionless coupling strength, once the material\nof choice is fixed, can be tuned viaan external magnetic\nfieldwhichcontrolstheprecessionfrequency \n. Thelight\nintensity can be controlled by the laser.\nWe start by considering blue detuning, this is shown\nin Fig. (5). The “phase diagram” is drawn for \u0001 = \n,\nand we set \u0014= \nand\u0011G= 0. We note that some of the\ntransitions are rather crossovers (“optomechanical limit\ncycles” vs.“optomagnonic limit cycles”). In addition, the\nother “phase boundaries” are only approximate, obtained\nfrom direct inspection of numerical simulations. These\nare not intended to be exact, and are qualitatively validfor departures of the set parameters, if not too drastic\n– for example, increasing \u0014will lead eventually to the\ndisappearance of the chaotic region.\nAs the diagram shows, there is a large range of pa-\nrameters that lead to magnetic switching, depicted in\nwhite. This area is approximately bounded by the con-\nditionB\u000bmax=\n.\u0001=GS, which in Fig. (5) corresponds\nto the diagonal since we took \u0001 = \n. This condition\nis approximate since it was derived in the fast cavity\nregime, see App. C. As discussed in Sec. III, magnetic\nswitching should be observable in experiments even for\nsmall light intensity in the case of blue detuning, pro-\nvided that all non-optical dissipation channels are small.\nThe caveat of low intensity is a slow switching time. For\nB\u000bmax=\n&\u0001=GS, the system can go into self oscilla-\ntions and even chaos. For optically induced fields much\nsmaller than the external magnetic field, B\u000bmax\u001c\nwe\nexpect trajectories of the spin in the xyplane, precessing\naround the external magnetic field along ezand therefore\nthe spin dynamics (after a transient) is effectively two-\ndimensional. This is depicted by the blue-shaded area\nin Fig.(5). These limit cycles are governed by the op-\ntomechanical attractor diagram presented in Ref. [40],\nas we show in App. E. There is large parameter region\nin which the optomagnonic limit cycles deviate from the\noptomechanical attractors. This is marked by orange in\nFig.(5). As the light intensity is increased, for \n=GS\u001c1\nthe limit cycles remain approximately confined to the xy\nplane but exhibit deviations from optomechanics. This\napproximate confinement of the trajectories to the xy\nplane at large B\u000bmax=\n(B\u000bmax=\n&0:5for\u0001 = \n)\ncan be understood qualitatively by looking at the ex-\npression of the induced magnetic field Boptdeduced in\nthe fast cavity limit, Eq. (10). Since we consider \u0001 = \n,\n\n=GS\u001c1impliesGS\u001d\u0001. In this limit, Bopt=\ncan\nbecome very small and the spin precession is around the\nezaxis. For moderate B\u000bmax=\nand\n=GS, the limit cy-\ncles are tilted and precessing around an axis determined\nby the effective magnetic field, a combination of the opti-\ncalinducedfieldandtheexternalmagneticfield. Bluede-\ntuning causes these limit cycles to occur in the southern\nhemisphere. Period doubling leads eventually to chaos.\nThe region where pockets of chaos can be found is rep-\nresented by red in the phase diagram. For large light\nintensity, such that B\u000bmax\u001d\n, the optical field domi-\nnates and the effective magnetic field is essentially along\ntheexaxis. The limit cycle is a precession of the spin\naround this axis.\nAccording to our results optomagnonic chaos is at-\ntained for values of the dimensionless coupling GS=\n\u0018\n1\u000010and light intensities G\u000b2\nmax=\n\u00180:1\u00001. This\nimplies a number of circulating photons similar to the\nnumber of locked spins in the material, which scales with\nthe cavity volume. This therefore imposes a condition\non the minimum circulating photon density in the cavity.\nFor YIG with characteristic frequencies \n\u00181\u000010GHz,7\ntheconditiononthecouplingiseasilyfulfilled(remember\nGS= 10GHz as calculated above). However the condi-\ntion on the light intensity implies a circulating photon\ndensity of\u0018108\u0000109photons/\u0016m3which is outside\nof the current experimental capabilities, limited by the\npower a typical microcavity can support (around \u0018105\nphotons/\u0016m3). On the other hand, magnetic switching\nand self-sustained oscillations of the optomechanical type\n(but taking place in the southern hemisphere) can be at-\ntained for low powers, assuming all external dissipation\nchannels are kept small. While self-sustained oscillations\nand switching can be reached in the fast-cavity regime,\nmorecomplexnonlinearbehaviorsuchasperioddoubling\nand chaos requires approaching sideband resolution. For\nYIG the examples in Figs. 3, 4 correspond to a preces-\nsion frequency \n\u00193\u0001109Hz(App. D), whereas \u0014can be\nestimated to be\u00181010Hz, taking into account the light\nabsorption factor for YIG ( \u00180:3cm\u00001) [35].\nFor red detuning \u0001<0, the regions in the phase dia-\ngram remain the same, except that instead of magnetic\nswitching, the solutions in this parameter range are fixed\npoints near the north pole. This can be seen by the sym-\nmetry of the problem: exchanging \u0001! \u0000 \u0001together\nwithex!\u0000exandez!\u0000ezleaves the problem un-\nchanged. The limit cycles and trajectories follow also\nthis symmetry, and in particular the limit cycles in the\nxyplane remain invariant.\nV. OUTLOOK\nThe observation of the spin dynamics predicted here\nwill be a sensitive probe of the basic cavity optomagnonic\nmodel, beyond the linear regime. Our analysis of the op-\ntomagnonic nonlinear Gilbert damping could be general-\nized to more advanced settings, leading to optomagnonic\nreservoir engineering (e.g. two optical modes connected\nby a magnon transition). Although the nonlinear dy-\nnamics presented here requires light intensities outside of\nthe current experimental capabilities for YIG, it should\nbe kept in mind that our model is the simplest case for\nwhich highly non-linear phenomena is present. Increas-\ning the model complexity, for example by allowing for\nmultiple-mode coupling, could result in a decreased light\nintensity requirement. Materials with a higher Faraday\nconstantwouldbealsobeneficial. Inthisworkwefocused\non the homogeneous Kittel mode. It will be an interest-\ning challenge to study the coupling to magnon modes at\nfinite wavevector, responsible for magnon-induced dissi-\npation and nonlinearities under specific conditions [41–\n43]. The limit cycle oscillations can be seen as “opto-\nmagnonic lasing”, analogous to the functioning principle\nof a laser where energy is pumped and the system set-\ntles in a steady state with a characteristic frequency, and\nalso discussed in the context of mechanics (“cantilaser”\n[44]). These oscillations could serve as a novel sourceof traveling spin waves in suitable geometries, and the\nsynchronization of such oscillators might be employed to\nimprove their frequency stability. We may see the de-\nsign of optomagnonic crystals and investigation of opto-\nmagnonic polaritons in arrays. In addition, future cav-\nity optomagnonics experiments will allow to address the\ncompletely novel regime of cavity-assisted coherent op-\ntical manipulation of nonlinear magnetic textures, like\ndomain walls, vortices or skyrmions, or even nonlinear\nspatiotemporal light-magnon patterns. 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Poot, and H. X. Tang, Nature\ncommunications 6, 5850 (2015).\n[37] T. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[38] In the magnetic literature, \u0011Gis denoted as \u000b[5].\n[39] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys-\nical review letters 88, 117601 (2002), 0110247.\n[40] F.Marquardt,J.G.E.Harris, andS.M.Girvin,Physical\nReview Letters 96, 103901 (2006).\n[41] A. M. Clogston, H. Suhl, L. R. Walker, and P. W. An-\nderson, J. Phys. Chem. Solids 1, 129 (1956).\n[42] H. Suhl, Journal of Physics and Chemistry of Solids 1,\n209 (1957).\n[43] G. Gibson and C. Jeffries, Physical Review A 29, 811\n(1984).\n[44] I. Bargatin and M. L. Roukes, Physical review letters 91,\n138302 (2003).\n[45] T. Liu, X. Zhang, H. X. Tang, and M. E. Flatté, (2016),\narXiv:1604.07052.\nAppendix A: Optomagnonic coupling Gfor plane waves\nIn this section we calculate explicitly the optomagnonic coupling presented in Eq.. (5) for the case of plane\nwaves mode functions for the electric field. We choose for definiteness the magnetization axis along the ^ zaxis, and\nconsider the case Gx\f\r6= 0. The Hamiltonian HMOis then diagonal in the the basis of circularly polarized waves,\neR=L=1p\n2(ey\u0007iez). The rationale behind choosing the coupling direction perpendicular to the magnetization axis,\nis to maximize the coupling to the magnon mode, that is to the deviations of the magnetization with respect to the\nmagnetization axis. The relevant spin operator is therefore ^Sx, which represents the flipping of a spin. In the case of\nplane waves, we quantize the electric field according to ^E+(\u0000)(r;t) = +(\u0000)iP\njejq\n~!j\n2\"0\"V^a(y)\nj(t)e+(\u0000)ikj\u0001r;whereV\nis the volume of the cavity, kjthe wave vector of mode jand we have identified the positive and negative frequency\ncomponents of the field as E!^E+,E\u0003!^E\u0000. The factor of \"0\"in the denominator ensures the normalization\n~!j=\"0\"hjj\u0001\nd3rjE(r)j2jji\u0000\"0\"h0j\u0001\nd3rjE(r)j2j0i, which corresponds to the energy of a photon in state jjiabove\nthe vacuumj0i. For two degenerate (R/L) modes at frequency !, using Eq. (2) we see that the frequency dependence\ncancels out and we obtain the simple form for the optomagnonic Hamiltonian HMO=~G^Sx(^ay\nL^aL\u0000^ay\nR^aR)with\nG=1\nSc\u0012F\n4p\". Therefore the overlap factor \u0018= 1in this case.9\nAppendix B: Rescaled fields and linearized dynamics\nTo analyze Eq. (6) it is convenient to re-scale the fields such that a=\u000bmaxa0,S=SS0and measure all times and\nfrequencies in \n. We obtain the rescaled equations of motion (time-derivatives are now with respect to t0= \nt)\n_a0=\u0000i(GS\n\nS0\nx\u0000\u0001\n\n)a0\u0000\u0014\n2\n(a0\u00001) (B1)\n_S0=\u0012G\u000b2\nmax\n\nja0j2ex\u0000ez\u0013\n\u0002S0+\u0011G\nS\u0010\n_S0\u0002S0\u0011\n(B2)\nIf we linearize the spin-dynamics (around the north-pole, e.g.), we should recover the optomechanics behavior. In\nthis section we ignore the intrinsic Gilbert damping term. We set approximately S0\u0019(S0\nx;S0\ny;1)Tand from Eq. (B1)\nwe obtain\n_S0\nx=S0\ny (B3)\n_S0\ny=\u0000G\u000b2\nmax\n\nja0j2\u0000S0\nx (B4)\nWe can now choose to rescale further, via S0\nx=\u0010\n\u000bmax=p\nS\u0011\nS00\nxand likewise for S0\ny. We obtain the following\nspin-linearized equations of motion:\n_S00\nx=S00\ny (B5)\n_S00\ny=\u0000Gp\nS\u000bmax\n\nja0j2\u0000S00\nx (B6)\n_a0=\u0000i(Gp\nS\u000bmax\n\nS00\nx\u0000\u0001\n\n)a0\u0000\u0014\n2\n(a0\u00001) (B7)\nThis means that the number of dimensionless parameters has been reduced by one, since the two parameters initially\ninvolving G, S, and\u000bmaxhave all been combined into\nGp\nS\u000bmax\n\n(B8)\nIn other words, for S0\nx;y=Sx;y=S\u001c1, the dynamics should only depend on this combination, consistent with the\noptomechanicalanalogyvalidinthisregimeasdiscussedinthemaintext(wherewearguedbasedontheHamiltonian).\nAppendix C: Switching in the fast cavity limit\nFrom Eq. (9) in the weak dissipation limit ( \u0011G\u001c1) we obtain\n_Sx=\nSy\n_Sy=\u0000SzBopt\u0000\nSx\u0000\u0011opt\nS_SxSz;\nfrom where we obtain an equation of motion for Sx. We are interested in studying the stability of the north pole once\nthe driving is turned on. Hence we set Sz=S,\nSx=\u0000\nSBopt\u0000\n2Sx\u0000\u0011opt\n_Sx;\nand we consider small deviations \u000eSxofSxfrom the equilibrium position that satisfies S0\nx=\u0000SBopt=\n, whereBopt\nis evaluated at S0\nx. To linear order we obtain\n\u000eSx=\u0000\n\u0012\n\n +S@Bopt\n@Sx\u0013\n\u000eSx+ 2GS\u0014\nBopt(\u0001 +GSB opt=\n)\nh\n(\u0014=\n)2+ (\u0001 +GSB opt=\n)2i2_\u000eSx:\nWe see that the dissipation coefficient for blue detuning ( \u0001>0) is always negative, giving rise to runaway solutions.\nTherefore the solutions near the north pole are always unstable under blue detuning, independent of the light intensity.10\nThese trajectories run to a fixed point near the south pole, which accepts stable solutions for \u0001>0(switching) or to\na limit cycle. Near the south pole, Sz=\u0000S,S0\nx=SBopt=\nand\n\u000eSx=\u0000\n\u0012\n\n\u0000S@Bopt\n@Sx\u0013\n\u000eSx\u00002GS\u0014\nBopt(\u0001\u0000GSB opt=\n)\nh\n(\u0014=\n)2+ (\u0001\u0000GSB opt=\n)2i2_\u000eSx:\nTherefore for \u0001> GSB opt=\nthere are stable fixed points, while in the opposite case there are also runaway\nsolutions that are caught in a limit cycle. For red detuning, \u0001!\u0000 \u0001and the roles of south and north pole are\ninterchanged.\nAppendix D: Nonlinear dynamics\nIn this section we give more details on the full nonlinear dynamics described in the main text. In Figs. 3 and (4) of\nthe main text we chose a relative coupling GS=\n = 3, around which a chaotic attractor is found. With our estimated\nGS\u00191010Hzfor YIG, this implies a precession frequency \n\u00193\u0001109Hz. In Fig. (3) the chaotic regime is reached at\n\u0014\u0019\n=2withG\u000b2\nmax=\n = 1, which implies \u000b2\nmax\u0019S=3, that is, a number of photons circulating in the (unperturbed)\n2.53\n-0.5\n-1Spin projectionSz/S\nGS/⌦Normalized coupling\nFigure 6. (Color online) Bifurcation density plot for G\u000b2\nmax=\n = 1and\u0014=\n = 1at\u0001 = \n(\u0011G= 0), as a function of the\nrelative coupling strength GS=\n. The dotted blue line indicates GS=\n = 3, for comparison with Fig. (4). As in the main\ntext, the points (obtained after the transient) are given by plotting the values of Szattained whenever the trajectory fulfills\nthe turning point condition _Sz= 0, for 20 different random initial conditions.11\n1.01.50-1-2Spin projectionGSz/⌦\n\u0000/⌦Detuning\nFigure 7. (Color online) Bifurcation density plot for GS=\n = 3,G\u000b2\nmax=\n = 1and\u0014=\n = 1(\u0011G= 0), as a function of the\ndetuning \u0001=\n. The dotted blue line indicates \u0001=\n = 1, for comparison with Fig. (4).\ncavity of the order of the number of locked spins and hence scaling with the cavity volume. Bigger values of the cavity\ndecay rate are allowed for attaining chaos at the same frequency, at the expense of more photons in the cavity, as can\nbe deduced from Fig. (4) where we took \u0014= \n. On the other hand we can think of varying the precession frequency\n\nby an applied external magnetic field and explore the nonlinearities by tuning GS=\nin this way (note that GSis\na material constant). This is done in Fig. (6). Alternatively, the nonlinear behavior can be controlled by varying the\ndetuning \u0001, as shown in Fig. (7).\nAppendix E: Relation to the optomechanical attractors\nIn this appendix we show that the optomagnonic system includes the higher order nonlinear attractors found in\noptomechanics as a subset in parameter space.\nIn optomechanics, the high order nonlinear attractors are self sustained oscillations with amplitudes Asuch that\nthe optomechanical frequency shift GAis a multiple of the mechanical frequency \n. Translating to our case, this\nmeansG\u000eS\u0018n\n. Since\u000eS=S\u0018Gj\u000bmaxj2=\n =B\u000bmax=\nwe obtain the condition\nGS\n\nB\u000bmax\n\n\u0018n (E1)\nfor observing these attractors. We can vary B\u000bmaxaccording to Eq. (E1). For \n=GS\u001c1we are in the limit of small\nB\u000bmax=\nand we expect limit cycles precessing along ezas discussed in Sec. (IV). In Fig. 8 the attractor diagram12\n5101520\n2015105GS/⌦GSx/⌦\n10302020301040GS/⌦GSx/⌦\nFigure 8. Attractor diagram for \u0001 = 1:5\nand\u0014=\n = 1with condition G2Sj\u000bmaxj2=n\n2. Top:n= 1, bottomn= 10. We\nplot theSxvalues attained at the turning points ( _Sx= 0) forSx>0. The diagram is symmetric for Sx<0as expected for\na limit cycle on the Bloch sphere. The diagram at the left coincides to a high degree of approximation with the predictions\nobtained for optomechanical systems (i.e. replacing the spin by a harmonic oscillator). In contrast, this is no longer the case\nfor the diagram on the right, which involves higher light intensities.13\nobtained by imposing condition (E1) is plotted. Since the trajectories are in the xyplane, we plot the inflection point\nof the coordinate Sx. We expect GSx=\nevaluated at the inflection point, which gives the amplitude of the limit\ncycle, to coincide with the optomechanic attractors for small B\u000bmax=\nand hence flat lines at the expected amplitudes\n(as calculated in Ref. [40]) as GS=\nincreases. Relative evenly spaced limit cycles increasing in number as larger\nvalues ofGS=\nare considered are observed, in agreement with Ref. [40]. Remarkable, these limit cycles attractors\nare found on the whole Bloch sphere, and not only near the north pole where the harmonic approximation is strictly\nvalid. These attractors are reached by allowing initial conditions on the whole Bloch sphere. For n= 1, (Fig. 8, top),\nswitching is observed up to GS=\n\u00184and then perfect optomechanic behavior. For higher values of n, deviations\nfrom the optomechanical behavior are observed for small GS=\n(implying large B\u000bmax=\naccording to Eq. (E1)) and\nlarge amplitude limit cycles, as compared to the size of the Bloch sphere. An example is shown in Fig. 8, bottom,\nforn= 10."    },    {        "title": "1211.3611v2.Spin_transport_and_tunable_Gilbert_damping_in_a_single_molecule_magnet_junction.pdf",        "content": "Spin transport and tunable Gilbert damping in a single-molecule magnet junction\nMilena Filipović,1Cecilia Holmqvist,1Federica Haupt,2and Wolfgang Belzig1\n1Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n2Institut für Theorie der Statistischen Physik, RWTH Aachen, D-52056 Aachen, Germany\n(Dated: October 24, 2019)\nWe study time-dependent electronic and spin transport through an electronic level connected\nto two leads and coupled with a single-molecule magnet via exchange interaction. The molecular\nspin is treated as a classical variable and precesses around an external magnetic field. We derive\nexpressions for charge and spin currents by means of the Keldysh nonequilibrium Green’s functions\ntechniqueinlinearorderwithrespecttothetime-dependentmagneticfieldcreatedbythisprecession.\nThe coupling between the electronic spins and the magnetization dynamics of the molecule creates\ninelastic tunneling processes which contribute to the spin currents. The inelastic spin currents, in\nturn, generate a spin-transfer torque acting on the molecular spin. This back-action includes a\ncontribution to the Gilbert damping and a modification of the precession frequency. The Gilbert\ndamping coefficient can be controlled by the bias and gate voltages or via the external magnetic\nfield and has a nonmonotonic dependence on the tunneling rates.\nPACS numbers: 73.23.-b, 75.76.+j, 85.65.+h, 85.75.-d\nI. INTRODUCTION\nSingle-molecule magnets (SMMs) are quantum mag-\nnets, i.e., mesoscopic quantum objects with a perma-\nnent magnetization. They are typically formed by\nparamagnetic ions stabilized by surrounding organic\nligands.1SMMs show both classical properties such\nas magnetization hysteresis2and quantum properties\nsuch as spin tunneling,3coherence,4and quantum phase\ninterference.2,5They have recently been in the center of\ninterest2,6,7in view of their possible applications as in-\nformation storage8and processing devices.9\nCurrently, a goal in the field of nanophysics is to\ncontrol and manipulate individual quantum systems, in\nparticular, individual spins.10,11Some theoretical works\nhave investigated electronic transport through a molecu-\nlarmagnetcontactedtoleads.12–19Inthiscase, thetrans-\nport properties are modified due to the exchange inter-\naction between the itinerant electrons and the SMM,20\nmaking it possible to read out the spin state of the\nmolecule using transport currents. Conversely, the spin\ndynamics and hence the state of an SMM can also be\ncontrolled by transport currents. Efficient control of the\nmolecule’s spin state can be achieved by coupling to fer-\nromagnetic contacts as well.21\nExperiments have addressed the electronic trans-\nport properties through magnetic molecules such as\nMn12and Fe 8,22,23which have been intensively stud-\nied as they are promising candidates for memory\ndevices.24Various phenomena such as large conduc-\ntance gaps,25switching behavior,26negative differen-\ntial conductance, dependence of the transport on mag-\nnetic fields and Coulomb blockades have been exper-\nimentally observed.22,23,27,28Experimental techniques,\nincluding, for instance, scanning tunneling microscopy\n(STM),22,23,29–31break junctions,32and three-terminal\ndevices,22,23,27have been employed to measure elec-\ntronic transport through an SMM. Scanning tunnelingspectroscopy and STM experiments show that quantum\nproperties of SMMs are preserved when deposited on\nsubstrates.29The Kondo effect in SMMs with magnetic\nanisotropyhasbeeninvestigatedboththeoretically14and\nexperimentally.33,34It has been suggested35and experi-\nmentally verified36that a spin-polarized tip can be used\nto control the magnetic state of a single Mn atom.\nIn some limits, the large spin Sof an SMM can be\ntreated as a classical magnetic moment. In that case,\nthe spin dynamics is described by the Landau-Lifshitz-\nGilbert (LLG) equation that incorporates effects of ex-\nternal magnetic fields as well as torques originating from\ndamping phenomena.37,38In tunnel junctions with mag-\nnetic particles, LLG equations have been derived using\nperturbativecouplings39,40andthenonequilibriumBorn-\nOppenheimer approximation.16Current-induced magne-\ntization switching is driven by a generated spin-transfer\ntorque (STT)41–44as a back-action effect of the elec-\ntronic spin transport on the magnetic particle.16,45–47\nA spin-polarized STM (Ref. 36) has been used to\nexperimentally study STTs in relation to a molecular\nmagnetization.48This experimental achievement opens\nnew possibilities for data storage technology and appli-\ncations using current-induced STTs.\nThe goal of this paper is to study the interplay be-\ntween electronic spin currents and the spin dynamics of\nan SMM. We focus on the spin-transport properties of\na tunnel junction through which transport occurs via a\nsingle electronic energy level in the presence of an SMM.\nThe electronic level may belong to a neighboring quan-\ntum dot (QD) or it may be an orbital related to the\nSMM itself. The electronic level and the molecular spin\nare coupled via exchange interaction, allowing for inter-\nactionbetweenthespinsoftheitinerantelectronstunnel-\ning through the electronic level and the spin dynamics of\nthe SMM. We use a semiclassical approach in which the\nmagnetization of the SMM is treated as a classical spin\nwhose dynamics is controlled by an external magneticarXiv:1211.3611v2  [cond-mat.mes-hall]  22 Jul 20132\nfield, while for the electronic spin and charge transport\nwe use instead a quantum description. The magnetic\nfield is assumed to be constant, leading to a precessional\nmotion of the spin around the magnetic field axis. The\nelectronic level is subjected both to the effects of the\nmolecular spin and the external magnetic field, generat-\ningaZeemansplitofthelevel. Thespinprecessionmakes\nadditional channels available for transport, which leads\nto the possibility of precession-assisted inelastic tunnel-\ning. During a tunnel event, spin-angular momentum may\nbe transferred between the inelastic spin currents and the\nmolecular spin, leading to an STT that may be used to\nmanipulate the spin of the SMM. This torque includes\nthe so-called Gilbert damping , which is a phenomenolog-\nically introduced damping term of the LLG equation,38\nand a term corresponding to a modification of the pre-\ncession frequency. We show that the STT and hence the\nSMM’s spin dynamics can be controlled by the external\nmagnetic field, the bias voltage across the junction, and\nthe gate voltage acting on the electronic level.\nThe paper is organized as follows: We introduce our\nmodel and formalism based on the Keldysh nonequilib-\nrium Green’s functions technique49–51in Sec. II, where\nwe derive expressions for the charge and spin currents in\nlinear order with respect to a time-dependent magnetic\nfield and analyze the spin-transport properties at zero\ntemperature. In Sec. III we replace the general magnetic\nfield of Sec. II by an SMM whose spin precesses in an\nexternal constant magnetic field, calculate the STT com-\nponents related to the Gilbert damping, and the modifi-\ncationoftheprecessionfrequency, andanalyzetheeffects\nof the external magnetic field as well as the bias and gate\nvoltages on the spin dynamics. Conclusions are given in\nSec. IV.\nII. CURRENT RESPONSE TO A TIME\nDEPENDENT MAGNETIC FIELD\nA. Model and Formalism\nFor the sake of clarity, we start by considering a junc-\ntion consisting of a noninteracting single-level QD cou-\npled with two normal, metallic leads in the presence of an\nexternal, time-dependent magnetic field (see Fig. 1). The\nleads are assumed to be noninteracting and unaffected\nby the external field. The total Hamiltonian describing\nthe junction is given by ^H(t) = ^HL;R+^HT+^HD(t).\nThe Hamiltonian of the free electrons in the leads reads\n^HL;R=P\nk;\u001b;\u00182fL;Rg\u000fk\u001b\u0018^cy\nk\u001b\u0018^ck\u001b\u0018, where\u0018denotes the\nleft (L) or right ( R) lead, whereas the tunnel cou-\npling between the QD and the leads can be written as\n^HT=P\nk;\u001b;\u00182L;R[Vk\u0018^cy\nk\u001b\u0018^d\u001b+V\u0003\nk\u0018^dy\n\u001b^ck\u001b\u0018]. The spin-\nindependent tunnel matrix element is given by Vk\u0018. The\noperators ^cy\nk\u001b\u0018(^ck\u001b\u0018)and ^dy\n\u001b(^d\u001b)are the creation (an-\nnihilation) operators of the electrons in the leads and\nthe QD, respectively. The subscript \u001b=\";#denotes\neVB(t)\nFIG. 1: (Color online) A quantum dot with a single electronic\nlevel\u000f0coupled to two metallic leads with chemical potentials\n\u0016Land\u0016Rinthepresenceofanexternaltime-dependentmag-\nnetic field~B(t). The spin-transport properties of the junction\naredeterminedbythebiasvoltage eV=\u0016L\u0000\u0016R, theposition\nof the level \u000f0, the tunnel rates \u0000Land\u0000R, and the magnetic\nfield.\nthe spin-up or spin-down state of the electrons. The\nelectronic level \u000f0of the QD is influenced by an ex-\nternal magnetic field ~B(t)consisting of a constant part\n~Bcand a time-dependent part ~B0(t). The Hamiltonian\nof the QD describing the interaction between the elec-\ntronic spin ^~ sand the magnetic field is then given by\n^HD(t) = ^Hc\nD+^H0(t), where the constant and time-\ndependent parts are ^Hc\nD=P\n\u001b\u000f0^dy\n\u001b^d\u001b+g\u0016B^~ s~Bcand\n^H0(t) =g\u0016B^~ s~B0(t). The proportionality factor gis the\ngyromagnetic ratio of the electron and \u0016Bis the Bohr\nmagneton.\nThe average charge and spin currents from the left lead\nto the electronic level are given by\nIL\u0017(t) =q\u0017\u001cd\ndt^NL\u0017\u001d\n=q\u0017i\n~\n\u0002^H;^NL\u0017\u0003\u000b\n;(1)\nwhere ^NL\u0017=P\nk;\u001b;\u001b0^cy\nk\u001bL(^\u001b\u0017)\u001b\u001b0^ck\u001b0Lis the charge and\nspin occupation number operator of the left contact. The\nindex\u0017= 0corresponds to the charge current, while\n\u0017=x;y;zindicates the different components of the spin-\npolarized current. The current coefficients q\u0017are then\nq0=\u0000eandq\u00176=0=~=2. In addition, it is useful to\ndefine the vector ^\u001b\u0017= (^1;^~ \u001b), where ^1is the identity\noperator and ^~ \u001bconsists of the Pauli operators with ma-\ntrix elements (^~ \u001b)\u001b\u001b0. Using the Keldysh nonequilibrium\nGreen’s functions technique, the currents can then be ob-\ntained as50,51\nIL\u0017(t) =\u00002q\u0017\n~Re\u0002\ndt0Tr\b\n^\u001b\u0017[^Gr(t;t0)^\u0006<\nL(t0;t)(2)\n+^G<(t;t0)^\u0006a\nL(t0;t)]\t\n;\nwhere ^Gr;a;<are the retarded, advanced, and lesser\nGreen’s functions of the electrons in the QD with the ma-\ntrix elements Gr;a\n\u001b\u001b0(t;t0) =\u0007i\u0012(\u0006t\u0007t0)hf^d\u001b(t);^dy\n\u001b0(t0)gi\nandG<\n\u001b\u001b0(t;t0) =ih^dy\n\u001b0(t0)^d\u001b(t)i, while ^\u0006r;a;<\nL(t;t0)\nare self-energies from the coupling between the QD\nand the left lead. Their nonzero matrix elements3\nare diagonal in the electronic spin space with re-\nspect to the basis of eigenstates of ^sz, given by\n\u0006r;a;<\nL(t;t0) =P\nkVkLgr;a;<\nkL(t;t0)V\u0003\nkL. The Green’s func-\ntionsgr;a;<\nkL(t;t0)are the retarded, advanced and lesser\nGreen’s functions of the free electrons in the left lead.\nThe retarded Green’s functions ^Gr\n0of the electrons in\nthe QD, in the presence of the constant magnetic field\n~Bc, are found using the equation of motion technique,52\nwhile the lesser Green’s functions ^G<\n0are obtained from\nthe Keldysh equation ^G<\n0=^Gr\n0^\u0006<^Ga\n0, where multipli-\ncation implies internal time integrations.51The time-\ndependent part of the magnetic field can be expressed\nas~B0(t) =P\n!(~B!e\u0000i!t+~B\u0003\n!ei!t), where~B!is a com-\nplex amplitude. This magnetic field acts as a time-\ndependent perturbation that can be expressed as ^H0(t) =P\n!(^H!e\u0000i!t+^Hy\n!ei!t), where ^H!is an operator in the\nelectronic spin space and its matrix representaton in the\nbasis of eigenstates of ^szis given by\n^H!=g\u0016B\n2\u0012\nB!zB!x\u0000iB!y\nB!x+iB!y\u0000B!z\u0013\n:(3)\nApplying Dyson’s expansion, analytic continuation rules\nand the Keldysh equation,51one obtains a first-order ap-\nproximation of the Green’s functions describing the elec-\ntrons in the QD that can be written as\n^Gr\u0019^Gr\n0+^Gr\n0^H0^Gr\n0; (4)\n^G<\u0019^Gr\n0^\u0006<^Ga\n0+^Gr\n0^H0^Gr\n0^\u0006<^Ga\n0+^Gr\n0^\u0006<^Ga\n0^H0^Ga\n0:\nThe expression for the currents in this linear approxima-\ntion is given by\nIL\u0017(t) =\u00002q\u0017\n~Re Tr\b\n^\u001b\u0017[^Gr\n0^\u0006<\nL+^G<\n0^\u0006a\nL (5)\n+^Gr\n0^H0^Gr\n0^\u0006<\nL+^Gr\n0^H0^G<\n0^\u0006a\nL+^G<\n0^H0^Ga\n0^\u0006a\nL]\t\n:\nEq. (5) is then Fourier transformed in the wide-\nband limit, in which the level width function, \u0000(\u000f) =\n\u00002 Imf\u0006r(\u000f)g, is constant, Ref\u0006r(\u000f)g= 0, and one can\nhence write the retarded self-energy originating from the\ndot-lead coupling as \u0006r;a(\u000f) =\u0007i\u0000=2. From this trans-\nformation, one obtains\nIL\u0017(t) =Idc\nL\u0017+X\n![IL\u0017(!)e\u0000i!t+I\u0003\nL\u0017(!)ei!t]:(6)\nUsing units in which ~= 1, the dc part of the\ncurrents51Idc\nL\u0017and the time-independent complex com-\nponentsIL\u0017(!)are given by\nIdc\nL\u0017=q\u0017\u0002d\u000f\n\u0019\u0000L\u0000R\n\u0000[fL(\u000f)\u0000fR(\u000f)] Tr Imf^\u001b\u0017^Gr\n0(\u000f)g(7)\nand\nIL\u0017(!) =\u0000iq\u0017\u0002d\u000f\n2\u0019\u0000L\u0000R\n\u0000n\n[fL(\u000f)\u0000fR(\u000f)] (8)\n\u0002Trf^\u001b\u0017[^Gr\n0(\u000f+!)^H!^Gr\n0(\u000f) + 2iImf^Gr\n0(\u000f)g^H!^Ga\n0(\u000f\u0000!)]g\n+X\n\u0018=L;R\u0000\u0018\n\u0000R[f\u0018(\u000f\u0000!)\u0000fL(\u000f)] Tr[^\u001b\u0017^Gr\n0(\u000f)^H!^Ga\n0(\u000f\u0000!)]o\n:In the above expressions, f\u0018(\u000f) = [e(\u000f\u0000\u0016\u0018)=kBT+ 1]\u00001is\nthe Fermi distribution of the electrons in lead \u0018, where\nkBis the Boltzmann constant. The retarded Green’s\nfunction ^Gr\n0(\u000f)is given by ^Gr\n0(\u000f) = [\u000f\u0000\u000f0\u0000\u0006r(\u000f)\u0000\n(1=2)g\u0016B^~ \u001b~Bc]\u00001.16\nThe linear response of the spin current with respect\nto the applied time-dependent magnetic field can be ex-\npressed in terms of complex spin-current susceptibilities\ndefined as\n\u001fL\n\u0017j(!) =@IL\u0017(!)\n@B!j; j =x;y;z: (9)\nThe complex components IL\u0017(!)are conversely given by\nIL\u0017(!) =P\nj\u001fL\n\u0017j(!)B!j. By taking into account that\n@^H!=@B!j= (1=2)g\u0016B^\u001bjand using Eq. (8), the current\nsusceptibilities can be written as\n\u001fL\n\u0017j(!) =\u0000iq\u0017g\u0016B\u0002d\u000f\n4\u0019\u0000L\u0000R\n\u0000n\n[fL(\u000f)\u0000fR(\u000f)](10)\n\u0002Trf^\u001b\u0017[^Gr\n0(\u000f+!)^\u001bj^Gr\n0(\u000f) + 2iImf^Gr\n0(\u000f)g^\u001bj^Ga\n0(\u000f\u0000!)]g\n+X\n\u0018\u0000\u0018\n\u0000R[f\u0018(\u000f\u0000!)\u0000fL(\u000f)]Tr[^\u001b\u0017^Gr\n0(\u000f)^\u001bj^Ga\n0(\u000f\u0000!)]o\n:\nThe components obey \u001fL\n\u0017j(\u0000!) =\u001fL\u0003\n\u0017j(!). In other\nwords, they satisfy the Kramers-Kronig relations53that\ncan be written in a compact form as\n\u001fL\n\u0017j(!) =1\ni\u0019P\u00021\n\u00001\u001fL\n\u0017j(\u0018)\n\u0018\u0000!d\u0018; (11)\nwithPdenoting the principal value.\nFor anyi;j;k =x;y;zsuch thatj6=kandj;k6=\ni, whereiindicates the direction of the constant part\nof the magnetic field ~Bc=Bc~ ei, the complex current\nsusceptibilities satisfy the relations\n\u001fL\njj(!) =\u001fL\nkk(!) (12)\nand\u001fL\njk(!) =\u0000\u001fL\nkj(!); (13)\nin addition to Eq. (11). The other nonzero compo-\nnents are\u001fL\n0i(!)and\u001fL\nii(!). In the absence of a constant\nmagnetic field, the only nonvanishing components obey\n\u001fL\nxx(!) =\u001fL\nyy(!) =\u001fL\nzz(!).\nFinally, the average value of the electronic spin in the\nQD reads~ s(t) =h^~ s(t)i= (1=2)P\n\u001b\u001b0~ \u001b\u001b\u001b0h^dy\n\u001b(t)^d\u001b0(t)i=\n\u0000(i=2)P\n\u001b\u001b0~ \u001b\u001b\u001b0^G<\n\u001b0\u001b(t;t)and the complex spin suscep-\ntibilities are defined as\n\u001fs\nij(!) =@si(!)\n@B!j: (14)\nThey represent the linear responses of the electronic spin\ncomponentstotheappliedtime-dependentmagneticfield\nand satisfy the relations similar to Eqs. (11), (12), and\n(13) given above.4\nB. Analysis of the spin and current responses\nWe start by analyzing the transport properties of the\njunction at zero temperature in response to the exter-\nnal time-dependent magnetic field ~B(t). The constant\ncomponent of the magnetic field ~Bcgenerates a Zeeman\nsplit of the QD level \u000f0, resulting in the levels \u000f\";#, where\n\u000f\";#=\u000f0\u0006g\u0016BBc=2in this section. The time-dependent\nperiodic component of the magnetic field ~B0(t)then cre-\nates additional states, i.e., sidebands, at energies \u000f\"\u0006!\nand\u000f#\u0006!(see Fig. 2). These Zeeman levels and side-\nbandscontributetotheelastictransportpropertiesofthe\njunction when their energies lie inside the bias-voltage\nwindow ofeV=\u0016L\u0000\u0016R.\nHowever, energylevelsoutsidethebias-voltagewindow\nmay also contribute to the electronic transport due to in-\nelastic tunnel processes generated by the time-dependent\nmagnetic field. In these inelastic processes, an electron\ntransmitted from the left lead to the QD can change its\nenergy by!and either tunnel back to the left lead or\nout into the right lead. If this perturbation is small, as is\nassumed in this paper where we consider first-order cor-\nrections, the transport properties are still dominated by\nthe elastic, energy-conserving tunnel processes that are\nassociated with the Zeeman levels.\nThe energy levels of the QD determine transport prop-\nerties such as the spin-current susceptibilities and the\nspin susceptibilities, which are shown in Fig. 3. The\nimaginary and real parts of the susceptibilities are plot-\nted as functions of the frequency !in Figs. 3(a) and 3(c).\nIn this case, the position of the unperturbed level \u000f0is\nsymmetric with respect to the Fermi surfaces of the leads\nand a peak or step in the spin-current and spin suscepti-\nbilities appears at a value of !, for which an energy level\nis aligned with one of the lead Fermi surfaces. In Figs.\n3(b) and 3(d), the susceptibilities are instead plotted as\nfunctions of the bias voltage, eV. Here, each peak or\nstep in the susceptibilities corresponds to a change in the\nnumber of available transport channels. The bias volt-\nage is applied in such a way that the energy of the Fermi\nsurface of the right lead is fixed at \u0016R= 0while the en-\nergy of the left lead’s Fermi surface is varied according\nto\u0016L=eV.\nIII. SPIN-TRANSFER TORQUE AND\nMOLECULAR SPIN DYNAMICS\nA. Model with a precessing molecular spin\nNow we apply the formalism of the previous section\nto the case of resonant tunneling through a QD in the\npresence of a constant external magnetic field and an\nSMM [see Fig. 4(a)]. An SMM with a spin Slives in\na(2S+ 1)-dimensional Hilbert space. We assume that\nthe spinSof the SMM is large and neglecting the quan-\ntum fluctuations, one can treat it as a classical vector\n\nµLµR=00↑+ω↑↑−ω↓+ω↓−ω↓FIG. 2: (Color online) Sketch of the electronic energy levels of\nthe QD in the presence of a time-dependent magnetic field. In\na static magnetic field, the electronic level \u000f0(solid black line)\nsplits into the Zeeman levels \u000f\";#(solid red and blue lines). If\nthe magnetic field in addition to the static component also in-\ncludes a time-dependent part with a characteristic frequency\n!, additionallevelsappearatenergies \u000f\"\u0006!(dottedredlines)\nand\u000f#\u0006!(dotted blue lines). Hence, there are six channels\navailable for transport.\nwhose end point moves on a sphere of radius S. In the\npresence of a constant magnetic field ~Bc=Bc~ ez, the\nmolecular spin precesses around the field axis according\nto~S(t) =S?cos(!Lt)~ ex+S?sin(!Lt)~ ey+Sz~ ez, where\nS?is the projection of ~Sonto thexyplane,!L=g\u0016BBc\nis the Larmor precession frequency and Szis the projec-\ntion of the spin on the zaxis [see Fig. 4(b)]. The spins\nof the electrons in the electronic level are coupled to the\nspin of the SMM via the exchange interaction J. The\ncontribution of the external magnetic field and the pre-\ncessional motion of the SMM’s spin create an effective\ntime-dependent magnetic field acting on the electronic\nlevel.\nThe Hamiltonian of the system is now given by ^H(t) =\n^HL;R+^HT+^HD(t) +^HS, where the Hamiltonians ^HL;R\nand ^HTare the same as in Sec. II. The Hamiltonian\n^HS=g\u0016B~S~Bcrepresents the interaction of the molecu-\nlar spin~Swith the magnetic field ~Bcand consequently\ndoes not affect the electronic transport through the junc-\ntion but instead contributes to the spin dynamics of the\nSMM. The Hamiltonian of the QD in this case is given by\n^HD(t) =^Hc\nD+^H0(t). Here, ^Hc\nD=P\n\u001b\u000f0^dy\n\u001b^d\u001b+g\u0016B^~ s~Bc\ne\u000b\nis the Hamiltonian of the electrons in the QD in the pres-\nence of the constant part of the effective magnetic field,\ngiven by~Bc\ne\u000b=\u0002\nBc+J\ng\u0016BSz\u0003\n~ ez. The second term of the\nQD Hamiltonian, ^H0(t) =g\u0016B^~ s~B0\ne\u000b(t), represents the in-\nteraction between the electronic spins of the QD, ^~ s, and\nthe time-dependent part of the effective magnetic field,\ngiven by~B0\ne\u000b(t) =JS?\ng\u0016B\u0002\ncos(!Lt)~ ex+ sin(!Lt)~ ey\u0003\n. The\ntime-dependent effective magnetic field can be rewritten\nas~B0\ne\u000b(t) =~B!Le\u0000i!Lt+~B\u0003\n!Lei!Lt, where~B!Lconsists\nof the complex amplitudes B!Lx=JS?=2g\u0016B,B!Ly=5\n0.00.51.01.52.02.5-6-4-2024\nw@e0DcijL@10-3mBD\nmL=e+wHaL\nmL=e¯+w\nImczzLReczzLImcxyLRecxyLImcxxLRecxxL\n0.00.51.01.52.0-1012\neV@e0DcijL@10-2mBD\ne¯-wHbL\ne¯\ne¯+w\ne-w\ne\ne+w\nImczzLReczzLImcxyLRecxyLImcxxLRecxxL\n0.00.51.01.52.02.5-2-101\nw@e0Dcijs@10-2mBe0-1DHcL\nImczzsReczzsImcxysRecxysImcxxsRecxxs\n0.00.51.01.52.0-4-20\neV@e0Dcijs@10-1mBe0-1DHdL\nImczzsReczzsImcxysRecxysImcxxsRecxxs\nFIG. 3: (Color online) (a) Frequency and (b) bias-voltage dependence of the spin-current susceptibilities. (c) Frequency and\n(d) bias-voltage dependence of the spin susceptibilities. In (a) and (c), the chemical potential of the left lead is \u0016L= 2\u000f0, while\nin (b) and (d) the frequency is set to != 0:16\u000f0. All plots are obtained at zero temperature with ~Bc=Bc~ ez, and the other\nparameters set to \u0016R= 0; \u000f\"= 1:48\u000f0; \u000f#= 0:52\u000f0;\u0000 = 0:02\u000f0, and \u0000L= \u0000R= 0:01\u000f0.\niJS?=2g\u0016B, andB!Lz= 0. The time-dependent pertur-\nbation can then be expressed as ^H0(t) = ^H!Le\u0000i!Lt+\n^Hy\n!Lei!Lt, where ^H!Lis an operator that can be written,\nusing Eq. (3) and the above expressions for B!Li, as\n^H!L=JS?\n2\u0012\n0 1\n0 0\u0013\n: (15)\nThe time-dependent part of the effective magnetic field\ncreates inelastic tunnel processes that contribute to the\ncurrents. The in-plane components of the spin current\nfulfill\nILx(!L) =\u0000iILy(!L) (16)\n=JS?\n2g\u0016B[\u001fL\nxx(!L) +i\u001fL\nxy(!L)];\nwhere~Bcis replaced by ~Bc\ne\u000b. Thezcomponent vanishes\nto lowest order in H0(t).54Therefore, the inelastic spin\ncurrent has a polarization that precesses in the xyplane.\nThe inelastic spin-current components, in turn, exert an\nSTT (Refs. 41-44) on the molecular spin given by\n~T(t) =\u0000[~IL(t) +~IR(t)]; (17)thus contributing to the dynamics of the molecular spin\nthrough\n_~S(t) =g\u0016B~Bc\u0002~S(t) +~T(t): (18)\nUsing expressions (6), (8), and (15), the torque of Eq.\n(17) can be calculated in terms of the Green’s functions\n^Gr\n0(\u000f)and ^Ga\n0(\u000f)as\nTi(t) =\u0000JS?\n2\u0002d\u000f\n2\u0019X\n\u0018\u0015\u0000\u0018\u0000\u0015\n\u0000[f\u0018(\u000f\u0000!L)\u0000f\u0015(\u000f)]\n(19)\n\u0002Imf(^\u001bi)#\"Gr\n0;\"\"(\u000f)Ga\n0;##(\u000f\u0000!L)e\u0000i!Ltg;\nwith\u0015=L;R. Here (^\u001bi)#\",Gr\n0;\"\"(\u000f), andGa\n0;##(\u000f)are\nmatrix elements of ^\u001bi,^Gr\n0(\u000f)and ^Ga\n0(\u000f)with respect to\nthe basis of eigenstates of ^sz. This STT can be rewritten\nin terms of the SMM’s spin vector as\n~T(t) =\u000b\nS_~S(t)\u0002~S(t) +\f_~S(t) +\r~S(t):(20)\nThe first term in this back-action gives a contribution to6\nzBceVS⊥(t)(a) (b) \nFIG. 4: (Color online) (a) Resonant tunneling in the pres-\nence of an SMM and an external, constant magnetic field.\nThe electronic level of Fig. 1 is now coupled with the spin\nof an SMM via exchange interaction with the coupling con-\nstantJ. The dynamics of the SMM’s spin ~Sis controlled by\nthe external magnetic field ~Bcthat also affects the electronic\nlevel. (b)PrecessionalmotionoftheSMM’sspininaconstant\nmagnetic field ~Bcapplied along the zaxis.\nthe Gilbert damping, characterized by the Gilbert damp-\ning coefficient \u000b. The second term acts as an effective\nconstant magnetic field and changes the precession fre-\nquency of the spin ~Swith the corresponding coefficient\n\f. The third term cancels the zcomponent of the Gilbert\ndamping term, thus restricting the STT to the xyplane.\nThe coefficient of the third term \ris related to \u000bby\n\r=\u000b =!LS2\n?=SSz. Expressing the coefficients \u000band\n\fin terms of the current susceptibilities \u001f\u0018\nxx(!L)and\n\u001f\u0018\nxy(!L)results in\n\u000b=\u0000JSz\ng\u0016B!LSX\n\u0018[Ref\u001f\u0018\nxx(!L)g\u0000Imf\u001f\u0018\nxy(!L)g];\n(21)\n\f=J\ng\u0016B!LX\n\u0018[Imf\u001f\u0018\nxx(!L)g+ Ref\u001f\u0018\nxy(!L)g]:(22)\nBy inserting the explicit expressions for Gr\n0;\"\"(\u000f)and\nGa\n0;##(\u000f\u0000!L), one obtains\n\u000b=J2S2\nz\n!LS\u0002d\u000f\n8\u0019X\n\u0018\u0015\u0000\u0018\u0000\u0015[f\u0018(\u000f\u0000!L)\u0000f\u0015(\u000f)]\u0002(23)\n1\n[(\u0000\n2)2+ (\u000f\u0000\u000f\")2][(\u0000\n2)2+ (\u000f\u0000\u000f#\u0000!L)2];\n\f=\u0000J\n!L\u0000\u0002d\u000f\n4\u0019X\n\u0018\u0015\u0000\u0018\u0000\u0015[f\u0018(\u000f\u0000!L)\u0000f\u0015(\u000f)]\u0002(24)\n(\u0000\n2)2+ (\u000f\u0000\u000f\")(\u000f\u0000\u000f#\u0000!L)\n[(\u0000\n2)2+ (\u000f\u0000\u000f\")2][(\u0000\n2)2+ (\u000f\u0000\u000f#\u0000!L)2];\nwhere\u000f\";#=\u000f0\u0006g\u0016BBc\ne\u000b=2 =\u000f0\u0006(!L+JSz)=2are the\nenergies of the Zeeman levels in this section. In the small\nprecession frequency regime, !L\u001ckBT,\r!0and in\nthe limit of Sz=S!1the expression for the coefficient\n\u000bis in agreement with Ref. 16.\n\nµLµR=0↑↓↑−ωL↓+ωLFIG. 5: (Color online) Sketch of the electronic energy levels\nof the QD in the presence of a molecular spin precessing with\nthe frequency !Laround an external, constant magnetic field.\nThe corresponding Zeeman levels are \u000f\";#. The precessional\nmotion of the molecular spin results in emission (absorption)\nof energy corresponding to a spin flip from spin up (down) to\nspin down (up). Hence, there are only four channels available\nfor transport.\nB. Analysis of the spin-transfer torque\nIn the case of resonant tunneling in the presence of\na molecular spin precessing in a constant external mag-\nnetic field, one also needs to take the exchange of spin-\nangular momentum between the molecular spin and the\nelectronic spins into account in addition to the effects\nof the external magnetic field. Due to the precessional\nmotion of the molecular spin, an electron in the QD emit-\nting (absorbing) an energy !Lalso undergoes a spin flip\nfrom spin up (down) to spin down (up), as indicated\nby the arrows in Fig. 5. As a result, the levels at en-\nergies\u000f\";#\u0006!Lare forbidden and hence do not con-\ntribute to the transport processes. Consequently, there\nare only four transport channels, which are located at\nenergies\u000f\";#\u0007!L. Also in this case, there are elastic\nand inelastic tunnel processes. Some of the possible in-\nelastic tunnel processes are shown in Fig. 6. These re-\nstrictions on the inelastic tunnel processes are also vis-\nible in Fig. 3(b), which identically corresponds to the\ncase of the presence of a precessing molecular spin with\n!L= 0:16\u000f0andJSz= 0:8\u000f0. Namely, from Eq. (16),\nwhich is equivalent to RefILx(!L)g= ImfILy(!L)g=\nJS?\n2g\u0016B[Ref\u001fL\nxx(!L)g\u0000Imf\u001fL\nxy(!L)g]andImfILx(!L)g=\n\u0000RefILy(!L)g=JS?\n2g\u0016B[Imf\u001fL\nxx(!L)g+ Ref\u001fxy(!L)g],\nand from the symmetries of the susceptibilities displayed\nin Fig. 3(b), it follows that there are no spin currents at\neV=\u000f\";#\u0006!L.\nAs was mentioned, the spin currents generate an STT\nacting on the molecular spin. A necessary condition\nfor the existence of an STT, and hence finite values of\nthe coefficients \u000band\fin Eqs. (23) and (24), is that\n~IL(t)6=\u0000~IR(t)[see Eq. (17)]. This condition is met\nby the spin currents generated, e.g., by the inelastic tun-\nnel processes shown in Figs. 6(b) and 6(c). These tunnel\nprocesses occur when an electron can tunnel into the QD,\nundergo a spin flip, and then tunnel off the QD into ei-7\nµLµL\nµLµLµL(a) \n(f) (e) (d) (c) (b) µL\nFIG. 6: (Color online) Sketch of the inelastic spin-tunneling\nprocesses in the QD in the presence of the precessing molec-\nular spin in the field ~Bc=Bc~ ezfor different positions of\nthe energy levels with respect to the chemical potentials of\nthe leads,\u0016Land\u0016R. Only transitions between levels with\nthe same color (blue or red) are allowed. Different colored\ncurved arrows (magenta, brown, or green) represent different\nprocesses.\nther lead. From these tunnel processes it is implied that\nthe Gilbert damping coefficient \u000band the coefficient \f\ncan be controlled by the applied bias or gate voltage as\nwell as by the external magnetic field. If a pair of QD\nenergy levels, coupled via spin-flip processes, lie within\nthe bias-voltage window, the spin currents instead fulfill\n~IL(t) =~IR(t), leading to a vanishing STT [see Fig. 6(d)].\nIn Figs. 6(e) and 6(f) the position of the energy levels of\nthe QD are symmetric with respect to the Fermi levels\nof the leads, \u0016Land\u0016R. When the QD level with energy\n\u000f\"is aligned with \u0016L, this simultaneously corresponds to\nthe energy level \u000f#being aligned with \u0016R[see Fig. 6(f)].\nAs a result, a spin-up electron can now tunnel from the\nleft lead into the level \u000f\", while a spin-down electron in\nthe level\u000f#can tunnel into the right lead. These addi-\ntional processes enhance the STT compared to that of\nthe case 6(e).\nThe two spin-torque coefficients \u000band\fexhibit a non-\nmonotonic dependence on the tunneling rates \u0000, as can\nbe seen in Figs. 7, 8, and 9. For \u0000!0, it is obvious\nthat\u000b;\f!0. In the weak coupling limit \u0000\u001c!L, the\ncoefficients \u000band\fare finite if the Fermi surface energy\nof the lead \u0018,\u0016\u0018fulfills either of the conditions\n\u000f#\u0014\u0016\u0018\u0014\u000f#+!L (25)\nor\u000f\"\u0000!L\u0014\u0016\u0018\u0014\u000f\" (26)\ninsuchawaythateachconditionissatisfiedbytheFermi\nenergy of maximum one lead. These conditions are re-\nlaxed for larger tunnel couplings as a consequence of the\nbroadening of the QD energy levels, which is also re-\nsponsible for the initial enhancement of \u000band\fwith in-\ncreasing \u0000. Notice, however, that \u000band\fare eventually\nsuppressed for \u0000\u001d!L, when the QD energy levels aresignificantly broadened and overlap so that spin-flip pro-\ncesses are equally probable in each direction and there\nis no net effect on the molecular spin. Physically, this\nsuppression of the STT can be understood by noticing\nthat for \u0000\u001d!La current-carrying electron perceives the\nmolecular spin as almost static due to its slow precession\ncompared to the electronic tunneling rates and hence the\nexchange of angular momenta is reduced. With increas-\ning tunneling rates, the coefficient \fbecomes negative\nbefore it drops to zero, causing the torque \f_~Sto oppose\nthe rotational motion of the spin ~S.\nIn Fig. 7, the Gilbert damping coefficient \u000band the\ncoefficient\fare plotted as functions of the applied bias\nvoltage at zero temperature. We analyze the case of the\nsmallestvalueof \u0000(redlines), assumingthat !L>0. For\nsmalleV, all QD energy levels lie outside the bias-voltage\nwindow and there is no spin transport [see Fig. 6(a)].\nHence\u000b;\f!0. AteV=\u000f#the tunnel processes in\nFig. 6(b) come into play, leading to a finite STT and the\ncoefficient\u000bincreases while the coefficient \fhas a local\nminimum. In the voltage region specified by Eq. (25) for\n\u0016L, the coefficient \u000bapproaches a constant value while\nthe coefficient \fincreases. By increasing the bias voltage\ntoeV=\u000f#+!Lthe tunnel processes in Fig. 6(c) occur,\nleading to a decrease of \u000band a local maximum of \f. For\n\u000f#+!L<eV <\u000f\"\u0000!L, the coefficients \u000b;\f!0[see Fig\n6(d)]. In the voltage region specified by Eq. (26) for \u0016L,\n\u000bapproaches the same constant value mentioned above\nwhile\fdecreases between a local maximum at eV=\n\u000f\"\u0000!Land a local minimum at eV=\u000f\", which approach\nthe same values as previously mentioned extrema. With\nfurther increase of eV, all QD energy levels lie within the\nbias-voltage window and the STT consequently vanishes.\nFigure 8 shows the spin-torque coefficients \u000band\fas\nfunctions of the position of the electronic level \u000f0. An\nSTT acting on the molecular spin occurs if the electronic\nlevel\u000f0is positioned in such a way that the inequalities\n(25) and (26) may be satisfied by some values of eV,\u000f0\nand!L. Again, we analyze the case of the smallest value\nof\u0000(red curve). For the particular choice of parameters\nin Fig. 8, there are four regions in which the inequalities\n(25) and (26) are satisfied. Within these regions, \u000bap-\nproaches a constant value while \fhas a local maximum\nas well as a local minimum. These local extrema occur\nwhen one of the Fermi surfaces is aligned with one of the\nenergy levels of the QD. For other values of \u000f0, both\u000b\nand\fvanish.\nThe coefficients \u000band\fare plotted as functions of the\nprecession frequency !Lin Fig. 9. Here, \u000f0=eV=2and\ntherefore the positions of the energy levels of the QD are\nsymmetric with respect to the Fermi levels of the leads,\n\u0016Land\u0016R. Once more, we focus first on the case of the\nsmallest value of \u0000(indicated by the red curve). The\nenergies of all four levels of the QD depend on !L, i.e.,\n~Bc. For!L>0, when the magnitude of the external\nmagnetic field is large enough, the tunnel processes in\nFig. 6(f) take place due to the above-mentioned sym-\nmetries. These tunnel processes lead to a finite STT,8\n0.00.51.01.52.01234567\neV@e0Da@10-4D\ne¯+wLHaL\ne¯\nee-wLG=5e0G=3e0G=0.2e0G=0.02e0\n0.00.51.01.52.02.5-8-6-4-2024\neV@e0Db@10-4D\ne¯+wLHbL\ne¯\nee-wLG=5e0G=3e0G=0.2e0G=0.02e0\nFIG. 7: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the applied bias voltage eV=\n\u0016L\u0000\u0016R, with\u0016R= 0, for different tunneling rates \u0000at zero temperature. Other parameters are \u0000L= \u0000R= \u0000=2,\u000f\"= 1:48\u000f0,\n\u000f#= 0:52\u000f0,S= 100,J= 0:01\u000f0,JSz= 0:8\u000f0, and!L= 0:16\u000f0. In the case of the smallest value of \u0000(red lines), \u000bapproaches\na constant value when \u0016Llies within the energy range specified by Eqs. (25) and (26). The coefficient \fhas one local minimum\nand one local maximum for the same energy range.\n-0.50.00.51.01.52.01234567\ne0@eVDa@10-4D\nmR=eHaL\nmR=e-wL\nmR=e¯mR=e¯+wL\nmL=e-wLmL=e\nmL=e¯+wL\nmL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV\n-0.50.00.51.01.52.0-6-4-2024\ne0@eVDb@10-4D\nmR=eHbL\nmR=e-wLmR=e¯+wL\nmR=e¯\nmL=e\nmL=e-wL\nmL=e¯+wL\nmL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV\nFIG. 8: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the position of the electronic\nlevel\u000f0for different tunneling rates \u0000at zero temperature. The applied bias voltage is eV=\u0016L\u0000\u0016R, with\u0016R= 0. Other\nparameters are \u0000L= \u0000R= \u0000=2,\u000f\"\u0000\u000f0= 0:24eV,S= 100,J= 0:005eV,JSz= 0:4eV, and!L= 0:08eV. In the case of the\nsmallest value of \u0000(red lines), there are four regions in which the Gilbert damping and the change of the precession frequency\noccur. In each of these regions \u000f0satisfies the inequalities (25) and (26), and \u000bapproaches a constant value, while \fhas one\nlocal maximum and one local minimum.\na maximum for the Gilbert damping coefficient \u000b, and\na negative minimum value for the \fcoefficient. As !L\nincreases, the inequalities of Eqs. (25) and (26) are sat-\nisfied and the tunnel processes shown in Fig. 6(e) may\noccur. Hence, there is a contribution to the STT, but\nas is shown in Eq. (23), the Gilbert damping decreases\nwith increasing precession frequency. At larger values of\n!L, resulting in \u000f#+!L=\u0016L, the Gilbert damping co-\nefficient has a step increase towards a local maximum,\nwhile the coefficient \fhas a local maximum, as a conse-\nquence of the enhancement of the STT due to additional\nspin-flip processes occurring in this case. For even larger\nvalue of!L, the conditions (25) and (26) are no longerfulfilled and both coefficients vanish. It is energetically\nunfavorable to flip the spin of an electron against the\nantiparallel direction of the effective constant magnetic\nfieldBc\ne\u000b. Hence, as !Lincreases, more energy is needed\nto flip the electronic spin to the direction of the field.\nThis causes \u000bto decrease with increasing !L. Addition-\nally, the larger the ratio !L=\u0000, the less probable it is that\nspin-angular momentum will be exchanged between the\nmolecular spin and the itinerant electrons. For !L= 0,\nthe molecular spin is static, i.e.,_~S= 0. In this case\n~T(t) =~0. The coefficient \u000bthen drops to zero while the\ncoefficient\freaches a negative local maximum which is\nclose to 0. Both \u000band\freach an extremum value for9\n-4-20240123\nwL@e0Da@10-4DHaL\nmL=e-wL\nmL=emL=e¯\nmL=e¯+wL\nG=5e0G=3e0G=0.2e0G=0.02e0\n-4-2024-6-4-20\nwL@e0Db@10-4DHbL\nmL=e¯+wLmL=emL=e¯\nmL=e-wL\nG=5e0G=3e0G=0.2e0G=0.02e0\nFIG. 9: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the precession frequency !L=\ng\u0016BBcof the spin ~Sof the SMM, with ~Bc=Bc~ ez, for different tunneling rates \u0000at zero temperature. The applied bias voltage\niseV=\u0016L\u0000\u0016R= 2\u000f0, with\u0016R= 0. The other parameters are the same as in Fig. 7. In the case of the smallest \u0000(red lines),\nthe coefficient \u000bhas a step increase towards a local maximum while the coefficient \fhas a local maximum or minimum at a\nvalue of!Lcorresponding to a resonance of \u0016Lwith one of the levels in the QD.\nlarge values of \u0000at this point. For !L<0and\u0000\u001cj!Lj\n(red lines), at the value of !Lfor which\u0016L=\u000f\"\u0000!L, the\ncoefficient\u000bhasastepincreasetowardsalocalmaximum\nwhilethecoefficient \fhasanegativelocalminimum. The\ncoefficient\u000bthen decreases with a further decrease of !L\nas long as\u000f#\u0014\u0016L\u0014\u000f\"\u0000!L. At the value of !Lfor\nwhich\u0016L=\u000f#,\u000bhas another step increase towards a\nlocal maximum while \fhas a maximum value. Accord-\ning to Eq. (23), the Gilbert damping also does not occur\nif~Sis perpendicular to ~Bc. In this case \f.0and the\nonly nonzero torque component \f_~S(t)acts in the oposite\ndirection than the molecular spin’s rotational motion.\nIV. CONCLUSIONS\nIn this paper we have first theoretically studied time-\ndependent charge and spin transport through a small\njunction consisting of a single-level quantum dot cou-\npled to two noninteracting metallic leads in the pres-\nence of a time-dependent magnetic field. We used the\nKeldysh nonequilibrium Green’s functions method to de-\nrive the charge and spin currents in linear order with\nrespect to the time-dependent component of the mag-\nnetic field with a characteristic frequency !. We then\nfocused on the case of a single electronic level coupled\nvia exchange interaction to an effective magnetic field\ncreated by the precessional motion of an SMM’s spin in\na constant magnetic field. The inelastic tunneling pro-\ncesses that contribute to the spin currents produce an\nSTT that acts on the molecular spin. The STT con-\nsists of a Gilbert damping component, characterized bythe coefficient \u000b, as well as a component, characterized\nby the coefficient \f, that acts as an additional effective\nconstant magnetic field and changes the precession fre-\nquency!Lof the molecular spin. Both \u000band\fdepend\non!Land show a nonmonotonic dependence on the tun-\nneling rates \u0000. In the weak coupling limit \u0000\u001c!L,\u000b\ncan be switched on and off as a function of bias and gate\nvoltages. The coefficient \fcorrespondingly has a local\nextremum. For \u0000!0, both\u000band\fvanish. Taking\ninto account that spin transport can be controlled by the\nbias and gate voltages, as well as by external magnetic\nfields, our results might be useful in spintronic applica-\ntions using SMMs. Besides a spin-polarized STM, it may\nbe possible to detect and manipulate the spin state of an\nSMM in a ferromagnetic resonance experiment56–59and\nthus extract information about the effects of the current-\ninduced STT on the SMM. Our study could be com-\nplemented with a quantum description of an SMM in a\nsingle-molecule magnet junction and its coherent prop-\nerties, as these render the SMM suitable for quantum\ninformation storage.\nAcknowledgments\nWe gratefully acknowledge discussions with Mihajlo\nVanevićandChristianWickles. 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Saitoh, Nature\nMater. 10, 655 (2011)."    },    {        "title": "1007.3508v1.Precessing_vortices_and_antivortices_in_ferromagnetic_elements.pdf",        "content": "Precessing vortices and antivortices in ferromagnetic elements\nA. Lyberatos,1S. Komineas,2and N. Papanicolaou3, 4\n1Department of Materials Science,University of Crete,PO BOX 2208,71003 Heraklion,Greece\n2Department of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece\n3Department of Physics, University of Crete, 71003 Heraklion, Crete, Greece\n4Institute for Theoretical and Computational Physics, University of Crete, Heraklion, Greece\nA micromagnetic numerical study of the precessional motion of the vortex and antivortex states\nin soft ferromagnetic circular nanodots is presented using Landau-Lifshitz-Gilbert dynamics. For\nsu\u000eciently small dot thickness and diameter, the vortex state is metastable and spirals toward the\ncenter of the dot when its initial displacement is smaller than a critical value. Otherwise, the vor-\ntex spirals away from the center and eventually exits the dot which remains in a state of in-plane\nmagnetization (ground state). In contrast, the antivortex is always unstable and performs damped\nprecession resulting in annihilation at the dot circumference. The vortex and antivortex frequencies\nof precession are compared with the response expected on the basis of Thiele's theory of collective\ncoordinates. We also calculate the vortex restoring force with an explicit account of the magne-\ntostatic and exchange interaction on the basis of the 'rigid' vortex and 'two-vortices side charges\nfree' models and show that neither model explains the vortex translation mode eigenfrequency for\nnanodots of su\u000eciently small size.\nPACS numbers: 75.70.Kw,75.75.Fk,75.75.Jn,75.78.Cd,75.78.Fg\nI. INTRODUCTION\nThe vortex state is one of the equilibrium states of thin\nsoft ferromagnetic elements of micrometer size and below\n(magnetic dots). The interplay between the magneto-\nstatic and exchange energy favours an in-plane, closed\n\rux domain structure with a 10 \u000020 nm central core,\nwhere the magnetization turns out of plane to avoid the\nhigh energetic cost of anti-aligned moments. Core rever-\nsal can be triggered by application of an in-plane pulsed\n\feld or pulsed current allowing the possibility of applica-\ntion of patterned thin \flm elements in data storage and\nmagnetic and magneto-electronic random access mem-\nory [1]. Core reversal is usually assumed to arise from\nthe spontaneous creation of a vortex-antivortex (VA) pair\n(vortex dipole) of opposite polarity with respect to the\noriginal vortex, followed by collision of the pair with the\noriginal vortex. A fundamental understanding of the dy-\nnamics of vortices and antivortices is therefore necessary\nto control the switching of the magnetization.\nThe basic excitation mode of the vortex or antivortex\nstate from its equilibrium position is in-plane gyrotropic\nmotion. It is a low frequency (GHz) mode corresponding\nto the displacement of the whole structure. The gener-\nalized dynamic force can be determined using Thiele's\ncollective-variable approach [2]-[3]. Theoretical [4]-[5]\nand experimental [6] studies of the dynamics of magnetic\nvortices in 2D \flms have shown a connection with the\ntopology of the magnetization structure . Magnetic vor-\ntices con\fned in circular dots can be described by ana-\nlytical models based on di\u000berent methods for accounting\nfor the magnetostatic interaction [7]-[9]. The vortex may\nbe 'rigid' or deform so that no magnetic charges appear\nat the side of the cylinder. The latter (two-vortices side\ncharge free model) provides a good description of the dy-\nnamic behavior of vortices in submicron-sized permalloydots, in particular the increase of vortex eigenfrequency\nwith dot aspect ratio L=R, whereLis the dot thickness\nandRis the dot radius. The basic assumption in these\ncalculations is that the vortex displacement lfrom equi-\nlibrium, at the dot center, is small l<<R .\nThe main objective here is to consider the response of\nthe vortex state to large perturbations, induced for in-\nstance by thermal activation. We focus our attention to\ndots of su\u000eciently small radius and thickness so that the\nvortex state is metastable . Using micromagnetic calcu-\nlations it is shown that vortex stability can be de\fned\nin terms of a critical displacement lcleading to an irre-\nversible transition to the ground state characterized by\nin-plane magnetization. We test the accuracy of the col-\nlective coordinate representation for vortex and antivor-\ntex dynamics and discuss the limitations of the approxi-\nmate analytical models [8],[9].\nII. VORTEX AND ANTIVORTEX PRECESSION\nThe dynamical behavior of a single vortex or antivor-\ntex trapped in a ferromagnetic nanodot was studied\nusing a \fnite-di\u000berence Landau-Lifshitz-Gilbert micro-\nmagnetic model. The material parameters for permal-\nloy (Ni 80Fe20) were used in the calculations: saturation\nmagnetization Ms= 800 emu/cc and exchange sti\u000b-\nness coe\u000ecient A= 1:3\u000210\u00006erg/cm. The mag-\nnetic anisotropy was neglected and the exchange length\nislex=p\nA=(2\u0019M2s) = 5:7 nm. The disk thickness is in\nthe rangeL= 5\u000010 nm , comparable to the exchange\nlength, so the magnetization dependence along the dot\nnormal can be neglected to a \frst approximation. The\ndisk radius is not large compared to the vortex core and\nis initially taken as R= 30 nm. Inspection of the phase\ndiagram of magnetic ground states for this material [10]arXiv:1007.3508v1  [cond-mat.mtrl-sci]  20 Jul 20102\nFIG. 1: a) Vortex snd b) antivortex structure in permalloy\ndot of thickness L= 5 nm and radius R= 30 nm.\nindicates that for the particular choice of disk dimensions,\nin-plane magnetization is the ground state of the system.\nThe integration of the LLG equation is carried out using\na damping parameter \u000b= 0:01, which is appropriate for\npermalloy [11].\nThe initial vortex or antivortex structure is de\fned as\nfollows: Using the coordinate system with the z axis par-\nallel to the dot cylindrical axis, the magnetization compo-\nnents aremz=\u0015cos \u0002(\u001a);mx+imy= sin \u0002 exp( i\u0014(\u001e\u0000\n\u001eo)), where m(r) =M(r)=Msis the reduced magneti-\nzation,\u001a;\u001eare cylindrical coordinates, \u0015=\u00061 is the\nvortex polarity, and \u0002( \u001a) is the magnetization angle to\nthe normal (z) direction. The vortex number is \u0014= 1\nfor a vortex and \u0014=\u00001 for an antivortex structure.\nThe constant \u001eo=\u0006\u0019=2 de\fnes the chirality i.e. the\ndirection of the curl of the vortex. The speci\fc choice\nof chirality is largely insigni\fcant for vortex dynamics,\nin contrast to the vortex number \u0014and polarity \u0015which\nplay an important role.\nThe initial pro\fle \u0002( \u001a) is assumed to have the modelform cos \u0002 = (cosh( c\u001a=lex))\u00001which is appropriate for\neasy-plane ferromagnets [12]. The parameter cde\fnes\nthe core radius b. The core radius is determined by the\ndot thickness Land the exchange length lexbut is inde-\npendent of the dot radius [13]. In the following, dot thick-\nness isL= 5 nm, unless speci\fed otherwise. Numerical\ncalculations of relaxed vortex and antivortex structures\nat the dot center were found to be in good agreement\nwith the model pro\fle for c'1. Typical examples of a\nvortex and antivortex structure are shown in Fig. 1. It\nshould be noted that other choices for the vortex pro-\n\fle are possible [14], for instance tan(\u0002 =2) =\u001a=bwas\nobtained by Usov et al. [15] using a variational proce-\ndure to minimize the exchange energy whereas the vor-\ntex core radius bwas determined by minimization also\nof the magnetostatic energy. This form is employed in\nanalytical calculations of vortex states in ferromagnetic\ndisks [7]-[9], however, micromagnetic modelling by Scholz\net al. [10], con\frmed by our calculations, has shown that\nit underestimates the core radius b, de\fned in Ref.[10] as\nthe \frstmz= 0 crossover from the vortex center.\nThe dynamical behavior of the magnetic vortex ( \u0014= 1)\ndepends on the initial displacement l(t= 0) =lofrom\nthe disk center. If the displacement is equal or smaller\nthan some critical value lc= 0:52R, the damped preces-\nsion of the vortex leads to relaxation to the disk center.\nFor positive vortex polarity \u0015= 1, an anticlockwise pre-\ncession is observed as shown in Fig. 2a. At the start of\nthe simulation the precession is not smooth as a result of\nthe internal relaxation (deformation) of the model vortex\nto minimize the energy during precession. If lo>lc, the\ndamped precession is clockwise and the distance from the\ndot center increases, as shown in Fig. 2b, until the vor-\ntex is annihilated and the magnetization is aligned along\nthe in-plane direction with quasi-uniform magnetization,\nthe so-called 'leaf' state [16] which is the ground state\nof the system. Irregularities in the precessional motion\narise from the uncertainty on the position of the vortex.\nThe antivortex instead is always unstable. For any\nchoice of initial displacement lo, the antivortex performs\ndamped precession to the edge of the disk and is an-\nnililated. For positive polarity, anticlockwise precession\nis observed. The sense of gyrotropic motion of a vortex\nor antivortex is switched on reversal of the polarity.\nIII. THE COLLECTIVE COORDINATE\nAPPROACH\nThe damped precession of the vortex or antivortex can\nbe described using Thiele's equation [2],[3] augmented by\na dissipative term.\nG\u0002dl\ndt+ 2QG\u0011dl\ndt\u0000@E(l)\n@l= 0 (1)\nwhere l= (lx;ly) is the position of the vortex center\nandE(lx;ly) is the potential energy of the shifted vor-3\n-0.6-0.4-0.20.00.20.40.6-0.6-0.4-0.20.00.20.40.6(a)\n   y/Rx/R\n-0.8-0.6-0.4-0.20.00.20.40.60.8-0.8-0.6-0.4-0.20.00.20.40.60.8\n   y/Rx/R(b)\nFIG. 2: Trajectory of a vortex of positive polarity \u0015= 1 in\nzero \feld for a time interval 0 < t < 2\u0002104. The initial\nposition vector of the vortex is a) lo= (0:52R;0) and b)\n(0:53R;0). The anisotropy is neglected and the damping is\n\u000b= 0:01.\ntex. The gyroforce G\u0002dl=dtdepends on the topolog-\nical structure of the magnetization and is proportional\nto the gyrovector G=\u0000G^z, where the gyroconstant is\nG= 2\u0019\u0014\u0015LMs=\rand\r= 1:76\u0002107rad Oe\u00001s\u00001is the\ngyromagnetic ratio. Q=\u00001\n2\u0014\u0015is the skyrmion number\nand\u0011is the dissipation constant.\nFor axially symmetric energy potential E=E(l) wherel=q\nl2x+l2y,@E=@lx=E0lx=land it is straightforward\nto show that\n_lx\u00002Q\u0011_ly=\u0000!ly (2)\n2Q\u0011_lx+_ly=!lx (3)\nwhere the angular frequency is\n!=1\nGl@E\n@l(4)\nThe vortex motion in complex form is given by\n(1 + 2Q\u0011i)\u0010\n_lx+i_ly\u0011\n=i!(lx+ily) (5)\nIntroducing polar coordinates lx+ily=lei\u001e\n_l=2Q\u0011\n1 +\u00112!(l)l (6)\n_\u001e=!(l)\n1 +\u00112(7)\nDividing and integrating over the time interval of the\ndamped precession, the time dependence of the preces-\nsion angle is\n\u001e(t) =2Q\n\u0011ln\u0012l\nlo\u0013\n(8)\nwherelois the initial displacement of the vortex center.\nThe clockwise or anticlockwise sense of gyration is there-\nfore dependent on the skyrmion number Q.\nMicromagnetic simulations of vortex motion were per-\nformed and the position of the vortex center ( lx;ly) was\ndetermined by a method of interpolation for the position\nof maximum mz. The precession angle \u001e= arctan(ly=lx)\nwas found to vary linearly with the logarithm of the vor-\ntex shift, in agreement with Eq.(8). Fig. 3 shows numeri-\ncal data for damped precession of a vortex of positive po-\nlarity (Q=\u00001=2) with initial displacement lo= 0:52R.\nThe relaxation to the disk center involves many revolu-\ntions (Fig. 2a) and the gradient \u00001=\u0011provides an accu-\nrate estimate of the dissipation constant \u0011= 0:013. It is\nevident that for permalloy nanodots, the damping in the\nvortex motion is weak and the angular frequency of pre-\ncession in Eq.(7) can be approximated using !'d\u001e=dt .\nFitting the numerical \u001e(t) curve to a 4th order poly-\nnomial, the time variation of the angular frequency !(t)\ncan be determined. The vortex shift l(t) exhibits oscilla-\ntions that are neglected by \ftting to a 4th order polyno-\nmial and the !(l) dependence is obtained using the !(t)\ncurve. The same procedure is employed for damped pre-\ncession leading to vortex annihilation ( lo= 0:53R). The4\nFIG. 3: Azimuthal angle \u001eof the vortex position as a function\nof the logarithm of the reduced vortex displacement s=l=R.\nThe initial radial position is lo= 0:52R.\nFIG. 4: Precession frequency of a vortex !=2\u0019as a function\nof the reduced diplacement s=l=Rfrom the center of the\ndot.\ncombined results for the dependence of the frequency of\nprecession!=2\u0019on reduced vortex shift s=l=Rare il-\nlustrated in Fig. 4.\nFor small displacement of the vortex center from its\nequilibrium position ( l= 0), the potential energy is\nE(l) =E(0) + (1=2)\u0014l2, where\u0014is the sti\u000bness coe\u000e-\ncient and the eigenfrequency is !o=\u0014=G [9]. At the\ncritical displacement lc, corresponding to a maximum in\nthe potential energy E(l) the precession frequency van-\nishes.\n0.00.10.20.30.40.50.60.70.80.1000.1020.1040.1060.1080.1100.1120.1140.116\n   Reduced energy density  e\nReduced vortex displacement  seMAX\nsc(a)\n0.00.10.20.30.40.50.60.70.80.90.000.020.040.060.080.100.12eeexReduced energy density  \nReduced vortex shift  sed(b)FIG. 5: (a) Reduced energy density of a permalloy dot\n\u000f=E=4\u0019M2\nsVas a function of normalized vortex displace-\nments. Results are shown for two sets of micromagnetic simu-\nlations (markers) and analytical (solid line) calculations using\nThiele's collective variable theory (Eq.10). The maximum en-\nergy density \u000fmaxoccurs at vortex displacement sc. (b) The\ncontribution of the magnetostatic and exchange terms to the\ntotal energy, obtained from micromagnetic calculations.\nThe motion of vortices and antivortices is driven by\nthe restoring force @E=@ l(Eq.1). The potential energy\nof the shifted vortex is axially symmetric E=E(l) and\ncan be written\nE(l) =E(0) +GZl\n0!(\u001a)\u001ad\u001a (9)\nA simpler form in terms of the reduced energy density\n\u000f=E=4\u0019M2\nsVover the dot volume Vis\n\u000f(l) =\u000f(0)\u00004QZs\n0!(s0)s0ds0(10)5\nwheres0=\u001a=R,s=l=Ris the reduced displacement\nof the vortex and the time associated with !is scaled\nby\u001co= 1=4\u0019\rMs. The function !(s0) was determined\nfrom micromagnetic simulations of the vortex precession\nand using Eq(10) the curve \u000f(l) expected from appli-\ncation of the theory of collective coordinates, was ob-\ntained, as shown in Fig. 5a. Superimposed is the energy\nevaluated directly from simulations of vortex relaxation.\nThiele's theory appears to provide a good description of\nthe vortex precession despite its limitations, for instance,\na) Thiele's approach is known to be an approximate de-\nscription of vortex dynamics in in\fnite \flms b) the vortex\nis here con\fned in a nanodot ( R= 5:3lex) and c) the vor-\ntex structure does not remain rigid during the relaxation\nprocess but is modi\fed as a result of the change in the\ndistribution of the demagnetizing \felds. The magneto-\nstatic and exchange contribution to the energy variation\n\u000f(l) was obtained from micromagnetic calculations and\nis shown in Fig. 5b. It should be noted that incorporat-\ning the demagnetizing energy to the total anisotropy, is\nstrictly valid for in\fnite thin \flms and results in mono-\ntonically decreasing energy \u000f(l), as reported in Ref. [17].\nThe oscillations in the energy variation during precession\nare related to the \fnite micromagnetic grid.\nThe potential energy of the vortex attains a maximum\nvalue at some critical value of the displacement sc= 0:52\ncorresponding to a zero crossover of the precession fre-\nquency!(Fig. 4). The stability of the vortex at the dot\ncenter arises from the magnetostatic energy, in particular\nthe volume magnetic charges resulting from vortex defor-\nmation and the surface charges at the side of the cylinder\n[9]. The face charges do not depend on ssince the charge\ndistribution on the top and bottom surfaces of the disk is\nunchanged with the vortex displacement. The exchange\nenergy decreases with increasing vortex shift s[9]. The\nmagnetostatic and exchange contributions to the restor-\ning force are in exact balance at the point of maximum\nenergy.\nA similar analysis was carried out for an antivortex\nstructure in a dot of identical dimensions. The potential\nenergy decreases monotonically with increasing displace-\nment s, as shown in Fig. 6. Application of the collec-\ntive coordinates treatment results in the solid curve in\nFig. 6 of slightly smaller curvature. The vortex insta-\nbility arises from the uncompensated magnetic charge\ndistribution within the antivortex core (Fig. 1b), so the\nmagnetostatic energy is reduced upon motion away from\nthe dot center. The precession frequency increases dur-\ning the relaxation process as shown in Fig. 7, as a result\nof the steeper energy gradient for large displacement s.\nAssuming identical position, it is evident that the an-\ntivortex precesses faster than the vortex as a result of\nthe larger magnetostatic energy gradient.\nFIG. 6: Energy density \u000f=E=4\u0019M2\nsVvs antivortex displace-\nments. The notation is similar to Fig. 5a.\nFIG. 7: Antivortex precession frequency !=2\u0019as a function of\nthe reduced diplacement s=l=Rfrom unstable equilibrium\nposition at the dot center.\nIV. DEPENDENCE OF VORTEX PRECESSION\nON DISK SIZE\nThe maximum potential energy Emax of the shifted\nmagnetic vortex, evaluated from plots such as Fig. 5a,\ndepends on the radius of the dot, where it is con\fned.\nIn Fig. 8, micromagnetic calculations of the reduced dot\nenergy density \u000fare shown as function of dot radius R,\nscaled by the exchange length. The curves correspond\nto the maximum vortex energy Emax and the minima\nassociated with the three equilibrium states of the mag-\nnetization (in-plane,perpendicular,vortex). The vortex6\nFIG. 8: Micromagnetic calculations of scaled dot energy den-\nsity\u000fvs dot radius R(in units of lex) for the three equilibrium\nstates of the magnetization and the vortex state of maximum\nenergy.\nFIG. 9: Barrier to vortex escape ( \u000fB), de\fned by the relation\n\u000fB=\u000fmax\u0000\u000fvortex as a function of dot radius (in units of\nlex).\nstate is unstable for small dots R < Rs, metastable for\nRs< R < R eqand a ground state for R > Reqwhere\nthe values Rs= 2:5lexandReq= 16lexare obtained\nfor the absolute and equilibrium single domain radius re-\nspectively. The corresponding variation of vortex barrier\nenergy\u000fB=Emax\u0000Evortex and displacement lc=scR\nfor vortex escape are shown in Figs. 9 and 10 respectively.\nFor small dots R= 3:5lex, the vortex is unstable and any\nshift from equilibrium at the dot center results in relax-\nation to the ground state (in-plane magnetization). The\nFIG. 10: Reduced displacement for maximum energy of the\nvortex state scas a function of dot radius (in units of lex).\ncritical size for vortex instability is larger than Rssince\nthe latter is de\fned assuming a random perturbation dif-\nferent than shifting the whole vortex. The vortex barrier\nenergy increases for larger dots and attains a maximum\nvalue, forR > 10lex, related to the vortex annihilation\n\feld [7]. The displacement lcfor vortex escape increases\nwith disk size to the maximum value imposed by the\ndisk perimeter ( lc=R!1), attained for sub-micron dots\nR>>lex. For su\u000eciently large dots, the vortex is within\nthe domain of attraction of the dot center, irrespective\nof the initial position.\nMicromagnetic calculations of the dependence of the\nfundamental vortex eigenfrequency !o=2\u0019, obtained for\nsmall perturbation s<< 1 as in Fig. 4, on the dot aspect\nratio\f=L=R are shown in Fig. 11. The dot thickness\nwas \fxed (L= 10 nm) and the radius Rwas allowed to\nvary. The eigenfrequency attains a maximum value at\n\f= 0:35 and vanishes for smaller radius \f'0:5 when\nthe vortex becomes unstable. The maximum value arises\nfrom the change in the relative contribution of the magne-\ntostatic and exchange energy to the sti\u000bness coe\u000ecient\n\u0016. For instance, the eigenfrequency assuming a 'rigid'\nvortex is [9]\n!o=\u0019\rMs\nQ\u0014\nF1(\f)\u00001\n(R=lex)2\u0015\n(11)\nwhereF1(x) =R\ndtt\u00001f(xt)J2\n1(t) corresponds to the av-\neraged in-plane dot demagnetizing factor, f(x) = 1\u0000\n[1\u0000exp(\u0000x)]=xandJ1is the Bessel function. Using\nthe approximation F1(\f)'(\f=2\u0019)[ln(8=\f)]\u00001=2], valid\nfor\f << 1, it can be shown that the eigenfrequency is\nmaximum at radius R= 4\u0019=L[ln(8=\f)\u00003=2]\u00001where\nall lengths are in units of lex. Previous studies were re-\nstricted to sub-micron sized dots where the second term7\nFIG. 11: Micromagnetic (markers) and analytical (solid lines)\ncalculations of vortex eigenfrequency !o=2\u0019as a function of\ndot aspect ratio \f=L=R a) the rigid vortex model and b)\nthe two-vortices model. The dot thickness is here L= 10 nm.\nof Eq.(11) could be neglected [9], so a monotonically in-\ncreasing eigenfrequency !o(\f) arising from the magneto-\nstatic energy only was reported.\nThe micromagnetic calculations are compared in\nFig. 11 with the curves obtained using the 'rigid' vor-\ntex and 'two-vortices' approximations for the magneto-\nstatic energy. For sub-micron sized disks with aspect\nratio\f < 0:05 corresponding to a radius R > 200 nm,\nthe micromagnetic calculations are in good quantitative\nagreement with the 'two vortices' model, as reported in\nRef.[9]. In this regime, the eigenfrequency is determined\nprimarily by the magnetostatic energy. The rigid vor-\ntex approximation fails to describe the dynamic behav-\nior since the magnetostatic energy can be decreased by\nelimination of the surface charges at the disk perime-\nter at the expense of some contribution from volume\nmagnetic charges arising from vortex deformation. Forsmaller disks R < 200 nm, the vortex eigenfrequency is\nbetween the predictions of the two models. A reduction\nin side charges occurs but is not complete as a result of\nthe large expense in exchange energy arising from vortex\ndeformation. Similar results can be obtained in principle\nfor thinner disks, however, the 'two vortices' approxima-\ntion is then valid for larger cylinders which are not easily\namenable to micromagnetic simulations.\nV. CONCLUSIONS\nMicromagnetic calculations were carried out of the pre-\ncessional behaviour of a single magnetic vortex or an-\ntivortex con\fned in a permalloy circular nanodot. The\nexistence of two domains of attraction for the vortex state\nare identi\fed arising from a maximum in the potential\nenergy of the shifted vortex. This e\u000bect is atrributed\nto the competition between the magnetostatic attractive\nand exchange repulsive forces on the shifted vortex. An-\ntivortices instead are always unstable and trace a spi-\nral trajectory of increasing distance from the dot center\nfollowed by annihilation at the dot envelope. The pre-\ncessional behaviour of vortices and antivortices is satis-\nfactorily described by Thiele's theory of collective coor-\ndinates, relating the angular frequency of precession to\nthe gradient of the potential energy. For small nano-\nsized dots, however, the 'rigid' vortex and 'two-vortices'\napproximation for the magnetostatic energy is not sat-\nisfactory and the exchange forces have a signi\fcant ef-\nfect on the translation mode vortex eigenfrequency. For\nantivortices, the development of an analytical model to\naccount for the magnetostatic interaction in circular dots\nis clearly needed to provide further insight on the results\nof our micromagnetic calculations.\nVortex stability is necessary in applications of nanos-\ntructured patterned media for data storage. Microfabri-\ncation downscaling implies that the displacement lcand\nassociated energy barrier may become useful characteri-\nzation parameters of the thermal stability of the recorded\ninformation.\n[1] B. Van Waeyenberge, A. Puzic, H. Stoll, K.W. Chou, T.\nTyliszczak, R. Hertel, M. F ahnle, H. Br uckl, K. Rott, G.\nWeiss, I. Neudecher, D. Weiss, C.H. Back and G. Sch utz,\nNature (London) 444, 461 (2006).\n[2] A.A. Thiele, PHys. Rev. Lett. 30, 230 (1973).\n[3] D.L. Huber, Phys. Rev. B 26, 3758 (1982).\n[4] N. Papanicolaou and T.N. Tomaras, Nucl. Phys. B 360,\n425 (1991).\n[5] S. Komineas and N. Papanicolaou, Physica (Amsterdam)\n99D , 81 (1996).\n[6] S.-B Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran,\nJ. St ohr and H.A. Padmore, Nature 304, 420 (2004).\n[7] K. Yu Guslienko, V. Novosad, Y. Otani, H. Shima andK. Fukamichi, Appl. Phys. Lett. 78, 3848 (2001).\n[8] K.L. Metlov and K. Yu Guslienko, J. Magn. Magn.\nMater. 242-245 , 1015 (2002).\n[9] K. Yu Guslienko, B.A. Ivanov, V. Novosad, Y. Otani, H.\nShima and K. Fukamichi, J. Appl. Phys. 91, 8037 (2002).\n[10] W. Scholz, K. Yu Guslienko, V. Novosad, D. Suess,\nT.Schre\r, R.W. Chantrell and J. Fidler, J. Magn. Magn.\nMater. 266, 155 (2003).\n[11] D.V. Berkov and J. Miltat, J. Magn. Magn. Mater. 320,\n1238 (2008).\n[12] S. Komineas and N. Papanicolaou, in Electromagnetic,\nmagnetostatic and exchange-interaction vortices in con-\n\fned magnetic structures , edited by E.O. Kamenetskii,8\n(Transworld Research Network, Kerala,2008).\n[13] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T.\nOno, Science 289, 034318 (2005).\n[14] P. Landeros, J. Escrig, D. Altbir, D. Laroze, J. d' Albu-\nquerque e Castro and P. Vargas, Phys. Rev. B 71, 094435\n(2005)\n[15] N.A. Usov and S.E. Peschany, J. Magn. Magn. Mater.118, L290 (1993).\n[16] K.L. Metlov and K. Yu Guslienko, Phys. Rev. B 70,\n052406 (2004)\n[17] D.D. Sheka, J.P. Zagorodny, J. Caputo, Y. Gaididei and\nF.G. Mertens, Phys. Rev. B 71, 134420 (2005)"    },    {        "title": "1412.1988v1.Calculating_linear_response_functions_for_finite_temperatures_on_the_basis_of_the_alloy_analogy_model.pdf",        "content": "arXiv:1412.1988v1  [cond-mat.mtrl-sci]  5 Dec 2014Calculating linear response functions for finite temperatu res on the basis of the alloy\nanalogy model\nH. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar, and D . K¨ odderitzsch\nDepartment Chemie/Phys. Chemie, Ludwig-Maximilians-Uni versit¨ at M¨ unchen,\nButenandtstrasse 5-13, D-81377 M¨ unchen, Germany\n(Dated: 8th December 2014)\nA scheme is presented that is based on the alloy analogy model and allows to account for thermal\nlattice vibrations as well as spin fluctuations when calcula ting response quantities in solids. Various\nmodels to deal with spin fluctuations are discussed concerni ng their impact on the resulting tem-\nperature dependent magnetic moment, longitudinal conduct ivity and Gilbert damping parameter.\nIt is demonstrated that using the Monte Carlo (MC) spin config uration as an input, the alloy ana-\nlogy model is capable to reproduce results of MC simulations on the average magnetic moment\nwithin all spin fluctuation models under discussion. On the o ther hand, response quantities are\nmuch more sensitive to the spin fluctuation model. Separate c alculations accounting for either the\nthermal effect due to lattice vibrations or spin fluctuations show their comparable contributions\nto the electrical conductivity and Gilbert damping. Howeve r, comparison to results accounting for\nboth thermal effects demonstrate violation of Matthiessen’ s rule, showing the non-additive effect of\nlattice vibrations and spin fluctuations. The results obtai ned for bcc Fe and fcc Ni are compared\nwith theexperimental data, showing rather good agreement f or thetemperature dependentelectrical\nconductivity and Gilbert damping parameter.\nI. INTRODUCTION\nFinite temperature has often a very crucial influence\non the response properties of a solid. A prominent ex-\nample for this is the electrical resistivity of perfect non-\nmagnetic metals and ordered compounds that only take\na non-zero value with a characteristic temperature ( T)\ndependence due to thermal lattice vibrations. While the\nHolstein transport equation1,2provides a sound basis for\ncorresponding calculations numerical work in this field\nhas been done so far either on a model level or for sim-\nplified situations.3–6In practice often the Boltzmann-\nformalism is adopted using the constant relaxation time\n(τ) approximation. This is a very popular approach in\nparticular when dealing with the Seebeck effect, as in\nthis case τdrops out.7,8The constant relaxation time\napproximation has also been used extensively when deal-\ning with the Gilbert damping parameter α.9–11Within\nthe description of Kambersky10,12the conductivity- and\nresistivity-like intra- and inter-band contributions to α\nshow a different dependency on τleading typically to\na minimum for α(τ) or equivalently for α(T).10,11A\nscheme to deal with the temperature dependent resistiv-\nity that is formally much more satisfying than the con-\nstant relaxation time approximation is achieved by com-\nbining the Boltzmann-formalism with a detailed calcula-\ntion of the phonon properties. As was shown by various\nauthors,13–16this parameter-free approach leads for non-\nmagneticmetalsingeneraltoaverygoodagreementwith\nexperimental data.\nAs an alternative to this approach, thermal lattice\nvibrations have also been accounted for within various\nstudies by quasi-static lattice displacements leading to\nthermallyinducedstructuraldisorderinthesystem. This\npoint of view provides the basis for the use of the al-\nloy analogy, i.e. for the use of techniques to deal withsubstitutional chemical disorder also when dealing with\ntemperature dependent quasi-static random lattice dis-\nplacements. An example for this are investigations on\nthe temperature dependence of the resistivity and the\nGilbert parameter αbased on the scattering matrix ap-\nproach applied to layered systems.17The necessary aver-\nageovermanyconfigurationsoflatticedisplacementswas\ntakenbymeansofthe supercelltechnique. Incontrastto\nthistheconfigurationalaveragewasdeterminedusingthe\nCoherent Potential Approximation (CPA) within invest-\nigations using a Kubo-Greenwood-like linear expression\nforα.18The same approach to deal with the lattice dis-\nplacements was also used recently within calculations of\nangle-resolved photo emission spectra (ARPES) on the\nbasis of the one-step model of photo emission.19\nAnother important contribution to the resistivity in\nthe case of magnetically ordered solids are thermally in-\nduced spin fluctuations.20Again, the alloy analogy has\nbeen exploited extensively in the past when dealing with\ntheimpactofspinfluctuationsonvariousresponsequant-\nities. Representing a frozen spin configuration by means\nof super cell calculations has been applied for calcula-\ntions of the Gilbert parameter for α17as well as the\nresistivity or conductivity, respectively.17,21,22Also, the\nCPA has been used for calculations of α23as well as the\nresistivity.20,24A crucial point in this context is obvi-\nously the modeling of the temperature dependent spin\nconfigurations. Concerning this, rather simple models\nhave been used,23but also quite sophisticated schemes.\nHere one should mention the transfer of data from Monte\nCarlo simulations based on exchange parameters calcu-\nlated in an ab-initio way25as well as work based on the\ndisordered local moment (DLM) method.24,26Although,\nthe standard DLM does not account for transversal spin\ncomponents it nevertheless allows to represent the para-\nmagnetic regime with no net magnetization in a rigor-2\nous way.Also, for the magnetically ordered regime below\nthe Curie-temperature it could be demonstrated that the\nuncompensated DLM (uDLM) leads for many situations\nstill to goodagreementwith experimentaldata on the so-\ncalled spin disorder contribution to the resistivity.20,24\nIn the following we present technical details and exten-\nsionsofaschemethatwasalreadyused beforewhendeal-\ning with the temperature dependence of response quant-\nities on the basis of Kubo’s response formalism. Various\napplications will be presented for the conductivity and\nGilbert damping parameter accounting simultaneously\nfor various types of disorder.\nII. THEORETICAL FRAMEWORK\nA. Configurational average for linear response\nfunctions\nMany important quantities in spintronics can be\nformulated by making use of linear response formal-\nism. Important examples for this are the electrical\nconductivity,27,28the spin conductivity29or the Gilbert\ndamping parameter.18,30Restricting here for the sake of\nbrevity to the symmetric part of the corresponding re-\nsponse tensor χµνthis can be expressed by a correlation\nfunction of the form:\nχµν∝Tr/angbracketleftbigˆAµℑG+ˆAνℑG+/angbracketrightbig\nc. (1)\nIt should be stressed that this not a real restriction as\nthe scheme described below has been used successfully\nwhen dealing with the impact of finite temperatures on\nthe anomalous Hall conductivity of Ni.31In this case the\nmore complex Kubo-Stˇ reda- or Kubo-Bastin formulation\nfor the full response tensor has to be used.32\nThe vector operator ˆAµin Eq. (1) stands for example\nin case of the electrical conductivity σµνfor the cur-\nrent density operator ˆjµ28while in case of the Gilbert\ndamping parameter αµνit stands for the torque oper-\natorˆTµ.9,18Within the Kubo-Greenwood-like equation\n(1) the electronic structure of the investigated system\nis represented in terms of its retarded Green function\nG+(r,r′,E). Within multiple scattering theory or the\nKKR (Korringa-Kohn-Rostoker)formalism, G+(r,r′,E)\ncan be written as:33–35\nG+(r,r′,E) =/summationdisplay\nΛΛ′Zm\nΛ(r,E)τmn\nΛΛ′(E)Zn×\nΛ′(r′,E)(2)\n−δmn/summationdisplay\nΛZn\nΛ(r,E)Jn×\nΛ′(r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(r,E)Zn×\nΛ′(r′,E)Θ(rn−r′\nn).\nHerer,r′refer to points within atomic volumes around\nsitesRm,Rn, respectively, with Zn\nΛ(r,E) =ZΛ(rn,E) =\nZΛ(r−Rn,E) being a function centered at site Rn. Ad-\nopting a fully relativistic formulation34,35for Eq. (2) one\ngets in a natural way access to all spin-orbit inducedproperties as for example the anomalous and spin Hall\nconductivity29,32,36or Gilbert damping parameter.18In\nthis case, the functions Zn\nΛandJn\nΛstand for the reg-\nular and irregular, respectively, solutions to the single-\nsite Dirac equation for site nwith the associated single-\nsite scattering t-matrix tn\nΛΛ′. The corresponding scat-\ntering path operator τnn′\nΛΛ′accounts for all scattering\nevents connecting the sites nandn′. Using a suitable\nspinor representation for the basis functions the com-\nbined quantum number Λ = ( κ,µ) stands for the relativ-\nistic spin-orbit and magnetic quantum numbers κandµ,\nrespectively.34,35,37\nAs was demonstrated by various authors27,28,38rep-\nresenting the electronic structure in terms of the Green\nfunction G+(r,r′,E) allows to account for chemical dis-\norder in a random alloy by making use of a suitable al-\nloy theory. In this case ∝an}bracketle{t...∝an}bracketri}htcstands for the configura-\ntional average for a substitutional alloy concerning the\nsite occupation. Corresponding expressions for the con-\nductivity tensor have been worked out by Velick´ y27and\nButler28usingthe single-siteCoherentPotentialApprox-\nimation (CPA) that include in particular the so-called\nvertex corrections.\nThe CPA can be used to deal with chemical but also\nwith any other type of disorder. In fact, making use of\nthe different time scales connected with the electronic\npropagation and spin fluctuations the alloy analogy is\nexploited when dealing with finite temperature magnet-\nism on the basis of the disordered local moment (DLM)\nmodel.26,39Obviously, the same approach can be used\nwhen dealing with response tensors at finite temperat-\nures. In connection with the conductivity this is often\ncalled adiabatic approximation.40Following this philo-\nsophy, the CPA has been used recently also when calcu-\nlating response tensors using Eq. ( 1) with disorder in the\nsystem caused by thermal lattice vibrations18,31as well\nas spin fluctuations.20,41\nB. Treatment of thermal lattice displacement\nA way to account for the impact of the thermal dis-\nplacement of atoms from their equilibrium positions, i.e.\nfor thermal lattice vibrations, on the electronic struc-\nture is to set up a representative displacement configura-\ntion for the atoms within an enlargedunit cell (super-cell\ntechnique). In this case one has to use either a very large\nsuper-cell or to take the average over a set of super-cells.\nAlternatively, one may make use of the alloy analogy for\nthe averaging problem. This allows in particular to re-\nstrict to the standard unit cell. Neglecting the correla-\ntion between the thermal displacements of neighboring\natoms from their equilibrium positions the properties of\nthe thermal averaged system can be deduced by making\nuse of the single-site CPA. This basic idea is illustrated\nby Fig.1. To make use of this scheme a discrete set\nofNvdisplacement vectors ∆ Rq\nv(T) with probability xq\nv\n(v= 1,..,Nv) is constructed for each basis atom qwithin3\nFigure 1. Configurational averaging for thermal lattice dis -\nplacements: the continuous distribution P(∆Rn(T)) for the\natomic displacement vectors is replaced by a discrete set of\nvectors ∆ Rv(T) occurring with the probability xv. The con-\nfigurational average for this discrete set of displacements is\nmade using the CPA leading to a periodic effective medium.\nthe standard unit cell that is conform with the local sym-\nmetry and the temperature dependent root mean square\ndisplacement ( ∝an}bracketle{tu2∝an}bracketri}htT)1/2according to:\n1\nNvNv/summationdisplay\nv=1|∆Rq\nv(T)|2=∝an}bracketle{tu2\nq∝an}bracketri}htT. (3)\nIn the general case, the mean square displacement along\nthe direction µ(µ=x,y,z) of the atom ican be either\ntaken from experimental data or represented by the ex-\npression based on the phonon calculations42\n∝an}bracketle{tu2\ni,µ∝an}bracketri}htT=3/planckover2pi1\n2Mi/integraldisplay∞\n0dωgi,µ(ω)1\nωcoth/planckover2pi1ω\n2kBT,(4)\nwhereh= 2π/planckover2pi1the Planck constant, kBthe Boltzmann\nconstant, gi,µ(ω) is a partial phonon density of states.42\nOn the other hand, a rather good estimate for the root\nmean square displacement can be obtained using Debye’s\ntheory. In this case, for systems with one atom per unit\ncell, Eq. ( 4) can be reduced to the expression:\n∝an}bracketle{tu2∝an}bracketri}htT=1\n43h2\nπ2MkBΘD/bracketleftbiggΦ(ΘD/T)\nΘD/T+1\n4/bracketrightbigg\n(5)\nwith Φ(Θ D/T) the Debye function and Θ Dthe Debye\ntemperature43. Ignoring the zero temperature term 1 /4\nand assuming a frozen potential for the atoms, the situ-\nationcanbe dealt with in full analogytothe treatmentof\ndisorderedalloysonthebasisoftheCPA.Theprobability\nxvfor a specific displacement vmay normally be chosen\nas 1/Nv. The Debye temperature Θ Dused in Eq. ( 5) can\nbe either taken fromexperimental data orcalculated rep-\nresenting it in terms of the elastic constants44. In general\nthe latter approach should give more reliable results in\nthe case of multicomponent systems.\nTo simplify notation we restrict in the following to sys-\ntems with one atom per unit cell. The index qnumbering\nsites in the unit cell can therefore be dropped, while the\nindexnnumbers the lattice sites.\nAssuming a rigid displacement of the atomic potential\nin the spirit of the rigid muffin-tin approximation45,46\nthe correspondingsingle-site t-matrix tlocwith respect to\nthe local frame of reference connected with the displaced\natomic position is unchanged. With respect to the globalframe of reference connected with the equilibrium atomic\npositions Rn, however, the corresponding t-matrix tis\ngiven by the transformation:\nt=U(∆R)tlocU(∆R)−1. (6)\nThe so-called U-transformation matrix U(s) is given in\nits non-relativistic form by:45,46\nULL′(s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|s|k)YL′′(ˆs).(7)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spherical\nBesselfunction, YL(ˆr) a realsphericalharmonics, CLL′L′′\na corresponding Gaunt number and k=√\nEis the\nelectronic wave vector. The relativistic version of the\nU-matrix is obtained by a standard Clebsch-Gordan\ntransformation.37\nThe various displacement vectors ∆ Rv(T) can be used\nto determine the properties of a pseudo-component of a\npseudo alloy. Each of the Nvpseudo-components with\n|∆Rv(T)|=∝an}bracketle{tu2∝an}bracketri}ht1/2\nTis characterized by a corresponding\nU-matrix Uvand t-matrix tv. As for a substitutional\nalloy the configurational average can be determined by\nsolving the multi-component CPA equations within the\nglobal frame of reference:\nτnn\nCPA=Nv/summationdisplay\nv=1xvτnn\nv (8)\nτnn\nv=/bracketleftbig\n(tv)−1−(tCPA)−1+(τnn\nCPA)−1/bracketrightbig−1(9)\nτnn\nCPA=1\nΩBZ/integraldisplay\nΩBZd3k/bracketleftbig\n(tCPA)−1−G(k,E)/bracketrightbig−1,(10)\nwhere the underline indicates matrices with respect to\nthe combined index Λ. As it was pointed out in the pre-\nvious work41, the cutoff for the angular momentum ex-\npansionin these calculations should be taken l≥lmax+1\nwith the lmaxvalue used in the calculations for the non-\ndistorted lattice.\nThe first of these CPA equations represents the re-\nquirement for the mean-field CPA medium that embed-\nding of a component vshould lead in the average to no\nadditional scattering. Eq. ( 9) gives the scattering path\noperator for the embedding of the component vinto the\nCPA medium while Eq. ( 10) gives the CPA scattering\npath operator in terms of a Brillouin zone integral with\nG(k,E) the so-called KKR structure constants.\nHaving solved the CPA equations the linear response\nquantity of interest may be calculated using Eq. ( 1)\nas for an ordinary substitutional alloy.27,28This im-\nplies that one also have to deal with the so-called ver-\ntex corrections27,28that take into account that one\nhas to deal with a configuration average of the type\n∝an}bracketle{tˆAµℑG+ˆAνℑG+∝an}bracketri}htcthat in general will differ from the\nsimpler product ∝an}bracketle{tˆAµℑG+∝an}bracketri}htc∝an}bracketle{tˆAνℑG+∝an}bracketri}htc.4\nC. Treatment of thermal spin fluctuations\nAs for the disorder connected with thermal displace-\nments the impact of disorder due to thermal spin fluc-\ntuations may be accounted for by use of the super-cell\ntechnique. Alternatively one may again use the alloy\nanalogy and determine the necessary configurational av-\nerage by means of the CPA as indicated in Fig. 2. As\nFigure 2. Configurational averaging for thermal spin fluc-\ntuations: the continuous distribution P(ˆen) for the orienta-\ntion of the magnetic moments is replaced by a discrete set of\norientation vectors ˆ efoccurring with a probability xf. The\nconfigurational average for this discrete set of orientatio ns is\nmade using the CPA leading to a periodic effective medium.\nfor the thermal displacements in a first step a set of rep-\nresentative orientation vectors ˆ ef(withf= 1,...,Nf) for\nthelocalmagneticmomentisintroduced(seebelow). Us-\ning the rigid spin approximation the spin-dependent part\nBxcoftheexchange-correlationpotentialdoesnotchange\nfor the local frame of reference fixed to the magnetic mo-\nment when the moment is oriented along an orientation\nvector ˆef. This implies that the single-site t-matrix tloc\nf\nin the local frame is the same for all orientation vectors.\nWith respect to the common global frame that is used\nto deal with the multiple scattering (see Eq. ( 10)) the\nt-matrix for a given orientation vector is determined by:\nt=R(ˆe)tlocR(ˆe)−1. (11)\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆe) that are determined by the vectors ˆ eor correspond-\ning Euler angles.37\nAgain the configurational average for the pseudo-alloy\ncan be obtained by setting up and solvingCPAequations\nin analogy to Eqs. ( 8) to (10).\nD. Models of spin disorder\nThe central problem with the scheme described above\nis obviously to construct a realistic and representative\nset of orientation vectors ˆ efand probabilities xffor each\ntemperature T. A rather appealing approach is to cal-\nculate the exchange-coupling parameters Jijof a sys-\ntem in an ab-initio way25,47,48and to use them in sub-\nsequent Monte Carlo simulations. Fig. 3(top) shows\nresults for the temperature dependent average reduced\nmagnetic moment of corresponding simulations for bcc-\nFe obtained for a periodic cell with 4096 atom sites. The0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC*\nKKR (MC*)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)\nMC\nMF-fit to MC (wMC(T))\nMF-fit to MC (w=const)\nExpt\nMF-fit to Expt (wExpt(T))\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC\nKKR (MC)\nKKR (DLM)\nFigure 3. Averaged reduced magnetic moment M(T) =\n/angbracketleftmz/angbracketrightT/|/angbracketleftm/angbracketrightT=0|along the z-axis as a function of the tem-\nperature T. Top: results of Monte Carlo simulations using\nscheme MC* (full squares) compared with results of sub-\nsequent KKR-calculations (open squares). Middle: results\nof Monte Carlo simulations using scheme MC (full squares)\ncompared with results using a mean-field fit with a constant\nWeiss field wMC(TC) (open diamonds) and a temperature de-\npendent Weiss field wMC(T) (open squares). In addition ex-\nperimental data (full circles) together with a correspondi ng\nmean-field fit obtained for a temperature dependent Weiss\nfieldwexp(T). Bottom: results of Monte Carlo simulations\nusing scheme MC (full squares) compared with results sub-\nsequent KKR-calculations using the MC (triangles up) and\na corresponding DLM (triangle down) spin configuration, re-\nspectively.\nfull line gives the value for the reduced magnetic mo-\nmentMMC∗(T) =∝an}bracketle{tmz∝an}bracketri}htT/m0projected on the z-axis for\nthe lastMonteCarlostep (ˆ zis the orientationofthetotal\nmoment, i.e. ∝an}bracketle{tm∝an}bracketri}htT∝bardblˆz; the saturated magnetic moment at\nT= 0 K is m0=|∝an}bracketle{tm∝an}bracketri}htT=0|). This scheme is called MC∗\nin the following. In spite of the rather large number of\nsites (4096) the curve is rather noisy in particular when\napproaching the Curie temperature. Nevertheless, the5\nspin configuration of the last MC step was used as an\ninput for subsequent SPR-KKR-CPA calculations using\ntheorientationvectors ˆ efwiththeprobability xf= 1/Nf\nwithNf= 4096. As Fig. 3(top) shows, the temperature\ndependent reducedmagnetic moment MKKR(MC∗)(T) de-\nduced from the electronic structure calculations follows\none-to-one the Monte Carlo data MMC∗(T). This is a\nvery encouraging result for further applications (see be-\nlow) as it demonstrates that the CPA although being a\nmean-field method and used here in its single-site formu-\nlation is nevertheless capable to reproduce results of MC\nsimulations that go well beyond the mean-field level.\nHowever, using the set of vectors ˆ efof scheme MC*\nalso for calculations of the Gilbert damping parameters\nαas a function of temperature led to extremely noisy\nand unreliable curves for α(T). For that reason an av-\nerage has been taken over many MC steps (scheme MC)\nleading to a much smoother curve for MMC(T) as can\nbe seen from Fig. 3(middle) with a Curie temperature\nTMC\nC= 1082 K. As this enlarged set of vectors ˆ efgot\ntoo large to be used directly in subsequent SPR-KKR-\nCPA calculations, a scheme was worked out to get a set\nof vectors ˆ efand probabilities xfthat is not too large\nbut nevertheless leads to smooth curves for M(T).\nThe first attempt was to use the Curie temperature\nTMC\nCtodeduceacorrespondingtemperatureindependent\nWeiss-field w(TC) on the basis of the standard mean-field\nrelation:\nw(TC) =3kBTC\nm2\n0. (12)\nThis leads to a reduced magnetic moment curve MMF(T)\nthat shows by construction the same Curie temperature\nas the MC simulations. For temperatures between T=\n0 K and TC, however, the mean-field reduced magnetic\nmoment MMF(T) is well below the MC curve (see Fig. 3\n(middle) ).\nAs an alternative to this simple approach we intro-\nduced a temperature dependent Weiss field w(T). This\nallows to describe the temperature dependent magnetic\nproperties using the results obtained beyond the mean-\nfield approximation. At the same time the calculation\nof the statistical average can be performed treating the\nmodel Hamiltonian in termsofthe mean field theory. For\nthis reason the reduced magnetic moment M(T), being\na solution of equation (see e.g.49)\nM(T) =L/parenleftbiggwm2\n0M(T)\nkBT/parenrightbigg\n, (13)\nwas fitted to that obtained from MC simulations\nMMC(T)withtheWeissfield w(T)asafittingparameter,\nsuch that\nlim\nw→w(T)M(T) =MMC(T), (14)\nwithL(x) the Langevin function.\nThe corresponding temperature dependent probability\nx(ˆe) for an atomic magnetic moment to be oriented alongˆeis proportionalto exp( −w(T)ˆz·ˆe/kBT) (see, e.g.49). To\ncalculate this value we used NθandNφpoints for a reg-\nular grid for the spherical angles θandφcorresponding\nto the vector ˆ ef:\nxf=exp(−w(T)ˆz·ˆef/kBT)/summationtext\nf′exp(−w(T)ˆz·ˆef′/kBT).(15)\nFig.4shows for three different temperatures the θ-\ndependent behavior of x(ˆe). As one notes, the MF-fit\n0 30 60 90 120 150 180\nθ00.050.10.150.20.250.3P(θ)MC\nMF-fit to MC (wMC(T))T = 200 K\n0 30 60 90 120 150 180\nθ00.050.10.150.2P(θ)MC\nMF-fit to MC (wMC(T))T = 400 K\n0 30 60 90 120 150 180\nθ00.050.1P(θ)MC\nMF-fit to MC (wMC(T))T = 800 K\nFigure 4. Angular distribution P(θ) of the atomic magnetic\nmoment mobtained from Monte Carlo simulations (MC) for\nthe temperature T= 200, 400, and 800 K compared with field\nmean-field (MF) data, xf, (full line) obtained by fitting using\na temperature dependent Weiss field w(T) (Eq.13).\nto the MC-results perfectly reproduces these data for all\ntemperatures. This applies of course not only for the\nangular resolved distribution of the magnetic moments\nshown in Fig. 4but also for the average reduced mag-\nnetic moment recalculated using Eq.( 13), shown in Fig.\n3. Obviously, the MF-curve MMF(MC)(T) obtained using\nthe temperature dependent Weiss field parameter w(T)\nperfectly reproduces the original MMC(T) curve. The\ngreat advantage of this fitting procedure is that it al-\nlows to replace the MC data set with a large number6\nNMC\nfof orientation vectors ˆ ef(pointing in principle into\nany direction) with equal probability xf= 1/NMC\nfby a\nmuch smaller data set with Nf=NθNφwithxfgiven\nby Eq. (15).\nAccordingly, the reduced data set can straight for-\nwardly be used for subsequent electronic structure cal-\nculations. Fig. 3(bottom) shows that the calcu-\nlated temperature dependent reduced magnetic moment\nMKKR−MF(MC)(T) agrees perfectly with the reduced\nmagnetic moment MMC(T) given by the underlying MC\nsimulations.\nThe DLM method has the appealing feature that it\ncombines ab-initio calculations and thermodynamics in\na coherent way. Using a non-relativistic formulation, it\nwas shown that the corresponding averaging over all ori-\nentations of the individual atomic reduced magnetic mo-\nments can be mapped onto a binary pseudo-alloy with\none pseudo-component having up- and downward orient-\nation of the spin moment with concentrations x↑and\nx↓, respectively.24,50For a fully relativistic formulation,\nwith spin-orbitcoupling included, this simplificationcan-\nnot be justified anymore and a proper average has to be\ntaken over all orientations.51As we do not perform DLM\ncalculationsbut use hereonly the DLM picture to repres-\nent MC data, this complication is ignored in the follow-\ning. Having the set of orientation vectors ˆ efdetermined\nby MC simulations the corresponding concentrations x↑\nandx↓can straight forwardly be fixed for each temper-\nature by the requirement:\n1\nNfNf/summationdisplay\nf=1ˆef=x↑ˆz+x↓(−ˆz), (16)\nwithx↑+x↓= 1. Using this simple scheme electronic\nstructure calculations have been performed for a binary\nalloy having collinear magnetization. The resulting re-\nduced magnetic moment MKKR−DLM(MC) (T) is shown in\nFig.3(bottom). As one notes, again the original MC\nresults are perfectly reproduced. This implies that when\ncalculating the projected reduced magnetic moment Mz\nthat is determined by the averaged Green function ∝an}bracketle{tG∝an}bracketri}ht\nthe transversal magnetization has hardly any impact.\nFig.3(middle) gives also experimental data for\ntheM(T).52While the experimental Curie-temperature\nTexp\nC= 1044 K52is rather well reproduced by the MC\nsimulations TMC\nC= 1082 K one notes that the MC-curve\nMMC(T) is well below the experimental curve. In partic-\nular,MMC(T) drops too fast with increasing Tin the\nlow temperature regime and does not show the T3/2-\nbehavior. The reason for this is that the MC simulations\ndo not properly account for the low-energy long-ranged\nspinwaveexcitationsresponsibleforthelow-temperature\nmagnetization variation. Performing ab-initio calcula-\ntions for the spin wave energies and using these data for\nthe calculation of M(T) much better agreement with ex-\nperiment can indeed be obtained in the low-temperature\nregime than with MC simulations.53\nAs the fitting scheme sketched above needs only thetemperature reduced magnetic moment M(T) as input\nit can be applied not only to MC data but also to ex-\nperimental data. Fig. 3shows that the mean field fit\nMMF(exp)(T) again perfectly fits the experimental re-\nduced magnetic moment curve Mexp(T). Based on this\ngood agreement this corresponding data set {ˆef,xf}has\nalso been used for the calculation of responsetensors (see\nbelow).\nAn additional much simpler scheme to simulate the\nexperimental Mexp(T) curve is to assume the individual\natomic moments to be distributed on a cone, i.e. with\nNθ= 1 and Nφ>>1.23In this case the opening angle\nθ(T) of the cone is chosen such to reproduce M(T). In\ncontrasttothestandardDLMpicture,thissimplescheme\nallows already to account for transversal components of\nthe magnetization. Corresponding results for response\ntensor calculations will be shown below.\nFinally, it should be stressed here that the various spin\nconfiguration models discussed above assume a rigid spin\nmoment, i.e. its magnitude does not change with temper-\nature nor with orientation. In contrast to this Ruban et\nal.54usealongitudinalspinfluctuation Hamiltonianwith\nthe corresponding parameters derived from ab-initio cal-\nculations. As a consequence, subsequent Monte Carlo\nsimulations based on this Hamiltonian account in par-\nticular for longitudinal fluctuations of the spin moments.\nA similar approach has been used by Drchal et al.55,56\nleading to good agreement with the results of Ruban et\nal. However, the scheme used in these calculations does\nnot supply in a straightforward manner the necessary\ninput for temperature dependent transport calculations.\nThis is different from the work of Staunton et al.57who\nperformed self-consistent relativistic DLM calculations\nwithout the restriction to a collinear spin configuration.\nThis approach in particular accounts in a self-consistent\nway for longitudinal spin fluctuations.\nE. Combined chemical and thermally induced\ndisorder\nThe various types of disorder discussed above may be\ncombined with each other as well as with chemical i.e.\nsubstitution disorder. In the most general case a pseudo-\ncomponent ( vft) is characterized by its chemical atomic\ntypet, the spin fluctuation fand lattice displacement\nv. Using the rigid muffin-tin and rigid spin approxim-\nations, the single-site t-matrix tloc\ntin the local frame is\nindependent from the orientation vector ˆ efand displace-\nment vector ∆ Rv, and coincides with ttfor the atomic\ntypet. With respect to the common global frame one\nhas accordingly the t-matrix:\ntvft=U(∆Rv)R(ˆef)ttR(ˆef)−1U(∆Rv)−1.(17)\nWith this the corresponding CPA equations are identical\nto Eqs. ( 8) to (10) with the index vreplaced by\nthe combined index ( vft). The corresponding pseudo-\nconcentration xvftcombines the concentration xtof the7\natomic type twith the probability for the orientation\nvector ˆefand displacement vector ∆ Rv.\nIII. COMPUTATIONAL DETAILS\nThe electronic structure of the investigated ferro-\nmagnets bcc-Fe and fcc-Ni, has been calculated self-\nconsistently using the spin-polarized relativistic KKR\n(SPR-KKR) band structure method.58,59For the ex-\nchangecorrelationpotential the parametrizationas given\nby Vosko et al.60has been used. The angular-momentum\ncutoff of lmax= 3 was used in the KKR multiple scatter-\ning expansion. The lattice parameters have been set to\nthe experimental values.\nIn a second step the exchange-coupling parameters\nJijhave been calculated using the so-called Lichten-\nstein formula.25Although the SCF-calculations have\nbeen done on a fully-relativistic level the anisotropy of\nthe exchange coupling due to the spin-orbit coupling has\nbeen neglected here. Also, the small influence of the\nmagneto-crystallineanisotropyfor the subsequent Monte\nCarlo (MC) simulations has been ignored, i.e. these have\nbeen based on a classical Heisenberg Hamiltonian. The\nMC simulations were done in a standard way using the\nMetropolis algorithm and periodic boundary conditions.\nThe theoretical Curie temperature TMC\nChas been de-\nduced from the maximum of the magnetic susceptibility.\nThe temperature dependent spin configuration ob-\ntained during a MC simulation has been used to con-\nstruct a set of orientations ˆ efand probabilities xfac-\ncording to the schemes MC* and MC described in sec-\ntionIIDto be used within subsequent SPR-KKR-CPA\ncalculations (see above). For the corresponding calcu-\nlation of the reduced magnetic moment the potential\nobtained from the SCF-calculation for the perfect fer-\nromagnetic state ( T= 0K) has been used. The calcu-\nlation for the electrical conductivity as well as the Gil-\nbertdampingparameterhasbeenperformedasdescribed\nelsewhere.41,61\nIV. RESULTS AND DISCUSSION\nA. Temperature dependent conductivity\nEq. (1) has been used together with the various\nschemes described above to calculate the temperature\ndependent longitudinal resistivity ρ(T) of the pure fer-\nromagnets Fe, Co and Ni. In this case obviously disorder\ndue to thermal displacements of the atoms as well as spin\nfluctuations contribute to the resistivity.\nTo give an impression on the impact of the thermal\ndisplacementsaloneFig. 5givesthe temperaturedepend-\nent resistivity ρ(T) of pure Cu (Θ Debye= 315 K) that\nis found in very good agreement with corresponding ex-\nperimental data.62This implies that the alloy analogy\nmodel that ignores any inelastic scattering events should0 100 200 300 400 500\nTemperature (K)01234ρxx (10-6Ω⋅cm)Expt\nTheory - alloy analogy\nTheory - LOVA\nCu\nFigure 5. Temperature dependent longitudinal resistivity of\nfcc-Cuρ(T) obtained by accounted for thermal vibrations as\ndescribed in section IIBcompared with corresponding ex-\nperimental data.62In addition results are shown based on\nthe LOVA (lowest order variational approximation) to the\nBoltzmann formalism.14\nin general lead to rather reliable results for the resistivity\ninduced by thermal displacements. Accordingly, com-\nparison with experiment should allow for magnetically\nordered systems to find out the most appropriate model\nfor spin fluctuations.\nFig.6(top) shows theoretical results for ρ(T) of bcc-\nFe due to thermal displacements ρv(T), spin fluctuations\ndescribed by the scheme MC ρMC(T) as well as the com-\nbination of the two influences ( ρv,MC(T)). First of all\none notes that ρv(T) is not influenced within the adop-\ntedmodelbytheCurietemperature TCbutisdetermined\nonly by the Debye temperature. ρMC(T), on the other\nhand, reaches saturation for TCas the spin disorder does\nnot increase anymore with increasing temperature in the\nparamagnetic regime. Fig. 6also shows that ρv(T) and\nρMC(T)arecomparableforlowtemperaturesbut ρMC(T)\nexceedsρv(T) more and more for higher temperatures.\nMost interestingly, however, the resistivity for the com-\nbined influence of thermal displacements and spin fluctu-\nationsρv,MC(T) does not coincide with the sum of ρv(T)\nandρMC(T) but exceeds the sum for low temperatures\nand lies below the sum when approaching TC.\nFig.6(bottom) shows the results of three differ-\nent calculations including the effect of spin fluctuations\nas a function of the temperature. The curve ρMC(T)\nis identical with that given in Fig. 6(top) based on\nMonte Carlo simulations. The curves ρDLM(MC) (T) and\nρcone(MC)(T) are based on a DLM- and cone-like repres-\nentation of the MC-results, respectively. For all three\ncases results are given including as well as ignoring the\nvertex corrections. As one notes the vertex corrections\nplay a negligible role for all three spin disorder models.\nThis is fully in line with the experience for the longitud-\ninal resistivity of disordered transition metal alloys: as\nlong as the the states at the Fermi level have domin-\nantly d-character the vertex corrections can be neglected\nin general. On the other hand, if the sp-character dom-8\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)vib\nfluct (MC)\nvib + fluct (MC)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)MC (VC)\nMC (NVC)\nDLM (VC)\nDLM (NVC)\ncone (VC)\ncone (NVC)\nFigure 6. Temperature dependent longitudinal resistivity of\nbcc-Feρ(T) obtained by accounted for thermal vibrations\nand spin fluctuations as described in section IIB. Top: ac-\ncounting for vibrations (vib, diamonds), spin fluctuations us-\ning scheme MC (fluct, squares) and both (vib+fluct, circles).\nBottom: accounting for spin fluctuations ˆ ef= ˆe(θf,φf) us-\ning the schemes: MC (squares) with 0 ≤θf≤π;0≤φf≤\n2π, DLM(MC) (triangles up) with θf1= 0,θf2=π, and\ncone(MC) (triangles down) θf=/angbracketleftθf/angbracketrightT;0≤φf≤2π. The\nfull and open symbols represent the results obtained with th e\nvertex corrections included (VC) and excluded (NV), respec t-\nively.\ninates inclusion of vertex corrections may alter the result\nin the order of 10 %.63,64\nComparing the DLM-result ρDLM(MC) (T) with\nρMC(T) one notes in contrast to the results for M(T)\nshown above (see Fig. 3(bottom)) quite an appreciable\ndeviation. This implies that the restricted collinear\nrepresentation of the spin configuration implied by the\nDLM-model introduces errors for the configurational\naverage that seem in general to be unacceptable, For\nthe Curie temperature and beyond in the paramagnetic\nregimeρDLM(MC) (T) andρMC(T) coincide, as it was\nshown formally before.20\nComparing finally ρcone(MC)(T) based on the conical\nrepresentationofthe MCspin configurationwith ρMC(T)\none notes that also this simplification leads to quite\nstrong deviations from the more reliable result. Never-\ntheless, one notes that ρDLM(MC) (T) agrees with ρMC(T)\nfor the Curie temperature and also accounts to some ex-\ntent for the impact of the transversal components of themagnetization.\nThe theoretical results for bcc-Fe (Θ Debye= 420 K)\nbased on the combined inclusion of the effects of thermal\ndisplacementsandspinfluctuationsusingtheMCscheme\n(ρv,MC(T)) are compared in Fig. 7(top) with experi-\nmental data ( ρexp(T)). For the Curie temperature ob-\n0 0.2 0.4 0.6 0.8 1 1.2 1.4\nT/TC020406080100120ρxx  (10-6Ω⋅cm)Expt: J. Bass and K.H. Fischer \nvib + fluct (MC)\nvib + fluct (exp)\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8\nT/TC01020304050ρxx  (10-6Ω⋅cm)Expt.: C.Y. Ho et al. (1983)\nvib\nvib (PM)\nfluct\nvib + fluct\nFigure 7. Top: Temperature dependent longitudinal res-\nistivity of bcc-Fe ρ(T) obtained by accounted for thermal\nvibrations and spin fluctuations using the scheme MC\n(vib+fluct(MC), squares) and a mean-field fit to the experi-\nmental temperature magnetic moment Mexp(vib+fluct(exp),\ndiamonds) compared with experimental data (circles).62Bot-\ntom: corresponding results for fcc-Ni. In addition results are\nshown accounting for thermal displacements (vib) only for\nthe ferromagnetic (FM) as well paramagnetic (PM) regime.\nExperimental data have been taken from Ref. 65.\nviously a very good agreement with experiment is found\nwhile for lower temperatures ρv,MC(T) exceeds ρexp(T).\nThis behavior correlates well with that of the temperat-\nure dependent reduced magnetic moment M(T) shown\nin Fig.3(middle). The too rapid decrease of MMC(T)\ncompared with experiment implies an essentially overes-\ntimated spin disorder at any temperature leading in turn\nto a too large resistivity ρv,MC(T). On the other hand,\nusing the temperature dependence of the experimental\nreducedmagneticmoment Mexp(T)tosetup thetemper-\nature dependent spin configuration as described above a\nvery satisfying agreement is found with the experimental\nresistivity data ρexp(T). Note also that above TCthe\ncalculated resistivity riches the saturation in contrast to\nthe experimental data where the continuing increase of9\nρexp(T) can be attributed to the longitudinal spin fluctu-\nations leading to a temperature dependent distribution\nof local magnetic moments on Fe atoms.54However, this\ncontribution was not taken into account because of re-\nstriction in present calculations using fixed value for the\nlocal reduced magnetic moments.\nFig.7(bottom) shows corresponding results for the\ntemperature dependent resistivity of fcc-Ni (Θ Debye=\n375 K). For the ferromagnetic (FM) regime that the\ntheoretical results are comparable in magnitude when\nonly thermal displacements ( ρv(T)) or spin fluctuations\n(ρMF(T)) are accounted for. In the later case the mean\nfieldw(T) has been fitted to the experimental M(T)-\ncurve. Taking both into account leads to a resistivity\n(ρv,MF(T)) that are well above the sum of the individual\ntermsρv(T) andρMF(T). Comparing ρv,MF(T) with ex-\nperimentaldata ρexp(T)ourfindingshowsthatthetheor-\netical results overshoots the experimental one the closer\none comes to the critical temperature. This is a clear\nindication that the assumption of a rigid spin moment\nis quite questionable as the resulting contribution to the\nresistivity due to spin fluctuations as much too small.\nIn fact the simulations of Ruban et al.54on the basis of\na longitudinal spin fluctuation Hamiltonian led on the\ncase of fcc-Ni to a strong diminishing of the averagelocal\nmagnetic moment when the critical temperature is ap-\nproachedfrom below (about 20% comparedto T= 0K).\nFor bcc-Fe, the change is much smaller (about 3 %) justi-\nfying on the case the assumption of a rigid spin moment.\nTaking the extreme point of view that the spin moment\nvanishescompletely abovethe criticaltemperature orthe\nparamagnetic (PM) regime only thermal displacements\nhave to be considered as a source for a finite resistivity.\nCorresponding results are shown in Fig. 7(bottom) to-\ngether with corresponding experimental data. The very\ngood agreementbetween both obviouslysuggeststhat re-\nmaining spin fluctuations above the critical temperature\nare of minor importance for the resistivity of fcc-Ni.\nB. Temperature dependent Gilbert damping\nparameter\nFig.8shows results for Gilbert damping parameter α\nof bcc-Fe obtained using different models for the spin\nfluctuations. All curves show the typical conductivity-\nlike behaviorfor low temperatures and the resistivity-like\nbehavior at high temperatures reflecting the change from\ndominating intra- to inter-band transitions.66The curve\ndenoted expt isbasedon aspin configurationtoted tothe\nexperimental Mexpt(T) data. Using the conical model to\nfitMexpt(T) as basis for the calculation of α(T) leads\nobviously to a rather good agreement with αM(expt)(T).\nHaving instead a DLM-like representation of Mexpt(T),\non the other hand, transverse spin components are sup-\npressed and noteworthy deviations from αM(expt)(T) are\nfound for the low temperature regime. Nevertheless, the\ndeviations are less pronounced than in the case of the0 200 400 600 800\nTemperature (K)02468α × 103fluct (MC)\nfluct (Expt)\nfluct (DLM)\nfluct (cone)\nFigure 8. Temperature dependent Gilbert damping α(T) for\nbcc-Fe, obtainedbyaccountedfor thermal vibrations andsp in\nfluctuations accounting for spin fluctuations using scheme\nMC (squares), DLM(MC) (triangles up), cone(MC) (triangles\ndown) and a MF fit to the experimental temperature reduced\nmagnetic moment (circles).\nlongitudinal resistivity (see Fig. 6(bottom)), where cor-\nresponding results are shown based on MMC(T) as a ref-\nerence. Obviously, the damping parameter αseems to\nbe less sensitive to the specific spin fluctuation model\nused than the resistivity. Finally, using the spin con-\nfiguration deduced from Monte Carlo simulations, i.e.\nbased on MMC(T) quite strong deviations for the result-\ningαM(MC)(T) fromαM(expt)(T) are found. As for the\nresistivity (see Fig. 6(bottom)) this seems to reflect the\ntoo fast drop of the reduced magnetic moment MMC(T)\nwith temperature in the low temperature regime com-\npared with temperature (see Fig. 3). As found before18\naccountingonly for thermal vibrations α(T) (Fig.6(bot-\ntom)) is found comparableto the casewhen only thermal\nspan fluctuations are allowed. Combing both thermal ef-\nfects does not lead to a curve that is just the sum of the\ntwoα(T) curves. As found for the conductivity (Fig. 6\n(top)) obviously the two thermal effects are not simply\nadditive. As Fig. 9(top) shows, the resulting damping\nparameter α(T) for bcc-Fe that accounts for thermal vi-\nbrationsaswellasspinfluctuationsisfoundinreasonable\ngood agreement with experimental data.18\nFig.9shows also corresponding results for the Gilbert\ndampingoffcc-Niasafunctionoftemperature. Account-\ning only for thermal spin fluctuations on the basis of the\nexperimental M(T)-curveleadsinthis casetocompletely\nunrealistic results while accounting only for thermal dis-\nplacements leads to results already in rather good agree-\nment with experiment. Taking finally both sources of\ndisorder into account again no simple additive behavior\nis found but the results are nearly unchanged compared\nto those based on the thermal displacements alone. This\nimplies that results for the Gilbert damping parameter\nof fcc-Ni hardly depend on the specific spin configura-\ntion model used but are much more governed by thermal\ndisplacements.10\n0 200 400 600 800\nTemperature (K)0246810α × 103vib\nvib + fluct (Expt)\nExpt 1\nExpt 2\n0 100 200 300 400 500\nTemperature (K)00.050.10.150.2αvib\nfluct (Expt)\nvib + fluct (Expt)\nExpt\nFigure 9. Top: Temperature dependent Gilbert damping\nα(T) for bcc-Fe, obtained byaccounted for thermal vibrations\nand spin fluctuations accounting for lattice vibrations onl y\n(circles) and lattice vibrations and spin fluctuations base d on\nmean-field fit to the experimental temperature reduced mag-\nnetic moment Mexpt(diamonds) compared with experimental\ndata (dashed and full lines).67,68Bottom: corresponding res-\nults for fcc-Ni. Experimental data have been taken from Ref.\n67.\nV. SUMMARY\nVarious schemes based on the alloy analogy that al-\nlow to include thermal effects when calculating responseproperties relevant in spintronics have been presented\nand discussed. Technical details of an implementation\nwithin the framework of the spin-polarized relativistic\nKKR-CPA band structure method have been outlined\nthat allow to deal with thermal vibrations as well as spin\nfluctuations. Various models to represent spin fluctu-\nations have been compared with each other concerning\ncorresponding results for the temperature dependence\nof the reduced magnetic moment M(T) as well as re-\nsponse quantities. It was found that response quantities\nare much more sensitive to the spin fluctuation model as\nthe reduced magnetic moment M(T). 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We show that, in the resultant equa-\ntion, the ratio of the Gilbert damping constant, α, and the coefficient, β, of the current-induced\ntorque, called non-adiabatic torque, depends on the relaxa tion time of the fluctuating field τc. The\nequality α=βholds when τcis very short compared to the time scale of magnetization dyn amics.\nWe apply our theory to current-induced magnetization rever sal in magnetic multilayers and show\nthat the switching time is a decreasing function of τc.\nSpin-transfer torque-induced magnetization dynamics\nsuch as current-induced magnetization reversal [1, 2, 3],\ndomain wall motion [4], and microwave generation [5]\nhave attracted a great deal of attention because of their\npotential applications to future nano-spinelectronic de-\nvices. In the absence of spin-transfer torque, magnetiza-\ntion dynamics is described by either the Landau-Lifshitz\n(LL) equation [6] or the Landau-Lifshitz-Gilbert (LLG)\nequation [7]. It is known that the LL and LLG equations\nbecome equivalent through rescaling of the gyromagnetic\nratio.\nHowever, this is not the case in the presence of spin-\ntransfer torque. For domain wall dynamics, the following\nLLG-type equation has been studied by several groups\n[8, 9, 10]:\n∂t/angbracketleftM/angbracketright+v·∇/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright\n+α\nM/angbracketleftM/angbracketright×∂t/angbracketleftM/angbracketright+β\nM/angbracketleftM/angbracketright×[(v·∇)/angbracketleftM/angbracketright],(1)\nwhereMrepresents the magnetization, vis the velocity,\nγis the gyromagnetic ratio and αis the Gilbert damping\nconstant. The second term on the left-hand side repre-\nsents the adiabatic contribution of spin-transfer torque.\nThe first and the second terms on the right-hand side are\nthe torque due to the effective magnetic field Hand the\nGilbert damping. The last term on the right-hand side\nof Eq. (1) represents the current-induced torque, called\n“non-adiabatic torque” or simply the βterm. The direc-\ntions of the adiabaticcontribution of spin-transfertorque\nand non-adiabatic torque are shown in Fig. 1 (a).\nAs shownbyThiaville et al., the value ofthe coefficient\nβstrongly influences the motion of the domain wall [8].\nHowever, the value of the coefficient βis still controver-\nsial, and different conclusions have been drawn from dif-\nferent approaches[9, 10, 11, 12, 13, 14, 15]. For example,\nBarnes and Maekawa showed that the value of βshould\nbeequaltothatoftheGilbertdampingconstant αtosat-\nisfy the requirement that the relaxation should cease at\nthe minimum of electrostatic energy, even under particle\nflow. Kohno et al.performed microscopic calculationsFIG. 1: (a) The direction of the magnetization M, the adia-\nbatic contribution of spin-transfer torque, ( v·∇)M, and the\nβterm,M×[(v·∇)M], are shown. The direction of the ve-\nlocityvis indicated by the dotted arrow. (b) The magnetic\nmultilayers, in which the pinned and the free layers are sepa -\nrated by a nonmagnetic spacer layer are schematically shown .\nThe magnetization vectors of the pinned and free layers are\nrepresented by S1andSs, respectively. The effective mag-\nnetic field to which S2is subject is represented by H. (c)\nThe direction of the magnetization of the free layer, S2, the\nspin-transfer torque ( S2×S1)×S2, and the non-adiabatic\ntorque,S2×S1, are shown. The direction of S1is indicated\nby the dotted arrow.\nof spin torques in disordered ferromagnets and showed\nthat theαandβterms arise from the spin relaxation\nprocesses and that α/negationslash=βin general [10]. Tserkovnyak et\nal.[11] derived the βterm using a quasiparticle approx-\nimation and showed that α=βwithin a self-consistent\npicture based on the local density approximation.\nIn the current-induced magnetization dynamics in the\nmagnetic multilayers shown in Fig. 1 (b) [16, 17, 18], the\nnon-adiabatic torque exerts a strong effect, and therefore\naffectsthedirect-currentvoltageofthe spintorquediode,\nas shown in Refs. [17, 18]. The magnetization dynam-\nics of the free layer, S2, has been studied by using the\nfollowing LLG-type equation,\n∂tS2−I\neg/planckover2pi1(S2×S1)×S2=γH×S2+α\nS2S2×∂tS2\n+ηIS2×S1, (2)\nwhereIis the charge current density, gis the amplitude\nof the spin torque introduced by Slonczewski [1], /planckover2pi1is2\nthe Dirac constant and ηrepresents the magnitude of\nthe “non-adiabatictorque” which is sometimes called the\nfield-like torque [17, 18].\nIn this paper, we study the magnetization dynamics\ninduced by spin-transfer torque in the framework of non-\nequilibrium thermodynamics. We derive the equation of\nmotion of the magnetization in the presence of a spin\ncurrentby usingthe local equilibrium assumption. In the\nresultant equation, the Gilbert damping term and the β\nterm are expressed as memory terms with the relaxation\ntime of the fluctuating field τc. We show that the value\nof the coefficient βis not equal to that of the Gilbert\ndamping constant αin general. However, we also show\nthat the equality α=βholds ifτc≪1/(γH). We apply\nour theory to the current-induced magnetization reversal\nin magnetic multilayersand showthat the switching time\nis a decreasing function of τc.\nLet us first briefly introduce the non-equilibrium sta-\ntistical theory of magnetization dynamics in the absence\nof spin current [19]. The LLG equation describing the\nmotion of magnetization Munder an effective magnetic\nfieldHis given by\n∂tM=γH×M+α\nMM×∂tM.(3)\nThe equivalent LL equation is expressed as\n∂tM=γ\n1+α2H×M−αγ\nM(1+α2)M×(M×H).(4)\nThe Langevin equations leading to Eqs. (3) and (4) by\ntaking the ensemble average of magnetization m, are\n∂tm=γHtot×m (5)\n∂tδH=−1\nτc(δH−χsm)+R(t), (6)\nwhere the total magnetic field Htotis the sum of the\neffective magnetic field Hand the fluctuating magnetic\nfieldδHandχsis the susceptibility ofthe local magnetic\nfieldinducedatthepositionofthespin. AccordingtoEq.\n(6) the fluctuating magnetic field δHrelaxes toward the\nreaction field χsmwith the relaxation time τc. The ran-\ndom field R(t) satisfies /angbracketleftR(t)/angbracketright= 0 and the fluctuation-\ndissipation relation, /angbracketleftRi(t)Rj(t′)/angbracketright=2\nτcχskBTδi,jδ(t−t′),\nwherekBis the Boltzmann constant, Tis the tempera-\nture,/angbracketleft···/angbracketrightdenotestheensembleaverage,and i,j= 1,2,3\nrepresents the Cartesian components. It was shown that\nEqs. (5) and (6) lead to Kawabata’s extended Landau-\nLifshitz equation [20] derived by the projection operator\nmethod [19]. In the Markovian limit, i.e.,τc≪1/(γH),\nwe can obtain the LLG equation (3) and the correspond-\ning LL equation (4) with α=γτcχsM[19].\nIn order to consider the flow of spins, i.e., spin cur-\nrent, we introduce the positional dependence. Since we\nare interested in the average motion, it is convenient to\nintroducethemeanvelocityofthecarrier, v. Theaveragemagnetization, /angbracketleftm(x,t)/angbracketright, is obtained by introducing the\npositional dependence and taking the ensemble average\nof Eq. (5). In terms of the mean velocity, the ensemble\naverage of the left-hand side of Eq. (5) leads to\n∂t/angbracketleftm/angbracketright+(v·∇)/angbracketleftm/angbracketright. (7)\nAssuming /angbracketleftδH×m/angbracketright ≈ /angbracketleftδH/angbracketright×/angbracketleftm/angbracketright, which is applicable\nwhen the thermal fluctuation is small compared to the\nmean value, we obtain\n∂t/angbracketleftm/angbracketright+(v·∇)/angbracketleftm/angbracketright=γ/angbracketleftHtot(x,t)/angbracketright×/angbracketleftm(x,t)/angbracketright.(8)\nThe mean magnetization density is expressed as\n/angbracketleftM(x,t)/angbracketright=ρ(x,t)/angbracketleftm(x,t)/angbracketright,i.e., by the product of the\nscalar and vectorial components both of which depend\non the position of the spin carrier at time t. The spin\ncarrier density satisfies the continuity equation,\n∂tρ(x,t)+∇·(vρ(x,t)) = 0. (9)\nBy multiplying the left-hand side of Eq. (8) by ρ(x,t)\nand using the continuity equation (9), the closed expres-\nsion for the mean magnetization is obtained as [21]\nρ(∂t/angbracketleftm/angbracketright+v·∇/angbracketleftm/angbracketright) =∂tρ/angbracketleftm/angbracketright+/angbracketleftm/angbracketright∇·vρ+ρv·∇/angbracketleftm/angbracketright\n=∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright, (10)\nwhere Div v/angbracketleftM/angbracketrightis defined by\nDivv/angbracketleftM/angbracketright=3/summationdisplay\ni=1∂vi/angbracketleftM/angbracketright\n∂xi=/angbracketleftM/angbracketright(∇·v)+(v·∇)/angbracketleftM/angbracketright.(11)\nBy multiplying the right-hand side of Eq. (8) by ρ(x,t)\nand using Eq. (10), we obtain\n∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γ(H+/angbracketleftδH/angbracketright)×/angbracketleftM/angbracketright.(12)\nEquation (12) takes the standard form of a time-\nevolution equation for extensive thermodynamical vari-\nables under flow [21]. The average of Eq. (6) with the\npositional dependence is given by\n∂t/angbracketleftδH(x,t)/angbracketright=−1\nτc[/angbracketleftδH(x,t)/angbracketright−χ/angbracketleftM(x(t),t)/angbracketright],(13)\nwherex(t) is the mean position at time tof the spin car-\nrier, which flows with velocity v=∂tx(t) andχ=χs/ρ\nis assumed to be a constant independent of the position.\nEquations (12) and (13) constitute the basis for the sub-\nsequent study of magnetization dynamics in the presence\nof spin-transfer torque.\nThe formal solution of Eq. (13) is expressed as\n/angbracketleftδH(x,t)/angbracketright=χ\nτc/integraldisplayt\n−∞ψ(t−t′)/angbracketleftM(x(t′),t′)/angbracketrightdt′,(14)\nwhere the memory kernel is given by ψ(t) = exp[−t/τc].\nUsing partial integration, we obtain\n/angbracketleftδH(x,t)/angbracketright=χ/angbracketleftM/angbracketright−/integraldisplayt\n−∞ψ(t−t′)χ/angbracketleft˙M(t′)/angbracketrightdt′,(15)3\nwhere the explicit expression for ˙M(t) =˙M(x(t),t) is\ngiven by the convective derivative,\n˙M(t) =∂tM(x(t),t)+(v·∇)M(x(t),t).(16)\nSubstituting Eq. (15) into Eq. (12), we obtain the equa-\ntion of motion for the mean magnetization density,\n∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright\n+γ/integraldisplayt\n−∞dt′ψ(t−t′)χ/angbracketleftM(t)/angbracketright×/angbracketleft˙M(t′)/angbracketright.(17)\nEquation (17) supplemented by Eq. (16) is the principal\nresultof this paper.\nWhen the relaxation time of the fluctuating field, τc, is\nvery short compared to the time scale of the magnetiza-\ntion dynamics, the memory kernel is decoupled and Eq.\n(17) can be written in the form of an LLG-type equation\nas\n∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright+α\nM/angbracketleftM/angbracketright×˙/angbracketleftM/angbracketright,(18)\nwhereα=γτcχMis the Gilbert damping constant. Sub-\nstituting the explicit form of the convective derivative,\nEq. (16), into Eq. (18) and using Eq.(11) we obtain the\nfollowing LLG-type equation:\n∂t/angbracketleftM/angbracketright+/angbracketleftM/angbracketright(∇·v)+(v·∇)/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright\n+α\nM/angbracketleftM/angbracketright×∂t/angbracketleftM/angbracketright+α\nM/angbracketleftM/angbracketright×[(v·∇)/angbracketleftM/angbracketright].(19)\nIf∇·v= 0, Eq. (19) reduces to Eq. (14) of Ref. [9],\nwhich is derived by replacing the time derivative of mag-\nnetization∂tMon both sides of the LLG equation (3)\nby the convective derivative ∂tM+v·∇·M. The term\n/angbracketleftM/angbracketright(∇·v) appears not on the right-hand side ofEq. (19)\nbut on the left-hand side, which means we cannot obtain\nEq. (19) using the same procedure used in Ref. [9]. As\nshown in Refs. [9, 22], Eq. (19) with /angbracketleftM/angbracketright(∇·v) = 0\nleads to a steady-state solution in the comoving frame,\n/angbracketleftM(t)/angbracketright=/angbracketleftM0(x−vt)/angbracketright, where /angbracketleftM0(x)/angbracketrightdenotes the\nstationary solution in the absence of domain wall mo-\ntion. However, if /angbracketleftM/angbracketright(∇·v)/negationslash= 0, the steady-state so-\nlution may break the Galilean invariance. The situa-\ntion/angbracketleftM/angbracketright(∇·v)/negationslash= 0 can be realized, for example, in\nmagnetic semiconductors [23, 24], where the spin carrier\ndensity is spatially inhomogeneous, i.e.,∇ρ/negationslash= 0.\nThe last term of Eq. (19) represents the non-adiabatic\ncomponent of the current-induced torque, which is also\nknown as the “ βterm”. By comparing Eq. (19) with\nEq. (1), one can see that the coefficient of the last term\nisequaltotheGilbert dampingconstant α. However,Eq.\n(19) is valid when the relaxation time of the fluctuating\nfield,τc, is very short compared to the time scale of the\nmagnetization dynamics. It should be noted that the\ngeneralformoftheequationdescribingthemagnetization\ndynamics is given by Eq. (17) where the last term on theright-hand side is the origin of the αandβterms. It is\npossible to projectthe torque representedby the memory\nfunction onto the direction of the αandβterms. This\nprojection leads to α/negationslash=βin general.\nIn order to observe the effect of τcon the magneti-\nzation dynamics we applied our theory to the current-\ninduced magnetization switching in the magnetic multi-\nlayer shown in Fig.1 (b). We assumed that the fixed and\nfree layers are single-domain magnetic layers acting as a\nlarge spin characterized by the total magnetization vec-\ntor defined as Si=/integraltext\ndV/angbracketleftMi/angbracketright, wherei= 1(2) for the\nfixed (free) layer and/integraltext\ndVdenotes the volume integra-\ntion over the fixed (free) layer. Both the magnetization\nvector of the fixed layer S1and the effective magnetic\nfield,H, acting on the free layer lie in the plane.\nIntegrating Eqs. (12) and (13) over the volume of the\nfree layer, we obtain the equations,\n∂tS2+/integraldisplay\ndSˆn·J=γ(H+/angbracketleftδH/angbracketright)×S2,(20)\n∂t/angbracketleftδH/angbracketright=−1\nτc(/angbracketleftδH/angbracketright−χVS2), (21)\nwhereJ=v⊗/angbracketleftM/angbracketrightis the spin current tensor/integraltext\ndSrep-\nresents the surface integration over the free layer, ˆnis\nthe unit normal vector of the surface, and χV=χ/Vis\ndefined by the volume of the free layer V.\nThe same procedure used to derive Eq. (17) yields\n∂tS2+/integraldisplay\ndSˆn·J=γH×S2\n+γ/integraldisplayt\n−∞dt′ψ(t−t′)χVS2(t)×∂t′S2(t′),(22)\nwhereψ(t) = exp[−t/τc].\nWhen the relaxation time of the fluctuating field is\nshort compared to the time scale of magnetization dy-\nnamics, the LLG-type equation in the presence of the\nspin-transfer torque is obtained as\n∂tS2+/integraldisplay\ndSˆn·J=γH×S2+α\nS2S2×∂tS2,(23)\nwhereα=γτcχVS2. By introducing the conventional\nform of the spin-transfer torque [1], we obtain the follow-\ning LLG-type equation:\n∂tS2−I\neg/planckover2pi1(S2×S1)×S2=γH×S2+α\nS2S2×∂tS2.(24)\nHowever, Eq. (24) is valid only when τc<1/(γH). As\nmentioned before, the torque represented by using the\nmemory function generally has a component parallel to\nthe non-adiabatic torque. In order to observe the ef-\nfect of the non-adiabatic torque induced by the memory\nfunction on the magnetization dynamics, we performed\nnumerical simulation using Eqs. (20) and (21).4\nFIG. 2: The z-component of the magnetization S2is plotted\nas a function of time for various values of τc. The initial\ndirection of the free layer is taken to lie in the direction of\nthe effective magnetic field, which is aligned to the zaxis.\nThe initial angle between S1andS2is taken to be 45◦. The\nGilbert damping constant αis 0.01.\nFor the simulation, we used the following conditions.\nAt the initial time of t= 0, we assumed that the mag-\nnetization of the free layer is aligned parallel to the ef-\nfective magnetic field Hand the angle between the mag-\nnetizations of the fixed and the free layers is 45◦. This\narrangement corresponds to the recent experiment on a\nmagnetic tunnel junction system [18]. We also assumed\nthat the fluctuation field has zero mean value at t= 0,\ni.e.,/angbracketleftδH(0)/angbracketright=0.\nIn Fig. 2, we plot the time dependence of the zcom-\nponent of the magnetization of the free layer, S2, under\nthe large-enough spin current to flip the magnetization\nof the free layer, Ig/planckover2pi1S2\n2S1/(eαγH) =−10. The value of\nτcis varied while the value of α= 0.01 is maintained.\nThe solid, dotted, and dot-dashed lines correspond to\nγHτc= 0.1,1.0, and 10.0, respectively. As shown in\nFig. 2, the time required for the magnetization of the\nfree layer to flip decreases with increasing τc, which can\nbe understood by considering the non-adiabatic torque\ninduced by the spin current. The non-adiabatic torque\ninduced by the spin current is obtained by projecting\nthe torque given by the last term of Eq. (22) onto the\ndirection of S2×S1, which results in the positive con-\ntribution to the spin-flip motion of S2. Since the last\nterm of Eq. (22) includes a memory function, the non-\nadiabatic torque induced by the spin current increases\nwith increasing τc. Therefore, the time required for S2\nto flip decreases with increasing τc. ForγHτc>10 we\nobserve no further decrease of the time required for S2\nto flip because the memory function is an integral of the\nvectorS2(t)×∂t′S2(t′) and the contributions from the\nmemory at t−t′≫1/(γH) is eliminated.\nIn conclusion, we derived the equation for the motion\nof magnetization in the presence of a spin current by us-\ning the local equilibrium assumption in non-equilibriumthermodynamics. We demonstrated that the value of the\ncoefficientβis not equal to that of the Gilbert damping\nconstantαin general. However, we also show that the\nequalityα=βholds ifτc≪1/(γH). We then applied\nour theory to current-induced magnetization reversal in\nmagnetic multilayersand showed that the switching time\nis a decreasing function of τc.\nThe authors would like to acknowledge the valuable\ndiscussions they had with S.E. Barnes, S. Maekawa, P.\nM. Levy, K. Kitahara, K. Matsushita, J. Sato and T.\nTaniguchi. This work was supported by NEDO.\n[1] J. C. Slonczewski, J. Magn. Magn. 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Ohno, Materials Today 9, 18 (2006)."    },    {        "title": "1308.0450v2.Spin_pumping_damping_and_magnetic_proximity_effect_in_Pd_and_Pt_spin_sink_layers.pdf",        "content": "arXiv:1308.0450v2  [cond-mat.mes-hall]  5 Apr 2016Spin pumping damping and magnetic proximity effect in Pd and P t spin-sink layers\nM. Caminale,1,2,∗A. Ghosh,2,†S. Auffret,2U. Ebels,2K. Ollefs,3F. Wilhelm,4A. Rogalev,4and W.E. Bailey1,5,‡\n1Fondation Nanosciences, F-38000 Grenoble, France\n2SPINTEC, Univ. Grenoble Alpes / CEA / CNRS, F-38000 Grenoble , France\n3Fakult¨ at f¨ ur Physik and Center for Nanointegration (CENI DE),\nUniversit¨ at Duisburg-Essen, 47057 Duisburg, Germany\n4European Synchrotron Radiation Facility (ESRF), 38054 Gre noble Cedex, France\n5Dept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: April 6, 2016)\nWe investigated the spin pumping damping contributed by par amagnetic layers (Pd, Pt) in both\ndirect and indirect contact with ferromagnetic Ni 81Fe19films. We find a nearly linear dependence\nof the interface-related Gilbert damping enhancement ∆ αon the heavy-metal spin-sink layer thick-\nnesses t Nin direct-contact Ni 81Fe19/(Pd, Pt) junctions, whereas an exponential dependence is o b-\nservedwhenNi 81Fe19and(Pd, Pt)areseparated by3nmCu. Weattributethequasi-l inear thickness\ndependence to the presence of induced moments in Pt, Pd near t he interface with Ni 81Fe19, quan-\ntified using X-ray magnetic circular dichroism (XMCD) measu rements. Our results show that the\nscattering of pure spin current is configuration-dependent in these systems and cannot be described\nby a single characteristic length.\nI. INTRODUCTION\nAs a novel means of conversion between charge- and\nspin-currents,spinHallphenomenahaverecentlyopened\nup new possibilities in magneto-electronics, with poten-\ntial applications in mesocale spin torques and electrical\nmanipulation of domain walls1–9. However, several as-\npects of the scattering mechanisms involved in spin cur-\nrent flow across thin films and interfaces are not entirely\nunderstood. Fundamental studies of spin current flow in\nferromagnet/non-magnetic-meal (F/N) heterostructures\nin the form of continuous films have attempted to iso-\nlate the contributions of interface roughness, microstruc-\nture and impurities10–12. magnet/non-magnetic-meal\n(F/N) heterostructures in the form of continuous films\nhave attempted to isolate the contributions of interface\nroughness, microstructure and impurities10–12. Proto-\ntypical systems in this class of studies are Ni 81Fe19/Pt\n(Py/Pt)3,5–7,13–18andNi 81Fe19/Pd(Py/Pd)8,11,14,16,19,20\nbilayers. Inthesesystems, PtandPdareemployedeither\nas efficient spin-sinks or spin/charge current transform-\ners, in spin pumping and spin Hall experiments, respec-\ntively. Pd and Pt are metals with high paramagnetic\nsusceptibility and when placed in contact with a ferro-\nmagnetic layer (eg. Py, Ni, Co or Fe) a finite magnetic\nmoment is induced at the interface by direct exchange\ncoupling21–24.\nThe role of the magnetic proximity effect on interface\nspin transport properties is still under debate. Zhang et\nal.25havereportedthat induced magnetic momentsin Pt\nand Pd films in direct contact with Py correlate strongly\nreduced spin Hall conductivities. This is ascribed to a\nspin splitting of the chemical potential and on the energy\ndependence of the intrinsic spin Hall effect. In standard\nspin pumping theory26, possible induced moments in N\nare supposed to be a priori included in calculations of\nthe spin-mixing conductance g↑↓of a F|N interface27,28,which tends to be insensitive to their presence.\nRecent theoretical works, on the other hand, propose\nthe need of a generalized spin pumping formalisms in-\ncluding spin flip and spin orbit interaction at the F |N\ninterface, in order to justify discrepancies between exper-\nimental and calculated values of mixing conductance29,30.\nAt present, it is still an open issue whether and how\nproximity-induced magnetic moments in F/N junctions\narelinked to the varietyofthe spin-transportphenomena\nreported in literature10,17,31.\nHere, we present an experimental study of the pro-\ntotypical systems: Py/Pd, Pt and Py/Cu/Pd, Pt het-\nerostructures. The objective of our study is to address\nthe role of proximity induced magnetic moments in spin\npumpingdamping. Tothisend, weemployedtwocomple-\nmentary experimental techniques. X-ray magnetic circu-\nlar dichroism (XMCD) is an element sensitive technique\nwhich allows us to quantify any static proximity-induced\nmagnetic moments in Pt and Pd. Ferromagnetic reso-\nnance(FMR) measurementsprovideindirectinformation\non the spin currents pumped out the Py layer by the pre-\ncessing magnetization, through the characterization of\nthe Pd, Pt thickness dependence of the interface-related\nGilbert damping α. In Fig. 3(Sec.IIIB), comparative\nmeasurements in Py/Cu/N and Py/N structures show a\nchange of the N thickness dependence of ∆ α(tN) from\nan exponential to a linear-like behavior. A change in\n∆α(tN) indicates a transformation in the spin scattering\nmechanism occurring at the interface, ascribed here to\nthe presence of induced moments in directly exchange\ncoupled F/N systems. Theoretical works predicted a\ndeviation from a conventional N-thickness dependence\nwhen interface spin-flip scattering is considered in the\npumping model29,30, howeverno functional form waspro-\nvided. For Py/N systems, we find that the experimental\nthickness dependence cannot be described by standard\nmodels16,26,32, but rather a linear function reproduces2\nthe data to a better degree of accuracy, by introducing\na different characteristic length. We speculate that the\nspatial extent of spin current absorption in F/N systems\nshows an inverse proportionality to interfacial exchange\ncoupling energy, obtained from XMCD, as proposed be-\nfore for spin polarized, decoupled interfaces in F 1/Cu/F 2\nheterostructures14.\nII. EXPERIMENT\nThe heterostructures were fabricated by DC mag-\nnetron sputtering on ion-cleaned Si/SiO 2substrates in\nthe form of substrate/seed/multilayer/cap stacks, where\nTa(5nm)/Cu(5nm) bilayerwasemployedas seed. Ta/Cu\nis employed to promote <111>growth in Py and subse-\nquent fcc layers (Pd, Pt), and Ta is known to not affect\nthe dampingstrongly17,32,33. Different stacksweregrown\nasmultilayer for each measurement.\nFor FMR measurements, we have multilayer =\nPy(tF)/N(tN), Py(tF)/Cu(3nm)/N( tN) with N = Pd,\nPt; an Al(3nm) film, oxidized in air, was used as\ncap. The smallest N layer thickness tNdeposited is\n0.4nm, the maximum interdiffusion length observed\nfor similar multilayers34. Samples with multilayer =\nPy(tN)/Cu(3nm) and no sink layer were also fabricated\nas reference for evaluation of the Gilbert damping en-\nhancement due tothe Pd orPtlayer. The tN-dependence\nmeasurementsofFMRweretakenforPythicknesses tF=\n5and 10nm. Results from the tF= 10nm data set are\nshown in Appendix A. Measurements of the FMR were\ncarried out at fixed frequency ωin the 4-24 Ghz range,\nby means of an in-house apparatus featuring an external\nmagnetic field up to 0.9T parallel to a coplanar waveg-\nuide with a broad center conductor width of 350 µm.\nFor XMCD measurements, given the low X-ray ab-\nsorption cross-sectionpresented by Pt and Pd absorption\nedges, a special set of samples was prepared, consisting\nof 20 repeats per structure in order to obtain sufficiently\nhighsignal-to-noiseratio. Inthiscase,wehave multilayer\n= [Py(5nm)/N] 20, with N = Pd(2.5nm) and Pt(1nm);\nCu(5nm)/Py(5nm)/Al(3nm) was deposited as cap. The\nPt and Pd thicknesses were chosen to yield a damping\nenhancement equal about to half of the respective satu-\nration value (as it will be shown later), i.e. a thicknesses\nfor which the F/N interface is formed but the damp-\ning enhancement is still increasing. XMCD experiments\nwere carried out at the Circular Polarization Beamline\nID-12 of the European Synchrotron Radiation Facility\n(ESRF)35. Measurementsweretakenintotalfluorescence\nyield detection mode, at grazing incidence of 10◦, with\neither left or right circular helicity of the photon beam,\nswitching a 0.9T static magnetic field at each photon en-\nergy value (further details on the method are in Ref.22).\nNo correction for self-absorption effects is needed; how-\never XMCD spectra measured at the L 2,3edges of Pd\nhave to be corrected for incomplete circular polarization\nrate of monochromatic X-rays which is 12% and 22% at/s49/s49/s46/s53/s52 /s49/s49/s46/s53/s54 /s49/s49/s46/s53/s56 /s49/s49/s46/s54/s48 /s49/s49/s46/s54/s50 /s49/s51/s46/s50/s54 /s49/s51/s46/s50/s56 /s49/s51/s46/s51/s48 /s49/s51/s46/s51/s50 /s49/s51/s46/s51/s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s116/s32/s88/s65/s83\n/s32/s65/s117/s32/s88/s65/s83\n/s80/s100/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s80/s116\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s32\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\n/s51/s46/s49/s52 /s51/s46/s49/s54 /s51/s46/s49/s56 /s51/s46/s50/s48 /s51/s46/s50/s50 /s51/s46/s51/s48 /s51/s46/s51/s50 /s51/s46/s51/s52 /s51/s46/s51/s54 /s51/s46/s51/s56/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s100/s32/s88/s65/s83\n/s32/s65/s103/s32/s88/s65/s83/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s45/s48/s46/s48/s57/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\nFIG. 1. (Color online) X-ray absorption (XAS, left axis) and\nmagnetic circular dichroism (XMCD, right axis) spectra at\nthe L-edges of Pt (top panel) and Pd (bottom panel) for\n[Py(5nm)/Pt(1nm)] 20and [Py(5nm)/Pd(2.5nm)] 20multilay-\ners. The dashed traces represent XAS spectra at L-edge of\nAg and Au used as background of Pd and Pt, respectively, to\nextract the values of induced magnetic moment reported in\nTab.I.\nL3and L2, respectively. The circular polarization rate is\nin excess of 95 % at the L 2,3edges of Pt.\nIII. RESULTS AND ANALYSIS\nIn order to study how the proximity-induced mag-\nnetic moments may affect the absorption of spin-currents\nthrough interfaces, the static moment induced in Pt, Pd\nlayers in direct contact with ferromagnetic Py in char-\nacterized first, by means of XMCD. The value of the\ninduced moment extracted for the two Py/N systems is\nusedtoestimatetheinterfacialexchangeenergyactingon\nthe two paramagnets. Afterwards, the dynamic response\nofthe magnetization is addressedby FMR measurements\nin Py/N(direct contact) and Py/Cu/N(indirect contact)\nheterostructures. From FMR measurements carried out\non both configurations as a function of N thickness, the\ndamping enhancement due to the presence of the spin-\nsink layers Pt and Pd is obtained from the frequency-\ndependence of the FMR linewidth. The relation between\nthe static induced moment and the spin pumping damp-\ning is discussed by comparing the results of the direct\nwith indirect contact systems.3\n-0.06-0.04-0.020.00XMCD\n11.5911.5811.5711.5611.55\nPhoton Energy (keV)1.0\n0.5\n0.0\n3210tCu (nm) Area\n 1 nm Cu\n 0.5 nm Cu\n 0 nm Cu\nFIG. 2. (Color online) XMCD spectra at the L 3edge of Pt\nfor [Py(5nm)/Cu(t Cu)/Pt(1nm)] 15, with t Cu= 0, 0.5 and\n1nm. As inset the ares of the peak is plot as a function of Cu\nthickness.\nA. XMCD: Probing the induced magnetic moment\nIn Fig.1we report X-ray absorption (XAS) and mag-\nnetic circular dichroism (XMCD) spectra at the L2,3\nedges of Pt (top panel) and Pd (bottom panel) taken on\nPy(5nm)/Pt(1nm) |20and Py(5nm)/Pd(2.5nm) |20, re-\nspectively. Rather intense XMCD signals have been de-\ntected at both Pt and Pd L 2,3edges, showing unambigu-\nously that a strong magnetic moment is induced by di-\nrect exchange coupling at the Py |N interface. The static\ninduced moment is expected to be ferromagnetically cou-\npled with the magnetization in Py21. From the integrals\nof XMCD spectra, the induced magnetic moment on the\nPt, Pd sitesis determined byapplying the sum rulesasin\nRef.22(and references therein). In Py/Pd(2.5nm) |20, Pd\natoms bear a moment of 0.12 µB/at, averaged over the\nwhole volume of the volume, with an orbital-to-spinratio\nmL/mS= 0.05. In Py/Pt(1nm) |20, a magnetic moment\n0.27µB/at is found on Pt, comparable to that reported\nfor Ni/Pt epitaxial multilayers23, with a relatively high\norbitalcharacter mL/mS= 0.18, ascomparedwithPdin-\nduced moment. The large difference in volume-averaged\ninduced moment per atom comes from the different film\nthickness, hence volume, for Pt and Pd. Assuming that\nthe induced magnetic moment is confined to the first\natomic layers at the interface with Py23,24, one could\nestimate 0.32 µB/at for Pd and 0.30 µB/at for Pt36.\nWhen a3nm thick Cu interlayeris introducedbetween\nPy and N, a two ordersofmagnitude smaller induced mo-\nment (0.0036 µB/at) was found for 2.5nm Pd22, while Pt\nshowed an XMCD signal of the order of the experimen-\ntal sensitivity, ∼0.5·10−3µB/at. In Fig. 2XMCD\nspectra at the L 3edge of Pt are shown for Cu interlayer\nthicknesses 0, 0.5 and 1nm. For 0.5nm Cu the integral\nof XMCD signal at the L 3edge shrinks to 30%, while\nfor 1nm it is reduced to zero within experimental error.\nThis result could be explained either by a 3d growth of\nthe Cu layer, allowing a fraction of the Pt layer to be in\ndirect contact with Py for Cu coverages of 0.5nm, or byNχmol37S37N0abulktN/angbracketleftM/angbracketrightMiJex\n[cm3/mol][1/eV·at][nm][nm][µB/at][µB/at][meV]\n10−4\nPd5.5±0.29.30.83±0.030.3892.50.1160.3242\nPt1.96±0.13.70.74±0.040.3921.00.270.30109\nTABLE I. Spin-sink layer N properties in Py/N heterostruc-\ntures: experimental molar susceptibility χmolat 20◦C; den-\nsity of states N0calculated from tabulated χmol; Stoner pa-\nrameter S from Ref.37; bulk lattice parameter a; layer thick-\nnessestN; volume averaged induced magnetic moment /angbracketleftM/angbracketright\nfrom XMCD measurement in Fig. 1; interface magnetic mo-\nmentMi36; Py|N interfacial exchange energy per interface\natomJex(Eq.1).\ndiffusion of magnetic Ni atoms in Cu on a scale shorter\nthan 1nm. The film then becomes continuous, and at\n1nm coverage, no direct exchange coupling takes place\nbetween Py and Pt layers. For FMR measurements pre-\nsented in the followingsection, a 3nm thick Cu interlayer\nis employed, reducing also any possible indirect exchange\ncoupling.\nFrom the values of induced moments in Pd and Pt,\nwe can make a step forward and estimate the interfacial\nexchange coupling energies for the two cases. Equating\ninteratomic exchange energy Jexand Zeeman energy for\nan interface paramagnetic atom, we have (see Appendix\nB1for the derivation)\nJex=1\n2/angbracketleftM/angbracketright\nµBN0StN\nti(1)\nwhere/angbracketleftM/angbracketrightis the thickness-averaged paramagnetic\nmoment, N0is the single-spin density of states (in\neV−1at−1), S is the Stoner factor and ti= 2∗a/√\n3\nis the polarized interface-layer thickness36. The 1/2fac-\ntor accounts for the fact that in XMCD measurements\nthe N layer has both interfaces in contact with F. Un-\nder the simplifying assumption that all the magnetic mo-\nment is confined to the interface N layer and assuming\nexperimental bulk susceptibility parameters for χv, we\nobtainJPd\nex= 42 meV for Pd and JPt\nex= 109 meV for\nPt (results and properties are summarized in Tab. I).\nHere the difference in estimated Jex, despite roughly\nequalMi, comes from the larger Stoner factor S for\nPd. A stronger interfacial exchange energy in Pt de-\nnotes a stronger orbital hybridization, yielding possibly\na higher orbital character of the interfacial magnetic mo-\nment in the ferromagnetic Py counterpart21. For com-\nparison, we consider the interatomic exchange param-\netersJexin ferromagnetic Py and Co, investigated in\nRef.14.Jexis estimated from the respective Curie tem-\nperatures TC, through Jex≃6kBTC/(m/µB)2, wherem\nis the atomic moment in µB/at (see Appendix B2). Ex-\nperimental Curie temperatures of 870K and 1388K give\nJCo\nex=293meV for Co and JPy\nex=393meV for Py, which\nare of the same order of the value calculated for Pt (de-\ntails about calculation in Appendix B2).4\n1.0\n0.5\n0.0\n14121086420\ntPt (nm) Py/Pt\n Py/Cu/Pt12\n10\n8\n6\n4\n2\n0\n468\n12468\n10\ntPt (nm) Py/Pt\n tc = 2.4 nm\n Py/Cu/Pt\n λα = 1.8 nm\n1.0\n0.5\n0.0∆α / ∆α0\n14121086420\ntPd (nm) Py/Pd\n Py/Cu/Pd6\n5\n4\n3\n2\n1\n0∆α (x103)\n468\n12468\n10\ntPd (nm) Py/Pd\n tc = 5.0 nm\n Py/Cu/Pd\n λα = 5.8 nma) b)\nc) d)\nFIG. 3. (Color online) Damping enhancement ∆ α, due to\npumped spin current absorption, as a function of thickness tN\nfor Py(5nm)/N and Py(5nm)/Cu(3nm)/N heterostructures,\nwith N = Pd( tN) (panels a,c), Pt( tN) (panels b,d). Solid\nlines result from a fit with exponential function (Eq. 2) with\ndecay length λα. Dashed lines represents instead a linear-\ncutoff behavior (Eq. 3) fortN< tc. Please notice in panels a,\nc the x-axis is in logarithmic scale. In panels b, c the dampin g\nenhancement is normalized to the respective saturation val ue\n∆α0.\nIn the following, the effect of these static induced mo-\nments on the spin pumping damping of the heterostruc-\ntures characterized will be discussed.\nB. FMR: damping enhancement\nThe main result of our work is now shown in Figure\n3. In Fig. 3the damping enhancement ∆ αis plotted as\na function of the spin-sink layer thickness tN, for Py/Pd,\nPy/Cu/Pd (panels a, c) and Py/Pt, Py/Cu/Pt (panels\nb, d). The enhancement ∆ αis compared with the damp-\ningαof a reference structure Py(5nm)/Cu, excluding\nthe sink layer N. Each value of αresults from established\nanalysis of the linewidth of 11 FMR traces13,14, employ-\ning ag-factor equal to 2.09 as a constant fit parameter\nfor all samples.\nIn Py/Cu/N systems (Fig. 3, green square markers),\n∆αrises with increasing tNthickness to similar satura-\ntion values ∆ α0= 0.0027, 0.0031 for Pd and Pt, but\nreachedondifferentlengthscales,giventhedifferentchar-Ng↑↓\neff(Py|Cu/N) λαg↑↓\neff(Py|N)tc\n[nm−2][nm] [nm−2][nm]\nPd 7.2 5.8±0.2145.0±0.3\nPt 8.3 2.4±0.1321.8±0.2\nTABLE II. Mixing conductance values extracted from the\ndamping enhancement ∆ αat saturation in Fig. 3, and re-\nspective length scales (see text for details).\nacteristic spin relaxation lengths of the two materials.\nFrom the saturation value, an effective mixing conduc-\ntanceg↑↓\neff(Py|Cu/N) = 7 .2−8.3nm−2is deduced in the\nframeworkof standardspin pumping picture13,17,19, with\nPy saturation magnetization µ0Ms= 1.04T. The fact\nthat the spin-mixing conductance is not material depen-\ndent indicates that similar Cu |N interfaces are formed.\nThe thickness dependence is well described by the expo-\nnentialfunction14,20\n∆α(tN) = ∆α0(1−exp(−2tN/λN\nα)) (2)\nas shown by the fit in Fig. 3a-b (continuous line). As\na result, exponential decay lengths λPt\nα= 1.8nm and\nλPd\nα= 5.8nm are obtained for Pt and Pd, respectively.\nWhen the Pt, Pd spin-sink layers come into direct con-\ntact with the ferromagnetic Py, the damping enhance-\nment ∆α(tN) changes dramatically. In Py/N systems\n(Fig.3a-b, trianglemarkers),the damping saturationval-\nues become ∆ αPt\n0= 0.0119 and ∆ αPd\n0= 0.0054 for Pt\nand Pd, respectively a factor ∼2 and∼4 larger as com-\npared to Py/Cu/N. Within the spin-pumping descrip-\ntion, a largerdamping enhancement implies a largerspin-\ncurrentdensitypumped outoftheferromagnetacrossthe\ninterface and depolarized in the sink.\nIn Py/N heterostructures, because of the magnetic\nproximity effect, few atomic layers in N are ferromagnet-\nically polarized, with a magnetic moment decaying with\ndistance from the Py |N interface. The higher value of\ndamping at saturation might therefore be interpreted as\nthe result of a magnetic bi-layer structure, with a thin\nferromagnetic N characterized by high damping αN\nhigh\ncoupled to a low damping αF\nlowferromagnetic Py38. To\ninvestigate whether damping is of bi-layer type, or truly\ninterfacial, in Fig 4we show the tFthickness dependence\nof the damping enhancement ∆ α, for a Py( tF)/Pt(4 nm)\nseries of samples. The power law thickness dependence\nadheres very closely to t−1\nF, as shown in the logarithmic\nplot. The assumption of composite damping for syn-\nchronous precession, as ∆ α(t1) = (α1t1+α2t2)/(t1+t2),\nshown here for t2= 0.25nm and 1.0nm, cannot follow\nan inverse thickness dependence over the decade of ∆ α\nobserved. Damping is therefore observed to have a pure\ninterfacial character.\nIn this case, the mixing conductances calculated from\nthe saturation values are g↑↓\neff(Py|Pd) = 14nm−2and\ng↑↓\neff(Py|Pt) = 32nm−2. From ab initio calculations\nwithin a standard spin-pumping formalism in diffusive\nfilms10,29, it is found g↑↓\neff(Py|Pd) = 23nm−2for Pd and5\n10 3 4 56789 20 30 40\ntF (nm)110\n2345678920∆α(10−3)Py(tF)/Pt(4)\ninterfacial, K=0.055\nbilayer, t2=0.25 nm\n             t2=1.0 nm\n26101520 30\ntF (nm)051015202530α(10−3)Py(t)/Pt(4nm)\nPy(t)\nFIG. 4. Logarithmic plot of the damping enhancement ∆ α\n(triangle markers) as a function of the Py layer thickness tF,\nin Py(t F)/Pt(4nm). Solid and dashed lines represents, re-\nspectively, fits according to the spin pumping ( interfacial )\nmodel ∆ α=Kt−1\nFand to a αlow(tF)/αhigh(t2)bilayermodel,\nwitht2= 0.25,1.0nm.Inset: Gilbert damping αfor Py(t F)\n(square markers) and Py(t F)/Pt(4nm) (round markers).\ng↑↓\neff(Py|Pt) = 22nm−2for Pt. Theoretical spin mixing\nconductance from a standard picture does reproduce the\nexperimental order of magnitude, but it misses the 2.3\nfactor between the Py |Pt and Py |Pd interfaces. Beyond\na standard pumping picture, Liu and coworkers29intro-\nduce spin-flipping scattering at the interface and calcu-\nlate from first principles, for ideal interfaces in finite dif-\nfusive films: g↑↓\neff(Py|Pd) = 15nm−2, in excellent agree-\nment with the experimental value here reported for Pd\n(Tab.II), andg↑↓\neff(Py|Pt) = 25nm−2. Zhang et al.10\nsuggest an increase up to 25% of the mixing conductance\ncan be obtained by introducing magnetic layers on the\nPt side. The results here reported support the emerging\nidea that a generalized model of spin pumping including\nspin-orbit coupling and induced magnetic moments at\nF|N interfaces may be required to describe the response\nof heterostructures involving heavy elements.\nThe saturation value of damping enhancement at ∆ α0\nas a function of the Cu interlayer thickness is shown in\nFig.5to follow the same trend of the XMCD signal\n(dashed line), reported from Fig. 2. Indeed, it is found\nthat the augmented ∆ α0in Py/N junctions is drasti-\ncally reduced by the insertion of 0.5nm Cu at the Py |N\ninterface17, and the saturation of the Py/Cu/N configu-\nration is already reached for 1nm of Cu interlayer. As\nsoonasacontinuousinterlayerisformedandnomagnetic\nmoment is induced in N, ∆ α0is substantially constant\nwith increasing Cu thickness.\nThe N-thickness dependence of ∆ α(tN) in Py/N sys-\ntems before saturation is addressed in the following. At\nvariance with the Py/Cu/N case, the thickness depen-\ndence of ∆ αis not anymore well described by an expo-1.0\n0.5\n0.0\n3210\ntCu (nm)1.0\n0.5\n0.0L3 XMCD area (norm.)Pt(3nm)\n XMCD Pt\n3210\ntCu (nm)1.0\n0.5\n0.0∆α0 (norm.)Pd(7nm)\n XMCD Pt\nFIG. 5. (Color online) Normalized damping enhancement ∆ α\n(left axis), due to spin pumping, as a function of interlayer\nthickness tCufor Py(5nm)/Cu( tCu)/N heterostructures, with\nN = Pd(7nm), N = Pt(3nm). The dashed line represents the\nXMCD signal (right axis) reported from inset in Fig. 2.\nnential behavior, as an exponential fit (with exponential\ndecay length as only free fit parameter) fails to repro-\nduce the increase of ∆ αtowards saturation (solid lines\nin Fig.3a-b). More rigorous fitting functions employed\nin spin pumping experiments, within standard spin trans-\nport theory16,26,32, cannot as well reproduce the experi-\nmental data (see Appendix. A). It is worth mentioning\nthat the same change of trend between the two configu-\nrations was observed for the same stacks with a 10nm\nthick Py layer (data shown in Appendix A, Fig.7). A\nchange of the functional dependence of ∆ αontNre-\nflects a change in the spin-depolarization processes the\npumped spin current undergoes, as for instance shown in\nRef.30when interfacial spin-orbit coupling is introduced\nin the spin-pumping formalism. Experimentally, a linear\nthickness dependence with sharp cutoff has been shown\nto characterize spin-current absorption in spin-sink lay-\ners exhibiting ferromagnetic order at the interface, as re-\nported for F 1/Cu/F 2(tF2) junctions with F = Py, Co,\nCoFeB14. Given the presence of ferromagnetic order in\nN at the interface of F/N structures, the data are tenta-\ntively fit with a linear function\n∆α= ∆α0tN/tN\nc (3)\nThis linear function better reproduce the sharp rise of\n∆α(dashed lines in Fig. 3a-b) and gives cutoff thick-\nnessestPt\nc= 2.4±0.2nm and tPd\nc= 5.0±0.3nm for Pt\nand Pd, respectively. The linearization is ascribed to the\npresence of ferromagnetic order in the paramagnetic Pd,\nPt spin-sink layers at the interface with the ferromag-\nnetic Py. The linear trend extends beyond the thickness\nfor which a continuous layer is already formed (less than\n1nm), and, especially for Pd, far beyond the distance\nwithin the non-uniform, induced moment is confined (up\nto0.9nm). InRef.14, thecutoff tcinF/Cu/Fheterostruc-\nturesis proposedto be on the orderofthe transversespin\ncoherence length λJin ferromagnetically ordered layers.\nλJcan be expressed in terms of the exchange splitting6\n6\n5\n4\n3\n2\n1\n0tc(nm)\n2520151050\n1/Jex(eV-1)PtPd\nPy\nCo\nFIG. 6. Effect of direct exchange strength on length scale of\nspin current absorption. Cutoff thickness tcextracted from\nthe ∆α(tN) data in Fig. 3as a function of reciprocal interfa-\ncial exchange energy 1 /Jexextracted from XMCD in Fig. 1.\nLabels are given in terms of Jex. The Co and Py points are\nfrom Ref.14.\nenergyJex,\nλJ=hvg\n2Jex(4)\nwherevgis the electronic group velocity at the Fermi\nlevel. This form, found from hot-electron Mott\npolarimetry1, is expressed equivalently for free electrons\nasπ/|k↑−k↓|, which is a scaling length for geometrical\ndephasing in spin momentum transfer2. Electrons which\nenter the spin-sink at E Fdo so at a distribution of angles\nwith respect to the interface normal, traverse a distribu-\ntionofpathlengths, andprecessbydifferentangles(from\nminority to majority or vice versa ) before being reflected\nback into the pumping ferromagnet. For a constant vg,\nit is therefore predicted that tcis inversely proportional\nto the exchange energy Jex.\nIn Figure 6we plot the dependence of the cutoff thick-\nnesstN\ncupontheinverseoftheestimatedexchangeenergy\nJex(Tab.I),asextractedfromtheXMCDmeasurements.\nA proportionality is roughly verified, as proposed for the\ntransverse spin coherence length across spin polarized in-\nterfaces. Under the simplistic assumption that tc=λJ,\nfrom the slope of the line we extract a Fermi velocity\nof∼0.1·106m/s (Eq.4), of the order of magnitude ex-\npectedforthematerialsconsidered39,40. Thesedatashow\nthat, up to a certain extent, length scale for spin-current\nscattering shares common physical origin in ferromag-\nnetic layers and paramagnetic heavy-metals, such as Pd\nand Pt, under the influence of magnetic proximity effect.\nThis unexpected results is observed in spite of the fact\nthat F 1/Cu/F 2and F/N systems present fundamental\ndifferences. In F/N structure, the induced moment in N\nis expected to be directly exchange coupled with the fer-\nromagneticcounterpart. Whereasin F 1/Cu/F 2, themag-\nnetic moment in F 2(off-resonance) are only weakly cou-\npled with the precession occurring in F 1(in-resonance),\nthrough spin-orbit torque and possible RKKY interac-tion. Magnetization dynamics in N might therefore be\nexpected with its own pumped spin current, albeit, to\nthe best of our knowledge, no experimental evidence\nof a dynamic response of proximity induced moments\nwas reported so far. From these considerations and the\nexperimental findings, counter-intuitively the proximity-\ninduced magnetic moments appear not to be involved in\nthe production of spin current, but rather to contribute\nexclusively with an additional spin-depolarization mech-\nanism at the interface.\nIV. CONCLUSIONS\nWe have investigated the effect of induced magnetic\nmoments in heavy metals at Py/Pt and Py/Pd inter-\nfaces on the absorption of pumped spin currents, by\nanalyzing ferromagnetic resonance spectra with varying\nPt, Pd thicknesses. Static, proximity-induced magnetic\nmoments amount to 0.32 and 0.3 µB/atom in Pd and\nPt, respectively, at the interface with Py, as deduced\nfrom XMCD measurements taken at the L 2,3edges. We\nhave shown that when the proximity induced moment\nin Pt and Pd is present, an onset of a linear-like thick-\nness dependence of the damping is observed, in con-\ntrast with an exponential trend shown by Py/Cu/Pd\nand Py/Cu/Pt systems, for which no induced moment\nis measured. These results point to the presence of an\nadditional spin-flip process occurringat the interface and\nto a change of the character of spin current absorption\nin the ultrathin Pd and Pt paramagnets because of the\ninterfacial spin polarization. The range of linear increase\nis proposed to be inversely proportional to the interfa-\ncial exchange energy in Py/Pt and Py/Pd, inferred from\nXMCD data.\nWEB acknowledges the Universit´ e Joseph Fourier and\nFondation Nanosciences for his research stay at SPIN-\nTEC. This work was supported in part by the U.S. NSF-\nECCS-0925829 and the EU EP7 NMP3-SL-2012-280879\nCRONOS. MC is financed by Fondation Nanosciences.\nAppendix A: N-thickness dependence\nIn order to confirm the results presented in the\nmanuscript, additional sample series with thicker Py\nlayer were fabricated and measured. The experimental\nresults for 10nm thick Py layer are shown in Fig.s 7and\n8forPdandPt, respectively. Wehavepresentedthedata\nhere, rather than including them with the other plots in\nFigure3, to keep the figures from being overcrowded. As\nexpected when doubling the ferromagnet thickness, the\nsaturation values ∆ α0are about half of those measured\nfor 5nm Py (Fig. 3). Confirming the data presented in\nthe manuscript, it is observedagaina changeof thickness\ndependence of ∆ α(tN), fromexponential for Py/Cu/N\n(solid lines; Eq. 2,λα= 4.8nm and 1.4nm for Pd and Pt7\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPd (nm)Py/Pd\nPy/Cu/Pd\n Eq. 3\n,  Eq. 2\n [16, 32] - ρPd=1.4E-7\n [16, 32] - ρPd->ρ(tPd)\nFIG. 7. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPd\nfor Py(10nm)/Pd and Py(10nm)/Cu(3nm)/Pd heterostruc-\ntures. Solid lines result from a fit with exponential functio n\n(Eq.2) with decay λα. Dashed lines represents instead a\nlinear-cutoff behavior (Eq. 3) fortPd< tc. Short-dash and\npoint-dash traces are fit to the data, employing equations\nfrom standard spin transport theory (see text for details)16,32.\nIn bottom panel, ∆ αis normalized to the respective satura-\ntion value.\nrespectively) to linear-like for Py/N (dashed lines; Eq. 3,\ntc= 5.3nm and 2nm for Pd and Pt respectively).\nThe experimental data are also fitted with a set of\nequations derived from standard theory of diffusive spin\ntransport16,26,32, describing the the dependence of ∆ α\non the thickness of adjacent metallic layers (either N or\nCu/N in our case) as follow\n∆α=γ¯h\n4πMstFMg↑↓\n1+g↑↓/gx\next(A1)\nwith (Eq. 7 in Ref.16, and Eq. 6 in Ref.32)\ngN\next=gNtanhtN/λN\nsd\ngCu/N\next=gCugCucothtN/λN\nsd+gNcothtCu/λCu\nsd\ngCucothtN/λN\nsdcothtCu/λCu\nsd+gN(A2)\nwheregx=σx/λx\nsd,σxandλx\nsdare the electrical conduc-\ntivity and spin diffusion length of the non magnetic layer\nx. For the thin Cu layer, we used a resistivity ρCu=\n1×107Ωm and a spin diffusion length λCu\nsd= 170nm32.\nFor the Pt and Pd layers, two fitting models in which the\nconductivity of the films is either constant or thickness\ndependent areconsidered, as recently proposed by Boone\nand coworkers16. The values of conductivity, as taken di-\nrectly from Ref.16, will influence the spin diffusion length\nλN\nsdand spin mixing conductance g↑↓resulting from the\nfit, but will not affect the conclusions drawn about the\noverall trend. When a constant resistivity is used (short-\ndash, blue lines), the model basically corresponds to the6\n5\n4\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPt (nm) Py/Pt\n Py/Cu/Pt\n Eq. 3\n, Eq. 2\n [16, 32] - ρPt=1.7E-7\n [16, 32] - ρPt->ρ(tPt)\nFIG. 8. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPtfor\nPy(10nm)/Pt and Py(10nm)/Cu(3nm)/Pt heterostructures.\nSolid lines result from a fit with exponential function (Eq. 2)\nwith decay λα. Dashed lines represents instead a linear-cutoff\nbehavior (Eq. 3) fortPt< tc. Short-dash and point-dash\ntraces are fit to the data, employing equations from standard\nspin transport theory (see text for details)16,32. In bottom\npanel, ∆ αis normalized to the respective saturation value.\nsimple exponential function in Eq. 2. It nicely repro-\nduces the data in the indirect contact case (Py/Cu/N)\nfor both Pd (Fig. 7) and Pt (Fig. 8), but it fails to fit\nthe direct contact (Py/N) configuration. When a thick-\nness dependent resistivity of the form ρN=ρb\nN+ρs\nN/tNis\nused (dash-point, cyan lines)16, in Py/Cu/N systems, no\nsignificant difference with the other functions is observed\nfor Pt, while for Pd a deviation from experimental trend\nis observed below 1.5nm. In Py/Nsystems, the fit better\ndescribes the rise at thicknesses shorter than the charac-\nteristic relaxation length, while deviates from the data\naround the saturation range.\nModels from standard spin transport theory cannot\nsatisfactorily describe the experimental data for the di-\nrect contact Py/N systems. For this reason a different\nmechanismforthe spin depolarizationprocesseshasbeen\nproposed, considering the presence of induced magnetic\nmoments in N in contact with the ferromagnetic layer.\nAppendix B: Interfacial interatomic exchange\n1. Paramagnets\nWe will show estimates for exchange energy based\non XMCD-measured moments in [Py/(Pt, Pd)] repeatsu-\nperlattices. Calculations of susceptibility are validated\nagainst experimental data for Pd and Pt. Bulk suscepti-\nbilities will be used to infer interfacial exchange parame-8\ntersJi\nex.\na. Pauli susceptibility For an itinerant electron sys-\ntem characterized by a density of states at the Fermi\nenergyN0, if an energy ∆ Esplits the spin-up and spin-\ndown electrons, the magnetization resulting from the\n(single-spin) exchange energy ∆ Eis\nM=µB/parenleftbig\nN↑−N↓/parenrightbig\n= 2µBN0S∆E(B1)\nwhereN0is the density of states in # /eV/at,Sis the\nStoner parameter, and 2∆ Eis the exchange splitting in\neV. Moments are then given in µB/at. Solving for ∆ E,\n∆E=M\n2µBN0S(B2)\nIf the exchange splitting is generated through the ap-\nplication of a magnetic field, ∆ E=µBH,\nµBH=M\n2µBN0S(B3)\nand the dimensionless volume magnetic susceptibility\ncan be expressed\nχv≡M\nH= 2µ2\nBN0S (B4)\nIn this expression, the prefactor can be evaluated\nthrough\nµ2\nB= 59.218 eV ˚A3(B5)\nso with N0[=]/eV/at, χvtakes units of volume per\natom, and is then also called an atomic susceptibility, in\ncm3/at, as printed in Ref37.\nb. Molar susceptiblity Experimentalvaluesaretabu-\nlated as molar susceptibilities. The atomic susceptibility\nχvcanbe contrastedwith the masssusceptibility χmand\nmolar susceptibility χmol\nχmass=χv\nρχmol=ATWT\nρχv(B6)\nwhere ATWT is the atomic weight (g/mol) and ρis\nthe density (g/cm3). These have units of χmass[=]cm3/g\nandχmol[=]cm3/mol. The molar susceptibility χmolis\nthen\nχmol= 2µ2\nBN0NAS (B7)\nin cm3/mol, where µBis the Bohr magneton, and\n2N0S=χmol\nNAµ2\nB(B8)Eq.B8providesaconvenentmethodtoestimateexper-\nimental unknowns, the density of states N0and Stoner\nparameter S, from measurements of χmol.\nExample: for Pd, the low-temperature measurement\n(differentfromtheroom-temperaturemeasurementinTa-\nbleI) isχmol∼7.0×10−4cm3/mol. In the denomina-\ntor, (NAµ2\nB) = 2.622 ×10−6Ry·cm3/mol, The value\n2N0Sconsistent with the experiment is 266/(Ry-at) or\n19.6/(eV-at). For the tabulated measurement of S= 9.3,\nthe inferred density of states is then N0= 1.05/eV/at.\nc. Interfacial exchange We canassumethat the Zee-\nman energy per interface atom is equal to its exchange\nenergy, through the Heisenberg form\nM2\np\nχvVat= 2Ji\nexsfsp (B9)\nwhereMpis the magnetization of the paramagnet,\nwith the atomic moment of the paramagnet mpin terms\nof its per-atom spin sp,\nMp=mp\nVatmp= 2µBsp (B10)\nVatis the volume of the paramagnetic site, sf,pare the\nper-atom spin numbers for the ferromagnetic and para-\nmagnetic sites, and Ji\nexis the (interatomic) exchange en-\nergy acting on the paramagnetic site from the ferromag-\nneticlayersonthe othersideoftheinterface. Interatomic\nexchangeenergyhasbeen distinguished fromintraatomic\n(Stoner) exchange involved in flipping the spin of a single\nelectron. Rewriting Eq B9,\nM2\np\nχvVat= 2Ji\nexsfMp\n2µBVat (B11)\nifsf= 1/2, appropriate for 4 πMs∼10 kG,\nJi\nex= 2µBMp\nχv(B12)\nand substituting for χvthrough Eq B4,\nJi\nex=Mp\nµBN0S(B13)\nIn the XMCD experiment, we measure the thickness-\naveraged magnetization as < M > in a [F/N]nsuper-\nlattice. We make a simplifying assumption that the ex-\nchange acts only on nearest-neighbors and so only the\nnear-interface atomic layer has a substantial magnetiza-\ntion. We can then estimate Mpfrom< M >through\n< M > t p= 2Mpti (B14)9\nwheretiis the polarized interface-layer thickness of\nN36. Since the interface exists on both sides of the N\nlayer, 2tiis the thickness in contact with F. Finally,\nJex=1\n2< M >\nµBN0Stp\nti(B15)\nThe exchange energy acting on each interface atom,\nfrom all neighbors, is JPt\nex= 109 meV for Pt and\nJPd\nex= 42 meV for Pd. Per nearest neighbor for an\nideal F/N(111) interface, it is JPy|Pt= 36 meV and\nJPy|Pd= 14 meV. Per nearest neighbor for an inter-\nmixed interface (6 nn), the values drop to 18 meV and 7\nmeV, respectively.\nSince explicit calculations for these systems are not\nin the literature, we can compare indirectly with theo-\nretical values. Dennler41showed that at a (3 d)F/(4d)N\ninterface (e.g. Co/Rh), there is a geometrical enhance-\nment in the moment induced in Nper nearest-neighbor\nofF. The 4d Natoms near the Finterface have larger\ninduced magnetic moments per nn of Fby a factor of\nfour. Specific calculations exist of JF|N(per neighbor)\nfor dilute Co impurities in Pt and dilute Fe impurties in\nPd42.JFe−Pd∼3 meV is calculated, roughly indepen-\ndent of composition up to 20% Fe. If this value is scaled\nup by a factor of four, to be consistent with the inter-\nface geometry in the XMCD experiment, it is ∼12 meV,\ncomparable with the value for Pd, assuming intermixing.\nTherefore the values calculated have the correct order of\nmagnitude.\n2. Ferromagnets\nThe Weiss molecular field,\nHW=βMs (B16)\nwhereβis a constant of order 103, can be used to give\nan estimate of the Curie temperature, as\nTC=µBgJJ(J+1)\n3kBHW (B17)\nDensity functional theory calculations have been used\ntoestimatethemolecularfieldrecently42,43; forspintype,\ntheJ(J+1) term is substutited with < s >2, giving an\nestimate of\nTC=2< s >2J0\n3kB(B18)where< s >is the number of spins on the atom as in\nEqB10; see the text by St¨ ohr and Siegmann44.< s >\ncan be estimated from m=1.07µBfor Py and 1.7 µBfor\nCo, respectively. Then\nJ0≃6kBTC\n(m/µB)2(B19)\nwith experimental Curie temperatures of 870 and 1388\nK, respectively, gives estimates of J0= 293 meV for Co\nandJ0= 393 meV for Py.\nNote that there is also a much older, simpler method.\nKikuchi45has related the exchange energies to the Curie\ntemperature for FCC lattices through\nJ= 0.247kBTC (B20)\nTaking 12 NN, 12 Jgives a total energy of 222 meV for\nPy (870 K) and 358 meV for FCC Co (1400K), not too\nfar off from the DFT estimates.\nd. Other estimates TheJ0exchange parameter is\ninteratomic, describing the interaction between spin-\nclusters located on atoms. Reversing the spin of one of\nthese clusters would change the energy J0. The Stoner\nexchange ∆ is different, since it is the energy involved in\nreversing the spin of a single electron in the electron sea.\nGenerally ∆ is understood to be greater than J0because\nit involvesmore coloumbrepulsion; interatomic exchange\ncan be screened more easily by spelectrons.\nThis exchange energy is that which is measured by\nphotoemissionandinversephotoemission. Measurements\nare quite different for Py and Co. Himpsel40finds an\nexchange splitting of ∆ = 270 meV for Py, which is not\ntoo far away from the Weiss J0value. For Co, however,\nthe value is between 0.9 and 1.2 eV, different by a factor\nof four. For Co the splitting needs to be estimated by a\ncombination of photoemission and inverse photoemission\nbecause the splitting straddles EF.46.\nFor comparison with the paramagnetic values of Ji\nex,\nwe use the J0estimates, since they both involve a bal-\nance between Zeeman energy (here in the Weiss field)\nand Heisenberg interatomic exchange. 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Lett. 64, 1059\n(1990)."    },    {        "title": "1610.06661v1.Spin_transport_and_dynamics_in_all_oxide_perovskite_La___2_3__Sr___1_3__MnO__3__SrRuO__3__bilayers_probed_by_ferromagnetic_resonance.pdf",        "content": "Spin transport and dynamics in all-oxide perovskite La 2=3Sr1=3MnO 3/SrRuO 3bilayers\nprobed by ferromagnetic resonance\nSatoru Emori,1,\u0003Urusa S. Alaan,1, 2Matthew T. Gray,1, 2Volker\nSluka,3Yizhang Chen,3Andrew D. Kent,3and Yuri Suzuki1, 4\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 USA\n2Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 USA\n3Department of Physics, New York University, New York, NY 10003, USA\n4Department of Applied Physics, Stanford University, Stanford, CA 94305 USA\n(Dated: November 11, 2021)\nThin \flms of perovskite oxides o\u000ber the possibility of combining emerging concepts of strongly\ncorrelated electron phenomena and spin current in magnetic devices. However, spin transport and\nmagnetization dynamics in these complex oxide materials are not well understood. Here, we ex-\nperimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite\nferromagnet/paramagnet bilayers of La 2=3Sr1=3MnO 3/SrRuO 3(LSMO/SRO) by broadband ferro-\nmagnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we\nestimate a short spin di\u000busion length of <\u00181 nm in SRO and an interfacial spin-mixing conductance\ncomparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we \fnd that anisotropic\nnon-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our\nresults demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for\nexamining spin-current transport in various perovskite heterostructures.\nI. INTRODUCTION\nManipulation and transmission of information by spin\ncurrent is a promising route toward energy-e\u000ecient mem-\nory and computation devices1. Such spintronic devices\nmay consist of ferromagnets interfaced with nonmagnetic\nconductors that exhibit spin-Hall and related spin-orbit\ne\u000bects2{4. The direct spin-Hall e\u000bect in the conductor\ncan convert a charge current to a spin current, which ex-\nerts torques on the adjacent magnetization and modi\fes\nthe state of the device5,6. Conversely, the inverse spin-\nHall e\u000bect in the conductor can convert a propagating\nspin current in the magnetic medium to an electric signal\nto read spin-based information packets7. For these device\nschemes, it is essential to understand the transmission of\nspin current between the ferromagnet and the conductor,\nwhich is parameterized by the spin-mixing conductance\nand spin di\u000busion length. These spin transport parame-\nters can be estimated by spin pumping at ferromagnetic\nresonance (FMR), in which a spin current is resonantly\ngenerated in the ferromagnet and absorbed in the adja-\ncent conductor8,9. Spin pumping has been demonstrated\nin various combinations of materials, where the magnetic\nlayer may be an alloy (e.g., permalloy) or insulator (e.g.,\nyttrium iron garnet) and the nonmagnetic conductor may\nbe a transition metal, semiconductor, conductive poly-\nmer, or topological insulator10{16.\nTransition metal oxides, particularly those with the\nperovskite structure, o\u000ber the intriguing prospect of\nintegrating a wide variety of strongly correlated elec-\ntron phenomena17,18with spintronic functionalities19,20.\nAmong these complex oxides, La 2=3Sr1=3MnO 3(LSMO)\nand SrRuO 3(SRO) are attractive materials for epitaxial,\nlattice-matched spin-source/spin-sink heterostructures.\nLSMO, a metallic ferromagnet known for its colossalmagnetoresistance and Curie temperature of >300 K,\ncan be an excellent resonantly-excited spin source be-\ncause of its low magnetization damping21{26. SRO, a\nroom-temperature metallic paramagnet with relatively\nhigh conductivity27, exhibits strong spin-orbit coupling28\nthat may be useful for emerging spintronic applications\nthat leverage spin-orbit e\u000bects2{4.\nA few recent studies have reported dc voltages at FMR\nin LSMO/SRO bilayers that are attributed to the in-\nverse spin-Hall e\u000bect in SRO generated by spin pump-\ning24{26. However, it is generally a challenge to separate\nthe inverse spin-Hall signal from the spin recti\fcation sig-\nnal, which is caused by an oscillating magnetoresistance\nmixing with a microwave current in the conductive mag-\nnetic layer29{31. Moreover, while the spin-mixing con-\nductance is typically estimated from the enhancement in\nthe Gilbert damping parameter \u000b, the quanti\fcation of \u000b\nis not necessarily straightforward in epitaxial thin \flms\nthat exhibit pronounced anisotropic non-Gilbert damp-\ning23,32{37. It has also been unclear how the Gilbert and\nnon-Gilbert components of damping in LSMO are each\nmodi\fed by an adjacent SRO layer. These points above\nhighlight the need for an alternative experimental ap-\nproach for characterizing spin transport and magnetiza-\ntion dynamics in LSMO/SRO.\nIn this work, we quantify spin transport parameters\nand magnetization damping in epitaxial LSMO/SRO bi-\nlayers by broadband FMR spectroscopy with out-of-plane\nandin-plane external magnetic \felds. Out-of-plane FMR\nenables straightforward extraction of Gilbert damping as\na function of SRO overlayer thickness, which is repro-\nduced by a simple \\spin circuit\" model based on di\u000busive\nspin transport38,39. We \fnd that the spin-mixing conduc-\ntance at the LSMO/SRO interface is comparable to other\nferromagnet/conductor interfaces and that spin current\nis absorbed within a short length scale of <\u00181 nm in thearXiv:1610.06661v1  [cond-mat.mtrl-sci]  21 Oct 20162\n42 44 46 48 50LSMO(10)\n  /SRO(18)LSMO(10)\n  log(intensity) (a.u.)\n2 (deg.)LSAT(002)  \nLSMO(002)  \nSRO(002)  \nFigure 1. 2 \u0012-!x-ray di\u000braction scans of a single-layer\nLSMO(10 nm) \flm and LSMO(10 nm)/SRO(18 nm) bilayer.\nconductive SRO layer. From in-plane FMR, we observe\npronounced non-Gilbert damping that is anisotropic and\nscales nonlinearly with excitation frequency, which is ac-\ncounted for by an existing model of two-magnon scat-\ntering40. This two-magnon scattering is also enhanced\nwith the addition of the SRO overlayer possibly due to\nspin pumping. Our \fndings reveal key features of spin\ndynamics and transport in the prototypical perovskite\nferromagnet/conductor bilayer of LSMO/SRO and pro-\nvide a foundation for future all-oxide spintronic devices.\nII. SAMPLE AND EXPERIMENTAL DETAILS\nEpitaxial \flms of LSMO(/SRO) were grown on\nas-received (001)-oriented single-crystal (LaAlO 3)0:3\n(Sr2AlTaO 6)0:7(LSAT) substrates using pulsed laser de-\nposition. LSAT exhibits a lower dielectric constant than\nthe commonly used SrTiO 3substrate and is therefore\nbetter suited for high-frequency FMR measurements.\nThe lattice parameter of LSAT (3.87 \u0017A) is also closely\nmatched to the pseudocubic lattice parameter of LSMO\n(\u00193.88 \u0017A). By using deposition parameters similar to\nthose in previous studies from our group41,42, all \flms\nwere deposited at a substrate temperature of 750\u000eC with\na target-to-substrate separation of 75 mm, laser \ruence\nof\u00191 J/cm2, and repetition rate of 1 Hz. LSMO was de-\nposited in 320 mTorr O 2, followed by SRO in 100 mTorr\nO2. After deposition, the samples were held at 600\u000eC\nfor 15 minutes in \u0019150 Torr O 2and then the substrate\nheater was switched o\u000b to cool to room temperature. The\ndeposition rates were calibrated by x-ray re\rectivity mea-\nsurements. The thickness of LSMO, tLSMO , in this study\nis \fxed at 10 nm, which is close to the minimum thickness\nat which the near-bulk saturation magnetization can be\nattained.\nX-ray di\u000braction results indicate that both the LSMO\n\flms and LSMO/SRO bilayers are highly crystalline\nand epitaxial with the LSAT(001) substrate, with high-\nresolution 2 \u0012-!scans showing distinct Laue fringes\naround the (002) Bragg re\rection (Fig. 1). In this study,\nthe maximum thickness of the LSMO and SRO layers\n770 780 790 800 810 main mode\n sec. mode\n  dIFMR/dH (a.u.)\n0H (mT) data\n fit(a) (b) \n770 780 790 800 810 data\n fit\n  dIFMR/dH (a.u.)\n0H (mT)Figure 2. Exemplary FMR spectra and \ftting curves: (a)\none mode of Lorentzian derivative; (b) superposition of a main\nmode and a small secondary mode due to slight sample inho-\nmogeneity.\ncombined is less than 30 nm and below the threshold\nthickness for the onset of structural relaxation by mis\ft\ndislocation formation41,42. The typical surface roughness\nof LSMO and SRO measured by atomic force microscopy\nis<\u00184\u0017A, comparable to the roughness of the LSAT sub-\nstrate surface.\nSQUID magnetometry con\frms that the Curie tem-\nperature of the LSMO layer is \u0019350 K and the room-\ntemperature saturation magnetization is Ms\u0019300 kA/m\nfor 10-nm thick LSMO \flms. The small LSMO thickness\nis desirable for maximizing the spin-pumping-induced en-\nhancement in damping, since spin pumping scales in-\nversely with the ferromagnetic layer thickness8,9. More-\nover, the thickness of 10 nm is within a factor of \u00192\nof the characteristic exchange lengthp\n2Aex=\u00160M2s\u00195\nnm, assuming an exchange constant of Aex\u00192 pJ/m in\nLSMO (Ref. 43), so standing spin-wave modes are not\nexpected.\nBroadband FMR measurements were performed at\nroom temperature. The \flm sample was placed face-\ndown on a coplanar waveguide with a center conductor\nwidth of 250 \u0016m. Each FMR spectrum was acquired at a\nconstant excitation frequency while sweeping the exter-\nnal magnetic \feld H. The \feld derivative of the FMR\nabsorption intensity (e.g., Fig. 2) was acquired using an\nrf diode combined with an ac (700 Hz) modulation \feld.\nEach FMR spectrum was \ftted with the derivative of the\nsum of the symmetric and antisymmetric Lorentzians, as\nshown in Fig. 2, from which the resonance \feld HFMR\nand half-width-at-half-maximum linewidth \u0001 Hwere ex-\ntracted. In some spectra (e.g., Fig. 2(b)), a small sec-\nondary mode in addition to the main FMR mode was\nobserved. We \ft such a spectrum to a superposition of\ntwo modes, each represented by a generalized Lorentzian\nderivative, and analyze only the HFMR and \u0001Hof the\nlarger-amplitude main FMR mode. The secondary mode\nis not a standing spin-wave mode because it appears\nabove or below the resonance \feld of the main mode\nHFMR with no systematic trend in \feld spacing. We\nattribute the secondary mode to regions in the \flm with3\n0.360.400.440.48\n  0Meff (T)\n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gop\n0 5 10 15 200.00.20.40.60.81.01.2\nLSMO/SRO\nLSMO\n  0HFMR (T)\nf (GHz)(a) (b) \n(c) \nFigure 3. (a) Out-of-plane resonance \feld\nHFMR versus excitation frequency ffor\na single-layer LSMO(10 nm) \flm and a\nLSMO(10 nm)/SRO(3 nm) bilayer. The\nsolid lines indicate \fts to the data using\nEq. 1. (b,c) SRO-thickness dependence of\nthe out-of-plane Land\u0013 e g-factor (b) and ef-\nfective saturation magnetization Me\u000b(c).\nThe dashed lines indicate the values aver-\naged over all the data shown.\nslightly di\u000berent Msor magnetic anisotropy. More pro-\nnounced inhomogeneity-induced secondary FMR modes\nhave been observed in epitaxial magnetic \flms in prior\nreports22,44.\nIII. OUT-OF-PLANE FMR AND ESTIMATION\nOF SPIN TRANSPORT PARAMETERS\nOut-of-plane FMR allows for conceptually simpler ex-\ntraction of the static and dynamic magnetic properties\nof a thin-\flm sample. For \ftting the frequency depen-\ndence ofHFMR, the Land\u0013 e g-factor gopand e\u000bective sat-\nuration magnetization Me\u000bare the only adjustable pa-\nrameters in the out-of-plane Kittel equation. The fre-\nquency dependence of \u0001 Hfor out-of-plane FMR arises\nsolely from Gilbert damping, so that the conventional\nmodel of spin pumping8,9,38,39can be used to analyze the\ndata without complications from non-Gilbert damping.\nThis consideration is particularly important because the\nlinewidths of our LSMO(/SRO) \flms in in-plane FMR\nmeasurements are dominated by highly anisotropic non-\nGilbert damping (as shown in Sec. IV). Furthermore, a\nsimple one-dimensional, time-independent model of spin\npumping outlined by Boone et al.38is applicable in the\nout-of-plane con\fguration, since the precessional orbit of\nthe magnetization is circular to a good approximation.\nThis is in contrast with the in-plane con\fguration with\na highly elliptical orbit from a large shape anisotropy\n\feld. By taking advantage of the simplicity in out-of-\nplane FMR, we \fnd that the Gilbert damping parame-\nter in LSMO is approximately doubled with the addition\nof a su\u000eciently thick SRO overlayer due to spin pump-\ning. Our results indicate that spin-current transmission\nat the LSMO/SRO interface is comparable to previously\nreported ferromagnet/conductor bilayers and that spin\ndi\u000busion length in SRO is <\u00181 nm.\nWe \frst quantify the static magnetic properties of\nLSMO(/SRO) from the frequency dependence of HFMR.\nThe Kittel equation for FMR in the out-of-plane con\fg-\nuration takes a simple linear form,\nf=gop\u0016B\nh\u00160(HFMR\u0000Me\u000b); (1)\nwhere\u00160is the permeability of free space, \u0016Bis the Bohrmagneton, and his the Planck constant. As shown in\nFig. 3(a), we only \ft data points where \u00160HFMR is at\nleast 0.2 T above \u00160Me\u000bto ensure that the \flm is sat-\nurated out-of-plane. Figures 3(b) and (c) plot the ex-\ntractedMe\u000bandgop, respectively, each exhibiting no\nsigni\fcant dependence on SRO thickness tSROto within\nexperimental uncertainty. The SRO overlayer therefore\nevidently does not modify the bulk magnetic proper-\nties of LSMO, and signi\fcant interdi\u000busion across the\nSRO/LSMO interface can be ruled out. The averaged\nMe\u000bof 330\u000610 kA/m (\u00160Me\u000b= 0:42\u00060:01 T) is close\ntoMsobtained from static magnetometery and implies\nnegligible out-of-plane magnetic anisotropy; we thus as-\nsumeMs=Me\u000bin all subsequent analyses. The SRO-\nthickness independence of gop, averaging to 2 :01\u00060:01,\nimplies that the SRO overlayer does not generate a signif-\nicant orbital contribution to magnetism in LSMO. More-\nover, the absence of detectable change in gopwith in-\ncreasingtSRO may indicate that the imaginary compo-\nnent of the spin-mixing conductance8,9is negligible at\nthe LSMO/SRO interface.\nThe Gilbert damping parameter \u000bis extracted from\nthe frequency dependence of \u0001 H(e.g., Figure 4(a)) by\n\ftting the data with the standard linear relation,\n\u0001H= \u0001H0+h\ngop\u0016B\u000bf: (2)\nThe zero-frequency linewidth \u0001 H0is typically attributed\nto sample inhomogeneity. We observe sample-to-sample\nvariation of \u00160\u0001H0in the range\u00191\u00004 mT with no\nsystematic correlation with tSRO or the slope in Eq. 2.\nMoreover, similar to the analysis of HFMR, we only \ft\ndata obtained at \u00150.2 T above \u00160Me\u000bto minimize spu-\nrious broadening of \u0001 Hat low \felds. The linear slope of\n\u0001Hplotted against frequency up to 20 GHz is therefore\na reliable measure of \u000bdecoupled from \u0001 H0in Eq. 2.\nFigure 4(a) shows an LSMO single-layer \flm and an\nLSMO/SRO bilayer with similar \u0001 H0. The slope, which\nis proportional to \u000b, is approximately a factor of 2 greater\nfor LSMO/SRO. Figure 4(b) summarizes the dependence\nof\u000bon SRO-thickness, tSRO. For LSMO single-layer\n\flms we \fnd \u000b= (0:9\u00060:2)\u000210\u00003, which is on the same\norder as previous reports of LSMO thin \flms21{23,26.\nThis low damping is also comparable to the values re-4\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n1/Gext 1/G↑↓m LSMO SRO\n0 5 10 15 201.01.52.02.53.0\nLSMO\n  0H (mT)\nf (GHz)LSMO/SRO(a) (b) (c) \nFigure 4. (a) Out-of-plane FMR linewidth \u0001 Hversus excitation frequency for LSMO(10 nm) and LSMO(10 nm)/SRO(3 nm).\nThe solid lines indicate \fts to the data using Eq. 2. (b) Gilbert damping parameter \u000bversus SRO thickness tSRO. The solid\ncurve shows a \ft to the di\u000busive spin pumping model (Eq. 5). (c) Schematic of out-of-plane spin pumping and the equivalent\n\\spin circuit.\"\nported in Heusler alloy thin \flms45,46and may arise from\nthe half-metal-like band structure of LSMO (Ref. 47).\nLSMO can thus be an e\u000ecient source of spin current\ngenerated resonantly by microwave excitation.\nWith a few-nanometer thick overlayer of SRO, \u000bin-\ncreases to\u00192\u000210\u00003(Fig. 4(b)). This enhanced damping\nwith the addition of SRO overlayer may arise from (1)\nspin scattering48,49at the LSMO/SRO interface or (2)\nspin pumping8,9where nonequilibrium spins from LSMO\nare absorbed in the bulk of the SRO layer. Here, we\nassume that interfacial spin scattering is negligible, since\n<\u00181 nm of SRO overlayer does not enhance \u000bsigni\fcantly\n(Fig. 4(b)). This is in contrast with the pronounced in-\nterfacial e\u000bect in ferromagnet/Pt bilayers48,49, in which\neven<1 nm of Pt can increase \u000bby as much as a fac-\ntor of\u00192 (Refs. 50{52). In the following analysis and\ndiscussion, we show that spin pumping alone is su\u000ecient\nfor explaining the enhanced damping in LSMO with an\nSRO overlayer.\nWe now analyze the data in Fig. 4(b) using a one-\ndimensional model of spin pumping based on di\u000busive\nspin transport38,39. The resonantly-excited magnetiza-\ntion precession in LSMO generates non-equilibrium spins\npolarized along ^ m\u0002d ^m=dt, which is transverse to the\nmagnetization unit vector ^ m. This non-equilibrium spin\naccumulation di\u000buses out to the adjacent SRO layer\nand depolarizes exponentially on the characteristic length\nscale\u0015s. The spin current density ~jsat the LSMO/SRO\ninterface can be written as38,53\n~jsjinterface =~2\n2e2^m\u0002d^m\ndt\u0010\n1\nG\"#+1\nGext\u0011; (3)\nwhere ~is the reduced Planck constant, G\"#is the inter-\nfacial spin-mixing conductance per unit area, and Gextis\nthe spin conductance per unit area in the bulk of SRO.\nIn Eq. 3, 1/ G\"#and 1/Gextconstitute spin resistors in\nseries such that the spin transport from LSMO to SRO\ncan be regarded analogously as a \\spin circuit,\" as il-\nlustrated in Fig. 4(c). In literature, these interfacialand bulk spin conductances are sometimes lumped to-\ngether as an \\e\u000bective spin-mixing conductance\" Ge\u000b\n\"#=\n(1=G\"#+ 1=Gext)\u00001(Refs. 10{13, 16, 20, 23, 26, 44). We\nalso note that the alternative form of the (e\u000bective) spin-\nmixing conductance g(e\u000b)\ne\u000b, with units of m\u00002, is related to\nG(e\u000b)\n\"#, with units of \n\u00001m\u00002, byg(e\u000b)\ne\u000b= (h=e2)G(e\u000b)\n\"#\u0019\n26 k\n\u0002G(e\u000b)\n\"#.\nThe functional form of Gextis obtained by solving the\nspin di\u000busion equation with appropriate boundary condi-\ntions38,39,53. In the case of a ferromagnet/nonmagnetic-\nmetal bilayer, we obtain\nGext=1\n2\u001aSRO\u0015stanh\u0012tSRO\n\u0015s\u0013\n; (4)\nwhere\u001aSROis the resistivity of SRO, tSROis the thick-\nness of the SRO layer, and \u0015sis the di\u000busion length of\npumped spins in SRO. Finally, the out\row of spin cur-\nrent (Eq. 3) is equivalent to an enhancement of Gilbert\ndamping9with respect to \u000b0of LSMO with tSRO = 0\nsuch that\n\u000b=\u000b0+gop\u0016B~\n2e2MstLSMO\u00141\nG\"#+ 2\u001aSRO\u0015scoth\u0012tSRO\n\u0015s\u0013\u0015\u00001\n:\n(5)\nThus, two essential parameters governing spin transport\nG\"#and\u0015scan be estimated by \ftting the SRO-thickness\ndependence of \u000b(Fig. 4(b)) with Eq. 5.\nIn carrying out the \ft, we \fx \u000b0= 0:9\u000210\u00003. We note\nthat\u001aSROincreases by an order of magnitude compared\nto the bulk value of \u00192\u000210\u00006\nm astSROis reduced to\na few nm; also, at thicknesses of 3 monolayers ( \u00191.2 nm)\nor below, SRO is known to be insulating54. We there-\nfore use the tSRO-dependent \u001aSRO shown in Appendix\nA while assuming \u0015sis constant. An alternative \ftting\nmodel that assumes a constant \u001aSRO, which is a common\napproach in literature, is discussed in Appendix A.\nThe curve in Fig. 4(b) is generated by Eq. 5 with G\"#=\n1:6\u00021014\n\u00001m\u00002and\u0015s= 0:5 nm. Given the scatter of5\n170175180185\n170\n175\n180\n185[010]\n[110]\n[100] \n  \n0HFMR (mT)\nLSMO\nLSMO/SRO\n0 5 10 15 200100200300400500\n  0HFMR (mT)\nf (GHz)H||[100]\nH||[110](a) (b) (c) \n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gip\n-6-4-20\n  0H||,4 (mT)\n(d) \n14 15330360 \n \n  \nFigure 5. (a) Angular dependence of HFMR at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to the data using Eq. 6. (b) Frequency dependence of HFMR for LSMO(10 nm)/SRO(7 nm) with \feld applied in the\n\flm plane along the [100] and [110] directions. Inset: close-up of HFMR versus frequency around 14-15 GHz. In (a) and (b), the\nsolid curves show \fts to the Kittel equation (Eq. 6). (c,d) SRO-thickness dependence of the in-plane cubic magnetocrystalline\nanisotropy \feld (c) and in-plane Land\u0013 e g-factor (d). The dashed lines indicate the values averaged over all the data shown.\nthe experimental data, acceptable \fts are obtained with\nG\"#\u0019(1:2\u00002:5)\u00021014\n\u00001m\u00002and\u0015s\u00190:3\u00000:9\nnm. The estimated ranges of G\"#and\u0015salso depend\nstrongly on the assumptions behind the \ftting model.\nFor example, as shown in Appendix A, the constant- \u001aSRO\nmodel yields G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nNevertheless, we \fnd that the estimated G\"#\nis on the same order of magnitude as those\nof various ferromagnet/transition-metal heterostruc-\ntures39,55,56, signifying that the LSMO/SRO interface\nis reasonably transparent to spin current. More impor-\ntantly, the short \u0015simplies the presence of strong spin-\norbit coupling that causes rapid spin scattering within\nSRO. This \fnding is consistent with a previous study on\nSRO at low temperature in the ferromagnetic state show-\ning extremely fast spin relaxation with Gilbert damping\n\u000b\u00181 (Ref. 28). The short \u0015sindicates that SRO may be\nsuitable as a spin sink or detector in all-oxide spintronic\ndevices.\nIV. IN-PLANE FMR AND ANISOTROPIC\nTWO-MAGNON SCATTERING\nIn epitaxial thin \flms, the analysis of in-plane FMR\nis generally more complicated than that of out-of-plane\nFMR. High crystallinity of the \flm gives rise to a non-\nnegligible in-plane magnetocrystalline anisotropy \feld,\nwhich manifests in an in-plane angular dependence of\nHFMR and introduces another adjustable parameter in\nthe nonlinear Kittel equation for in-plane FMR. More-\nover, \u0001Hin in-plane FMR of epitaxial thin \flms often\ndepends strongly on the magnetization orientation and\nexhibits nonlinear scaling with respect to frequency due\nto two-magnon scattering, a non-Gilbert mechanism for\ndamping23,32{37. We indeed \fnd that damping of LSMO\nin the in-plane con\fguration is anisotropic and domi-\nnated by two-magnon scattering. We also observe ev-idence of enhanced two-magnon scattering with added\nSRO layers, which may be due to spin pumping from\nnonuniform magnetization precession.\nFigure 5(a) plots HFMR of a single-layer LSMO \flm\nand an LSMO/SRO bilayer as a function of applied \feld\nangle within the \flm plane. For both samples, we observe\nclear four-fold symmetry, which is as expected based on\nthe epitaxial growth of LSMO on the cubic LSAT(001)\nsubstrate. Similar to previous FMR studies of LSMO on\nSrTiO 3(001)57,58, the magnetic hard axes (corresponding\nto the axes of higher HFMR) are alongh100i. The in-\nplane Kittel equation for thin \flms with in-plane cubic\nmagnetic anisotropy is59,\nf=gip\u0016B\nh\u00160\u0002\nHFMR +Hjj;4cos(4\u001e)\u00031\n2\u0002\n\u0014\nHFMR +Me\u000b+1\n4Hjj;4(3 + cos(4\u001e))\u00151\n2\n;\n(6)\nwheregipis the Land\u0013 e g-factor that is obtained from in-\nplane FMR data, Hjj;4is the e\u000bective cubic anisotropy\n\feld, and\u001eis the in-plane \feld angle with respect to\nthe [100] direction. Given that LSMO is magnetically\nvery soft (coercivity on the order of 0.1 mT) at room\ntemperature, we assume that the magnetization is par-\nallel to the \feld direction, particularly with \u00160H\u001d10\nmT. In \ftting the angular dependence (e.g., Fig. 5(a))\nand frequency dependence (e.g., Fig. 5(b)) of HFMR to\nEq. 6, we \fx Me\u000bat the values obtained from out-of-\nplane FMR (Fig. 3(b)) so that Hjj;4andgipare the\nonly \ftting parameters. For the two samples shown in\nFig. 5(a), the \fts to the angular dependence and fre-\nquency dependence data yield consistent values of Hjj;4\nandgip. For the rest of the LSMO(/SRO) samples, we\nuse the frequency dependence data with Hjj[100] and\nHjj[110] to extract these parameters. Figures 5(c) and\n(d) show that Hjj;4andgip, respectively, exhibit no sys-\ntematic dependence on tSRO, similar to the \fndings from6\nout-of-plane FMR (Figs. 3(b),(c)). The in-plane cubic\nmagnetocrystalline anisotropy in LSMO(/SRO) is rela-\ntively small, with \u00160Hjj;4averaging to\u00192.5 mT.gipav-\nerages out to 1 :99\u00060:02, which is consistent with gop\nfound from out-of-plane FMR.\nWhile the magnetocrytalline anisotropy in\nLSMO(/SRO) is found to be modest and indepen-\ndent oftSRO, we observe much more pronounced\nin-plane anisotropy and tSRO dependence in linewidth\n\u0001H, as shown in Figs. 6(a) and (b). Figure 6(a)\nindicates that the in-plane dependence of \u0001 His four-\nfold symmetric for both LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm). \u0001 His approximately a factor of 2\nlarger when the sample is magnetized along h100icom-\npared to when it is magnetized along h110i. One might\nattribute this pronounced anisotropy to anisotropic\nGilbert damping60, such that the sample magnetized\nalong the hard axes h100imay lead to stronger damp-\ning. However, we \fnd no general correlation between\nmagnetocrystalline anisotropy and anisotropic \u0001 H: As\nwe show in Appendix B, LSMO grown on NdGaO 3(110)\nwith pronounced uniaxial magnetocrystalline anisotropy\nexhibits identical \u0001 Hwhen magnetized along the easy\nand hard axes. Moreover, whereas Gilbert damping\nshould lead to a linear frequency dependence of \u0001 H,\nfor LSMO(/SRO) the observed frequency dependence\nof \u0001His clearly nonlinear as evidenced in Fig. 6(b).\nThe pronounced anisotropy and nonlinear frequency\ndependence of \u0001 Htogether suggest the presence of a\ndi\u000berent damping mechanism.\nA well-known non-Gilbert damping mechanism in\nhighly crystalline ultrathin magnetic \flms is two-magnon\nscattering23,32{37,40,61,62, in which uniformly precessing\nmagnetic moments (a spin wave, or magnon mode, with\nwavevector k= 0) dephase to a k6= 0 magnon mode with\nadjacent moments precessing with a \fnite phase di\u000ber-\nence. By considering both exchange coupling (which re-\nsults in magnon energy proportional to k2) and dipolar\ncoupling (magnon energy proportional to \u0000jkj) among\nprecessing magnetic moments, the k= 0 andk6= 0\nmodes become degenerate in the magnon dispersion re-\nlation61as illustrated in Fig. 6(c).\nThe transition from k= 0 tok6= 0 is activated by\ndefects that break the translational symmetry of the\nmagnetic system by localized dipolar \felds40,61,62. In\nLSMO(/SRO), the activating defects may be faceted such\nthat two-magnon scattering is more pronounced when\nthe magnetization is oriented along h100i. One possibil-\nity is that LSMO thin \flms naturally form pits or islands\nfaceted alongh100iduring growth. However, we are un-\nable to consistently observe signs of such faceted defects\nin LSMO(/SRO) samples with an atomic force micro-\nscope (AFM). It is possible that these crystalline defects\nare smaller than the lateral resolution of our AFM setup\n(<\u001810 nm) or that these defects are not manifested in sur-\nface topography. Such defects may be point defects or\nnanoscale clusters of distinct phases that are known to\nexist intrinsically even in high-quality crystals of LSMO(Ref. 63).\nAlthough the de\fnitive identi\fcation of defects that\ndrive two-magnon scattering would require further in-\nvestigation, we can rule out (1) atomic step terraces\nand (2) mis\ft dislocations as sources of anisotropic two-\nmagnon scattering. (1) AFM shows that the orienta-\ntion and density of atomic step terraces di\u000ber randomly\nfrom sample to sample, whereas the anisotropy in \u0001 H\nis consistently cubic with larger \u0001 HforHjjh100ithan\nHjjh110i. This is in agreement with the recent study\nby Lee et al. , which shows anisotropic two-magnon scat-\ntering in LSMO to be independent of regularly-spaced\nparallel step terraces on a bu\u000bered-oxide etched SrTiO 3\nsubstrate23. (2) Although Woltersdorf and Heinrich have\nfound that mis\ft dislocations in Fe/Pd grown on GaAs\nare responsible for two-magnon scattering33, such dis-\nlocations are expected to be virtually nonexistent in\nfully strained LSMO(/SRO) \flms on the closely-latticed\nmatched LSAT substrates41,42.\nWe assume that the in-plane four-fold anisotropy and\nnonlinear frequency dependence of \u0001 Hare entirely due\nto two-magnon scattering. For a sample magnetized\nalong a given in-plane crystallographic axis hhk0i=h100i\norh110i, the two-magnon scattering contribution to \u0001 H\nis given by40\n\u0001Hhhk0i\n2m = \u0000hhk0i\n2m sin\u00001sp\nf2+ (fM=2)2\u0000fM=2p\nf2+ (fM=2)2+fM=2;(7)\nwherefM= (gip\u0016B=h)\u00160Msand \u0000hhk0i\n2m is the two-\nmagnon scattering parameter. The angular dependence\nof \u0001His \ftted with33\n\u0001H= \u0001H0+h\ngip\u0016B\u000bf\n+ \u0001Hh100i\n2m cos2(2\u001e) + \u0001Hh110i\n2m cos2(2[\u001e\u0000\u0019\n4]):(8)\nSimilarly, the frequency dependence of \u0001 Hwith the sam-\nple magnetized along [100] or [110], i.e., \u001e= 0 or\u0019=4,\nis well described by Eqs. 7 and 8. In principle, it should\nbe possible to \ft the linewidth data with \u0001 H0,\u000b, and\n\u00002mas adjustable parameters. In practice, the \ft car-\nried out this way is overspeci\fed such that wide ranges\nof these parameters appear to \ft the data. We there-\nfore impose a constraint on \u000bby assuming that Gilbert\ndamping for LSMO(/SRO) is isotropic: For each SRO\nthicknesstSRO,\u000bis \fxed to the value estimated from\nthe \ft curve in Fig. 4(c) showing out-of-plane FMR data.\n(This assumption is likely justi\fed, since the damping\nfor LSMO(10 nm) on NdGdO 3(110) with strong uniaxial\nmagnetic anisotropy is identical for the easy and hard\ndirections, as shown in Appendix B.) To account for the\nuncertainty in the Gilbert damping in Fig. 4(c), we vary \u000b\nby\u000625% for \ftting the frequency dependence of in-plane\n\u0001H. Examples of \fts using Eqs. 7 and 8 are shown in\nFig. 6(a),(b).\nFigure 6(d) shows that the SRO overlayer enhances\nthe two-magnon scattering parameter \u0000 2mby up to a7\n0 5 10 15 2004812\nLSMOLSMO/SRO\n  0H (mT)\nf (GHz)(a) (b) \n(c) (d) \n  \nFMR \nfreq. \nk f \nk=0 k≠0 \n0 5 10 15 200102030\n  02m (mT)\ntSRO (nm)H||[100]\nH||[110]\n0510\n0\n5\n10\nLSMO\nLSMO/SRO[110][010]\n[100]\n   0H (mT)\nFigure 6. (a) In-plane angular dependence of\nlinewidth \u0001H at 9 GHz for LSMO(10 nm) and\nLSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to Eq. 8. (b) Frequency depen-\ndence of \u0001H for LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm) with Happlied along the [100]\ndirection. The solid curves indicate \fts to\nEq. 7. The dashed and dotted curves indicate\nestimated two-magnon and Gilbert damping\ncontributions, respectively. (c) Schematic of a\nspin wave dispersion curve (when the magne-\ntization is in-plane and has a \fnite component\nparallel to the spin wave wavevector k) and two-\nmagnon scattering. (d) Two-magnon scattering\ncoe\u000ecient \u0000 2m, estimated for the cases with H\napplied along the [100] and [110] axes, plotted\nagainst SRO thickness tSRO. The dashed curve\nis the same as that in Fig. 4(c) scaled to serve\nas a guide for the eye for \u0000 2mwith H along\n[100].\nfactor of\u00192 forHjj[100]. By contrast, for Hjj[110], al-\nthough LSMO/SRO exhibits enhanced \u0001 Hcompared to\nLSMO, the enhancement in \u0000 2mis obscured by the un-\ncertainty in Gilbert damping. In Table I, we summa-\nrize the Gilbert and two-magnon contributions to \u0001 H\nfor LSMO single layers and LSMO/SRO (averaged val-\nues for samples with tSRO>4 nm) with Hjj[100] and\nHjj[110]. Comparing the e\u000bective spin relaxation rates,\n(gip\u0016B=h)\u00160Ms\u000band (gip\u0016B=h)\u00160\u00002m, reveals that two-\nmagnon scattering dominates over Gilbert damping.\nWe now speculate on the mechanisms behind the\nenhancement in \u0000 2min LSMO/SRO, particularly for\nHjj[100]. One possibility is that SRO interfaced with\nLSMO directly increases the rate of two-magnon scat-\ntering, perhaps due to formation of additional defects at\nthe surface of LSMO. If this were the case we might ex-\npect a signi\fcant increase and saturation of \u0000 2mat small\ntSRO. However, in reality, \u0000 2mincreases for tSRO>1 nm\n(Fig. 6(d)), which suggests spin scattering in the bulk\nof SRO. We thus speculate another mechanism, where\nk6= 0 magnons in LSMO are scattered by spin pump-\nTable I. Spin relaxation rates extracted from in-plane FMR\n(106s\u00001)\nLSMO LSMO/SRO*\nGilbert:gip\u0016B\nh\u00160Ms\u000b 11\u00062 23\u00064\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[100]) 290\u000650 550\u0006100\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[110]) 140\u000660 250\u000660\n* Averaged over samples with tSRO>4 nm.ing into SRO. As shown by the guide-for-the-eye curve\nin Fig. 6(d), the tSROdependence of \u0000 2m(forHjj[100])\nmay be qualitatively similar to the tSROdependence of\n\u000bmeasured from out-of-plane FMR (Fig. 4(c)); this cor-\nrespondence would imply that the same spin pumping\nmechanism, which is conventionally modeled to act on\nthek= 0 mode, is also operative in the degenerate k6= 0\nmagnon mode in epitaxial LSMO. Indeed, previous stud-\nies have electrically detected the presence of spin pump-\ning fromk6= 0 magnons by the inverse spin-Hall e\u000bect in\nY3Fe5O12/Pt bilayers64{66. However, we cannot conclu-\nsively attribute the observed FMR linewidth broadening\nin LSMO/SRO to such k6= 0 spin pumping, since it\nis unclear whether faster relaxation of k6= 0 magnons\nshould necessarily cause faster relaxation of the k= 0\nFMR mode. Regardless of its origin, the pronounced\nanisotropic two-magnon scattering introduces additional\ncomplexity to the analysis of damping in LSMO/SRO\nand possibly in other similar ultrathin epitaxial magnetic\nheterostructures.\nV. SUMMARY\nWe have demonstrated all-oxide perovskite bilayers\nof LSMO/SRO that form spin-source/spin-sink systems.\nFrom out-of-plane FMR, we deduce a low Gilbert damp-\ning parameter of \u00191\u000210\u00003for LSMO. The two-fold en-\nhancement in Gilbert damping with an SRO overlayer\nis adequately described by the standard model of spin\npumping based on di\u000busive spin transport. We ar-\nrive at an estimated spin-mixing conductance G\"#\u0019\n(1\u00002)\u00021014\n\u00001m\u00002and spin di\u000busion length \u0015s<\u00181\nnm, which indicate reasonable spin-current transparency\nat the LSMO/SRO interface and strong spin scattering8\nwithin SRO. From in-plane FMR, we reveal pronounced\nnon-Gilbert damping, attributed to two-magnon scatter-\ning, which results in a nonlinear frequency dependence\nand anisotropy in linewidth. The magnitude of two-\nmagnon scattering increases with the addition of an SRO\noverlayer, pointing to the presence of spin pumping from\nnonuniform spin wave modes. Our \fndings lay the foun-\ndation for understanding spin transport and magneti-\nzation dynamics in epitaxial complex oxide heterostruc-\ntures.\nACKNOWLEDGEMENTS\nWe thank Di Yi, Sam Crossley, Adrian Schwartz, Han-\nkyu Lee, and Igor Barsukov for helpful discussions, and\nTianxiang Nan and Nian Sun for the design of the copla-\nnar waveguide. This work was funded by the National\nSecurity Science and Engineering Faculty Fellowship of\nthe Department of Defense under Contract No. N00014-\n15-1-0045.\nAPPENDIX A: SPIN PUMPING AND SRO\nRESISTIVITY\nWhen \ftting the dependence of the Gilbert damping\nparameter\u000bon spin-sink thickness, a constant bulk re-\nsistivity for the spin sink layer is often assumed in lit-\nerature. By setting the resistivity of SRO to the bulk\nvalue\u001aSRO= 2\u000210\u00006\nm and \ftting the \u000b-versus-tSRO\ndata (Fig. 4(c) and reproduced in Fig. 7(a)) to Eq. 5,\nwe arrive at G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nThe \ft curve is insensitive to larger values of G\"#because\nthe bulk spin resistance 1/ Gext, with the relatively large\nresistivity of SRO, dominates over the interfacial spin re-\nsistance 1/G\"#(see Eqs. 4 and 5). As shown by the dot-\nted curve in Fig. 7, this simple constant- \u001aSROmodel ap-\npears to mostly capture the tSRO-dependence of \u000b. This\nmodel of course indicates \fnite spin pumping at even\nvery small SRO thickness <\u00181 nm, which is likely non-\nphysical since SRO should be insulating in this thickness\nregime54. Indeed,\u0015sestimated with this model should\nprobably be considered a phenomenological parameter:\nAs pointed out by recent studies, strictly speaking, a\nphysically meaningful estimation of \u0015sshould take into\naccount the thickness dependence of the resistivity of the\nspin sink layer39,56,67, especially for SRO whose thickness\ndependence of resistivity is quite pronounced.\nFigure 7(b) plots the SRO-thickness dependence of the\nresistivity of SRO \flms deposited on LSAT(001) mea-\nsured in the four-point van der Pauw geometry. The\ntrend can be described empirically by\n\u001aSRO=\u001ab+\u001as\ntSRO\u0000tth; (9)\nwhere\u001ab= 2\u000210\u00006\nm is the resistivity of SRO in the\nbulk limit,\u001as= 1:4\u000210\u000014\nm2is the surface resistivity\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n0 10 20 3010-61x10-51x10-4\n  SRO (m)\ntSRO (nm)(a) (b) Figure 7. (a) Gilbert damping parameter \u000bversus SRO\nthicknesstSRO. The solid curve is a \ft taking into account\nthetSROdependence of SRO resistivity, whereas the dotted\ncurve is a \ft assuming a constant bulk-like SRO resistivity.\n(b) Resistivity of SrRuO 3\flms on LSAT(001) as a function\nof thickness.\n0 5 10 15 20012345\nhard\neasy\n  0H (mT)\nf (GHz)\n0 5 10 15 200123450H (mT)\n  \nf (GHz)hard\neasy(b) (a) \nFigure 8. Frequency dependence of in-plane FMR\nlinewidth \u0001 Hof LSMO(10 nm) on (a) LSAT(001) and (b)\nNdGaO 3(110), with the magnetization along the magnetic\neasy and hard axes. The solid curves are \fts to Eq. 7 with\nthe Gilbert damping parameter \u000b\fxed to 0:9\u000210\u00003.\ncoe\u000ecient, and tth= 1 nm is the threshold thickness\nbelow which the SRO layer is essentially insulating. The\nvalue oftthagrees with literature reporting that SRO\nis insulating at thickness of 3 monolayers ( \u00191.2 nm) or\nbelow54. Given the large deviation of \u001aSROfrom the bulk\nvalue, especially at small tSRO, the trend in Fig. 7(b)\nsuggests that taking into account the tSRO dependence\nof\u001aSROis a sensible approach.\nAPPENDIX B: IN-PLANE DAMPING OF LSMO\nON DIFFERENT SUBSTRATES\nIn Fig. 8, we compare the frequency dependence of \u0001 H\nfor 10-nm thick LSMO \flms deposited on di\u000berent sub-\nstrates: LSAT(001) and NdGaO 3(110). (NdGaO 3is an\northorhombic crystal and has ap\n2-pseudocubic param-\neter of\u00193.86 \u0017A, such that (001)-oriented LSMO grows\non the (110)-oriented surface of NdGaO 3.) As shown\nin Sec. IV, LSMO on LSAT(001) exhibits cubic mag-\nnetic anisotropy within the \flm plane with the h110iand\nh100ias the easy and hard axes, respectively. LSMO\non NdGaO 3(110) exhibits uniaxial magnetic anisotropy\nwithin the \flm plane with [1 \u001610] and [001] of NdGaO 3as9\nthe easy and hard axes, respectively.68Whereas LSMO\non LSAT(001) shows distinct magnitudes of damping\nwhen the \flm is magnetized along the easy and hard\naxes (Figs. 8(a) and 6(a)), in LSMO on NdGaO 3(110)damping is identical for both the easy and hard axes\n(Figs. 8(b)). These results demonstrate that higher\ndamping (wider linewidth) is in general not linked to the\nmagnetic hard axis of LSMO.\n\u0003satorue@stanford.edu\n1R. L. Stamps, S. Breitkreutz, J. \u0017Akerman, A. V. Chu-\nmak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen,\nS. A. Majetich, M. Kl aui, I. L. Prejbeanu, B. Dieny, N. M.\nDempsey, and B. Hillebrands, J. Phys. D. Appl. Phys. 47,\n333001 (2014).\n2A. Ho\u000bmann, IEEE Trans. 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We argue for an expre ssion in which the coupling of mag-\nnetization fluctuations to particle-hole transitions is we ighted by the spin-dependent part of the\ntheory’s exchange-correlation potential, a quantity whic h has large spatial variations on an atomic\nlength scale. Our SDFT result for αGis closely related to the previously proposed spin-torque\ncorrelation-function expression.\nPACS numbers:\nI. INTRODUCTION\nThe Gilbert parameter αGcharacterizes the damping\nof collective magnetization dynamics1. The key role of\nαGin current-driven2and precessional3magnetization\nreversal has renewed interest in the microscopic physics\nof this important material parameter. It is generally\naccepted that in metals the damping of magnetization\ndynamics is dominated3by particle-hole pair excitation\nprocesses. The main ideas which arise in the theory of\nGilbert damping have been in place for some time4,5. It\nhas however been difficult to apply them to real materi-\nals with the precision required for confident predictions\nwhich would allow theory to play a larger role in design-\ning materials with desired damping strengths. Progress\nhas recently been achieved in various directions, both\nthrough studies6of simple models for which the damp-\ning can be evaluated exactly and through analyses7of\ntransition metal ferromagnets that are based on realis-\ntic electronic structure calculations. Evaluation of the\ntorquecorrelationformula5forαGusedinthelatercalcu-\nlations requires knowledge only of a ferromagnet’s mean-\nfield electronic structure and of its Bloch state lifetime,\nwhich makes this approach practical.\nRealistic ab initio theories normally employ spin-\ndensity-functional theory9which has a mean-field theory\nstructure. In this article we use time-dependent spin-\ndensity functional theory to derive an explicit expression\nfor the Gilbert damping coefficient in terms of Kohn-\nSham theory eigenvalues and eigenvectors. Our final\nresult is essentially equivalent to the torque-correlation\nformula5forαG, but has the advantages that its deriva-\ntion is fully consistent with density functional theory,\nthat it allows for a consistent microscopic treatments of\nboth dissipative and reactive coefficients in the Landau-\nLiftshitz Gilbert (LLG) equations, and that it helps\nestablish relationships between different theoretical ap-\nproaches to the microscopic theory of magnetization\ndamping.\nOur paper is organized as follows. In Section II\nwe relate the Gilbert damping parameter αGof a fer-\nromagnet to the low-frequency limit of its transversespin response function. Since ferromagnetism is due\nto electron-electron interactions, theories of magnetism\nare always many-electron theories, and it is necessary to\nevaluate the many-electron response function. In time-\ndependent spin-density functional theory the transverse\nresponse function is calculated using a time-dependent\nself- consistent-field calculation in which quasiparticles\nrespond both to external potentials and to changesin the\ninteraction-induced effective potential. In Section III we\nuse perturbation theory and time-dependent mean-field\ntheory to express the coefficients which appear in the\nLLG equations in terms of the Kohn-Sham eigenstates\nand eigenvaluesof the ferromagnet’sground state. These\nformal expressions are valid for arbitrary spin-orbit cou-\npling, arbitrary atomic length scale spin-dependent and\nscalarpotentials, and arbitrarydisorder. By treating dis-\norder approximately, in Section IV we derive and com-\npare two commonly used formulas for Gilbert damping.\nFinally, in Section V we summarize our results.\nII. MANY-BODY TRANSVERSE RESPONSE\nFUNCTION AND THE GILBERT DAMPING\nPARAMETER\nThe Gilbert damping parameter αGappears in the\nLandau-Liftshitz-Gilbert expression for the collective\nmagnetization dynamics of a ferromagnet:\n∂ˆΩ\n∂t=ˆΩ×Heff−αGˆΩ×∂ˆΩ\n∂t. (1)\nIn Eq.( 1) Heffis an effective magnetic field which\nwe comment on further below and ˆΩ = (Ω x,Ωy,Ωz) is\nthe direction of the magnetization. This equation de-\nscribes the slow dynamics of smooth magnetization tex-\ntures and is formally the first term in an expansion in\ntime-derivatives.\nThe damping parameter αGcan be measured by per-\nforming ferromagnetic resonance (FMR) experiments in\nwhich the magnetization direction is driven weakly away\nfrom an easy direction (which we take to be the ˆ z-\ndirection.). To relate this phenomenological expression2\nformally to microscopic theory we consider a system in\nwhich external magnetic fields couple only11to the elec-\ntronic spin degree of freedom and associate the magneti-\nzation direction ˆΩ with the direction of the total electron\nspin. Forsmalldeviationsfromtheeasydirection,Eq.(1)\nreads\nHeff,x= +∂ˆΩy\n∂t+αG∂ˆΩx\n∂t\nHeff,y=−∂ˆΩx\n∂t+αG∂ˆΩy\n∂t. (2)\nThe gyromagnetic ratio has been absorbed into the unitsof the field Heffso that this quantity has energy units\nand we set /planckover2pi1= 1 throughout. The corresponding formal\nlinear response theory expression is an expansion of the\nlong wavelength transverse total spin response function\nto first order12in frequency ω:\nS0ˆΩα=/summationdisplay\nβ[χst\nα,β+ωχ′\nα,β]Hext,β (3)\nwhereα,β∈ {x,y},ω≡i∂tis the frequency, S0is the to-\ntal spin ofthe ferromagnet, Hextis the external magnetic\nfield and χis the transverse spin-spin response function:\nχα,β(ω) =i/integraldisplay∞\n0dtexp(iωt)/an}bracketle{t[Sα(t),Sβ(t)]/an}bracketri}ht=/summationdisplay\nn/bracketleftbigg/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht\nωn,0−ω−iη+/an}bracketle{tΨ0|Sβ|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sα|Ψ0/an}bracketri}ht\nωn,0+ω+iη/bracketrightbigg\n(4)\nHere|Ψn/an}bracketri}htis an exact eigenstate of the many-body Hamiltonian and ωn,0is the excitation energy for state n. We use\nthis formal expression below to make some general comments abou t the microscopic theory of αG. In Eq.( 3) χst\nα,βis\nthe static ( ω= 0) limit of the response function, and χ′\nα,βis the first derivative with respect to ωevaluated at ω= 0.\nNotice that we have chosen the normalization in which χis the total spin response to a transverse field; χis therefore\nextensive.\nThe keystep in obtainingthe Landau-Liftshitz-Gilbert\nform for the magnetization dynamics is to recognize that\nin the static limit the transverse magnetization responds\nto an external magnetic field by adjusting orientation to\nminimize the total energy including the internal energy\nEintand the energy due to coupling with the external\nmagnetic field,\nEext=−S0ˆΩ·Hext. (5)\nIt follows that\nχst\nα,β=S2\n0/bracketleftBigg\n∂2Eint\n∂ˆΩαˆΩβ/bracketrightBigg−1\n. (6)\nWe obtain a formal equation for Heffcorresponding to\nEq.( 2) by multiplying Eq.( 3) on the left by [ χst\nα,β]−1and\nrecognizing\nHint,α=−1\nS0/summationdisplay\nβ∂2Eint\n∂ˆΩα∂ˆΩβˆΩβ=−1\nS0∂Eint\n∂ˆΩα(7)\nas the internal energy contribution to the effective mag-\nnetic field Heff=Hint+Hext. With this identification\nEq.( 3) can be written in the form\nHeff,α=/summationdisplay\nβLα,β∂tˆΩβ (8)\nwhere\nLα,β=−S0[i(χst)−1χ′(χst)−1]α,β=iS0∂ωχ−1\nα,β.(9)According to the Landau-Liftshitz Gilbert equation then\nLx,y=−Ly,x= 1 and\nLx,x=Ly,y=αG. (10)\nExplicit evaluation of the off-diagonal components of L\nwill in general yield very small deviation from the unit\nresult assumed by the Landau-Liftshitz-Gilbert formula.\nThe deviation reflects mainly the fact that the magneti-\nzation magnitude varies slightly with orientation. We do\nnot comment further on this point because it is of little\nconsequence. Similarly Lx,xis not in general identical\ntoLy,y, although the difference is rarely large or impor-\ntant. Eq.( 10) is the starting point we use later to derive\napproximate expressions for αG.\nIn Eq.( 9) χα,β(ω) is the correlation function for an\ninteracting electron system with arbitrary disorder and\narbitrary spin-orbit coupling. In the absence of spin-\norbit coupling, but still with arbitrary spin-independent\nperiodic and disorder potentials, the ground state of a\nferromagnet is coupled by the total spin-operator only to\nstates in the same total spin multiplet. In this case it\nfollows from Eq.( 4) that\nχst\nα,β= 2/summationdisplay\nnRe/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht]\nωn,0=δα,βS0\nH0\n(11)\nwhereH0is a static external field, which is necessary\nin the absence of spin-orbit coupling to pin the magne-\ntization to the ˆ zdirection and splits the ferromagnet’s3\nground state many-body spin multiplet. Similarly\nχ′\nα,β= 2i/summationdisplay\nnIm[/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht]\nω2\nn,0=iǫα,βS0\nH2\n0.\n(12)\nwhereǫx,x=ǫy,y= 0 and ǫx,y=−ǫy,x= 1, yielding\nLx,y=−Ly,x= 1 and Lx,x=Ly,y= 0. Spin-orbit\ncoupling is required for magnetization damping8.\nIII. SDF-STONER THEORY EXPRESSION FOR\nGILBERT DAMPING\nApproximate formulas for αGin metals are inevitably\nbased on on a self-consistent mean-field theory (Stoner)\ndescriptionofthemagneticstate. Ourgoalistoderivean\napproximate expression for αGwhen the adiabatic local\nspin-densityapproximation9isused forthe exchangecor-\nrelation potential in spin-density-functional theory. The\neffective Hamiltonian which describes the Kohn-Sham\nquasiparticle dynamics therefore has the form\nHKS=HP−∆(n(/vector r),|/vector s(/vector r)|)ˆΩ(/vector r)·/vector s,(13)\nwhereHPis the Kohn-Sham Hamiltonian of a paramag-\nnetic state in which |/vector s(/vector r)|(the local spin density) is set to\nzero,/vector sis the spin-operator, and\n∆(n,s) =−d[nǫxc(n,s)]\nds(14)\nis the magnitude of the spin-dependent part of the\nexchange-correlation potential. In Eq.( 14) ǫxc(n,s) is\nthe exchange-correlation energy per particle in a uni-\nform electron gas with density nand spin-density s.\nWe assume that the ferromagnet is described using\nsome semi-relativistic approximation to the Dirac equa-\ntion like those commonly used13to describe magnetic\nanisotropy or XMCD, even though these approximations\nare not strictly consistent with spin-density-functional\ntheory. Within this framework electrons carry only a\ntwo-componentspin-1/2degreeoffreedomandspin-orbit\ncoupling terms are included in HP. Sincenǫxc(n,s)∼\n[(n/2 +s)4/3+ (n/2−s)4/3], ∆0(n,s)∼n1/3is larger\nclosertoatomic centersand farfrom spatiallyuniform on\natomic length scales. This property figures prominently\nin the considerations explained below.\nIn SDFT the transverse spin-response function is ex-\npressed in terms of Kohn-Sham quasiparticle response to\nboth external and induced magnetic fields:\ns0(/vector r)Ωα(/vector r) =/integraldisplayd/vectorr′\nVχQP\nα,β(/vector r,/vectorr′) [Hext,β(/vectorr′)+∆0(/vectorr′)Ωβ(/vectorr′)].\n(15)\nIn Eq.( 15) Vis the system volume, s0(/vector r) is the magni-\ntude of the ground state spin density, ∆ 0(/vector r) is the mag-\nnitude of the spin-dependent part of the ground stateexchange-correlation potential and\nχQP\nα,β(/vector r,/vectorr′) =/summationdisplay\ni,jfj−fi\nωi,j−ω−iη/an}bracketle{ti|/vector r/an}bracketri}htsα/an}bracketle{t/vector r|j/an}bracketri}ht/an}bracketle{tj|/vectorr′/an}bracketri}htsβ/an}bracketle{t/vectorr′|i/an}bracketri}ht,\n(16)\nwherefiis the ground state Kohn-Sham occupation fac-\ntor for eigenspinor |i/an}bracketri}htandωij≡ǫi−ǫjis a Kohn-\nSham eigenvalue difference. χQP(/vector r,/vectorr′) has been normal-\nized so that it returns the spin-density rather than total\nspin. Like the Landau-Liftshitz-Gilbert equation itself,\nEq.( 15) assumes that only the direction of the mag-\nnetization, and not the magnitudes of the charge and\nspin-densities, varies in the course of smooth collective\nmagnetization dynamics14. This property should hold\naccurately as long as magnetic anisotropies and exter-\nnal fields are weak compared to ∆ 0. We are able to use\nthis property to avoid solving the position-space integral\nequation implied by Eq.( 15). Multiplying by ∆ 0(/vector r) on\nboth sides and integrating over position we find15that\nS0Ωα=/summationdisplay\nβ1\n¯∆0˜χQP\nα,β(ω)/bracketleftbig\nΩβ+Hext,β\n¯∆0/bracketrightbig\n(17)\nwhere we have taken advantage of the fact that in FMR\nexperiments Hext,βandˆΩ are uniform. ¯∆0is a spin-\ndensity weighted average of ∆ 0(/vector r),\n¯∆0=/integraltext\nd/vector r∆0(/vector r)s0(/vector r)/integraltext\nd/vector rs0(/vector r), (18)\nand\n˜χQP\nα,β(ω) =/summationdisplay\nijfj−fi\nωij−ω−iη/an}bracketle{tj|sα∆0(/vector r)|i/an}bracketri}ht/an}bracketle{ti|sβ∆0(/vector r)|j/an}bracketri}ht\n(19)\nis the response function of the transverse-part of the\nquasiparticleexchange-correlationeffective field response\nfunction, notthe transverse-part of the quasiparticle\nspin response function. In Eq.( 19), /an}bracketle{ti|O(/vector r)|j/an}bracketri}ht=/integraltext\nd/vector rO(/vector r)/an}bracketle{ti|/vector r/an}bracketri}ht/an}bracketle{t/vector r|j/an}bracketri}htdenotes a single-particle matrix ele-\nment. Solving Eq.( 17) for the many-particle transverse\nsusceptibility (the ratio of S0ˆΩαtoHext,β) and inserting\nthe result in Eq.( 9) yields\nLα,β=iS0∂ωχ−1\nα,β=−S0¯∆2\n0∂ωIm[˜χQP−1\nα,β].(20)\nOur derivation of the LLG equation has the advantage\nthat the equation’s reactive and dissipative components\nare considered simultaneously. Comparing Eq.( 15) and\nEq.( 7) we find that the internal anisotropy field can also\nbe expressed in terms of ˜ χQP:\nHint,α=−¯∆2\n0S0/summationdisplay\nβ/bracketleftbig\n˜χQP−1\nα,β(ω= 0)−δα,β\nS0¯∆0/bracketrightbig\nΩβ.(21)\nEq.( 20) and Eq.( 21) provide microscopic expressions\nfor all ingredients that appear in the LLG equations4\nlinearized for small transverse excursions. It is gener-\nally assumed that the damping coefficient αGis inde-\npendent of orientation; if so, the present derivation is\nsufficient. The anisotropy-field at large transverse ex-\ncursions normally requires additional information about\nmagnetic anisotropy. We remark that if the Hamiltonian\ndoes not include a spin-dependent mean-field dipole in-\nteraction term, as is usually the case, the above quantity\nwill return only the magnetocrystalline anisotropy field.\nSince the magnetostatic contribution to anisotropy is al-\nways well described by mean-field-theory it can be added\nseparately.\nWe conclude this section by demonstrating that the\nStoner theory equations proposed here recover the exact\nresults mentioned at the end of the previous section for\nthe limit in which spin-orbit coupling is neglected. We\nconsider a SDF theory ferromagnet with arbitrary scalar\nand spin-dependent effective potentials. Since the spin-\ndependent part of the exchange correlation potential is\nthen the only spin-dependent term in the Hamiltonian it\nfollows that\n[HKS,sα] =−iǫα,β∆0(/vector r)sβ (22)\nand hence that\n/an}bracketle{ti|sα∆0(/vector r)|j/an}bracketri}ht=−iǫα,βωij/an}bracketle{ti|sβ|j/an}bracketri}ht.(23)\nInserting Eq.( 23) in one of the matrix elements of\nEq.( 19) yields for the no-spin-orbit-scattering case\n˜χQP\nα,β(ω= 0) =δα,βS0¯∆0. (24)The internal magnetic field Hint,αis therefore identically\nzero in the absence of spin-orbit coupling and only exter-\nnal magnetic fields will yield a finite collective precession\nfrequency. Inserting Eq.( 23) in both matrix elements of\nEq.( 19) yields\n∂ωIm[˜χQP\nα,β] =ǫα,βS0. (25)\nUsing both Eq.( 24) and Eq.( 25) to invert ˜ χQPwe re-\ncover the results proved previously for the no-spin-orbit\ncase using a many-body argument: Lx,y=−Ly,x= 1\nandLx,x=Ly,y= 0. The Stoner-theory equations de-\nrived here allow spin-orbit interactions, and hence mag-\nnetic anisotropy and Gilbert damping, to be calculated\nconsistently from the same quasiparticle response func-\ntion ˜χQP.\nIV. DISCUSSION\nAs long as magnetic anisotropy and external magnetic\nfields are weak compared to the exchange-correlation\nsplitting in the ferromagnet we can use Eq.( 24) to ap-\nproximate ˜ χQP\nα,β(ω= 0). Using this approximation and\nassuming that damping is isotropic we obtain the follow-\ning explicit expression for temperature T→0:\nαG=Lx,x=−S0¯∆2\n0∂ωIm[˜χQP−1\nx,x] =π\nS0/summationdisplay\nijδ(ǫj−ǫF)δ(ǫi−ǫF)/an}bracketle{tj|sx∆0(/vector r)|i/an}bracketri}ht/an}bracketle{ti|sx∆0(/vector r)|j/an}bracketri}ht\n=π\nS0/summationdisplay\nijδ(ǫj−ǫF)δ(ǫi−ǫF)/an}bracketle{tj|[HP,sy]|i/an}bracketri}ht/an}bracketle{ti|[HP,sy]|j/an}bracketri}ht.(26)\nThe second form for αGis equivalent to the first and follows from the observation that for m atrix elements between\nstates that have the same energy\n/an}bracketle{ti|[HKS,sα]|j/an}bracketri}ht=−iǫα,β/an}bracketle{ti|∆0(/vector r)sβ|j/an}bracketri}ht+/an}bracketle{ti|[HP,sα]|j/an}bracketri}ht= 0 (for ωij= 0). (27)\nEq. ( 26) is valid for any scalar and any spin-dependent potential. It is clear however that the numerical value of αG\nin a metal is very sensitive to the degree of disorder in its lattice. To s ee this we observe that for a perfect crystal\nthe Kohn-Sham eigenstates are Bloch states. Since the operator ∆0(/vector r)sαhas the periodicity of the crystal its matrix\nelements are non-zero only between states with the same Bloch wav evector label /vectork. For the case of a perfect crystal\nthen\nαG=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′δ(ǫ/vectorkn′−ǫF)δ(ǫ/vectorkn−ǫF)/an}bracketle{t/vectorkn′|sx∆0(/vector r)|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn′/an}bracketri}ht\n=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′δ(ǫ/vectorkn′−ǫF)δ(ǫ/vectorkn−ǫF)/an}bracketle{t/vectorkn′|[HP,sy]|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|[HP,sy]|/vectorkn′/an}bracketri}ht. (28)\nwherenn′are band labels and s0is the ground state\nspin per unit volume and the integral over /vectorkis over theBrillouin-zone (BZ).5\nClearly αGdiverges16in a perfect crystal since\n/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn/an}bracketri}htis generically non-zero. A theory of\nαGmust therefore always account for disorder in a crys-\ntal. The easiest way to account for disorder is to replace\ntheδ(ǫ/vectorkn−ǫF) spectral function of a Bloch state by a\nbroadened spectral function evaluated at the Fermi en-\nergyA/vectorkn(ǫF). If disorder is treated perturbatively this\nsimpleansatzcan be augmented17by introducing impu-\nrity vertex corrections in Eq. ( 28). Provided that the\nquasiparticlelifetimeiscomputedviaFermi’sgoldenrule,these vertex corrections restore Ward identities and yield\nan exact treatment of disorder in the limit of dilute im-\npurities. Nevertheless, this approach is rarely practical\noutside the realm of toy models, because the sources of\ndisorder are rarely known with sufficient precision.\nAlthough appealing in its simplicity, the δ(ǫ/vectorkn−ǫF)→\nA/vectorkn(ǫF) substitution is prone to ambiguity because it\ngives rise to qualitatively different outcomes depending\non whether it is applied to the first or second line of Eq.\n( 28):\nα(TC)\nG=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′A/vectork,n(ǫF)A/vectork,n′(ǫF)/an}bracketle{t/vectorkn′|[HP,sy]|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|[HP,sy]|/vectorkn′/an}bracketri}ht,\nα(SF)\nG=π\ns0/integraldisplay\nBZd/vectork\n(2π)3/summationdisplay\nnn′A/vectork,n(ǫF)A/vectork,n′(ǫF)/an}bracketle{t/vectorkn′|sx∆0(/vector r)|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn′/an}bracketri}ht.\n(29)\nα(TC)\nGis the torque-correlation (TC) formula used in\nrealistic electronic structure calculations7andα(SF)\nGis\nthe spin-flip (SF) formula used in certain toy model\ncalculations18. The discrepancy between TC and SF ex-\npressions stems from inter-band ( n/ne}ationslash=n′) contributions\nto damping, which may now connect states with dif-\nferentband energies due to the disorder broadening of\nthe spectral functions. Therefore, /an}bracketle{t/vectorkn|[HKS,sα]|/vectorkn′/an}bracketri}htno\nlonger vanishes for n/ne}ationslash=n′and Eq. ( 27) indicates that\nα(TC)\nG≃α(SF)\nGonly if the Gilbert damping is dominated\nby intra-band contributions and/or if the energy differ-\nence between the states connected by inter-band transi-\ntions is small compared to ∆ 0. When α(TC)\nG/ne}ationslash=α(SF)\nG,\nit isa priori unclear which approach is the most accu-\nrate. One obvious flaw of the SF formula is that it pro-\nducesaspuriousdampinginabsenceofspin-orbitinterac-\ntions; this unphysical contribution originates from inter-\nband transitions and may be cancelled out by adding\nthe leading order impurity vertex correction19. In con-\ntrast, [HP,sy] = 0 in absence of spin-orbit interaction\nandhencetheTCformulavanishesidentically, evenwith-\nout vertex corrections. From this analysis, TC appears\nto have a pragmatic edge over SF in materials with weak\nspin-orbitinteraction. However, insofarasit allowsinter-\nband transitions that connect states with ωi,j>∆0,\nTC is not quantitatively reliable. Furthermore, it canbe shown17that when the intrinsic spin-orbit coupling\nis significant (e.g. in ferromagnetic semiconductors), the\nadvantage of TC over SF (or vice versa) is marginal, and\nimpurity vertex corrections play a significant role.\nV. CONCLUSIONS\nUsing spin-density functional theory we have derived\na Stoner model expression for the Gilbert damping co-\nefficient in itinerant ferromagnets. This expression ac-\ncounts for atomic scale variations of the exchange self\nenergy, as well as for arbitrary disorder and spin-orbit\ninteraction. By treating disorder approximately, we have\nderived the spin-flip and torque-correlationformulas pre-\nviously used in toy-model and ab-initio calculations, re-\nspectively. Wehavetracedthediscrepancybetweenthese\nequations to the treatment of inter-band transitions that\nconnect states which are not close in energy. A better\ntreatment of disorder, which requires the inclusion of im-\npurity vertex corrections, will be the ultimate judge on\nthe relativereliabilityofeitherapproach. Whendamping\nis dominated by intra-band transitions, a circumstance\nwhich we believe is common, the two formulas are identi-\ncal and both arelikely to provide reliable estimates. This\nwork was suported by the National Science Foundation\nunder grant DMR-0547875.\n1For a historical perspective see T.L. Gilbert, IEEE Trans.\nMagn.40, 3443 (2004).\n2Foranintroductoryreviewsee D.C. RalphandM.D.Stiles,\nJ. Magn. Mag. Mater. 320, 1190 (2008).3J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag-\nnetic Structures III: Fundamentals of Nanomagnetism\n(Springer-Verlag, New York, 2005).\n4V. Korenman and R. E. Prange, Phys. Rev. B 6, 27696\n(1972).\n5V. Kambersky, Czech J. Phys. B 26, 1366 (1976).\n6Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004); E.M. Hankiewicz, G. Vig-\nnale and Y. Tserkovnyak, Phys. Rev. B 75, 174434 (2007);\nY. Tserkovnyak et al., Phys. Rev. B 74, 144405 (2006) ;\nH.J. Skadsem, Y. Tserkovnyak, A. Brataas, G.E.W. Bauer,\nPhys. Rev. B 75, 094416 (2007); H. Kohno, G. Tatara\nand J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006);\nR.A. Duine et al., Phys. Rev. B 75, 214420 (2007). Y.\nTserkovnyak, A. Brataas, and G.E.W. Bauer, J. Magn.\nMag. Mater. 320, 1282 (2008).\n7K. Gilmore, Y.U.IdzerdaandM.D. Stiles, Phys.Rev.Lett.\n99, 27204 (2007); V. Kambersky, Phys. Rev. B 76, 134416\n(2007).\n8For zero spin-orbit coupling αGvanishes even in presence\nof magnetic impurities, provided that their spins follow th e\ndynamics of the magnetization adiabatically.\n9O. Gunnarsson, J. Phys. F 6, 587 (1976).\n10Z. Qian, G. Vignale, Phys. Rev. Lett. 88, 056404 (2002).\n11In doing so we dodge the subtle difficulties which compli-\ncate theories of orbital magnetism in bulk metals. See for\nexample J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys.\nRev. Lett. 99, 197202 (2007); I. Souza and D. Vanderbilt,\nPhys. Rev. B 77, 054438 (2008) and work cited therein.\nThis simplification should have little influence on the the-\nory of damping because the orbital contribution to the\nmagnetization is relatively small in systems of interest an dbecause it in any event tends to be collinear with the spin\nmagnetization.\n12For most materials the FMR frequency is by far the small-\nest energy scale in the problem. Expansion to linear order\nis almost always appropriate.\n13See for example A.C. Jenkins and W.M. Temmerman,\nPhys. Rev. B 60, 10233 (1999) and work cited therein.\n14This approximation does not preclude strong spatial varia-\ntions of|s0(/vector r)|and|∆0(/vector r)|at atomic lenghtscales; rather it\nis assumed that such spatial profiles will remain unchanged\nin the course of the magnetization dynamics.\n15For notational simplicity we assume that all magnetic\natoms are identical. Generalizations to magnetic com-\npounds are straight forward.\n16Eq. ( 26) is valid provided that ωτ <<1. While this con-\ndition is normally satisfied in cases of practical interest, it\ninvariably breaks down as τ→ ∞. Hence the divergence\nof Eq. ( 26) in perfectcrystals is spurious.\n17I. Garate and A.H. MacDonald (in preparation).\n18J. Sinova et al., Phys. Rev. B 69, 85209 (2004). In order to\nget the equivalence, trade hzby ∆0and use ∆ 0=JpdS0,\nwhereJpdis the p-d exchange coupling between GaAs va-\nlence band holes and Mn d-orbitals. In addition, note that\nour spectral function differs from theirs by a factor 2 π.\n19H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan\n75, 113706 (2006)."    },    {        "title": "1902.07563v1.CoFeB_MgO_CoFeB_structures_with_orthogonal_easy_axes__perpendicular_anisotropy_and_damping.pdf",        "content": "CoFeB/MgO/CoFeB structures with orthogonal easy\naxes: perpendicular anisotropy and damping\nH. G lowi\u0013 nskia, A. _Zywczakb, J. Wronac, A. Kryszto\fka, I. Go\u0013 scia\u0013 nskaa,\nT. Stobieckid,e, J. Dubowika,\u0003\naInstitute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17,\n60-179 Poznan, Poland\nbAGH University of Science and Technology, Academic Centre of Materials and\nNanotechnology, Al. Mickiewicza 30, 30-059 Krakow, Poland\ncSingulus Technologies AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany\ndAGH University of Science and Technology, Department of Electronics, Al. Mickiewicza\n30, 30-059 Krakow, Poland\neAGH University of Science and Technology, Faculty of Physics and Applied Computer\nScience, Al. Mickiewicza 30, 30-059, Krakw Poland\nAbstract\nWe report on the Gilbert damping parameter \u000b, the e\u000bective magnetization\n4\u0019Meff, and the asymmetry of the g-factor in bottom-CoFeB(0.93 nm)/MgO(0.90{\n1.25 nm)/CoFeB(1.31 nm)-top as-deposited systems. Magnetization of CoFeB\nlayers exhibits a speci\fc noncollinear con\fguration with orthogonal easy axes\nand with 4\u0019Meffvalues of +2 :2 kG and\u00002:3 kG for the bottom and top\nlayers, respectively. We show that 4 \u0019Meffdepends on the asymmetry g?\u0000gk\nof theg-factor measured in the perpendicular and the in-plane directions re-\nvealing a highly nonlinear relationship. In contrast, the Gilbert damping is\npractically the same for both layers. Annealing of the \flms results in collinear\neasy axes perpendicular to the plane for both layers. However, the linewidth\n\u0003Corresponding author\nEmail address: dubowik@ifmpan.poznan.pl (J. Dubowik)\nPreprint submitted to Journal of Physics: Condensed Matter February 21, 2019arXiv:1902.07563v1  [cond-mat.mes-hall]  20 Feb 2019is strongly increased due to enhanced inhomogeneous broadening.\nKeywords: ferromagnetic resonance, perpendicular magnetic anisotropy,\nmagnetization precession damping\nPACS: 75.30.Gw, 75.70.Tj, 75.78.-n, 76.50.+g\n1. Introduction\nCoFeB/MgO/CoFeB systems are extensively employed in magnetic tun-\nnel junctions (MTJs), which are important for modern spintronic devices\nsuch as read-heads and magnetic random-access memory [1]. In these ap-\nplications the two key features are the perpendicular magnetic anisotropy\n(PMA) with PMA constant K?and magnetization damping with inhomoge-\nneous (extrinsic) and Gilbert (intrinsic) contributions to the ferromagnetic\nresonance (FMR) linewidth.\nThe FMR linewidth is usually enhanced in Ta/CoFeB/MgO stacks for\nwhich the values of PMA and the Gilbert damping parameter \u000bare scattered\n[2, 3, 4]. Recent experimental results [4, 5] indicate that there is no correlation\nbetweenK?and\u000bin these systems. Speci\fcally, \u000bis approximately constant\nwhile the PMA tends to improve on annealing. However, systems with a high\nPMA have often an increased linewidth due to an inhomogeneous broadening\n[6, 7] so that an extrinsic contribution to the linewidth may be as high as\n400{500 Oe [8] despite \u000bis of 0.01 { 0.02 in these systems. An increase in\nlinewidth is attributed to an angular dispersion of the easy PMA axis, which\nresults in a high inhomogeneous broadening attributed to the zero-frequency\nlinewidth \u0001 H0[6].\nIt has been shown that PMA in CoFe/Ni multilayers is linearly propor-\n2tional to the orbital-moment asymmetry [7, 9] in accordance with the Bruno's\nmodel [see Ref. [7] for discussion]. On the other hand, substantial PMA in\nTa/CoFeB/MgO systems [2] has been considered as related to an inhomoge-\nneous concentration of the anisotropy at the interface [10] so that the Bruno's\nmodel may be not valid in this case. Based on our experimental results, we\naim to shed some light on possible correlation between asymmetry of the\ng-factor and the e\u000bective magnetization 4 \u0019Meff, which are the magnetic\nparameters measured directly in a broadband FMR experiment. According\nto well known Kittel's formula, a departure from the free electron g-factor\nis proportional to \u0016L=\u0016S[11] so that we can discuss the asymmetry of the\ng-factor as well as on the asymmetry of the orbital moment on equal footing.\nHere, we prefer to use asymmetry in g-factor for evaluating the relationship\nbetween orbital moment and PMA.\nAs far as we know, FMR has not yet been thoroughly investigated in\n\"full\" Ta/CoFeB/MgO/CoFeB/Ta MTJ structures. In particular, a depen-\ndence of PMA on the asymmetry in the g-factor has not yet been proved\nin CoFeB/MgO/CoFeB systems. In this paper, we aim to independently\ncharacterize each CoFeB layer separated by a MgO tunnel barrier in terms\nof the\u000bparameter and 4 \u0019Meff. By analyzing FMR measurements in the\nin-plane and out-of-plane con\fgurations, we \fnd that PMA correlates with\ntheg-factor asymmetry in a highly nonlinear relationship.\n2. Experimental methods\nThe samples were sputtered in an Ar atmosphere using a Singulus Timaris\nPVD Cluster Tool. The CoFeB magnetic \flms were deposited by dc-sputtering\n3from a single Co 40Fe40B20target, whereas the MgO barriers were deposited\nby rf-sputtering directly from a sintered MgO target. The samples were de-\nposited on an oxidized silicon wafer with 5 Ta/ 20 Ru /Ta 3 bu\u000ber layers\nand capped with 5 Ta/ 5 Ru (numbers indicate the nominal thickness in\nnanometres). The studied structures consist of two ferromagnetic CoFeB\n(0.93 nm { bottom and 1.31 nm { top) \flms separated by a MgO barrier of\ndi\u000berent thicknesses (0.90, 1.1, and 1.25 nm). It is important to note that we\ninvestigated the as-deposited samples so that the CoFeB layers were amor-\nphous [3, 12]. The e\u000bect of annealing treatment (330oC for 1 hr) on magnetic\nproperties of the system will be discussed at the end of the paper.\nHysteresis loops of the samples were measured by vibrating sample mag-\nnetometer (VSM) with the perpendicular and in-plane magnetic \felds. The\nsaturation magnetization Msof 1200 G in the as-deposited state was deter-\nmined from magnetic moment per unit area vs. CoFeB thickness dependen-\ncies [13]. To investigate anisotropy and damping in studied samples, vector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) spectra of the S21\nparameter were analyzed [14]. VNA-FMR was performed at a constant fre-\nquency (up to 40 GHz) by sweeping an external magnetic \feld, which was\napplied either in-plane or perpendicular to the sample plane. These two con-\n\fgurations will be referred to as the in-plane and out-of-plane con\fgurations.\nExperimental data were \ftted using the Kittel formula\n!\n\rk=q\n(Hr+Ha) (Hr+Ha+ 4\u0019Meff) (1)\nfor the in-plane con\fguration and\n!\n\r?= (Hr\u00004\u0019Meff) (2)\n4for the out-of-plane con\fguration, where != 2\u0019fis the angular microwave\nfrequency,Hrthe resonance \feld, \rk;?=gk;?\u0016B=~the gyromagnetic ratio,\ngkandg?are the spectroscopic g-factors for the in-plane and out-of-plane\ncon\fgurations, respectively, ~the reduced Planck constant, \u0016Bthe Bohr\nmagneton, and Hathe in-plane uniaxial anisotropy \feld. 4 \u0019Meff= 4\u0019Ms\u0000\nH?is the e\u000bective magnetization , where Msis the saturation magnetization,\nandH?= 2K?=Msis the perpendicular anisotropy \feld and K?is the\nperpendicular anisotropy constant. For the in-plane easy axis 4 \u0019Meff>0\nwhereas for the perpendicular to the plane easy axis 4 \u0019Meff<0. According\nto Eqs. (1) and (2), 4 \u0019Meff=\u00002Keff=Ms, whereKeffis the e\u000bective\nanisotropy constant de\fned as K?\u00002\u0019M2\ns[15].\n3. Results and discussion\nFigure 1 (e) presents hysteresis loops of the sample with a 1.25 nm thick\nMgO barrier measured in the out-of-plane (red line) and in-plane con\fgu-\nration (black line). The shape of the loops in both directions is nearly the\nsame for each con\fguration as the saturation \felds (of Hs\u00192 kOe) for both\nlayers have nearly the same magnitude with the opposite signs in 4 \u0019Meff.\nEach hysteresis loop is a sum of the loops typical for the easy and hard axis\nand, as explained below, we can infer from magnetization reversals which\nlayer possesses PMA.\nLet us assume that the bottom CoFeB layer (B) has an in-plane easy axis\nand the top layer (T) has a perpendicular to the plane easy axis so that their\nmagnetization directions are orthogonal at remanence. Three con\fgurations\nof a magnetic \feld Happlied for the magnetization measurements are shown\n5-10 -5 0 5 10-101\n  Normalized moment\nH (kOe)HTT\nBH\ne.a.e.a.\nB\nHe.a.e.a. T\nBe.a.\ne.a.a) b) c)\nB\nT\nB+T\nd)\n 10 51\n-1\n-10 -5 0e)\nH (kOe)Figure 1: (a)-(c) Con\fgurations used for the magnetic measurements with a magnetic\n\feld applied perpendicular or parallel to the \flm plane. (d) Example of schematic pictures\nof the magnetization reversals of a CoFeB/MgO/CoFeB structure for con\fguration (a).\n(e) Hysteresis loops of a CoFeB/MgO/CoFeB structure measured in con\fgurations (a)\n- black line and (b) - red line. The inset shows schematically the model reversals for\ncon\fgurations (a)-black and (b)-red\n.\nin Figs. 1 (a) - (c). These con\fgurations enable magnetization reversals to be\nobserved with Horiented parallel- (a) (perpendicular- (b)) to the easy axis of\nB (T) layer, respectively, or perpendicular to both easy axes (c). Further, we\nwill refer to these con\fgurations as (a), (b), and (c) con\fgurations. As it is\nschematically shown in Fig. 1 (d), an apparent magnetization reversal of B+T\nfor the con\fguration (a) is a sum of independent magnetization reversals of\nB and T. For the perfectly asymmetric structure with 4 \u0019MB\neff=\u00004\u0019MT\neff\nwith the same thickness (i.e. with the same magnetic moments MSVT;B) the\n6apparent magnetization reversals taken in con\fgurations (a) and (b) would\noverlay. However, as it is seen in Fig. 1 (e) they do not completely overlay\nso that the curve taken in the con\fguration (b) lies a bit higher than that\ntaken in (a). As it is shown in the inset of (e), a simple model explains that\nthe T layer (i.e. the with nominal thickness tof 1.3 nm) possesses an easy\naxis perpendicular to the plane, while the B layer with t= 0:93 nm has an\nin-plane easy axis.\nIn the model, the magnetization reversals in each layer can be approxi-\nmated with a normalized relation [16] M(H;S) = arctan[H=H s\u0002tan(\u0019S=2)]=\narctan[H=H max\u0002tan(\u0019S=2)], where Hsof 2 kOe is a saturation \feld for\nthe hard direction and Sis de\fned as a ratio of remanence to the satura-\ntion moment. For Hkparallel to the easy axis, S= 1 (B layer in Fig. 1\n(d)) and for H?perpendicular to the easy axis (T layer in Fig. 1 (d)),\nS= 0:66 as well as Hmax= 10 kOe are arbitrary chosen for the sake of\nsimplicity. The apparent magnetization curve for con\fguration (a) is a sum\n[tB\u0002M(H;S = 1) +tT\u0002M(H;S = 0:66)]=(tB+tT). For the con\fguration\n(b),tTandtBare reversed in the sum. In order to satisfy the experimental\ndata shown in (e), a ratio tB=tT= 0:79. It is easily seen that if the B layer\nhad an in-plane easy axis and the T layer had an easy axis perpendicular to\nthe plane, a curve taken in con\fguration (b) would lie lower than that taken\nin con\fguration (a). Hence, the thin B layer is that with the in-plane easy\naxis.\nFigures 2 (a) and (b) show typical VNA-FMR spectra of the CoFeB/MgO(1.25\nnm)/CoFeB system measured (see Figs. 1) in con\fguration (a) and (b) , re-\nspectively. Two FMR peaks associated with the bottom and top CoFeB lay-\n76 8ImS21(a.u.)\nH(kOe)topbottom(a) 20GHz\nin-planeconfiguration\n4 6 8 10(b)ImS21(a.u.)\nH(kOe)top\nbottom20GHz\nout-of-planeconfiguration\n( )\n()Figure 2: Typical VNA-FMR spectrum of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure with resonance peaks from bottom (B) and top (T) layers measured in the in-\nplane (a) and out-of-plane (b) con\fgurations. Solid red lines represent the Lorentzian\n\fts to the experimental data. (c) Dependence of the FMR \feld on the polar angle \u0002\nof applied \feld in X band (9.1 GHz). The easy axis of magnetization of the B is in the\nin-plane orientation. For the T layer, the out-of-plane direction becomes the easy axis.\ners are clearly visible. To determine the resonance \feld Hrand the linewidth\n\u0001Hat constant frequency with a high precision, the spectra were \ftted with\nLorentzians (marked by solid lines in Fig. 2 (a) and (b)). Figure 2 (c) shows\ndependencies of the X-band (9.1 GHz) resonance \felds of the B and T layers\non the polar angle between the \flm normal and the direction of an applied\n\feld. It is clearly seen that the T layer has 4 \u0019Meff<0 (i.e., a perpendicular\neasy axis) and the B layer with 4 \u0019Meff>0 has an in-plane easy axis. From\nFigs. 2 (a) and (b), we can clearly see that the intensity (area under the FMR\n8peak) of the T layer is higher than that of the B layer. This additionally\ncon\frms that the bottom layer has the lower magnetic moment than that of\nthe top layer.\nA typicalHrvs.fdependence, observed for the CoFeB/MgO(1.25 nm)/CoFeB\nsystem is shown in Fig. 3 (a) and (b) for the in-plane (a) and out-of-plane\n(b) con\fguration, respectively. The observed data points are \ftted using\nEqs. (1) and (2). The values of 4 \u0019Meff, obtained from the \ftting are found\nto be of +2 :2 kG and\u00002:3 kG for the bottom and top layers, respectively.\nThefversusHrdata for the B layer were \ftted assuming Haof 30 Oe as\ncon\frmed by VSM measurements (not shown) in the con\fguration presented\nin Fig. 1(c). The values of gkof the top and bottom layers are equal to 2.04\nand 2.08, respectively, in contrast, the values of g?for these layers are 2.06\nand 2.22. One can notice the di\u000berences in values of g?resulting from clear\ndi\u000berences in the slopes of the f(Hr) dependencies (see, Fig. 3 (b)) for the\nbottom (\r?= 2:88 MHz/Oe) and top ( \r?= 3:11 MHz/Oe) layer, respec-\ntively.\nTo sum up, VSM and FMR measurements con\frmed the presence of or-\nthogonal easy axes in our CoFeB/MgO/CoFeB systems and showed that the\nthickness ratio tB=tT= 0:79 is slightly higher than the ratio of nominal thick-\nness (tB\nnom=tT\nnom= 0:71). The thinner B layer has an in-plane easy axis while\nthe T layer has a perpendicular easy axis. However, keeping in mind our for-\nmer studies of a dead magnetic layer (DML) in the Ta/CoFeB/MgO (B) and\nMgO/CoFeB/Ta (T) structures [13] deposited in the same Timaris system,\nwe estimated DMLB'0:23 nm and DMLT'0:4. With such asymmetric\nDMLs the e\u000bective thickness tB\neff'0:7 nm andtT\neff'0:9 nm which satis\fes\n90 5 1005101520\n0 5 10010203040\nf (GHz)\nHr(kOe)\n(b)(a)\nin-plane configuration\nout-of-plane configuration\nf (GHz)\nHr(kOe)1.25 nm MgO\nbottom\ntop0 2 46 8 10048121620\ntop 1.25 nm MgO \n 1.25 nm MgO \n 0.96 nm MgO\n 0.96 nm MgO\n 0.85 nm MgO\n 0.85 nm MgOf (GHz)\nH (kOe)bottomFigure 3: FMR dispersion relations of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure measured in the in-plane con\fguration (a) and out-of-plane con\fguration (b).\nThe solid lines show the \fts given in accordance with Eqs. (1) and (2). Inset in (a) shows\nthat the \ftting parameter practically do not depend on the MgO thickness.\ntB=tT= 0:78. VNA-FMR measurements, which o\u000ber a greater precision than\nVSM measurements, give 4 \u0019Meff=\u00002:3 kG (K?= 10:4\u0002106erg/cm3) and\n4\u0019Meff= +2:2 kG (K?= 7:7\u0002106erg/cm3) for the T and B layers, respec-\ntively. All \ftting parameters for a CoFeB/MgO(1.25 nm)/CoFeB structure\nare juxtaposed in Table 1. As it is shown in the inset of Fig. 3 (a), the thick-\nness of MgO spacer within a range of 0.9 { 1.25 nm had almost no in\ruence\non the \ftting parameters, therefore, the values of \ftting parameters 4 \u0019Meff,\ng,\u000b, and \u0001H0are typical for all samples with various MgO thickness.\n10Table 1: Parameters determined from VNA-FMR spectra for the as-deposited\nCoFeB(0.93 nm)/MgO (1.25 nm)/CoFeB(1.31 nm) for the in-plane and out-of-plane con-\n\fgurations: the in-plane anisotropy \feld ( Ha), the e\u000bective magnetization (4 \u0019M eff), spec-\ntroscopicg-factors for in-plane and out-of-plane con\fguration, Gilbert damping ( \u000b), the\nfrequency-independent FMR linewidth (\u0001 H0). The values of the \ftting parameters do\nnot depend on the MgO thickness. The values of g?are marked by asterisks.\nIn-plane con\fguration\nHa(Oe) 4\u0019Meff(kG)gk,g? \u000b \u0001H0(Oe)\ntop 0 -2.29\u00060.05 2.04\u00060.02 0.018\u00060.002 102\u000622\nbottom 30 2.22\u00060.15 2.08\u00060.03 0.017\u00060.002 69\u000623\nOut-of-plane con\fguration\ntop { -2.3\u00060.01 2.22\u00060.01?0.018\u00060.001 95\u000613\nbottom { 2.19\u00060.04 2.06\u00060.02?0.017\u00060.003 160\u000630\nAlthough it is counter-intuitive that the thinner B layer possesses an in-\nplane easy axis, the same feature has been reported for other Ta/CoFeB(1\nnm)/MgO systems deposited in the same Timaris equipment [17]. Similar ef-\nfect has been recently observed in a substrate/MgO/CoFeB/Ta/CoFeB/MgO\nstructure, where the thicker CoFeB layer exhibits a strong PMA in con-\ntrast to the relatively weak PMA in the thinner CoFeB layer [18, 19]. It is\npossible that the growth mode of the MgO layer in contact with an amor-\nphous CoFeB layer might be responsible. The perpendicular anisotropy in\nthese systems originates from the CoFe/MgO interface [20]. The structure\nof the unannealed CoFeB layers is amorphous regardless of underlying lay-\ners, whereas the MgO barrier deposited on the amorphous CoFeB has an\namorphous structure of up to four monolayers (that is about 0.9 nm) [21].\n11Hence, there are subtle di\u000berences between the CoFeB/MgO (bottom) and\nMgO/CoFeB (top) interfaces; the interface of the bottom CoFeB layer is\nmainly amorphous whereas the interface of the top layer is crystalline, be-\ncause the barrier thickness of the investigated samples is above the transition\nfrom amorphous to crystalline phase. Therefore, di\u000berent structures for the\nCoFeB/MgO interfaces may result in di\u000berent values of anisotropy constant.\nAnother explanation is that the measured dependence Keff\u0002teffvs.teff\nin \flms with PMA is often strongly nonlinear due to either intermixing at\ninterfaces [22] or magnetoelastic e\u000bects [15], with Keff\u0002teffexhibiting a\nmaximum as a function of decreasing teffand with the PMA eventually\nbeing lost for small teffof, for example, 0.7 nm.\nThe values of gfactor yield the ratio of the orbital \u0016Land spin\u0016Smag-\nnetic moments in accordance with equation [9, 11]\n\u0016L\n\u0016S=g\u00002\n2; (3)\nwhere\u0016S=\u0016B. Hence, the di\u000berence between orbital moments \u0001 \u0016Lalong\nthe easy and hard direction in the in-plane [Fig. 1 (a)] and out-of-plane [Fig. 1\n(b)] con\fgurations is proportional to ( g?\u0000gk) and reads \u0001 \u0016L=\u0016B(g?\u0000\ngk)=2. \u0001\u0016Lis of 0.09\u0016Band\u00000:01\u0016Bfor the T and B layer, respectively.\nIn CoFe/Ni multilayers [7], the PMA has been shown to be proportional\nto the orbital moment anisotropy in accordance to Bruno model [23]. How-\never, in the case of the CoFeB/MgO systems this direct relationship between\nthe orbital moment asymmetry and the perpendicular anisotropy is not ful-\n\flled. As can be seen in Table 1, ( g?\u0000gk)\u00190 for the B layer corresponds to\n4\u0019Meff= 2:2 kG. Hence, while ( g?\u0000gk) is negligible, a decrease in 4 \u0019Meff\ndue to PMA from 4 \u0019MS= 15 kG to 2.2 kG is substantial. In contrast,\n12(g?\u0000gk)\u00190:18 is exceptionally large for the T layer, while 4 \u0019Meffmerely\ndecreases to - 2.3 kG. In accordance with the earlier report [24], this con\frms\nthat any relationship between the orbital moment asymmetry and the per-\npendicular anisotropy in CoFeB/MgO systems is highly nonlinear. Of course,\nother factors controlled by annealing such as disorder at interfaces and over-\nor underoxidized interfaces would also play a signi\fcant role in PMA [20].\nFuture work con\frming such a nonlinear relationship for a broad range of\ntCoFeB might resolve this issue.\nAt present, there is no doubt that PMA in MgO/CoFeB structures is\nan interface e\u000bect and it is correlated with the presence of oxygen atoms\nat the interface despite the weak spin-orbit coupling [20, 25]. The origin\nof PMA is attributed to hybridization of the O-p with Co(Fe)-d orbitals at\nthe interface [20] and/or to a signi\fcant contribution of thickness dependent\nmagnetoelastic coupling [15]. A deviation of the g-factor from the 2.0 value\nis expressed by g'2\u00004\u0015=\u0001 , where\u0015 < 0 is the spin-orbit constant for\nFe(Co) and \u0001 is the energy levels splitting in the ligand \feld [11]. While\nthe deviation of the g-factor is inversely proportional to \u0001, PMA (and hence\n4\u0019Meff) is proportional to the enhanced spin-orbit-induced splitting around\nthe Fermi level [20]. This may result in a complex relationship between PMA\nandg-factor anisotropy.\nThe Gilbert damping parameter \u000bis evaluated from the dependence of\nthe linewidth \u0001 Hon the resonance frequency as shown in Fig. 4 for the\nin-plane (a) and the out-of-plane (b) con\fgurations. The lines are linear \fts\nto\n\u0001H=\u000b4\u0019f\n\rk;?+ \u0001H0; (4)\n13/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s98/s41/s40/s97/s41\n/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56\n/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41Figure 4: Linewidth as a function of frequency measured in the in-plane con\fguration (a)\nand out-of-plane con\fguration (b). The \u000bdamping parameter is obtained using Eq. (4).\nThe thickness of MgO was 1.25 nm.\nwhere \u0001H0is the inhomogeneous broadening related to CoFeB layer quality.\nThe values of \u000band \u0001H0are shown in Table 1. The top and the bottom layers\nshow almost the same \u000bof 0.017 - 0.018. This suggests that the damping has\nno relation to PMA. While \u0001 H0for the top layer is almost the same for both\ncon\fgurations, \u0001 H0for the bottom layer at the (b) con\fguration is nearly\ntwice as large as that for the (a) con\fguration. Such a behavior suggests\nthat the layer B is rather inhomogeneous with a large angular dispersion of\nmagnetization across the layer [26, 27].\nSpin pumping to Ta layers (which are a part of the bu\u000ber and cap-\n14ping layers, as shown in Fig. 1 (e)) may also in\ruence the damping in\nCoFeB/MgO/CoFeB systems since magnetization precession induces a spin\ncurrent to the adjacent nonmagnetic Ta layers that result in an enhanced\ndamping [8]. This is an interface e\u000bect and hence scales inversely propor-\ntional to the CoFeB layer thickness. Because the bottom layer with an in-\nplane easy axis is thinner than the top layer with a perpendicular easy axis,\nthe spin pumping e\u000bect a\u000bects it more. To estimate spin pumping e\u000bect the\nstandard equation [28] without back\row is used\n\u0001\u000b=g\u0016Bg#\"\n4\u0019Msteff; (5)\nwhereteffis the e\u000bective thickness of CoFeB and g#\"is the mixing con-\nductance. The measured damping of both layers is of 0.017 - 0.018, while\ndamping of a bulk CoFeB is around 0.004 [12]. Therefore, an increase of \u0001 \u000b\ndue to spin pumping is of 0.014 which gives the mixing conductance g#\"= 0:8\nand 1\u00021015cm\u00002for the e\u000bective thickness 0.7 nm and 0.9 nm of B and\nT layer, respectively. The value of mixing conductance g#\"for Ta/CoFeB\ninterface found in the literature lies in a broad range from 1 :67\u00021014to\n1:4\u00021015cm\u00002[29, 30, 31, 32]. Taking into account our simpli\fcation (the\nlack of back\row), this estimation gives the maximal values of mixing conduc-\ntance. Hence, we can conclude that spin pumping substantially in\ruences\nthe damping in our structures. It is worth mentioning that the measured \u000b\nof 0.017 - 0.018 for CoFeB/MgO/CoFeB systems agrees with \u000b= 0:015 for\nthe Ta/CoFeB(1)/MgO structure reported in [3].\nFinally, we would like to make a further comment on postdeposition an-\nnealing of our CoFeB/MgO/CoFeB systems. We found that annealing at\n330oC for 1 hr, beside increasing Msto 1500 G, enhances also PMA so that\n15both layers possess easy axes perpendicular to the plane. 4 \u0019Meffattains\n-1 kG and -4 kG for the B and T layers, respectively. We found that an\nincrease in K?of 7:7\u0002106erg/cm3equally contributes to both layers and,\nfor example, K?= 17\u0002106erg/cm3for the T layer. On the other hand, the\nlinewidth \u0001 Hstrongly broadens to \u0018400 Oe and\u0018700 Oe for the B layer\nand the T layer, respectively. These values are in agreement with recently\nreported values for a similar systems [17]. Moreover, as it is shown in Fig. 5,\n\u0001Hdoes not follow the linear dependence described by Eq. (4). Therefore,\nit is impossible to determine \u000bprecisely for the annealed systems. Such a\nbehavior of \u0001 Hand the decreased remanence with respect to the saturation\nmagnetization (see, [17]) both con\frm a strong angular dispersion of the easy\nPMA axis in both layers. It has been observed that with increasing PMA\nthe dispersion of anisotropy also increases [6, 7, 27]. As a result, dispersion\nin PMA leads to a large two magnon scattering contribution to the linewidth\nfor in-plane magnetization and to an enhanced Gilbert damping [6]. While\nthe magnetic parameters practically do not depend on the MgO thickness in\nas-deposited structures, the annealed structures show a substantial spread in\n4\u0019Meffas it is shown in Fig. 6, which may imply some di\u000berent CoFeB/MgO\ninterfaces due to, for example, boron di\u000busion [30, 33].\n4. Conclusion\nWe investigated the CoFeB/MgO/CoFeB as-deposited systems with the\nin-plane and out-of-plane orthogonal easy axes due to the substantial dif-\nference in PMA for the bottom (B) and the top (T) CoFeB layers, respec-\ntively. The T and the B layer had comparable Gilbert damping \u000bsuggesting\n16/s53 /s49/s48 /s49/s53 /s50/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s32/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 5: Linewidth as a function of frequency measured in the in-plane con\fguration for\nthe annealed structure. The thickness of MgO was 1.25 nm.\nthat there is no correlation between the Gilbert damping and PMA. We\nalso showed that 4 \u0019Meffcorrelates with the asymmetry in the g-factor (and\nhence with \u0001 \u0016L) and this correlation is highly nonlinear. Annealing enhances\nPMA in both layers but it has detrimental e\u000bect on the linewidth, however.\nTherefore, despite the Gilbert parameter shows no correlation with PMA, it\nseems that there is some correlation between the linewidth (see Eq. 4) and\nPMA in the annealed systems through a combined e\u000bect between dispersion\nof local anisotropy easy axes in crystallites with a high PMA.\n17/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s107/s79/s101/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 6: FMR dispersion relations of CoFeB/MgO(0.9 { 1.25 nm)/CoFeB annealed struc-\nture measured in the in-plane con\fguration.\nAcknowledgments\nWe acknowledge support from the the project \\Marie Sk lodowska-Curie\nResearch and Innovation Sta\u000b Exchange (RISE)\" Contract No. 644348 with\nthe European Commission, as part of the Horizon2020 Programme, and\npartially by the project NANOSPIN PSPB-045/2010 under a grant from\nSwitzerland through the Swiss Contribution to the enlarged European Union.\n18References\n[1] B. 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Kurinec,\n\\Study of boron di\u000busion in MgO in CoFeB/MgO \flm stacks using par-\nallel electron energy loss spectroscopy,\" Applied Physics Letters , vol. 94,\nno. 8, p. 082110, 2009.\n24"    },    {        "title": "1710.10833v2.Probe_of_Spin_Dynamics_in_Superconducting_NbN_Thin_Films_via_Spin_Pumping.pdf",        "content": "1\n \n \nProbe of Spin Dynamics in Superconducting NbN Thin\n \nFilms via \n \nSpin Pumping\n \n \nYunyan\n \nYao\n1,2\n,\n \nQi\n \nSong\n1,2\n,\n \nYota\n \nTakamura\n3,4\n,\n \nJuan\n \nPedro\n \nCascales\n3\n,\n \nWei\n \nYuan\n1,2\n,\n \nYang\n \nMa\n1,2\n,\n \nYu\n \nYun\n1,2\n,\n \nX.\n \nC.\n \nXie\n1,2\n,\n \nJagadeesh\n \nS.\n \nMoodera\n3,5\n,\n \nand\n \nWei\n \nHan\n1,2*\n \n1\nInternational Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China\n.\n \n2\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, China\n.\n \n3\nPlasma Science and Fusion\n \nCenter and Francis Bitter Magnet Laboratory, Massachusetts \nInstitute of Technology, Cambridge, MA 02139, USA\n.\n \n4\nSchool of Engineering, Tokyo Institute of Technology, Tokyo 152\n-\n8550, Japan.\n \n5\nDepartment of Physics, Massachusetts Institute of Technology, Camb\nridge, MA 02139, USA\n \n*Correspondence to: \nweihan@pku.edu.cn (W.H.)\n \n \n \nAbstract\n:\n \nThe emerging field of superconductor (SC) spintronics has attracted \nintensive attentions recently\n. \nMany fantastic spin dependent properties in SC\ns\n \nhave been discovered, including large \nmagnetoresistance, long spin lifetimes and the giant spin Hall effect\n, \netc\n. Regarding the spin \ndynamic\ns\n \nin \ns\nuperconducting \nthin \nfilms, \nfew studies has been reported yet. \nHere, we report the \ninvestig\nation of the spin d\nynamics in a\n \ns\n-\nwave superconducting NbN film via spin pumping from \nan adjacent insulating ferromagnet GdN\n \nfilm\n. \nA\n \nprofound coherence peak of the Gilbert damping \nof GdN \nis observed slightly below the superconducting critical temperature of the NbN\n, \nwhich \nag\nrees well\n \nwith recent\n \ntheoretical \nprediction\n \nfor \ns\n-\nwave SCs \nin the presence of impurity spin\n-\norbit \nscattering\n. \nThis observation is also \na manifestation of the dynamic\n \nspin injection\n \ninto \nsuperconducting NbN thin film.\n \nOur results \ndemonstrate that spin pumping could be used \nt\no probe \nthe dynamic spin susceptibility\n \nof \nsuperconducting thin films, thus pave the \nway for future \ninvestigation of spin dynamics \nof\n \ninterfacial and \ntwo \ndimensional\n \ncrystalline \nSC\ns\n.\n \n 2\n \n \nI.\n \nINTRODUCTION\n \nThe \ninterplay between s\nuperconductivity and \nspintronics \nhas\n \nbeen intensively investigated \nin \nthe last decade\n \n[\n1\n-\n3\n]\n.\n \nIn the content of superconductivity, the \nferromagnetic \nmagnetization \nhas\n \nbeen found to \nplay\n \nan important role in the superconducting critical temperature (T\nC\n) in the \nferromagnet\n \n(FM)\n/\nsuperconductor (\nSC\n)\n \njunctions\n \n[\n4\n-\n9\n]\n.\n \nBesides, \nunusual\n \nspin\n-\npolarized\n \nsupercurrent \nhas been observed \nin \nferromagnetic \nJosephson junctions\n \n[\n2\n,\n3\n,\n10\n-\n13\n]\n. \nFurthermore, \nnon\n-\nabelian Majorana fermions have been proposed for \np\nx\n \n+ ip\ny\n \nsuperconductor surfaces driven \nby magnetic proximity effect, whic\nh has the potential for topological quantum computation \n[\n14\n]\n. \nIn the content of spintronics\n \n[\n15\n,\n16\n]\n, \nthe co\noper pairs \nin \nSCs\n \nhave exhibited many fa\nntas\ntic spin\n-\ndependent properties\n \nthat \nare promising \nfor future\n \ninformation technologies.\n \nFor instance, l\narge \nmagnetoresistance has been \nachi\ne\nved\n \nin the spin valve with \nSCs \nbetween two ferromagnetic films\n \n[\n7\n,\n17\n]\n, \nextremely \nl\nong spin lifetimes \nand large spin Hall \neffect have \nbeen discovered for the \nquasiparticles\n \nof\n \nSCs\n, which exceeds several orders compared to the normal states\n \nabove T\nC\n \n[\n18\n,\n19\n]\n. \nDespite these inte\nnsive studies of the spin\n-\ndependent \npr\noperties \nin the FM/SC junctions\n, \nthe spin dynamics of \nthe \nsupercond\nu\nc\nting thin films\n \nhas not been \nreported\n \nyet.\n \n \nVery \nintriguingly\n, \nit has been\n \nproposed \nrecently \nt\no investigate\n \nthe\n \nspin dynamics in \nsuperconducting \nfilms via spin pumping\n \n[\n20\n]\n, a well\n-\nestablished technique to \nperform d\nynamic\n \nspin injection and to probe the dynamic spin susceptibility in various materials, including metal, \nse\nmiconductors, Rashba 2DEGs, and topological insulators, etc\n \n[\n21\n-\n30\n]\n.\n \n \nIn this letter\n, we report the \nexperimental investigation \nof \nthe spin dynamics in \ns\n-\nwav\ne \nsuperconducting NbN thin film\ns\n \nvia spin pumping. \nA profou\nnd coherence peak of the Gilbert \ndamping\n \nof GdN\n \nis observed slightly below the superconducting critical temperature\n \n(T\nC\n)\n \nof the \nNbN \nin the \nNbN/GdN/NbN\n \ntrilayer samples\n, which indicates \ndynamic spin injection into the \nN\nbN \nthin film\n. Besides, \nthe \ninterface\n-\nenhanced Gilbert damping\n \nprobes \ndynamic spin susceptibility\n \nin \nthe \nNbN thin films \nin the presence of impurity spin\n-\norbit\n \nscattering\n, \nwhich is consistent w\nith \nthe\n \nrecent theoretical study\n \n[\n20\n]\n. \nOur experimental results further \ndemonstrat\ne that\n \nspin pumping \ncould \nbe\n \na \npowerful tool to study the spin dynamics\n \nin \nthe emerging two dimensional SCs\n \n[\n31\n-\n33\n]\n. \n \nII. EXPERIMENTAL DETAILS\n \nThe NbN (t)/GdN (d)/NbN (t) trilayer samples \nare\n \ngrown on Al\n2\nO\n3\n \n(~5 nm)\n-\nbuffered thermally \noxidized Si substrates by d.c. reactive magnetron sputtering at 300 \n°\nC in an ultrahigh vacuum 3\n \n \nchamber. The \nNbN layers \nare\n \ndeposited from a pure Nb target (99.95%) in Ar and N\n2\n \ngas mixture \nat a pressure of 2.3 mTorr (20% N\n2\n), and \nGdN films \nare\n \ndeposited from a pure Gd target (99.9%) \nin Ar and N\n2\n \ngas mixture at a pressure of 2.8 mTorr (6% N\n2\n).\n \nThe NbN and GdN layers \nare \nof \ntextured crystalline quality with a preferred direction along (111)\n-\norientation, as evidenced by X\n-\nray diffraction results (\nF\nig. S1)\n \n[\n34\n]\n. After the growth, a thin Al\n2\nO\n3\n \nlayer (~ 10 nm) was deposited \nin situ\n \nas a capping layer to avoid sample degradation with air exposure. The Curie temperature of \nthe GdN film was determined via the offset\n-\nmagnetization as a function of the temperature us\ning \na Magnetic Properties Measurement System (MPMS; Quantum Design).\n \nThe T\nC\n \nof the SC NbN \nthin films was measured by four\n-\nprobe resistance technique as a function of the temperature in a \nPhysical Properties Measurement System (PPMS; Quantum Design) using s\ntandard ac lock\n-\nin \ntechnique at low frequency of 7 Hz.\n \nThe FMR spectra of the multilayer samples were measured \nusing the coplanar wave guide technique with a vector network analyzer (VNA, Agilent E5071C) \nin the variable temperature insert of PPMS. The samp\nles were attached to the coplanar wave guide \nusing insulating silicon paste. During the measurement, the amplitudes of forward complex \ntransmission coefficients (S\n21\n) were recorded as a functio\nn of the\n \nin\n-\nplane\n \nmagnetic field from ~ \n4\n000 to 0 Oe at various\n \ntemperatures under different microwave frequencies\n \nand microwave power \nof 5 dBm\n.  \n \nIII. RESULTS and DISCUSSION\n \nFig\n.\n \n1\n(a)\n \nillustrates \nspin pumping and \nthe interfacial \ns\n-\nd\n \nexchange coupling between \nspins\n \nin \nthe NbN layer and \nm\nagnetic moment\ns\n \nin the GdN laye\nr\n. \nDue to the interfacial \ns\n-\nd\n \nexchange \ninteraction (\nJ\nsd\n), the time\n-\ndependent magnetization in the GdN layer pumps a quasiparticles\n-\nmediated spin current into the NbN layer, and the spin\n-\nflip scattering of quasiparticles \naccompanies a quantum process of mag\nnon annihilation in the GdN layer, giving rise to enhanced \nGilbert damping\n \n[\n20\n]\n. Hence, the interfacial s\n-\nd exchange coupling provides the route to detect the \nspin dynami\ncs of the NbN layer by measuring the magnetization dynamics of GdN\n.\n \nThe spin \npumping is performed by measuring Gilbert damping of the GdN \nfilm \nin the NbN (t)/GdN (d)/NbN \n(t) trilayer heterostructures\n \nvia ferromagnetic resonance (FMR) technique \n[\n35\n]\n. \nNbN is a\n \ns\n-\nwave \nSC\n \nwith short coherence length of ~ 5 nm\n \nand \nspin diffusion length of ~ 7 nm\n \n[\n19\n,\n36\n]\n. \nGdN is an \ninsulating FM\n. \nA 10 nm \nNbN film is used f\nor the NbN (t)/GdN (d)/NbN (t) \nsamples to justify the \nassumption that the spin backflow from SC is small in the theoretical study \n[\n20\n]\n.\n \nIn a typical sample \nof N\nbN (10)/GdN (5)/NbN (10)\n \n(with thickness in nm)\n, the Curie temperature (T\nCurie\n) of GdN is 4\n \n \ndetermined to be ~ 38 K \nfrom \nthe temperature dependence of \nmagnetic moments\n \n(Fig. 1\n(b)\n), and \nT\nC\n \nof NbN is \nobtained to be \n~ 10.8 K\n \nvia\n \nresistivity vs. temperature \nmeasurement \n(Fig. 1\n(c)\n)\n. \nThe \nT\nC\n \nis slightly affected by the \nexternal in\n-\nplane magnetic\n \nfield of 4000 Oe\n \ndue to the large critical \nfield.\n \n \nFig. 1\n(d)\n \nshows a typical FMR signal (S\n21\n) vs.\n \nthe magnetic field measured \nat \nT\n \n= \n10 K, with \na microwave excitation \nfrequency (\nf\n) of 15 GHz. \nThe half linewidth (\nH\n\n) could be obtained by \nthe Lorentz fitting of the \nFMR \nsignal\n \nfollowing the relationship\n \n[\n37\n]\n:\n \n2\n21 0\n2 2\n( )\n( ) ( )\nres\nH\nS S\nH H H\n\n\n  \n \n \n \n \n \n(1)\n \nwhere S\n0\n \nis the coefficient for the transm\nitted microwave power, \nH\n \nis the external magnetic field, \nand \nres\nH\nis the resonance magnetic field. Gilbert damping (\n\n) is determined using numerical \nfitting of\n \nH\n\nvs. \nf\n \n(Fig. 1\n(e)\n) \nbased \non the spin\n-\nrelaxation\n \nmechanism \n(\nFig. S2\n)\n \n[\n34\n,\n38\n]\n:\n \n0\n2\n4 2\n1 (2 )\nf f\nH H A\nf\n  \n  \n   \n\n \n \n \n \n(2)\n \nwhere \n0\nH\n\n \nis related to the inhomogeneous properties,  \n\n \nis the gyromagnetic ratio, \nA\n \nis t\nhe \nspin\n-\nrelaxation coefficient, and \n\n \nis the \nspin\n-\nrelaxation time constant\n. \n \nThe Gilbert damping \nof GdN \nis studied\n \nas a function of the temperature for two trilayer samples\n \nwith the same interface\n:\n \nNbN (2)/GdN (5)/NbN (2) and NbN (10)/GdN (5)/NbN (10)\n. \nThe 2 nm \nnm Nb\nN layer\n \nis not superconducting\n \ndown to 2 K \nwhile the \nNbN (10)/GdN (5)/NbN (10)\n \nexhibits \na T\nC\n \nof ~ 10.8 K \n(Fig. 2(a)\n). \nThe non\n-\nsuperconducting \nfeature of 2 nm NbN\n \nsample \nand lower \nT\nC\n \nof t\nhe 10 nm NbN \nsample \ncompared to over 16 K\n \nfor \nsingly crystalline \nNbN\n \ncould be \nattributed\n \nto \nreduced thickness, \npolycrystalline property, and magnetic proximity effect\n \n[\n1\n,\n36\n,\n39\n]\n. \nInterestingly, \na profound\n \ncoherence peak of the Gilbert damping \nis observed\n \nin the NbN (10)/GdN (5)/NbN (10)\n, \nbut not on \nNbN (2)/GdN (5)/NbN (2), \nas shown in Fig. 2\n(b)\n. \nThis feature is also evident \nfrom the \ntemperature dependence of the half\n \nlinewidth\n \n(Fig. S\n3\n)\n \n[\n34\n]\n.\n \nThe peak of the Gilbert damping in \nthe NbN (10)/GdN (5)/NbN (10) is observed at ~ 8.5 K, which is slightly below the T\nC\n \n(~ 10.8\n \nK) \nof the 10 nm NbN \nlayer\ns\n. These results indicate the successful dynamic spin injection into the 10 \nnm NbN \nsuperconducting \nlayer, which authenticates a charge\n-\nfree method to injec\nt spin\n-\npolarized \ncarriers into\n \nSC\ns\n \nbeyond previous repor\nts of electrical spin injection\n \n[\n18\n,\n40\n-\n42\n]\n. Furthermore, the 5\n \n \nobservation of the profound coherence peak \nat \nT\n \n= \n~\n \n0.8\n \nT\nC\n \nis expe\ncted \nbased on \nrecent theoretical \nstudies \nof the spin dynamic\ns for \ns\n-\nwave \ns\nupercond\nu\nc\nting thin films\n \nvia spin pumping\n \n[\n20\n]\n. \nSince \nthe thickness of NbN layer (d = 10 nm) i\ns longer than \nits\n \nspin diffusion length of ~ 7 nm, the spin \nbackflow effect, which \nis expected to\n \nreduce\n \nthe damping peak,\n \nis negligible\n \nhere\n \n[\n20\n,\n21\n]\n. \nAccording to the theory, \nGilbert damping is related to the interfacial \ns\n-\nd\n \nexchange interaction and \nthe imaginary part of the dynamic spin susceptibility of the SC\n \n[\n20\n]\n.\n \n \n \n \n \n \n2\nIm ( )\nR\nsd q\nq\nJ x\n \n\n\n \n \n \n \n \n(3)\n \nFor the \ns\n-\nwave superconducting NbN thin films, the superconducting gap \n∆\n \nforms\n \nbelow T\nC\n. At \nthe temperature slightly below T\nC\n, two coherence peaks of the density states exist around the edge \nof the superconducting gap following the BCS theory\n \n[\n43\n]\n, and these peaks in turn give rise to the \nenhancement of t\nhe dynamic spin susceptibility\n \nin the pre\nse\nnce of \nimpurity spin\n-\norbit scattering\n.\n \nQuantitatively\n,\n \nthe ratio of the peak value of Gil\nb\nert damping over \nthat \nat \nT\n \n~ \nT\nC\n \nis\n \n~ 1.8\n, which \nalso\n \nagree\ns\n \nwell wit\nh the theoretical calculation\n \nusing \nprevious\n \nexperimental values of spin \ndiffusion length (~ 7 nm), phase\n \ncoherence length (~ 5 nm), and mean free path (~ 0.3 nm)\n \nfor \nNbN\n \nthin films \n[\n19\n,\n20\n,\n36\n]\n. \nAs the temperature further decreases, the number of quasiparticles \ndecreases \nrapidly \nas the \n∆\n \ngrows, \ngiving rise to the fast decrease of Gilbert damping\n \nbelow \nT\n \n=\n \n~ \n7\n \nK\n. \n \nNext, the thickness of th\ne GdN laye\nr is varied to further study spin pumping\n \nand the spin \ndynamics in NbN layer. For\n \nboth \ns\namples of NbN (10)/GdN (d)/NbN (10) with d \n= \n10\n \nnm and d \n=\n \n30 nm, a profound coherence peak of Gilbert damping is observed slightly below the \nsuperconducting \ntemperat\nure of NbN, as shown in Figs. 3\n(a)\n \nand 3\n(b)\n \n(red circles). While for all \nthe samples of NbN (2)/GdN (d)/NbN (2), no such coherence peak of the Gilbert damping is \nnoticeable (green circles\n \nin Figs. 3(a) and 3(b)\n). The role of the effective \nmagnetization\n \nin the \nobserved \ncoherence peak\n \nhas been ruled out since it exhibits similar temperature dependence for \nNbN (2\n)/GdN (d\n)/NbN (2) and NbN (10)/G\ndN (d\n)/NbN (10) sampl\nes (\nF\nig. S\n4\n)\n \n[\n34\n]\n. \nClearly, \na \nmore profound damping peak for \nthe d = 10 nm sample\n \nis \nobserved\n \ncompared to \nd = 5 nm and \nd = \n30 nm \nsamples. \nFig. 3\n(c)\n \nshows t\nhe ratio of the peak Gilbert damping over the value at \nT\n \n= \n~ \nT\nC\n \nas a function of the GdN thickness\n, which\n \nis in the range from \n1.8 to 2.8\n. \nThe T\nC\n \nexhibit little \nvariation as a function of the GdN thickness\n \n(Fig. S5)\n \n[\n34\n]\n. \nFor a deeper understanding of the \nunderlying mechanism\n \nto account for the thickness dependence of the \nratio,\n \nfurther\n \ntheoretical and 6\n \n \nexperim\ne\nn\nt\nal\n \nstudies \nw\nould be essential\n. One possible cause might be related to the interface \nproximity exchange effect and/or \nthe \npresence of magnetic loose spins leading to scattering at the \ninterface, which could a\nffect the spin diffusion length, \ncoherence length\n, and mean free path\n \nof the \nN\nbN layer. \n \nTo further confirm \nthat \nthe observed coherence peak of the Gilbert damping aris\nes\n \nfrom the\n \ns\n-\nd\n \nexchange interaction\n \nat the SC/FM interface\n, \nthe interface\n-\ninduced Gilbert damping\n \n(\nS\n\n)\n \nis \nstudied as a function of temperatur\ne. \nS\n\n \nis obtained from the thickness dependence of \ntotal \nGilbert \ndamping \nat each temperature\n \n[\n34\n]\n, as shown in \nFig. 4\n(a)\n. \nT\nhe \ncontribution \nfrom GdN itself is \nsignificantly small\n \ncompared to\n \nS\n\n. \nThe unpe\nrturbed peak at ~ 8.5 K (Fig. 4\n(b)\n) unambiguously \ndemonstrates that the origin of the coherence peak in the Gilbert damping is indeed due to the \ninterfacial \ns\n-\nd\n \nexchange interaction between the magnetization of GdN and the spins \nof the \nquasiparticles\n \nin superconducting NbN thin films\n. \nThe \nslightly upturn of the interface\n-\ninduced \nGilbert damping starting at \nT\n \n= 11 K might be associated with the fluctuation superconductivity \nand/or the higher superconducting transition temperature for some regions of the NbN films than \nthe zero resistance tem\nperature. \nFor comparison, the interface\n-\ninduced Gilbert damping in the \nsamples of NbN (2)/GdN (d)/NbN (2) does not \nexhibit\n \nany \nsignature\n \nof coherence peak \nbetween \n4 and 15 K\n \n(\nF\nig. S6\n)\n \n[\n34\n]\n.\n \n \nNoteworthy is that our results are essentially different from previous reports of spin pumping \ninto \nsuperconducting\n \nNb films using the ferromagnetic metal permalloy\n \n(Py)\n, where a monotonic \ndecrease of\n \nthe Gilbert damping is reported when the temperature decreases\n \n[\n44\n,\n45\n]\n. \nThis feature \nis also \nconfirmed\n \nin our studies \nby measuring the temperature dependence of\n \nGilbert damping of \nPy in \nNb\n(100)/Py(20\n)/Nb(100)\n \n(F\nig. S7)\n \n[\n34\n]\n. \nSince \nP\ny\n \nis a \nFM\n \nmetal, the interface \nexchange \ninteraction\n \nwould be \nsignificant\n \nto \nstrongly\n \nor even \ncompletely\n \nsuppress\n \nthe\n \nsuperconducting gap\n \nof Nb\n \nat the \nNb/Py interface \n[\n17\n,\n20\n,\n43\n]\n. \nHence, no coherence peak of Gilbert damping is expected\n \nbased on the \ntheoretical\n \nstudy \nthat assumes the completely suppression of the gap at the interface\n \n[\n45\n]\n. \nBesid\nes, the strong suppression of the superconducting gap at the Nb/Py interface is in good \nagreement of previous tunnel spectroscopy measurements of vanishing superconducting gap at the \ninterface between Nb and Ni\n \n[\n4\n6\n]\n. \nHowever, \nfor \nFM insulating GdN, charge carriers from NbN do \nnot penetrate into GdN, thus not weakening the \nsuperconductivity\n \nat the interface\n, resulting the \nsurvival of the\n \nsuperconducting gap \n[\n47\n,\n48\n]\n. \n 7\n \n \nThe experimental \ndemonstration of the dynamic spin \nsusceptibility\n \nin \nsuperconducting\n \nthin \nfilms\n \nvia\n \nspin pumping \ncould be \nessential\n \nfor the field of superconductors.\n \nFor bulk SCs,\n \ndynamic \nspin susceptibility has been studied from the \ntemperature \ndependent \nspin \nrel\na\nxation rate via\n \nthe\n \nnuclear magnetic resonance\n \n[\n43\n,\n49\n,\n50\n]\n. \nIt also provides an avenue for  identifying unconventional \nsuperconducting paring mechanisms\n \n[\n51\n]\n, while limited to mostly bulk SCs due to low signal\n-\nto\n-\nnoise ratio\n \n[\n43\n,\n49\n-\n51\n]\n. S\npin pumping \nmethod \nhas much better \nsignal\n-\nto\n-\nnoise ratio\n \nso it could be \nvery \nsensitive\n \nto \nprobe the dynamic spin susceptibility\n \nof \nsuperconducting \nthin films\n.\n \nFurthermore, \nthe \nspin pumping offers a special \ntechnique \nto probe the \npair breaking strength, \nthe impurity s\npin\n-\norbit scattering, and the \nmagnetic proximity effect in the SC/FM junctions, which is \nalso \nof \nconsiderable interest\n \nfor the field of SC \nspintronics\n \n[\n1\n-\n3\n]\n. \n \nIV. CON\nCLUSION\n \nIn conclusion, \nthe \nspin \ndynamic\ns\n \nof \nsuperconducting \nNbN \nfilm\ns\n \nare\n \ninvestigated\n \nvia spin \npumping from \nan adjacent FM insulating layer\n. \nA profound coherence peak of the Gilbert damping \nis observed below T\nC\n,\n \nwhich indicates the dynamic spin injection \ninto the superconducting NbN \nfilms.\n \nOur results \npaves the way for\n \nfuture investigation of interface impurity spin\n-\norbit scattering, \nand pair breaking strength in FM/SC junctions, as w\nell as the spin dynamics in the \ninterfacial and \ntwo dimensional \ncrystalline SCs.\n \nIt may also be useful for the search for Majorana fermions in the \nSC/ferromagnetic insulator heterostructures.\n \n \n \nAcknowledgments\n \nWe acknowledge the fruitful discussion with Yuan Li and Tao Wu. Y.Y., Q.S., W.Y., Y.M., \nY.Y., X.C.X., and W.H.\n \nacknowledge the financial support from National Basic Research \nPrograms of China (973 program Grant Nos. 2015CB921104 and 2014CB920902) and National \nNatural Science Foundation of China (NSFC Grant No. 11574006). Y.T., J.P.C, and J.S.M. \nacknowledge the gra\nnts NSF DMR\n-\n1700137 and ONR N00014\n-\n16\n-\n1\n-\n2657. Y.T. also \nacknowledges the JSPS Overseas Research Fellowships and the Fundacion Seneca (Region de \nMurcia) posdoctoral fellowship (19791/PD/15). W.H. also acknowledges the support by the 1000 \nTalents Program for\n \nYoung Scientists of China.\n \n 8\n \n \nReferences\n:\n \n[1]\n \nA. I. Buzdin, Proximity effects in superconductor\n-\nferromagnet heterostructures.\n \nRev. Mod. \nPhys.\n \n77\n, 935 (2005).\n \n[2]\n \nJ. Linder and J. W. A. Robinson, Superconducting spintronics.\n \nNat. Phys.\n \n11\n, \n307 (2015).\n \n[3]\n \nM. Eschrig, Spin\n-\npolarized supercurrents for spintronics.\n \nPhys. 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Phys.\n \n101\n, 083901 \n(2007).\n \n \n \n \n \n \n 12\n \n \nFigure 1\n \n \nFig. 1. \nSpin pumping into the \nsuperconducting\n \nNbN thin films. \n(a)\n \nSchematic of the\n \ninterfacial\n \ns\n-\nd\n \nexchange interaction (\nJ\nsd\n) between the spins in NbN layer and the rotating magnetization of GdN \n13\n \n \nlayer under the ferromagnetic resonance\n \n(FMR)\n \nconditions. \n(b) \nThe magnetic moment as a \nfunction of the temperature. Inset: The magnetic hysteresis loop at \nT\n \n= \n5 K. \n(c) \nThe four\n-\nprobe \nresistance as a function of the temperature\n \nwith \nin\n-\nplane \nmagnetic field at 0 and 4000 Oe\n. \n(d) \nThe \ntypical \nFMR \nsignal\n \nmeasured at \nT\n \n= \n10 K and \nf\n \n= \n15 GHz. The red line indicates the Lorentz fitting \ncurve to obtain the half linewidth based on equation (1). \n(e)\n \nThe half linewidth\n \nas a function of the \nmicrowave\n \nfrequency at \nT\n \n= \n10K. The red solid line is \nthe fitting c\nurve based on spin\n-\nrelaxation \nmodel. The results in Fig. 1\n(b\n-\nd)\n \nare obtained on the NbN (10)/GdN (5)/NbN (10) sample.\n \n \n \n 14\n \n \nFigure 2\n \n \n \n \n \nFig. 2. \nSpin dynamics \nof \nthe \nsuperconducting\n \nNbN thin films probed via \nspin pumping\n. The \nnormalized four\n-\nprobe resistance\n \n(\na\n)\n \nand Gilbert damping (\nb\n) \nas a function of the temperature for \nthe samples of NbN (2)/GdN (5)/NbN (2) and NbN (10)/GdN (5)/NbN (10), respectively. \n \n \n15\n \n \n \n \nFigure 3\n \n \nFig. 3. \nThe GdN thickness effect on spin dynamics \nof \nthe superconducting \nNbN thin films\n. \n(a\n-\nb)\n \nThe Gilbert damping as a function of the temperature for the samples of NbN (\nt\n)/GdN (10)/NbN \n(\nt), \nand NbN (\nt\n)/GdN (30)/NbN (\nt\n). Insets\n: The normalized four\n-\nprobe resistance as a function of \nthe temperature for the samples of NbN (10)/GdN (10)/NbN (10) and NbN (10)/GdN (30)/NbN \n(10). \n(c)\n \nThe \nratio of \nthe \npeak Gilbert damping \nover \nthe value\n \nat \nT\n \n= \n~ T\nC\n \n(\npeak C\n/ (~ T )\n \n) \nas a \nfun\nction of the GdN layer thickness.\n \n16\n \n \nFigure 4\n \n \n \nFig. 4. \nThe interface\n-\ninduced Gilbert damping\n \nat the \nNbN/GdN interface\n. \n(a)\n \nThe Gilbert damping \nas a function of the GdN thickness for the samples of NbN (10)/GdN (d)/NbN (10) \nat \nT\n \n= \n11\n,\n \n8.5\n, \nand 6 K\n, respect\nively\n. \n(b)\n \nThe interface\n-\ninduced Gilbert damping as a function of temperature\n \nfor \nNbN (10)/GdN (d)/NbN (10)\n. \n \n \n \n1\n \nSupplementary Materials for:\n \n \nProbe of Spin Dynamics in Superconducting NbN Thin Films via \n \nSpin Pumping\n \n \nYunyan\n \nYao\n1,2\n,\n \nQi\n \nSong\n1,2\n,\n \nYota\n \nTakamura\n3,4\n,\n \nJuan\n \nPedro\n \nCascales\n3\n,\n \nWei\n \nYuan\n1,2\n,\n \nYang\n \nMa\n1,2\n,\n \nYu\n \nYun\n1,2\n,\n \nX.\n \nC.\n \nXie\n1,2\n,\n \nJagadeesh\n \nS.\n \nMoodera\n3,5\n,\n \nand\n \nWei\n \nHan\n1,2*\n \n1\nInternational Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China.\n \n2\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, Chi\nna.\n \n3\nPlasma Science and Fusion Center and Francis Bitter Magnet Laboratory, Massachusetts \nInstitute of Technology, Cambridge, MA 02139, USA.\n \n4\nSchool of Engineering, Tokyo Institute of Technology, Tokyo 152\n-\n8550, Japan.\n \n5\nDepartment of Physics, Massachusetts\n \nInstitute of Technology, Cambridge, MA 02139, USA\n \n*Correspondence to: \nweihan@pku.edu.cn (W.H.)\n \n 2\n \nS1.\n \nDetermination\n \nof\n \nGilbert\n \ndamping\n \nA\n \nnonlinear\n \nbehavior\n \nof\n \nhalf\n \nlinewidth\n \n(\nH\n\n)\n \nvs.\n \nmicrowave\n \nfrequency\n \n(\nf\n)\n \nis\n \nobserved\n \non\n \nthe\n \nNbN\n \n(t)/GdN\n \n(d)/NbN\n \n(t)\n \nsamples,\n \nas\n \nshown\n \nin\n \nFig.\n \nS2(a\n-\nc).\n \nTo\n \nour\n \nbest\n \nknowledge,\n \nt\nhis\n \nnonlinear\n \nbehavior\n \ncould\n \nbe\n \nattributed\n \nto\n \ntwo\n \nmechanisms,\n \nnamely\n \nspin\n-\nrelaxation\n \n[\n38\n,\n52\n]\n,\n \nand\n \ntwo\n-\nmagnon\n \nscattering\n \n[\n53\n,\n54\n]\n.\n \nFor\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism,\n \nH\n\n \nis\n \nrelated\n \nto\n \na\n \ntemperature\n-\ndependent\n \nspin\n-\nrelaxation\n \ntime\n \nconstant\n \n(\n\n),\n \nand\n \ncan\n \nbe\n \nexpressed\n \nby\n \n[\n38\n]\n:\n \n \n \n0\n2\n4 2\n1 (2 )\nf f\nH H A\nf\n  \n  \n    \n\n \n \n(S1)\n \nwhere\n \n0\nH\n\n \nis\n \nassociated\n \nwith\n \nthe\n \ninhomogeneous\n \nproperties,\n \n\n \nis\n \nthe\n \ngyromagnetic\n \nratio,\n \nand\n \nA\n \nis\n \nthe\n \nspin\n-\nrelaxation\n \ncoefficient.\n \nWhile,\n \nthe\n \nrelationship\n \nof\n \nH\n\n \nand\n \nf\n \nbased\n \non\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \ncan\n \nbe\n \nexpressed\n \nby\n \n[\n54\n,\n55\n]\n:\n \n \n\n\n\n\n2\n2\n1\n2\n2\n(2 ) 2 2\nsin\n(2 ) 2 2\neff eff\neff eff\nf M M\nH\nf M M\n    \n    \n\n \n \n \n \n(S2)\n \nwhere\n \n\n \nis\n \nthe\n \ntwo\n-\nmagnon\n \nscattering\n \ncoefficient,\n \nand\n \neff\nM\nis\n \nthe\n \neffective\n \nmagnetization.\n \nClearly,\n \nas\n \nshown\n \nin\n \nFig.\n \nS2(a\n-\nc),\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \nfails\n \nto\n \ngive\n \nreasonable\n \nfittings\n \n(blue\n \nlines)\n \nto\n \nour\n \nexperimental\n \nresults\n \n(black\n \nsquares).\n \nWhileas,\n \nour\n \nexperimental\n \nresults\n \nagree\n \nwell\n \nwith\n \nthe\n \nfitting\n \ncurves\n \nbased\n \non\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism\n \n(red\n \nlines).\n \nBesides,\n \nit\n \nis\n \nobserved\n \nthat\n \nthe\n \nnonlinearity\n \nof\n \nH\n\n \nvs.\n \nf\n \nincreases\n \nas\n \nthe\n \nGdN\n \nlayer\n \nthickness\n \nincreases,\n \nwhich\n \nis\n \nalso\n \nopposite\n \nto\n \nthe\n \ntwo\n-\nmagnon\n \nscat\ntering\n \nmechanism\n.\n \nSince\n \ntwo\n-\nmagnon\n \nscattering\n \narises\n \nfrom\n \nthe\n \ndefects\n \nat\n \nthe\n \ninterface,\n \na\n \nmore\n \nconspicuously\n \nnonlinear\n \nbehavior\n \nis\n \nexpected\n \nas\n \nthe\n \nFM\n \nfilms\n \nbecome\n \nthinner\n \n[\n55\n-\n57\n]\n.\n \nAnother\n \nfeature\n \nthat\nH\n\n \ndecreases\n \nas\n \nf\n \nincreases\n \n(Fig.\n \nS2(c))\n \ncannot\n \nbe\n \nexplained\n \nby\n \nthe\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \neither.\n \nHence,\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism\n \nis\n \nused\n \nto\n \nanalyze\n \nthe\n \nexperimental\n \nresults\n \nof\n \nH\n\n \nvs.\n \nf\n,\n \nand\n \nGilbert\n \ndamping\n \ncould\n \nbe\n \nobtained\n \nsubsequently.\n \n \nS2.\n \nRole\n \nof\n \nthe\n \neffective\n \nmagnetization\n \n 3\n \nIn\n \nprevious\n \nstudies,\n \nit\n \nha\ns\n \nbeen\n \nsuggested\n \nthat\n \nthe\n \nenhancement\n \nof\n \nthe\n \nGilbert\n \ndamping\n \ncould\n \nbe\n \nalso\n \nassociated\n \nwith\n \nthe\n \nchange\n \nof\n \neff\nM\n \n[\n35\n,\n58\n]\n.\n \nTo\n \nrule\n \nout\n \nthe\n \neffect\n \nof\n \neff\nM\n,\n \nwe\n \nsystemically\n \nstudy\n \nthe\n \ntemperature\n \ndependence\n \nof\n \neff\nM\nbetween\n \nthe\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nand\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples.\n \nThe\n \n4\neff\nM\n\n \nis\n \nbased\n \non\n \nthe\n \nKittel\n \nformula\n \n[\n59\n]\n:\n \n \n \n \n \n \n \n1\n2\n( )[ ( 4 )]\n2\nres res eff\nf H H M\n\n\n\n \n \n \n \n               \n(S3)\n \nwhere\n \nf\n \nis\n \nthe\n \nmicrowave\n \nfrequency,\n \nand\n \nres\nH\nis\n \nthe\n \nresonance\n \nmagnetic\n \nfield.\n \nAs\n \nshown\n \nin\n \nfig.\n \nS4(a),\n \nthe\n \nfitting\n \ncurve\n \n(red\n \nline)\n \nagrees\n \nwell\n \nwith\n \nthe\n \nexperimental\n \nresults.\n \nSimilar\n \ntemperature\n \ndependences\n \nof\n \n4\neff\nM\n\nfor\n \nthe\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nand\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nare\n \nobserved\n \n(Figs.\n \nS4(c\n-\nd)).\n \nThis\n \nobservation\n \nconfirms\n \nthat\n \nthe\n \nobserved\n \ncoherence\n \npeak\n \nof\n \nthe\n \nGilbert\n \ndamping\n \nat\n \nT\n \n=\n \n~\n \n8.5\n \nK\n \nis\n \nnot\n \nrelated\n \nto\n \nthe\n \neffective\n \nmagnetization.\n \n \nS3.\n \nDetermination\n \nof\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nPrevious\n \nstudies\n \nhave\n \nidentified\n \nthe\n \nmajor\n \nsources\n \nthat\n \ncontribute\n \nto\n \nthe\n \nGilbert\n \ndamping\n \nof\n \nthe\n \nFM\n \nfilms,\n \nincluding\n \nbulk\n \ndamping\n \n(\nB\n\n)\n \nand\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \n(\nS\n\n)\n \nthat\n \nis\n \ndue\n \nto\n \nspin\n \npumping\n \nand\n \nother\n \ninterface\n \neffects\n \n[\n20\n-\n22\n]\n.\n \nAs\n \nshown\n \nin\n \nFig.\n \n4(a)\n \nand\n \nFig.\n \nS6(a),\n \na\n \nslightly\n \nincrease\n \nof\n \nthe\n \nGilbert\n \ndamping\n \nis\n \nobserved\n \nas\n \nthe\n \nthickness\n \nof\n \nthe\n \nGdN\n \nlayer\n \nincreases\n \nfrom\n \n20\n \nto\n \n30\n \nnm.\n \nThis\n \nfeature\n \ncould\n \nbe\n \nassociated\n \nwith\n \nnon\n-\nhomogeneous\n \nfield\n \nexcitation\n \nwhen\n \nthe\n \nw\nidth\n \nof\n \ncoplanar\n \nwave\n \nguide\n \nis\n \nvery\n \nsmall\n \ncompared\n \nto\n \nthe\n \nsamples\n \n[\n60\n,\n61\n]\n,\n \nand\n \nthe\n \neddy\n \ncurrent\n \nloss\n \neffect\n \n[\n62\n]\n.\n \nBoth\n \nof\n \nthese\n \neffects\n \ncontribute\n \nto\n \nthe\n \nGilbert\n \ndamping\n \nthat\n \nis\n \nproportional\n \nto\n \nd\n2\n.\n \nHence,\n \nthe\n \ntotal\n \nGilbert\n \ndamping\n \ncan\n \nbe\n \nexpressed\n \nby:\n \n2\n1\n( ) '\nB S\nd\nd\n   \n  \n \n \n \n \n(S4)\n \nAnd\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \nda\nmping\n \nand\n \nbulk\n \ndamping\n \ncould\n \nbe\n \nobtained\n \nsubsequently.\n \nBased\n \non\n \nthe\n \nfitted\n \nresults,\n \nB\n\n \nis\n \nsignificantly\n \nsmall\n \ncompared\n \nto\n \nS\n\n.\n \nThe\n \n~10%\n \nerror\n \nbar\n \nof\n \nS\n\n \nat\n \neach\n \ntemperature\n \nmakes\n \nit\n \nhard\n \nto\n \nobtain\n \nthe\n \naccurate\n \nvalues\n \nof\n \nB\n\n,\n \nwhich\n \nrequire\n \nfuture\n \nstudies.\n \nComparing\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nfor\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nand\n \nNbN\n 4\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nsamples\n \n(Fig.\n \n4(b)\n \nand\n \nFi\ng.\n \nS6(b)),\n \nthe\n \nprofound\n \ncoherence\n \npeak\n \nis\n \nonly\n \nobserved\n \non\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nat\n \n~\n \n8.5\n \nK.\n \nThis\n \nobservation\n \nunambiguously\n \ndemonstrates\n \nthat\n \nthe\n \norigin\n \nof\n \nthe\n \ncoherence\n \npeak\n \nin\n \nthe\n \nGilbert\n \ndamping\n \narises\n \nfrom\n \nthe\n \ninterfacial\n \ns\n-\nd\n \nexchange\n \ninter\naction\n \nbetween\n \nthe\n \nmagnetic\n \nmoments\n \nof\n \nGdN\n \nand\n \nthe\n \nspins\n \nof\n \nthe\n \nquasiparticles\n \nof\n \nNbN\n \nin\n \nits\n \nsuperconducting\n \nstate.\n \nS4.\n \nThe\n \nimpact\n \nof\n \nspin\n \npumping\n \nexperiments\n \non\n \nthe\n \nT\nC\n \nBeyond\n \nthe\n \nmagnetic\n \nfield\n \neffect\n \non\n \nthe\n \nsuperconducting\n \nfilms\n \n(Fig.\n \n1(c)),\n \nthe\n \nspin\n \npumping\n \nexperiments\n \nwith\n \nvarious\n \nmicrowave\n \nexcitations\n \ndo\n \nnot\n \naffect\n \nthe\n \nsuperconducting\n \nfilms\n \neither.\n \nAs\n \nshown\n \nin\n \nFigs.\n \nS8(a)\n \nand\n \nS8(b),\n \nthe\n \nT\nC\n \nexhibit\n \na\n \nsmall\n \nvariation\n \np\nrobed\n \non\n \nthe\n \ntypical\n \nsample\n \nNbN\n \n(10)/GdN\n \n(5)/NbN\n \n(10).\n \n \n \n \n 5\n \nFigure\n \nS\n1\n \n \n \n \n \nFigure\n \nS\n1.\n \nC\nrystalline\n \nstructure\n \nof\n \nthe\n \n(111)\n-\ntextured\n \nGdN\n \nand\n \nNbN\n \nfilms.\n \n(a\n-\nb)\n,\n \nXRD\n \nresults\n \nmeasured\n \non\n \nthe\n \nNbN\n \n(10)/GdN\n \n(50)/NbN\n \n(10)\n \nsample.\n \nTwo\n \nmain\n \nobservable\n \npeaks\n \ncorrespond\n \nto\n \nGdN\n \n(111)\n \nand\n \nNbN\n \n(111).\n \n \n \n \n6\n \nFigure\n \nS\n2\n \n \n \n \nFig\nure\n \nS\n2\n.\n \nComparison\n \nof\n \nthe\n \nspin\n-\nrelaxation\n \nand\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanisms\n.\n \n(a\n-\nc)\n \nThe\n \nexperimental\n \nresults\n \nof\n \n\nH\n \nvs.\n \nf\n \nat\n \nT\n \n=\n \n10\n \nK\n \nand\n \nthe\n \nfitting\n \ncurves\n \n(red\n/green\n \nlines\n \nfor\n \nthe\n \nspin\n \nrelaxation/\ntwo\n-\nmagnon\n \nscattering)\n \non\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nwith\n \nd\n \n=\n \n5,\n \n10,\n \nand\n \n30\n \nnm\n \nrespectively.\n \n \n7\n \n \nFigure\n \nS\n3\n \n \n \n \nFigure\n \nS\n3.\n \nHalf\n \nlinewidth\n \nas\n \na\n \nfunction\n \nof\n \ntemperature\n \nfor\n \nNbN\n \n(10\n)/GdN\n \n(\n5)/NbN\n \n(10\n)\n \n(a)\n \nand\n \nNbN\n \n(10)/GdN\n \n(10)/NbN\n \n(10)\n \n(b)\n \nsamples.\n \n \n \n \n8\n \nFigure\n \nS\n4\n \n \n \n \nFigure\n \nS\n4.\n \nC\nharacterization\n \nof\n \neffective\n \nmagnetization.\n \n(a)\n \nThe\n \ndetermination\n \nof\n \neff\nM\n \nvia\n \nKittel\n \nf\normula\n \n(red\n \nline)\n \nfrom\n \nthe\n \nexperimental\n \nresults\n \nof\n \nf\n \nvs.\n \nres\nH\n.\n \n(b\n-\nd)\n,\n \n4\neff\nM\n\n \nas\n \na\n \nfunction\n \nof\n \ntemperature\n \nfor\n \nthe\n \nNbN\n \n(\n2\n)/GdN\n \n(d)/N\nbN\n \n(2\n)\n \n(\ngreen\n \ncircles)\n \nand\n \nNbN\n \n(\n10\n)/GdN\n \n(d)/N\nbN\n \n(10\n)\n \nsamples\n \n(red\n \ncircles)\n \nwith\n \nd\n \n=\n \n5\n \nnm\n \n(\nb\n)\n,\n \n10\n \nnm\n \n(\nc\n),\n \nand\n \n30\n \nnm\n \n(\nd\n),\n \nrespectively.\n \n \n \n9\n \n \nFigure\n \nS5\n \n \n \n \n \n \n \n \nFigure\n \nS\n5.\n \nT\nC\n \nof\n \nNbN\n \nas\n \na\n \nfunction\n \nof\n \nthe\n \nGdN\n \nthickness\n.\n \nThese\n \nresults\n \nare\n \nobtained\n \non\n \nthe\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n.\n \n \n \n \n10\n \nFigure\n \nS\n6\n \n \n \nFigure\n \nS6.\n \nI\nnterface\n-\ninduced\n \nGilbert\n \ndamping\n \nin\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n.\n \n(a)\n \nThe\n \nGilbert\n \ndamping\n \nconstant\n \nas\n \na\n \nfunction\n \nof\n \nthe\n \nGdN\n \nthickness\n \nat\n \nT\n \n=\n \n11,\n \n8,\n \nand\n \n6\n \nK\n,\n \nrespectively\n.\n \nSolid\n \nlines\n \nare\n \nthe\n \nfitting\n \ncurves\n \nbased\n \non\n \nthe\n \nequation\n \n(S4)\n.\n \n(b)\n \nThe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nas\n \na\n \nfunction\n \nof\n \ntemperature.\n \n \n \n11\n \nFigure\n \nS7\n \n \n \n \n \n \nFigure\n \nS\n7.\n \nTemperature\n \ndependence\n \nof\n \nGilbert\n \ndamping\n \nof\n \nPy\n \nin\n \nthe\n \nNb\n \n(100)/Py\n \n(20)/Nb\n \n(100)\n \nsample.\n \n \n \n12\n \nFigure\n \nS8\n \n \n \n \n \n \nFigure\n \nS\n8\n.\n \nThe\n \nimpact\n \nof\n \nspin\n \npumping\n \nexperiments\n \non\n \nthe\n \nT\nC\n \non\n \nthe\n \ntypical\n \nNbN\n \n(10\n)/GdN\n \n(5\n)/NbN\n \n(10)\n \nsample\n.\n \n(a)\n \nThe\n \nfour\n-\nprobe\n \nresistance\n \nvs.\n \ntemperature\n \nunder\n \n4000\n \nOe\n \nwith\n \nvarious\n \nmicrowave\n \nexcitation\n \nfrequencies.\n \n(b)\n \nThe\n \nfour\n-\nprobe\n \nresistance\n \nvs.\n \ntemperature\n \naround\n \nthe\n \nFMR\n \nresonance\n \nconditions\n \nof\n \nGdN\n.\n \n \n \n \n"    },    {        "title": "1507.03075v1.Realization_of_the_thermal_equilibrium_in_inhomogeneous_magnetic_systems_by_the_Landau_Lifshitz_Gilbert_equation_with_stochastic_noise__and_its_dynamical_aspects.pdf",        "content": "Realization of the thermal equilibrium in inhomogeneous\nmagnetic systems by the Landau-Lifshitz-Gilbert equation with\nstochastic noise, and its dynamical aspects\nMasamichi Nishino1\u0003and Seiji Miyashita2;3\n1Computational Materials Science Center,\nNational Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan\n2Department of Physics, Graduate School of Science,\nThe University of Tokyo, Bunkyo-Ku, Tokyo, Japan\n3CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan\n(Dated: July 14, 2015)\n1arXiv:1507.03075v1  [cond-mat.mtrl-sci]  11 Jul 2015Abstract\nIt is crucially important to investigate e\u000bects of temperature on magnetic properties such as\ncritical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz-\nGilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties.\nApproaches of Langevin noises have been developed to introduce the temperature e\u000bect into the\nLLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state,\nthe system parameters must satisfy some condition known as the \ructuation-dissipation relation.\nIn inhomogeneous magnetic systems in which spin magnitudes are di\u000berent at sites, the condition\nrequires that the ratio between the amplitude of the random noise and the damping parameter\ndepends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous mag-\nnetic systems, we systematically showed agreement between the stationary state of the stochastic\nLLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in\nvarious magnetic systems including dipole-dipole interactions. We demonstrated how violations of\nthe condition result in deviations from the true equilibrium state. We also studied the characteris-\ntic features of the dynamics depending on the choice of the parameter set. All the parameter sets\nsatisfying the condition realize the same stationary state (equilibrium state). In contrast, di\u000berent\nchoices of parameter set cause seriously di\u000berent relaxation processes. We show two relaxation\ntypes, i.e., magnetization reversals with uniform rotation and with nucleation.\nPACS numbers: 75.78.-n 05.10.Gg 75.10.Hk 75.60.Ej\n2|||||||||||||||||||||||||-\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation1has been widely used in the study of dy-\nnamical properties of magnetic systems, especially in micromagnetics. It contains a relax-\nation mechanism by a phenomenological longitudinal damping term. The Landau-Lifshitz-\nBloch (LLB) equation2contains, besides the longitudinal damping, a phenomenological\ntransverse damping and the temperature dependence of the magnetic moment are taken\ninto account with the aid of the mean-\feld approximation. Those equations work well in\nthe region of saturated magnetization at low temperatures.\nThermal e\u000bects are very important to study properties of magnets, e.g., the amount of\nspontaneous magnetization, hysteresis nature, relaxation dynamics, and the coercive force in\npermanent magnets. Therefore, how to control temperature in the LLG and LLB equations\nhas been studied extensively. To introduce temperature in equations of motion, a coupling\nwith a thermal reservoir is required. For dynamics of particle systems which is naturally\nexpressed by the canonical conjugated variables, i.e., ( q;p), molecular dynamics is performed\nwith a Nose-Hoover (NH) type reservoir3{5or a Langevin type reservoir6. However, in the\ncase of systems of magnetic moments, in which dynamics of angular momenta is studied, NH\ntype reservoirs are hardly used due to complexity7. On the other hand, the Langevin type\nreservoirs have been rather naturally applied2,8{18although multiplicative noise19requires the\nnumerical integration of equations depending on the interpretation, i.e., Ito or Stratonovich\ntype.\nTo introduce temperature into a LLG approach by a Langevin noise, a \ructuation-\ndissipation relation is used, where the temperature is proportional to the ratio between\nthe strength of the \ructuation (amplitude of noise) and the damping parameter of the\nLLG equation. For magnetic systems consisting of uniform magnetic moments, the ratio is\nuniquely given at a temperature and it has been often employed to study dynamical prop-\nerties, e.g., trajectories of magnetic moments of nano-particles8, relaxation dynamics in a\nspin-glass system20or in a semiconductor21. The realization of the equilibrium state by\nstochastic LLG approaches by numerical simulations is an important issue, and it has been\ncon\frmed in some cases of the Heisenberg model for uniform magnetic moments.22,23\n3In general cases, however, magnetic moments in atomic scale have various magnitudes of\nspins. This inhomogeneity of magnetization is important to understand the mechanisms of\nnucleation or pinning.24{28To control the temperature of such systems, the ratio between the\namplitude of noise and the damping parameter depends on the magnetic moment at each\nsite. In order to make clear the condition for the realization of the canonical distribution\nas the stationary state in inhomogeneous magnetic systems, we review the guideline of the\nderivation of the condition in the Fokker-Planck equation formalism in the Appendix A.\nSuch a generalization of the LLG equation with a stochastic noise was performed to study\nproperties of the alloy magnet GdFeCo29, in which two kinds of moments exist. They ex-\nploited a formula for the noise amplitude, which is equivalent to the formula of our condition\nA (see Sec II). They found surprisingly good agreements of the results between the stochas-\ntic LLG equation and a mean-\feld approximation. However, the properties in the true\ncanonical distribution is generally di\u000berent from those obtained by the mean-\feld analysis.\nThe LLG and LLB equations have been often applied for continuous magnetic systems or\nassemblies of block spins in the aim of simulation of bulk systems, but such treatment of the\nbulk magnets tend to overestimate the Curie temperature11, and it is still under develop-\nment to obtain properly magnetization curves in the whole temperature region2,11,17,18. The\nin\ruence of coarse graining of block spin systems on the thermal properties is a signi\fcant\ntheme, which should be clari\fed in the future. To avoid such a di\u000eculty, we adopt a lattice\nmodel, in which the magnitude of the moment is given at each magnetic site.\nWithin the condition there is some freedom of the choice of parameter set. In the present\npaper, in particular, we investigate the following two cases of parameter sets, i.e., case A,\nin which the LLG damping constant is the same in all the sites and the amplitude of the\nnoise depends on the magnitude of the magnetic moment at each site, and case B, in which\nthe amplitude of the noise is the same in all the sites and the damping constant depends on\nthe magnitude of the moment. (see Sec II.). We con\frm the realization of the equilibrium\nstate, i.e., the canonical distribution in various magnetic systems including critical region by\ncomparison of magnetizations obtained by the LLG stochastic approach with those obtained\nby standard Monte Carlo simulations, not by the mean-\feld analysis. We study systems\nwith not only short range interactions but also dipole-dipole interactions, which causes\nthe demagnetizing \feld statically. We \fnd that di\u000berent choices of the parameter set which\nsatis\fes the \ructuation-dissipation relation give the same stationary state (equilibrium state)\n4even near the critical temperature. We also demonstrate that deviations from the relation\ncause systematic and signi\fcant deviations of the results.\nIn contrast to the static properties, we \fnd that di\u000berent choices of parameter set cause\nserious di\u000berence in the dynamics of the relaxation. In particular, in the rotation type\nrelaxation in isotropic spin systems, we \fnd that the dependences of the relaxation time on\nthe temperature in cases A and B show opposite correlations as well as the dependences of the\nrelaxation time on the magnitude of the magnetic moment. That is, the relaxation time of\nmagnetization reversal under an unfavorable external \feld is shorter at a higher temperature\nin case A, while it is longer in case B. On the other hand, the relaxation time is longer for\na larger magnetic moment in case A, while it is shorter in case B. We also investigate the\nrelaxation of anisotropic spin systems and \fnd that the metastability strongly a\u000bects the\nrelaxation at low temperatures in both cases. The system relaxes to the equilibrium state\nfrom the metastable state by the nucleation type of dynamics. The relaxation time to the\nmetastable state and the decay time of the metastable state are a\u000bected by the choice of\nthe parameter set.\nThe outline of this paper is as follows. The model and the method in this study are ex-\nplained in Sec II. Magnetization processes as a function of temperature in uniform magnetic\nsystems are studied in Sec III. Magnetizations as a function of temperature for inhomoge-\nneous magnetic systems are investigated in Sec. IV, in which not only exchange interactions\n(short-range) but also dipole interactions (long-range) are taken into account. In Sec. V\ndynamical aspects with the choice of the parameter set are considered, and the dependences\nof the relaxation process on the temperature and on the magnitude of magnetic moments\nare also discussed. The relaxation dynamics via a metastable state is studied in Sec. VI.\nSec. VII is devoted to summary and discussion. In Appendix A the Fokker-Planck equation\nfor inhomogeneous magnetic systems is given both in Stratonovich and Ito interpretations,\nand Appendix B presents the numerical integration scheme in this study.\nII. MODEL AND METHOD\nAs a microscopic spin model, the following Hamiltonian is adopted,\nH=\u0000X\nhi;jiJi;jSi\u0001Sj\u0000X\niDA\ni(Sz\ni)2\u0000X\nihi(t)Sz\ni+X\ni6=kC\nr3\nik\u0010\nSi\u0001Sk\u00003(rik\u0001Si)(rik\u0001Sk)\nr2\nik\u0011\n:(1)\n5Here we only consider a spin angular momentum Sifor a magnetic moment Miat each\nsite (iis the site index) and regard Mi=Siignoring the di\u000berence of the sign between\nthem and setting a unit: g\u0016B= 1 for simplicity, where gis the g-factor and \u0016Bis the Bohr\nmagneton30. Interaction Ji;jbetween the ith andjth magnetic sites indicates an exchange\ncoupling,hi;jidenotes a nearest neighbor pair, DA\niis an anisotropy constant for the ith\nsite,hiis a magnetic \feld applied to the ith site, and the \fnal term gives dipole interactions\nbetween the ith andkth sites whose distance is ri;k, whereC=1\n4\u0019\u00160is de\fned using the\npermeability of vacuum \u00160.\nThe magnitude of the moment Miis de\fned as Mi\u0011jMij, which is not necessarily\nuniform but may vary from site to site. In general, the damping parameter may also have\nsite dependence, i.e., \u000bi, and thus the LLG equation at the ith site is given by\nd\ndtMi=\u0000\rMi\u0002He\u000b\ni+\u000bi\nMiMi\u0002dMi\ndt; (2)\nor in an equivalent formula:\nd\ndtMi=\u0000\r\n1 +\u000b2\niMi\u0002He\u000b\ni\u0000\u000bi\r\n(1 +\u000b2\ni)MiMi\u0002(Mi\u0002He\u000b\ni); (3)\nwhere\ris the gyromagnetic constant. Here He\u000b\niis the e\u000bective \feld at the ith site and\ndescribed by\nHe\u000b\ni=\u0000@\n@MiH(M1;\u0001\u0001\u0001;MN;t) (4)\n, which contains \felds from the exchange and the dipole interactions, the anisotropy, and\nthe external \feld.\nWe introduce a Langevin-noise formalism for the thermal e\u000bect. There have been several\nways for the formulation to introduce a stochastic term into the LLG equation. The stochas-\ntic \feld can be introduced into the precession term and/or damping term8,9,11. Furthermore,\nan additional noise term may be introduced10,12. In the present study we add the random\nnoise to the e\u000bective \feld He\u000b\ni!He\u000b\ni+\u0018iand we have\nd\ndtMi=\u0000\r\n1 +\u000b2\niMi\u0002(He\u000b\ni+\u0018i)\u0000\u000bi\r\n(1 +\u000b2\ni)MiMi\u0002(Mi\u0002(He\u000b\ni+\u0018i)); (5)\nwhere\u0018\u0016\niis the\u0016(=1,2 or 3 for x,yorz) component of the white Gaussian noise applied at\ntheith site and the following properties are assumed:\nh\u0018\u0016\nk(t)i= 0;h\u0018\u0016\nk(t)\u0018\u0017\nl(s)i= 2Dk\u000ekl\u000e\u0016\u0017\u000e(t\u0000s): (6)\n6We call Eq. (5) stochastic LLG equation. We derive a Fokker-Planck equation6,8for\nthe stochastic equation of motion in Eq. (5) in Stratonovich interpretation, as given in\nappendix A,\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni) (7)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nHere we demand that the distribution function at the stationary state ( t!1 ) of the\nequation of motion (Eq. (7)) agrees with the canonical distribution of the system (Eq. (1))\nat temperature T, i.e.,\nPeq(M1;\u0001\u0001\u0001;MN)/exp\u0010\n\u0000\fH(M1;\u0001\u0001\u0001;MN)\u0011\n; (8)\nwhere\f=1\nkBT.\nConsidering the relation\n@\n@MiPeq(M1;\u0001\u0001\u0001;MN) =\fHe\u000b\niPeq(M1;\u0001\u0001\u0001;MN); (9)\nwe \fnd that if the following relation\n\u000bi\nMi\u0000\rDi\f= 0 (10)\nis satis\fed at each site i, the canonical distribution in the equilibrium state is assured.\nWhen the magnetic moments are uniform, i.e., the magnitude of each magnetic moment\nis the same and Mi=jMij=M, the parameters \u000biandDiare also uniform \u000bi=\u000band\nDi=Dfor a given T. However, when Miare di\u000berent at sites, the relation (10) must be\nsatis\fed at each site independently. There are several ways of the choice of the parameters\n\u000biandDito satisfy this relation. Here we consider the following two cases: A and B.\nA: we take the damping parameter \u000bito be the same at all sites, i.e., \u000b1=\u000b2=\u0001\u0001\u0001=\u000bN\u0011\n\u000b. In this case the amplitude of the random \feld at the ith site should be\nDi=\u000b\nMikBT\n\r/1\nMi: (11)\nB: we take the amplitude of the random \feld to be the same at all sites, i.e., D1=D2=\n\u0001\u0001\u0001=DN\u0011D. In this case the damping parameter at the ith site should be\n\u000bi=D\rMi\nkBT/Mi: (12)\n700.20.40.60.81\n0123456m\nTFIG. 1: (color online) Comparison of the temperature dependence of min the stationary state\nbetween the stochastic LLG method and the Langevin function (green circles). Crosses and boxes\ndenotemin case A ( \u000b= 0:05) and case B ( D= 1:0), respectively. In the stochastic LLG\nsimulation \u0001 t= 0:005 was set and 80000 time steps (40,000 steps for equilibration and 40,000\nsteps for measurement) were employed. The system size N=L3= 103was adopted.\nWe study whether the canonical distribution is realized in both cases by comparing data\nobtained by the stochastic LLG method with the exact results or with corresponding data\nobtained by Monte Carlo simulations. We set the parameters \r= 1 andkB= 1 hereafter.\nIII. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN HOMOGE-\nNEOUS MAGNETIC SYSTEMS\nA. Non-interacting magnetic moments\nAs a \frst step, we check the temperature e\u000bect in the simplest case of non-interacting\nuniform magnetic moments, i.e., Ji;j= 0,DA\ni= 0,C= 0 in Eq. (1) and Mi=M(or\nSi=S), where\u000bandDhave no site i-dependence. In this case the magnetization in a\nmagnetic \feld ( h) at a temperature ( T) is given by the Langevin function:\nm=1\nNhNX\ni=1Sz\nii=M \ncoth\u0010hM\nkBT\u0011\n\u0000kBT\nhM!\n: (13)\nWe compare the stationary state obtained by the stochastic LLG method and Eq. (13).\n8We investigate m(T) ath= 2 forM= 1. Figure 1 shows m(T) when\u000b= 0:05 is \fxed (case\nA) and when D= 1:0 is \fxed (case B). We \fnd a good agreement between the results of\nthe stochastic LLG method and the Langevin function in the whole temperature region as\nlong as the relation (10) is satis\fed. Numerical integration scheme is given in Appendix B.\nThe time step of \u0001 t= 0:005 and total 80000 time steps (40000 steps for equilibration and\n40000 steps for measurement) were adopted.\nB. Homogeneous magnetic moments with exchange interactions\nNext, we investigate homogenous magnetic moments ( Mi=jMij=M) in three di-\nmensions. The following Hamiltonian ( C= 0,Ji;j=J,DA\ni=DA, andh(t) =hin Eq.\n(1)):\nH=\u0000X\nhi;jiJSi\u0001Sj\u0000X\niDA(Sz\ni)2\u0000X\nihSz\ni (14)\nis adopted.\nThere is no exact formula for magnetization ( m) as a function of temperature for this\nsystem, and thus a Monte Carlo (MC) method is applied to obtain reference magnetization\ncurves for the canonical distribution because MC methods have been established to obtain\n\fnite temperature properties for this kind of systems in the equilibrium state. Here we\nemploy a MC method with the Metropolis algorithm to obtain the temperature dependence\nof magnetization.\nIn order to check the validity of our MC procedure, we investigated magnetization\ncurves as functions of temperature (not shown) with system-size dependence for the three-\ndimensional classical Heisenberg model ( DA= 0 andh= 0 in Eq. (14)), and con\frmed that\nthe critical temperature agreed with past studies31, wherekBTc= 1:443Jfor the in\fnite\nsystem size with M= 1.\nWe givem(T) for a system of M= 2 with the parameters J= 1,h= 2 andDA= 1:0\nfor cases A and B in Fig 2. The system size was set N=L3= 103and periodic boundary\nconditions (PBC) were used. Green circles denote mobtained by the Monte Carlo method.\nAt each temperature ( T) 10,000 MC steps (MCS) were applied for the equilibration and\nfollowing 10,000\u000050,000 MCS were used for measurement to obtain m. Crosses and boxes\ndenotemin the stationary state of the stochastic LLG equation in case A ( \u000b= 0:05) and\nin case B ( D= 1:0), respectively. Here \u0001 t= 0:005 was set and 80000 steps (40000 for\n900.511.52\n0 5 10 15 20 25m\nTFIG. 2: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo\nmethod (green circles) and the stochastic LLG method in the homogeneous magnetic system with\nM= 2. Crosses and boxes denote case A with \u000b= 0:05 and case B with D= 1:0, respectively.\ntransient and 40000 for measurement) were used to obtain the stationary state of m. The\nm(T) curves show good agreement between the MC method and the stochastic LLG method\nin both cases. We checked that the choice of the initial state for the MC and the stochastic\nLLG method does not a\u000bect the results. The dynamics of the stochastic LLG method leads\nto the equilibrium state at temperature T.\nIV. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN INHOMO-\nGENEOUS MAGNETIC SYSTEMS\nA. Inhomogeneous magnetic moments with exchange interactions\nHere we study a system which consists of two kinds of magnitudes of magnetic moments.\nThe Hamiltonian (14) is adopted but the moment Mi=jMijhasi-dependence. We investi-\ngate a simple cubic lattice composed of alternating M= 2 andM= 1 planes (see Fig. 3 (a)),\nwhereJ= 1,h= 2 andDA= 1:0 are applied. We consider two cases A and B mentioned\nin Sec. II.\nThe reference of m(T) curve was obtained by the MC method and is given by green\ncircles in Figs. 3 (b) and (c). In the simulation, at each temperature ( T) 10,000 MCS were\napplied for the equilibration and following 10,000 \u000050,000 MCS were used for measurement.\n10(a)\n00.511.5\n05 1 0 1 5 2 0m\nT(b)\n00.511.5\n05 1 0 1 5 2 0m\nT(c)FIG. 3: (color online) (a) A part of the system composed of alternating M= 2 (red long\narrows) and M= 1 (short blue arrows) layers. (b) Comparison of temperature ( T) dependence of\nmbetween the Monte Carlo method (green circles) and the stochastic LLG method for \u000b= 0:05.\n\u0001t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were employed.\nCrosses denote mwhenDi=D(Mi)\u0011\u000b\nMikBT\n\rwas used. Triangles and Diamonds are mfor\nDi=D(1) =\u000bkBT\n\rfor alliandDi=D(2) =\u000b\n2kBT\n\rfor alli, respectively. (c) Comparison\nof temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the\nstochastic LLG method for D= 1:0. \u0001t= 0:005 and 80,000 steps (40,000 for transient time and\n40,000 for measurement) were employed. Crosses denote mwhen\u000bi=\u000b(Mi)\u0011D\rMi\nkBTwas used.\nTriangles are mfor\u000bi=\u000b(Mi= 1) =D\r\u00021\nkBTfor alliand Diamonds are mfor\u000bi=\u000b(2) =D\r\u00022\nkBT\nfor alli.\n11The system size N=L3= 103was adopted with PBC. In case A, \u000b(= 0:05) is common for\nall magnetic moments in the stochastic LLG method and Mi(orSi) dependence is imposed\nonDiasDi=D(Mi)\u0011\u000b\nMikBT\n\r. In case B, D= 1:0 is common for all magnetic moments in\nthe stochastic LLG method and \u000bi=\u000b(Mi)\u0011D\rMi\nkBT. Crosses in Figs. 3 (b) and (c) denote\nmby the stochastic LLG method for cases A and B, respectively. For those simulations\n\u0001t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were\nemployed at each temperature. In both Figs. 3 (b) and (c), we \fnd good agreement between\nm(T) by the stochastic LLG method (crosses) and m(T) by the MC method (green circles).\nNext, we investigate how the results change if we take wrong choices of parameters. We\nstudym(T) when a uniform value Di=Dfor case A ( \u000bi=\u000bfor case B) is used for all\nspins, i.e., for both Mi= 1 andMi= 2. IfD(Mi= 2) =\u000b\n2kBT\n\ris used for all spins, m(T)\nis shown by Diamonds in Fig. 3 (b), while if D(Mi= 1) =\u000bkBT\n\ris applied for all spins,\nm(T) is given by triangles in Fig. 3 (b). In the same way, we study m(T) for a uniform\nvalue of\u000b. In Fig. 3 (c) triangles and diamonds denote m(T) when\u000bi=\u000b(Mi= 1) and\n\u000bi=\u000b(Mi= 2) are used, respectively. We \fnd serious di\u000berence in m(T) when we do not\nuse correct Mi-dependent choices of the parameters. The locations of triangle (diamond) at\neach temperature Tare the same in Figs. 3 (a) and (b), which indicates that if the ratio\n\u000b=D is the same in di\u000berent choices, the same steady state is realized although this state is\nnot the true equilibrium state for the inhomogeneous magnetic system. Thus we conclude\nthat to use proper relations of Mi-dependence of Dior\u000biis important for m(T) curves of\ninhomogeneous magnetic systems and wrong choices cause signi\fcant deviations.\nB. Critical behavior of Inhomogeneous magnetic moments\nIn this subsection, we examine properties near the critical temperature. Here we adopt\nthe case ofh= 0 andDA= 0 in the same type of lattice with M= 1 and 2 as Sec. IV A. We\ninvestigate both cases of the temperature control (A and B). The Hamiltonian here has O(3)\nsymmetry and mis not a suitable order parameter. Thus we de\fne the following quantity\nas the order parameter31:\nma=q\nm2\nx+m2\ny+m2\nz; (15)\n1200.511.5\n0123456ma\nTFIG. 4: (color online) Comparison of temperature ( T) dependence of mabetween the MC method\n(green circles) and the stochastic LLG method for the system of inhomogeneous magnetic moments.\nN=L3= 203. PBC were used. In the MC method 10,000 MCS and following 50,000 MCS were\nused for equilibration and measurement at each temperature, respectively. The stochastic LLG\nmethod was performed in case A with \u000b= 0:05 (croses) and in case B with D= 1:0 (diamonds).\nHere \u0001t= 0:005 was applied and 240,000 steps were used (40,000 for transient and 200,000 for\nmeasurement).\nwhere\nmx=1\nNhNX\ni=1Sx\nii; my=1\nNhNX\ni=1Sy\nii;andmz=m=1\nNhNX\ni=1Sz\nii: (16)\nIn Fig. 4, green circles denote temperature ( T) dependence of magiven by the MC\nmethod. The system size N=L3= 203with PBC was adopted and in MC simulations\n10,000 MCS and following 50,000 MCS were employed for equilibration and measurement,\nrespectively at each temperature. The magnetizations of maobtained by the stochastic LLG\nmethod for case A (crosses) and case B (diamonds) are given in Fig. 4. Here \u000b= 0:05 and\nD= 1:0 were used for (a) and (b), respectively. \u0001 t= 0:005 was set and 240,000 steps\n(40,000 for transient and 200,000 for measurement) were applied.\nIn both cases ma(T) curve given by the stochastic LLG method shows good agreement\nwith that obtained by the MC method. Thus, we conclude that as long as the relation (10)\nis satis\fed, the temperature dependence of the magnetization is reproduced very accurately\neven around the Curie temperature, regardless of the choice of the parameter set.\n1300.511.5\n0123456m\nTFIG. 5: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo\nmethod (green circles) and the stochastic LLG method. Crosses and diamonds denote case A\nwith\u000b= 0:05 and case B with D= 1:0, respectively. A reduction of mfrom fully saturated\nmagnetization is observed at around T= 0 due to the dipole interactions. As a reference, mby\nthe MC method without the dipole interactions ( C= 0) is given by open circles.\nC. Inhomogeneous magnetic moments with exchange and dipole interactions\nWe also study thermal e\u000bects in a system with dipole interactions. We use the same\nlattice as in the previous subsections. The system is ( Ji;j=J,DA\ni=DA, andhi(t) =hin\nEq. (1)) given by\nH=\u0000X\nhi;jiJSi\u0001Sj\u0000X\niDA(Sz\ni)2\u0000X\nihSz\ni+X\ni6=kC\nr3\nik\u0010\nSi\u0001Sk\u00003(rik\u0001Si)(rik\u0001Sk)\nr2\nik\u0011\n:(17)\nHere a cubic lattice with open boundary conditions (OBC) is used. Since Jis much larger\nthanC=a3(J\u001dC=a3) for ferromagnets, where ais a lattice constant between magnetic\nsites. However, we enlarge dipole interaction as C= 0:2 witha= 1 forJ= 1 to highlight\nthe e\u000bect of the noise on dipole interactions. We set other parameters as h= 0:1,DA= 0:1.\nStudies with realistic situations will be given separately.\nWe study cases A ( \u000b= 0:05) and B ( D= 1:0) for this system. We depict in Fig. 5\nthe temperature ( T) dependences of mwith comparison between the MC (green circles) and\nstochastic LLG methods. Crosses and diamonds denote m(T) for cases A and B, respectively.\nDipole interactions are long-range interactions and we need longer equilibration steps, and\n14we investigate only a small system with N=L3= 63. In the MC method 200,000 MCS\nwere used for equilibration and 600,000 steps were used for measurement of m, and for\nthe stochastic LLG method \u0001 t= 0:005 was set and 960,000 steps (160,000 and 800,000\ntime steps for equilibration and measurement, respectively) were consumed. A reduction of\nmfrom fully saturated magnetization is observed. As a reference, mby the MC method\nwithout the dipole interactions ( C= 0) is given by open circles in Fig. 5. This reduction of\nmis caused by the dipole interactions.\nWe \fnd that even when dipole interactions are taken into account in inhomogeneous\nmagnetic moments, suitable choices of the parameter set leads to the equilibrium state.\nFinally, we comment on the comparison between the LLG method and the Monte Carlo\nmethod. To obtain equilibrium properties of spin systems, the Monte Carlo method is more\ne\u000ecient and powerful in terms of computational cost. It is much faster than the stochastic\nLLG method to obtain the equilibrium m(T) curves, etc. For example, it needs more than\n10 times of CPU time of the MC method to obtain the data for Fig. 5. However, the MC\nmethod has little information on the dynamics and the stochastic LLG method is used to\nobtain dynamical properties because it is based on an equation of motion of spins. Thus, it\nis important to clarify the nature of stochastic LLG methods including the static properties.\nFor static properties, as we saw above, the choice of the parameter set, e.g., cases A and\nB, did not give di\u000berence. However, the choice gives signi\fcant di\u000berence in dynamical\nproperties, which is studied in the following sections.\nV. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER\nSET IN ISOTROPIC SPIN SYSTEMS ( DA=0)\nNow we study the dependence of dynamics on the choice of parameter set. The temper-\nature is given by\nkBT=\rDiMi\n\u000bi; (18)\nwhich should be the same for all the sites. In general, if the parameter D(amplitude of\nthe noise) is large, the system is strongly disturbed, while if the parameter \u000b(damping\nparameter) is large, the system tends to relax fast. Therefore, even if the temperature is\nthe same, the dynamics changes with the values of Dand\u000b. When the anisotropy term\nexists, i.e.,DA6= 0, in homogeneous systems ( Mi=M) given by Eq. (14), the Stoner-\n15Wohlfarth critical \feld is hc= 2MDAatT= 0. If the temperature is low enough, the\nmetastable nature appears in relaxation. On the other hand, if Tis rather high orDA= 0,\nthe metastable nature is not observed. In this section we focus on dynamics of isotropic spin\nsystems, i.e.,DA=0.\nA. Relaxation with temperature dependence\nIn this subsection we investigate the temperature dependence of magnetization relaxation\nin cases A and B. We adopt a homogeneous system ( Mi=M= 2) withDA= 0 in Eq. (14).\nInitially all spins are in the spin down state and they relax under a unfavorable external\n\feldh= 2. The parameter set M= 2,\u000b= 0:05,D= 0:05 givesT= 2 by the condition\n(Eq. (10)). Here we study the system at T= 0:2;1;2, and 10. We set \u000b= 0:05 in case A\nand the control of the temperature is performed by D, i.e.D= 0:005;0:025;0:05, and 0:25,\nrespectively. In case B we set D= 0:05, and the control of the temperature is realized by\n\u000b, i.e.,\u000b= 0:5;0:1;0:05, and 0:01, respectively.\nWe depict the temperature dependence of m(t) for cases A and B in Figs. 6 (a) and (b),\nrespectively. Here the same random number sequence was used for each relaxation curve.\nRed dash dotted line, blue dotted line, green solid line, and black dashed line denote T= 0:2,\nT= 1,T= 2 andT= 10, respectively. Relaxation curves in initial short time are given in\nthe insets.\nIn case A, as the temperature is raised, the initial relaxation speed of mbecomes faster\nand the relaxation time to the equilibrium state also becomes shorter. This dependence is\nascribed to the strength of the noise with the dependence D/T, and a noise with a larger\namplitude disturbs more the precession of each moment, which causes faster relaxation.\nOn the other hand, in case B, the relaxation time to the equilibrium state is longer at\nhigher temperatures although the temperature dependence of the initial relaxation speed of\nmis similar to the case A. In the initial relaxation process all the magnetic moments are\nin spin-down state ( Sz\ni'\u00002). There the direction of the local \feld at each site is given\nbyHe\u000b\ni'JP\njSz\nj+h=\u00002\u00026 + 2 =\u000010, which is downward and the damping term\ntends to \fx moments to this direction. Thus, a large value of the damping parameter at a\nlow temperature T(\u000b/1\nT) suppresses the change of the direction of each moment and the\ninitial relaxation speed is smaller. However, in the relaxation process thermal \ructuation\n16-2-1012\n0 50 100 150 200m\ntime(a)\n-2.2-2-1.8-1.6-1.4-1.2-1\n012345678\n-2-1012\n0 50 100 150 200m\ntime-2.2-2-1.8-1.6-1.4-1.2-1\n012345678(b)FIG. 6: (color online) (a) Time dependence of the magnetization ( m(t)) in case A, where \u000b= 0:05\nfor a homogeneous system with M= 2. Red dash dotted line, blue dotted line, green solid line,\nand black dashed line denote T= 0:2,T= 1,T= 2 andT= 10, respectively. Inset shows the time\ndependence of m(t) in the initial relaxation process. (b) Time dependence of the magnetization\n(m(t)) in case B, where D= 0:05 for a homogeneous system with M= 2. Correspondence between\nlines and temperatures is the same as (a).\ncauses a deviation of the local \feld and then a rotation of magnetic moments from \u0000z\ntozdirection advances (see also Fig. 11 ). Once the rotation begins, the large damping\nparameter accelerates the relaxation and \fnally the relaxation time is shorter.\nB. Relaxation with spin-magnitude dependence\nNext we study the dependence of relaxation on the magnitude of magnetic moments\nin cases A and B. Here we adopt a homogeneous system ( Mi=M) without anisotropy(\nDA= 0) atT= 2 andh= 2. The initial spin con\fguration is the same as the previous\nsubsection. Because\nD/T\nM;and\u000b/M\nT; (19)\nraising the value of Mis equivalent to lowering temperature in both cases A and B and it\ncauses suppression of relaxation in case A, while it leads to acceleration of relaxation in case\nB. Because Ma\u000bects the local \feld from the exchange energy at each site, changing the\nvalue ofMunder a constant external \feld his not the same as changing Tand it may show\n17-1.5-1-0.500.511.5\n01 0 2 0 3 0 4 0 5 0m\ntime(a)\n-1.5-1-0.500.511.5\n02468 1 0m\ntime(b)FIG. 7: (color online) Comparison of the time dependence of mbetween cases A and B by the\nstochastic LLG method. Red and blue lines denote cases A and B, respectively. (a) \u000b= 0:05 for\ncase A and D= 1:0 for case B, (b) \u000b= 0:2 for case A and D= 1:0 for case B.\nsome modi\fed features.\nIn the relation (19), T= 0:2, 1, 2, 10 at M= 2 (Fig.6 (a) and (b)) are the same as\nM= 20, 4, 2, 0.4 at T= 2, respectively. We studied the relaxation ratio de\fned as m(t)=M\nwithMdependence at T= 2 for these four values of M, and compared with the relaxation\ncurves of Fig.6 (a) and (b). We found qualitatively the same tendency between relaxation\ncurves with Mdependence and those with 1 =Tdependence in both cases. A di\u000berence was\nfound in the initial relaxation speed (not shown). When M > 2, the initial relaxation at\nT= 2 is slower than that of the corresponding TatM= 2. The downward initial local\n\feld at each site is stronger for larger Mdue to a stronger exchange coupling, which also\nassist the suppression of the initial relaxation.\nIt is found that the relaxation time under a constant external \fled becomes longer as\nthe value of Mis raised in case A, while it becomes shorter in case B. This suggests that\ndi\u000berent choices of the parameter set lead to serious di\u000berence in the relaxation dynamics\nwithMdependence.\n18VI. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER\nSET IN ANISOTROPIC SPIN SYSTEMS ( DA6= 0)\nA. Di\u000berent relaxation paths to the equilibrium in magnetic inhomgeneity\nIf the anisotropy term exists DA6= 0 but the temperature is relatively high, metastable\nnature is not observed in relaxation. We consider the relaxation dynamics when Mihas\nidependence in this case. We study the system (alternating M= 2 andM= 1 planes)\ntreated in Sec. IV A. We set a con\fguration of all spins down as the initial state and observe\nrelaxation of min cases A and B. In Sec. IV A we studied cases A ( \u000b=0.05) and B ( D=1.0)\nfor the equilibrium state and the equilibrium magnetization is m'0:95 atT= 5. We\ngive comparison of the time dependence of mbetween the two cases in Fig. 7 (a), with the\nuse of the same random number sequence. The red and blue curves denote cases A and\nB, respectively. We \fnd a big di\u000berence in the relaxation time of mand features of the\nrelaxation between the two cases.\nThe parameter values of \u000bandDare not so close between the two cases at this tempera-\nture (T= 5), i.e.,D(M= 1) = 0:25 andD(M= 2) = 0:125 for case A and \u000b(M= 1) = 0:2\nand\u000b(M= 2) = 0:4 for case B. Thus, to study if there is a di\u000berence of dynamics even\nin close parameter values of \u000bandDbetween cases A and B at T= 5, we adopt common\n\u000b= 0:2, whereD(M= 1) = 1 and D(M= 2) = 0:5, as case A and common D= 1:0, where\n\u000b(M= 1) = 0:2 and\u000b(M= 2) = 0:4, as case B. We checked that this case A also gives the\nequilibrium state. In Fig. 7 (b), the time dependence of mfor both cases is given. The red\nand blue curves denote cases A and B, respectively. There is also a di\u000berence (almost twice)\nof the relaxation time of mbetween cases A and B. Thus, even in close parameter region of\n\u000bandD, dynamical properties vary depending on the choice of the parameters.\nB. Relaxation with nucleation mechanism\nIn this subsection we study a system with metastability. We adopt a homogeneous\nsystem (M= 2) withJ= 1,DA= 1 andh= 2. Here the Stoner-Wohlfarth critical \feld\nishc= 2MDA=4, and if the temperature is low enough, the system has a metastable state\nunderh= 2.\nAt a high temperature, e.g., T= 10 (\u000b= 0:05,D= 0:25), the magnetization relaxes\n19(a)\n-2-1012\n0 80 160 240 320m\ntime\n-2-1012\n0 50 100 150 200 250 300 350m\ntime(b)\n-2-1012\n0 50 100 150 200 250 300 350m\ntime(c)FIG. 8: (color online) (a) Dashed line shows m(t) for\u000b= 0:05,D= 0:25, andT= 10. Blue\nand green solid lines give m(t) for\u000b= 0:05 atT= 3:5 (case A) and D= 0:25 atT= 3:5 (case\nB), respectively. These two lines were obtained by taking average over 20 trials with di\u000berent\nrandom number sequences. The 20 relaxation curves for cases A and B are given in (b) and (c),\nrespectively.\nwithout being trapped as depicted in Fig 8(a) with a black dotted line. When the tempera-\nture is lowered, the magnetization is trapped at a metastable state. We observe relaxations\nin cases A and B, where \u000b= 0:05 for case A and D= 0:25 for case B are used. In Figs. 8(b)\nand (c), we show 20 samples (with di\u000berent random number sequences) of relaxation pro-\ncesses atT= 3:5 for case A ( \u000b= 0:05,D= 0:0875) and case B ( D= 0:25,\u000b= 0:143),\nrespectively. The average lines of the 20 samples are depicted in Fig 8(a) by blue and green\nsolid lines for cases A and B, respectively. In both cases, magnetizations are trapped at a\nmetastable state with the same value of m(m'\u00001:55). This means that the metastabil-\nity is independent of the choice of parameter set. Relaxation from the metastable state to\nthe equilibrium is the so-called stochastic process and the relaxation time distributes. The\nrelaxation time in case A is longer. If the temperature is further lowered, the escape time\nfrom the metastable state becomes longer. In Figs. 9 (a) and (b), we show 20 samples of\nrelaxation at T= 3:1 for cases A and B, respectively. There we \fnd the metastable state\nmore clearly.\nHere we investigate the initial relaxation to the metastable state at a relatively low\ntemperature. In Figs. 10 (a) and (b), we depict the initial short time relaxation of 20\nsamples at T= 2 in cases A ( \u000b= 0:05,D= 0:05) and B(D= 0:25,\u000b= 0:25), respectively.\nThe insets show the time dependence of the magnetization in the whole measurement time.\n20-2-1012\n0 200 400 600 800m\ntime(a)\n-2-1012\n0 200 400 600 800m\ntime(b)FIG. 9: (a) and (b) illustrate 20 relaxation curves for \u000b= 0:05 atT= 3:1 (case A) and D= 0:25\natT= 3:1 (case B), respectively. Metastability becomes stronger than T= 3:5. No relaxation\noccurs in all 20 trials in (a), while \fve relaxations take place in 20 trials in (b).\nWe \fnd that the relaxation is again faster in case B.\nThe metastability also depends on Mas well asDAand largeMgives a strong metastabil-\nity. Here we conclude that regardless of the choice of the parameter set, as the temperature\nis lowered, the relaxation time becomes longer due to the stronger metastability, in which\nlargerD(larger\u000b) gives faster relaxation from the initial to the metastable state and faster\ndecay from the metastable state.\nFinally we show typical con\fgurations in the relaxation process. When the anisotropy\nDAis zero or weak, the magnetization relaxation occurs with uniform rotation from \u0000z\ntozdirection, while when the anisotropy is strong, the magnetization reversal starts by a\nnucleation and inhomogeneous con\fgurations appear with domain wall motion. In Figs. 11\nwe give an example of the magnetization reversal of (a) the uniform rotation type (magneti-\nzation reversal for DA= 0 withD= 0:05,T= 2,\u000b= 0:1,M= 4) and of (b) the nucleation\ntype (magnetization reversal for DA= 1 withD= 0:25,T= 3:1,\u000b= 0:161,M= 2 ).\nVII. SUMMARY AND DISCUSSION\nWe studied the realization of the canonical distribution in magnetic systems with the\nshort-range (exchange) and long-range (dipole) interactions, anisotropy terms, and magnetic\n\felds by the Langevin method of the LLG equation. Especially we investigated in detail the\n21-2.2-2-1.8-1.6-1.4-1.2-1.0\n012345678m\ntime(a)\n-2-1012\n0 200 400 600 800\ntime\n-2.2-2-1.8-1.6-1.4-1.2-1\n012345678m\ntime(b)\n-2-1012\n0 200 400 600 800\ntimeFIG. 10: Initial relaxation curves of magnetization. Insets show m(t) in the whole measurement\ntime. (a) and (b) illustrate 20 relaxation curves for \u000b= 0:05 atT= 2 (case A) and D= 0:25 at\nT= 2 (case B), respectively.\n(b)(a)\nFIG. 11: (a) Typical uniform rotation type relaxation observed in the isotropic spin system. (b)\nTypical nucleation type relaxation observed in the anisotropic spin system.\nthermal equilibration of inhomogeneous magnetic systems. We pointed out that the spin-\nmagnitude dependent ratio between the strength of the random \feld and the coe\u000ecient of the\ndamping term must be adequately chosen for all magnetic moments satisfying the condition\n(10). We compared the stationary state obtained by the present Langevin method of the\n22LLG equation with the equilibrium state obtained by the standard Monte Carlo simulation\nfor given temperatures. There are several choices for the parameter set, e.g., A and B. We\nfound that as long as the parameters are suitably chosen, the equilibrium state is realized as\nthe stationary state of the stochastic LLG method regardless of the choice of the parameter\nset, and the temperature dependence of the magnetization is accurately produced in the\nwhole region, including the region around the Curie temperature.\nWe also studied dynamical properties which depend on the choice of the parameters. We\nshowed that the choice of the parameter values seriously a\u000bects the relaxation process to\nthe equilibrium state. In the rotation type relaxation in isotropic spin systems under an\nunfavorable external \feld, the dependences of the relaxation time on the temperature in\ncases A and B exhibited opposite correlations as well as the dependences of the relaxation\ntime on the magnitude of the magnetic moment. The strength of the local \feld in the initial\nstate strongly a\u000bects the speed of the initial relaxation in both cases.\nWe also found that even if close parameter values are chosen in di\u000berent parameter sets\nfor inhomogeneous magnetic systems, these parameter sets cause a signi\fcant di\u000berence of\nrelaxation time to the equilibrium state. In the nucleation type relaxation, the metastability,\nwhich depends on DAandM, strongly a\u000bects the relaxation in both cases A and B. Lowering\ntemperature reinforces the metastability of the system and causes slower relaxation. The\nrelaxation to the metastable state and the decay to the metastable state are a\u000bected by the\nchoice of the parameter set, in which larger Dcauses fast relaxation at a \fxed T.\nIn this study we adopted two cases, i.e., A and B in the choice of the parameter set.\nGenerally more complicated dependence of MiorTon the parameters is considered. How\nto chose the parameter set is related to the quest for the origin of these parameters. It\nis very important for clari\fcation of relaxation dynamics but also for realization of a high\nspeed and a low power consumption, which is required to development of magnetic devices.\nStudies of the origin of \u000bhave been intensively performed32{41. To control magnetization\nrelaxation at \fnite temperatures, investigations of the origin of Das well as\u000bwill become\nmore and more important. We hope that the present work gives some useful insight into\nstudies of spin dynamics and encourages discussions for future developments in this \feld.\n23Acknowledgments\nThe authors thank Professor S. Hirosawa and Dr. S. Mohakud for useful discussions.\nThe present work was supported by the Elements Strategy Initiative Center for Magnetic\nMaterials under the outsourcing project of MEXT and Grant-in-Aid for Scienti\fc Research\non Priority Areas, KAKENHI (C) 26400324.\n24Appendix A: Fokker-Planck equation\nThe LLG equation with a Langevin noise (Eq. (5)) is rewritten in the following form for\n\u0016component ( \u0016= 1;2 or 3 forx;yorz) of theith magnetic moment,\ndM\u0016\ni\ndt=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t) +g\u0016\u0017\ni(Mi)\u0018\u0017\ni(t): (A1)\nHeref\u0016\niandg\u0016\u0017\niare given by\nf\u0016\ni=\u0000\r\n1 +\u000b2\ni\u0014\n\u000f\u0016\u0017\u0015M\u0017\niHe\u000b;\u0015\ni+\u000bi\nMi\u000f\u0016\u0017\u0015\u000f\u0015\u001a\u001bM\u0017\niM\u001a\niHe\u000b;\u001b\ni\u0015\n(A2)\nand\ng\u0016\u0015\ni=\u0000\r\n1 +\u000b2\ni\u0014\n\u000f\u0016\u0017\u0015M\u0017\ni+\u000bi\nMi(\u0000M2\ni\u000e\u0016\n\u0015+M\u0016\niM\u0015\ni)\u0015\n; (A3)\nwhereHe\u000b;\u0015\nican have an explicit time ( t) dependence, and \u000f\u0016\u0017\u0015denotes the Levi-Civita\nsymbol. We employ the Einstein summation convention for Greek indices ( \u0016,\u0017\u0001\u0001\u0001).\nWe consider the distribution function F\u0011F(M1;\u0001\u0001\u0001;MN;t) in the 3N-dimensional\nphase space ( M1\n1;M2\n1;M3\n1;\u0001\u0001\u0001;M1\nN;M2\nN;M3\nN). The distribution function F(M1;\u0001\u0001\u0001;MN;t)\nsatis\fes the continuity equation of the distribution:\n@\n@tF(M1;\u0001\u0001\u0001;MN;t) +NX\ni=1@\n@M\u000b\ni\u001a\u0000d\ndtM\u000b\ni\u0001\nF\u001b\n= 0: (A4)\nSubstituting the relation (A1), the following di\u000berential equation for the distribution func-\ntionFis obtained.\n@\n@tF(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\nin\u0000\nfi+g\u000b\f\ni\u0018\f\ni\u0001\nFo\n: (A5)\nRegarding the stochastic equation (A1) as the Stratonovich interpretation, making use\nof the stochastic Liouville approach42, and taking average for the noise statistics (Eq. (6)),\nwe have a Fokker-Planck equation.\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\nf\u000b\niP\u0000Dig\u000b\f\ni@\n@M\u001b\ni(g\u001b\f\niP)\u001b\n; (A6)\nwhereP\u0011P(M1;\u0001\u0001\u0001;MN;t) is the averaged distribution function hFi.\nSubstituting the relation\n@\n@M\u001b\nig\u001b\f\ni=\u0000\r\u000bi\nMi(1 +\u000b2\ni)4M\f\ni (A7)\n25and Eq. (A3) into g\u000b\f\ni(@\n@M\u001b\nig\u001b\f\ni), we \fnd\ng\u000b\f\ni(@\n@M\u001b\nig\u001b\f\ni) = 0: (A8)\nThus Eq.(A6) is simpli\fed to\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\u0000\nf\u000b\ni\u0000Dig\u000b\f\nig\u001b\f\ni@\n@M\u001b\ni\u0001\nP\u001b\n: (A9)\nSubstituting Eqs. (A2) and (A3), we have a formula in the vector representation.\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) = (A10)\nX\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\nMi\u0002He\u000b\ni+\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nSince@\n@Mi\u0001(Mi\u0002He\u000b\ni) = 0, it is written as\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni) (A11)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nIn the case that Eq. (A1) is given under Ito de\fnition, we need Ito-Stratonovich trans-\nformation, and the corresponding equation of motion in Stratonovich interpretation is\ndM\u0016\ni\ndt=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)\u0000Dig\u0015\u0017\ni(Mi)@g\u0016\u0017\ni(Mi)\n@M\u0015\ni+g\u0016\u0017\ni(Mi)\u0018\u0017\ni(t): (A12)\nThen the Fokker-Planck equation in Ito interpretation is\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\u0000\nf\u000b\ni\u0000Dig\u0015\u0017\ni@g\u000b\u0017\ni\n@M\u0015\ni\u0000Dig\u000b\f\nig\u001b\f\ni@\n@M\u001b\ni\u0001\nP\u001b\n:\nSinceg\u0015\u0017\ni@g\u000b\u0017\ni\n@M\u0015\ni=\u00002\r2\n1+\u000b2\niM\u000b\ni, the vector representation is given by\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni)\n\u00002\rDiMi\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\n(A13)\n26Appendix B: Numerical integration for stochastic di\u000berential equations\nIn stochastic di\u000berential equations, we have to be careful to treat the indi\u000berentiability\nof the white noise. In the present paper we regard the stochastic equation, e.g., Eq. (5), as\na stochastic di\u000berential equation in Stratonovich interpretation:\ndM\u0016\ni=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)dt+g\u0016\u0017\ni\u00101\n2\u0000\nMi(t) +Mi(t+dt)\u0001\u0011\ndW\u0017\ni(t); (B1)\nwheredW\u0017\ni(t) =Rt+dt\ntds\u0018\u0017\ni(s), which is the Wiener process. This equation is expressed by\ndM\u0016\ni=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)dt+g\u0016\u0017\ni(Mi(t))\u000edW\u0017\ni(t); (B2)\nwhere\u000eindicates the usage of the Stratonovich de\fnition.\nA simple predictor-corrector method called the Heun method8,19, superior to the Euler\nmethod, is given by\nM\u0016\ni(t+ \u0001t) =M\u0016\ni(t)\n+1\n2[f\u0016\ni(^M1(t+ \u0001t);\u0001\u0001\u0001;^MN(t+ \u0001t);t+ \u0001t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)]\u0001t\n+1\n2[g\u0016\u0017\ni(^Mi(t+ \u0001t)) +g\u0016\u0017\ni(Mi(t))]\u0001W\u0017\ni; (B3)\nwhere \u0001W\u0017\ni\u0011W\u0017\ni(t+ \u0001t)\u0000W(t) and ^M\u0016\ni(t+ \u0001t) is chosen in the Euler scheme:\n^M\u0016\ni(t+ \u0001t) =M\u0016\ni(t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)\u0001t+g\u0016\u0017\ni(Mi(t))\u0001W\u0017\ni: (B4)\nThis scheme assures an approximation accuracy up to the second order of \u0001 Wand \u0001t. Sev-\neral numerical di\u000berence methods19for higher-order approximation, which are often compli-\ncated, have been proposed.\nHere we adopt a kind of middle point method equivalent to the Heun method.\nM\u0016\ni(t+ \u0001t) =M\u0016\ni(t)\n+f\u0016\ni(M1(t+ \u0001t=2);\u0001\u0001\u0001;MN(t+ \u0001t=2);t+ \u0001t=2)\u0001t\n+g\u0016\u0017\ni(Mi(t+ \u0001t=2))\u0001W\u0017\ni; (B5)\nwhereM\u0016\ni(t+ \u0001t=2) is chosen in the Euler scheme:\nM\u0016\ni(t+ \u0001t=2) =M\u0016\ni(t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)\u0001t=2 +g\u0016\u0017\ni(Mi(t))\u0001~Wi\u0017; (B6)\n27where \u0001 ~Wi\u0017\u0011W\u0017\ni(t+ \u0001t=2)\u0000W\u0017\ni(t). Considering the following relations,\nh\u0001~Wi\u0017\u0001W\u0017\nii=\n[W\u0017\ni(t+ \u0001t=2)\u0000W\u0017\ni(t)][W\u0017\ni(t+ \u0001t)\u0000W\u0017\ni(t)]\u000b\n=Di\u0001t; (B7)\nh\u0001W\u0017\nii= 0 andh\u0001~Wi\u0017i= 0, this method is found equivalent to the Heun method. 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Zhao3, Xiaoxi Liu4, and Yan Zhou1\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China\n2School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China\n3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n4Department of Information Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan\nReliable transport of magnetic skyrmions is required for any future skyrmion-based information processing devices. Here we\npresent a micromagnetic study of the in-plane current-driven motion of a skyrmion in a ferromagnetic nanotrack with spatially\nsinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque coefficients. It is found that the skyrmion moves in\na sinusoidal pattern as a result of the spatially varying Gilbert damping and/or non-adiabatic spin-transfer torque in the nanotrack,\nwhich could prevent the destruction of the skyrmion caused by the skyrmion Hall effect. The results provide a guide for designing\nand developing the skyrmion transport channel in skyrmion-based spintronic applications.\nIndex Terms —magnetic skyrmions, racetrack memories, micromagnetics, spintronics.\nI. I NTRODUCTION\nMagnetic skyrmions are quasiparticle-like domain-wall\nstructures with typical sizes in the sub-micrometer regime [1]–\n[7]. They are theoretically predicted to exist in magnetic metals\nhaving antisymmetric exchange interactions [8], and confirmed\nby experiments [9], [10] just after the turn of the twenty-\nfirst century. Isolated skyrmions are expected to be used to\nencode information into bits [11], which might lead to the\ndevelopment of novel spintronic applications, such as the\nracetrack memories [12]–[19], storage devices [20]–[22], and\nlogic computing devices [23].\nThe write-in and read-out processes of skyrmions in thin\nfilms are realizable and controllable at low temperatures [24]–\n[26]. A recent experiment has realized the current-induced\ncreation and motion of skyrmions in Ta/CoFeB/TaO trilayers\nat room temperature [27]. Experimental investigations have\nalso demonstrated the increased stability of skyrmions in mul-\ntilayers [28]–[30], which makes skyrmions more applicable to\npractical room-temperature applications.\nHowever, the skyrmion experiences the skyrmion Hall effect\n(SkHE) [31], [32], which drives it away from the longitudinal\ndirection when it moves in a narrow nanotrack. As a con-\nsequence, in the high-speed operation, the transverse motion\nof a skyrmion may result in its destruction at the nanotrack\nedges [18], [33]–[36]. Theoretical and numerical works have\nproposed several intriguing methods to reduce or eliminate the\ndetrimental transverse motion caused by the SkHE. For ex-\nample, one could straightforwardly enhance the perpendicular\nmagnetic anisotropy near the nanotrack edges to better confine\nthe skyrmion motion [33]. An alternative solution is to trans-\nport skyrmions on periodic substrates [37]–[40], where the\nskyrmion trajectory can be effectively controlled. Moreover, by\nconstructing antiferromagnetic skyrmions [34], [35] and anti-\nThe first two authors contributed equally to this work. Corre-\nsponding authors: X. Liu (email: liu@cs.shinshu-u.ac.jp) and Y . Zhou\n(email: zhouyan@cuhk.edu.cn).ferromagnetically exchange-coupled bilayer skyrmions [18],\n[36], the SkHE can be completely suppressed. Recently, it is\nalso found that the skyrmionium can perfectly move along\nthe driving force direction due to its spin texture with a zero\nskyrmion number [41], [42].\nIn this paper, we propose and demonstrate that a skyrmion\nguide with spatially sinusoidally varying Gilbert damping\nand/or non-adiabatic spin-transfer torque (STT) coefficients\ncan be designed for transporting skyrmions in a sinusoidal\nmanner, which is inspired by a recent study on the magnetic\nvortex guide [43], where the vortex core motion is controlled\nvia spatially varying Gilbert damping coefficient. The results\nprovide a guide for designing and developing the skyrmion\ntransport channel in future spintronic devices based on the\nmanipulation of skyrmions.\nII. M ETHODS\nOur simulation model is an ultra-thin ferromagnetic nan-\notrack with the length land the width w, where the thick-\nness is fixed at 1nm. We perform the simulation using\nthe standard micromagnetic simulator, i.e., the 1.2 alpha 5\nrelease of the Object Oriented MicroMagnetic Framework\n(OOMMF) [44]. The simulation is accomplished by a set of\nbuilt-in OOMMF extensible solver (OXS) objects. We employ\nthe OXS extension module for modeling the interface-induced\nantisymmetric exchange interaction, i.e., the Dzyaloshinskii-\nMoriya interaction (DMI) [45]. In addition, we use the updated\nOXS extension module for simulating the in-plane current-\ninduced STTs [46]. The in-plane current-driven magnetization\ndynamics is governed by the Landau-Lifshitz-Gilbert (LLG)\nequation augmented with the adiabatic and non-adiabatic\nSTTs [44], [47]\ndM\ndt=\u0000\r0M\u0002Heff+\u000b\nMS(M\u0002dM\ndt) (1)\n+u\nM2\nS(M\u0002@M\n@x\u0002M)\u0000\fu\nMS(M\u0002@M\n@x);arXiv:1607.04983v3  [cond-mat.mes-hall]  15 Dec 20162\nFig. 1. (a) The magnetic damping coefficient \u000b(x)and non-adiabatic STT\ncoefficient\f(x)as functions of xin the nanotrack. (b) Trajectories of current-\ndriven skyrmions with \f=\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6.\nDot denotes the skyrmion center. Red cross indicates the skyrmion destruction.\n(c) Skyrmion Hall angle \bas a function of xfor skyrmion motion with \f=\n\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6. The dashed lines indicate\n\b =\u000614\u000e. (e) Real-space top-views of skyrmion motion with \f=\u000b=2 =\n0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6.wandvdenote the nanotrack\nwidth and velocity direction, respectively. The dashed line indicates the central\nline of the nanotrack. The skyrmion is destroyed at t= 870 ps when\f=\n2\u000b= 0:6. The out-of-plane magnetization component is represented by the\nred (\u0000z)-white ( 0)-green ( +z) color scale.\nwhere Mis the magnetization, MSis the saturation magne-\ntization,tis the time, \r0is the Gilbert gyromagnetic ratio,\n\u000bis the Gilbert damping coefficient, and \fis the strength of\nthe non-adiabatic STT. The adiabatic STT coefficient is given\nbyu, i.e., the conduction electron velocity. The effective field\nHeffis expressed as\nHeff=\u0000\u0016\u00001\n0@E\n@M; (2)\nwhere\u00160is the vacuum permeability constant. The average\nenergy density Econtains the exchange, anisotropy, demag-\nnetization, and DMI energies, which is given as\nE=A[r(M\nMS)]2\u0000K(n\u0001M)2\nM2\nS\u0000\u00160\n2M\u0001Hd(M) (3)\n+D\nM2\nS(Mz@Mx\n@x+Mz@My\n@y\u0000Mx@Mz\n@x\u0000My@Mz\n@y);\nwhereA,K, andDare the exchange, anisotropy, and DMI\nenergy constants, respectively. nis the unit surface normal\nvector, and Hd(M)is the demagnetization field. Mx,My\nandMzare the three Cartesian components of M.\nThe model is discretized into tetragonal volume elements\nwith the size of 2nm\u00022nm\u00021nm, which ensures a\ngood compromise between the computational accuracy and ef-\nficiency. The magnetic parameters are adopted from Refs. [14],\nFig. 2. (a)vx, (b)vy, and (c) \bas functions of \u000band\fgiven by Eq. (11)\nand Eq. (12), respectively. vxandvyare reduced by u.\n[23]:\r0= 2:211\u0002105m/(A\u0001s),A= 15 pJ/m,D= 3mJ/m2,\nK= 0:8MJ/m3,MS= 580 kA/m. In all simulations, we\nassumeu= 100 m/s andw= 50 nm. The skyrmion is initially\nlocated at the position of x= 100 nm,y= 25 nm.\nThe Gilbert damping coefficient \u000bis defined as a function\nof the longitudinal coordinate xas follows [Fig. 1(a)]\n\u000b(x) =\u000bamp\u0001f1 + sin [2\u0019(x=\u0015\u000b)]g+\u000bmin; (4)\nwhere\u000bamp= (\u000bmax\u0000\u000bmin)=2is the amplitude of the \u000b\nfunction.\u000bmaxand\u000bminstand for the maximum and mini-\nmum values of the \u000bfunction, respectively. \u0015\u000bdenotes the\nwavelength of the \u000bfunction. It is worth mentioning that the\nspatially varying \u000bcan be achieved by gradient doping of\nlanthanides impurities in ferromagnets [43], [48], [49]. Exper-\niments have found that \u000bis dependent on the interface [50].\nThus it is also realistic to construct the varying \u000bby techniques\nsuch as interface engineering. Indeed, as shown in Ref. [51],\nlocal control of \u000bin a ferromagnetic/non-magnetic thin-film\nbilayer has been experimentally demonstrated by interfacial\nintermixing induced by focused ion-beam irradiation.\nIn a similar way, the non-adiabatic STT coefficient \fis\nalso defined as a function of the longitudinal coordinate xas\nfollows [Fig. 1(a)]\n\f(x) =\famp\u0001f1 + sin [2\u0019(x=\u0015\f)\u0000']g+\fmin;(5)\nwhere\famp= (\fmax\u0000\fmin)=2is the amplitude of the \f\nfunction.\fmaxand\fminstand for the maximum and minimum\nvalues of the \ffunction, respectively. \u0015\fand'denote the\nwavelength and phase of the \ffunction, respectively. Since\nthe value of \fdepends on the material properties [52], it is\nexpected to realize the spatial varying \fby constructing a\nsuperlattice nanotrack using different materials, similar to the\nmodel given in Ref. [43]. Note that the effect of varying \f\nhas also been studied in spin torque oscillators [53].\nIII. R ESULTS\nA. Nanotrack with spatially uniform \u000band\f\nWe first recapitulate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially uniform \u000band\f. As\nshown in Fig. 1(b), the skyrmion moves along the central line\nof the nanotrack when \f=\u000b= 0:3. However, due to the\nSkHE, it shows a transverse shift toward the upper and lower\nedges when \f= 2\u000b= 0:6and\f=\u000b=2 = 0:15, respectively.3\nThe skyrmion is destroyed by touching the upper edge when\n\f= 2\u000b= 0:6att= 870 ps.\nThe skyrmion Hall angle \b, which characterizes the trans-\nverse motion of the skyrmion caused by the SkHE, is defined\nas\n\b = tan\u00001(vy=vx): (6)\nFigure 1(c) shows \bas a function of xfor the skyrmion motion\nwith\f=\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6. It\ncan be seen that \b = 0\u000ewhen\f=\u000b= 0:3, indicating\nthe moving skyrmion has no transverse motion [Fig. 1(d)].\nWhen\f=\u000b=2 = 0:15,\bincreases from\u000015\u000eto0\u000e,\nindicating the moving skyrmion has a transverse shift toward\nthe lower edge which is balanced by the transverse force due to\nthe SkHE and the edge-skyrmion repulsive force [Fig. 1(d)].\nWhen\f= 2\u000b= 0:6,\bdecreases from 15\u000eto3\u000ewithin\n870 ps, indicating the moving skyrmion shows a transverse\nmotion toward the upper edge. At t= 870 ps, the skyrmion\nis destroyed as it touches the upper edge of the nanotrack\n[Fig. 1(d)]. It should be noted that the skyrmion profile is\nrigid before it touches the nanotrack edge. In order to better\nunderstand the transverse motion caused by the SkHE, we also\nanalyze the in-plane current-driven skyrmion motion using the\nThiele equation [54]–[57] by assuming the skyrmion moves in\nan infinite film, which is expressed as\nG\u0002(v\u0000u) +D(\fu\u0000\u000bv) =0; (7)\nwhere G= (0;0;\u00004\u0019Q)is the gyromagnetic coupling vector\nwith the skyrmion number\nQ=1\n4\u0019Z\nm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\ndxdy: (8)\nm=M=M Sis the reduced magnetization and Dis the\ndissipative tensor\nD= 4\u0019\u0012DxxDxy\nDyxDyy\u0013\n: (9)\nu= (u;0)is the conduction electron velocity, and vis the\nskyrmion velocity. For the nanoscale skyrmion studied here,\nwe have\nQ=\u00001;Dxx=Dyy= 1;Dxy=Dyx= 0: (10)\nHence, the skyrmion velocity is given as\nvx=u(\u000b\f+ 1)\n\u000b2+ 1; vy=u(\f\u0000\u000b)\n\u000b2+ 1: (11)\nThe skyrmion Hall angle \bis thus given as\n\b = tan\u00001(vy=vx) = tan\u00001\u0012\f\u0000\u000b\n\u000b\f+ 1\u0013\n: (12)\nBy calculating Eq. (11), we show vxas functions of \u000band\f\nin Fig. 2(a). vxranges between 0:5uand1:21u, indicating the\nskyrmion always moves in the +xdirection. When \u000b= 0:42\nand\f= 1,vxcan reach the maximum value of vx= 1:21u.\nSimilarly, we show vyas functions of \u000band\fin Fig. 2(b).\nvyranges between\u00000:5uandu, indicating the skyrmion can\nmove in both the \u0006ydirections. When \u000b < \f ,vy>0, the\nskyrmion shows a positive transverse motion, while when \u000b>\n\f,vy<0, the skyrmion shows a negative transverse motion.\nFig. 3. (a) Trajectories of current-driven skyrmions with \u000bamp =\n0:315;0:225;0:215.\u0015\u000b= 2wand\f= 0:3. (b) \bas a function of x\nfor skyrmion motion with \u000bamp= 0:315;0:225;0:215.\u0015\u000b= 2wand\n\f= 0:3. (c) Trajectories of current-driven skyrmions with \u0015\u000b=w;2w;4w.\n\u000bamp= 0:225 and\f= 0:3. (d)\bas a function of xfor skyrmion motion\nwith\u0015\u000b=w;2w;4w.\u000bamp= 0:225 and\f= 0:3.\nFig. 4. (a) Trajectories of current-driven skyrmions with \famp =\n0:315;0:225;0:215.\u0015\f= 2w,'= 0, and\u000b= 0:3. (b) \bas a function\nofxfor skyrmion motion with \famp= 0:315;0:225;0:215.\u0015\f= 2w,\n'= 0 , and\u000b= 0:3. (c) Trajectories of current-driven skyrmions with\n\u0015\f=w;2w;4w.\famp= 0:225,'= 0, and\u000b= 0:3. (d)\bas a function\nofxfor skyrmion motion with \u0015\f=w;2w;4w.\famp= 0:225,'= 0, and\n\u000b= 0:3.\nBy calculating Eq. (12), we also show \bas functions of \u000b\nand\fin Fig. 2(c), where \bvaries between \b = 45\u000eand\n\b =\u000045\u000e. Obviously, one has \b = 0\u000e,\b<0\u000e, and \b>0\u000e\nfor\u000b=\f,\u000b>\f , and\u000b<\f , respectively, which agree with\nthe simulation results for the nanotrack when the edge effect\nis not significant, i.e., when the skyrmion moves in the interior\nof the nanotrack. For example, using Eq. (12), the skyrmion\nhas\b = 14\u000eand\b =\u000014\u000efor\f= 2\u000b= 0:6and\f=\n\u000b=2 = 0:15, respectively, which match the simulation results\natt\u00180ps where the edge effect is negligible [Fig. 1(c)].\nB. Nanotrack with spatially varying \u000bor\f\nWe first demonstrate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially varying \u000band spatially\nuniform\f, i.e.,\u000bis a function of x, as in Eq. (4), and \f=\n0:3. Figure 3(a) shows the trajectories of the current-driven\nskyrmions with different \u000b(x)functions where \u0015\u000b= 2wand\n\f= 0:3. For\u000bmax= 0:75,\u000bmin= 0:12, i.e.,\u000bamp= 0:315,\nthe skyrmion moves in the rightward direction in a sinusoidal\npattern. For \u000bmax= 0:6,\u000bmin= 0:15, i.e.,\u000bamp= 0:225, the\nmaximum transverse shift of skyrmion is reduced in compared\nto that of\u000bamp= 0:315. For\u000bmax= 0:45,\u000bmin= 0:2, i.e.,\n\u000bamp= 0:125, the amplitude of the skyrmion trajectory further4\nFig. 5. Trajectories of current-driven skyrmions with '= 0\u00182\u0019.\u000bamp=\n\famp= 0:225 and\u0015\u000b=\u0015\f= 2w.\ndecreases. \bas a function of xcorresponding to Fig. 3(a) for\ndifferent\u000b(x)functions are given in Fig. 3(b). Figure 3(c)\nshows the trajectories of the current-driven skyrmions with\ndifferent\u0015\u000bwhere\u000bamp= 0:225and\f= 0:3.\bas a function\nofxcorresponding to Fig. 3(c) for different \u0015\u000bare given in\nFig. 3(d).\nWe then investigate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially uniform \u000band spatially\nvarying\f, i.e.,\fis a function of x, as in Eq. (5), and \u000b=\n0:3. Figure 4(a) shows the trajectories of the current-driven\nskyrmions with different \f(x)functions where \u0015\f= 2w,'=\n0and\u000b= 0:3. The results are similar to the case with spatially\nvarying\u000b. For\fmax= 0:75,\fmin= 0:12, i.e.,\famp= 0:315,\nthe skyrmion moves in the rightward direction in a sinusoidal\npattern. For \fmax= 0:6,\fmin= 0:15, i.e.,\famp= 0:225, the\nmaximum transverse shift of skyrmion is reduced in compared\nto that of\famp= 0:315. For\fmax= 0:45,\fmin= 0:2, i.e.,\n\famp= 0:125, the amplitude of the skyrmion trajectory further\ndecreases. \bas a function of xcorresponding to Fig. 4(a) for\ndifferent\f(x)functions are given in Fig. 4(b). Figure 4(c)\nshows the trajectories of the current-driven skyrmions with\ndifferent\u0015\fwhere\famp= 0:225and\u000b= 0:3.\bas a function\nofxcorresponding to Fig. 4(c) for different \u0015\fare given in\nFig. 4(d).\nFrom the skyrmion motion with spatially varying \u000bor\nspatially varying \f, it can be seen that the amplitude of\ntrajectory is proportional to \u000bampor\famp. The wavelength of\ntrajectory is equal to \u0015\u000b;\f, while the amplitude of trajectory is\nproportional to \u0015\u000b;\f.\balso varies with xin a quasi-sinusoidal\nmanner, where the peak value of \b(x)is proportional to \u000bamp,\n\famp, and\u0015\u000b;\f. As shown in Fig. 2(c), when \fis fixed at a\nvalue between \u000bmaxand\u000bmin, larger\u000bampwill lead to larger\npeak value of \b(x). On the other hand, a larger \u0015\u000b;\fallows\na longer time for the skyrmion transverse motion toward a\ncertain direction, which will result in a larger amplitude of\ntrajectory as well as a larger peak value of \b(x).\nFig. 6. \bas a function of xfor skyrmion motion with '= 0\u00182\u0019.\n\u000bamp=\famp= 0:225 and\u0015\u000b=\u0015\f= 2w.\nC. Nanotrack with spatially varying \u000band\f\nWe also demonstrate the in-plane current-driven skyrmion\nmotion in a nanotrack with both spatially varying \u000band\f,\ni.e., both\u000band\fare functions of x, as given in Eq. (4) and\nEq. (5), respectively.\nFigure 5 shows the trajectories of the current-driven\nskyrmions with spatially varying \u000band\fwhere\u000bamp=\n\famp= 0:225and\u0015\u000b=\u0015\f= 2w. Here, we focus on the effect\nof the phase difference between the \u000b(x)and\f(x)functions.\nFor'= 0 and'= 2\u0019, as the\u000b(x)function is identical to\nthe\f(x)function, the skyrmion moves along the central line\nof the nanotrack. For 0<'< 2\u0019, as\u000b(x)could be different\nfrom\f(x)at a certainx, it is shown that the skyrmion moves\ntoward the right direction in a sinusoidal pattern, where the\nphase of trajectory is subject to '. Figure 6 shows \bas a\nfunction of xcorresponding to Fig. 5 for '= 0\u00182\u0019where\n\u000bamp=\famp= 0:225 and\u0015\u000b=\u0015\f= 2w. It shows that\n\b = 0\u000ewhen'= 0 and'= 2\u0019, while it varies with xin\na quasi-sinusoidal manner when 0<'< 2\u0019. The amplitude\nof trajectory as well as the peak value of \b(x)reach their\nmaximum values when '=\u0019.\nIV. C ONCLUSION\nIn conclusion, we have shown the in-plane current-driven\nmotion of a skyrmion in a nanotrack with spatially uniform\n\u000band\f, where \bis determined by \u000band\f, which can vary\nbetween \b = 45\u000eand\b =\u000045\u000ein principle. Then, we\nhave investigated the in-plane current-driven skyrmion motion\nin a nanotrack with spatially sinusoidally varying \u000bor\f.\nThe skyrmion moves on a sinusoidal trajectory, where the\namplitude and wavelength of trajectory can be controlled by\nthe spatial profiles of \u000band\f. The peak value of \b(x)is\nproportional to the amplitudes and wavelengths of \u000b(x)and\n\f(x). In addition, we have demonstrated the in-plane current-\ndriven skyrmion motion in a nanotrack having both spatially\nsinusoidally varying \u000band\fwith the same amplitude and\nwavelength. The skyrmion moves straight along the central5\nline of the nanotrack when \u000b(x)and\f(x)have no phase\ndifference, i.e., '= 0. 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We then derive the Langevin\nequation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic\nsoliton behaves as a classical massive particle immersed in a viscous medium. By considering a\nthermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract\nthe average drift velocity of a soliton. The di\u000busion coe\u000ecient is inversely proportional to a small\ndamping constant \u000b, which can yield a drift velocity of tens of m/s under a temperature gradient\nof 1 K/mm for a domain wall in an easy-axis antiferromagnetic wire with \u000b\u001810\u00004.\nPACS numbers: 75.78.-n, 66.30.Lw, 75.10.Hk\nIntroduction. |Ordered magnetic materials exhibit\nsolitons and defects that are stable for topological rea-\nsons [1]. Well-known examples are a domain wall (DW)\nin an easy-axis magnet or a vortex in a thin \flm. Their\ndynamics have been extensively studied because of fun-\ndamental interest as well as practical considerations such\nas the racetrack memory [2]. A ferromagnetic (FM) soli-\nton can be driven by various means, e.g., an external\nmagnetic \feld [3] or a spin-polarized electric current [4].\nRecently, the motion of an FM soliton under a temper-\nature gradient has attracted a lot of attention owing to\nits applicability in an FM insulator [5{8]. A temperature\ngradient of 20 K/mm has been demonstrated to drive a\nDW at a velocity of 200 \u0016m/s in an yttrium iron garnet\n\flm [9].\nAn antiferromagnet (AFM) is of a great current inter-\nest in the \feld of spintronics [10{12] due to a few advan-\ntages over an FM. First, the characteristic frequency of\nan AFM is several orders higher than that of a typical\nFM, e.g., a timescale of optical magnetization switching\nis an order of ps for AFM NiO [13] and ns for FM CrO 2\n[14], which can be exploited to develop faster spintronic\ndevices. Second, absence of net magnetization renders\nthe interaction between AFM particles weak, and, thus,\nleads us to prospect for high-density AFM-based devices.\nDynamics of an AFM soliton can be induced by an elec-\ntric current or a spin wave [15{17].\nA particle immersed in a viscous medium exhibits a\nBrownian motion due to a random force that is required\nto exist to comply with the \ructuation-dissipation theo-\nrem (FDT) [18, 19]. An externally applied temperature\ngradient can also be a driving force, engendering a phe-\nnomenon known as thermophoresis [20]. Dynamics of an\nFM and an AFM includes spin damping, and, thus, in-\nvolves thermal \ructuations at a \fnite temperature [21].\nThe corresponding thermal stochastic \feld in\ruences dy-\nnamics of a magnetic soliton [8, 22, 23], e.g., by assisting\na current-induced motion of an FM DW [24].\nFIG. 1. (Color online) A thermal stochastic force caused by\na temperature gradient pushes an antiferromagnetic domain\nwall to a colder region. The di\u000busion coe\u000ecient of the domain\nwall is inversely proportional to a small damping constant,\nwhich may give rise to a sizable drift velocity.\nIn this Rapid Communication, we study the Brown-\nian motion of a soliton in an AFM under a tempera-\nture gradient. We derive the stochastic Landau-Lifshitz-\nGilbert (LLG) equation for an AFM with the aid of the\nFDT, which relates the \ructuation of the staggered and\nnet magnetization to spin damping. We then derive the\nLangevin equation for the soliton's center of mass by em-\nploying the collective coordinate approach [16, 25]. We\ndevelop the Hamiltonian mechanics for collective coordi-\nnates and conjugate momenta of a soliton, which sheds\nlight on stochastic dynamics of an AFM soliton; it can be\nconsidered as a classical massive particle moving in a vis-\ncous medium. By considering a thermodynamic ensemble\nof solitons, we obtain the Fokker-Planck equation, from\nwhich we extract the average drift velocity. As a case\nstudy, we compute the drift velocity of a DW in a quasi\none-dimensional easy-axis AFM.\nThermophoresis of a Brownian particle is a multi-\nfaceted phenomenon, which involves several competing\nmechanisms. As a result, a motion of a particle depends\non properties of its environment such as a medium or\na temperature T[26]. For example, particles in pro-\ntein (e.g., lysozyme) solutions move to a colder region\nforT > 294 K and otherwise to a hotter region [20, 27].arXiv:1503.07854v2  [cond-mat.mes-hall]  8 Jul 20152\nThermophoresis of an AFM soliton would be at least as\ncomplex as that of a Brownian particle. We focus on one\naspect of it in this Rapid Communication; the e\u000bect of\nthermal stochastic force on dynamics of the soliton. We\ndiscuss two other possible mechanisms, the e\u000bects of a\nthermal magnon current and an entropic force [5], later\nin the Rapid Communication.\nMain results .|Before pursuing details of derivations,\nwe \frst outline our three main results. Let us consider\na bipartite AFM with two sublattices that can be trans-\nformed into each other by a symmetry transformation\nof the crystal. Its low-energy dynamics can be devel-\noped in terms of two \felds: the unit staggered spin\n\feldn\u0011(m1\u0000m2)=2 and the small net spin \feld\nm\u0011(m1+m2)=2 perpendicular to n. Here, m1and\nm2are unit vectors along the directions of spin angular\nmomentum in the sublattices.\nStarting from the standard Lagrangian description of\nthe antiferromagnetic dynamics [28], we will show below\nthat the appropriate theory of dissipative dynamics of\nantiferromagnets at a \fnite temperature is captured by\nthe stochastic LLG equation\ns(_n+\fn\u0002_m) =n\u0002(h+hth); (1a)\ns(_m+\fm\u0002_m+\u000bn\u0002_n) =n\u0002(g+gth)\n+m\u0002(h+hth);(1b)\nin conjunction with the correlators of the thermal\nstochastic \felds gthandhth,\nhgth\ni(r;t)gth\nj(r0;t0)i= 2kBT\u000bs\u000e ij\u000e(r\u0000r0)\u000e(t\u0000t0);(2a)\nhhth\ni(r;t)hth\nj(r0;t0)i= 2kBT\fs\u000e ij\u000e(r\u0000r0)\u000e(t\u0000t0);(2b)\nwhich are independent of each other [29]. This is our \frst\nmain result. Here, \u000band\fare the damping constants\nassociated with _nand _m,g\u0011\u0000\u000eU=\u000enandh\u0011\u0000\u000eU=\u000em\nare the e\u000bective \felds conjugate to nandm,U[n;m]\u0011\nU[n]+R\ndVjmj2=2\u001fis the potential energy ( \u001frepresents\nthe magnetic susceptibility), and s\u0011~S=Vis the spin\nangular momentum density ( Vis the volume per spin)\nper each sublattice. The potential energy U[n(r;t)] is\na general functional of n, which includes the exchange\nenergyR\ndVA ij@in\u0001@jnat a minimum [28].\nSlow dynamics of stable magnetic solitons can often\nbe expressed in terms of a few collective coordinates\nparametrizing slow modes of the system. The center of\nmassRrepresents the proper slow modes of a rigid soli-\nton when the translational symmetry is weakly broken.\nTranslation of the stochastic LLG equation (1) into the\nlanguage of the collective coordinates results in our sec-\nond main result, a Langevin equation for the soliton's\ncenter of mass R:\nMR+ \u0000_R=\u0000@U=@R+Fth; (3)\nwhich adds the stochastic force Fthto Eq. (5) of Tveten\net al. [17]. The mass and dissipation tensors are symmet-\nric and proportional to each other: Mij\u0011\u001aR\ndV(@in\u0001@jn) and \u0000 ij\u0011Mij=\u001c;where\u001c\u0011\u001a=\u000bs is the relaxation\ntime,\u001a\u0011\u001fs2is the inertia of the staggered spin \feld\nn. The correlator of the stochastic \feld Fthobeys the\nEinstein relation\nhFth\ni(t)Fth\nj(t0)i= 2kBT\u0000ij\u000e(t\u0000t0): (4)\nA temperature gradient causes a Brownian motion of\nan AFM soliton toward a colder region. In the absence\nof a deterministic force, the average drift velocity is pro-\nportional to a temperature gradient V/kBrTin the\nlinear response regime. The form of the proportional-\nity constant can be obtained by a dimensional analy-\nsis. Let us suppose that the mass and dissipation ten-\nsors are isotropic. The Langevin equation (3) is, then,\ncharacterized by three scalar quantities: the mass M,\nthe viscous coe\u000ecient \u0000, and the temperature T, which\nde\fne the unique set of natural scales of time \u001c\u0011M=\u0000,\nlengthl\u0011pkBTM= \u0000, and energy \u000f\u0011kBT. Using\nthese scales to match the dimension of a velocity yields\nV=\u0000c\u0016(kBrT);where\u0016\u0011\u0000\u00001is the mobility of an\nAFM soliton and cis a numerical constant. The explicit\nsolution of the Fokker-Planck equation, indeed, shows\nc= 1. This simple case illustrates our last main result;\na drift velocity of an AFM soliton under a temperature\ngradient in the presence of a deterministic force Fis given\nby\nV=\u0016F\u0000\u0016(kBrT): (5)\nFor a DW in an easy-axis one-dimensional AFM, the\nmobility is \u0016=\u0015=2\u000bs\u001b, where\u0015is the width of the\nwall and\u001bis the cross-sectional area of the AFM. For\na numerical estimate, let us take an angular momen-\ntum density s= 2~nm\u00001, a width\u0015= 100 nm, and a\ndamping constant \u000b= 10\u00004following the previous stud-\nies [17, 30]. For these parameters, the AFM DW moves\nat a velocity V= 32 m/s for the temperature gradient of\nrT= 1 K/mm.\nStochastic LLG equation. |Long-wave dynamics of an\nAFM on a bipartite lattice at zero temperature can de-\nscribed by the Lagrangian [28]\nL=sZ\ndVm\u0001(n\u0002_n)\u0000U[n;m]: (6)\nWe use the potential energy U[n;m]\u0011R\ndVjmj2=2\u001f+\nU[n] throughout the Rapid Communication, which re-\nspects the sublattice exchange symmetry ( n!\u0000n;m!\nm). Minimization of the action subject to nonlinear con-\nstraintsjnj= 1 and n\u0001m= 0 yields the equations of mo-\ntion for the \felds nandm. Damping terms that break\nthe time reversal symmetry can be added to the equa-\ntions of motion to the lowest order, which are \frst order\nin time derivative and zeroth order in spatial derivative.\nThe resultant phenomenological LLG equations are given3\nby\ns(_n+\fn\u0002_m) =n\u0002h; (7a)\ns(_m+\fm\u0002_m+\u000bn\u0002_n) =n\u0002g+m\u0002h (7b)\n[16, 30, 31]. The damping terms can be derived from the\nRayleigh dissipation function\nR=Z\ndV(\u000bsj_nj2+\fsj_mj2)=2; (8)\nwhich is related to the energy dissipation rate by \u0000_U=\n2R. The microscopic origin of damping terms does not\nconcern us here but it could be, e.g., caused by thermal\nphonons that deform the exchange and anisotropy inter-\naction.\nAt a \fnite temperature, thermal agitation causes \ruc-\ntuations of the spin \felds nandm. These thermal \ruc-\ntuations can be considered to be caused by the stochas-\ntic \felds gthandhthwith zero mean, which are con-\njugate to nandm, respectively; their noise correlators\nare then related to the damping coe\u000ecients by the FDT.\nThe standard procedure to construct the noise sources\nyields the stochastic LLG equation (1). The correlators\nof the stochastic \felds are obtained in the following way\n[18, 32]. Casting the linearized LLG equation (7) into the\nformfh;gg= ^\r\nf_n;_mgprovides the kinetic coe\u000ecients\n^\r. Symmetrizing the kinetic coe\u000ecients ^ \rproduces the\ncorrelators (2) of the stochastic \felds consistent with the\nFDT.\nLangevin equation. |For slow dynamics of an AFM,\nthe energy is mostly dissipated through the temporal\nvariation of the staggered spin \feld ndue toj_mj2'\n(\u000b\u001c)2jnj2\u001cj_nj2(from Eq. (7)), which allows us to set\n\f= 0 to study long-term dynamics of the magnetic soli-\nton [17]. At this point, we switch to the Hamiltonian for-\nmalism of an AFM [33], which sheds light on the stochas-\ntic dynamics of a soliton. The canonical momentum \feld\n\u0019conjugate to the staggered spin \feld nis\n\u0019\u0011\u000eL=\u000e _n=sm\u0002n: (9)\nThe stochastic LLG equations (1) can be interpreted as\nHamilton's equations,\n_n=\u000eH=\u000e \u0019=\u0019=\u001a; _\u0019=\u0000\u000eH=\u000en\u0000\u000eR=\u000e _n+gth;(10)\nwith the Hamiltonian\nH\u0011Z\ndV\u0019\u0001_n\u0000L=Z\ndVj\u0019j2\n2\u001a+U[n]: (11)\nLong-time dynamics of magnetic texture can often be\ncaptured by focusing on a small subset of slow modes,\nwhich are parametrized by the collective coordinates\nq=fq1;q2;\u0001\u0001\u0001g. A classical example is a DW in a one-\ndimensional easy-axis magnet described by the position\nof the wall Xand the azimuthal angle \b [3, 33]. An-\nother example is a skyrmion in an easy-axis AFM \flm,which is described by the position R= (X;Y ) [34, 35].\nTranslation from the \feld language into that of collective\ncoordinates can be done as follows. If the staggered spin\n\feldnis encoded by coordinates qasn(r;t) =n[r;q(t)],\ntime dependence of nre\rects evolution of the coordi-\nnates: _n= _qi@n=@qi. With the canonical momenta p\nde\fned by\npi\u0011@L\n@_qi=Z\ndV@n\n@qi\u0001\u0019; (12)\nHamilton's equations (10) translate into\nM_q=p;_p+ \u0000_q=F+Fth; (13)\nwhere F\u0011\u0000@U=@qis the deterministic force and Fth\ni\u0011R\ndV@ qin\u0001gthis the stochastic force. Hamilton's equa-\ntions (13) can be derived from the Hamiltonian in the\ncollective coordinates and conjugate momenta,\nH\u0011pTM\u00001p=2 +U(q); (14)\nwith the Poisson brackets fqi;pjg=\u000eij;fqi;qjg=\nfpi;pjg= 0. An AFM soliton, thus, behaves as a classi-\ncal particle moving in a viscous medium.\nWe focus on a translational motion of a rigid AFM\nsoliton by choosing its center of mass as the collective\ncoordinates q=R;n(r;t) =n(r\u0000R(t)). Eliminat-\ning momenta from Hamilton's equations (13) yields the\nLangevin equation for the soliton's center of mass:\n\u001cR+_R=\u0016F+\u0011; (15)\nwhere \u0011\u0011\u0016Fthis the stochastic velocity. Here the mo-\nbility tensor of the soliton \u0016\u0011\u0000\u00001relates a deterministic\nforce to a drift velocity h_Ri=\u0016Fat a constant temper-\nature [36]. The mobility is inversely proportional to a\ndamping constant, which can be a small number for an\nAFM, e.g., \u000b\u001810\u00004for NiO [37]. The correlator (2) of\nthermal stochastic \felds is translated into the correlator\nof the stochastic velocity,\nh\u0011i(t)\u0011j(t0)i= 2kBT\u0016ij\u000e(t\u0000t0)\u00112Dij\u000e(t\u0000t0):(16)\nFrom Eq. (16), we see that di\u000busion coe\u000ecient and the\nmobility of the soliton respect the Einstein-Smoluchowski\nrelation:D=\u0016kBT, which is expected on general\ngrounds. It can also be explicitly veri\fed as follows.\nA system of an ensemble of magnetic solitons at ther-\nmal equilibrium is described by the partition function\nZ\u0011R\n\u0005i[dpidxi=2\u0019~] exp(\u0000H=k BT), which provides the\nautocorrelation of the velocity, h_xi_xji=2 =M\u00001\nijkBT=2\n(the equipartition theorem). In the absence of an ex-\nternal force, multiplying \u001cxi+ _xi=\u0011i(15) byxjand\nsymmetrizing it with respect to indices iandjgive the\nequation,\u001cd2hxixji=dt2+dhxixji=dt= 2\u001ch_xi_xji, where\nthe \frst term can be neglected for long-term dynamics\nt\u001d\u001c. This equation in conjunction with the autocorre-\nlation of the velocity allows us to obtain the di\u000busion co-\ne\u000ecientDijin Eq. (16),hxixji= 2kBT\u001cM\u00001\nijt= 2Dijt,4\nwithout prior knowledge about the correlator (2) of the\nstochastic \felds.\nAverage dynamics. |An AFM soliton exhibits Brown-\nian motion at a \fnite temperature. The following Fokker-\nPlanck equation for an ensemble of solitons in an inho-\nmogeneous medium describes the evolution of the density\n\u001a(R;t) at timet\u001d\u001c:\n@\u001a\n@t+r\u0001j= 0;withj\u0011\u0016F\u001a\u0000Dr\u001a\u0000DT(kBrT);(17)\nwhereDT\u0011\u0016\u001ais the thermophoretic mobility (also\nknown as the thermal di\u000busion coe\u000ecient) [20, 38]. A\nsteady-state current density j=\u0016F\u001a0\u0000DT(kBrT) with\na constant soliton density \u001a(r;t) =\u001a0solves the Fokker-\nPlanck equation (17), from which the average drift veloc-\nity of a soliton can be extracted [39]:\nV=\u0016F\u0000\u0016(kBrT): (18)\nLet us take an example of a DW in a quasi one-\ndimensional easy-axis AFM with the energy U[n] =R\ndV(Aj@xnj2\u0000Kn2\nz)=2. A DW in the equilibrium\nisn(0)= (sin\u0012cos \b;sin\u0012sin \b;cos\u0012) with cos \u0012=\ntanh[(x\u0000X)=\u0015], where\u0015\u0011p\nA=K is the width of\nthe wall. The position Xand the azimuthal angle \b\nparametrize zero-energy modes of the DW, which are en-\ngendered by the translational and spin-rotational symme-\ntry of the system. Their dynamics are decoupled, \u0000 X\b=\n0, which allows us to study the dynamics of Xseparately\nfrom \b. The mobility of the DW is \u0016=\u0015=2\u000bs\u001b, where\u001b\nis the cross-sectional area of the AFM. The average drift\nvelocity (18) is given by\nV=\u00001\n2\u000bkB\u0015rT\ns\u001b: (19)\nDiscussion |The deterministic force Fon an AFM\nsoliton can be extended to include the e\u000bect of an elec-\ntric current, an external \feld, and a spin wave [15{17]. It\ndepends on details of interaction between the soliton and\nthe external degrees of freedom, whose thorough under-\nstanding would be necessary for a quantitative theory for\nthe deterministic drift velocity \u0016F. The Brownian drift\nvelocity V(18) is, however, determined by local property\nof the soliton. We have focused on the thermal stochastic\nforce as a trigger of thermophoresis of an AFM soliton in\nthis Rapid Communication. There are two other possi-\nble ingredients of thermophoresis of a magnetic soliton.\nOne is a thermal magnon current, scattering with which\ncould exert a force on a soliton [40]. The other is an\nentropic force, which originates from thermal softening\nof the order-parameter sti\u000bness [5]. E\u000bects of these two\nmechanisms have not been studied for an AFM soliton;\nfull understanding of its thermophoresis is an open prob-\nlem.\nIn order to compare di\u000berent mechanisms of thermally-\ndriven magnetic soliton motion, let us address a closelyrelated problem of thermophoresis of a DW in a quasi\none-dimensional FM wire with an easy- xz-plane easy-\nz-axis [3], which has attracted a considerable scrutiny\nrecently. To that end, we have adapted the approach de-\nveloped in this Rapid Communication to the FM case,\nwhich leads to the conclusion that a DW drifts to a\ncolder region by a Brownian stochastic force at the ve-\nlocity given by the same expression for an AFM DW,\nVB=\u0000kB\u0015rT=2\u000bs\u001b [41]. A thermal magnon cur-\nrent pushes a DW to a hotter region at the velocity\nVM=kBrT=6\u00192s\u0015m, where\u0015m\u0011p\n~A=sT is the\nthermal-magnon wavelength [7]. According to Schlick-\neiser et al. [5], an entropic force drives a DW to a hotter\nregion at the velocity VE=kBrT=4sa, whereais the\nlattice constant. The Brownian stochastic force, there-\nfore, dominates the other forces for a thin wire, \u001b\u001c\u0015a=\u000b\n(supposing rigid motion) [42].\nWithin the framework of the LLG equations that are\n\frst order in time derivative, the thermal noise is white\nas long as slow dynamics of a soliton is concerned,\ni.e., the highest characteristic frequency of the natural\nmodes parametrized by the collective coordinates is much\nsmaller than the temperature scale, ~!\u001ckBT. The\nthermal noise could be colored in general [26], e.g., for\nfast excitations of magnetic systems, which may be ex-\namined in the future. In addition, local energy dissi-\npation (8) allowed us to invoke the standard FDT at\nthe equilibrium to derive the stochastic \felds. It would\nbe worth pursuing to understand dissipative dynamics\nof general magnetic systems, e.g., with nonlocal energy\ndissipation with the aid of generalized FDTs at the out-\nof-equilibrium [43].\nWe have studied dynamics of an AFM soliton in the\nHamiltonian formalism. Hamiltonian's equations (13) for\nthe collective coordinates and conjugate momenta can be\nderived from the Hamiltonian (14) with the conventional\nPoisson bracket structure. By replacing Poisson brackets\nwith commutators, the coordinates and conjugate mo-\nmenta can be promoted to quantum operators. This may\nprovide a one route to study the e\u000bect of quantum \ruc-\ntuations on dynamics of an AFM soliton [44].\nAfter the completion of this work, we became aware\nof two recent reports. One is on thermophoresis of an\nAFM skyrmion [35], whose numerical simulations sup-\nport our result on di\u000busion coe\u000ecient. 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Usp. 23,\n21 (1980).\n[29] Employing the quantum FDT would change the corre-\nlator of the stochastic \felds to hhth\ni(r;!)hth\nj(r0;!0)i=\n[2\u0019\u000eij\u000bs~!=tanh( ~!=2kBT)]\u000e(r\u0000r0)\u000e(!\u0000!0) in the fre-\nquency space [18]. We focus on slow dynamics of an AFM\nsoliton in the manuscript, ~!\u001ckBT, which allows us to\nreplace ~!=tanh( ~!=2kBT) with 2kBT, yielding Eq. (2).\n[30] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nPhys. Rev. B 90, 094408 (2014).\n[31] B. A. Ivanov and D. D. Sheka, Phys. Rev. Lett. 72, 404\n(1994); N. Papanicolaou, Phys. Rev. B 51, 15062 (1995);\nH. V. Gomonay and V. M. Loktev, Phys. Rev. B 81,\n144427 (2010); A. C. Swaving and R. A. Duine, Phys.\nRev. B 83, 054428 (2011).\n[32] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79,\n014402 (2009).\n[33] H. J. Mikeska, J. Phys. C: Solid St. Phys. 13, 2913 (1980);\nF. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n[34] I. Rai\u0014 cevi\u0013 c, D. Popovi\u0013 c, C. 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Also for our\nexample|a DW in a 1D easy-axis AFM|the conserva-\ntive force and torque exerted by magnons vanish [17].\n[41] Unlike 1D domain walls, Brownian motions are drasti-\ncally distinct between 2D FM and AFM solitons due to\nthe gyrotropic force, which signi\fcantly slows down fer-\nromagnetic di\u000busion [23, 35].\n[42] A DW in a wire with a large crosssection \u001b\u001da2forms\na 2D membrane. Its \ructuations foment additional soft\nmodes of the dynamics, which needs to be taken into\naccount to understand the dynamics of such a DW [25].\n[43] M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett.\n103, 010602 (2009); U. Seifert and T. Speck, Europhys.\nLett. 89, 10007 (2010).\n[44] S.-Z. Lin and L. N. Bulaevskii, Phys. Rev. B 88, 060404\n(2013).\n[45] P. Yan, Y. Cao, and J. Sinova, arXiv:1504.00651."    },    {        "title": "2203.03225v2.Ultrafast_optical_observation_of_spin_pumping_induced_dynamic_exchange_coupling_in_ferromagnetic_semiconductor_metal_bilayer.pdf",        "content": "Ultrafast optical observation of spin-pumping induced dynamic exchange coupling in\nferromagnetic semiconductor/metal bilayer\nX. Liu,1,\u0003P. Liu,1,\u0003H. C. Yuan,2J. Y. Shi,2H. L. Wang,3S. H. Nie,3F.\nJin,2Z. Zheng,2X. Z. Yu,3J. H. Zhao,3,yH. B. Zhao,2,zand G. L upke1,x\n1Department of Applied Science, The College of William and Mary, Williamsburg, Virginia, 23187, USA\n2Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education),\nShanghai Ultra-precision Optical Manufacturing Engineering Research Center,\nDepartment of Optical Science and Engineering, Fudan University, Shanghai, 200433, China\n3State Key Laboratory of Supperlattices and Microstructures,\nInstitute of Semiconductors, Chinese Academy of Sciences, Beijing, 100083, China\n(Dated: May 10, 2022)\nSpin angular momentum transfer in magnetic bilayers o\u000bers the possibility of ultrafast and low-loss\noperation for next-generation spintronic devices. We report the \feld- and temperature- dependent\nmeasurements on the magnetization precessions in Co 2FeAl/(Ga,Mn)As by time-resolved magneto-\noptical Kerr e\u000bect (TRMOKE). Analysis of the e\u000bective Gilbert damping and phase shift indicates\na clear signature of an enhanced dynamic exchange coupling between the two ferromagnetic (FM)\nlayers due to the reinforced spin pumping at resonance. The temperature dependence of the dynamic\nexchange-coupling reveals a primary contribution from the ferromagnetism in (Ga,Mn)As.\nINTRODUCTION\nThere has been growing interest in the ultrafast op-\ntical manipulation of magnetic dynamics in ferromag-\nnetic heterostructures due to its potential applications\nin advanced functional spintronic devices. The spin-\npumping (SP) e\u000bect, in which a spin-precessing ferro-\nmagnetic layer transfers its angular momentum into an-\nother layer by a chargeless spin current, brings a new\nmechanism for spin controlling and hence plays an im-\nportant role in the design of future spintronic devices.[1]\nSince Heinrich, B. et al \frst reported the spin-pumping\ne\u000bect as increased damping of the source layer in fer-\nromagnetic resonance (FMR) experiments [2] and a few\nFMR experiments on SP e\u000bect have been performed on\ntransition-metal multilayers [3{6], topological insulators\n[7, 8] and semiconductors [9{11]. In addition, Danilov\nA. et al. demonstrated that the mutual SP e\u000bect modi-\n\fes the precession dynamics in a pseudo spin-valve where\nmagnetization precessions are excited simultaneously in\ntwo FM layers by femtosecond laser pulses.[12] However,\nno SP e\u000bect has ever been observed yet for the het-\nerostructure of a Heusler alloy and a ferromagnetic (FM)\nsemiconductor. Importantly, the hard and soft ferromag-\nnetic phases in such materials can potentially exhibit a\ndynamic exchange coupling that is completely indepen-\ndent of the static exchange coupling due to spin pumping.\nThis could o\u000ber a possibility of ultrafast low-power con-\ntrol of spin current for next-generation spintronic devices.\nIn this study, we investigate the magnetization preces-\nsion dynamics of the Heusler alloy Co 2FeAl/FM semi-\nconductor (Ga,Mn)As heterostructure as a function of\napplied \feld and temperature by time-resolved magneto-\noptical Kerr e\u000bect (TRMOKE). Analysis of the \feld-\ndependent e\u000bective Gilbert damping indicates a clear\nsignature of the enhanced dynamic exchange couplingbetween the two FM layers due to a reinforced spin\npumping. In addition, curvatures of the phase shift as a\nfunction of applied \feld elucidate the dynamic exchange-\ncoupling model where the counter-precessing precessions\nare damped signi\fcantly at the resonant frequency of\nthe two FM layers. The magnetization precession in the\nCo2FeAl layer transfers a pure spin current directly into\nthe ferromagnetic semiconductor (Ga,Mn)As layer with-\nout a nonmagnetic metal spacer. On the other hand,\nthe temperature-dependent results manifest a strong con-\ntribution from the ferromagnetism of (Ga,Mn)As to the\ndynamic exchange-coupling e\u000bect. These results provide\nvaluable insight into the topic of dynamic exchange cou-\npling and the detection of spin current. Furthermore,\nthey suggest a new pathway of ultrafast spin manipula-\ntion in metal/semiconductor bilayer systems at low power\nand therefore promote the development and design of fu-\nture spintronic devices.\nRESULTS AND DISCUSSION\nThe Co 2FeAl/(Ga,Mn)As bilayer sample is grown\non GaAs (001) substrates by molecular-beam epitaxy\n(MBE). The thickness of Co2FeAl and Ga1-xMnxAs\n(x=0.07) layer is 10 nm and 150 nm, respectively. The\nsample is capped with 2-nm thick Al layer to avoid ox-\nidation and contamination. The hard FM Co 2FeAl ex-\nhibits an in-plane uniaxial magnetic anisotropy with an\neasy axis along the [110] direction (Fig. 1(a)), whereas\nthe easy axis of the soft FM (Ga,Mn)As is along the [1-\n10] direction below its Curie temperature Tc= 50 K.[13]\nRe\rection high-energy electron di\u000braction (RHEED)\npatterns, high-resolution double-crystal x-ray di\u000brac-\ntion (DCXRD) measurements, and high-resolution cross-\nsectional transmission electron microscopy (HRTEM) re-arXiv:2203.03225v2  [cond-mat.mtrl-sci]  7 May 20222\nFIG. 1. (a) Schematic of TRMOKE measurement geom-\netry, depicting the structure of the sample and the mag-\nnetization M precessing around the e\u000bective \feld He\u000b in\nCo2FeAl/(Ga,Mn)As bilayer in a canted magnetization con-\n\fguration with H applied along hard axis [1-10]. (b)\nTRMOKE data from Co 2FeAl/(Ga,Mn)As bilayer under the\ndi\u000berent intensity of the applied \feld. (c) FFT analysis for\nmagnetization precession frequency under the di\u000berent inten-\nsities of the applied \feld, where the solid lines represent the\nFFT peaks \ftted by Lorentz functions.\nveal high-quality, single-crystalline, epitaxial growth of\nthe Co2FeAl and (Ga,Mn)As thin \flms.[13] At low tem-\nperatures ( T < T c), a ferromagnetic alignment of lo-\ncal Mn moments in the (Ga,Mn)As layer is expected,\nwhereas at high temperatures ( T >T c) the Mn ions ex-\ntending a few nanometers from the interface remain spin-\npolarized due to the ferromagnetic proximity e\u000bect.[13]\nFigure 1(a) shows the experimental geometry\nof TRMOKE measurements. Field-dependent\nTRMOKE measurements are performed on the\nCo2FeAl/(Ga,Mn)As bilayer sample from 7 K to\n300 K utilizing 100-fs pump and probe pulses at 800\nnm wavelength with a repetition rate of 80 MHz. The\nexternal magnetic \feld is set along the easy-hard axis\n[100] of the Co 2FeAl layer and the pump \ruence is\nset atI= 5\u0016J/cm2. The probe pulses utilize the\nbalanced detection technique with a half-wave plate and\nWollaston prism to investigate the transient magnetic\nstate change along longitudinal and polar directions.\nFigure 1b displays the TRMOKE data at 10 K with\nan in-plane magnetic \feld scanning from 366 Oe to\n550 Oe. The precession signals can be well \ftted by a\ndamped-harmonic function with a linear background:\n\u0012k=a0+b0t+Aexp (\u0000t=\u001c) sin (2\u0019ft+'0) , where\na0+b0trepresents the linear background, Ais the\nprecession amplitude, \u001cis the relaxation time, f is the\nprecession frequency and '0is the phase. The mag-\nnetization precession decays with di\u000berent relaxation\ntimes, with the fastest decay at 458 Oe. This indicates\nthat a dynamic exchange coupling may occur between\nthe magnetization precession in the Co 2FeAl layer and\nthe (Ga,Mn)As layer. Here, only one frequency can\nbe extracted from FFT analyses, as seen in Fig. 1(c),\ncoherent spin precession of (Ga,Mn)As decays fast and\nthus vanishes very shortly.\nThe magnetization precession in the\nCo2FeAl/(Ga,Mn)As bilayer system is described bythe following modi\fed Landau-Lifshitz-Gilbert(LLG)\nequation with an additional spin-torque term:\ndM\ndt=\u0000\rM\u0002He\u000b+\u000b0M\u0002dM\ndt+\u000bsp\u0012\nM\u0002dM\ndt\u0000M0\u0002dM0\ndt\u0013\nwhere Mis the magnetization direction of the Co 2FeAl\nlayer,\ris the gyromagnetic ratio, \u000b0is the intrinsic\nGilbert damping constant, and He\u000bis the e\u000bective mag-\nnetic \feld in the Co 2FeAl layer including the external\nmagnetic \feld, the demagnetization \feld, the anisotropy\n\feld, and the exchange-coupling \feld. The last term de-\nscribes the spin torque which acts on both layers as a\nbidirectional e\u000bect, in which \u000bsprepresents the contribu-\ntion of spin pumping to the damping and M0denotes the\nmagnetization of (Ga,Mn)As. Then, the e\u000bective Gilbert\ndamping can be obtained from the relaxation time \u001c, us-\ning [14, 15]\n\u000b= 2=(\u001c\r(Ha+Hb))\nwhereHaandHbare determined in our previous anal-\nysis, which includes the out-of-plane, in-plane uniaxial,\ncrystalline cubic, unidirectional and rotatable magnetic\nanisotropies.[15]\nFigure 2(a) shows the temperature-dependent Gilbert\ndamping as a function of the external \feld. At T=\n10 K, the damping of the magnetization precession is\nmost pronounced with an external \feld H= 458 Oe. Be-\nlow the Curie temperature ( Tc= 50 K) of (Ga,Mn)As,\nthe damping peak \frst shifts from 450 Oe to 650 Oe\nwith the temperature increasing from 10 K to 35 K, as\nsummarized in Fig. 2(b) insert, and then gradually be-\ncomes inconspicuous and \fnally disappears at T=Tc.\nSuch a temperature dependency clearly shows that the\ndamping peak only exists when the ferromagnetism of\n(Ga,Mn)As is well-developed. Meanwhile, the strongest\nGilbert damping extracted across all the \felds as a func-\ntion of temperature (Fig. 2(b)) shows a transition tem-\nperature close to Tc. This manifests the crucial role of\nspontaneous (Ga,Mn)As magnetization in the damping\nof magnetization precession of Co 2FeAl.\nThe ultrafast pump excitation causes a transient en-\nhancement of exchange coupling,[15] which induces a dy-\nnamic exchange-coupling torque acting on both Fe(Co)\nspins and Mn spins. In such a case, as shown in Fig. 2(c),\nthe magnetizations of both FM layers are suddenly pulled\ntowards each other and start to precess with opposite an-\ngular momentum along their own equilibrium directions.\nAt the resonance, i.e., fCFA =fGMA , the precessing\nmagnetization of Co 2FeAl \\pumps\" a spin current Iidi-\nrectly into the (Ga,Mn)As layer, which exerts a torque\nonto the (Ga,Mn)As magnetization and thereby counter-\nacts its precession. Meanwhile, this spin current Iicar-\nries an out\row of angular momentum from the Co 2FeAl3\nFIG. 2. (a) E\u000bective Gilbert damping constant as a func-\ntion of externally applied \feld at di\u000berent temperatures from\n10 K to 50 K. (b) Red rectangular solids denote the peak\n(or strongest if not peak) Gilbert damping from 10 K to\n50 K. Blue circle solids are the corresponding \felds as a\nfunction of temperature. (c) Illustration of photo-excited\nexchange-coupling torque and spin-pumping generated dy-\nnamic exchange-coupling mode for damping.\nlayer and leads to damping to its magnetization preces-\nsion. In other words, the spin current reinforces the M\ndamping for both FM layers at the resonance. Techni-\ncally, there should also be a spin current Ijinjecting into\nthe Co 2FeAl layer from the magnetization precession of\n(Ga,Mn)As. [2] However, such a spin current should be\nmuch smaller than that from the Co 2FeAl layer.\nIn addition to the discussion on damping, the dynamic\nexchange coupling between the two FM layers can also be\nevinced by the \feld-dependency of the precession phase.\nFig. 3(a) shows that from 10 K to 35 K, the phase drops\ndown dramatically around certain \feld windows that\ncorrespond to the Gilbert damping peaks, which move\nto higher \feld ranges as temperature increases. When\nT > 30 K, the dramatic phase shift becomes less con-\ntrastive and then completely disappears when T=Tc.\nWe notice that similar features of phase-shifting are re-\nported in the FMR experiments on similar hard/soft FM\nsystems.[16, 17] The observed 30 °{ 40 °phase shift at 10\nK, as shown in Fig. 3(c), is comparable with those of the\ndynamic exchange-coupling spin-valve structures.[7, 16{\n19]\nFIG. 3. Phase of magnetization precession as a function of\nexternally applied \feld at di\u000berent temperatures from 10 K to\n50 K (a), and zoomed speci\fcally at 10 K (b). (c) TRMOKE\ndata as a function of normalized precession phase under dif-\nferent applied \felds. The red straight line is a guideline of\nthe phase shift.\nCONCLUSION\nIn summary, we have studied the dynamics of the\nmagnetization precession of Co 2FeAl/(Ga,Mn)As het-\nerostructure as a function of applied \feld and temper-\nature. The pronounced peaks in e\u000bective Gilbert damp-\ning of Co 2FeAl magnetization precession reveal the en-\nhanced dynamic exchange coupling between the two FM\nlayers due to spin pumping. The corresponding \feld-\ndependency of phase shift, which corresponds to that\nof the Gilbert damping, con\frms the counter-precessing\nexchange-coupled model where both precessions are\ndamped signi\fcantly at the resonance. In addition, the\ntemperature-dependent results manifest a strong contri-\nbution from the ferromagnetism in (Ga,Mn)As to the dy-\nnamic exchange-coupling e\u000bect. These results provide\nvaluable insight into the topic of dynamic exchange cou-\npling and the detection of spin current. Mover, they sug-\ngest a novel route of ultrafast low-power spin manipu-\nlation in metal/semiconductor bilayer system and hence\npromote the research of the future spintronic devices.4\nMETHODS\nMOKE experiments. The magnetization of the\nexchange-coupled Co 2FeAl/(Ga,Mn)As bilayer is mea-\nsured using a longitudinal MOKE setup. The sample\nis illuminated with p-polarized light and the re\rected s-\npolarized light is detected with a photodiode. The mag-\nnetic \feld is applied along the in-plane [110] or [-110]\ncrystallographic directions. The measurements are con-\nducted from 5 K to above room temperature.\nTRMOKE experiments. For the pump-probe\nTRMOKE measurements, a Ti:sapphire oscillator laser\nsystem is employed, which produces 150-fs pulses at 800-\nnm wavelength with a repetition rate of 80 MHz. The\nprobe(pump) \ruence is \fxed at \u00180.5(5)\u0016J/cm2. The\nprobe pulses ( \u0015= 800 nm) use the balanced detection\napproach with a half-wave plate and Wollaston prism to\ninvestigate the transient magnetic state change along lon-\ngitudinal and polar directions. The measurements are\nconducted from 5 K to above room temperature.\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.\nACKNOWLEDGMENTS\nThe work at the College of William and Mary was\nsponsored by the DOE through Grant No. DEFG02-\n04ER46127. The work at the Department of Optical\nScience and Engineering, Fudan University, was sup-\nported by the National Natural Science Foundation of\nChina with Grant No. 11774064, National Key Re-\nsearch and Development Program of China (Grant No.\n2016YFA0300703), and National Key Basic Research\nProgram (No. 2015CB921403). The work at the State\nKey Laboratory of Superlattices and Microstructures, In-\nstitute of Semiconductors, Chinese Academy of Sciences,\nwas supported by National Natural Science Foundation\nof China with Grant No. U1632264.\nAUTHOR CONTRIBUTIONS\nX. L., P. L., H. C. Y, J. H. Z., H. B. Z., and G. L.\ndesigned and analyzed the experiments. H. L. W., S.\nH. N., J. Y. S., and X. Z. Y. prepared the samples and\ncarried out characterizations using MOKE, RHEED, and\nSQUID measurements. H. C. Y., J. Y. S., X. L., P. L.,\nand F. J. performed the TRMOKE experiments. X. L.,\nP. L., H. C. Y. and J. Y. S. conducted the data analysis\nand simulations. All authors discussed the results. X. L.,P. L., J. H. Z., H. B. Z., and G. L. wrote the manuscript\nwith contributions from all authors.\nCOMPETING INTERESTS\nThe authors declare no competing \fnancial interests.\n\u0003These authors contributed equally to the work.\nyemail: jhzhao@red.semi.ac.cn\nzemail: hbzhao@fudan.edu.cn\nxemail: gxluep@wm.edu\n[1] Tserkovnyak, Y. et al. Enhanced Gilbert damping in thin\nferromagnetic \flm. Phys. Rev. Lett. 88, 117601 (2002).\n[2] Heinrich, B. et al. Dynamic exchange coupling in mag-\nnetic bilayers. Phys. Rev. Lett. 90, 187601 (2003).\n[3] Stenning, G. B. G. et al. Magnetization dynamics in an\nexchange-coupled NiFe/CoFe bilayer studied by x-ray de-\ntected ferromagnetic resonance. New J. Phys. 17, 013019\n(2015).\n[4] van der Laan, G. et al. Time-resolved X-ray detected fer-\nromagnetic resonance of spin currents. J. Electron Spec-\ntrosc. 220, 137-146 (2017).\n[5] Saitoh, E. et al. Conversion of spin current into charge\ncurrent at room temperature: Inverse spin-Hall e\u000bect.\nAppl. Phys. Lett. 88, 182509 (2006).\n[6] Costache, M. V. et al. Electrical detection of spin pump-\ning due to the precessing magnetization of a single ferro-\nmagnet. Phys. Rev. Lett. 97, 216603 (2006).\n[7] Figueroa, A. I. et al. Spin pumping through a topo-\nlogical insulator probed by x-ray detected ferromagnetic\nresonance. J. Magn. Magn. Mater. 400, 178-183 (2016).\n[8] Shiomi, Y. et al. Spin-electricity conversion induced by\nspin injection into topological insulators. Phys. Rev. Lett.\n113, 196601 (2014).\n[9] Shikoh, E. et al. Spin-pump-induced spin transport in\np-type Si at room temperature. Phys. Rev. Lett. 110,\n127201 (2013).\n[10]\u0014Zuti\u0013 c, I. & Dery, H. Spintronics: Taming spin currents.\nNat. Mater. 10, 647 (2011).\n[11] Ando, K. et al. Electrically tunable spin injector free\nfrom the impedance mismatch problem. Nat. Mater. 10,\n655 (2011).\n[12] Danilov, A. P. et al. Optically excited spin pumping\nmediating collective magnetization dynamics in a spin\nvalve structure. Phys. Rev. B 98, 060406 (2018).\n[13] Nie, S. H. et al. Ferromagnetic interfacial interaction\nand the proximity e\u000bect in a Co2FeAl/(Ga,Mn)As bilaye.\nPhys. Rev. Lett. 111, 027203 (2013).\n[14] Zhao, H. B. et al. Ultrafast magnetization dynamics of\nepitaxial Fe \flms on AlGaAs (001). Appl. Phys. Lett. 86,\n152512 (2005).\n[15] Liu, X. et al. Ultrafast enhancement of interfa-\ncial exchange coupling in ferromagnetic bilayer. ,\narXiv:2203.00293 ().\n[16] Marcham, M. K. et al. Phase-resolved x-ray ferromag-\nnetic resonance measurements of spin pumping in spin\nvalve structures. Phys. Rev. B 87, 180403 (2013).\n[17] Baker, A. A. et al. Anisotropic absorption of pure spin\ncurrents. Phys. Rev. Lett 116, 047201 (2016).5\n[18] Li, J. et al. Direct detection of pure ac spin current by\nx-ray pump-probe measurements. Phys. Rev. Lett 117,\n076602 (2016).\n[19] Baker, A. A. et al. Spin pumping in magnetic trilayer\nstructures with an MgO barrier. Sci. Rep. 6, 35582(2016)."    },    {        "title": "0810.1340v2.Transverse_spin_diffusion_in_ferromagnets.pdf",        "content": "Transverse spin di\u000busion in ferromagnets\nYaroslav Tserkovnyak,1E. M. Hankiewicz,2and Giovanni Vignale3\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Institut f ur Theoretische Physik und Astrophysik,\nUniversit at W urzburg, 97074 W urzburg, Germany\n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n(Dated: October 29, 2018)\nWe discuss the dissipative di\u000busion-type term of the form m\u0002r2@tmin the phenomenological\nLandau-Lifshitz equation of ferromagnetic precession, which describes enhanced Gilbert damping of\n\fnite-momentum spin waves. This term arises physically from itinerant-electron spin \rows through\na perturbed ferromagnetic con\fguration and can be understood to originate in the ferromagnetic\nspin pumping in the continuum limit. We develop a general phenomenology as well as provide\nmicroscopic theory for the Stoner and s-dmodels of ferromagnetism, taking into account disorder\nand electron-electron scattering. The latter is manifested in our problem through the Coulomb drag\nbetween the spin bands. The spin di\u000busion is identi\fed in terms of the transverse spin conductivity,\nin analogy with the Einstein relation in the kinetic theory.\nPACS numbers: 75.30.Ds,72.25.-b,76.50.+g,75.45.+j\nI. INTRODUCTION\nThe problem of spin di\u000busion through conducting fer-\nromagnetic medium attracted much attention over sev-\neral decades.1,2,3,4,5,6Semiclassical spin transport in the\npresence of a weak magnetic \feld Hcan be captured by\nthe conventional di\u000busion equation (neglecting spin re-\nlaxation):\n@tS=H\u0002S+Dr2S; (1)\nwhereDis the di\u000busion coe\u000ecient and His the total\ne\u000bective \feld (omitting the gyromagnetic ratio), includ-\ning the applied and exchange contributions. The \frst\nterm on the right-hand side describes spin precession in\nthe local \feld while the second term stands for the ordi-\nnary di\u000busion of spin density S. Equation (1) is, how-\never, not applicable to most realistic ferromagnets, whose\nspin interactions are characterized by a large exchange\nenergy \u0001 xc. In particular, when \u0001 xcis comparable to\nthe Fermi energy (which is the case in transition met-\nals), the spin precession in the exchange \feld cannot be\ntreated in the di\u000busive transport framework. Further-\nmore, the time-dependent exchange \feld induces spin-\npumping currents7,8inside the ferromagnet with spatially\ninhomogeneous magnetization dynamics, which can con-\nsiderably modify the self-consistent magnetic equation of\nmotion. Here, we wish to elucidate the central role of\nsuch self-consistent dissipative spin currents, which gov-\nern the di\u000busion-like terms in the magnetic equation of\nmotion in the limit of strong ferromagnetic exchange cor-\nrelations.\nThis paper is a follow up to our previous work,9pro-\nviding additional technical details and o\u000bering a broader\nphenomenological base. Apart from assuming strong\nexchange correlations limit, our phenomenological ap-\nproach and the main results of the paper should not be\nsensitive to the microscopic details and do not rely on\nthe speci\fc model of the ferromagnetic material (suchas the Stoner or an s-dmodel, for example). The main\ngoals of this paper are as follows: (i) to put the results of\nRef. 9 into a broader phenomenological perspective, (ii)\nto explicitly show that two quite di\u000berent models|the\nspin-polarized itinerant electron liquid (treated in Ref. 9)\nand thes-dmodel|lead to the same phenomenology and\ncan be treated in parallel, and (iii) to make direct contact\nwith the spin-pumping theory.7,8\nTo be speci\fc, let us consider a continuous ferromag-\nnetic medium, with the e\u000bective \feld and spin density\ninitially pointing along the zaxis. For weak excitations\nclose to this state, we may try expanding the ensuing\ntransverse spin-current density as2,6\nji=\u0000D0z\u0002@iS\u0000D00@iS; (2)\nwhich enters in the continuity equation:\n@tS=H\u0002S\u0000X\ni=x;y;z@iji: (3)\nIn the limit of vanishing ferromagnetic correlations, we\nrecover Eq. (1) by setting D0!0 andD00!Din Eq. (2).\nHereafter, we are focusing exclusively on the transverse\nspin dynamics and spin currents. The longitudinal spin\n\rows are conventionally described in terms of the ordi-\nnary di\u000busion for spin-up and spin-down electrons with\nspin-dependent di\u000busion coe\u000ecients and spin-\rip scatter-\ning between the up and down spin bands.10Understand-\ning the transverse spin \rows and dynamics requires more\ncare, in part due to the inherently quantum-mechanical\nbehavior in the case of a strong exchange \feld. When\nthe magnetic excitation is driven by the self-consistent\ntransverse \feld h=z\u0002H\u0002z, there should also be \feld-\ndriven contributions to the transverse spin current (2),\nsuch as ji/@ih.\nThe problem in fact simpli\fes in the limit of strong\nexchange correlations. We will in the following employ\na mean-\feld view of ferromagnetism, where the collec-\ntive spin dynamics are driven by the exchange \feld, H=arXiv:0810.1340v2  [cond-mat.mes-hall]  17 Mar 20092\n\u0000\u0001xcm(r;t) (setting \u0016h= 1 throughout), parametrized\nby the local and instantaneous spin-density orientation,\nm=S=S, which has to be solved for self-consistently.\nSince we are only interested in the transverse spin dy-\nnamics, we set the magnitude of the spin density Sto be\nspatially and time independent. In the limit of large \u0001 xc,\nthe spin currents can be parametrized by m(r;t). We can\nthus proceed phenomenologically and expand jiin spa-\ntial and time derivatives of m(r;t). For a static magnetic\npro\fle m(r), we have the familiar exchange spin \row\nj0\ni=\u0000Am\u0002@im (4)\n(whereAis the material-dependent exchange-sti\u000bness\nconstant), which is the only \frst-order form allowed by\nspin-rotational and time-reversal symmetries. To avoid\nunnecessary complications, we will assume isotropic fer-\nromagnet throughout this paper. Dynamics allow for\ndissipative spin-current contributions that break time-\nreversal symmetry:\nj00\ni=\u0000\u0011m\u0002@i@tm: (5)\nFocusing on linear deviations of mfrom the equilibrium,\nm(0)=z, we omit terms such as @im\u0002@tm.\nAccording to the time-reversal property, the spin-\ncurrent density (4) corresponds to the D0term in Eq. (2),\nwhile the spin-current density (5) is analogous to the D00\nterm, although the latter two are certainly not identi-\ncal. In fact, we wish to emphasize the striking di\u000ber-\nence between the di\u000busive picture for the spin currents,\nEq. (2), on one side and Eqs. (4) and (5) on the other\nside, where we expand spin currents phenomenologically\nin terms of the time-dependent magnetic texture m(r;t).\nThe latter approximation is speci\fc to the limit of strong\nexchange correlations, where the nondissipative spin cur-\nrent, Eq. (4), is determined by the instantaneous mag-\nnetic pro\fle, while the dissipative spin current, Eq. (5),\ncan be interpreted as quasiparticle spin pumping by the\ncollective magnetic dynamics,8rather than ordinary spin\ndi\u000busion. It is also instructive to draw analogy between\ncoe\u000ecients Aand\u0011in Eqs. (4), (5) and the shear modu-\nlus and shear viscosity, respectively, in elasticity theory.\nIn the next section, we develop further the phenomeno-\nlogical grounds for Eqs. (4) and (5), before proceeding\nwith microscopic calculations for the dissipative coe\u000e-\ncient\u0011in Secs. III and IV. In Sec. V, we discuss a spin-\npumping interpretation of dissipative spin current (5),\nbefore summarizing our work in Sec. VI.\nII. PHENOMENOLOGY\nA. Landau-Lifshitz theory\nThe conventional starting point for studying ferro-\nmagnetic precession is the nondissipative Landau-Lifshitz\n(LL) equation11\n@tmjLL=H\u0003\u0002m; (6)where we de\fne the e\u000bective \feld H\u0003as the functional\nderivative of the free energy:\nH\u0003\u0011@mF[m]=S: (7)\nIn this Landau-Lifshitz phenomenology, which is applica-\nble well below the Curie temperature, only the position-\ndependent direction of the magnetization is taken to\nbe a dynamic variable, parametrizing the Free energy\nF[m(r)]. The angular-momentum density S=Smis\nassumed to be related to the magnetization by a con-\nstant conversion factor, the e\u000bective gyromagnetic ratio.\n(Abusing terminology, we say spin density synonymously\nwith angular-momentum density .) Since in the common\ntransition-metal ferromagnets the gyromagnetic ratio is\nnegative, we wrote Eq. (7) with an extra minus sign in\ncomparison to the standard de\fnition, where mis taken\nto be the direction of the magnetization rather than the\nspin density. The right-hand side of Eq. (6) is the phe-\nnomenological reactive torque on the spatially-resolved\nmagnetic precession, which generalizes the simple Larmor\nprecession of Eq. (1). Note that the dissipation power\nP[m(r;t)]\u0011\u0000SZ\nd3rH\u0003\u0001@tm (8)\nclearly vanishes according to Eq. (6). We also eas-\nily verify that the time reversal (under which t!\u0000t,\nm!\u0000m, and H\u0003!\u0000H\u0003) leaves Eq. (6) unchanged,\nas it should in the absence of dissipation. The only dis-\nsipative term we can write in the quasistationary limit\n(i.e., up to the \frst order in @t), assuming spatially uni-\nform and isotropic ferromagnet, is the so-called Gilbert\ndamping:12\n@tmjLLG=H\u0003\u0002m\u0000\u000bm\u0002@tm; (9)\nwhere\u000bis a material-dependent dimensionless (Gilbert)\nconstant. A typical experimental value for \u000bturns out\nto be often of the order of 10\u00002in various metallic ferro-\nmagnets, which means that it takes roughly 2 \u0019=\u000b\u001810\nprecession cycles for an out-of-equilibrium magnetiza-\ntion to relax to a static equilibrium direction along H\u0003.\nThe Gilbert damping breaks time-reversal symmetry and\ncauses a \fnite dissipation power:\nP[m(r;t)] =\u000bSZ\nd3r(@tm)2: (10)\nAs a side comment, we note that an alternative, so-called\nLandau-Lifshitz damping term m\u0002H\u0003\u0002mis mathemati-\ncally identical to the Gilbert damping m\u0002@tmin Eq. (9),\nup to an extra factor of (1 + \u000b2) on the left-hand side of\nthe equation.\nThe e\u000bective \feld H\u0003is in practice dominated by the\napplied magnetic \feld, magnetic crystal anisotropies, and\nmagnetostatic (dipole-dipole) interactions. In the pres-\nence of spatial inhomogeneities, there is also exchange\ncontribution to the free energy, which to the leading3\n(quadratic) order in magnetic inhomogeneities can be\nwritten as11\nFxc=A\n2Z\nd3r\u0002\n(@xm)2+ (@ym)2+ (@zm)2\u0003\n:(11)\nThe corresponding e\u000bective \feld is\nHxc=\u0000(A=S)r2m; (12)\nand the associated term in LL Eq. (6) is\n@tmjxc= (A=S)m\u0002r2m: (13)\nThis equation can also be formally written as\nS@tmjxc=\u0000X\ni=x;y;z@ij0\ni;\nj0\ni=\u0000Am\u0002@im; (14)\nwhich simply recovers our equilibrium spin current (4).\nWe emphasize that this spin current does not depend on\nmagnetic dynamics.\nTo summarize these preliminary considerations, the\nphenomenological LL equation describes collective mag-\nnetic precession driven by local e\u000bective \felds as well as\nequilibrium spin currents. At this point, there is, how-\never, a conspicuous asymmetry in the treatment of the\ndissipative correction to the LL equation, i.e., Gilbert\ndamping (9), which depends only on the local magnetic\ndynamics and thus does not involve spin currents. To\novercome this \\discrepancy,\" we expand the dissipative\nterms to second order in spatial derivatives, generalizing\nGilbert term to\n@tmjdiss=\u0000\u000bm\u0002@tm+ (\u0011=S)m\u0002r2@tm;(15)\nwhere\u0011is a new phenomenological parameter, charac-\nterizing spin-wave damping. Assuming spatial-inversion\nsymmetry (under which @i!\u0000@iandm!m), pre-\nvents us from writing any phenomenological terms linear\nin spatial derivatives. Recall also that we are always as-\nsuming small perturbations with respect to a uniform\nequilibrium magnetization, so that all spatial and time\nderivatives must hit a single m(for example, a dissipa-\ntive term of the formP\ni[@im\u0001(m\u0002@tm)]@imis dis-\nregarded since it is higher order in small deviations of\nm). Additional quadratic terms would be allowed phe-\nnomenologically if, e.g., we developed our linearized the-\nory with respect to an equilibrium magnetic texture, such\nas a domain wall or magnetic spiral. Some of such terms\nwere discussed in Ref. 13, which is beyond our present\nscope. Finally, we note that we wrote Eq. (15) with no\ndirect coupling to the e\u000bective \feld H\u0003. We justify this\nby assuming that the ferromagnetic correlations are char-\nacterized by a very large energy scale \u0001 xc, so that mi-\ncroscopic processes responsible for dissipation are driven\nby the collective variable m, rather than directly by H\u0003.\nIn transition-metal ferromagnets, the internal exchange\nenergy is of the order of eV, while the e\u000bective \feld H\u0003corresponds to microwave frequencies (i.e., at least three\norders of magnitude smaller than the exchange energy).\nThis means that when we excite magnetic dynamics by\nan external \feld, the microscopic degrees of freedom re-\nspond not to the small driving \feld but rather the much\nlarger self-consistent exchange \feld parametrized by the\ntime-dependent m. For the same reason, the spin current\nin Eq. (14) depends only on the magnetic pro\fle m(r),\nirrespective of how it is created by applied \felds.\nThe total dissipation power corresponding to Eq. (15)\nnow becomes\nP[m(r;t)] =Z\nd3r\u0002\n\u000bS(@tm)2+\u0011(@i@tm)2\u0003\n:(16)\nSimilarly to Eq. (14), we can also write the \u0011term in\nEq. (15) in the form of the divergence of the spin-current\ndensity\nj00\ni=\u0000\u0011@i(m\u0002@tm); (17)\nreproducing Eq. (5). We thus identi\fed two contribu-\ntions to the spin-current density: usual exchange spin\ncurrent (14) and dissipative spin current (17), which\nwe will later interpret as the dynamically-driven spin\npumping.7,8Spin current (17) can thus damp down\nspin-wave excitations even in the absence of any spin-\nrelaxation scattering.23The latter is, however, believed\nto be the culprit for a \fnite Gilbert damping \u000b,14which\nrelaxes uniform magnetic precession by transferring its\nangular momentum to the atomic lattice.\nIn the presence of dissipative currents (17), the relative\nlinewidth of the spin-wave resonance15is proportional to\n\u000b+ (\u0011=S)q2, for the wave vector q. In the absence of\nthe Gilbert damping \u000b, thus, the spectral width of the\nspin-wave excitation would vanish in the long-wavelength\nlimit.1\nB. Mermin ansatz for spin current\nWe now wish to establish a microscopic procedure for\nevaluating the dissipative component of the spin current,\nEq. (17). Ref. 9 adapted Mermin ansatz16for this pur-\npose, which we will reproduce below. Microscopically,\nthe spin-current density jiis carried by conducting elec-\ntrons responding to the mean-\feld exchange interaction\n^Hxc=\u0000\u0001xcm(r;t)\u0001^\u001b=2 (18)\nin the self-consistent single-electron Hamiltonian (which\ncould stem, e.g., either from the coupling to the localized\ndelectrons in the s\u0000dmodel or the itinerant electron\nStoner/LDA exchange). ^\u001bis the vector of Pauli matrices,\nwhich de\fnes the electron spin operator.\nLet us for the moment view exchange interaction (18)\nas an external parametric driving \feld, not concerning\nwith a self-consistent determination of m(r;t). In par-\nticular, we may allow for an instantaneous deviation of4\nthe electron spin density sfrom the exchange-\feld direc-\ntionm. This will allow us for a trick to \fnd the ensuing\nspin \rows, which is what we are after. The spin-density\ncontinuity equation corresponding to Hamiltonian (18) is\n@ts= \u0001 xcz\u0002(sm\u0000s)\u0000@iji: (19)\nThe equilibrium orientation of mis taken to be along the\nzaxis and we assume small-angle excitations, which do\nnot modulate the magnitude of the spin density, s=jsj.\nshere is the spin density of the conducting electrons,\nwhich in, e.g., the s\u0000dmodel has to be distinguished\nfrom the total spin density Sthat enters Eq. (3).\nWe next use the Mermin ansatz to relate the spin-\ncurrent density jito the spin density s:\nji=\u001b?\u0001xc@i(m\u0000s=s); (20)\nwhere\u001b?is the transverse spin conductivity, to be evalu-\nated later by the Kubo formula. Eq. (20) is analogous to\nOhm's law for electric current density, with the expres-\nsion on the right-hand side reminiscent of the gradient\nof the electrochemical potential. The physical reasoning\nbehind ansatz (20) is simple: there should be no dis-\nsipative spin currents in the static con\fguration, which\ncorresponds to s(r) =sm(r). The advantage in writing\nthe spin current in this form is that \u001b?will now have\nto be evaluated in the limit of ( q;!)!0. Combining\nEqs. (19) and (20) will then give us the spin current to\nthe linear order in qand!: exactly what we need to re-\nlate it to Eq. (17) and read out \u0011. In fact, it is su\u000ecient\nto \fnd \u0001 xc(m\u0000s=s)\u0019\u0000z\u0002@tmfrom Eq. (19), which\nis valid to the linear order in !and zeroth order in q,\nbefore putting it into Eq. (20) to \fnally \fnd\nji=\u0000\u001b?@i(z\u0002@tm): (21)\nComparing this with Eq. (17), we immediately identify \u0011\nwith the transverse spin conductivity:\n\u0011=\u001b?: (22)\nEquation (22) can be interpreted as an analog of the Ein-\nstein relation for transverse spin di\u000busion in strong fer-\nromagnets.\nC. Transverse spin conductivity\nAs is the case with the charge conductivity, it is con-\nvenient to evaluate the transverse spin conductivity in\nthe velocity gauge. Namely, we eliminate the spin \\po-\ntential,\" corresponding to small magnetization deviations\n\u000em=m\u0000zin Eq. (18), by the SU(2) gauge transfor-\nmation\n^ (r;t)!ei\u0001xcRt\n\u00001dt0\u000em(r;t0)\u0001^\u001b=2^ 0(r;t); (23)\nat the expense of introducing the SU(2) vector potential\n^Ai=\u0000\u0001xcZt\n\u00001dt0@im(r;t0)\u0001^\u001b=2; (24)which enters the kinetic part of the single-particle Hamil-\ntonian as\n^Hk=X\ni(pi\u0000^Ai)2=2m\u0003; (25)\nwherepi=\u0000i@iandm\u0003is the electron's e\u000bective mass\n(assuming exchange-split parabolic bands). It is easy to\nverify that the e\u000bective \feld driving the spin current in\nvelocity gauge (25), ^Ei=\u0000@t^Ai, is the same as the \fc-\ntitious \feld ^Ei=\u0000@i^Vin original length gauge (18).\nOne caveat is in order: Eqs. (23)-(25) are only valid for\nan Abelian exchange potential, which would be the case\nif only one vector component of \u000em(r;t) was modulated\n(e.g.,\u000emxor\u000emy) in space and time. Such scenario is\nsu\u000ecient for our purpose, in order to establish the trans-\nverse spin conductivity entering Eq. (20).\nFourier transforming the electric \feld ^Eiin time,R\ndtei!t, the usual relationship is obtained: ^Ei(!) =\ni!^Ai(!). We now proceed to construct the semiclassi-\ncal transport equation for the spin current driven by a\nspatially homogeneous \fctitious \feld Ei= Tr[ ^Ei^\u001b] =\n\u0001xc@im, to deduce the long-wavelength conductivity de-\n\fned by Ohm's law24\nji=\u001b?Ei: (26)\nThe semiclassical spin-current response, in the presence\nof the exchange splitting \u0001 xc, with disorder and electron-\nelectron scattering is given by17\n@tji+ \u0001 xcz\u0002ji=nEi\n4m\u0003\u0000ji\u00121\n\u001cdis\n?+1\n\u001cee\n?\u0013\n;(27)\nwherenis the total equilibrium (conducting) electron\ndensity. The second term on the right-hand side of\nEq. (27) describes spin-current relaxation, due to dis-\norder and electron-electron scattering. Note that even\nin Galilean-invariant systems, spin-independent Coulomb\ninteraction between electrons causes relaxation of a ho-\nmogeneous spin current, in contrast to the ordinary cur-\nrent. Solving Eq. (27) at low frequencies, we recover\nEq. (26) for the current component along Ei, with3,9\n\u001b?=n\n4m\u0003\u001c?\n1 + (\u001c?\u0001xc)2; (28)\nwhere the total transverse spin scattering rate is de\fned\nby\n1\n\u001c?=1\n\u001cdis\n?+1\n\u001cee\n?: (29)\nIn particular, in the limit of weak spin polarization and\nno electron-electron interactions, \u001c?should reduce to the\nordinary momentum scattering time \u001c, and\u001b?to the\nquarter of the Drude conductivity n\u001c=m\u0003.5\nIII. MICROSCOPIC CALCULATION\nA. Spin-current autocorrelator\nIn order to substantiate the preceding phenomenology,\nwe need to establish the microscopic expressions for the\ninvolved scattering times, \u001cdis\n?and\u001cee\n?. In the velocity\ngauge discussed in the previous section, the transverse\nspin conductivity is given, according to the Kubo for-\nmula, by the spin-current autocorrelation function:9\n\u001b?=\u00001\n4m\u00032Vlim\n!!0=mhhP\nl^\u001bxlpxl;P\nl^\u001bxlpxlii!\n!;(30)\nwhere the summation is over all electrons in volume V\nand\nhh^A;^Bii!=\u0000iZ1\n0dtei(!+i0+)th[^A(t);^B(0)]i(31)\nrepresents the Fourier-transformed retarded (Kubo)\nlinear-response function for the expectation value of the\nobservable ^Aunder the action of a classical \feld that\ncouples linearly to the observable ^B.=min Eq. (30) is\ninserted out of convenience, since the linear in !response\nfunction is guaranteed to be imaginary. (The zeroth-\norder in!correlator includes also the omitted \\diamag-\nnetic piece\" of the spin current in the velocity gauge.)\nAssuming isotropic disorder (and for the moment no\nelectron-electron interactions), the ladder vertex correc-\ntions to the conductivity vanish and we only need to eval-\nuate the bubble diagram de\fned by the (single-particle)\nspin-dependent Green's functions\nGR;A\n\u001b(p;!) =1\n!\u0000p2=2m\u0003+\u0016\u001b\u0006i=2\u001c\u001b; (32)\nwhere\u001b=\";#(=\u0006) is the spin index along the zaxis,\n\u0016\u001b=\u0016+\u001b\u0001xc=2 is the spin- \u001belectron Fermi energy, \u0016is\nthe chemical potential, and \u001c\u001bis the spin-dependent dis-\norder scattering time. In the Born approximation for\ndilute white-noise disorder, the scattering rate is pro-\nportional to the electron density of states, and we can\nwrite\u001c\u001b=\u001c\u0017=\u0017\u001b, where\u001cparametrizes the strength of\nthe scattering potential, \u0017\u001bis the spin-\u001bband density of\nstates, and \u0017= (\u0017\"+\u0017#)=2. A straightforward calcula-\ntion then leads to9\n\u001b?=n\n4m\u00031\n\u001cdis\n?\u00012xc; (33)\nin the strong exchange coupling limit, where\n1\n\u001cdis\n?=4\n3\u0016\"+\u0016#\nn\u001c(\u0017\u00001\n\"+\u0017\u00001\n#)(34)\nidenti\fes the disorder contribution to e\u000bective transverse\nspin scattering rate (29).B. Spin-force autocorrelator\nIn the presence of electron-electron interactions, it\nis convenient to express the spin- current autocorrelator\n(30) in terms of the spin- force autocorrelator. To this\nend, we use the equation of motion for the operators\nde\fning Kubo formula (30) to \fnd\n\u001b?=\u00001\n4m\u00032\u00012xcVlim\n!!0=mhhP\nl^\u001bxlFxl;P\nl^\u001bxlFxlii!\n!;\n(35)\nwhereFxl= _pxl=\u0000i[pxl;^H] is the force operator along\nthexaxis for the lth electron. Evaluated with respect to\na uniform magnetization, m=z, the force operator Fxl\nconsists of two pieces: the disorder force and the electron-\nelectron interaction force. Evaluating correlator (35) in\nthe clean limit to second order in Coulomb interactions,\none \fnds for the transverse spin scattering rate:5,9\n1\n\u001cee\n?= \u0007(p)m\u0003a2\nBr4\ns(kBT)2; (36)\nwhereaBis the Bohr radius, Ttemperature, kBBoltz-\nmann constant, rsthe dimensionless Wigner-Seitz radius,\nand \u0007(p) is a dimensionless function of the spin polariza-\ntionp= (n\"\u0000n\")=n(nsbeing spin-selectron density),\nwhich was discussed in Refs. 5,9. Notice that scattering\nrate (36) has the Landau quasiparticle scaling with tem-\nperature. The \fnite-frequency modi\fcation of scatter-\ning rate (36) is, furthermore, accomplished by replacing\n(2\u0019kBT)2!(2\u0019kBT)2+!2.\nC. Spin-density autocorrelator\nIt is also possible to calculate the transverse spin dif-\nfusion directly, as a linear spin-density response to the\ntransverse magnetic \feld. We will carry that out in\nSec. IV for two popular mean-\feld models of ferromag-\nnetism in metals: the Stoner and the s\u0000dmodels. In ad-\ndition to o\u000bering an alternative approach to the problem,\nthis derivation provides a justi\fcation for the preceding\nheuristic utilization of the Mermin ansatz.\nStarting with the mean-\feld Hamiltonian for itinerant\nelectrons\n^H=p2\n2m\u0003+U(r)\u0000\u0016\u0000\u0001xc^\u001bz=2; (37)\nand directly solving for the self-consistent spin-density\nresponse to a small driving magnetic \feld, we will derive\nin the next section the following general relation:\n\u0011=\u00012\nxc\nq2lim\n!!0=m~\u001f+\u0000(q;!)\n!; (38)\nvalid at long wavelengths, q!0. The axially-symmetric\n(Kubo) spin-response function is de\fned by\n~\u001f+\u0000(q;!) =\u00001\n2hhs+(r;t);s\u0000(r0;0)iiq;!; (39)6\nwheres\u0006=sx\u0006isyis the transverse spin density of\nitinerant electrons. The disorder potential U(r) entering\nEq. (37) is, as before, taken to obey the Gaussian white-\nnoise correlations:\nhU(r)U(r0)i=1\n2\u0019\u0017\u001c\u000e(r\u0000r0); (40)where\u0017= (\u0017\"+\u0017#)=2 is the spin-averaged density of\nstates at the Fermi level and \u001cis the characteristic scat-\ntering time.\nWriting the spin density s(r) = Tr [ ^\u001b^\u001a(r)]=2 in terms of the electron density matrix \u001a\u000b\f(r) = \ty\n\f(r)\t\u000b(r) in spin\nspace, we proceed to evaluate ~ \u001f+\u0000in the standard imaginary-time formalism. At temperature T, we have:\n~\u001f+\u0000(q;i\nn) =\u0000T\n2VX\npp0;mG#(p+q;p0+q;i!m+i\nn)G\"(p0;p;i!m); (41)\nwhere\nG\u001b(p;p0;i!m) =\u00001\nVZ\nd3rd3r0Z1=T\n0d\u001ce\u0000ip\u0001r+ip0\u0001r0+i!m\u001c\n\t\u001b(r;\u001c)\ty\n\u001b(r0;0)\u000b\n(42)\nis the \fnite-temperature single-particle Matsubara Green's function. \n n= 2n\u0019T is the bosonic and !m= (2m+1)\u0019T\nfermionic Matsubara frequencies, where nandmare integer indices.\nThe disorder-averaged Green's function is given by\nhG\u001b(p;p0;i!m)i=\u000epp0\ni!m\u0000\"p\u001b+isign(!m)=2\u001c\u001b; (43)\nwhere\"p\u001b=p2=2m\u0003\u0000\u0016\u0000\u001b\u0001xc=2. The analytic continuation of the Matsubara Green's functions into the retarded\n(advanced) Green's functions is accomplished by replacing i!m!!\u0006i0+and sign(!m)!\u0006 . According to our\nconvention (40), \u001c\u001b=\u001c\u0017=\u0017\u001b. Taking into account the vertex ladder corrections (as shown in Fig. 1), we obtain for\nthe disorder-averaged response function:\n~\u001f+\u0000(q;i\nn) =\u0000T\n2VX\nmP\npG#(p+q;i!m+i\nn)G\"(p;i!m)\n1\u0000(\u0018=V)P\npG#(p+q;i!m+i\nn)G\"(p;i!m); (44)\nwhere\u0018= 1=2\u0019\u0017\u001c and by the Green's functions with a single wave-vector argument here we understand disorder-\naveraged propagators (43). Inserting Eq. (43) into Eq. (44) and performing an analytic continuation onto the real\nfrequencies, it is straightforward to calculate ~ \u001f+\u0000(q;!). Setting the temperature to zero and taking the !!0 limit,\nwe \fnd:\n=m~\u001f+\u0000(q;!) =!\n4\u0019<e~\u001fRA\u0000~\u001fAA\n(1\u0000\u0018~\u001fRA)(1\u0000\u0018~\u001fAA); (45)\nwhere ~\u001fXY(q) =R\ndpGX\n#(p)GY\n\"(p\u0000q) andR\ndp\u0011R\nd3p=(2\u0019)3in three dimensions. All energies entering these\nGreen's functions are set at the Fermi level. To the lowest order in 1 =\u001c\u0001, we now obtain:\n=m~\u001f+\u0000(q;!) =!\n8\u0019\u0014Z\ndpA#(p)A\"(p\u0000q) + 4\u0018=mZ\ndpGA\n\"(p\u0000q)A#(p)Z\ndpGA\n\"(p\u0000q)<eGR\n#(p)\u0015\n; (46)\nwhereA\u001b=\u00002=mGR\n\u001bis the spectral function.\nThe second term in Eq. (46) is the vertex ladder correc-\ntion, which is necessary for Eq. (46) to give a meaningful\nresult. In particular, the vertex correction cancels the\nspuriousq= 0 contribution of the \frst term, which would\ngive\u000b\u00181=\u001c\u0001xc. Finally, in the limit of q\u001c\u0019\u0001xc=vF,we arrive at:\n=m~\u001f+\u0000(q;!) =\u0016\"+\u0016#\n3m\u0003\u001c(\u0017\u00001\n\"+\u0017\u00001\n#)\u00014xc!q2: (47)\nUsing Eq. (38), this \fnally gives:\n\u0011=\u0016\"+\u0016#\n3m\u0003\u001c(\u0017\u00001\n\"+\u0017\u00001\n#)\u00012xc; (48)7\nwhich agrees with Eqs. (22), (28), and (34) in the relevant\nhere limit of \u001c\u00001\n?\u001c\u0001xc.\nIV. MEAN-FIELD FERROMAGNETISM\nA. Time-dependent LDA\nIn a spin-density-functional theory (s-DFT),5,18the\nmany-body problem of itinerant ferromagnetism is re-\nduced to the single-electron Hamiltonian\n^H(t) =p2\n2m\u0003+U(r)\u0000\u0016\n\u0000[\u0001xcm(r;t) +!0z+h(r;t)]\u0001^\u001b=2:(49)\n!0\u001c\u0001xcis the ferromagnetic Larmor precession fre-\nquency in the presence of a uniform magnetic \feld ap-\nplied along the zaxis. \u0001 xcm(r;t) is the self-consistent\nexchange \feld, such that Hamiltonian (49) produces the\ncorrect spin-density response. Since we are ultimately in-\nterested in the equation of motion for the collective ferro-\nmagnetic dynamics, the spin-density response is all that\nis needed. In the local-density approximation (LDA) of\nthe s-DFT, the exchange \feld follows the local and in-\nstantaneous magnetization direction m(r;t).h(r;t) is\nthe external rf driving \feld, which we will treat pertur-\nbatively.\nThe time-dependent portion of the Hamiltonian is thus\ngiven by\n^H0(t) =\u0000Z\nd3r[\u0001xc\u000em(r;t) +h(r;t)]\u0001^\u001b=2:(50)\nSince\u000em=\u000es=S(where\u000edenotes small deviations from\nequilibrium), we have for the transverse spin component\ns+=sx+isy:\ns+(q;!) = ~\u001f+\u0000(q;!)\u0014\nh+(q;!) +\u0001xc\nSs+(q;!)\u0015\n:(51)\nThe self-consistent response function to the rf \feld,\n\u001f+\u0000=s+=h+, is thus\n\u001f\u00001\n+\u0000(q;!) = ~\u001f\u00001\n+\u0000(q;!)\u0000\u0001xc\nS: (52)\nIn the LDA approximation, the problem thus trivially\nreduces to calculating the spin-spin response function for\na noninteracting Hamiltonian with a \fxed exchange \feld.\nLet us in general write\n~\u001f+\u0000(q;!) =S\n[!r(q;!) + \u0001 xc\u0000!]\u0000i\u000b(q;!)!;(53)\nin terms of functions !rand\u000bthat are to be determined.\nThe self-consistent response function then becomes:\n\u001f+\u0000(q;!) =S\n[!r(q;!)\u0000!]\u0000i\u000b(q;!)!: (54)Atq= 0, obviously !r(!)\u0011!0and\u000b(!)\u00110. This fol-\nlows in general from the spin conservation in the presence\nof Coulomb interactions and arbitrary spin-independent\npotentialU(r). In this paper, we are most interested in\ntheq-dependent damping function \u000b(q;!), which can be\nidenti\fed by a microscopic evaluation of ~ \u001f+\u0000(q;!). In\nthe limit of strong exchange correlations, \u0001 xc\u001d!r, we\nimmediately obtain from Eq. (53):\n\u000b(q;!!0)\u0019\u00012\nxc\nSlim\n!!0=m~\u001f+\u0000(q;!)\n!: (55)\nIn inversion-symmetric systems, the leading in qspin-\nwave contribution to Gilbert damping is \u000b(q;!!0) =\n(\u0011=S)q2, so that self-consistent response function (54)\ncorresponds to the dissipative term\n@tmjdiss= (\u0011=S)m\u0002r2@tm (56)\nin Landau-Lifshitz Eq. (6) of motion for the magnetic\nspin direction m(r;t). This is the desirable result and,\naccording to Eq. (55), the microscopic expression for \u0011\ngives Eq. (38) of the previous section. In the next section,\nwe will demonstrate that Eq. (55) is generic to mean-\feld\ntreatment of conducting ferromagnets.\nB.s\u0000dmodel in RPA\nIt is also instructive to pursue a more basic descrip-\ntion starting with a ferromagnetic lattice of localized d\nelectrons exchange-coupled to itinerant selectrons. The\ncorresponding Hamiltonian is\n^H(t) =^H0\u0000X\ni[JSi\u0001s(ri;t) +Si\u0001h(ri;t)];(57)\nwhere Siare localdspins and ^H0consists of the de-\ncoupled Hamiltonian for itinerant electrons, dc Zeeman\nHamiltonian of the delectrons, as well as the d\u0000dex-\nchange and possible dipolar interactions. his the applied\nrf \feld, which we take for simplicity to couple to the lo-\ncalized spin only. As long as the average exchange \feld\nexperienced by the selectrons is su\u000eciently strong and\nthe magnetization is dominated by the delectrons, we\ncan disregard their direct rf coupling for our purpose.\nIf also the Fermi wavelength is long in comparison to\nthedlattice spacing, we will treat the electronic band\nstructure in the e\u000bective-mass approximation, and also\ncoarse grain the local spins:P\niSi!R\nd3rS(r) and\nSi!(V=N)S(r), whereN=V is the density of dsites.\nLet us compute the spin-density response function for\nthedlattice:\n\u001f+\u0000(r;r0;t) =\u00001\n2hhS+(r;t);S\u0000(r0;0)ii: (58)\nFor this purpose, it is convenient to de\fne bosonic\nmagnon operators:\nap=1p\n2DNX\nie\u0000ip\u0001riSi+; (59)8\n8\nF:\nq,iΩn=\nq,iΩn+\np+q p/prime+qp/primep\n↓i(ωm+Ωn)↑iωm\n+\n +···\nFIG. 1: In this RPA summation for the magnon propagator, Eq. (64), the magnon self-energy is provided by the s-electron\nspin-response bubble, Eq. (41). Straight double lines denote disorder-averaged s-electron Green’s functions and dashed lines\ndescribe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the d-electron\nspin propagators.\nEq. (61) with ˆH0for the selectrons, resulting in the\nlongitudinal mean-field exchange field ∆ xc=JS, where\nS=DN/V is the averaged d-electron spin density. The\nspin-flop term of Eq. (61) can be expanded in terms of\nthe electronic field operators Ψ σas:\nˆH/prime\nxc=−J\nV/radicalbigg\nDN\n2/summationdisplay\npp/prime/bracketleftBig\na†\npΨ†\n↑(p/prime)Ψ↓(p/prime+p)\n+apΨ†\n↓(p/prime)Ψ↑(p/prime−p)/bracketrightBig\n. (62)\nIn the imaginary-time (Matsubara) formalism, the re-\nsponse function (58) can be written as:\nχ+−(q,τ)=−SF(q,τ), (63)\nin terms of the magnon Green’s function\nF(q,τ)=−T/angbracketleftbig\naq(τ)a†\nq(0)/angbracketrightbig\n, (64)\nusing the definition (59). Here, Tis the time-ordering\nsymbol. For the noninteracting Hamiltonian (60),\nF0(q,iΩn)=1\niΩn−εq, (65)\nwhere F(q,iΩn)=/integraltext1/T\n0dτeiΩnτF(q,τ) and Ω n=2nπT,\nas previously, is the bosonic Matsubara frequency. In or-\nder to calculate Fin the presence of the s−dexchange\n(62), we will sum up the bubble diagrams (which con-\nstitute an RPA approximation) shown in Fig. 1. It is\neasy to recognize that each bubble (which is the magnon\nself-energy in this approximation) is just the itinerant\nelectron spin-density response function that was already\ncalculated in Sec. III C.\nSumming up the diagrams in Fig. 1, we thus obtain\nF−1(q,iΩn)=F−1\n0(q,iΩn)−Σ(q,iΩn), (66)\nwhere the self-energy (the s-electron spin-response func-\ntion)\nΣ(q,iΩn)=−J2S˜χ+−(q,iΩn) (67)\nfollows from Eq. (44). Combining Eqs. (63), (66), and\n(67), we finally obtain (after analytic continuation onto\nreal frequencies):\nχ−1\n+−(q,ω)=εq−ω\nS−J2˜χ+−(q,ω). (68)Eq. (68) is actually quite trivial: It can be also obtained\nby treating the d-orbital magnetization dynamics and\nthe associated s−dtorque in a mean-field approxima-\ntion analogous to the preceding discussion of the Stoner\nmodel. Inverting Eq. (68), we can write it in the form\n(54), after identifying\nωr(q,ω)=εq−SJ2/Rfracture˜χ+−(q,ω) (69)\nand\nα(q,ω)=SJ2/Ifracturm˜χ+−(q,ω)\nω, (70)\nwhich is, in fact, exactly the same as Eq. (55) in the\nlow-frequency limit, using ∆ xc=JS. We should em-\nphasize that although χ+−is defined in this section for\na different physical system than χ+−in Sec. IV A (and\nthus, not surprisingly, is found to be somewhat different),\nthe itinerant-electron response ˜ χ+−is the same through-\nout the paper. The reason why the q2magnetic damping\nα(q,ω) is identified in terms of the same quantity ˜ χ+−in\nthe two different models of ferromagnetism can be traced\nto our phenomenological identification of this damping in\nterms of the conducting-electron transverse conductivity,\nEq. (22). The latter is governed by the mean-field struc-\nture of the exchange field, irrespective of the microscopic\norigin of the ferromagnetic order.\nLet us also note in the passing that, unlike the idealized\nStoner model considered in the previous section, the s−d\nmagnetic damping may have a finite q= 0 value even in\nthe absence of any additional spin-dependent terms in the\nHamiltonian. When the gyromagnetic ratios of the two\nelectron species differ (ultimately stemming from some\nform of spin-orbit interaction), the total spin does no\nlonger precess undamped in the uniform field, and the\nuniform transverse spin component can decohere in the\npresence of ordinary scalar disorder. Since Eq. (70) cor-\nresponds to the magnetic field coupled to the delectrons\nonly, we implicitly set the selectron gfactor to zero.\nV. SPIN-PUMPING INTERPRETATION\nIt is illuminating to interpret the key result of this\npaper for the transverse spin diffusion of the form (17)\nFIG. 1: In this RPA summation for the magnon propagator, Eq. (64), the magnon self-energy is provided by the s-electron\nspin-response bubble, Eq. (41). Straight double lines denote disorder-averaged s-electron Green's functions and dashed lines\ndescribe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the d-electron\nspin propagators.\nwhich obey the canonical commutation relations,\n[ap;ay\np0] =\u000epp0, close to the fully-magnetized ground\nstate. To be speci\fc, let us take the Heisenberg model for\nexchange coupling, so that in the ground state, Siz=D,\nthed-orbital spin [assuming the applied dc magnetic \feld\nto point along the \u0000zdirection, as in Eq. (49)]. The d-\norbital Hamiltonian for magnon excitations close to the\nground state can thus be written as:\n^H0=X\np\"pay\npap: (60)\nIn terms of the magnon operators, we, furthermore,\nrewrite the s\u0000dexchange interaction as\n^Hxc=\u0000J\nVr\nDN\n2X\np\u0002\nay\nps+(p) +aps\u0000(\u0000p)\u0003\n\u0000JX\niSizsz(ri); (61)\nwheres\u0006(p) =R\nd3re\u0000ip\u0001rs\u0006(r) is the Fourier-\ntransformed transverse s-electron spin density. Approx-\nimatingSiz\u0019D, we can combine the second term in\nEq. (61) with ^H0for theselectrons, resulting in the\nlongitudinal mean-\feld exchange \feld \u0001 xc=JS, where\nS=DN=V is the averaged d-electron spin density. The\nspin-\rop term of Eq. (61) can be expanded in terms of\nthe electronic \feld operators \t \u001bas:\n^H0\nxc=\u0000J\nVr\nDN\n2X\npp0h\nay\np\ty\n\"(p0)\t#(p0+p)\n+ap\ty\n#(p0)\t\"(p0\u0000p)i\n: (62)\nIn the imaginary-time (Matsubara) formalism, re-\nsponse function (58) can be written as:\n\u001f+\u0000(q;\u001c) =\u0000SF(q;\u001c); (63)\nin terms of the magnon Green's function\nF(q;\u001c) =\u0000T\naq(\u001c)ay\nq(0)\u000b\n; (64)\nusing de\fnition (59). Here, Tis the time-ordering sym-\nbol. For noninteracting Hamiltonian (60),\nF0(q;i\nn) =1\ni\nn\u0000\"q; (65)whereF(q;i\nn) =R1=T\n0d\u001cei\nn\u001cF(q;\u001c) and \nn= 2n\u0019T,\nas previously, is the bosonic Matsubara frequency. In or-\nder to calculateFin the presence of s\u0000dexchange (62),\nwe will sum up the bubble diagrams (which constitute an\nRPA approximation) shown in Fig. 1. It is easy to recog-\nnize that each bubble (which is the magnon self-energy\nin this approximation) is just the itinerant electron spin-\ndensity response function that was already calculated in\nSec. III C.\nSumming up the diagrams in Fig. 1, we thus obtain\nF\u00001(q;i\nn) =F\u00001\n0(q;i\nn)\u0000\u0006(q;i\nn); (66)\nwhere the self-energy (the s-electron spin-response func-\ntion)\n\u0006(q;i\nn) =\u0000J2S~\u001f+\u0000(q;i\nn) (67)\nfollows from Eq. (44). Combining Eqs. (63), (66), and\n(67), we \fnally obtain (after analytic continuation onto\nreal frequencies):\n\u001f\u00001\n+\u0000(q;!) =\"q\u0000!\nS\u0000J2~\u001f+\u0000(q;!): (68)\nEquation (68) is actually quite trivial: it can be also ob-\ntained by treating the d-orbital magnetization dynamics\nand the associated s\u0000dtorque in a mean-\feld approxima-\ntion analogous to the preceding discussion of the Stoner\nmodel. Inverting Eq. (68), we can write it in form (54),\nafter identifying\n!r(q;!) =\"q\u0000SJ2<e~\u001f+\u0000(q;!) (69)\nand\n\u000b(q;!) =SJ2=m~\u001f+\u0000(q;!)\n!; (70)\nwhich is, in fact, exactly the same as Eq. (55) in the\nlow-frequency limit, using \u0001 xc=JS. We should em-\nphasize that, although \u001f+\u0000is de\fned in this section for\na di\u000berent physical system than \u001f+\u0000in Sec. IV A (and\nthus, not surprisingly, is found to be somewhat di\u000berent),\nthe itinerant-electron response ~ \u001f+\u0000is the same through-\nout the paper. The reason why the q2magnetic damping\n\u000b(q;!) is identi\fed in terms of the same quantity ~ \u001f+\u0000in9\nthe two di\u000berent models of ferromagnetism can be traced\nto our phenomenological identi\fcation of this damping in\nterms of the conducting-electron transverse conductivity,\nEq. (22). The latter is governed by the mean-\feld struc-\nture of the exchange \feld, irrespective of the microscopic\norigin of the ferromagnetic order.\nLet us also note in the passing that, unlike the idealized\nStoner model considered in the previous section, the s\u0000d\nmagnetic damping may have a \fnite q= 0 value even in\nthe absence of any additional spin-dependent terms in the\nHamiltonian. When the gyromagnetic ratios of the two\nelectron species di\u000ber (ultimately stemming from some\nform of spin-orbit interaction), the total spin no longer\nprecesses undamped in the uniform \feld, and the uniform\ntransverse spin component can decohere in the presence\nof ordinary scalar disorder. Since Eq. (70) corresponds\nto the magnetic \feld coupled to the delectrons only, we\nimplicitly set the selectrongfactor to zero.\nV. SPIN-PUMPING INTERPRETATION\nIt is illuminating to interpret the key result of this\npaper for the transverse spin di\u000busion of form (17) in\nterms of the spin pumping associated with a nonuni-\nform magnetic dynamics in ferromagnetic bulk.8The\nferromagnetic spin pumping was originally proposed in\nthe context of magnetic multilayers with sharp normal-\nmetaljferromagnetic interfaces. This paper shows that\nanalogous processes also take place in the continuous fer-\nromagnetic medium.\nTo illustrate the direct connection between the trans-\nverse spin di\u000busion and the spin pumping, we consider\na periodic stack of alternating F and N layers forming a\ntwo-component superlattice in the xdirection.8We treat\nthe model depicted in Fig. 2, in which an F jN bilayer\nforms the unit cell with thickness b=L+d, where the\nnormal-metal spacer of width Lseparates the magnetic\n\flms of thickness\nd\u001d\u0015sc=vF=\u0019\u0001xc: (71)\nThe latter approximation allows us to neglect the trans-\nverse spin-current coherence between two interfaces of\nthe same magnetic layer.8Translational invariance is as-\nsumed for simplicity in the lateral directions. We con-\nsider here collective spin-wave excitations, taking both\nthe static and dynamic exchange couplings into account.7\nThe static (RKKY-like) exchange interaction between\nneighboring ferromagnetic layers is mediated by the dis-\nsipationless spin currents \rowing through the normal-\nmetal spacer.19We will parametrize the strength of this\ncoupling by the corresponding precession frequency !xc\nof a single ferromagnetic \flm that is exchange-coupled to\na pinned \flm. In the presence of the magnetic dynam-\nics, additional dissipative spin currents set in. Their ori-\ngin lies in the spin pumping by the individual magnetic\nlayers into the adjacent normal spacers, which at low\nfrequencies is given by8Ipump\ns = (\u0016h=4\u0019)~g\"#\nNjFm\u0002@tm.\nLb=L+d\nd\nxFIG. 2: A schematic view of the superlattice considered in the\ntext: an FjN bilayer is repeated along the xaxis, with either\nferromagnetic or antiferromagnetic alignment of the consecu-\ntive magnetic layers. The system is translationally invariant\nalong the two remaining axes.\n~g\"#\nNjFis the dimensionless spin-mixing conductance per\nunit area of the F jN interface (which is assumed to be\nreal-valued, for simplicity). This interfacial spin pump-\ning induces nonlocal spin transfer in magnetoelectronic\ncircuits, which can in general be treated as a source term\nentering spin transport equations in normal and mag-\nnetic layers. In a collinear superlattice of Fig. 2, the\nproblem simpli\fes considerably, because the spin-current\nvector Ipump\ns/m\u0002@tmis transverse with respect to\nthe magnetic alignment (in both ferromagnetic and anti-\nferromagnetic cases), within the linear-response regime.\nThis means that the spin current pumped by one fer-\nromagnetic layer is either scattered back by the normal\nspacer and reabsorbed, or transmitted and absorbed by a\nneighboring layer, with no possibility to reach more dis-\ntant neighbors, subject to condition (71). Spin relaxation\nin normal spacers would only cause an overall increase\nin the e\u000bective Gilbert damping parameter of a uniform\nmagnetic precession, and will thus be omitted, since our\nprimary interest here is nonlocal damping e\u000bects. The\nproblem of dynamic exchange between two adjacent fer-\nromagnetic layers thus e\u000bectively reduces to the analo-\ngous e\u000bect in magnetic bilayers, which was studied in de-\ntail in Ref. 7. In particular, the net spin pumping through\na given normal spacer is /m1\u0002@tm1\u0000m2\u0002@tm2, which\nre\rects the dynamic spin injection in the opposite direc-\ntions by the adjacent magnetic layers m1andm2. Notice\nthat the total pumping vanishes in the case of a perfectly\nsynchronous precession, m1(t) =m2(t).\nLet us now put the static and dynamic exchange inter-\nactions into the equation of motion for small-angle spin\ndynamics of a multilayer with respect to an all-parallel\ncon\fguration, ui(t) =mi(t)\u0000z. For long-wavelength ex-\ncitations, it may be approximated as a continuous func-\ntionu(x;t) of the coordinate xnormal to the interfaces.\nFor the uniaxial e\u000bective \feld H\u0003=\u0000!0z, the spin-wave\ndynamics obey the di\u000berential equation\n@tu=\u0002\n!0u\u0000!xcb2@2\nxu+\u000b@tu\u0000\u000b0b2@2\nx@tu\u0003\n\u0002z;(72)\nwhere we made the following de\fnitions: !xc=Jxc=Sd\nand\u000b0=<eA\"#\nFjNjF=4\u0019Sd. Here,Jxcis the exchange10\ncoupling between two consecutive magnetic layers and\n1=A\"#\nFjNjF= 2=~g\"#\nNjF+e2L\u001a=\u0019 (73)\nis the e\u000bective pumping resistance of the spacer. The \frst\nterm on the right-hand side of Eq. (73) parametrizes the\npumping strength of the individual interfaces, as was dis-\ncussed above, and the second term is the ordinary ohmic\nresistance of the normal spacer (neglecting any spin relax-\nation), which backscatters the pumped currents and thus\nsuppresses the dynamic exchange. The second spatial\nderivatives in Eq. (72) re\rect simply the di\u000berence of the\nstatic and dynamic exchange spin currents through two\nconsecutive normal spacers (which themselves require a\n\fnite misalignment of the adjacent magnetic layers) in\nthe continuum limit. The static Heisenberg coupling can\nbe interpreted as the superlattice equivalent of the bulk\nexchange-sti\u000bness parameter Aof Eq. (11), which for the\nsuperlattice becomes A=Jxcb. Both!xcand\u000b0are sen-\nsitive to the normal-interlayer thickness L, vanishing in\nthe limitL! 1 . It follows from Eq. (72) that the\nsmall-momentum, q\u001cb\u00001, spin-wave excitations of the\nsuperlattice, propagating perpendicular to the interfaces,\nu/exp[i(qx\u0000!t)], obey the dispersion relation\n!(q) =!0+ (bq)2!xc\n1 +i[\u000b+ (bq)2\u000b0]: (74)\nWhenq!0,!(q) reduces to the Larmor frequency !0\nof the individual magnetic layers because the static and\ndynamic exchange couplings vanish when the consecutive\nmagnetic layers move coherently in phase. Equation (74)\nholds up to momenta comparable to b\u00001, whenbqhas to\nbe replaced by 2 sin( bq=2).\nThe matters are quite di\u000berent for an\nantiferromagnetically-aligned superlattice, which is\nthe ground state when, for example, Jxc<0 and\nH\u0003= 0. In this case, we have a more complex\ndispersion:\n!(q) =\u0000!xc\u0006p\n(bq)2(1 +\u000b2)\u00004\u000b2+i[2\u000b+ (bq)2\u000b0]\n1 +\u000b2+ 4\u000b\u000b0+ (bq)2\u000b02;\n(75)\nwhere plus and minus signs refer, respectively, to the\nmodes with antisymmetric and symmetric dynamics in\nthe adjacent layers for overdamped motion, and to the\nright- and left-propagating modes when the real part\nof!(q) is signi\fcant. Note that now !xc<0, so that\n=m! > 0, as required for a stable con\fguration. In the\nabsence of bulk magnetization damping, \u000b= 0, Eq. (75)\nreduces to\n!(q) =\u0006(bq)!xc\n1\u0006i(bq)\u000b0; (76)\nwith linear dispersion and damping at small q. Equa-\ntions (75) and (76) can also be generalized to large mo-\nmenta by replacing bqwith 2 sin(bq=2). Notice that in\nEqs. (72), (74), and (76), the dynamic coupling modi-\n\fes the damping similarly to the way the static couplinga\u000bects the excitation frequency of the magnetic superlat-\ntice. Crystal and shape anisotropies on top of the simple\ne\u000bective \felds assumed above might become important\nin real structures, and their inclusion is straightforward.\nLet us now compare the damping ( bq)2\u000b0in Eq. (74)\nwith\u000b(q) = (\u001b?=S)q2corresponding to Eq. (33), which\nis the analogous quantity for the bulk. Keeping only\nthe mixing conductance contribution to Eq. (73) and\napproximating8~g\"#\u0019p2\nF=2\u0019in terms of the character-\nistic Fermi momentum pFin the normal metal, we have\nfor theq-dependent part of the damping:\n\u000b(q) = (bq)2\u000b0\u0018(b=\u0015F)2\nSdq2; (77)\nup to a numerical constant. At the same time, the bulk\n\u000b(q), corresponding to Eq. (33), can be written as\n\u000b(q)\u0018(\u0015sc=\u0015F)2\nSlq2; (78)\nwhich establishes a loose formal correspondence between\nthe two results. Here, l=vF\u001cis the mean free path, \u0015F\nthe Fermi wavelength, and the ferromagnetic coherence\nlength\u0015scwas de\fned in Eq. (71).\nComparing Eqs. (77) and (78), we interpret the length\nscaleb$\u0015scto describe the longest distance over which\nferromagnetic regions can communicate via spin trans-\nfer. The length scale d$\u0015characterizes momentum\nscattering relevant for spin transfer, which in the case\nof the superlattice with sharp interfaces corresponds to\nthe magnetic \flm width d: Approximating ~ g\"#\u0019p2\nF=2\u0019\nabove, we e\u000bectively took the normal spacers to be bal-\nlistic and, because of Eq. (71), the spin transfer does\nnot penetrate deep into the ferromagnetic layers, mak-\ning possible disorder scattering there irrelevant for our\nproblem.\nVI. DISCUSSION AND OUTLOOK\nEstimating the numerical value of the dimensionless q2\ndamping, according to Eq. (28),\n\u000b(q) =\u001b?q2\nS\u0018\u0012\u0016F=\u0001xc\npF=q\u00132\u001c?\u0001xc\n1 + (\u001c?\u0001xc)2; (79)\nwe can see that it will most likely be at most compa-\nrable or smaller than the typical q= 0 Gilbert damp-\ning\u000b\u001810\u00002, in metallic ferromagnets. Damping (79)\nmay, however, become dominant in weak ferromagnets,\nsuch as diluted magnetic semiconductors. We are not\naware of systematic experimental investigations of the\nq2damping in metallic ferromagnets. q2scaling of rel-\native linewidth was reported in Ref. 20 for the iron-rich\namorphous Fe 90\u0000xNixZr10alloys. However, we are not\ncertain whether the strong damping observed there can\nbe attributed to the mechanism discussed in our paper.\nAnother intriguing context where the physics discussed\nhere can play out to be important is the current-driven11\nnonlinear ferromagnetic dynamics in mesoscopic as well\nas bulk magnetic systems. The q2magnetic damping\ndescribed by Eq. (15) can be physically thought of the\nviscous-like spin transfer between magnetic regions pre-\ncessing slightly out-of-phase. The obvious consequence\nof this is the enhanced damping of the inhomogeneous\ndynamics and thus the synchronization of collective mag-\nnetic precession. This phenomenon was predicted in\nRef. 7 and unambiguously observed in Ref. 21, in the case\nof the coupled dynamics of a magnetic bilayer: When the\ntwo layers are tuned to similar resonance conditions, only\nthe symmetric mode corresponding to the synchronized\ndynamics produces a strong response, while the antisym-\nmetric mode is strongly suppressed. It is thus natural to\nsuggest that the q2viscous magnetic damping in the con-\ntinuum limit may have far-reaching consequences for the\ncurrent-driven nonlinear power spectrum as that mea-\nsured in Ref. 22. This needs a further investigation.\nThe role of electron-electron interactions was mani-\nfested in our theory through the spin Coulomb drag,\nwhich enhances the e\u000bective transverse spin scattering\nrate (29). This becomes particularly important, in com-\nparison to the disorder contribution to the transverse spin\nscattering, in the limit of weak magnetic polarization.9We \fnally emphasize that the study in this paper was\nlimited exclusively to weak linearized perturbations of\nthe magnetic order with respect to a uniform equilib-\nrium state. When the equilibrium or out-of-equilibrium\nmagnetic state is macroscopically nonuniform, as is the\ncase with, e.g., the magnetic spin spirals, domain walls,\nvortices, and other topological states, the longitudinal as\nwell as transverse spin currents become relevant for the\nmagnetic dynamics. The longitudinal spin currents lead\nto additional contributions to the spin-transfer torques,\nmodifying the magnetic equation of motion. Such spin\ntorques leading to the dissipative q2damping terms were\ndiscussed in Ref. 13. These latter contributions to the\nmagnetic damping are likely to dominate in strongly-\ntextured magnetic systems.\nAcknowledgments\nWe are grateful to Gerrit E. W. Bauer and Arne\nBrataas for stimulating discussions. This work was sup-\nported in part by the Alfred P. Sloan Foundation (YT)\nand NSF Grant No. DMR-0705460 (GV).\n1B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898\n(1969).\n2A. J. Leggett, J. Phys. C: Sol. State Phys. 3, 448 (1970).\n3A. Singh, Phys. Rev. B 39, 505 (1989); A. Singh and\nZ. Te\u0014 sanovi\u0013 c, ibid.39, 7284 (1989).\n4V. L. Sobolev, I. Klik, C. R. Chang, and H. L. Huang, J.\nAppl. Phys. 75, 5794 (1994); A. E. Meyerovich and K. A.\nMusaelian, Phys. Rev. Lett. 72, 1710 (1994); D. I. Golosov\nand A. E. Ruckenstein, ibid.74, 1613 (1995); Y. Takahashi,\nK. Shizume, and N. Masuhara, Phys. Rev. B 60, 4856\n(1999).\n5Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002).\n6V. P. Mineev, Phys. Rev. B 69, 144429 (2004); ibid.72,\n144418 (2005).\n7Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 67, 140404(R) (2003).\n8Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n9E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B 78, 020404(R) (2008).\n10T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n11E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part\n2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n12T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n13J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008).\n14E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B 75, 174434 (2007).\n15B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 (1993).16N. D. Mermin, Phys. Rev. B 1, 2362 (1970).\n17I. D'Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000).\n18K. Capelle, G. Vignale, and B. L. Gy or\u000by, Phys. Rev. Lett.\n87, 206403 (2001).\n19J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n20J. A. Fernandez-Baca, J. W. Lynn, J. J. Rhyne, and G. E.\nFish, J. Appl. Phys. 61, 3406 (1987).\n21B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n22I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley,\nJ. C. Sankey, D. C. Ralph, and R. A. Buhrman, Phys. Rev.\nB76, 024418 (2007).\n23It is natural to also wonder about a possible additional\nspin current of the form ji/@i@tm, which does not\nbreak time-reversal symmetry. Such spin current leads to a\nwave-vector-dependent correction to the e\u000bective gyromag-\nnetic ratio, which is very small in practice. It parallels the\nstructure of the spin pumping in magnetic nanostructures,\nwhich consists of the dominant dissipative piece of the form\nm\u0002@tmand a smaller piece of the form @tm. The latter\nmerely causes a slight rescaling of the gyromagnetic ratio.\nWhile the dissipative piece of the spin pumping has been\nunambiguously established in a number of experiments,8\nthe small correction to the gyromagnetic ratio is yet to be\nobserved.\n24We need to remark here that the above gauge transfor-\nmation does not a\u000bect the transverse spin current in the\nlinearized theory."    },    {        "title": "1207.6686v1.Ultrafast_optical_control_of_magnetization_in_EuO_thin_films.pdf",        "content": "1  \nUltrafast optical control of magnetization in EuO thin films \nT. Makino1,*, F. Liu2,3, T. Yamasaki4, Y. Kozuka2, K. Ueno5,6, A. Tsukazaki2, T. Fukumura6,7, Y. Kong3, and \nM. Kawasaki1,2 \n1 Correlated Electron Research Group (CERG) and Cr oss-Correlated Materials Research Group (CMRG), RIKEN Advanced Science Institut e, \nWako 351-0198, Japan \n2 Quantum Phase Electronics Ce nter and Department of Applied Physics, Un iversity of Tokyo, Tokyo 113-8656, Japan, \n3 School of Physics, Nankai Un iversity, Tianjin 300071, China  \n4 Institute for Materials Research, T ohoku University, Sendai, 980-8577, Japan \n5Graduate School of Arts and  Sciences, University of  Tokyo, Tokyo 153-8902, Japan \n6PRESTO, Japan Science and Technology Agency, Tokyo 102-0075, Japan, \n7Department of Chemistry, Univers ity of Tokyo, Tokyo 113-0033, Japan, \n \nAll-optical pump-probe detection of magnetization precession has been performed for ferromagnetic EuO \nthin films at 10 K. We demonstrate that the circ ularly-polarized light can be used to control the \nmagnetization precession on an ultrafast time scale. This takes place within the 100 fs duration of a single \nlaser pulse, through combined contribution from two nonthermal photomagnetic effects, i.e., enhancement \nof the magnetization and an inverse Faraday effect. From the magnetic field dependences of the frequency \nand the Gilbert damping parameter, the intrinsic Gilbert damping coefficient is evaluated to be α ≈ 3×10-3. \nPACS numbers: 78.20.Ls, 42 .50.Md, 78.30.Hv, 75.78.J \n 2 Optical control of the spin in magnetic materials has \nbeen one of the major issues in the field of spintronics, \nmagnetic storage technology, and quantum computing1. \nOne type of the spin controls is based on the directional manipulation in the spin moments\n2. This yields in \nobservations of spin precession (reorientation) in \nantiferromagnets and ferromagne ts when magnetization is \ncanted with respect to an external field3–14. In many \nprevious reports, the spin precession has been driven with \nthe thermal demagnetization induced with the photo-irradiation. Far more intriguing is the ultrafast \nnonthermal control of magnetization by light\n8,10,14, which \ninvolves triggering and suppression of the precession. The precession-related anisotropy is expected to be \nmanipulated through laser-induced modulation of \nelectronic state because the anisotropy field originates \nfrom the magnetorcrystalline anisotropy based on the \nspin-orbit coupling. Recently, the spin precession with the \nnon-thermal origin has been observed in bilayer manganites due to a hole-concentration-dependent \nanisotropic field in competing magnetic phases\n15. Despite \nthe success in triggering the reorientation by ultrafast laser pulses, the authors have not demonstrated the possibility \nof the precessional stoppage. \n On the other hand, photomagnetic switch of the \nprecession has been reported in ferrimagnetic  garnets with \nuse of helicity in light\n8,10. The authors attributed the \nswitching behavior to long-lived photo-induced \nmodification of the magnetocrystalline anisotropy16 \ncombined with the inverse Faraday effects17,18. The \nunderlying mechanism for the former photo-induced effect \nis believed to be redistribution in doped ions16. This is too \nunique and material-dependent, which is not observed in wide variety of magnets. For establishing the universal \nscheme of such “helicity-controllable” precession, it \nshould be more useful to rely on more generalized mechanisms such as the carrier-induced ferromagnetism \nand the magnetic polarons\n19. A ferromagnet should be a \nbetter choice than a ferrimagnet or an antiferromagnet, e.g., for aiming a larger-amplitude modulation by making \nuse of its larger polarization-rotation angle per unit length . \nWe have recently reported  the optically-induced \nenhancement of magnetization in ferromagnetic EuO \nassociated with the optical transition from the 4 f to 5 d \nstates\n20. This enhancement was attributed to the \nstrengthened collective magnetic ordering, mediated with \nthe magnetic polarons. The helicity-controllable \nprecession is expected to be observed in EuO by combining the photo-induced magnetization \nenhancement20 with the inverse Faraday effect17,18 because \nthe magnetization is related to the magnetic anisotropy. \nThe occurrence of the inverse Faraday effects is expected \nbecause of the high crystalline symmetry in EuO17,18. The \nmagnetic properties of EuO are represented by the \nsaturation magnetization of 6.9 μB/Eu, the Curie \ntemperature of 69 K, and the strong in-plane \nanisotropy21,22. \n In this article, we report observation of the \nphotomagnetic switch of the spin precession with the \nnonthermal origin in a EuO thin film  for the first time to \nthe best of our knowledge. Due to the above-mentioned \nreasons, our findings deserve the detailed investigations \nsuch as the dependence on the circularly polarized lights, the frequency of precession, the Gilbert damping constants, \nand the magnitudes of the photo-induced anisotropic field. \n EuO films were deposited on YAlO\n3 substrate using a \npulsed laser deposition system with a base pressure lower \nthan 8×10-10 Torr22. The EuO films were then capped with \nAlO x films in-situ . EuO and AlO x layers have thicknesses \nof 310 and 30 nm, respectively. The film turned out to be \ntoo insulating to be quantified by a conventional transport measurement method. The all-optical experiments have \nbeen performed using a standard optical set-up with a \nTi:sapphire laser combined with a regenerative amplifier (accompanied with optical parametric amplifier). The \nwavelength, width, and repetition rate of the output pulse \nwere 650 nm, ≈100 fs, and 1 kHz, respectively.  The \npump and probe pulses were both incident on the film at \nangles of θ\nH ≈ 45 degree from the direction normal to the \nfilm plane as shown in inset of Fig. 1. The direction of the probe beam is slightly deviated from that of the pump so \nas to ensure the sufficient spatial separation of the \nreflected beams. The angle between the sample plane and the external field is approximately 45 degree. The \npolarization rotation of the reflected probe pulses due to \nthe Kerr effect was detected using a Wollaston prism and a balanced photo-receiver. The pump fluence was \napproximately 0.5 mJ/cm\n2. A magnetic field was applied \nusing a superconducting electromagnet cryostat. The maximum applied magnetic field was μ\n0H ≈ 3 T. All the \nmeasurements were performed at 10 K. \n Figure 1 shows a magneto-optical Kerr signal as a \nfunction of the pump-probe delay time for a EuO film at \nμ\n0H = 3.2 T under the irradiation of right-circularly 3 polarized ( σ+) light. Its time trace is composed of \ninstantaneous increase and d ecay of the Kerr rotation, and \nsuperimposed oscillation20. The oscillatory structure \ncorresponds to the precession of magnetization. A solid (black) curve in Fig. 1 shows the result of fit to the \nexperimental data using an exponentially decaying \nfunction and a damped oscillatory function. The precession is observed even wi th the linearly polarized \nlight, which is consistent with the fact that EuO is a \nferromagnet at this temperature. \n \nFIG. 1 (color online). Time-resolved Kerr signals recorded \nfor a EuO thin film at a magnetic field of 3.2 T, and a \ntemperature of 10 K for ri ght circularly-polarized ( σ\n+) \nlight. The inset schematically shows the experimental arrangement. Experimental data are shown by (red) \nsymbols, while the result of fit was shown by a full (blue) \nline. \n \nFor the detailed discussion of the precession properties, \nwe subtracted the non-oscillatory part from the Kerr signal as a background. The results are shown in Fig. 2 for nine \nmagnetic fields and for σ\n+ and left-circularly polarization \n(σ¯). The subtracted data were then fitted with the damped \nharmonic function in the form of Aexp(−t/τ) sin(2πft+φ), \nwhere A and φ are the amplitude and the phase of \noscillation, respectively. The amplitude of the precession was not found to depend on the plane of the linear \npolarization of the pump pulse. There is a linear \nrelationship between the amplitude of precession and the pump fluence for the excitation intensity range measured. \nIt is also noticed in Fig. 2 that the precession amplitudes \nare different each other for the two helicities ( σ\n+ and σ¯) \neven at the same magnetic fields. The magnetic field \ndependence of the amplitude is summarized in Fig. 3(d). The minimum precession amplitude appears at around μ0H \n= +0.4 T for the σ¯, while the minimum is observed at \nμ0H = −0.4 T for the σ+ as indicated by the shaded regions. \nTo explain such disappearance of the precession and the triggering of precession even  with a linearly-polarized \nlight, it is necessary to take two effects into account. One \nof the effects that we seek should be odd with respect to the helicity of light. An effective magnetic field through \nthe inverse Faraday effect is plausible to interpret this \nphenomenon because this satisfies the above requirements \n[H\nF\n// (black arrows) in Figs. 3(a) and 3(b)]. While the \nnormal Faraday effect causes difference in the refractive \nindices for the left and right circularly polarized lights propagating in a magnetized medium, it is also possible to \ninduce the inverse process where circularly polarized \nlights create a magnetization or an effective field\n17,18. The \nfield associated with the inverse Faraday effect changes its \nsign when the circular polarization is changed from \nleft-handed to right-handed. \n \nFIG. 2 (color online). A series of precession signals under various \nmagnetic fields for right- and left-circularly polarized ( σ+ and σ¯) \nlights. Solid circles show the e xperimental data for which the \nnon-oscillatory background is s ubtracted, while solid curves \nrepresent the calculated data as described in the text.  \n \nThe other effect involved is considered to be the \nphotoinduced enhancement of the anisotropic field \n(magnetization) associated with the 4 f →5d optical \ntransition [ ΔM (purple arrows) in Figs. 3(a) and 3(b)]20. \nOur previous work quantified the photoinduced \n4 enhancement of the magnetization to be ΔM/M ≈  0.1%20. \nThe amplitude of precession is determined from \ncombination of ΔM with the component of the \ninverse-Faraday field ( HF\n//) approximately projected onto \nthe easy-axis direction. For example, no precession is \ntriggered for μ0H of +0.4 T ( −0.4 T) and σ¯ (σ+), which is \ndue to the balance of these two effects [Fig. 3(a)]. On the other hand, constructive contribution of these effects leads \nto a change in the direction of the magnetization [two \ndashed lines and a red arrow in Fig. 3(b)], which enhances the precession amplitude. The strength of the \nphotoinduced field H\nF can be estimated to be \napproximately 0.2 T at the laser fluence of 0.5 mJ/cm2. \nThe derivation was based on Eq. (17) of Ref. 10. For more \nquantitative discussion for the suppression and \nenhancement of precession, the effect of the perpendicular component of inverse Faraday field is necessary to be \ntaken into account. Such analysis is not performed here \nbecause this goes beyond the scope of our work. \n \nFIG. 3 (color online). Graphical  illustrations of the magnetic \nprecession; its suppression (a) and enhancement (b). M is a \nmagnetization (green), H the external magnetic field (blue), Heff the \neffective magnetic field (red), ΔM a photo-induced magnetization \nenhancement (purple), and the HF\n// the inverse Faraday field (black). \nThe situations of suppression correspond to the conditions of 0.4 T for σ¯ and −0.4 T for σ+. The situations of enhancement are for \nopposite cases. Magnetic field depe ndences of the magnetization \nprecession related quantities for σ+ and σ¯; precession frequency f \n(c), amplitude (d), and effective Gilbert damping αeff (e) (f). \n \nFor the derivation of the precession-related parameters, \nwe plot the frequency ( f) and the amplitude of the \nmagnetization precession for two different helicities as a \nfunction of H with closed symbols in Figs. 3(c) and 3(d). \nTo deduce the Landé g-factor g, we calculated f(H) using \na set of Kittel equations for taking the effect of tilted \ngeometry into account as12,23: \n12 f HH                             ( 1 )  \n2\n1e f f cos( ) cosH HH M               ( 2 )  \n2e f f cos( ) cos 2H HH M               ( 3 )  \nHere, γ is the gyromagnetic ratio ( gμB/h), μB the Bohr \nmagneton,  h Planck’s constant, and θH an angle between \nthe magnetic field and direction normal to the plane. Meff \nis the effective demagnetizing field given as Meff = MS-2K\n⊥/MS, where MS is the saturation magnetization and K⊥ is \nthe perpendicular magnetic anisotropy constant. θ is an \nequilibrium angle for the magnetization, which obeys the \nfollowing equation: \neff sin 2 (2 / )sin( )H HM                ( 4 )  \nA solid (black) line in Fig. 3( c) corresponds to the result \nof the least-square fit for the frequency f. The values of \nparameters are g ≈ 2 and μ0Meff  ≈ 2.4 T. The g value is \nconsistent with the one derived from the static ferromagnetic resonance measurement\n24. \n \nHaving evaluated the precession-related parameters \nsuch as g and Meff, we next discuss H dependence of an \neffective Gilbert damping parameter αeff. This quantity is \ndefined as: \neff1\n2f                              ( 5 )  \nFigures 3(e) and 3(f) show the effective Gilbert \ndamping parameter αeff derived from the decay time \nconstant ( τ) for σ+ and σ¯, respectively. Despite relatively \nstrong ambiguity shown with ba rs in Figs. 3(e) and 3(f), \nthe damping parameters αeff is not independent of the \nmagnetic field. It is rather a ppropriate to interpret that for \n5 αeff for low fields are larger than those at higher fields. \nSuch dependence on magnetic field is consistent with \nthose in general observed for a wide range of the \nferrimagnets and ferromagnet s. Two-magnon scattering \nhas been adopted for the explanation of this trend25. When \nthe magnitude or direction of the magnetic anisotropy \nfluctuates microscopically, magnons can couple more efficiently to the precessional motion\n25. Such may cause \nan additional channel of relaxation. Due to the suppressed \ninfluence of the abovementioned two-magnon scattering, the higher-field data correspon d to an intrinsic Gilbert \ndamping constant α ≈ 3×10\n-3, as shown with a dashed \n(black) line in Figs. 3(e)  and 3(f). This value is \ncomparable with that reported in Fe26,27,28,29 and \nsignificantly larger than that of yttrium iron garnet, which \nis known for intrinsically low magnetic damping8,10,14. \n \nIn conclusion, we have reported the observation of \nmagnetization precession and the dependence on light helicity in ferromagnetic EuO films. We attribute it to the \nphoto-induced magnetization enhancement combined with \nthe inverse Faraday effect. The magnetic field dependence of the precession properties al lowed us the evaluation of \nthe Gilbert damping constant to be ≈3×10\n-3. \nAcknowledgements—the authors thank K. Katayama, \nM. Ichimiya, and Y. Takagi for helpful discussion. This \nresearch is granted by the Japan Society for the Promotion \nof Science (JSPS) throug h the “Funding Program for \nWorld-Leading Innovative R&D on Science and \nTechnology (FIRST Program),” initiated by the Council \nfor Science and Technology Policy (CSTP) and in part supported by KAKENHI (Grant Nos. 23104702 and \n24540337) from MEXT, Japan (T. M.). \n \nREFERENCES \n* tmakino@riken.jp  \n1 A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. \nPisarev, A. M. Balbashov, and Th. Rasing, Nature \n435, 655 (2005).  \n2 A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod. \nPhys. 82, 2731 (2010).  \n3 C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, \nD. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 (1999).  \n4 Q. Zhang, A. V. Nurmikko, A. Anguelouch, G. Xiao, \nand A. Gupta, Phys. Rev. Lett. 89, 177402 (2002).  \n5 M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, \nL. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. \nRev. Lett.  88, 227201 (2002).  6 R. J. 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B 84, 134425 (2011)  \n "    },    {        "title": "1412.3783v1.Deviation_From_the_Landau_Lifshitz_Gilbert_equation_in_the_Inertial_regime_of_the_Magnetization.pdf",        "content": "arXiv:1412.3783v1  [cond-mat.mtrl-sci]  11 Dec 2014Deviation From the Landau-Lifshitz-Gilbert equation in th e Inertial regime of the\nMagnetization\nE. Olive and Y. Lansac\nGREMAN, UMR 7347, Universit´ e Fran¸ cois Rabelais-CNRS, Pa rc de Grandmont, 37200 Tours, France\nM. Meyer, M. Hayoun, and J.-E. Wegrowe\nLaboratoire des Solides Irradi´ es, ´Ecole Polytechnique, CEA-DSM, CNRS, F-91128 Palaiseau, Fr ance\n(Dated: February 7, 2018)\nWe investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic\nresonance (FMR) context. Analytical predictions and numer ical simulations of the complete equa-\ntions within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the\nusual precession resonance, the inertial model gives a seco nd resonance peak associated to the nuta-\ntion dynamics provided that the damping is not too large. The analytical resolution of the equations\nof motion yields both the precession and nutation angular fr equencies. They are function of the in-\nertial dynamics characteristic time τ, the dimensionless damping αand the static magnetic field H.\nA scaling function with respect to ατγHis found for the nutation angular frequency, also valid for\nthe precession angular frequency when ατγH≫1. Beyond the direct measurement of the nutation\nresonance peak, we show that the inertial dynamics of the mag netization has measurable effects on\nboth the width and the angular frequency of the precession re sonance peak when varying the applied\nstatic field. These predictions could be used to experimenta lly identify the inertial dynamics of the\nmagnetization proposed in the ILLG model.\nPACS numbers:\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation is a ki-\nnetic equation that does not contain acceleration terms,\ni.e. that does not contain inertia. The corresponding\ntrajectory is reduced to a damped precession around\nthe axis defined by the effective field. The measurement\nof this precession is usually performed by the mean of\nferromagnetic resonance (FMR). The power absorbed by\nthe system is then measured at steady state while adding\nan oscillatory field to the effective field, and tuning the\nfrequency close to the resonance frequency. However,\nthe validity of the LLG equation is limited to large\ntime scales1, or low frequency regimes (similarly to the\nDebye model of electric dipoles2). Indeed, the precession\nwith damping described by the LLG equation is a\ndiffusion process in a field of force, for which the angular\nmomentum has reached equilibrium. Accordingly, if\nthe measurements are performed at fast enough time\nscales, or high enough frequencies, inertial terms should\nbe expected to play a role in the dynamics, which is no\nlonger reduced to a damped precession3–9. A nutation\ndynamics is therefore expected, giving a second resonant\npeak at the nutation frequency, and this new absorption\nshould be measurable with dedicated spectroscopy (e.g.\nusing infrared spectroscopy).\nDespite its fundamental importance, a systematic\nexperimental investigation of possible inertial effects of\nthe uniform magnetization has however been overlooked.\nIn order to evidence experimentally the consequences\nof inertia in the dynamics of a uniform magnetization,\nit is first necessary to establish the characteristics\nthat would allow to discriminate inertia from spuriouseffects in spectroscopy experiments. We propose in this\npaper some simple theoretical and numerical tools than\ncan be used by experimentalists in order to evidence\nunambiguouslytheeffectsofinertiaofthemagnetization.\nThe LLG equation reads :\ndM\ndt=γM×/bracketleftbigg\nHeff−ηdM\ndt/bracketrightbigg\n(1)\nwhereMisthemagnetization, Hefftheeffectivemagnetic\nfield,ηthe Gilbert damping, and γthe gyromagnetic ra-\ntio. If the description is extended to the fast degrees\nof freedom (i.e. the degrees of freedom that includes the\ntime derivative of the angularmomentum), a supplemen-\ntary inertial term should be added with the correspond-\ningrelaxationtime τ. FromthisInertialLandau-Lifshitz-\nGilbert (ILLG) model, the new equation reads3–7:\ndM\ndt=γM×/bracketleftbigg\nHeff−η/parenleftbiggdM\ndt+τd2M\ndt2/parenrightbigg/bracketrightbigg\n(2)\nOne of the main consequences of the new dynami-\ncal equation is the emergence of the second resonance\npeak associated to the nutation at high frequencies, as\nreported in our previous study7. In the literature the\nnutation dynamics of magnetic moments has been in-\nvestigated using various theoretical approaches though\nnot yet evidenced experimentally. B¨ ottcher and Henk\nstudied the significance of nutation in magnetization dy-\nnamics of nanostructures such as a chain of Fe atoms,\nand Co islands on Cu(111)8. They found that the nu-\ntation is significant on the femtosecond time scale with\na typical damping constant of 0.01 up to 0.1. Moreover,2\nthey concluded that nutation shows up preferably in low-\ndimensional systems but with a small amplitude with\nrespect to the precession. Zhu et al.predicted a nuta-\ntion dynamics for a single spin embedded in the tunnel-\ning barrierbetween twosuperconductors10. This unusual\nspin dynamics is caused by coupling to a Josephson cur-\nrent. They argue that this prediction might be directly\ntested for macroscopic spin clusters. The nutation is also\ninvolved in the dynamics of a single spin embedded in\nthe tunnel junction between ferromagnets in the pres-\nence of an alternating current11. In an atomistic frame-\nwork, Bhattacharjee et al.showed that first-principle\ntechniques used to calculate the Gilbert damping factor\nmay be extended to calculate the moment of inertia ten-\nsor associated to the nutation9.\nOur previous work7was focussed on the short time\nnutation dynamics generated by the ILLG equation, and\nwas limited to fixed values of the inertial characteristic\ntime scale τ, the dimensionless damping αand the static\nfieldH. In this paper we present a combined analytical\nand numerical simulation study of the ILLG equation\nwith new results. In particular we derive analytical re-\nsults in the small inclination limit that can be used in\nferromagnetic resonance (FMR) experiments, and which\nallow to predict both the precession and nutation reso-\nnance angular frequencies. We also investigate the ILLG\nequation while varying the three parameters α,τandH,\nand scaling functions are found. Finally, we present im-\nportant indications for experimental investigations of the\ninertial dynamics of the magnetization. Indeed, a conse-\nquence of the ILLG equation is the displacement of the\nwell-known FMR peak combined with a modified shape\nwith respect to that given by the LLG equation. This\ndisplacement could not be without consequences on the\ndetermination of the gyromagnetic factor γby ferromag-\nnetic resonance.\nThe paper is organized as follows. In section II we\nshow analytical solutions of the precession and nutation\ndynamics for the uniform magnetization in a static ap-\nplied field H. The small inclination limit is investigated\nin order to reproduce the usual experimental FMR con-\ntext. In section III we describe the numerical simula-\ntions of the magnetization inertial dynamics in both a\nstatic anda smallperpendicular sinusoidalmagneticfield\n(Heff=H+h⊥(ω)). The resonance curves are computed\nand, provided that the damping is not too large, a nu-\ntation resonance peak appears in addition to the usual\nferromagnetic resonance peak associated to the magne-\ntization precession. In section IV the behavior of the\nILLG equation is investigated in details while varying\nthe characteristic time τof the inertial dynamics, the di-\nmensionless damping αand the static field H. A very\ngood agreement is found between the analytical and nu-\nmerical simulation results, and a scaling function with\nrespect to ατγHis found. In section V we propose ex-\nperiments in the FMR context that should evidence the\ninertial dynamics of the magnetization described in the\nILLGmodel. Inparticular,whenthestaticfieldisvaried,the ILLG precession resonance peak has different behav-\niors compared to the usual LLG precession peak with\nshifted resonance angular frequency and modified shape.\nWe show that the differences between LLG and ILLG\nprecession peaks are more pronounced in large damping\nmaterials and increase with the static field. Finally, we\nderive the conclusions in section VI.\nII. ANALYTICAL SOLUTIONS FOR THE ILLG\nEQUATION\nThe magnetization position is described in spherical\ncoordinates ( Ms,θ,φ), where Msis the radius coordinate\nfixed at a constant value for the uniformly magnetized\nbody,θis the inclination and φis the azimuthal angle.\nIn a static magnetic field Hˆ zapplied in the zdirection,\ni.e.H=H(cosθer−sinθeθ) in the spherical basis\n(er,eθ,eφ), Eq. (2) gives the following system :\n¨θ=−1\nτ˙θ−1\nτ1˙φsinθ+˙φ2sinθcosθ\n−ω2\nτ1sinθ (3a)\n¨φsinθ=1\nτ1˙θ−1\nτ˙φsinθ−2˙φ˙θcosθ (3b)\nwhere the characteristic times are τandτ1=ατ,ω2=\nγHis the Larmor angular frequency, and α=γηMsis\nthe dimensionless damping.\nUsingthedimensionlesstime t′=t/τ, Eqs. (3)become\nθ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ\n−/tildewideω2/tildewideτ1sinθ (4a)\nφ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ(4b)\nwhere\nθ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,\nand\n/tildewideτ1=τ\nτ1=1\nα\n/tildewideω2=ω2τ=τγH\nIn the following subsections we extract analytical re-\nsults that can be used to predict the positions in the\nangular frequency domain of the precession and nutation\nresonance peaks. We will consider the small inclination\nlimit which holds in the FMR context.3\nA. Precession : exact and approximate solutions\nTo determine the precession dynamics of the iner-\ntial model we search for the long time scale solution\nφ′(t′) =φ′\nprec, whereφ′\nprecis the constant precession ve-\nlocity. Since the damping progressivelyshifts the magne-\ntization to the zaxis, we investigate the small inclination\nlimit where φ′(t′) =φ′\nprecshould hold. With sin θ∼θ\nand cosθ∼1, Eqs. (4) therefore reads :\nθ′′+θ′+/tildewideω2\n0θ= 0 (5a)\nφ′\nprec=/tildewideτ1θ′\nθ+2θ′(5b)\nwhere the natural angular frequency of the overdamped\nharmonic oscillator θ(t′) defined by Eq. (5a) is given by\n/tildewideω0=/radicalBig\n/tildewideτ1(φ′prec+/tildewideω2)−φ′2prec (6)\nThe characteristic equation associated to the differential\nequation Eq. (5a) is β2+β+/tildewideω2\n0= 0 which gives in the\naperiodic regime the two solutions\nβ±=−1±/radicalbig\n1−4/tildewideω2\n0\n2(7)\nSince|β+|<|β−|, the inclination of the magnetization\nbehaves at long time scales as\nθ(t′)∼eβ+t′,\nwhich inserted in Eq. (5b) gives\nφ′\nprec=/tildewideτ1β+\n1+2β+(8)\nIn original time units, the precession velocity ˙φprecis\ntherefore the solution of\n˙φprec=β+(˙φprec)\nατ/parenleftBig\n1+2β+(˙φprec)/parenrightBig (9)\nwhere the function β+(˙φprec) is given by\nβ+(˙φprec) =−1+/radicalbigg\n1−4τ/parenleftBig˙φprec+γH\nα−τ˙φ2prec/parenrightBig\n2(10)\nEquation 9 may be numerically solved to extract the\nprecession velocity, and therefore the precession reso-\nnance peak when a sinusoidal magnetic field h⊥(ω) is\nsuperimposed perpendicular to the static field Hˆ z.\nForτ≪10−11sandα≤0.1, theprecessionvelocity ˙φprec\nfor small applied static fields may be accurately evalu-\nated from a quadratic equation : in this case /tildewideω2\n0≪1 and\nEq. (7) leads to β+≈ −/tildewideω2\n0. Eq. (8) therefore gives a\ncubic equation in φ′\nprecwhere the cubic term −2αφ′3\nprecis negligeable. In this case the solution of the resulting\nquadratic equation is in original time units\n˙φprec=−b−/radicalbig\nb2+12τγH/α\n6τ(11)\nwithb= 2τγH−α−1/α. We choose the negative so-\nlution of the quadratic equation in order to agree with\nthe negative velocity ˙φLLG=−γH/(1+α2) given by the\nLLG model.\nB. Nutation : angular frequency\nUnlike the precession, the nutation properties should\nbederivedconsideringintermediatetime scaleswherethe\nprecession has not yet reached a constant velocity. Eqs.\n(4) should therefore be reconsidered. To derive the nuta-\ntion properties, it is convenient to examine the angular\nvelocityθ′. For simplicity we note θ′=/tildewideωθandφ′=/tildewideωφ.\nEqs. (4) therefore rewrite\n/tildewideω′\nθ=−/tildewideωθ−/tildewideτ1/tildewideωφsinθ+/tildewideω2\nφsinθcosθ\n−/tildewideω2/tildewideτ1sinθ (12a)\n/tildewideω′\nφsinθ=/tildewideτ1/tildewideωθ−/tildewideωφsinθ−2/tildewideωφ/tildewideωθcosθ(12b)\nWe derive Eq. (12a) with respect to time t′which gives\n/tildewideω′′\nθ=−/tildewideω′\nθ+(2/tildewideωφcosθ−/tildewideτ1)/tildewideω′\nφsinθ−/tildewideτ1/tildewideωφ/tildewideωθcosθ\n+/tildewideω2\nφ/tildewideωθ(cos2θ−sin2θ)−/tildewideω2/tildewideτ1/tildewideωθcosθ\nwhere the term /tildewideω′\nφsinθmay be replaced with the expres-\nsion in Eq. (12b). We therefore obtain\n/tildewideω′′\nθ+/tildewideω′\nθ+/parenleftbig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1cosθ/parenrightbig\n/tildewideωθ=\n/tildewideτ1/tildewideωφsinθ+3/tildewideτ1/tildewideωφ/tildewideωθcosθ−2/tildewideω2\nφcosθsinθ\n−(3cos2θ+sin2θ)/tildewideω2\nφ/tildewideωθ (13)\nEq. (13)shouldbecloselyrelatedtothenutationdynam-\nics since it describes the /tildewideωθoscillator. This assumption\nwill be confirmed in section IVA2 for a broad range of\nparameters. Eq. (13) defines the damped oscillator /tildewideωθ\nwhich is non-linearly coupled to the /tildewideωφoscillator. This\nexpression shows that, in the absence of coupling and in\nthe smallinclination limit θ≪1rad, the/tildewideωθoscillatoros-\ncillatesatthenaturalangularfrequency/radicalbig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1. We\ntherefore deduce an approximate expression for the nu-\ntation angularfrequency in the weak coupling case which\nis given by the expression\n/tildewideωweak\nnu=/radicalBig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1 (14)\nwhich in original time units gives\nωweak\nnu=√1+ατγH\nατ(15)4\nFrom Eq. (15) we deduce the following asymptotic\nbehaviors : when τ≪1/αγHthenωweak\nnu∼1/ατ, and\nwhenτ≫1/αγHthenωweak\nnu∼1/√ατ.\nBecause of the non-linear coupling terms in the right-\nhand side of Eq. (13), the true position of the nuta-\ntion resonancepeak in FMR experiments may differ from\nthe approximate angular frequency defined by Eq. (15).\nHoweverthe simulation of the resonancecurves with a si-\nnusoidal magnetic field h⊥(ω) superimposed perpendic-\nular to the static field Hˆ zwill show in section IVA2\nthat the non-linear coupling terms only slightly shift the\nnutation resonance peak from the approximate angular\nfrequency.\nIII. NUMERICAL SIMULATIONS OF THE\nRESONANCE CURVES IN THE ILLG MODEL\nWe apply a fixed magnetic field H=Hˆ zalong the\nzdirection, and a small sinusoidal magnetic field h⊥=\nh⊥cosωtˆ xin thexdirection. In the spherical basis the\ncomponents of the total magnetic field Heff=H+h⊥in\nEq. (2) are\nHeff\nr=Hcosθ+h⊥sinθcosφcosωt\nHeff\nθ=−Hsinθ+h⊥cosθcosφcosωt\nHeff\nφ=−h⊥sinφcosωt.\nwhich lead to the following dynamical equations for the\nspherical angles ( θ,φ) of the magnetization\n¨θ=−1\nτ˙θ−1\nτ1˙φsinθ+˙φ2sinθcosθ\n−ω2\nτ1sinθ+ω3\nτ1cosθcosφcosωt(16a)\n¨φsinθ=1\nτ1˙θ−1\nτ˙φsinθ−2˙φ˙θcosθ\n−ω3\nτ1sinφcosωt (16b)\nwhereω3=γh⊥is the angular frequency associated to\nthe sinusoidal field.\nUsing the dimensionless time t′=t/τ, Eqs. (16) be-\ncome\nθ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ\n−/tildewideω2/tildewideτ1sinθ+/tildewideω3/tildewideτ1cosθcosφcos/tildewideωt′(17a)\nφ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ\n−/tildewideω3/tildewideτ1sinφcos/tildewideωt′(17b)\nwhere\nθ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,and\n/tildewideτ1=τ\nτ1=1\nα\n/tildewideω2=ω2τ=τγH\n/tildewideω3=ω3τ=τγh⊥\n/tildewideω=ωτ\nWe useγ= 1011rad.s−1.T−1, and we vary the charac-\nteristic time τfor three different values of the dimension-\nless damping α= 0.1, 0.01 and 0 .5. We investigate sev-\neralvalues of the static magnetic field from H= 0.2Tup\ntoH= 200T. We numerically integrate Eqs. (17) using\neither a double precision second order Runge-Kutta algo-\nrithm or a double precision five order Gear algorithm12.\nTypically, we use time steps 10−7< δt′<10−3depend-\ning on the values of τandω.\nThe resonance curves are obtained by investigating\nthe magnetization response to the small oscillating field\nh⊥(ω) =h⊥cosωtˆ xapplied perpendicular to the static\nfieldH=Hˆ z. We analyse the permanent dynami-\ncal regime where the magnetization components oscil-\nlate around well defined mean values. For fixed values\nof the oscillating field angular frequency ωand oscil-\nlating field amplitude h⊥, we compute the mean value\n< M⊥>(averaged over time) of the transverse magneti-\nzationM⊥(t) =/radicalBig\nM2x(t)+M2y(t), fromwhichweextract\nfor fixed values of ωthe transverse susceptibility defined\nbyχ⊥=d < M ⊥> /dh ⊥. We choose values of the\noscillating field amplitude h⊥= 10−1,10−2,10−3,10−4\nand 10−5T, and we plot < M⊥>with respect to h⊥\nfor each ω. As an example, we show the case α= 0.1,\nτ= 2×10−10s,H= 2Tandω= 1.2×1011rad.s−1.\nThe inset of Fig. 1 shows that the response is linear\n< M⊥>=χ⊥h⊥wherefrom we extract the transverse\nsusceptibility χ⊥using a linear fitting. We repeat the\nsame procedure for each oscillating field angular fre-\nquencyωwhich gives the resonance curve χ⊥(ω) of the\ntransverse susceptibility shown in Fig. 1. Two peaks\nclearly appear, the usual FMR peak associated to the\nprecession velocity, and the nutation peak associated to\nthe nutation dynamics originatingfrom the inertial term.\nIV. RESULTS\nA. Effects of τ\nWe now examine the ILLG model when varying the\ncharacteristic time τ. For different values of the parame-\nterτ, we show in Fig. 2 the typical profiles of the trans-\nverse susceptibility χ⊥versus the angular frequency ωof\nthe applied oscillating field. The four resonance curves\nplotted in figure 2 are obtained by numerical simulations\nwithH= 2Tandα= 0.1. They show how the nu-\ntation resonance peak position depend on the value of\nτ. Asτis increased, the nutation peak moves towards5\n01×10112×10113×1011\nω (rad.s-1)1234567χ⊥10-510-410-310-2\nh⊥10-510-410-310-210-1< Μ⊥>Precession peak\nNutation peak\nFigure 1: Resonance curves of the transverse susceptibil-\nityχ⊥(ω) with respect to the oscillating field angular fre-\nquencyω. The resonance curves are computed within the\nILLG model with τ= 2×10−10s, for dimensionless damping\nα= 0.1 and for an applied static field H= 2T. Two reso-\nnance peaks are observed : the precession resonance at lower\nangular frequency which is the usual FMR and the nutation\nresonance at higher angular frequency. Inset : Example of\nthe calculation of χ⊥such that < M⊥>=χ⊥h⊥obtained for\nω= 1.2×1011rad.s−1.\nthe precession peak with an increasing intensity which is\nan order of magnitude smaller than the precession one\nforτ= 10−11s. Note that the transverse susceptibil-\nity at the resonance follows a power law of the form\nχ⊥(ωILLG\nnu)∝1/ωILLG\nnu, whereωILLG\nnuis defined as the nu-\ntation resonance angular frequency. A similar power law\nis reported for the precession peak obtained for different\nstatic fields H(see section IVB).\nWe now compare the analytical and numerical sim-\nulation results concerning the positions in the angular\nfrequency domain of both the precession and nutation\nresonance peaks.\n1. Precession peak\nWe define ωprec=|˙φprec|as the angular frequency of\nthe precession. When computed from the exact expres-\nsions (9) and (10) we will refer to ωexact\nprec, and when com-\nputed from the approximate expression (11) we will refer\ntoωapprox\nprec. Finally, we will denote by ωILLG\nprecthe angular\nfrequency of the precession resonance peak obtained in\nthe numerical simulations of the ILLG model. Eq. (9)\nmaybeeasilynumericallysolvedtofindthesolution ˙φprec\nfor several values of αandτ. The behavior with respect\ntoτofωprecobtained either analytically or from the sim-\nulated FMR curves is shown in Fig. 3. There is an excel-10 -4 10 -3 10 -2 10 -1 10 0\n10 11 10 12 10 13 10 14 10 15 χ\n⊥\nω ( rad.s -1 )τ=1×10 -113UHFHVVLRQ \n1utation \nτ=1×10 -12\nτ=1×10 -13\nτ=1×10 -14\nFigure 2: Resonance curves of the transverse susceptibilit y\nshowing the displacement of the nutation peak caused by the\nvariation of τ:τ= 10−11s (open circles), 10−12s (filled\ncircles), 10−13s (crosses), and 10−14s (open squares). These\ncurves are simulated using the ILLG model with α= 0.1,\nandH= 2T. Note that the precession peak positions are\nonly slightly affected. The dotted line shows the power law\nfitted on χ⊥∝1/ωILLG\nnu, whereωILLG\nnuis the resonance angular\nfrequency of the nutation.\nlent agreement between the analytical prediction ωexact\nprec\nand the precession resonance peak ωILLG\nprecobtained in nu-\nmericalsimulations. WealsoshowinFig.3theprecession\nangular frequency ωapprox\nprec. Forτ <10−11sandα= 0.1,\nit nicely agrees with the exact analytical value and with\nthenumericalsimulationresults, but the approximateso-\nlution becomes no longer valid for τ >10−11s. To quan-\ntify the validity of the approximate solution we compute,\nforτ= 10−12sand for three different dampings, the\nrelative difference\nδana\nprec=ωapprox\nprec−ωexact\nprec\nωexactprec×100\nWe show in the inset of Fig. 3 the evolution of δana\nprecwith\nrespect to the applied static field H. ForH <20T\nthe relative difference remains less than 0 .1% for small\ndamping α= 0.01, and remains less than 3% for mod-\nerate damping α= 0.1. For large damping α= 0.5 the\napproximate solution remains valid for small fields, but\nfor 12T < H < 20Tthe error becomes larger than 10%.\n2. Nutation peak\nFigure 4 displays both the analytical prediction of the\nnutation angular frequency ωweak\nnugiven by Eq. (15) and\nthe angular frequency ωIILG\nnuof the nutation resonance6\n10-1610-1510-1410-1310-1210-1110-1010-9\nτ (s)01×10112×1011ωprec (rad.s-1)\n0 5 10 15 20 25 30\nH (T)10-410-310-210-1100101102δprecana (%)α=0.5\nα=0.1\nα=0.01\nFigure 3: (Color online) Comparaison of the analytical and\nnumerical simulation results for the precession angular fr e-\nquency obtained for α= 0.1 andH= 2T. Filled circles\n(black) are the precession angular frequency ωexact\nprec, open cir-\ncles (red) are the position of the precession resonance peak s\nωILLG\nprec, stars (orange) are the approximate precession angular\nfrequencies ωapprox\nprecvalid for small values of τ. Thedashed line\n(black) is the LLG precession angular frequency, i.e.without\ninertial term. Inset : relative difference δana\nprecfor three differ-\nent dampings.\nobtained in the numerical simulations. The agreement\nis excellent for τ <10−11s, and indicates that the non-\nlinearcouplingtermsofEq. (13)donotsignificantlyshift\nthe angular frequency of the nutation resonance from the\napproximate angular frequency ωweak\nnu. On the contrary,\nin the range 10−11s < τ < 10−8s, the simulated nuta-\ntion resonance angular frequency is slightly higher than\nωweak\nnu, as shown in the upper inset of Fig. 4. In the lower\ninset ofFig. 4 we show the relative difference δnubetween\nthe approximate nutation angular frequency ωweak\nnuand\nthe nutation resonance angular frequency ωILLG\nnuof the\nnumerical simulations, i. e.\nδnu=ωILLG\nnu−ωweak\nnu\nωILLGnu×100\nWe therefore see that in the range 10−11s < τ <10−8s,\nthe approximate nutation angular frequency remains\nless than 15% close to the simulated nutation resonance\nangular frequency.\nB. Scaling and overview of the ILLG equation\nIn the preceding section we investigated the behav-\nior of the ILLG model when varying the characteristic\ntime scale τwhich drives the inertial dynamics. We also10-1610-1510-1410-1310-1210-1110-1010-9\nτ (s)1011101210131014101510161017ωnu (rad.s-1)\n10-1110-1010-9\nτ (s)10111012ωnu (rad.s-1)\n10-1310-1210-1110-1010-910-8\nτ (s)010\nδnu (%)\nFigure 4: (Color online) Comparaison of the analytical and\nnumerical simulation results for the nutation angular fre-\nquency obtained for α= 0.1 andH= 2T. Filled cir-\ncles (black) are the approximate nutation angular frequenc ies\nωweak\nnuand open circles (red) are the positions ωILLG\nnuof the\nsimulated nutation resonance peaks. Upper inset : enlarge-\nment showing the effect of the non-linear coupling terms of\nEq. (13). Lower inset : relative difference δnubetween ωweak\nnu\nandωILLG\nnu.\nvary the static field Hand the dimensionless damping\nα. Increasing Hmoves both the precession and nuta-\ntion resonance peaks to higher angular frequencies, with\nsmaller and broadened peaks, while increasing the di-\nmensionless damping moves both peaks to lower angular\nfrequencies with still smaller and broadened peaks. Note\nthat the ILLG precession resonances obtained when the\nstatic field His varied show that the transverse suscep-\ntibility follows a power law χ⊥∝1/ωIILG\nprec(not shown).\nThis law is the same as the one resulting from the LLG\nmodel13.\nEq. (15) suggests a scaling function\nωnu\nγH=√1+x\nx\nwherex=ατγH. Scaling curves obtained for different\nvalues of τ,αandHare shown in the inset of Fig. 5\nwhere both the precession and nutation resonance an-\ngular frequencies are dispayed with respect to ατγH.\nFig. 5 is an enlargement of the intermediate region of\nthe inset where we added the points obtained by the nu-\nmerical simulations for H= 2Tandα= 0.1. The\ntwo asymptotic behaviors of the nutation are highlighted\nwith the dashed lines in agreement with Eq. (15) :\nwhenατγH≪1 thenωweak\nnu/γH= 1/ατγH, and when\nατγH≫1 thenωweak\nnu/γH= 1/√ατγH. Remarquably,\nwe see that the precession peak position divided by γH\nalso scales as ωprec/γH∼1/√ατγHwhenατγH≫1.7\n0.01 0.1 1 10 100\nατγH0.010.1110100ωnu/γH , ωprec/γH\n10-610-410-210010210-2100102104106\n1/ατγH\n1/(ατγH)1/2\nFigure 5: (Color online) Scaling curves : nutation ωnuand\nprecession ωprecpeak positions in the angular frequency do-\nmain divided by γHwith respect to ατγH. Open circles\n(red) are the nutation and precession resonance peak posi-\ntions obtained in the numerical simulations for α= 0.1 and\nH= 2T. Other points are ωweak\nnucomputed from Eq. (15),\nandωexact\npreccomputed from Eq. (9). Different values of the\nstatic field Hand the dimensionless damping αare reported :\nH= 0.2Tandα= 0.1 (red open diamonds), H= 2T(blue\nopen squares for α= 0.1 and blue filled squares for α= 0.01),\nH= 20Tandα= 0.1 (green open triangles), H= 200T\nandα= 0.1 (black crosses). The dashed lines are the two\nasymptotic behaviors of the nutation in agreement with Eq.\n(15). Inset : same scaling curves (without red open circles)\ndisplayed on larger scales.\nThe two asymptotic behaviors intersect at ατγH= 1\nandω/γH= 1. This point corresponds to the max-\nimum value of the LLG precession angular frequency\nωLLG/γH= 1/(1 +α2) which is obtained in the limit\ncase of no damping α= 0.\nThe inset of Figure 5 indicates that only one resonance\npeakisexpectedwhen ατγH→ ∞. Inthiscase,boththe\nnutation and the precession contribute to a unique peak.\nOn the contrary, for finite ατγHthey remain separated.\nThere are two different well-defined peaks in the investi-\ngated range ( ατγH≤100). For ατγH≪1 the preces-\nsion peak is close to the usual LLG precession peak, and\nthe nutation peak shifts rapidly ( ωweak\nnu/γH∼1/ατγH)\nto high angular frequencies. In other words, the nutation\noscillator defined by Eq. (13) is independent of the pre-\ncession for ατγH≪1, whereas both synchronize at the\nsame frequency for ατγH→ ∞.\nAccurate predictions about the precession and nutation\npeak positions in the angular frequency domain can be\nmade, as long as the non-linear coupling terms of Eq.\n(13) remain weak or compensate each other.V. TOWARDS EXPERIMENTAL EVIDENCE\nOF THE INERTIAL DYNAMICS OF THE\nMAGNETIZATION\nThroughout the preceding sections we studied the new\nproperties of the inertial dynamics of the magnetization\nwithin the ILLG model. We specifically considered the\nFMR framework where a small perpendicular sinusoidal\nfield is applied implying that the small inclination limit\nholds. We now focus on possible simple experiments in\nsuch FMR framework that should highlight the inertial\ndynamics of the magnetization.\nThe first direct evidence would of course be the measure\nof the nutation resonance peak at frequencies larger than\nthe precession resonance peak. Since the expected nuta-\ntion resonance peak is given by Eq. (15), the evolution\nwith the static field Hmay be used to discriminate the\nnutation resonnce peak from possible other higher fre-\nquency peaks.\nHowever the amplitude of the nutation resonance peak is\nsmallerthan forthe precessionpeak, and itmaybe tricky\nin unfavorable situations to measure such a peak, for ex-\nample in materials with small characteristic time τ. Fur-\nthermore, for large dimensionless damping αboth peaks\nhave smaller amplitude and are rounded. It may even\nappear that the nutation resonance peak of the magneti-\nzationinthe ILLGmodel disappearsforalargedamping,\nlikethe resonantpeakofthe classicaldrivendamped har-\nmonic oscillator. For exampleFig. 6showsthat for mate-\nrials with a large damping ( α= 0.5) the resonance peaks\nare smaller and rounded compared to smaller damping\n(α= 0.1), and the nutation resonance peak disappears\nforH≤5T.\nIt is therefore necessary to find measurable characteris-\nticsofthemagnetizationinertialdynamicsotherthanthe\ndirect measure of the nutation resonance peak. Actually,\nwe show in the following that beyond the nutation reso-\nnance peak, the inertial dynamics has measurable effects\non the precession resonance peak. Indeed, as shown in\nFig. 7, the shape of the precession peak and its position\nintheangularfrequencydomainaremodifiedbytheiner-\ntial dynamics. And the effects are shown to be more pro-\nnounced for large damping materials and for large static\nmagnetic fields H. To show these effects we compare the\nprecession resonance angular frequencies ωILLG\nprecandωLLG\nprec\nobtained in the numerical simulations of both the ILLG\nand non-inertial LLG models. We use two different di-\nmensionless damping α= 0.1 andα= 0.5, and vary the\namplitude Hof the static magnetic field. For the ILLG\nmodel, we choose, as in Ref. 4, a rough estimation of the\ncharacteristic time scale τ= 10−12s.\nA. Angular frequency of the precession resonance\npeak\nWefirstlookatthepositionoftheprecessionresonance\npeakin the angularfrequencydomain. Fig.8(a)and 8(b)8\n02×10124×10126×1012\nω (rad.s-1)1×10-21×10-11×100χ⊥α=0.5\nFigure 6: (Color online) Resonance curves of the transverse\nsusceptibility χ⊥(ω) with respect to the oscillating field angu-\nlar frequency ω. The resonance curves are computed within\nthe ILLG model with τ= 10−12s, for a large dimension-\nless damping α= 0.5 and for various applied static fields :\nH= 2T(black filled circles), H= 5T(green filled trian-\ngles),H= 20T(blue open circles) and H= 50T(red open\ntriangles). Both resonance peaks clearly appear for H= 20T\nandH= 50T. ForH= 5TandH= 2Tthe nutation reso-\nnance peak dissapears due to the large damping.\n00.10.20.3\n1×10 12 2×10 12 3×10 12 20 T\n00.20.40.6\n0 1×10 12 2×10 12 10 T2 T\n0123\n0 1×10 12 \n00.050.1\n3×10 12 4×10 12 5×10 12 6×10 12 50 T2 T 2 T \nχ\n⊥ILLG LLG ILLG LLG χ\n⊥\nILLG LLG \nILLG LLG \nω (rad.s -1 ) ω (rad.s -1 )\nFigure 7: Precession resonance curves of the transverse sus -\nceptibility simulated for differentvalues ofthestatic fiel dH=\n2T, 10T, 20T, and 50 T. ILLG model (filled circles) and\nnon-inertial LLGmodel (opencircles). γ= 1011rad.s−1.T−1,\nα= 0.1, andτ= 10−12s.\ndisplay the evolution of the resonance angular frequencyωprecwith respect to Hobtained for α= 0.1 andα=\n0.5 within the numerical simulations of both the ILLG\nand LLG models. As expected the resonance angular\n0 10 20 30 40 50 60\nH (T)0123456ωprec (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)0123456ωprec (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)01020304050δ prec (%)a) b) α=0.1\nc)α=0.5\nα=0.5\nα=0.1LLGLLG\nILLG\nILLG\nFigure 8: (Color online) (a) and (b) Precession resonance an -\ngular frequency with respect to the applied static field. Re-\nsults obtainedin thenumerical simulations of theILLGmode l\n(withτ= 10−12s) and non-inertial LLG model, for dimen-\nsionless damping (a) α= 0.1 (blue open circles for LLG and\nred filled circles for ILLG) and (b) α= 0.5 (blue open squares\nfor LLG and red filled squares for ILLG). (c) Relative differ-\nenceδprecbetween LLG and ILLG precession resonance an-\ngular frequencies for α= 0.1 (green filled circles) and α= 0.5\n(green filled squares).\nfrequency of the LLG precession is linear with Hsince\nωLLG\nprec=γH/(1+α2) whereas the behavior is not linear in\nHfor the ILLG model. In Fig. 8(c) we plot the relative\ndifference\nδprec=ωLLG\nprec−ωILLG\nprec\nωLLGprec×100\nbetweenbothresonanceangularfrequencies. Therelative\ndistance between both precession peaks increases with H\nand with the dimensionless damping α.\nB. Width of the precession resonance peak\nWe now examine the evolution with Hof the shape\nof the precession resonance peak obtained in the sim-\nulations of the ILLG and LLG models. For α= 0.1,\nthe full width at half maximum (FWHM) is shown in\nFig. 9(a) while Fig. 9(b) displays the FWHM divided\nby the resonance angular frequency. For large damping\nα= 0.5 we change the criterion since the reduced\namplitude of the resonant peak does not allow anymore\nto compute the FWHM. We therefore compute the\nbandwith defined by the width of the peak at Amax/√\n29\n0 10 20 30 40 50 60\nH (T)00.10.20.3FWHM / ωprec\n0 10 20 30 40 50 60\nH (T)0.00.51.01.52.0FWHM (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)00.51Bandwidth / ωprec\n0 10 20 30 40 50 60\nH (T)012345Bandwidth (1012 rad.s-1)a)\nb)\nd)c)α=0.1\nα=0.1\nα=0.5\nα=0.5LLGLLG\nLLGLLGILLGILLG\nILLGILLG\nFigure 9: (Color online) (a) Full width at half maximum\n(FWHM) for the precession resonance peak for α= 0.1 within\nthe LLG (blue open circles) and the ILLG (red filled circles)\nmodels. (b)FWHMdividedeither by ωLLG\nprec(blueopencircles)\nor byωILLG\nprec(red filled circles). (c) Bandwidth of the preces-\nsion resonance peak for α= 0.5 within the LLG (blue open\nsquares)andILLG(redfilledsquares)models. (d)Bandwidth\ndivided either by ωLLG\nprec(blue open squares) or by ωILLG\nprec(red\nfilled squares).\nThe numerical simulations of the ILLG model are computed\nwithτ= 10−12s\nwhereAmaxis the maximum value of the peak. The\nbandwidth for α= 0.5 is shown in Fig. 9(c) and the\nbandwidth divided by the resonance angular frequency\nis plotted in Fig. 9(d). The numerical simulations of the\nILLG and LLG models lead to different behaviors for\nthe shape of the precession resonance peak. In the LLG\nmodel the FWHM and the bandwidth exhibit a linear\nevolution with the applied static field which results in\na constant evolution when divided by the resonance\nangular frequency. Very different behaviors are observed\nwithin the ILLG model where no linear evolution of the\nFWHM or the bandwidth is measured.\nFigs. 8 and 9 show that high applied static fields in\nlarge damping materials produce large differences be-\ntween the positions and shapes of the precession reso-\nnance peaks originating from the LLG and ILLG mod-\nels. Therefore, applying high static fields in large damp-\ning materials better allows to differentiate the precession\npeak originating from the ILLG and LLG models.\nAlthough the theory is clear and allows in principle to\ndifferentiate inertialfromnon-inertialdynamicswhen ex-\naminingboth precessionresonancepeaks, the experimen-\ntal investigations are rather more complex. Indeed, the\nexperimental demonstration of inertial effects first ne-\ncessitate to identify and control the different contribu-tions to the effective field (anisotropy, dipolar interac-\ntion, magnetostriction, ...) other than the applied static\nfield.\nVI. CONCLUSION\nThe magnetization dynamics in the ILLG model that\ntakes into account inertial effects has been studied from\nboth analytical and numerical points ofview. Within the\nFMR context, a nutation resonance peak is expected in\naddition to the usual precession resonance peak.\nAnalytical solutions of the inertial precession and nuta-\ntion angular frequencies are presented. The analytical\nsolutions nicely agree with the numerical simulations of\nthe resonance curves in a broad range of parameters.\nAt first, we investigated the effects of the time scale τ\nwhich drives the additional inertial term introduced in\nEq. (2)comparedtotheusualLLGequationEq. (1). We\nalso varied the dimensionless damping αand the static\nmagnetic field H, and a scaling function with respect to\nατγHis found for the nutation angular frequency. Re-\nmarquably, the same scaling holds for the precession an-\ngular frequency when ατγH≫1.\nIn the second part of the paper we focussed on the sig-\nnatures of the inertial dynamics which could be detected\nexperimentallywithintheFMRcontext. Weshowedthat\nbeyondthemeasureofthenutationresonancepeakwhich\nwould be a direct signature of the inertial dynamics, the\nprecession is modified by inertia and the ILLG preces-\nsion resonance peak is different from the usual LLG pre-\ncession peak. Indeed, whereas a linear evolution with\nrespect to His expected for the LLG precession reso-\nnance angular frequency, the ILLG precession resonance\nangular frequency is clearly non-linear. Furthermore, the\nshape of the precession resonance peak is different in the\nLLG and ILLG models. Again, the width variation of\nthe precession resonance peak is non-linear in the ILLG\ndynamics as opposed to the linear evolution with Hin\nthe LLG dynamics. We also showed that the difference\nbetween both LLG and ILLG precession peaks is more\npronounced when the damping is increased and when τ\nis increased. For example the discrepancy between the\nLLG and ILLG precession resonance angular frequencies\natH= 20Tforτ= 1psis expected to be of the order\nof 20% for α= 0.1 and 30% for α= 0.5. Therefore, large\ndamping materials are better candidates to experimen-\ntallyevidencetheinertialdynamicsofthemagnetization.\nFinally, a specific behavior of the amplitude of the mag-\nnetic susceptibility as a function of the nutation reso-\nnance angular frequency ωnuis predicted, of the form\nχ⊥(ωnu)∝ω−1\nnu(analogousto that ofthe usual FMR sus-\nceptibility). This law could be a useful criterion in order\nto discriminate the nutation peak among the other exci-\ntations that could also occur close to the infrared region\n(100 GHz up to 100 THz) in a ferromagnetic material.10\n1W.F. Brown Thermal Fluctuations of a Single-Domain\nParticle, Phys. Rev. 130, 1677 (1963).\n2R. Kubo, M. Toda, N. Hashitzume, Statistical physics II,\nNonequilibrium Statistical Mechanics , Springer Series in\nSolid-State Sciences 31, Berlin 1991 (second edition), Ed.\nP. Fulde, Chap 3, Paragraph 3.4.3, p. 131.\n3M.-C. Ciornei, Role of magnetic inertia in damped\nmacrospin dynamics , Ph. D. thesis, Ecole Polytechnique,\nPalaiseau France 2010.\n4M.-C. Ciornei, J. M. Rub´ ı, and J.-E. Wegrowe, Magnetiza-\ntion dynamics in the inertial regime : Nutation predicted\nat short time scales , Phys. Rev. B 83, 020410(R) (2011).\n5M. F¨ ahnle, D. Steiauf, and Ch. Illg, Generalized Gilbert\nequation including inertial damping : Derivation from an\nextended breathing Fermi surface model , Phys. Rev. B 84,\n172403 (2011).\n6J.-E. Wegrowe, C. Ciornei Magnetization dynamics, Gyro-\nmagnetic Relation, and Inertial Effects , Am J. Phys. 80,\n607 (2012).\n7E. Olive, Y. Lansac, and J.-E. wegrowe, Beyond ferromag-\nnetic resonance : the inertial regime of the magnetization ,Appl. Phys. Lett. 100, 192407 (2012).\n8D. B¨ ottcher, and J. Henk Significance of nutation in mag-\nnetization dynamics of nanostructures , Phys. Rev. B 86,\n020404(R) (2012).\n9S. Bhattacharjee, L. Nordstr¨ om, and J. Fransson Atomistic\nspin dynamic method with both damping and moment of\ninertia effects included from first principles , Phys. Rev.\nLett.108, 057204 (2012).\n10J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky\nNovel spin dynamics in a Josephson junction , Phys. Rev.\nLett.92, 107001 (2004).\n11J. Fransson, and J. Xi. Zhu Spin dynamics in a tunnel\njunction between ferromagnets , New J. Phys. 10, 013017\n(2008).\n12C. W. Gear, Numerical initial value problems in ordinary\ndifferential equations , Prentice Hall, Englewood Cliffs (N.\nJ.) 1971.\n13A. G. Gurevich and G. A. Melkov, Magnetization Oscilla-\ntion and Waves , CRC Press, 1996, p. 19."    },    {        "title": "1504.06042v1.Magnetization_damping_in_noncollinear_spin_valves_with_antiferromagnetic_interlayer_couplings.pdf",        "content": "arXiv:1504.06042v1  [cond-mat.mes-hall]  23 Apr 2015Magnetizationdamping innoncollinearspinvalveswithant iferromagnetic interlayer couplings\nTakahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1\n1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan\n2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: October 29, 2018)\nWe study the magnetic damping in the simplest of synthetic an tiferromagnets, i.e. antiferromagnetically\nexchange-coupled spin valves in which applied magnetic fiel ds tune the magnetic configuration to become\nnoncollinear. We formulate the dynamic exchange of spin cur rents in a noncollinear texture based on the spin-\ndiffusiontheorywithquantum mechanicalboundaryconditionsa ttheferrromagnet|normal-metal interfacesand\nderive the Landau-Lifshitz-Gilbert equations coupled by t he static interlayer non-local and the dynamic ex-\nchange interactions. We predict non-collinearity-induce d additional damping that can be sensitively modulated\nbyanapplied magnetic field. The theoretical results compar e favorablywithpublished experiments.\nI. INTRODUCTION\nAntiferromagnets (AFMs) boast many of the functionali-\ntiesofferromagnets(FM)thatareusefulinspintroniccirc uits\nanddevices: Anisotropicmagnetoresistance(AMR),1tunnel-\ning anisotropicmagnetoresistance(TAMR),2current-induced\nspintransfertorque,3–8andspincurrenttransmission9–11have\nall been found in or with AFMs. This is of interest because\nAFMshaveadditionalfeaturespotentiallyattractivefora ppli-\ncations. InAFMsthetotalmagneticmomentis(almost)com-\npletely compensated on an atomic length scale. The AFM\norder parameter is, hence, robust against perturbations su ch\nas external magnetic fields and do not generate stray fields\nthemselveseither. AspintronictechnologybasedonAFM el-\nementsisthereforeveryattractive.12,13Drawbacksarethedif-\nficulty to controlAFMs by magnetic fields and much higher\n(THz)resonancefrequencies,14–16whicharedifficulttomatch\nwith conventional electronic circuits. Man-made magnetic\nmultilayers in which the layer magnetizations in the ground\nstate isorderedin anantiparallelfashion,17i.e. so-calledsyn-\nthetic antiferromagnets,donot su ffer fromthis drawbackand\nhave therefore been a fruitful laboratory to study and modu-\nlate antiferromagnetic couplings and its consequences,18but\nalso found applications as magnetic field sensors.19Trans-\nport in these multilayers including the giant magnetoresis -\ntance (GMR)20,21are now well understood in terms of spin\nand charge diffusive transport. Current-induced magnetiza-\ntionswitchinginF|N|Fspinvalvesandtunneljunctions,22has\nbeen a game-changer for devices such as magnetic random\naccess memories(MRAM).23A keyparameterof magnetiza-\ntiondynamicsisthemagneticdamping;asmalldampinglow-\nersthethresholdofcurrent-drivenmagnetizationswitchi ng,24\nwhereasalargedampingsuppresses“ringing”oftheswitche d\nmagnetization.25\nMagnetization dynamics in multilayers generates “spin\npumping”, i.e. spin current injection from the ferromagnet\ninto metallic contacts. It is associated with a loss of an-\ngular momentum and an additional interface-related magne-\ntization damping.26,27In spin valves, the additional damp-\ning is suppressed when the two magnetizations precess in-\nphase, while it is enhanced for a phase di fference ofπ(out-\nof-phase).27–30This phenomenon is explained in terms of a\n“dynamic exchange interaction”, i.e. the mutual exchange o fnon-equilibriumspin currents,which shouldbe distinguis hed\nfrom(butcoexistswith)theoscillatingequilibriumexcha nge-\ncoupling mediated by the Ruderman-Kittel-Kasuya-Yosida\n(RKKY) interaction. The equilibriumcoupling is suppresse d\nwhenthespacerthicknessisthickerthantheelasticmean-f ree\npath,31,32while the dynamiccouplingise ffective onthe scale\noftheusuallymuchlargerspin-flipdi ffusionlength.\nAntiparallel spin valves provide a unique opportunity to\nstudy and control the dynamic exchange interaction between\nferromagnets through a metallic interlayer for tunable mag -\nnetic configurations.33,34An originallyantiparallel configura-\ntionisforcedbyrelativelyweakexternalmagneticfieldsin toa\nnon-collinearconfigurationwith a ferromagneticcomponen t.\nFerromagneticresonance(FMR)andBrillouinlightscatter ing\n(BLS) are two useful experimentalmethodsto investigateth e\nnature and magnitude of exchange interactions and magnetic\ndamping in multilayers.35Both methods observe two reso-\nnances, i.e. acoustic (A) and optical (O) modes, which are\ncharacterizedbytheirfrequenciesandlinewidths.36,37\nTimopheev etal.observedaneffectoftheinterlayerRKKY\ncoupling on the FMR and found the linewidth to be a ffected\nby the dynamic exchange coupling in spin valves with one\nlayerfixed by the exchange-biasof an inert AFM substrate.38\nThey measured the FMR spectrum of the free layer by tun-\ning the interlayer coupling (thickness) and reported a broa d-\nening of the linewidth by the dynamic exchange interaction.\nTaniguchi et al.addressed theoretically the enhancement of\nthe Gilbert damping constant due to spin pumping in non-\ncollinear F|N|F trilayer systems, in which one of the magne-\ntizations is excited by FMR while the other is o ff-resonant,\nbutadoptaroleasspinsink.39Thedynamicsofcoupledspin\nvalvesinwhichbothlayermagnetizationsarefreetomoveha s\nbeencomputedby oneof us29and bySkarsvåg et al.33,49but\nonly for collinear (parallel and antiparallel) configurati ons.\nCurrent-induced high-frequency oscillations without app lied\nmagnetic field in ferromagnetically coupled spin valves has\nbeenpredicted.40\nInthepresentpaper,wemodelthemagnetizationdynamics\nof the simplest of synthetic antiferromagnets, i.e. the ant i-\nferromagnetically exchange-coupled spin valve in which th e\n(in-plane) ground state magnetizations are for certain spa cer\nthicknesses ordered in an antiparallel fashion by the RKKY\ninterlayercoupling.41We focusonthecoupledmagnetization2\nmodes in symmetric spin valves in which in contrast to pre-\nvious studies, both magnetizations are free to move. We in-\nclude static magnetic fields in the film plane that deform the\nantiparallelconfigurationintoacantedone. Microwaveswi th\nlongitudinal and transverse polarizations with respect to an\nexternalmagneticfieldthenexciteAandOresonancemodes,\nrespectively.31,42–46We develop the theory for magnetization\ndynamics and damping based on the Landau-Lifshitz-Gilbert\nequationwithmutualpumpingofspincurrentsandspintrans -\nfer torques based on the spin di ffusion model with quantum\nmechanical boundary conditions.27,47,48We confirm28,49that\nthe additional damping of O modes is larger than that of the\nA modes. We report that a noncollinear magnetization con-\nfigurationinducesadditionaldampingtorquesthat to the be st\nof ourknowledgehavenotbeen discussedin magneticmulti-\nlayers before.50The external magnetic field strongly a ffects\nthe dynamics by modulating the phase of the dynamic ex-\nchange interaction. We compute FMR linewidths as a func-\ntion of applied magnetic fields and find good agreement with\nexperimental FMR spectra on spin valves.31,32The dynam-\nics of magnetic multilayers as measured by ac spin trans-\nfer torque excitation30reveals a relative broadening of the O\nmodes linewidths that is well reproduced by our spin valve\nmodel.\nIn Sec. II we present our model for noncollinear spin\nvalves based on spin-di ffusiontheory with quantum mechan-\nical boundary conditions. In Sec. III, we consider the mag-\nnetization dynamics in antiferromagnetically coupled non -\ncollinear spin valves as shown in Fig. 1(b). We derive the\nlinearized magnetization dynamics, resonance frequencie s,\nand lifetimes of the acoustic and optical resonance modes in\nSec. IV. We discuss the role of dynamicspin torqueson non-\ncollinear magnetization configurations in relation to exte rnal\nmagnetic field dependence of the linewidth. In Sec. V, we\ncompare the calculated microwave absorption and linewidth\nwith published experiments. We summarize the results and\nendwiththeconclusionsinSec. VI.\nII. SPINDIFFUSIONTRANSPORTMODEL\nWe consider F1|N|F2 spin valves as shown in Fig. 1(a), in\nwhichthemagnetizations MjoftheferromagnetsF j(j=1,2)\nare coupled by a antiparallel interlayer exchange interact ion\nand tilted towards the direction of an external magnetic fiel d.\nApplied microwaves with transverse polarizations with re-\nspect to an external magnetic field cause dynamics and, via\nspinpumping,spincurrentsandaccumulationsinthenormal -\nmetal (NM) spacer. The longitudinal component of the spin\naccumulation diffuses into and generates spin accumulations\ninF thatwe showtobesmall later,butdisregardinitially. L et\nusdenotethepumpedspincurrent JP\nj,whileJB\njisthediffusion\n(back-flow)spin currentdensity inducedby a spin accumula-NM zy\nx\nF1 F2 0 dN\nJ1PJ2P\nJ2BJ1Bµsθθ\n(c) Acoustic mode (d) Optical mode (a)\nm1 m2\nH\n*OUFSMBZFS\u0001DPVQMJOH \n(b)\n(b) H\nhx hy\nFIG.1. (a) Sketch of the sample withinterlayer exchange-co uplings\nillustrating the spin pumping and backflow currents. (b) Mag netic\nresonance in an antiferromagnetically exchange-coupled s pin valve\nwith a normal-metal (NM) film sandwiched by two ferromagnets\n(F1,F2)subjecttoamicrowave magneticfield h. Themagnetization\nvectors (m1,m2) are tilted by an angle θin a static in-plane mag-\nneticfield Happliedalongthe y-axis. Thevectors mandnrepresent\nthesum anddifference ofthe twolayer magnetizations, respectively.\n(c) and (d): Precession-phase relations for the acoustic an d optical\nmodes.\ntionµsjin NM,bothat theinterfaceF j, with27,51\nJP\nj=Gr\nemj×/planckover2pi1∂tmj, (1a)\nJB\nj=Gr\ne/bracketleftBig/parenleftBig\nmj·µsj/parenrightBig\nmj−µsj/bracketrightBig\n, (1b)\nwheremj=Mj//vextendsingle/vextendsingle/vextendsingleMj/vextendsingle/vextendsingle/vextendsingleis the unit vector along the magnetic\nmoment of F j(j=1,2). The spin current througha FM |NM\ninterface is governed by the complex spin-mixing conduc-\ntance (per unit area of the interface) G↑↓=Gr+iGi.27The\nreal component Grparameterized one vector component of\nthe transverse spin-currentspumped and absorbed by the fer -\nromagnets. The imaginary part Gican be interpreted as an\neffective exchange field between magnetization and spin ac-\ncumulation, which in the absence of spin-orbit interaction is\nusuallymuchsmallerthantherealpart,forconductingaswe ll\nasinsultingmagnets.523\nThediffusionspin-currentdensityin NMreads\nJs,z(z)=−σ\n2e∂zµs(z), (2)\nwhereσ=ρ−1is the electrical conductivity and µs(z)=\nAe−z/λ+Bez/λthe spin accumulationvectorthat is a solution\nofthespindiffusionequation∂2\nzµs=µs/λ2,whereλ=√Dτsf\nis the spin-diffusion length, Dthe diffusion constant, and τsf\nthe spin-flip relaxation time. The vectors AandBare de-termined by the boundary conditions at the F1 |NM (z=0)\nand F2|NM (z=dN) interfaces: Js,z(0)=JP\n1+JB\n1≡Js1and\nJs,z(dN)=−JP\n2−JB\n2≡−Js2. Theresultingspin accumulation\ninN reads\nµs(z)=2eλρ\nsinh/parenleftBigdN\nλ/parenrightBig/bracketleftBigg\nJs1cosh/parenleftBiggz−dN\nλ/parenrightBigg\n+Js2cosh/parenleftbiggz\nλ/parenrightbigg/bracketrightBigg\n,(3)\nwithinterfacespin currents\nJs1=ηS\n1−η2δJP\n1+η2/parenleftBig\nm2·δJP\n1/parenrightBig\n1−η2(m1·m2)2m1×(m1×m2), (4a)\nJs2=−ηS\n1−η2δJP\n2+η2/parenleftBig\nm1·δJP\n2/parenrightBig\n1−η2(m1·m2)2m2×(m2×m1). (4b)\nHere\nδJP\n1=JP\n1+ηm1×(m1×JP\n2), (5a)\nδJP\n2=JP\n2+ηm2×(m2×JP\n1), (5b)\nS=sinh(dN/λ)/grandη=gr/[sinh(dN/λ)+grcosh(dN/λ)]\nare the efficiency of the back flow spin currents, and gr=\n2λρGris dimensionless. The first terms in Eqs. (4a) and (4b)\nrepresent the mutual pumping of spin currents while the sec-\nondtermsmaybeinterpretedasa spincurrentinducedbythe\nnoncollinear magnetization configuration, including the b ack\nflowfromthe NMinterlayer.\nIII. MAGNETIZATIONDYNAMICSWITH DYNAMIC\nSPINTORQUES\nWe consider the magnetic resonance in the non-collinear\nspin valve shown in Fig. 1. The magnetization dynamics are\ndescribedbytheLandau-Lifshitz-Gilbert(LLG)equation,\n∂tm1=−γm1×Heff1+α0m1×∂tm1+τ1,(6a)\n∂tm2=−γm2×Heff2+α0m2×∂tm2−τ2.(6b)\nThe first term in Eqs. (6a) and (6b) represents the torque in-\nducedbytheeffectivemagneticfield\nHeff1(2)=H+h(t)−4πMsm1(2)zˆz+Jex\nMsdFm2(1),(7)\nwhich consists of an in-plane applied magnetic field H,\na microwave field h(t), and the demagnetization field\n−4πMsm1(2)zˆzwith saturation magnetization Ms. The inter-\nlayer exchange field is Jex/(MsdF)m2(1)with areal density of\ntheinterlayerexchangeenergy Jex<0(forantiferromagnetic-\ncoupling) and F layer thickness dF. The second term is the\nGilbert dampingtorque that governsthe relaxationcharact er-\nized byα0itowards an equilibrium direction. The third term,τ1(2)=γ/planckover2pi1/(2eMsdF)Js1(2), is the spin-transfertorque induced\nby the absorption of the transverse spin currents of Eqs. (4a )\nand (4b), andγandα0are the gyromagnetic ratio and the\nGilbert dampingconstant of the isolated ferromagneticfilm s,\nrespectively. SometechnicaldetailsofthecoupledLLGequ a-\ntionsarediscussedinAppendixA.Introducingthetotalmag -\nnetizationdirection m=(m1+m2)/2andthedifferencevector\nn=(m1−m2)/2,theLLG equationscanbewritten\n∂tm=−γm×(H+h)\n+2πγMs(mzm+nzn)׈z\n+α0(m×∂tm+n×∂tn)+τm, (8a)\n∂tn=−γn×/parenleftBigg\nH+h+Jex\nMsdFm/parenrightBigg\n+2πγMs(nzm+mzn)׈z\n+α0(m×∂tn+n×∂tm)+τn,(8b)\nwhere the spin-transfer torques τm=(τ1+τ2)/2 andτn=\n(τ1−τ2)/2become\nτm/αm=m×∂tm+n×∂tn\n+2ηm·(n×∂tn)\n1−ηCm+2ηn·(m×∂tm)\n1+ηCn,(9a)\nτn/αn=m×∂tn+n×∂tm\n−2ηm·(n×∂tm)\n1+ηCm−2ηn·(m×∂tn)\n1−ηCn,(9b)\nandC=m2−n2,while\nαm=α1gr\n1+grcoth(dN/2λ), (10a)\nαn=α1gr\n1+grtanh(dN/2λ), (10b)\nwithα1=γ/planckover2pi12/(4e2λρMsdF).4\nIV. CALCULATIONANDRESULTS\nWe consider the magnetization dynamics excited by lin-\nearly polarized microwaves with a frequency ωand in-plane\nmagnetic field h(t)=(hx,hy,0)eiωtthat is much smaller than\nthe saturation magnetization. For small angle magnetizati on\nprecession the total magnetization and di fference vector may\nbe separated into a static equilibrium and a dynamic com-\nponent as m=m0+δmandn=n0+δn, respectively,\nwherem0=(0,sinθ,0),n0=(cosθ,0,0),C=−cos2θ,\nands=−ˆzsin2θ. The equilibrium (zero torque) conditions\nm0×H=0 andn0×(H+Jex/(MsdF)m0)=0 lead to the\nrelation\nsinθ=H/Hs, (11)\nwhereHs=−Jex/(MsdF)=|Jex|/(MsdF) is the saturation\nfield. TheLLGequationsread\n∂tδm=−γδm×H−γm0×h\n+2πγMs(δmzm0+δnzn0)׈z\n+α0(m0×∂tδm+n0×∂tδn)+δτm,(12a)\n∂tδn=−γδn×H−γn0×h\n+2πγMs(δnzm0+δmzn0)׈z\n−γHs(m0×δn−n0×δm)\n+α0(m0×∂tδn+n0×∂tδm)+δτn,(12b)\nwithlinearizedspin-transfertorques\nδτm/αm=m0×∂tδm+n0×∂tδn\n−ηsin2θ\n1+ηcos2θ∂tδnzm0+ηsin2θ\n1−ηcos2θ∂tδmzn0,(13a)\nδτn/αn=m0×∂tδn+n0×∂tδm\n+ηsin2θ\n1−ηcos2θδmzm0−ηsin2θ\n1+ηcos2θδnzn0,(13b)\nTo leading order in the small precessing components δmand\nδn,theLLG equationsinfrequencyspace become\nδmx=γhxγ(Hs+4πMs)+iω/parenleftBig\nα0+αm(1+η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nA−i∆Aωsin2θ,(14a)\nδny=−γhxγ(Hs+4πMs)+iω/parenleftBig\nα0+αn(1−η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nA−i∆Aωcosθsinθ,\n(14b)\nδmz=−γhxiω\nω2−ω2\nA−i∆Aωsinθ, (14c)\nδnx=−γhy4πγMs+iω/parenleftBig\nα0+αn(1−η)\n1+ηcos2θ/parenrightBig\nω2−ω2\nO−i∆Oωcosθsinθ,(15a)\nδmy=γhy4πγMs+iω/parenleftBig\nα0+αm(1+η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nO−i∆Oωcos2θ, (15b)\nδnz=γhyiω\nω2−ω2\nO−i∆Oωcosθ. (15c)The A modes (δmx,δny,δmz) are excited by hx, while the O\nmodes (δnx,δmy,δnz) couple to hy. The poles inδm(ω)and\nδn(ω)define the resonance frequencies and linewidths that\ndo not depend on the magnetic field since we disregard\nanisotropyandexchange-bias.\nA. Acoustic andOpticalmodes\nAn antiferromagnetically exchange-coupled spin valves\ngenerallyhave non-collinearmagnetizationconfiguration sby\nthepresenceofexternalmagneticfields. For H<Hs(0<θ<\nπ/2),theacousticmode:\nωA=γH/radicalbig\n1+(4πMs/Hs), (16)\n∆A=α0γ/parenleftBig\nHs+4πMs+Hssin2θ/parenrightBig\n+αmγ(Hs+4πMs)+αA(θ)γHs,(17)\nandthe opticalmode:\nωO=γ/radicalBig\n(4πMs/Hs)(H2s−H2), (18)\n∆O=α0γ/parenleftBig\n4πMs+Hscos2θ/parenrightBig\n+αn4πγMs+αO(θ)γHs, (19)\nwhere\nαA(θ)=α1grsin2θ\n1+grtanh(dN/2λ)+2grsin2θ/sinh(dN/λ),(20)\nαO(θ)=α1grcos2θ\n1+grtanh(dN/2λ)+2grcos2θ/sinh(dN/λ).(21)\nThe additional broadeningin ∆Ais proportional toαmand\nαAwhile that in∆Oscales withαnandαOin. Figure 2 (a)\nshowsαmandαnas a function of spacer layer thickness, in-\ndicating thatαnis always larger than αm, and thatαn(αm)\nstrongly increases (decreases) with decreasing N layer thi ck-\nness, especially for dN<λand large gr. Figure 2(b) shows\nthedependenceofαAandαOonthetiltedangleθfordifferent\nvaluesof dN. Asθincreases,αAincreasesfrom0to αmwhile\nαOdecreasesαmto 0. The additional damping can be ex-\nplained by the dynamic exchange. When two magnetizations\nin spin valves precess in-phase, each magnet receives a spin\ncurrent that compensates the pumped one, thereby reducing\nthe interface damping. When the magnetizations precess out\nof phase, theπphase difference between both spin currents\nmeans that the moduli have to be added, thereby enhancing\nthedamping.\nWhen the magnetizations are tilted by an angle θas\nsketched in Fig. 1, we predict an additional damping torque\nexpressedbythesecondtermsofEqs.(4a)and(4b). Figure3\nshows the ratiosαA(θ)/αmandαO(θ)/αnas a function ofθ\nandgrfor different values ofλ/dN, thereby emphasizing the\nadditional damping in the presence of noncollinear magneti -\nzations. In Fig. 3(a,b)with λ/dN=1, i.e. for a spin-di ffusive\ninterlayer,the additionaldampingof both A- and O-modesis\nsignificant in a large region of parameter space. On the other5\n0.1 1 10 100λ/d N012345αm/α1, αn/α1/α1 /α12\n1\n0.5\n0 30 60 90 \nθ (degree)0.00.10.20.30.40.5αA, αOαO αAλ/d N=1 \n2\n4\n10(b)(a)\nαnαmgr=5\nFIG.2. (a)αm(dashed line)andαn(solidline) as a functionof λ/dN\nfor different values of the dimensionless mixing conductance gr. (b)\nαAC(dashed line)andαOP(solidline)as afunction oftiltangle θfor\ngr=5and different values ofλ/dN.\nhand, in Fig. 3(c,d) with λ/dN=10, i.e. for an almost spin-\nballistic interlayer, the additional damping is more impor tant\nfortheA-mode,whiletheO-modeisa ffectedonlyclosetothe\ncollinear magnetization. In the latter case the intrinsic d amp-\ningα0dominates,however.\nB. In-phaseandOut-of-phasemodes\nWhentheappliedmagneticfieldislargerthanthesaturation\nfield (H>Hs), both magnetizations point in the ˆydirection,\nand theδm(A) andδn(O) modes morph into in-phase and\n180◦out-of-phase (antiphase) oscillations of δm1andδm2,\nrespectively. The resonance frequency53and linewidth of the\nin-phasemodefor H>Hs(θ=π/2)are\nωA=γ/radicalbig\nH(H+4πMs), (22)\n∆A=2(α0+αm)γ(H+2πMs), (23)\nwhilethoseoftheout-of-phasemodeare\nωO=γ/radicalbig\n(H−Hs)(H−Hs+4πMs),(24)\n∆O=2(α0+αn)γ(H−Hs+2πMs).(25)\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 \n\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 \tB\n \tC\n \n\tD\n \tE\nHS\nHS\nθ (degree) θ (degree)\n\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \nθ (degree)\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \nθ (degree)\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 HS\n\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 HS0.1 0.3 0.5 0.7 0.9 \n0.1 0.2 0.3 0.4 0.5 0.6 \n0.55 0.65 0.75 0.85 0.95 \n0.01 0.02 0.03 0.04 0.05 \nFIG. 3. (a,c)αA(θ)/αmand (b,d)αO(θ)/αnas a function ofθandgr\nfor different values ofλ/dN. (a,b) withλ/dN=1, (c,d) withλ/dN=\n10\nFigure 4(a) shows the calculated resonance frequencies of\nthe A andO modesas a functionof an appliedmagneticfield\nHwhile 4(b) displays the linewidths for α1/α0=1, which\nis representative for ferromagnetic metals, such as permal -\nloy (Py) with an intrinsic magnetic damping of the order of\nα0=0.01andacomparableadditionaldamping α1duetospin\npumping. A value gr=4/5 corresponds toλ=20/200nm,\nρ=10/2.5µΩcmfor N=Ru/Cu,54,55Gr=2/1×1015Ω−1m−2\nfor the N|Co(Py) interface56,57, anddF=1nm, for example.\nThecolorsinthefigurerepresentdi fferentrelativelayerthick-\nnessesdN/λ. The linewidth of the A mode in Fig. 4(b) in-\ncreases with increasing H, while that of the O mode starts\nto decrease until a minimum at the saturation field H=Hs.\nFigure 4(c) shows the linewidths for α1/α0=10, which de-\nscribes ferromagnetic materials with low intrinsic dampin g,\nsuch as Heusler alloys58and magnetic garnets.59In this case,\nthe linewidth of the O modeis much largerthanthat of the A\nmode,especiallyforsmall dN/λ.\nIn the limit of dN/λ→0 is easily established experimen-\ntally. The expressions of the linewidth in Eqs. (17) and (19)\narethengreatlysimplifiedto ∆A=γ(Hs+4πMs+Hssin2θ)α0\nand∆O=γ/parenleftBig\n4πMs+Hscos2θ/parenrightBig\nα0+(4πγMs)grα1,and∆A≪\n∆Owhengrα1≫α0. The additional damping, Eq. (10b) re-\nduces toαm→0 andαn→2[γ/planckover2pi1/(4πMsdF)(h/e2)Gr] when\nthemagnetizationsarecollinearandintheballisticspint rans-\nport limit.27In contrast to the acoustic mode, the dynamic\nexchange interaction enhances damping of the optical mode.\n∆O≫∆AhasbeenobservedinPy |Ru|Pytrilayerspinvalves32\nandCo|Cu multilayers30, consistentwiththepresentresults.\nFor spin valves with ferromagnetic metals,\nthe interface backflow spin-current [(1b)] reads\nJB\nj=(Gr/e)/bracketleftBig\nξF/parenleftBig\nmj·µsj/parenrightBig\nmj−µsj/bracketrightBig\n,whereξF=6\n012ω/(4 πγ Ms)ωAωO\n02468∆/(4 πMsα0) dN/λ=\n0.3\n1(a)\n(b) \n(c)\n0 0.5 1 1.5\nH/H s050dN/λ=0.01\n0.1\n0.3\n1∆/(4 πγ Msα0)0.01γ\nA mode  O mode 0.1\nFIG. 4. (a) Resonance frequencies of the A and O modes as a func -\ntion of magnetic field for Hs/(4πMs)=1. (b), (c) Linewidths of the\nA (dashed line) and the O (solid line) modes for Hs/(4πMs)=1,\ngr=5, and different values of dN/λ. (b)α1/α0=1 and (c)\nα1/α0=10.\n1−(G/2Gr)(1−p2)(1−ηF) (0≤ξF≤1),Gis the\nN|F interface conductance per unit area, and pthe conduc-\ntance spin polarization.51Here the spin diffusion efficiency\nis\n1\nηF=1+σF\nGλFtanh(dF/λF)\ncosh(dF/λF), (26)\nwhereσF,λF, anddFare the conductivity, the spin-flip dif-\nfusion length, and the layer thickness of the ferromagnets,\nrespectively. For the material parameters of a typical fer-\nromagnet with dF=1 nm, the resistivity ρF=10µΩcm,\nG=2Gr=1015Ω−1m−2,λF=10nm,and p=0.7,ξF=0.95,\nwhichjustifiesdisregardingthiscontributionfromtheout set.\nV. COMPARISONWITH EXPERIMENTS\nFMRexperimentsyieldtheresonantabsorptionspectraofa\nmicrowavefield ofa ferromagnet. Themicrowaveabsorptiont\ndP/dH ϕ=90 o\n20 o\n0o×5\n×5ϕHh(t)(a)\n5\nH (kOe) 22 4 6 0 0  0.4  0.8  1.2 H/H s\nCo(3)|Ru(1)|Co(3) Co(3)|Ru(1)|Co(3) \nExperiment (Ref. 11) Calculation \n 0  2  4  6  8  0  0.2  0.4  0.6  0.8  1 ∆/(4 πγ Msα₀)H/H s\nA mode \nO mode \n 0  2000  4000  6000 \nField (Oe) (b)\nExperiment (Ref. 10): [Co(1)|Cu(1)] 10 Co(1) Calculation: Co(1)|Cu(1)|Co(1) \nFIG.5. (a)Derivativeofthemicrowaveabsorptionspectrum dP/dH\nat frequencyω/(2π)=9.22 GHz for different anglesϕbetween the\nmicrowavefieldandtheexternalmagneticfieldfor Hs/(4πMs)=0.5,\nω/(4πγMs)=0.35,dN/λ=0.1,dF/λ=0.3α0=α1=0.02, and\ngr=4. The experimental data have been adopted from Ref. 31.\n(b) Computed linewidths of the A and O modes of a Co |Cu|Co spin\nvalve (dashed line) compared with experiments on a Co |Cu multi-\nlayer (solidline).30\npowerP=2/angbracketlefth(t)·∂tm(t)/angbracketrightbecomesinourmodel\nP=1\n4γ2Ms(Hs+4πMs)∆A\n(ω−ωA)2+(∆A/2)2h2\nxsin2θ\n+1\n4γ2Ms(4πMs)∆O\n(ω−ωO)2+(∆O/2)2h2\nycos2θ. (27)\nPdepends sensitively on the character of the resonance, the\npolarization of the microwave, and the strength of the ap-\nplied magnetic field. In Figure 5(a) we plot the normalized\nderivative of the microwave absorption spectra dP/(P0dH)\nat different anglesϕbetween the microwave field h(t) and\nthe external magnetic field H, where P0=γMsh2and\nh(t)=h(sinϕ,cosϕ,0)eiωt. Here we use the experimen-\ntal values Hs=5kOe, 4πMs=10kOe,dN=1nm,\ndF=3nm, and microwave frequency ω/(2π)=9.22GHz\nas found for a symmetric Co |Ru|Co trilayer.31λ=20nm for7\nRu,α0=α1=0.02, andgr=4 is adopted (correspond-\ning toGr=2×1015Ω−1m−2).Whenh(t) is perpendicularto\nH(ϕ=90◦), only the A mode is excited by the transverse\n(δmx,δmz) component. When h(t) is parallel to H(ϕ=0◦),\nthe O mode couples to the microwave field by the longitudi-\nnalδmycomponent. For intermediate angles ( ϕ=20◦), both\nmodes are excited at resonance. We observe that the opti-\ncal mode signal is broader than the acoustic one, as calcu-\nlated. The theoretical resonance linewidths of the A and O\nmodes as well as the absorption power as a function of mi-\ncrowave polarization reproduce the experimental results f or\nCo(3.2nm)|Ru(0.95nm)|Co(3.2nm)well.31\nFigure 5(b) shows the calculated linewidths of A and\nO modes as a function of an applied magnetic field for a\nCo(1nm)|Cu(1nm)|Co(1nm) spin valve. The experimental\nvaluesλ=200nm andρ=2.5µΩcm for Cu,α0=0.01\nand 4πMs=15kOe for Co, and gr=5 (corresponding to\nGr=1015Ω−1m−2) for the interface have been adopted.57\nWe partially reproduce the experimental data for magnetic\nmultilayers; for the weak-field broadenings of the observed\nlinewidthsagreementisevenquantitative. Theremainingd is-\ncrepanciesintheappliedmagneticfielddependencemightre -\nflect exchange-dipolar49and/or multilayer30spin waves be-\nyondourspinvalvemodelinthe macrospinapproximation.\nVI. CONCLUSIONS\nIn summary, we modelled the magnetization dynamics\nin antiferromagnetically exchange-coupled spin valves as a\nmodel for synthetic antiferromagnets. We derivethe Landau -\nLifshitz-Gilbert equations for the coupled magnetization s in-\ncluding the spin transfer torques by spin pumping based on\nthe spin diffusion model with quantum mechanical boundary\nconditions. We obtain analytic expressionsfor the linewid ths\nof magnetic resonance modes for magnetizations canted byapplied magneticfields and achieve goodagreementwith ex-\nperiments. We findthatthelinewidthsstronglydependonthe\ntype of resonance mode (acoustic and optical) as well as the\nstrength of magnetic fields. The magnetic resonance spectra\nreveal complex magnetization dynamics far beyond a simple\nprecessionevenin the linear responseregime. Our calculat ed\nresults compare favorably with experiments, thereby provi ng\ntheimportanceofdynamicspincurrentsinthesedevices. Ou r\nmodel calculation paves the way for the theoretical design o f\nsyntheticAFMmaterialthatisexpectedtoplayaroleinnext -\ngenerationspin-baseddata-storageandinformationtechn olo-\ngies.\nVII. ACKNOWLEDGMENTS\nTheauthorsthanksK.Tanaka,T.Moriyama,T.Ono,T.Ya-\nmamoto, T. Seki, and K. Takanashi for valuable discussions\nand collaborations. This work was supported by Grants-in-\nAidforScientificResearch(GrantNos. 22540346,25247056,\n25220910,268063)fromtheJSPS,FOM(StichtingvoorFun-\ndamenteel Onderzoek der Materie), the ICC-IMR, EU-FET\nGrant InSpin 612759, and DFG Priority Programme 1538\n“Spin-CaloricTransport”(BA 2954 /2).\nAppendixA: CoupledLandau-Lifshitz-Gilbertequationsin\nnoncollinearspinvalves\nBoth magnets and interfaces in our NM |F|NM spin valves\nare assumed to be identical with saturation magnetization Ms\nandGrthe real part of the spin-mixing conductance per unit\narea (vanishing imaginary part). When both magnetizations\nare allowed to precess as sketched in Fig. 1 (a), the LLG\nequationsexpandedtoincludeadditionalspin-pumpandspi n-\ntransfertorquesread\n∂mi\n∂t=−γmi×Heffi+α0imi×∂mi\n∂t\n+αSPi/bracketleftBigg\nmi×∂mi\n∂t−ηmj×∂mj\n∂t+η/parenleftBigg\nmi·mj×∂mj\n∂t/parenrightBigg\nmi/bracketrightBigg\n+αnc\nSPi(ϕ)mi×/parenleftBig\nmi×mj/parenrightBig\n, (A1)\nαnc\nSPi(ϕ)=αSPiη2\n1−η2(mi·mj)2/bracketleftBigg\nmj·mi×∂mi\n∂t+η/parenleftBigg\nmi·mj×∂mj\n∂t/parenrightBigg\n(mj·mi)/bracketrightBigg\n, (A2)\nwhereγandα0iare the gyromagnetic ratio and the Gilbert\ndamping constant of the isolated ferromagnetic films labele d\nbyiand thickness dFi. Asymmetric spin valves due to the\nthickness differencedFisuppress the cancellation of mutual\nspin-pumpinA-mode,whichmaybeadvantagetodetectboth\nmodesintheexperiment. Thee ffectivemagneticfield\nHeffi=Hi+h(t)+Hdii(t)+Hexj(t) (A3)consistsoftheZeemanfield Hi,amicrowavefield h(t),thedy-\nnamic demagnetization field Hdii(t), and interlayer exchange\nfieldHexj(t). The Gilbert damping torque parameterized\nbyα0igoverns the relaxation towards an equilibrium direc-\ntion. The third term in Eq. (A1) represents the mutual spin\npumping-induced damping-like torques in terms of damping\nparameter8\nαSPi=γ/planckover2pi12Gr\n2e2MsdFiηS\n1−η2, (A4)\nwhere\nη=gr\nsinh(dN/λ)+grcosh(dN/λ)(A5)\nandgr=2λρGrisdimensionless. ThefourthterminEq. (A1)\nis the damping Eq. (A2) that depends on the relative angleϕbetween the magnetizations. When mjis fixed along the\nHidirection, i.e. a spin-sink limit, Eq. (A1) reduces to the\ndynamicstiffnessin spinvalveswithoutanelectricalbias.60\nWhen the magnetizations are noncollinear as in Fig. 1, we\nhave to take into account the additional damping torques de-\nscribedbythe secondtermsin Eqs.(4a) and(4b ,). 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Lakshmanan2\n1Centre for Nonlinear Science and Engineering, School of Ele ctrical and Electronics Engineering, SASTRA Deemed Univer sity,\nThanjavur - 613 401, India\n2Department of Nonlinear Dynamics, School of Physics, Bhara thidasan University, Tiruchirapalli - 620 024, India\nThe magnetization dynamics of two parallelly coupled spin t orque oscillators, destabilization of steady states and re moval of\nmultistability, are investigated by taking into account th e influence of field-like torque. It is shown that the existenc e of such torque\ncan cancel the effect of damping and can, therefore, cause th e oscillators to exhibit synchronized oscillations in resp onse to direct\ncurrent. Further, our results show that the presence of field -like torque enhances the power and Q-factor of the synchron ized\noscillations. The validity of the above results is confirmed by numerical and analytical studies based on the stochastic Landau-\nLifshitz-Gilbert-Slonczewski equation.\nIndex Terms —nonlinear dynamics,spintronics,synchronization\nI. I NTRODUCTION\nSynchronization phenomenon in spin torque oscillators\n(STOs) has been the subject of active research in recent year s\ndue to its potential applications to generate microwave pow er\nin nanoscale devices [1]–[5]. A number of significant effort s\nhave been made to study magnetization dynamics and synchro-\nnization of STOs driven by spin polarized current [6], injec tion\nlocking [7], external ac excitation [8], [9], spin waves [10 ],\nmagnetic fields [11]–[13], electrical couplings [14], [15] and\nthrough self-emitted microwave currents [16]. The synchro -\nnization of STOs greatly enhances the output microwave\npower when compared with the low output power of an\nindividual STO. Also it is more desirable for an enhancement\nof efficiency, quality factor and oscillation frequency of t he\npractical STO devices such as high density microwave sig-\nnal processors and chip-to-chip communication system [14] ,\n[17]–[20]. Moreover, synchronization of STOs has also been\nidentified in new applications such as wireless communicati on,\nbrain-inspired computing and microwave assisted magnetic\nreading [21]–[25]. In particular, it has been observed that\nan STO with the configuration of perpendicularly magnetized\nfree layer and in-plane magnetized pinned layer is suitable\nfor high emission of power, narrow line width and wide\nfrequency tunability [20], [26], [27]. The oscillation pro perties\nof this STO have also been investigated both experimentally\nand theoretically in Refs. [27], [28]. Further, the existen ce\nand stability of the synchronized state and the conditions t o\nsynchronize the individual precessions have also been stud ied\nin an array of Nserially connected identical STOs coupled\nthrough current has been demonstrated in Ref. [29]. Recentl y,\nthe mutual synchronization between two parallelly connect ed\nSTOs, coupled by current, has also been identified [30].\nIn this connection, some of the important issues in under-\nstanding the nonlinear dynamics of the system of coupled\nSTOs are the formation of steady states, multistability and\nthe decrease of frequency with respect to current. The oc-currence of steady states and multistable states prevents t he\nsystem to exhibit stable synchronized oscillations for all initial\nconditions. Removing these steady states and multistabili ty be-\nhaviour and making the system to exhibit stable synchronize d\noscillations for all initial conditions are important task s and\nhave not yet been fully clarified as far our understanding goe s.\nAlso, a decrease in the frequency of synchronized oscillati ons\nwhile increasing the current limits the enhancement of fre-\nquency beyond some specific value which is also a problem\nto be overcome with minimal efforts.\nIn this paper, we study the existence of steady states and\nmultistable states in the absence of field-like torque, thei r\nremoval and the mutual synchronization of the macrospin\ndynamics of a system of two parallelly coupled STOs in the\npresence of field-like torque [31]–[39]. By solving the asso ci-\nated stochastic Landau-Lifshitz-Gilbert-Slonczewski(s LLGS)\nequation with the configuration of perpendicularly magneti zed\nfree layer and in-plane magnetized pinned layer(as introdu ced\nin Sec.II), the analytical formula for the frequency of syn-\nchronized oscillations is derived in Sec.III. The existenc e of\nsteady states and multistable states is confirmed and the imp act\nof field-like torque on the STOs for various strengths of cou-\npling is observed. In the absence of field-like torque the two\nSTOs show the existence of steady states and synchronized\noscillations. The presence of field-like torque removes the\nsteady states and makes the system to oscillate with in-phas e\nsynchronization(Sec. III and Appendix). Further, the freq uency\nof synchronized oscillations is also enhanced in the presen ce\nof field-like torque. The onset of steady states in the absenc e of\nsuch a torque and the onset of stable synchronized oscillati ons\ndue to it are also analytically verified.\nII. M ODEL DESCRIPTION OF TWO PARALLELLY COUPLED\nSTO S\nWe consider a system that consists of two parallelly coupled\nspin torque oscillators. The schematic diagram of the syste mIEEE TRANSACTIONS ON MAGNETICS JOURNALS 2\nFig. 1. The schematic view of two parallely coupled spin torq ue oscillators.\nthat consists of two parallely coupled spin torque oscillat ors is\nshown in Fig.1. Each oscillator consists of a perpendicular ly\nmagnetized free layer, where the direction of magnetizatio n\nis allowed to change and an in-plane magnetized pinned\nlayer where the direction of magnetization is fixed along the\npositive x-direction. Both free and pinned layers are separ ated\nby a nonmagnetic conducting layer. The two free layers are\nlabeled as j= 1,2and the material parameters of the two\noscillators are kept identical for simplicity. The unit vec tor\nalong the direction of free layer’s magnetization is given b y\nmj= (mjx,mjy,mjz). The z axis is kept perpendicular to the\nplane of the free layer and ex,eyandezare unit vectors along\npositive x,y and z directions respectively. The unit vector along\nthe direction of magnetization of the pinned layers is given by\nP(=ex). The magnetization of the free layers (j= 1,2)is\ngoverned by the following sLLGS equation,\ndmj\ndt=−γmj×Heff,j+αmj×dmj\ndt\n+γHSjmj×(mj×P)+γβHSjmj×P, j= 1,2.(1)\nHereHeff,j is the effective field, given by Heff,j=\n[Ha+ (Hk−4πMs)mjz]ez+Hth,j, which includes ex-\nternally applied field Ha, crystalline anisotropy field Hk,\nshape anisotropy field (or demagnetizing field) 4πMsand the\nthermal noise given by [40]–[42]\nHth,j=√\nDGj, D=2αkBT\nγMsµ0V△t(2)\nIn the above, Gjis the Gaussian random number generator\nvector of the jthoscillator with components (Gjx,Gjy,Gjz),\nwhich satisfies the statistical properties < Gjm(t)>= 0 and\n< Gjm(t)Gjn(t′)>=δmnδ(t−t′)for allm,n=x,y,z .\nMsis the saturation magnetization, γis the gyromagnetic\nratio,αis the Gilbert damping parameter, βis the strength\nof the field-like torque, kBis the Boltzmann constant, Tis\nthe temperature, Vis the volume of the free layer, △tis the\nstep size of the time scale used in the simulation, µ0is the\nmagnetic permeability at free space and HSjis the strength\nof the spin-transfer torque, given by\nHSj=¯hηIj\n2eMsV(1+λmj.P). (3)In Eq.(3) ¯h=h/2π(h- Planck’s constant), eis the electron\ncharge,ηandλare dimensionless parameters which determine\nthe magnitude and the angular dependence of the spin transfe r\ntorque respectively. Ijis the total current passing through the\nfree layer which is given by [30]\nIj=I0+Icoupling\nj=I0+I0χ[mjx(t)−mj′x(t)],(4)\nwherej,j′= 1,2, j/negationslash=j′andIcoupling\nj is the current\ninjected from the free layer j′toj. In Eq.(4) I0is the current\nflowing through the free layer when there is no coupling\nbetween the oscillators. The second term in Eq.(4) describe s\nthe current flowing through the connection between the two\nSTOs and χis the coupling strength which characterizes the\nenergy loss in the connector. The oscillating electric curr ent\ngenerated by the STO is proportional to [2Vi/(RP+RAP)][1+\n△R(mj.P)/(RP+RAP)]as pointed out in [30], which\nimplies that the electric current generated by the oscillat or\ndepends upon the component of the free layer’s magnetizatio n\nalong the pinned layer’s magnetization direction. Here Viis\nthe external voltage, RPandRAP=RP+△Rare the\nresistances of the STO when the magnetization of the free\nlayer is parallel and antiparallel to the magnetization of t he\npinned layer, respectively.\nIII. E FFECT OF FIELD -LIKE TORQUE\nA. Destabilization of steady state due to aribitrary initia l\nconditions (covering both the hemispheres of magnetizatio n)\nby field-like torque\nTo understand the dynamics of the magnetization of the free\nlayer, Eq.(1) is numerically solved by Runge-Kutta 4th orde r\nstep-halving method for the material parameters [27], [28] ,\n[30]Ms= 1448.3emu/c.c., Hk= 18.6kOe,η= 0.54,λ\n=η2,γ= 17.64 Mrad/(Oe s), α= 0.005, µ0= 1 and V=\nπ×60×60×2nm3. Throughout our study HaandTare\nfixed as 2.0 kOe and 300 K respectively.\nTo study the dependance of the nature of the evolution of\nm1andm2on the initial conditions on the sphere formed by\nthe unit vector maround the origin, we have plotted the time\nevolution of m1x,m2xandm1zandm2zin Figs.2(a,c,e) and\n(b,d,f) respectively for I0= 5.0 mA and χ= 0.6. Figs.2(a)\nand (b) confirm the oscillations of m1andm2around the\npositive z-direction in the absence of field-like torque. Th e\ninitial conditions of the two STNOs have been chosen from\nthe northern hemisphere ( 0.99< m1z,m2z<1.00). The\nrandom fluctuations in Fig.2(b) is due to the thermal noise.\nNext, when the initial conditions of the two STNOs are taken\nfrom the two different hemispheres( 0.99< m1z<1.00,\n−0.99> m2z>−1.00), the system shows steady state\nmotion which we can observe from Figs.2(c) and (d) in the\nabsence of field-like torque. This is due to the fact that when\nthe two magnetization vectors evolve in the two different\nhemispheres the term I0χ[mjx(t)−mj′x(t)](see Eq.(4))\ncan become negative and consequently the current passing\nthrough the oscillators gets reduced. On the other hand,\nwhen field-like torque is additionally present(as shown in\nFigs.2(e) and (f) with χ=0.6), even with initial conditions taken\nfrom two different hemispheres, both the oscillators exhib itIEEE TRANSACTIONS ON MAGNETICS JOURNALS 3\nFig. 2. (Color online) Time evolution of m1x,m2x(a) andm1z,m2z(b) for the initial conditions from same hemispheres( 0.99< m1z,m2z<1.00).\nTime evolution of m1x,m2xandm1z,m2zwhenβ= 0(c,d) and β= 0.6(e,f) for the initial conditions from different hemisphere s(0.99< m1z<1.00,\n−0.99> m2z>−1.00). HereI0= 5.0 mA, T= 300 K and χ= 0.6. The inset figures in (a) and (e) show the synchronizatio n ofm1x(black solid line)\nandm2x(red solid circle). Similarly, the inset in (b) and (f) show t he synchronization of m1z(black solid line) and m2z(red solid line).\nsynchronized oscillations. The synchronization between t he\ntwo oscillators is shown in the insets of Figs.2(e) and (f).\nIn addition to the above, the LLGS equation with random\ntorque is solved for 200 trials in order to average the dynami cs.\nFor this purpose, we have also plotted the averaged values of\nmagnetization components < m1x>,< m 2x>,< m 1z>\nand< m2z>in Figs.3. Figs.3(a) and (b) show the averaged\ndynamics of the xandzcomponents of the magnetizations\nin the absence of field-like torque for the initial condition s\nfrom the same hemisphere. Due to the randomness of the\nphase, the average value of mxbecomes close to zero and\nthis clearly shows the significance of LLGS equation with\nthermal noise at finite temperature. The averaged dynamics\ncorresponding to steady state motion of the two oscillators\nfor the inital conditions from different hemispheres have b een\nplotted in Figs.3(c) and (d) in the absence of field-like torq ue.\nFurther, Figs.3(e) and (f) show the average dynamics of thesynchronized oscillations between the two oscillators due to\nthe presence of field-like torque corresponding to the initi al\nconditions similar to Figs.3(c) and (d). Thus when the initi al\nconditions are taken from different hemispheres, Figs.3(c ) and\n(d) imply that synchronized oscillations are not possible a nd\nonly steady states can exist in the absence of field-like torq ue,\nwhile Figs.3(e) and (f) confirm that synchronized oscillati ons\nindeed can be induced by the presence of field-like torque.\nB. Probability of synchronizations and steady state for ar-\nbitrary initial conditions\nThe dynamics of the coupled spin torque oscillators is more\ncomplicated than that of a single oscillator. In Appendix we\nshow that the dynamics of the two oscillators can be altered\nwhen there is a lack of simultaneity between the currents\npassing through the individual oscillators and the externa lIEEE TRANSACTIONS ON MAGNETICS JOURNALS 4\nFig. 3. (Color online) Averaged time evolution of m1x,m2x(a) andm1z,m2z(b) from 200 distinct initial conditions from same hemisphe res(0.99<\nm1z,m2z<1.00). Average time evolution of m1x,m2xandm1z,m2zwhenβ= 0(c,d) and β= 0.6(e,f) from 200 distinct initial conditions taken from\ndifferent hemispheres( 0.99< m1z<1.00,−0.99> m2z>−1.00). HereI0= 5.0 mA, T= 300 K and χ= 0.6.\nmagnetic field when they are switched off at different times\nwith even nanosecond differences.\nFrom the above studies we understand that there is a\ndefinite probability for the oscillators to reach steady sta tes\nin different hemispheres, and therefore it is essential to v erify\ntheir existence and the possibility of their removal by suit able\nmeans. Here by probability we mean only the possibility of\ninitial conditions reaching a particular final state(synch ronized\nstate/steady state) and we do not associate this with the\nprobability concept related to the randomness of the therma l\nfield. Hence, Eq.(1) is numerically solved for 100 numbers\nof randomly chosen initial conditions, chosen from both the\nhemispheres, and the corresponding probability to reach st eady\nstate (SS) and synchronized oscillation(SYN) state are com -\nputed. The values of SS and SYN are plotted against current in\nFigs. 4(a) and 4(b) for β= 0andβ= 0.61respectively, when\nχ= 0.5. Fig.4(a) shows that in the absence of field-like torque,\nthere is a nonzero probability of existence for both the stea dystate and synchronized oscillations beyond a critical curr ent\nstrength, whereas in the presence of positive field-like tor que\nthe system exhibits synchronized oscillations only, as sho wn in\nFig.4(b). Also we wish to point out here that by multistabili ty\nwe imply in this paper the possibility of the coexistence of\nsteady states and synchronized oscillatory states for arbi trary\nglobal initial conditions. In order to understand the impac t of\nfield-like torque, in Figs.4(c) and 4(d), we have depicted th e\nbifurcation diagrams of the system corresponding to Eq.(1)\nin the absence and presence of field-like torque respectivel y.\nIn the absence of field-like torque ( β= 0) the system shows\n(Fig.4(c)) multistability when the current I0exeeds the critical\ncurrentIc\n0. In the multistable region both the steady state and\nsynchronized oscillatory state are stable. Now by introduc ing\nthe field-like torque, we have plotted the bifurcation diagr am\nas a function of I0in Fig.4(d) for β= 0.61. It shows that the\nfield-like torque facilitates the emergence of stable synch ro-\nnized oscillatory state by destabilizing the steady-state throughIEEE TRANSACTIONS ON MAGNETICS JOURNALS 5\n 0 0.5 1\n02468(a)Probability\nI0(mA)SYN\nSS\n 0 0.5 1\n02468(b)Probability\nI0(mA)SYN\nSS\n-1-0.5 0 0.5 1\n 0 1 2 3 4 5 6 7 8χ = 0.5(c)\nHBm1x , m2x \nI0 (mA)-0.8-0.4 0 0.4 0.8\n 0 1 2 3 4 5 6 7 8χ = 0.5 (d)\nHBm1x , m2x \nI0 (mA)\nFig. 4. (Color online) Probabilities of synchronized oscil lations (PSOs) and\nsteady state (PSS) in the (a) absence ( β= 0) and (b) presence ( β= 0.61) of\nfield-like torque. The bifurcation diagrams of the system sp ecified by Eq.(1)\nare plotted in (c) the absence ( β= 0) and (d) presence ( β= 0.61) of field-\nlike torque. The red line( m1x) and black open circle( m2x) represent the\nmaxima(m1x,m2x>0) and minima( m1x,m2x<0) of the stable synchro-\nnized oscillatory state and the blue line( m1x) and the magenta square( m2x)\nindicate the stable steady state. ‘HB’ represents the Hopf b ifurcation point.\nThe other parameters are χ= 0.5 and T= 300 K.\nHopf bifurcation. By increasing the strength of the current , the\nexistence of the monostable synchronized oscillatory stat e can\nbe seen in Fig.4(d) for I0> Ic\n0.\nC. Removal of steady state by field-like torque\nTo analyze the impact of field-like torque on coupling\nstrength, we plot the SYN and SS for 100 randomly chosen\ninitial conditions for I0= 8mA. Figure 5(a) shows that in\nthe absence of field-like torque the probability fo SYN(SS)\nreduces(increases) from 1(0) when the coupling strength is\nincreased. This evidences that the system does not exhibit\nsynchronized oscillations for all initial conditions beyo nd\nsome critical value of coupling strength in the absence of fie ld-\nlike torque. From Fig.5(b) it is observed that the oscillato rs do\nnot get synchronized for all initial conditions in the absen ce\nof field-like torque. However, beyond certain critical valu e\nof positive field-like torque both the oscillators oscillat e syn-\nchronously for all initial conditions, which is confirmed fr om\nFig.5(b) where the SYN reaches 1 when the strength of field-\nlike torque is increased beyond the critical value ( βc= 0.33).\nWe have also depicted the bifurcation diagram with respect\ntoβforχ= 0.6 and I0= 8 mA in Fig.5(c). It is evident\nfrom the figure that the field-like torque term destabilizes t he\nsteady state and leads to only the synchronized oscillatory\nstate when β > β c. The magnetization trajectories of the\nsystem underlying Eq.(1) corresponding to β= 0 and 0.34\nare plotted as Figs.5(d) and 5(e) respectively. These figure s\nconfirm the existence of a stable steady state and the out-\nof-plane synchronized oscillatory state in the absence and\npresence of field-like torque respectively.\nFig. 5. (a) Probabilities of synchronized oscillations(re d) and steady state\n(black) against coupling strength in the absence of field-li ke torque at I0\n= 8 mA. (b) Probabilities of synchronized oscillations(red ) and steady state\n(black) against field-like torque at χ= 0.6,I0= 8 mA and T= 300 K.\nThe vertical lines correspond to the critical values χc= 0.29 and βc= 0.33\nobtained from Eq.(8) and Eq.(9) respectively. (c) The bifur cation diagram of\nthe system corresponding to Eq.(1) with χ= 0.6 and I0= 8 mA. The red\nline(m1x) and black open circle( m2x) represent the maxima and minima\nof the stable synchronized oscillatory state and the blue li ne(m1x) and the\nmagenta square( m2x) indicate the stable steady state. The magnetization\ntrajectories of the two oscillators are shown for (d) β= 0 and (e) β= 0.34.\nD. Steady states and critical values of and βandχfor\nsynchronized oscillations\nThe Eq.(1) can be transformed into spherical\npolar coordinates using the transformations mj=\n(sinθjcosφj,sinθjsinφj,cosθj)as follows:\n(1+α2)dθj\ndt=\n−2παFsinθj+√\nDGjx(αcosφjcosθj−sinφj)\n+√\nDGjy(αsinφjcosθj+cosφj)\n−γHSj[(α−β)sinφj+(1+αβ)cosθjcosφj], (5)\n(1+α2)sinθjdφj\ndt=\n2πFsinθj−√\nDGjx(αsinφj+cosθjcosφj)\n+√\nDGjy(αcosφj−cosθjsinφj)\n+γHSj[(1+αβ)sinφj−(α−β)cosθjcosφj], (6)\nwhereF= (γ/2π)[Ha+√\nDGz+(Hk−4πMs)cosθj].\nThe steady state solution of the system (1) is found around\nφ∗\n1=φ∗\n2≈3π/2, and\nθ∗\n1≈sin−1/parenleftbiggHS0\nHa+P/parenrightbigg\n, θ∗\n2≈π−sin−1/parenleftbiggHS0\nHa−P/parenrightbigg\n,\nwhereHS0= ¯hηI0/2eMsVandP=Hk−4πMs. Here, the\nthermal noise is not included for simplicity. From the linea r\nstability analysis, in the absence of field-like torque the s teady\nstate is found to be stable when [45]\n2/summationdisplay\ni=1/parenleftbigg∂fi\n∂θi+∂gi\n∂φi/parenrightbigg\nθ∗\n1,θ∗\n2,φ∗\n1,φ∗\n2<0. (7)\nHere,fiandgiare derived from Eqs.(5) and (6) as ˙θi=\nfi(θ1,θ2,φ1,φ2),˙φi=gi(θ1,θ2,φ1,φ2), i= 1,2.From the\ncondition (7), the critical value of coupling strength χcaboveIEEE TRANSACTIONS ON MAGNETICS JOURNALS 6\nwhich the system exhibits stable steady state solution in th e\nabsence of field-like torque ( β= 0), is derived as\nχc=λ+α\n2HS0[2PU−2Haτ−−Pτ+], (8)\nwhereτ±= (/radicalbig\n1−T+±/radicalbig\n1−T−),T±=H2\nS0/(Ha±P)2\nandU= (T++T−−1).\nHowever, in the presence of field-like torque, the critical\nvalue of βcabove which the steady state loses the stability,\nso that the synchronized state is the only stable state, can b e\nfound to be\nβc=αP[τ+−2U]−2HS0(λ−χ)+2Haατ−\n2HS0α(λ−χ)+Haτ−+Pτ+.(9)\nThe values of χcandβcmatch well with the numerical\nvalues, as confirmed by the vertical lines in Figs. 5(a,b).\nE. Stability of synchronized oscillations in the presence o f\nfield-like torque\nIn the absence of field-like torque and thermal noise the\nstability of the synchronized oscillations has already bee n\nconfirmed by Taniguchi et al [30]. However, here(Eq.(11)),\nin the presence of positive field-like torque and thermal noi se\nthe stability of the synchronized oscillations is confirmed by\nperturbing φ1asφ1=φ2+δφafter synchronization is reached,\nand the time evolution of δφis analysed over nperiods of\noscillations. By substituting φ1= 2πft+δφ(t), φ2= 2πft\nandθ1=θ2=θin Eq.(6) and after averaging over n\noscillations we can obtain [30]\n1\nnT/integraldisplaynT\n0dδφ\ndtdt≈ −χγHS0(1+αβ)\n(1+α2)nT/integraldisplaynT\n0δφ. (10)\nThe solution of Eq.(10) is given by\nδφ(t)≈δφ(0)exp/parenleftbigg\n−χγHS0(1+αβ)nT\n(1+α2)/parenrightbigg\n, (11)\nindicating the small deviation (δφ)betweenφ1andφ2expo-\nnentially decreases to zero as the number of oscillations( n)\nincreases. This implies that the presence of field-like torq ue\nand thermal noise do not affect the stability of the synchro-\nnized oscillatory state of the two parallelly coupled spin t orque\noscillators as long as (1 +αβ)>0. Further, from Eq.(11)\none may also note that when n→ ∞ the phase difference\nbetween oscillations of the two oscillators approaches zer o\ncorresponding to in-phase synchronized oscillations. Thi s has\nalso been verified numerically by using the algorithm given i n\nRef. [43].\nF . Frequency, power and Q-factor of synchronized oscilla-\ntions\nThe in-phase synchronization and its stability between the\ntwo oscillators in the presence of field-like torque have bee n\nconfirmed in Figs.2(c), 3(c) & 4(b) and Eq.(11) respectively .\nIn the synchronized state, the values of θ1andθ2are the\nsame and can be approximated to a constant value [30], [44]\nsince the amplitude of the oscillations of m1zandm2zare\nsmall as shown in Figs.2(d) and 3(d). Also, φ1= 2πft andφ1−φ2= 2nπ,n=0,±1,±2. . . . Here, fis the frequency of\nthe synchronized oscillations derived from Eq.(6) as\nf(θ) =/parenleftbigg1\n1+α2/parenrightbigg\n/bracketleftBigg\nF+(β−α)γ¯hηI0cosθ\n4πeMsVλsin2θ/parenleftBigg\n1−1/radicalbig\n1−λ2sin2θ/parenrightBigg/bracketrightBigg\n.(12)\nThe frequency and power spectral density(PSD) of the syn-\nchronized oscillations against current in the absence ( β= 0)\nand presence ( β= 0.61andβ=−0.61) of field-like torque\nhave been plotted in Figs. 6(a) and 6(b) respectively for χ=\n0.5. The solid line in Fig.6(a) corresponds to numerically c om-\nputed frequency and the open circles correspond to analytic ally\ncomputed frequency from Eq.(12). From Fig.6(a) it is observ ed\nthat the frequency of the synchronized oscillations is enha nced\nby positive field-like torque and decreased by negative field -\nlike torque. Fig. 6(a) shows that the frequency obtained fro m\nthe analytical expression(open circles) and numerical com pu-\ntation(solid lines) matches well and this evidently sugges ts the\nvalidity of the analytical results. The small deviation app earing\nin the frequency for positive field-like torque at about 3 mA\nis due to the drop in the mean value of θjaround 3 mA.\n 4 5 6 7\n2 4 6 7.5(a)\nβ = 0.61\nβ = 0\nβ = −0.61f (GHz)\nI0 (mA)04812\n 6  6.2  6.4  6.6  6.8β = 0β = -0.61\nβ = 0.61(b)PSD (arb. unit)\nf (GHz)\nFig. 6. (Color online) (a) The frequency of synchronized osc illations in the\nabsence and presence of field-like torque when χ= 0.5 and T= 300 K.\nThe solid line and open circle correspond to the frequency co mputed by\nnumerical and analytical (Eq.(12)) calculations, respect ively. (b) The power\nspectral density of the oscillations in the absence ( β= 0 ) and presence\n(β=0.61 and -0.61) of field-like torque when I0= 2.0 mA, T= 300 K and\nχ= 0.5.\nIn order to elucidate the experimental consequences of\nenhancement of the frequency and power of synchronized\noscillations due to field-like torque, we have plotted the\nspectral power in the frequency domain in Fig.6(b) for β= 0,\nβ= 0.61andβ=−0.61whenI0= 2.0 mA, χ= 0.5 and\nT= 300 K. It is evident from Fig.6(b) that the frequency\nof the synchronized oscillations is enhanced by the positiv e\nfield-like torque. Also, the power and Q-factor are enhanced\nby negative field-like torque. For instance the frequency is\nincreased by 0.241 GHz when βis increased from 0 to 0.61.\nThe power is enhanced by more than 2.5 times when βis\nnegatively increased from 0 to -0.61 along with the incremen t\nof Q-factor from 447.51( β= 0) to 672.61 ( β= -0.61). On the\nother hand the power is decreased by increasing the value of\nβfrom 0 to 0.61 with a slight decrement in Q-factor from\n447.51(β= 0) to 411.33 ( β= 0.61). Thus, the negative field-\nlike torque enhances the power with large increment in Q-\nfactor and positive field-like torque increases the frequen cy\nwith slight decrement in Q-factor.IEEE TRANSACTIONS ON MAGNETICS JOURNALS 7\nFig. 7. (Color online) Time evolution of m1x,m2x(a & c) and m1z,m2z(b & d) for β=0 (a & b) and β=0.2 (c & d) when the currents passing through\nthe first, second oscillators and the applied field are instan taneously switched off at 500 ns and switched on at 1500 ns. He reI0= 2.0 mA, T= 300 K and\nχ= 0.5. The inset figures show the synchronization of m1x(black solid circle) and m2x(red solid line).\nIV. C ONCLUSION\nIn conclusion, the existence of steady state and its removal\nin the system of two parallelly coupled spin torque oscillat ors\nby field-like torque has been investigated theoretically, w ith\na physical configuration of perpendicularly magnetized fre e-\nlayer and in-plane magnetized pinned layer. The numerical\nsimulation of the LLGS equation has revealed that the exis-\ntence of field-like torque can cancel out the damping effect\nand thus can induce synchronized oscillations with respect to\napplied current. One can also note that the existence of stea dy\nstate behavior in coupled STOs can be efficiently removed\nby introducing the field-like torque. The frequency of the\nsynchronized oscillations gets enhanced by positive field- like\ntorque. Also, the power and Q-factor are enhanced by the\nnegative-field like torque.\nV. A PPENDIX\nA. Destabilization of steady staets due to small time delays\nin switching of current and field\nIn this appendix we wish to point out even when the initial\nconditions are chosen in the same hemisphere, multistable\nstates can arise due to nanoscale level time delays in switch ing\noff the current and field. Investigations on pulse fields by\nKikuchiet al. [46] and Flovik et al. [47] suggest that the out-\nof-plane magnetic field can be produced on magnetic layers fo rthe duration of nano and picco second by nonsized coil using\ncurrent or by laser pulses through inverse Faraday effect. A s\nan example, in this Appendix, we consider a situation where\ninitially the currents to the first and second oscillators ar e\nswitched on at τon\nI1,1andτon\nI2,1, respectively, and the field at\nτon\nHa,1. After the oscillators attain synchronized oscillations, the\ncurrents and field are switched off at τoff\nI1,1,τoff\nI2,1andτoff\nHa,1.\nAfter some time they are again switched on at τon\nI1,2,τon\nI2,2and\nτon\nHa,2, respectively. Figs.7 & 8 show the time evolution of mx\nandmzcomponents of the two oscillators in the presence\nof thermal noise field for the initial conditions chosen for\n0.99< m1z,m2z<1.00, whenI0= 2.0 mA, χ= 0.5. To\nconfirm the synchronized oscillations, the m1xandm2xare\nplotted as inset figures for small time window. Figs.7(a,b) &\n(c,d) have been plotted for β=0 andβ=0.2, respectively, when\nτon\nI1,1=τon\nI2,1=τon\nHa,1= 0s,τoff\nI1,1=τoff\nI2,1=τoff\nHa,1= 500ns\nandτon\nI1,2=τon\nI2,2=τon\nHa,2= 1500ns. Figs.7(a) & (c) show\nthat irrespective of whether the field-like torque is presen t\nor not, both the oscillators reach steady state after 500 ns\nand regain synchronized oscillations after 1500 ns. The fina l\nsynchronized oscillations are similar as in Ref. [30] for β= 0.\nTo show the impact of field-like torque on retrieving the\nmagnetizations from steady states to synchronized oscilla tions\nFigs.8 are plotted for τon\nI1,1=τon\nI2,1=τon\nHa,1= 0s,\nτoff\nI1,1= 504ns, τoff\nI2,1= 500ns, τoff\nHa,1= 496nsandIEEE TRANSACTIONS ON MAGNETICS JOURNALS 8\nFig. 8. (Color online) Time evolution of m1x,m2x(a & c) and m1z,m2z(b & d) for β=0 (a & b) and β=0.2 (c & d) when the currents passing through\nthe first oscillator, second oscillator and applied field are cut off at 504 ns, 500 ns and 496 ns respectively and switched o n simultaneously at 1500 ns. Here\nI0= 2.0 mA, T= 300 K and χ= 0.5. The inset figures show the synchronization of m1x(black solid circle) and m2x(red solid line).\nτon\nI1,2=τon\nI2,2=τon\nHa,2= 1500 ns. It is observed that in\nthe absence of field-like torque, the oscillations of the two\noscillators damp out after 500 ns to the steady states at\ndifferent hemispheres, formed by the unit vector maround the\norigin, and continue in the same steady states even after the\ncurrents and field are applied at 1500 ns as shown in Figs.8(a)\nand (b). For the present case, m1zandm2zreach steady states\nat north and south poles respectively. Occasionally, the th ermal\nnoise leads both the oscillators to steady states at norther n\nhemisphere and they exhibit synchronized oscillations aft er the\ncurrents and field are switched on at 1500 ns. This is shown in\nFig.9, where we can observe that in the presence of thermal\nnoisem2zreturns to north pole after 500 ns and oscillates\nafter 1500 ns. In the absence of thermal noise m2zmoves\nto the steady state at south pole after 500 ns and continues\nthere even after the currents and field are switched on at 1500\nns. On the other hand in the presence of positive field-like\ntorque, both the oscillators always attain the steady state at the\nnorthern hemisphere after the currents and field are switche d\noff around 500 ns and reach synchronized oscillations after\nthe currents and field are switched on at 1500 ns as shown in\nFigs.8(c) and (d). From Figs.7(a) & 8(a) it is understood tha t\nthe lack of simultaneity in switching off the currents and fie ld\ntransforms the system from getting synchronized oscillato ry\nstate to steady state. In realistic applications the couple d\noscillators may be switched off and on many times. Every time\nthe system is switched off, the currents passing through theindividual oscillators might be cut off at slightly differn t times\nwith at least few nanosecond differences between them due to\nvarious disturbances or at the same time. In these situation s,\nthe system of coupled oscillators may exhibit synchronized\noscillations or steady state motion as shown in Figs.7(a) &\n8(a) respectively. However, the presence of field-like torq ue\ndestabilizes the steady state at the southern hemisphere an d\nmakes the magnetization vectors of the two oscillators to\nstay in the northern hemisphere and exhibit synchronized\noscillations after the currents and field are switched on as\nconfirmed in Figs.8(c) & (d). It has also been verified that\nthe positive field-like torque destabilizes the steady stat e and\nmakes synchronized oscillations even when τon\nI1,1,τon\nI2,1,τon\nHa,1\nandτon\nI1,2,τon\nI2,2,τon\nHa,2differ by nanoseconds. From Figs.8(b) &\n(d) it is also verified that the system reaches steady state wh en\nthe magnetization vectors evolve in opposite hemispheres a nd\nthat the thermal noise has no impact on it.\nTo prove the strong destabilization of steady states by\nfield-like torque the average values of the zcomponents of\nmagnetizations are plotted in Figs.10(a) and (b) from 100 tr ials\nin the absence and presence of field-like torque respectivel y.\nFrom Fig.10(a) it can be understood that when the field\nand currents are switched off at 500 ns with nanoscale time\ndifference between them, some of the magnetizations of the\nfirst and second oscillators are settled near north pole of th e\nsphere and the remaining magnetizations of the two oscillat ors\nsettle near south pole. When the currents and field are switch edIEEE TRANSACTIONS ON MAGNETICS JOURNALS 9\nFig. 9. (Color online) Time evolution of m2zwith(blue) and without(black)\nthermal noise in the absence of field-like torque when I0= 2.0 mA, T= 300\nK andχ= 0.5.\nFig. 10. (Color online) Averaged time evolution of m1z,m2zwhenβ= 0\n(a) andβ= 0.2(b) from 100 trials for the same initial conditon taken for\nthe Figs.8(b) and (d) when the currents passing through the fi rst oscillator,\nsecond oscillator and applied field are cut off at 504 ns, 500 n s and 496 ns\nrespectively and switched on simultaneously at 1500 ns.. He reI0= 2.0 mA,\nT= 300 K and χ= 0.5.\non simultaneously at time 1500 ns, the values of < m1z>\nand< m2z>slightly increase from their corresponding\nvalues between 500 ns and 1500 ns. This is due to the fact\nthat the thermal fluctuations occasionally drive both of the\nmagnetizations into the northern hemisphere and make them\nto oscillate synchronously after 1500 ns (Refer Fig.9). Hen ce,\nfew of the cases from out of the 100 trials make synchronized\noscillations after 1500 ns which tend to increase the values of\n< m1z>and< m2z>after 1500 ns. On the other hand when\nthe field-like torque is present the magnetizations of the tw o\noscillators are driven into the northern hemisphere and exh ibit\nsynchronized oscillations for all the 100 trials as shown in\nFig.10(b). Also, we checked that a negative field-like torqu e\ndoes not produce synchronized oscillations when the curren ts\nand field are switched on again after switching off at differe nt\ntimes.\nWe also wish to point out that two things can happen in\nthe absence of field-like torque as seen from Figs.7 and 8.\nFirst, due to the lack of simultaneity in switching off/on th e\ncurrents passing through the individual oscillators and fie ld the\nmagnetizations of the two oscillators are driven into the st eady\nstates near the poles at opposite hemispheres(occasionall y the\nmagnetizations are kept in the northern hemisphere due to\nthermal flucutation) formed by mand continue there even\nafter the currents and field are switched on again. Second, if\nthe magnetizations are settled in the steady states at diffe rent\nhemispheres, the synchronized oscillations are not possib le by\napplying field Haand currents I1andI2. When the field-\nlike torque is additionally present, the magnetizations ar e keptin the northern hemispheres only and avoid steady states at\nopposite hemisperes. Also, even if the magnetizations are\nin steady states at opposite hemispheres the synchronized\noscillations can be induced by the field-like torque.\nACKNOWLEDGEMENTS\nThe work of V .K.C. forms part of a research project\nsponsored by CSIR Project No. 03/1444/18/EMR II. M.L.\nwishes to thank the Department of Science and Technology\nfor the award of a SERB Distinguished Fellowship under\nGrant No.SB/DF/04/2017 in which R. Arun is supported by\na Research Associateship.\nREFERENCES\n[1] J. Grollier, V . Cros, and A. Fert, Phys. Rev. B 73, 060409(R) (2006).\n[2] S. Urazhdin, P. Tabor, V . Tiberkevich and A. Slavin, Phys . Rev. Lett.\n105, 104101 (2010).\n[3] F.B. Mancoff, N. D. Rizzo, B. N. Engel and S. Tehrani, Natu re437,\n393 (2005).\n[4] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russ ek, and J.\nA. 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B 96, 140411(R) (2017)."    },    {        "title": "1408.0341v1.Tunnel_magnetoresistance_and_spin_transfer_torque_switching_in_polycrystalline_Co2FeAl_full_Heusler_alloy_magnetic_tunnel_junctions_on_Si_SiO2_amorphous_substrates.pdf",        "content": "1 \n Tunnel magnetoresistance and spin -transfer -torque switching in \npolycrystalline  Co2FeAl full -Heusler  alloy magnetic tunnel junctions  on \nSi/SiO 2 amorphous substrate s \nZhenchao Wen, Hiroaki  Sukegawa, Shinya Kasai, Koichiro Inomata , and Seiji Mitani  \nNational Institute for Materials Science (NIMS), 1 -2-1 Sengen, Tsukuba 305 -0047, Japan  \n \nAbstract:  \nWe studied polycrystalline B2-type Co2FeAl (CFA)  full-Heusler alloy  based magnetic \ntunnel junctions ( MTJs ) fabricated on a Si/SiO 2 amorphous substrate . Polycrystalline  CFA \nfilms with a (001)  orientation, a high  B2 ordering, and  a flat surface  were achieved using a \nMgO buffer layer . A tunnel magnetoresistance (TMR) ratio up to  175% was obtained  for an \nMTJ with a  CFA/MgO/CoFe  structure on a 7.5 -nm-thick  MgO buffer . Spin-transfer  torque \ninduced magnetization  switching  was achieved in the MTJs with a 2-nm-thick \npolycrystalline  CFA film  as a switching  layer. Using a thermal activation model , the \nintrinsic  critical current  density  (Jc0) was determined  to be 8.2 × 106 A/cm2, which is lower  \nthan 2.9 × 107 A/cm2, the value  for epitaxial CFA -MTJs  [Appl. Phys. Lett.  100, 182403 \n(2012) ]. We found that t he Gilbert damping constant  () evaluated using  ferromagnetic  \nresonance measurements  for the polycrystalline CFA film was ~0.015 and was almost \nindependent of the CFA thickness (2~18 nm) . The low Jc0 for the polycrystalline MTJ was \nmainly attributed to the low  of the CFA layer compared with the  value in the  epitaxial \none (~0.04).   \n  \n \n \n  2 \n I. INTRODUCTION  \nHalf-metallic ferromagnets (HMF s) draw  great interest because of the perfect  spin \npolarization  of conduction electrons at the Fermi level , which is considered to enhanc e the  \nspin-dependent transport efficiency  of high-performance spintronic devices . [1–3] Cobalt-\nbased full -Heusler alloys with the chemical formula Co 2YZ (where Y is a transition metal \nand Z is a main group element) , are extensively studied as a type of HMF s owing  to their \nhigh Curie  temperature  of approximately 1000 K , high spin polarizat ion, and low damping \nconstant . [4, 5]  They exhibit great  potential for application s in spintronic s, including  \ncurrent -perpendicular -to-plane giant magnetoresistance ( CPP-GMR ) read head s, [6, 7]  \nmagnetoresistive random access memories (MRAM s), [8] and spin transistors such as  spin-\nfunctional metal -oxide -semiconductor field -effect transistor s (spin -MOSFET s). [9, 10]  In \nparticular , magnetic tunnel junctions (MTJs) with  Co-based full -Heusler alloy  electrodes  \nhave been shown tremendous ly increasing tunnel magnetoresistance ( TMR ) ratios during \nthe last decade  since Inomata et al.  [11] demonstrated a TMR ratio of 16% using \nCo2Cr0.6Fe0.4Al/AlO x/CoFe MTJs  at room temperature (RT) . [12–21] Recently , a \nremarkable  TMR ratio of approximately  2000 % at 4.2 K  (354% at RT ) was achieved  using  \nepitaxial Co2MnSi /MgO /Co2MnSi (001)  MTJs, demonstrating the half -metallicity of Co -\nbased Heusler alloys and strong 1 coherent tunneling effect in the MgO /Heusler MTJ s \n[21].  \nThe Co2FeAl  (CFA)  alloy  is of particular  interest because of  its high spin polarization (a \nhalf-metallic electronic structure) [22] and low effective  damping  constant () ~ 0.001  [23], \nwhich are benefi cial for enhancing the TMR ratio and lowering the magnetization 3 \n switching  current of spin-transfer torque (STT) . CFA fil ms prepared using sputtering \ndeposition generally have a disordered B2 structure  (swapping between Y and Z sites) rather \nthan an ordered L21 structure  owing  to the thermodynamic  stability of CFA  [24]. \nNevertheless , the spin polarization  calculated for the L21 structure is conserved even for the \nB2 structure  [25]. Importantly , a CFA film with a ( 001) orientation  has a large in-plane \nlattice spacing ( d(200)/2 = d(110) = 0.203 nm) compared with other half -metallic Heusler \nalloy s such as Co2FeAl 0.5Si0.5 (d(110) = 0. 201 nm) and Co2MnSi  (d(110) = 0. 198 nm). \nTherefore, a nearly perfect CFA/MgO (001) heterostructure is easily achieved by the \nmagnetron sputtering method , and this is  favorable  for enhancing  the coherent tunneling \neffect  [13, 18]. Recently , TMR  ratios as high as  360% at RT (785% at 10 K) were \ndemonstrated in  epitaxial CFA-based MTJs with a sputter -deposited  MgO barrier  [17–19]. \nThe large  TMR ratio originated  from the high spin polarization of the CFA layer and the \nstrong contribution of the coherent tunneling effect through 1 Bloch states in the CFA and \nthe MgO barrier. Moreover, (001) -textured CFA films can be grown on MgO -buffered \nSi/SiO 2 amorphous substrate s, and a relatively large  TMR ratio of 166% at RT ( 252% at 48 \nK) was achieved in  a (001) -textured CFA/MgO/CoFe MTJ  [20]. Such polycrystalline  full-\nHeusler MTJs on amorphous substrate s are desired  because of  their compatibility with  \npractical industrial  applications of full -Heusler spintronic devices, while s ingle -crystal \nMgO(001) substrates  have a limit ed scope of application.  \nFurthermore, STT -induced magnetization switching  (STT switching) , a key techn ology  \nfor writing information in spintronics devices , was realized using  MTJs  with a n epitaxial  \nCFA ultrathin (~1.5 nm)  layer  as a switching  (free) layer  [26]. However, a large  critical  4 \n switching current density  (Jc0) of 2.9 × 107 A/cm2 was observed  owing  to the enhancement \nof , which was ~0.04,  of the epitaxial CFA film. In addition , STT switching can be \ndisturbed  by the stabilization of intermediate  magnetic states possibly because of  the \npresence of in -plane magnetocrystalline anisotropy , which is generally seen in epitaxial \nmagnetic films (e.g. , 4-fold anisotropy  for cubic  (001) films)  [27]. Therefore, reducing the \nundesirable  magnetic anisotropy using polycrystalline Heusler alloy  films is effective for \nhighly efficient STT switching . \nIn this work, we systematically  studied MTJs with (001) -textur ed polycrystalline CFA \nfilms  on Si/SiO 2 amorphous substrate s. A MgO buffer was introduced  in the MTJs for \nachiev ing (001)  texture with B2-ordering structure  of CFA layers  on the amorphous  \nsubstrates . The (001) -texture, B2 order, and surface morphology  of the polycrystalline  CFA \nfilms and the TMR effect in the entire  MTJ stacks were characterized  for varying MgO \nbuffer  thickness . Also, the MgO -barrier  thickness and resistance -area product ( RA) \ndependence of the TMR ratios were  investigated  for the polycrystalline CFA -MTJs.  \nFurthermore,  STT switching  was examined  in low -RA MTJs with a thin polycrystalline  \nCFA film (2.0 nm)  as a free layer.  Jc0 of 8.2  × 106 A/cm2 was demonstrated by a thermal \nactivation model for switching current ; this is far lower than the value for epitaxial CFA -\nMTJs  [26]. The  values for the polycrystalline  CFA films , obtained  using  a waveguide -\nbased ferromagnetic resonance method , were almost constant against  the CFA thickness , \nand a relatively low  of ~0.015 was demonstrated. We attributed  the reduction in the Jc0 of \nthe STT switching  to the reduced   of the polycrystalline CFA films .  5 \n  \nII. EXPERIMENT  \nAll multilayer stack s were deposited on thermally oxidized Si/SiO 2 amorphous \nsubstrate s at RT  using an ultra-high vacuum magnetron sputtering system with a base \npressure lower than 4  10-7 Pa. MgO  layers were deposited from a sintered MgO target by \nRF sputtering  with an RF power density of 2.19 W/cm2 and an Ar pressure of 10 mTorr . \nCFA layers were deposited from  a stoichiometric Co 50Fe25Al25 (at.%) alloy target  using  DC \npower . The structur al properties  and surface morphology of CFA film s on MgO buffer s \nwere characterized  by out-of-plane  (2θ-ω scan) X-ray diffraction (XRD) with Cu Kα \nradiation  (λ = 0.15418 nm)  and atomic force microscop y (AFM), respectively. MTJ stack s \nwith the structure of CFA/MgO /Co75Fe25/IrMn/Ru (unit: nm)  were deposited on MgO -\nbuffered thermally oxidized Si /SiO 2 amorphous substrate s and patterned into junctions with \nactive area of 5  10 m2 by conventional lithography  methods with Ar ion milling . For \nSTT switching, spin-valve  MTJ s with the structure of MgO  (7.5)/ Cr (40)/CFA  (2)/MgO  \n(0.6–0.8)/Co 75Fe25 (5)/Ru  (0.8)/Co 75Fe25 (5)/IrMn  (15)/Ta  (5)/Ru  (10) (unit: nm)  were \nprepared on the amorphous substrates and nanofabricated into 100 -nm-scale d ellipse s. The \nactual  areas of the MTJ nano -pillars were obtained according to  the ratio of the  RA to the  \njunction resistance ; the RA was characteriz ed by current-in-plane tunneling (CIPT)  \nmeasurement  [28] before patterning . The MTJ  stack s were post-annealed in a vacuum  \nfurnace for 30 minutes  under  a magnetic field of 5 kOe. The magneto -transport  properties \nwere measured  using a DC 2- or 4- probe method . The magnetic damping constant  of the \npolycrystalline CFA film was measured by waveguide -based ferromagnetic resonance  6 \n (FMR) . The films were patterned into rectangular shape elements of 600  × 20 μm2 using \nUV lithography  together with  Ar ion milling, and then coplanar waveguides made of Au \nwere fabricated on them. The FMR signal obtained as change of the real part of S21 signal , \nwas determined  using  a network analyzer. An external magnetic field along the longitudinal \naxis was varied from 0 to  1.9 kOe, while  the excitation power was fixed as 0 dBm. All \nmeasurements were performed at RT. \nIII. RESULTS  \nA. Effect of MgO buffer on polycrystalline CFA films and TMR  \nBefore MTJ multilayer  films  were grown , MgO buffer w as deposited  on a Si/SiO 2 \namorphous  substrate  in order to  establish the (001) -texture of polycrystalline CFA films  by \ntaking advantage of  the unique (001) -texture property of MgO  layer s on an amorphous \nsubstrate . CFA films were subsequently  grown on the  MgO buffer layer, and the structural \nproperties of the CFA films depend ing on the MgO buffer thickness w ere investigated  by \nXRD . The out-of-plane XRD patterns  of 30-nm-thick CFA films are shown  in Fig.  1(a); the \nCFA films were annealed at Ta = 400  °C, and the thickness es of the MgO  buffers ( tMgO) are \n2.5, 5.0, 7.5, and 10.0 nm . In addition to the peaks (denoted  by “s”) from Si/SiO 2 substrates, \nMgO(002), CFA(002) , and (004) peaks were observed  along with the absence of other \noriented peaks, demonstrating  the (001) -texture established  in the  stacks . Figure 1(b) shows  \nthe tMgO dependence of the  integrated  intensity of CFA(002) peaks for as-deposited CFA \nfilms , and CFA films annealed at  400 °C and 480  °C. With  increasing tMgO, the intensity of \nthe peaks initially  increases for all samples  owing to the improved (001) -texture of the \nMgO buffer layer , reaching  a maximum  at tMgO = 7.5 nm. The reduction in intensity at tMgO 7 \n = 10 nm  may be caused by  the degraded surface morphology  of the MgO buffer layer. In \naddition, the XRD intensity increases w ith increasing annealing temperature, which \nindicates  that the increasing temperature  improve s the B2 order and (001) -texture  of the \nCFA films . In the XRD  2θ-ω scan with the diffraction vector along CFA[111], (111) \nreflection was not detected , indicating  that the CFA film s have a B2-ordering structure  with \nswapping between Fe and Al atoms while Co atoms occupy the regular sites.  The degree of \nB2 order ing was estimated according to  the ratio of the integrated  intensity of  the CFA(002) \nand (004) peaks. The peaks were fitted by Voigt profiles , and t he ratio of their integrated  \nintensit ies, i.e., the ratio of  I(002) to I(004), is shown in Fig. 1(c).  The maximum \nI(002)/I(004) value was obtained  at tMgO = 7.5 nm for all  of the samples : as-deposit ed and  \nanneal ed at 400  °C or 480 °C. This value is comparatively large for  CFA film s annealed at \n480 °C, indicating  the improvement of  the B2 ordering and  the mosaicity  of the CFA films  \ndue to  annealing at high temperature . The degree of B2 order ing, SB2, can be  evaluated  \nusing  the ratio s according to the following equation : [29] \n      √                   \n                   ,                   (1) \nwhere  [I(002)/ I(004)] exp. is the ratio of the integrated intensity  of the (002) peak to that of \nthe (004) peak  as determined by experiments, and [I(002)/ I(004)] cal. is the ideal ratio of the \ntwo peaks. For tMgO = 7.5 nm, t he ordering parameter SB2 is calculated to be 0.89, 0.95, and \n0.98 for CFA film s as deposited , anneal ed at 400  °C, and 480  °C, respectively . The results \ndemonstrate  a high B2 order  and an excellent  (001) -texture were established in the  \npolycrystalline  CFA film s on MgO -buffered Si/SiO 2 amorphous  substrates .  8 \n For stacking  MTJ multilayers with the polycrystalline CFA films , the tMgO dependence \nof the surface morphology  of the CFA films  was investigated . Figure 2 shows the average \nsurface roughness (Ra) and peak -to-valley ( P-V) value  as a function of tMgO for 30 -nm-thick \nCFA films annealed at 400  °C and 480  °C, respectively. For the samples annealed at \n400 °C, flat surface s with Ra ~ 0.1 nm and P-V ranging from  1.3 to 1.5 nm were  observed \nfor all values of tMgO. The inset of Fig.  2 shows an example of AFM image s of the samples \n(annealed at 400 °C on a 7.5 -nm-thick MgO buffer ). The results indicate  the feasib ility of \nstacking MTJs with a thin MgO ba rrier. In addition , the samples annealed at 480 °C with \nhigher Ra and P-V values were observed as well as a large tMgO dependence.  \nThe whole MTJ  stacks  with the structure of MgO  (tMgO)/CFA  (30)/MgO  (tbarr)/Co75Fe25 \n(5)/IrMn  (15)/Ru  (10) (unit: nm)  were then fabricated  on Si /SiO 2 substrate s with var ying \nMgO buffer thickness  tMgO (2.510.0 nm) and MgO barrier thickness tbarr (1.5, 1.8, and 2.0 \nnm). The stacks were annealed at 370 °C in the presence of  a magnetic field  of 5 kOe . \nFigure 3 shows TMR  ratios as a function of tMgO for the polycrystalline  CFA -MTJs  \nmeasured a t RT  using CIPT . The 30-nm-thick CFA films were post-annealed at Ta = \n400 °C and 480  °C in order to improve  the B2 ordering . The TMR ratios obtained in MTJs \nwith Ta = 400  °C were higher than those  of MTJs with Ta = 480  °C, which can be attributed  \nto a better  CFA/MgO -barrier  interface due to the flat CFA  surface  annealed at  400 °C, \nalthough a  higher degree of B2 ordering was observed for Ta = 480  °C, as shown  in Fig s. 1 \nand 2 . With increas ing tMgO, the TMR ratio increases and is nearly saturated  at tMgO > 5 nm,  \nindicating  that high-quality CFA films with B2 order and (001) -texture were established \nwith the more than  5-nm-thick MgO buffer layer, which is consistent  with the XRD 9 \n analyses . A slight  reduction in the  TMR ratio was observed at tMgO = 10 nm, which could  \nbe caused by the reduction in the degree of (001) -orientation of the MgO buffer . The MTJs \nwith a 1.8-nm-thick MgO barrier  exhibit  larger  TMR ratio s than those with 1.5- and 2.0 -\nnm-thick MgO barri ers, which could be due to the  plastic relaxation of the MgO barrier  [30] \nand/or the oscillatory behavior of the TMR ratio as a function of MgO thickness  [18].  \nB. MgO barrier thickness and RA dependences of  TMR  \nIn order to realize  STT switching in the polycrystalline  CFA -MTJs, it is expected that  \nintroduction of a conductive underlayer, reduction in the free -layer thickness , and control of \nthe RA of the barrier layer  are required . The MgO barrier thickness and RA dependence s of \nthe TMR  ratio w ere investigated  using spin-valve  MTJs with the structure of MgO  buffer \n(7.5)/Cr  (40)/CFA  (30)/MgO  (tbarr: 1.2-2.0)/Co 75Fe25 (5)/IrMn  (15)/Ta  (5)/Ru  (10) (unit: nm)  \non a Si/SiO 2 substrate . A Cr under layer was selected  as the conductive electrode because  Cr \nhas a very small l attice mismatch  with CFA  (~0.6%)  and can further facilitate  the ordering \nstructure of full -Heusler alloys  [13]. The Cr layer s for the samples  as deposited, annealed at \n400 °C, and 600 °C were prepared  on the 7.5-nm-thick MgO buffer  for the MTJ stacks . The \nentire stacks were annealed at 370 °C in the presence of a magnetic field  of 5000 Oe , and \nthen their TMR ratios and RA values were characterized using  CIPT  measurement .  \nFigure 4 (a) shows t he dependence of the TMR ratios  on the nominal thickness of the \nMgO  barrier ( tbarr) for the MTJs  with different Cr annealing  conditions . For the sample s that \nwere as-deposited and annealed at 400 °C, the TMR ratio increases with tbarr, and TMR \nratios greater than 100% were achieved for the whole range of tbarr. This means  that the 10 \n (001) -texture and  B2 order ing of CFA films can be maintained on the MgO/Cr buffer layers.  \nWe obtained the largest  TMR ratio of 175% for the 400 °C annealed sample  with tbarr = \n1.95 nm ; this TMR ra tio is higher than 16 6%, which was  observed in the MTJ without the \nCr buffer , which  indicates that the Cr buffer with optim al condition s promotes CFA(001) \ngrowth  and improves the effective tunneling spin polarization. On the other hand , the \nsamples annealed at  a high temperature  (600 °C) exhibit ed smaller TMR ratios  (80–120% ). \nThis was  attributed  to the rough  surface  (Ra = 0.4 nm  and P-V = 3.2 nm) of the CFA film \non the Cr layer  annealed at 600 °C, which can lead  to a declined crystalline  orientation of \nthe MgO(001)  barrier . Furthermore, oscillation  behavior  of the TMR ratios as a function of \ntbarr was observed for all of the structure s. The TMR oscillation behavior is typically \nobserved in epitaxial  MTJs such as Fe/MgO /Fe [31], Co 2MnSi/MgO/Co 2MnSi  [32], \nCo2Cr0.6Fe0.4Al/MgO/ Co2Cr0.6Fe0.4Al [33], and CFA/MgO /CoFe  [18] MTJs, while it is \nabsent  in polycrystalline  MTJs such as  CoFeB/MgO /CoFeB  [34] MTJs . More remarkable \noscillation amplitude in epitaxial full-Heusler alloy -based  MTJs than that of epitaxial \nFe/MgO /Fe MTJs was observed, which may be related to the electronic  structures of full -\nHeusler alloy electrode s and the full-Heusler /MgO interface ; however, the origin has not \nbeen understood yet . The unexpected  oscillation behavior  in the polycrystalline CFA -MTJs \nmay be also attributed to the unique electronic  structure of CFA and the interface. In \naddition, the flat buffer layer with a good crystallinity  enabled us to achieve a well -defined \nlayer -by-layer growth for the CFA layer a nd the MgO barrier, which may be advantageous  \nfor observ ing the oscillatory  behavior . The oscillation  period was approximately 0.2 nm  in \nnominal thickness , which seems to be shorter than that for the epitaxial  CFA/MgO/CoFe 11 \n (0.32 nm, short -period)  [18]. Further investigation is needed to clarify the origin of the \nbehavior.  \nThe RA as a function of the MgO barrier thickness  is plotted  in Fig. 4( b). We observed  \na typi cal behavior of an exponential  increase with increasing  tbarr. According to the Wenzel -\nKramer -Brillouin (WKB) approximation , the relationship between RA and tbarr can be \nexpressed as follow s: \n                  √   \n        ,                            (2) \nwhere h, m, and ϕ are Planck’s constant , the effective electron mass  assum ed as a free \nelectron mass (9.11  × 10−31kg) here, and the barrier height energ y of the tunnel barrier , \nrespectively  [31]. A similar barrier height of 0.7 eV was obtained for all three samples by \nthe fitting of RA-tbarr curves. This value is greater  than the reported values for Fe/MgO /Fe \ngrown using molecular -beam-epitaxy  (MBE) (0.39 eV)  [31], sputtered \nCoFeB/MgO /CoFeB  (0.29 –0.39 eV)  [35–37], and CoFeB/MgO  (electron -beam \nevaporated )/CoFeB  (0.48 eV)  [38] MTJs . The reasons  may be due to  the different  densities \nof oxygen vacancy defect s in the MgO barriers and/or the deviation of the actual MgO \nthickness from the nominal one . \nFigure 5 shows TMR ratios in a low -RA regime for the polycrystalline CFA -MTJs with \n2-nm-thick CFA film as a free layer. The MTJs were annealed at 225 ° C for 30 minutes  in \norder to reduce the influence of Cr layer to CFA . A TMR ratio of 40 –60% was achieved \nwith a n RA of 7–20  m2 (nominal MgO thickness: 0.6–0.8 nm) for the 100-nm-scale d 12 \n elliptical MTJs with the thin CFA layer,  which is favorable for achieving STT switching in \nthe MTJ stacks with a thin CFA free layer and the low RA value.  In addition , the TMR ratio \nof the poly crystalline MTJs with 2 -nm-thick CFA film is comparable to that with a thick  \n(30 nm) CFA film at a low RA value, as shown in Fig. 5, which indicates that the (001) -\ntexture and B2 ordering can be maintained in the thin 2 -nm-thick CFA films.  \nC. STT -induced magnetization switching  \nSTT-induced magnetization switching was performed in spin-valve  MTJs with the \nstructure of Si/SiO 2-substr ate/MgO  (7.5)/Cr  (40)/CFA  (2)/MgO  (0.6–0.8)/Co 75Fe25 (5)/Ru  \n(0.8)/Co 75Fe25 (5)/IrMn  (15)/Ta (5)/Ru  (10) (unit: nm). A schematic of the structure of a \npolycrystalline  CFA -MTJ nano pillar  is shown in Fig.  6(a). The synthetic  antiferromagnetic \ncoupling exchange bias of CoFe/Ru/CoFe/IrMn was employed  to reduce the offset \nmagnetic field  of hysteresis  loops . Figure 6(b) indicate s the tunneling resistance  of an MTJ \nnanopillar as a function of the applied magnetic  field ( H) measured with a DC bias voltage \nof 1 mV . A TMR ratio of 43% was observed  in the MTJ with a thin CFA free layer  (2.0 nm) \nand MgO barrie r (0.75 nm). Sharp switching between parallel  (P) and antiparallel (AP) \nmagnetic configurations was observed. The RA of the MTJ was determined using CIPT \nmeasurement s to be 13 m2, and the active area of the MTJ nanopillar was calculated  to \nbe 1.24 ×  10−2 m2. The hysteresis offset field ( Hoffset) and the coerciv ity field ( Hc) were \ndetermined using the R-H loops to be −11 and 2 6 Oe, respectively.  Figure 6(c) shows the \nrepresent ative resistance -current ( R-I) loops of  the CFA -MTJ nanopillar measured by a DC \ncurrent with a sweep rate of 1.2 ×  104 A/s at different magnetic fields of 0, −11 and −20 13 \n Oe, respectively . The positive current indicates that electrons  flow from the bottom \nelectrode to the top electrode . Magnetic switching between P (low-resistance)  and AP \n(high -resistance) states was achieved owing  to the current. When current was applied in the \nnegative ( positive ) direction , the P (AP) state can be  obtained  from A P (P) state , \ncorrespond ing to magnetization reversal of the CFA free layer. Also, the critical switching \ncurrents  (Ic) in both directions shift in the negative  direction  with the decrease of magnetic  \nfield from 0 to −20 Oe . These results indicate  typical behavior s of STT-induced \nmagnetization switching .  \nSince the STT  switching  by the DC current  is a thermally activated process  [39–41], \nwe use a thermal activation model for switching currents deduced from R-I loops to \nevaluate the intrinsic  critical switching current density ( Jc0) and thermal stability  factor  0 \n(= KuV/kBT) for the MTJ , where  Ku is the uniaxial magnetic anisotropy, V is the volume of \nthe free layer, kB is the Boltzmann constant,  and T is the absolute temperature.  In the \nthermal activation model , the sweep  current I(t) is assumed to increase linearly with time t, \ni.e., I(t) = vt, and the cumulative probability distribution function  P(t) of the switching \ncurrent in Hoffset can be expressed  as \n            (      \n  {   [  (    \n   )]        }),     (3) \nwhere f0 is the effective attempt frequency (=109 Hz), Ic0 is the intrinsic switching current,  \nand v is a constant sweep rate  of the sweep current in the measurement of R-I loops  [41]. \nThe distribution of the critical switching current Ic was obtained by repeating the \nmeasurement of the R-I loops for 300 times.  Figure 6(d) shows typical R-I loops at Hoffset = 14 \n −11 Oe for the polycrystalline CFA -MTJ nanopillar. The mean  critical current s in the \npositive  (Ic,PAP) and negative (Ic,APP) directions  are determined  to be 530 an d 400 μA , \ncorresponding  to the critical current density of 4.3 × 106 (Jc,PAP) and 3.2  × 106 A/cm2 \n(Jc,APP), respectively . Figure s 6 (e) and ( f) show the switching probability for Ic,PAP and \nIc, APP as a function of the sweep current.  Using the  constant sweep rate  v = 1.2 ×  10−4 A/s \nin the measurement of R-I loops , the switching probability  was fitted using  Eq. (3) , as \nshown by the solid lines. As a result, the intrinsic  current density of Jc0,PAP (Jc0,APP) = 9.1 \n× 106 A/cm2 (7.3 × 106 A/cm2) and  thermal stability of 0,PAP (0,APP) = 3 0.0 (28.4) were \nachieved for the polycrystalline  CFA -MTJ  nanopillar .  \nD. Gilbert damping  of the polycrystalline CFA film  \nThe Gilbert damping parameter , , is a critical parameter for  determining  the critical \ncurrent density of STT switching. In order to  examine  for the polycrystalline CFA film , \nwaveguide -based FMR  was performe d in a single ferromagnetic layered sample, consisting \nof SiO 2-substrate//MgO(7.5 nm)/Cr(40 nm)/CFA(2 -18 nm) structure . Typical FMR spectra  \nwith varied external magnetic field s (Hext) for a sample of 18-nm-thick  CFA are shown in \nthe inset of Fig. 7(a). A clear shift in the resonant frequency can be seen as Hext increases \nfrom 500 to 1 500 Oe . The peak intensity is relatively small at low magnetic field , possibly \nbecause of  the anisotropy distribution inside the film.  \nFigure s 7(a) and (b) show  the Hext dependence of the resonant frequency  (f0), \ndemagnetization  field (Hd), and magnetic  damping parameter ( H), respectively,  estimated \nby fitting  each spectrum using an analytical solution  [42]. Here, we assume the 15 \n gyromagnetic ratio  () as 2 × 0.00297 GHz/Oe  and neglect the in-plane magnetic \nanisotropic field . Both Hd and H exhibit a weak dependence on Hext possibly due to the \nanisotropy distribution, which become s saturated at a high magnetic field range.  The f0 was \nfitted using  the simpl ified Kittel formula : \nd ext ext H H H f  \n20\n            (4) \nThen, we obtained Hd = 12154  ± 17.9 Oe, which agree s well  with the values  obtained in the \nindividual resonant spectrum  for Hext > 300 Oe . The magnetic field dependence of  can be  \nexcluded by the  fitting  equation , \n0 exp H H kext H \n,             (5) \nwhere k and H0 are fitting parameters , and   is estimated to be 0.01 48 ± 0.0003 . Figure \n7(c) summarize s CFA thickness  dependence of  the saturation  magnetization , Ms, and \nGilbert damping constant , , of polycrystalline CFA  films . The weak thickness dependence \nof Ms and  indicat es that  the CFA film guarantees  a good quality even in a thin thickness \nregime  at around 2 nm .  \nIV. DISCUSSION  \nUsing polycrystalline  CFA full -Heusler thin films , MTJs with the structure of \nCFA /MgO/Co 75Fe25 were successfully fabricated on a SiO 2 amorphous  substrate . The \neffects of the  MgO buffer on the structural properties of the polycrystalline  CFA films and \nthe TMR in the MTJs were investigated. Optimized (001) -texture, B2 order, and surface 16 \n morphology  of the CFA films was demonstrated  on a 7.5-nm-thick MgO buffer , which \ncontributes a large  TMR ratio in the whole polycrystalline MTJ stacks.  In order to achieve \nSTT switching, Cr underlayer s were  utilized  as a conductive electrode on the optimized \nMgO buffer . The Cr layer was known to have very small l attice mismatch (~0.6%) with \nCFA  and facilitate  the ordering structure of full -Heusler alloys.  A TMR ratio of 175% was \nachieved in the  polycrystallin e CFA -MTJs  on MgO/Cr -buffered Si/SiO 2 substrate.  The \nbuffer layer dependence of the structural properties of polycrystalline CFA films and the \nTMR ratios in entire MTJ stacks are significant for practical spintronic applications of full -\nHeusler alloy materials.  A proper buffer layer with minim al diffusion , and enhanc ed (001) -\ntexture and order ing parameter  of polycrystalline  full-Heusler alloys is required for further \nincreasing the TMR ratio s of the MTJs .  \nSTT switching was performed  in the CFA -MTJs with polycrystalline CFA as a free \nlayer.  The average  intrinsic  current density Jc0 = (Jc0,PAP + Jc0,APP)/2 = 8.2 × 106 A/cm2 \nfor the polycrystalline CFA -MTJ  is generally comparable to  that reported for CoFeB -MTJs  \nwith in -plane magnetization  [43, 44]; however, it is much lower than the value of 2.9 × 107 \nA/cm2 for epitaxial  CFA -MTJs  [26]. Based on Slonczewski ’s model of STT switching  [45, \n46], the simplif ied Jc0, ignoring external magnetic field s, is given by , \n         \n      \n     ,                         (6) \nwhere e is the electron charge,   is reduced Planck’s constant , Ms is the saturation \nmagnetization, t is the thickness of the free layer,    is the spin -transfer efficiency , and Heff \nthe effective field acting on the free layer, including the magnetocrystalline  anisotropy field , 17 \n demagneti zation field, stray field . The polycrystalline CFA free layer ( t = 2.0 nm) is thicker  \nthan the epitaxial one  (t = 1.5 nm). A similar  Ms (~1000 emu/cm3) at RT was observed for \nboth polycrystalline  and epitaxial  CFA films. The   of the polycrystalline CFA -MTJ s \nshould be smaller than that of the epitaxial  CFA -MTJ s because  the TMR ratio  of the \npolycrystalline CFA -MTJ s (43%)  is lower than t hat of the epitaxial  CFA -MTJ s (60%)  [26]. \nAccording ly, the low Jc0 in the polycrystalline CFA -MTJ s can be mainly attributed to the \nsmall   of the polycrystalline CFA free layer  (~0.015)  compar ed with that of the epitaxial  \nCFA films  (~0.04)  [26]. For the polycrystalline CFA -MTJ s, annealing was performed with \na low temperature of 225 °C and a short time of 30 minutes , whereas the epitaxial  CFA -\nMTJs  were annealed at 360 °C for 1 hour ; this could be a factor in the reduction in  [47]. \nAs a result, a low er Jc0 was obtained  in the polycrystalline CFA -MTJ s than that in the \nepitaxial  ones. However , the  value is still greater  than that reported for a 50-nm-thick \nCFA film on a MgO  layer  annealed at 600  C (~0.001 ) [23]; this may be attributed  to the \ninter-diffusion of Cr atoms into the CFA layer  and the residual magnetic moments on the \nCr surface . Consequently, a proper buffer material is strongly required for CFA full -\nHeusler MTJs in order to reduce the  of the free layer  and thus the  Jc0 of STT switching.  \nAnother factor  for reduc ing Jc0 is the magnetic anisotropy of the free layer. In perpendicular \nanisotropy CoFeB/MgO/CoFeB tunnel junction s, a low current density for STT switching  \nwas demonstrated owing to the perpendicular magnetic anisotropy (PMA)  [48-53]. To \nevaluate the contribution of interface PMA, magnetization measurements at in -plane and \nout-of-plane of the polycrystalline CFA film was performed.  An effective anisotropy \nenergy density ( Keff) of −5 × 106 erg/cm3 was obtained  for a 2 -nm-thick  CFA/MgO 18 \n structure  where the negative Keff indicates the CFA layer is in -plane magnetized . In general, \nKeff can be simply expressed by the equation of Keff = Kv + Ki/t, where Kv is the volume \nanisotropy energy density which can  be treated as demagnetization energy density  (2πMs2) \nfor simplicity , Ki is the interface anisotropy energy density , and t is the thickness of the \nCFA layer . As a result, the value of Ki can be calculated to be 0.25 erg/cm2, indicating that \nan interface PMA  is induced at the CFA/MgO interface . In the CFA/MgO -MTJs, the \ninterface PMA  can cause a reduce d effective anisotropy , and may also play a role for the \nreduction in Jc0. For further decreasing Jc0, a chosen buffer material, out -of-plane \nmagnetization  of the CFA free layer  [54–56], and/or advanced  fabricating techniques for \nhigh-quality CFA -MTJs are required.  \nV. CONCLUSION  \nIn conclusion, TMR ratios and STT -induced magnetization switching were studied in \n(001) -textur ed polycrystalline CFA full-Heusler based MTJs  on Si /SiO 2 amorphous \nsubstrate s. CFA films with a good  (001) -texture and high B2 order were achieved on MgO -\nbuffer ed Si/SiO 2 amorphous substrate s. The MgO barrier thickness and RA dependence s of \nthe TMR ratio in the polycrystalline CFA -MTJs  were also studied. Moreover, STT \nswitching was achieved in the MTJs with a thin polycrystalline  CFA film  (2.0 nm) as a free \nlayer.  The Jc0 of 8.2 × 106 A/cm2 was demonstrated for the polycrystalline  CFA-MTJs with \nin-plane magnetization using  a thermal activation model for a cumulative switching \nprobability distribution  with a sweep current . 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(Color online) (a) Out-of-plane ( 2θ-ω scan) XRD patterns for polycrystalline CFA full -\nHeusler alloy films on MgO -buffered Si/SiO 2 amorphous substrates with varied MgO buffer \nthickness, tMgO: 2.5, 5.0, 7.5, and 10.0 nm. (b) Normalized  integrated  intensity of CFA(002) peak , \nand (c) Ratios of (002) to (004) peaks as a function of tMgO for as -deposited  (as-dep.)  samples and \nthose annealed at  400 °C and 480  °C. \n20 40 60 8010210410610810101012Intensity (Log Scale)\n \n2(degree )2.5 nm5.0 nm7.5 nmTa = 400 oC\nssCFA(004)MgO(002)CFA(002)\n10.0 nm\n2 4 6 8 100.00.10.20.30.4\n  I(002)/I(004)\ntMgO (nm) as-dep.\n 400 oC\n 480 oC(a) (b) (c)\n2 4 6 8 100.20.40.60.81.01.2\n  Normalized intensity\ntMgO (nm) as-dep.\n 400 oC\n 480 oCCFA(002)23 \n  \nFIG. 2. (Color online) Surface morphology  of polycrystalline CFA full -Heusler  alloy films on \nMgO -buffered Si/SiO 2 amorphous substrates with respect to  the thickness of the MgO buffer. T he \nCFA films were 30 -nm-thick  and were deposited at RT and post -annealed at 400 °C and 480 °C, \nrespectively. Inset: AFM image of the surface of the CFA film annealed at 400 °C. \n \n \n \nFIG. 3. (Color online) TMR ratios as a function of MgO buffer thickness tMgO for polycrystalline \nCFA /MgO/CoFe MTJs with different thicknesses of the MgO barrier, tbarr: 1.5 nm, 1.8 nm, and 2 .0 \nnm. The 30 -nm-thick CFA bottom electrodes were directly deposited on the MgO buffer layer and \n2 4 6 8 100.00.20.40.60.81.0\nRa = 0.1 nm, P-V = 1.3 nm Ra, 480 C\n P-V, 480 C\n  Ra (nm)\ntMgO (nm) Ra, 400 C\n P-V, 400 C\n0246\nP-V (nm)\n2 4 6 8 1050100150200\n Ta:  400 C, tbarr:  1.5 nm\n Ta:  480 C, tbarr:  1.8 nm\n  TMR (%)\ntMgO (nm) Ta: 400 C,  tbarr: 1.8 nm\n Ta: 400 C,  tbarr: 2.0 nm24 \n post-anneal ed at 400 °C and 480 °C after deposition at RT. The MTJ  stacks  were annealed at 370 °C \nbefore the CIPT measurement.  \n \n \n \nFIG. 4. (Color online) The thickness of MgO barrier, tbarr, dependence of (a) TMR ratios  and (b) RA \nat RT for polycrystalline CFA /MgO/CoFe MTJs with as -deposited, 400 °C and 600 °C annealed \nMgO(7.5 nm)/Cr(40 nm) buffer layers on SiO 2 amorphous substr ates, characterized using CIPT  \nmeasurement . \n \n \n1.2 1.4 1.6 1.8 2.01011021031041051061.2 1.4 1.6 1.8 2.0050100150200\ntbarr (nm)\n  RA (m2)MgO/Cr(40-nm, as-dep.)\nMgO/Cr(40-nm, 400 C)\nMgO/Cr(40-nm, 600 C)\n  TMR (%)\ntbarr (nm) MgO/Cr(40-nm, as-dep.)\n MgO/Cr(40-nm, 400 C) \n MgO/Cr(40-nm, 600 C)SiO2//MgO/Cr/CFA/MgO/CoFe/IrMn(a)\n(b)25 \n  \nFIG. 5. (Color online) RA dependence of TMR ratios for polycrystalline CFA -MTJs with a 2-nm-\nthick  CFA layer (“thin-CFA ”) as a bottom electrode . The squared symbol  indicates the TMR ratio \nfor a polycrystalline CFA -MTJ with a 30-nm-thick CFA layer ( “thick -CFA ”). \n \n \nFIG. 6. (a) Schematic illustration of the structure of polycrystalline  CFA -MTJ nano pillar.  (b) R-H \nloops  for a polycrystalline CFA -MTJ  nanopillar . Wide arrows show the magnetic configuration s of \nbottom (free) and top (reference) electrodes of the MTJ, and narrow arrows indicate sweep direction \nof the applied magnetic field.  (c) R-I loops for the MTJ at  magnetic fields of 0, –11 and –20 Oe, \nrespectivel y. Arrows indicate sweep direction of the applied current.  (d) Representative R-I loops of \nthe CFA -MTJ at applied magnetic  field of –11 Oe. (e) and ( f) Switching  probabilities for Ic,P->AP and \n0 5 10 15 200204060\n  TMR (%)\nRA (m2) \"thin-CFA\"\n \"thick-CFA\"\n26 \n Ic,AP->P obtained by  repeating R-I measurements for 3 00 times.  Solid lines are fitting curve s given by \nEq. ( 3). All measurements were performed at RT.  \n \nFIG. 7. The Hext dependence of (a) resonant frequency f0, (b) demagnetization  field Hd, and  \nmagnetic damping parameter H estimated by fitting the FMR spectr um at each magnetic field  for \na sample of a 18-nm-thick polycrystalline CFA film. Inset of (a) is t ypical FMR spectra at Hext of \n500, 1000, and 1500 Oe. (c) CFA thickness  t dependence of saturation magnetization  Ms and \ndamping constant 0 of the polycrystalline  CFA film.  \n102030\n  Frequency (GHz)\n0 500 1000 1500 20000.000.020.040.06\n  \n\nHex (Oe)51015\nHd (kOe) 8 12 16-20-100\n0.5 kOe 1 kOe\nFrequency (GHz)\n  Real [S21]\n1.5 kOe\n3 6 912 15 180.000.040.08\n  \n\nt (nm)05001000\nMs (emu/cm3)(b)\n(c)(a)"    },    {        "title": "1901.01941v1.Giant_anisotropy_of_Gilbert_damping_in_epitaxial_CoFe_films.pdf",        "content": "Giant anisotropy of Gilbert damping in epitaxial CoFe \flms\nYi Li,1, 2Fanlong Zeng,3Steven S.-L. Zhang,2Hyeondeok Shin,4Hilal Saglam,2, 5Vedat Karakas,2, 6Ozhan\nOzatay,2, 6John E. Pearson,2Olle G. Heinonen,2Yizheng Wu,3, 7,\u0003Axel Ho\u000bmann,2,yand Wei Zhang1, 2,z\n1Department of Physics, Oakland University, Rochester, MI 48309, USA\n2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n3State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China\n4Computational Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA\n5Department of Physics, Illinois Institute of Technology, Chicago IL 60616, USA\n6Department of Physics, Bogazici University, Bebek 34342, Istanbul, Turkey\n7Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China\n(Dated: January 8, 2019)\nTailoring Gilbert damping of metallic ferromagnetic thin \flms is one of the central interests in\nspintronics applications. Here we report a giant Gilbert damping anisotropy in epitaxial Co 50Fe50\nthin \flm with a maximum-minimum damping ratio of 400 %, determined by broadband spin-torque\nas well as inductive ferromagnetic resonance. We conclude that the origin of this damping anisotropy\nis the variation of the spin orbit coupling for di\u000berent magnetization orientations in the cubic lattice,\nwhich is further corroborate from the magnitude of the anisotropic magnetoresistance in Co 50Fe50.\nIn magnetization dynamics the energy relaxation rate\nis quanti\fed by the phenomenological Gilbert damping\nin the Landau-Lifshits-Gilbert equation [1], which is a\nkey parameter for emerging spintronics applications [2{\n6]. Being able to design and control the Gilbert damp-\ning on demand is crucial for versatile spintronic device\nengineering and optimization. For example, lower damp-\ning enables more energy-e\u000ecient excitations, while larger\ndamping allows faster relaxation to equilibrium and more\nfavorable latency. Nevertheless, despite abundant ap-\nproaches including interfacial damping enhancement [7{\n9], size e\u000bect [10, 11] and materials engineering [12{14],\nthere hasn't been much progress on how to manipulate\ndamping within the same magnetic device. The only\nwell-studied damping manipulation is by spin torque [15{\n18], which can even fully compensate the intrinsic damp-\ning [19, 20]. However the requirement of large current\ndensity narrows its applied potential.\nAn alternative approach is to explore the intrinsic\nGilbert damping anisotropy associated with the crys-\ntalline symmetry, where the damping can be continu-\nously tuned via rotating the magnetization orientation.\nAlthough there are many theoretical predictions [21{25],\nmost early studies of damping anisotropy are disguised\nby two-magnon scattering and linewidth broadening due\nto \feld-magnetization misalignment [26{29]. In addition,\nthose reported e\u000bects are usually too weak to be consid-\nered in practical applications [30, 31].\nIn this work, we show that a metallic ferromagnet can\nexhibit a giant Gilbert damping variation by a factor\nof four along with low minimum damping. We inves-\ntigated epitaxial cobalt-iron alloys, which have demon-\nstrated new potentials in spintronics due to their ultralow\ndampings [32, 33]. Using spin-torque-driven and induc-\ntive ferromagnetic resonance (FMR), we obtain a four-\nfold (cubic) damping anisotropy of 400% in Co 50Fe50thin\n\flms between their easy and hard axes. For each angle,the full-range frequency dependence of FMR linewidths\ncan be well reproduced by a single damping parame-\nter\u000b. Furthermore, from \frst-principle calculations and\ntemperature-dependent measurements, we argue that\nthis giant damping anisotropy in Co 50Fe50is due to the\nvariation of the spin-orbit coupling (SOC) in the cu-\nbic lattice, which di\u000bers from the anisotropic density of\nstate found in ultrathin Fe \flm [30]. We support our\nconclusion by comparing the Gilbert damping with the\nanisotropic magnetoresistance (AMR) signals. Our re-\nsults reveal the key mechanism to engineer the Gilbert\ndamping and may open a new pathway to develop novel\nfunctionality in spintronic devices.\nCo50Fe50(CoFe) \flms were deposited on MgO(100)\nsubstrates by molecular beam epitaxy at room temper-\nature, under a base pressure of 2 \u000210\u000010Torr [34]. For\nspin-torque FMR measurements, i) CoFe(10 nm)/Pt(6\nnm) and ii) CoFe(10 nm) samples were prepared. They\nwere fabricated into 10 \u0016m\u000240\u0016m bars by photolithog-\nraphy and ion milling. Coplanar waveguides with 100-\nnm thick Au were subsequently fabricated [18, 35]. For\neach layer structure, 14 devices with di\u000berent orienta-\ntions were fabricated, as shown in Fig. 1(a). The geom-\netry de\fnes the orientation of the microwave current, \u0012I,\nand the orientation of the biasing \feld, \u0012H, with respect\nto the MgO [100] axis (CoFe [1 10]).\u0012Iranges from 0\u000e\nto 180\u000ewith a step of 15\u000e(D1 to D14, with D7 and D8\npointing to the same direction). For each device we \fx\n\u0012H=\u0012I+ 45\u000efor maximal recti\fcation signals. In addi-\ntion, we also prepared iii) CoFe(20 nm) 40 \u0016m\u0002200\u0016m\nbars along di\u000berent orientations with transmission copla-\nnar waveguides fabricated on top for vector network an-\nalyzer (VNA) measurements. See the Supplemental Ma-\nterials for details [36].\nFig. 1(b) shows the angular-dependent spin-torque\nFMR lineshapes of CoFe(10 nm)/Pt devices from dif-\nferent samples (D1 to D4, hard axis to easy axis) atarXiv:1901.01941v1  [cond-mat.mtrl-sci]  7 Jan 20192\nFIG. 1. (a) Upper: crystalline structure, axes of bcc Co 50Fe50\n\flm on MgO(100) substrate and de\fnition of \u0012Hand\u0012I.\nLower: device orientation with respect to the CoFe crystal\naxis. (b) Spin-torque FMR lineshapes of i) CoFe(10 nm)/Pt\ndevices D1 to D4 measured. (c) Resonances of D1 and D4\nfrom (b) for \u00160Hres<0. (d) Resonances of iii) CoFe(20\nnm) for\u0012H= 45\u000eand 90\u000emeasured by VNA FMR. In (b-d)\n!=2\u0019= 20 GHz and o\u000bset applies.\n!=2\u0019= 20 GHz. A strong magnetocrystalline anisotropy\nas well as a variation of resonance signals are observed.\nMoreover, the linewidth increases signi\fcantly from easy\naxis to hard axis, which is shown in Fig. 1(c). We have\nalso conducted rotating-\feld measurements on a sec-\nond CoFe(10 nm)/Pt device from a di\u000berent deposition\nand the observations can be reproduced. This linewidth\nanisotropy is even more pronounced for the CoFe(20 nm)\ndevices without Pt, measured by VNA FMR (Fig. 1d).\nFor the CoFe(10 nm) devices, due to the absence of the\nPt spin injector the spin-torque FMR signals are much\nweaker than CoFe/Pt and completely vanish when the\nmicrowave current is along the easy axes.\nFigs. 2(a-b) show the angular and frequency de-\npendence of the resonance \feld Hres. In Fig. 2(a), the\nHresfor all four sample series match with each other,\nwhich demonstrates that the magnetocrystalline proper-\nties of CoFe(10 nm) samples are reproducible. A slightly\nsmallerHresfor CoFe(20 nm) is caused by a greater e\u000bec-\ntive magnetization when the thickness increases. A clear\nfourfold symmetry is observed, which is indicative of the\ncubic lattice due to the body-center-cubic (bcc) texture\nof Co 50Fe50on MgO. We note that the directions of the\nhard axes has switched from [100] and [010] in iron-rich\nalloys [33] to [110] and [1 10] in Co 50Fe50, which is con-\nω/2πμ0Hres  (T) μ0Hres  (T) [110]\n[110][100][010](a) (b) CoFe(10 nm)/Pt \nω/2π=2045o90 o135o\n135o180o 225oCoFe(10 nm)/Pt \nCoFe(10 nm) CoFe(20 nm) θH:\n[100]\n[110][010]FIG. 2. (a) Resonance \feld \u00160Hresas a function of \u0012Hat\n!=2\u0019= 20 GHz for di\u000berent samples. Diamonds denote the\nrotating-\feld measurement from the second CoFe(10 nm)/Pt\ndevice. The black curve denotes the theoretical prediction.\n(b)\u00160Hresas a function of frequency for the CoFe(10 nm)/Pt\ndevices. Solid curves denote the \fts to the Kittel equation.\nsistent with previous reports [37, 38].\nThe magnetocrystalline anisotropy can be quanti-\n\fed from the frequency dependence of \u00160Hres. Fig.\n2(b) shows the results of CoFe(10 nm)/Pt when HB\nis aligned to the easy and hard axes. A small uniax-\nial anisotropy is found between [1 10] (0\u000eand 180\u000e) and\n[110] (90\u000e) axes. By \ftting the data to the Kittel equa-\ntion!2=\r2=\u00162\n0(Hres\u0000Hk)(Hres\u0000Hk+Ms), where\n\r= 2\u0019(geff=2)\u000128 GHz/T, we obtain geff= 2:16,\n\u00160Ms= 2:47 T,\u00160H[100]\nk= 40 mT,\u00160H[010]\nk= 65 mT\nand\u00160H[110]\nk=\u00160H[110]\nk=\u000043 mT. Taking the disper-\nsion functions from cubic magnetocrystalline anisotropy\n[39, 40], we obtain an in-plane cubic anisotropy \feld\n\u00160H4jj= 48 mT and a uniaxial anisotropy \feld \u00160H2jj=\n12 mT. Fig. 2(a) shows the theoretical predictions from\nH4jjandH2jjin black curve, which aligns well with all\n10-nm CoFe samples.\nWith good magnetocrystalline properties, we now turn\nto the energy relaxation rate. Fig. 3(a) shows the full-\nwidth-half-maximum linewidths \u00160\u0001H1=2of the spin-\ntorque FMR signals at !=2\u0019= 20 GHz. Again, a fourfold\nsymmetry is observed for CoFe(10 nm)/Pt and CoFe(10\nnm), with the minimal (maximal) linewidth measured\nwhen the \feld lies along the easy (hard) axes. For\nCoFe(10 nm) devices, we did not measure any spin-torque\nFMR signal for HBalong the hard axes ( \u0012H= 45\u000e, 135\u000e\nand 225\u000e). This is due to the absence of the Pt spin\ninjector as well as the near-zero AMR ratio when the rf\ncurrent \rows along the easy axes, which will be discussed\nlater. For all other measurements, the linewidths of CoFe\ndevices are smaller than for CoFe/Pt by the same con-\nstant, independent of orientation (upper diagram of Fig.\n3a). This constant linewidth di\u000berence is due to the spin\npumping contribution to damping from the additional Pt\nlayer [41, 42]. Thus we can deduce the intrinsic damp-\ning anisotropy from CoFe(10 nm)/Pt devices, with the3\nω/2π 105, 195 deg 75, 165 deg 120, 210 deg 135, 225 deg(HA) 45, 135 deg (HA) \n60, 150 deg \n90, 180 deg(EA) θHCoFe(10 nm)/Pt \n(b) = -\n=-\n[100] [110] [110] [010](a) ω/2π=20 \nω/2π θH\n0, 90 deg 15, 75 deg 22.5, 67.5 deg 30, 60 deg 42.5, 50 deg \n40, 52.5 deg \n37.5, 55 deg CoFe(20 nm) (VNA) \n(c)CoFe(10 nm)/Pt CoFe(10 nm) \n90 deg (EA) \nfor CoFe \nFIG. 3. (a) \u00160\u0001H1=2as a function of \u0012Hat!=2\u0019= 20 GHz\nfor the CoFe(10 nm) series in Fig. 2(a). Top: Addtional\nlinewidth due to spin pumping of Pt. The green region de-\nnotes the additional linewidth as 4 :5\u00060:7 mT. (b-c) \u00160\u0001H1=2\nas a function of frequency for (b) CoFe(10 nm)/Pt and (c)\nCoFe(20 nm) samples. Solid lines and curves are the \fts to\nthe data.\ndamping shifted from CoFe(10 nm) devices by a constant\nand is much easier to measure.\nIn Fig. 3(b-c) we show the frequency dependence of\n\u00160\u0001H1=2of CoFe(10 nm)/Pt devices from spin-torque\nFMR and CoFe(20 nm) devices from VNA FMR. For\nboth the easy and hard axes, linear relations are ob-\ntained, and the Gilbert damping \u000bcan be extracted\nfrom\u00160\u0001H1=2=\u00160\u0001H0+ 2\u000b!=\r with the \fts shown\nas solid lines. Here \u00160\u0001H0is the inhomogeneous broad-\nening due to the disorders in lattice structures. In Fig.\n3(b) we also show the linewidths of the CoFe(10 nm)\ndevice along the easy axis ( \u0012H= 90\u000e), which has a\nsigni\fcant lower linewidth slope than the easy axis of\nCoFe(10 nm)/Pt. Their di\u000berences yield a spin pump-\ning damping contribution of \u0001 \u000bsp= 0:0024. By using\n\u0001\u000bsp=\r\u0016hg\"#=(4\u0019MstM), we obtain a spin mixing con-\nductance of g\"#(CoFe/Pt) = 25 nm\u00002, which is compa-\nrable to similar interfaces such as NiFe/Pt [43, 44]. For\n\u0012Hbetween the easy and hard axes, the low-frequency\nlinewidth broadenings are caused by the deviation of\nmagnetization from the biasing \feld direction, whereas\nat high frequencies the \feld is su\u000ecient to saturate the\nmagnetization. In order to \fnd the damping anisotropy,\nwe \ft the linewidths to the angular model developed bySuhl [45, 46], using a single \ft parameter of \u000band the\nextractedH2jjandH4jjfrom Fig. 2. The solid \ftting\ncurves in Fig. 3(b) nicely reproduce the experimental\npoints.\nThe obtained damping anisotropy for all the samples\nare summarized in Fig. 4, which is the main result of\nthe paper. For CoFe(10 nm)/Pt samples, \u000bvaries from\n0.0056 along the easy axis to 0.0146 along the hard axis.\nBy subtracting the spin pumping \u0001 \u000bspfrom both values,\nwe derive a damping anisotropy of 380%. For CoFe(20\nnm) samples measured by VNA FMR, \u000bvaries from\n0.0054 to 0.0240, which yields an anisotropy of 440% and\nreproduces the large anisotropy from spin-torque FMR.\nThis giant damping anisotropy implies, technologically,\nnearly four times smaller critical current to switch the\nmagnetization in a spin-torque magnetic random access\nmemory, or to excite auto-oscillation in a spin-torque os-\ncillator, by simply changing the magnetization orienta-\ntion from the hard axis to the easy axis within the same\ndevice. In addition, we emphasize that our reported\ndamping anisotropy is not subject to a dominant two-\nmagnon scattering contribution, which would be mani-\nfested as a nonlinear linewidth softening at high frequen-\ncies [28, 31]. For this purpose we have extended the fre-\nquency of spin-torque FMR on CoFe(10 nm)/Pt up to 39\nGHz, see the Supplemental Materials for details [36]. We\nchoose CoFe(10 nm)/Pt samples because they provide\nthe best signals at high frequencies and the additional Pt\nlayer signi\fcantly helps to excite the dynamics. Linear\nfrequency dependence of linewidth persists throughout\nthe frequency range and \u0001 H0is unchanged for the two\naxes, with which we can exclude extrinsic e\u000bects to the\nlinewidths. We also note that our result is substantially\ndi\u000berent from the recent report on damping anisotropy\nin Fe/GaAs [30], which is due to the interfacial SOC and\ndisappears quickly as Fe becomes thicker. In compari-\nson, the Gilbert damping anisotropy in Co 50Fe50is the\nintrinsic property of the material, is bonded to its bulk\ncrystalline structure, and thus holds for di\u000berent thick-\nnesses in our experiments.\nIn order to investigate the dominant mechanism for\nsuch a large Gilbert damping anisotropy, we perform\ntemperature-dependent measurements of \u000band the re-\nsistivity\u001a. Fig. 5(a) plots \u000bas a function of 1 =\u001afor\nthe CoFe(10 nm)/Pt and CoFe(20 nm) samples and for\nHBalong the easy and hard axes. The dominant lin-\near dependence reveals a major role of conductivitylike\ndamping behavior. This is described by the breathing\nFermi surface model for transition-metal ferromagnets,\nin which\u000bcan be expressed as [23, 24, 47{49]:\n\u000b\u0018N(EF)j\u0000\u0000j2\u001c (1)\nwhereN(EF) is the density of state at the Fermi level, \u001c\nis the electron relaxation time and \u0000\u0000=h[\u001b\u0000;Hso]iE=EF\nis the matrix for spin-\rip scatterings induced by the SOC\nHamiltonian Hsonear the Fermi surface [48, 49]. Here \u001c4\n(b) CoFe(10 nm) CoFe(20 nm) CoFe(20 nm) CoFe(10 nm)/Pt - ∆α sp \nCoFe(10 nm ) 400 %\n100 %\nFIG. 4. Renormalized damping and its anisotropy for\nCoFe(10 nm) and CoFe(20 nm), measured from spin-torque\nFMR and VNA FMR, respectively. For CoFe(20 nm)/Pt sam-\nples, \u0001\u000bsphas been subtracted from the measured damping.\nis proportional to the conductivity (1 =\u001a) from the Drude\nmodel, with which Eq. (1) gives rise to the behaviors\nshown in Fig. 5(a).\nFor the origin of damping anisotropy, we \frst check\nthe role of N(EF) by ab-initio calculations for di\u000berent\nordered cubic supercells, which is shown in the Supple-\nmental Materials [36]. However, a negligible anisotropy\ninN(EF) is found for di\u000berent magnetization orienta-\ntions. This is consistent with the calculated anisotropy\nin Ref. [30], where less than 0.4% change of N(EF) was\nobtained in ultrathin Fe \flms. The role of \u001ccan also be\nexcluded from the fact that the resistivity di\u000berence be-\ntween the easy and hard axes is less than 2% [36]. Thus\nwe deduce that the giant damping anisotropy of 400% is\ndue to the change of j\u0000\u0000j2, or the SOC, at di\u000berent crys-\ntalline directions. In particular, unlike the single element\nFe, disordered bcc Fe-Co alloy can possess atomic short-\nrange order, which gives rise to local tetragonal crystal\ndistortions due to the di\u000berent lattice constants of Fe and\nCo [2{4]. Such local tetragonal distortions will preserve\nglobal cubic symmetry but can have large e\u000bects on the\nSOC. We emphasize that our CoFe samples, which did\nnot experience annealing, preserve the random disorder.\nOur \frst principle calculations also con\frm the role of lo-\ncal tetragonal distortions and its enhancement on SOC,\nsee the Supplemental Materials for details [36].\nThe anisotropy of the SOC in Co 50Fe50can be re\rected\nby its AMR variation along di\u000berent crystalline orienta-\ntions. The AMR ratio can be de\fned as:\nAMR(\u0012I) =\u001ak(\u0012I)\n\u001a?(\u0012I)\u00001 (2)\nwhere\u001ak(\u0012I) and\u001a?(\u0012I) are measured for the biasing\n\feld parallel and perpendicular to the current direction,\nrespectively. The main contribution of AMR is the asym-\nmetrics-delectron scatterings where the s-orbitals are\nmixed with magnetization-containing d-orbitals due toSOC [53, 54]. Since both the damping and AMR origi-\nnate from SOC and, more precisely, are proportional to\nthe second order of SOC, a large damping anisotropy is\nexpected to be accompanied by a large AMR anisotropy\nand vice versa. Furthermore, due to the fourfold sym-\nmetry, the AMR should be invariant when the current\ndirection is rotated by 90 degrees, as the AMR is a func-\ntion of\u0012Ias de\fned in Eq. (1). Thus the damping and\nAMR should exhibit similar angular dependence on \u0012H\nand\u0012I, respectively.\nIn Fig. 5(b) we compare renormalized \u000b(\u0012H) with\nCoFe(20 nm) CoFe(10 nm)/Pt : (a)\n300 K 8 K F(θI)/F max (b) \n,10 nm \n20 nm 10 nm \n20 nm \nFIG. 5. (a) \u000b(T) as a function of 1 =\u001a(T).T= 8 K, 30 K, 70\nK, 150 K and 300 K for CoFe(10 nm)/Pt and T= 8 K and\n300 K for CoFe(20 nm). Dashed and dotted lines are guides\nto eyes. (b) Renormalized \u000b(\u0012H) and AMR( \u0012I) andF(\u0012I) for\nCoFe(10 nm)/Pt and CoFe(20 nm). Circles, crosses and +\ndenote\u000b, AMR and F, respectively.\nAMR(\u0012I) for 10-nm and 20-nm CoFe samples, where the\nAMR values are measured from Hall bars with di\u000berent\n\u0012I. The AMR ratio is maximized along h100iaxes and\nminimized alongh110iaxes, with a large anisotropy by a\nfactor of 10. This anisotropy is also shown by the inte-\ngrated spin-torque FMR intensity for CoFe(10 nm)/Pt,\nde\fned asF(\u0012I) = \u0001H1=2Vmax\ndc [17, 18] and plotted in\nFig. 5(b). The large AMR anisotropy and its symme-\ntry clearly coincide with the damping anisotropy mea-\nsured in the same samples, which con\frms our hypoth-\nesis of strong SOC anisotropy in CoFe. Thus we con-\nclude that the damping anisotropy is dominated by the\nvariation of SOC term in Eq. (1). In parallel, we also\ncompare\u000b(\u0012H) and AMR( \u0012I) for epitaxial Fe(10 nm)\nsamples grown on GaAs substrates [36]. Experimentally\nwe \fnd the anisotropy less is than 30% for both damping\nand AMR, which helps to explain the presence of weak\ndamping anisotropy in epitaxial Fe [30].5\nWe compare our results with prior theoretical works on\ndamping anisotropy [23, 24]. First, despite their propor-\ntional relationship in Fig. 5(a), the giant anisotropy in\n\u000bis not re\rected in 1 =\u001a. This is because the s-dscatter-\ning, which dominates in the anisotropic AMR, only con-\ntributes a small portion to the total resistivity. Second,\nneither the anisotropy of damping nor AMR are sensitive\nto temperature. This is likely because the thermal excita-\ntions at room temperature ( \u00180:025 eV) are much smaller\nthan the spin-orbit coupling ( \u00180:1 eV [47]). Third, the\ndamping tensor has been expressed as a function of M\nanddM=dt[24]. However in a fourfold-symmetry lat-\ntice and considering the large precession ellipticity, these\ntwo vectors are mostly perpendicular to each other, point\ntowards equivalent crystalline directions, and contribute\nequivalently to the symmetry of damping anisotropy.\nIn summary, we have experimentally demonstrated\nvery large Gilbert damping anisotropy up to 400% in\nepitaxial Co 50Fe50thin \flms which is due to their bulk,\ncubic crystalline anisotropy. We show that the damping\nanisotropy can be explained by the change of spin-orbit\ncoupling within the breathing Fermi surface model, which\ncan be probed by the corresponding AMR change. Our\nresults provide new insights to the damping mechanism\nin metallic ferromagnets, which are important for opti-\nmizing dynamic properties of future magnetic devices.\nWe are grateful for fruitful discussions with Bret Hein-\nrich. W.Z. acknowledges supports from the U.S. Na-\ntional Science Foundation under Grants DMR-1808892,\nMichigan Space Grant Consortium and DOE Visit-\ning Faculty Program. Work at Argonne, including\ntransport measurements and theoretical modeling, was\nsupported by the U.S. Department of Energy, Of-\n\fce of Science, Materials Science and Engineering Di-\nvision. Work at Fudan, including thin \flm growth\nand fabrication, was supported by the Nat'l Key Ba-\nsic Research Program (2015CB921401), Nat'l Key Re-\nsearch and Development Program (2016YFA0300703),\nNSFC (11734006,11474066,11434003), and the Program\nof Shanghai Academic Research Leader (17XD1400400)\nof China. O.O. and V.K. acknowledge supports\nfrom Bogazici University Research Fund (17B03D3),\nTUBITAK 2214/A and U.S. Department of State Ful-\nbright Visiting Scholar Program.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] S. 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B 10, 4626 (1974).7\nSupplemental Materials:Giant anisotropy of Gilbert damping in epi-\ntaxial CoFe \flms\nbyYi Li, Fanlong Zeng, Steven S.-L. Zhang, Hyeondeok Shin, Hilal Saglam, Vedat Karakas, Ozhan Ozatay, John E.\nPearson, Olle G. Heinonen, Yizheng Wu, Axel Ho\u000bmann and Wei Zhang\nCrystallographic quality of Co 50Fe50\flms\nFIG. S-1. Crystallographic characterization results of CoFe \flms. (a) RHEED pattern of the CoFe(10 nm) \flm. (b) XRD\nof the CoFe(10 nm) and (20 nm) \flms. (c) X-ray re\rectometry measured for the CoFe(20 nm) \flm. (d) AFM scans of the\nCoFe(20 nm) \flm. (e) Rocking curves of the CoFe(20 nm) \flm for [100] and [110] rotating axes.\nFig. S-1 shows the crystallographic characterization for the epitaxial CoFe samples. Re\rection high-energy electron\ndi\u000braction (RHEED) shows very clear and sharp patterns which shows high quality of the epitaxal \flms. X-ray\ndi\u000braction (XRD) yields clear CoFe(002) peaks at 2 \u0012= 66:5\u000e. X-ray re\rectometry scan of the CoFe (20 nm) \flm\nshows a good periodic pattern and the \ft gives a total thickness of 19.84 nm. Atomic-force microscopy (AFM) scans\nfor 10\u0016m\u000210\u0016m and 100 nm\u0002100 nm scales show smooth surface with a roughness of 0.1 nm. Lastly XRD rocking\ncurves for [100] and [110] rotating axes show a consistent linewidth of 1.45\u000e, which indicates isotropic mosaicity of\nthe CoFe \flms.\nAs a result of the crystallographic characterizations, we believe our MBE-grown CoFe samples are epitaxial, have\nsmooth surfaces and exhibit excellent crystalline quality. Moreover, we can exclude the source of inhomogeneity\nfrom misorientation of crystallities (mosaicity) due to isotropic rocking curves. This means the inhomogeneous FMR\nlinewidth broadening is isotropic, as is consistent with the experiments.\nDevice geometries for Spin-torque FMR and VNA FMR measurements.\nFig. S-2 shows the device geometry for Spin-torque FMR and VNA FMR measurements. For spin-torque FMR,\nwe have prepared CoFe(10 nm)/Pt, CoFe(10 nm) and Fe(10 nm) devices. A second CoFe(10 nm)/Pt sample is also\nprepared for rotating-\feld measurements. For VNA FMR, we have prepared CoFe(20 nm) samples. All the CoFe\n\flms are grown on MgO(100) substrates; the Fe \flm is grown on a GaAs(100) substrate. Au (100 nm) coplanar\nwaveguides are subsequently fabricated on top of all devices. For VNA FMR samples, an additional SiO 2(100 nm) is8\nFIG. S-2. (a) Spin-torque FMR devices of CoFe(10 nm)/Pt samples. (b) Illustration of the Spin-torque FMR circuit. (c) Front\nand (d) back view of the VNA FMR devices for CoFe(20 nm) samples.\ndeposited between CoFe and Au for electric isolation. The CoFe(20 nm) bars is only visible from the back view in\nFig. S-2(d).\nSpin-torque FMR lineshapes\nFigure S-3 shows the full lineshapes of (a) CoFe(10 nm)/Pt(6 nm), (b) CoFe(10 nm) and (c) Fe(10 nm) devices\nmeasured at !=2\u0019= 20 GHz. The Fe \flms were deposited on GaAs substrates by MBE growth. (a) and (b) are used\nto extract the resonance \felds and linewidths in Figs. 2(a) and 3(a) of the main text. (c) is used to examine the\ncorrelation between damping anisotropy and AMR anisotropy.\nSpin-torque FMR linewidths as a function of frequency for CoFe(10 nm) devices.\nFigure S-4(a) shows the spin-torque FMR linewidths for CoFe(10 nm) devices. Because there is no spin torque\ninjection from Pt layer, the FMR signals are much weaker than CoFe(10 nm)/Pt and the extracted linewidths are\nmore noisy. The excitation of the dynamics is due to the magnon charge pumping e\u000bect [1] or inhomogeneities of the\nOersted \felds. No signal is measured for the rf current \rowing along the easy axis (magnetic \feld along the hard\naxis, see Fig. S-3b), because of the negligible AMR ratio.\nFigure S-4(b) shows the angular dependence of the extracted Gilbert damping for CoFe(10 nm)/Pt and CoFe(10\nnm). The former is extracted from Fig. 3(b) of the main text. The latter is extracted from Fig. S-4(a). The blue\ndata points for CoFe(10 nm)/Pt are obtained from the resonances at negative biasing \felds. Those data are used in\nFig. 4 of the main text.9\nFIG. S-3. Spin-torque FMR lineshapes of (a) CoFe(10 nm)/Pt, (b) CoFe(10 nm) and (c) Fe(10 nm) devices measured at\n!=2\u0019= 20 GHz. \u0012H\u0000\u0012Iis \fxed to 45\u000e.\nFIG. S-4. (a) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm) devices. Solid lines and curves are the \fts to the experiments.\n\u0012H\u0000\u0012Iis \fxed to 45\u000e. (b)\u000bas a function of \u0012Hfor CoFe(10 nm)/Pt and CoFe(10 nm) devices.\nSpin-torque FMR for CoFe(10 nm)/Pt up to 39 GHz.\nFig. S-5 shows the spin-torque FMR lineshapes and linewidths up to 39 GHz for CoFe(10 nm)/Pt devices along\nthe easy and hard axes ( \u0012H= 90\u000eand45\u000e). At!=2\u0019= 32:1 GHz (Fig. S-5a), the spin-torque FMR amplitude is\n0.1\u0016V for the easy axis and 0.02 \u0016V for the hard axis. 10 seconds of time constant is used to obtained the signals.\nThroughout the frequency range, linewidths demonstrate good linear dependence on frequency as shown Fig. S-5(b).\nFor the hard axis the signal has reached the noise bottom limit at 32.1 GHz. For the easy axis the noise bottom limit\nis reached at 39 GHz. The two linear \fts yield \u000b= 0:0063 and\u00160\u0001H0= 1:8 mT for the easy axis and \u000b= 0:00153\nand\u00160\u0001H0= 1:5 mT for the hard axis. The two damping parameters are close to the values obtained below 20 GHz\nin the main text. Also the inhomogeneous linewidth \u00160\u0001H0nicely match between easy and hard axes.10\nFIG. S-5. High-frequency ST-FMR measurement of i) CoFe(10 nm)/Pt for the biasing \feld along the easy axis ( \u0012H= 90\u000e) and\nhard axis (\u0012H= 45\u000e). Left: lineshapes of ST-FMR at !=2\u0019= 32:1 GHz. Right: linewidth as a function of frequency. Lines\nare linear \fts to the data by setting both \u000band \u0001H0as free parameters.\nLow-temperature FMR linewidths and dampings for CoFe(10 nm)/Pt and CoFe(20 nm).\nFIG. S-6. (a-b) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm)/Pt devices at di\u000berent temperatures. (c) Extracted\ndamping at di\u000berent temperatures, same as in Fig. 4 of the main text.\nFigure S-6 shows the frequency dependence of linewidths for extracting temperature-dependent Gilbert damping\nin Fig. 5(a) of the main text.\nFor CoFe(10 nm)/Pt samples, we plot both \u000band resistivity \u001ameasured at di\u000berent temperatures in Fig. S-6(c).\nThe measurements of \u001awere conducted with a biasing magnetic \feld of 1 Tesla parallel to the current direction, so\nthat the AMR in\ruence is excluded. Also the resistivity variation between the easy and hard axes is very small, about\n1%, which is much smaller than the damping anisotropy.\nWe have also conducted the low-temperature VNA FMR of the new CoFe(20 nm) samples at 8 K, in addition to\nthe room-temperature measurements. The linewidths data are shown in Fig. S-6(d) for both easy and hard axes.\nThe extracted damping are: \u000b= 0:0054 (EA, 300 K), 0.0061 (EA, 8 K), 0.0240 (HA, 300 K) and 0.0329 (HA, 8 K).\nThose values are used in Fig. 4(b) and Fig. 5(a) of the main text.\nFor CoFe(10 nm) the damping anisotropy decreases from 380 % at 300 K to 273 % at 30 K by taking out the spin\npumping damping enhancement (an unexpected reduction of alpha happens at 8 K for the hard axis). For CoFe(2011\nnm) the damping anisotropy increases from 440 % at 300 K to 540 % at 8 K. Thus a clear variation trend of damping\nanisotropy in CoFe \flms remains to be explored.\nFirst-principle calculation of N(EF)anisotropy for Co 50Fe50\nFIG. S-7. Density of states as a function of energy. EFis the Fermi level.\nFirst-principle calculations were done using QUANTUM ESPRESSO for a cubic lattice of Co 50Fe50of CsCl, Zintl\nand random alloy structures. Supercells consisting of 4 \u00024\u00024 unit cells were considered with a total of 128 atoms (64\ncobalt and 64 iron atoms). The calculations were done using plane-wave basis set with a 180 Ry kinetic energy cut-o\u000b\nand 1440 Ry density cut-o\u000b. For both Co and Fe atoms, fully relativistic PAW pseudopotentials were used. Figure\nS-7 shows the density of states (DOS) of the CsCl form for di\u000berent magnetization orientations \u0012in thexy-plane.\nClearly, DOS exhibits no anisotropy ( <0:1% variation at E=EF). No anisotropy was found in the Zintl form, either.\nThus, we conclude that the Gilbert damping anisotropy in Co 50Fe50cannot be caused by a variation of N(EF) with\nrespect to magnetization direction in ideal ordered structures.\nSOC induced by atomic short-range order (ASRO)\nIn our experiment, because the Co 50Fe50\flms were grown by MBE at low temperatures, they do not form the\nordered bcc B2 structure but instead exhibit compositional disorder. Transition metal alloys such as CoPt, NiFe, and\nCoFe tend to exhibit ASRO [2{4]. The ASRO in CoFe is likely to give rise to local tetragonal distortions because of the\ndi\u000berent lattice constants of bcc Fe and (metastable) bcc Co at 2.856 \u0017A and 2.82 \u0017A, respectively. Such local tetragonal\ndistortions will preserve global cubic (or four-fold in-plane) symmetry, but can have large e\u000bects on the SOC, with\nconcomitant e\u000bect on spin-orbit induced magnetization damping. For example, \frst-principle calculations using the\ncoherent-potential approximation for the substitutionally disordered system shows that a tetragonal distortion of 10%\nin the ratio of the tetragonal axes aandcgives rise to an magnetocrystalline anisotropy energy (MAE) density [2, 3]\nof about 1 MJ/m3. These results are consistent with our observed MAE in Co 50Fe50.\nTo con\frm this mechanism, we performed DFT-LDA calculations on 50:50 CoFe supercells consisting of a total\nof 16 atoms for CsCl, zintl, and random alloy structures; in the random alloy supercell, Co or Fe atoms randomly\noccupied the atomic positions in the supercell. Note that all CoFe geometries are fully relaxed, including supercell\nlattice vectors.\n1. Structural relaxation including spin-orbit coupling (SOC) shows local tetragonal distortions for random alloy\nsupercell. Among the three di\u000berent CoFe phases, tetragonal c/a ratio for the supercell in optimized geometry\nis largest (1.003) in the random alloy supercell with SOC, which means local tetragonal distortions are more12\nFIG. S-8. Density of states (DOS) for (a) CsCl, (b) Zintl, and (c) alloy form of CoFe with SOC (black solid) and without SOC\n(red solid).\nTABLE I. Relaxed atomic positions (including SOC) of the alloy structure. In the ideal CsCl or Zintl structures, the atomic\npositions are all multiples of 0.25 in units of the lattice vector components.\nAtom x-position y-position z-position\nCo 0.003783083 0.000000000 0.000000000\nFe -0.001339230 0.000000000 0.500000000\nFe -0.002327721 0.500000000 0.000000000\nFe 0.002079922 0.500000000 0.500000000\nFe 0.502327721 0.000000000 0.000000000\nFe 0.497920078 0.000000000 0.500000000\nCo 0.496216917 0.500000000 0.000000000\nFe 0.501339230 0.500000000 0.500000000\nCo 0.250000000 0.250000000 0.254117992\nFe 0.250000000 0.250000000 0.752628048\nFe 0.250000000 0.750000000 0.247371952\nCo 0.250000000 0.750000000 0.745882008\nCo 0.750000000 0.250000000 0.250415490\nCo 0.750000000 0.250000000 0.746688258\nCo 0.750000000 0.750000000 0.253311742\nCo 0.750000000 0.750000000 0.749584510\ndominant in random alloy compared to CsCl and Zintl structures. [c/a values : CsCl (0.999), Zintl (0.999),\nAlloy (1.003)]. In addition, the alloy system exhibited local distortions of Co and Fe position relative to their\nideal positions. In contrast, in CsCl and Zintl structures the Co and Fe atoms exhibited almost imperceptible\ndistortions. Table shows the relaxed atomic positions in the alloys structure in units of the lattice vectors. In\nthe ideal (unrelaxed) system, the positions are all at multiples of 0.25; the relaxed CsCl and Zintl structures no\ndeviations from these positions larger than 1 part in 106\n2. SOC changes the density of states (DOS) at the Fermi energy, notably for the random alloy but notfor the CsCl\nand Zintl structures. Figure S-8 shows DOS for (a) CsCl, (b) Zintl, and (c) random alloy structure with SOC\n(black lines) and without it red lines). We can see signi\fcant DOS di\u000berence for the random alloy supercell\nwith SOC where tetragonal distortions occurred, while almost no changes are observed in the CsCl and Zintl\nstructures.\n3. The local distortions in the alloy structure furthermore gave rise to an energy anisotropy with respect to the\nmagnetization direction. The energy (including SOC) of the relaxed alloy structure for di\u000berent directions of\nthe magnetization is shown in Fig. S-9. While the supercell was rather small, because of the computational\nexpense in relaxing the structure with SOC, so that no self-averaging can be inferred, the \fgure demonstrates\nan induced magnetic anisotropy that arises from the SOC and local distortions. No magnetic anisotropy was\ndiscernible in the CsCl and Zintl structures.\nAs a result from the DFT calculation, we attribute the large SOC e\u000bect in damping anisotropy of Co 50Fe50to local\ntetragonal distortions in disordered Co and Fe alloys. These distortions give rise to SOC-induced changes of DOS at\nthe Fermi level, as well as magnetic anisotropy energy with respect to the crystallographic axes.13\nFIG. S-9. Change in total energy (per supercell) of the alloy structure as function of the magnetization direction.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas and A. Ferguson, Nature\nNano. 10, 50 (2015)\n[2] S. Razee, J. Staunton, B. Ginatempo, E. Bruno, and F. Pinski, Phys. Rev. B 64, 014411 (2001).\n[3] Y. Kota and A. Sakuma, Appl. Phys. Express 5, 113002 (2012).\n[4] I. Turek, J. Kudrnovsk\u0013 y, and K. Carva, Phys. Rev. B 86, 174430 (2012)."    },    {        "title": "1103.5858v3.Spin_motive_forces_due_to_magnetic_vortices_and_domain_walls.pdf",        "content": "Spin motive forces due to magnetic vortices and domain walls\nM.E. Lucassen,1,\u0003G.C.F.L. Kruis,2R. Lavrijsen,2H.J.M. Swagten,2B. Koopmans,2and R.A. Duine1\n1Institute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Department of Applied Physics, Center for NanoMaterials and COBRA Research Institute,\nEindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 9, 2021)\nWe study spin motive forces, i.e.,spin-dependent forces, and voltages induced by time-dependent\nmagnetization textures, for moving magnetic vortices and domain walls. First, we consider the\nvoltage generated by a one-dimensional \feld-driven domain wall. Next, we perform detailed calcu-\nlations on \feld-driven vortex domain walls. We \fnd that the results for the voltage as a function of\nmagnetic \feld di\u000ber between the one-dimensional and vortex domain wall. For the experimentally\nrelevant case of a vortex domain wall, the dependence of voltage on \feld around Walker breakdown\ndepends qualitatively on the ratio of the so-called \f-parameter to the Gilbert damping constant, and\nthus provides a way to determine this ratio experimentally. We also consider vortices on a magnetic\ndisk in the presence of an AC magnetic \feld. In this case, the phase di\u000berence between \feld and\nvoltage on the edge is determined by the \fparameter, providing another experimental method to\ndetermine this quantity.\nPACS numbers: 75.60.Ch, 72.15.Gd, 72.25.Ba\nI. INTRODUCTION\nOne of the recent developments in spintronics is the\nstudy of spin motive forces1and spin pumping2. These\ne\u000bects lead to the generation of charge and spin currents\ndue to time-dependent magnetization textures. The idea\nof spin motive forces due to domain walls is easily un-\nderstood on an intuitive level: if an applied current in-\nduces domain-wall motion,3{8Onsager's reciprocity the-\norem tells us that a moving domain wall will induce a\ncurrent. This idea was already put forward in the eight-\nies by Berger9. In the case of a domain wall driven by\nlarge magnetic \felds ( i.e.,well above the so-called Walker\nbreakdown \feld), a fairly simple approach to the prob-\nlem is justi\fed where one goes to a frame of reference in\nwhich the spin quantization axis follows the magnetiza-\ntion texture.10This transformation gives rise to a vec-\ntor potential from which e\u000bective electric and magnetic\n\felds are derived. Experimentally, the domain-wall in-\nduced voltage has recently been measured above Walker\nbreakdown11, and the results are consistent with this ap-\nproach. It has also been shown that the induced voltage\nwell above Walker breakdown is determined from a topo-\nlogical argument that follows from the properties of the\nabove-mentioned vector potential.12\nThe above approach only captures the reactive con-\ntribution to the spin-motive forces. When the velocity\nof the domain wall is below or just above Walker break-\ndown, a theory is needed that includes more contribu-\ntions to the spin motive forces. Renewed interest has\nshed light on the non-adiabatic and dissipative contribu-\ntions to the spin motive forces13{15that are important in\nthis regime. In this paper, we study this regime.\nThe article is organized as follows. In Section II we\nsummarize earlier results that give a general framework\nto compute electrochemical potentials for given time-dependent magnetization textures. In Section III we con-\nsider an analytical model for a one-dimensional domain\nwall and numerically determine the form of the spin accu-\nmulation and the electrochemical potential. The results\nagree with the known results for the potential di\u000berence\ninduced by a moving one-dimensional domain wall.13In\nSection IV we turn to two-dimensional systems and study\na vortex domain wall in a permalloy strip. We use a\nmicro-magnetic simulator to obtain the magnetization\ndynamics, and numerically evaluate the reactive and dis-\nsipative contributions to the voltage below and just above\nWalker breakdown and compare with experiment.11An-\nother example of a two-dimensional system is a vortex\non a disk which we treat in Section V. For small enough\ndisks, the magnetic con\fguration is a vortex. Both ex-\nperimentally and theoretically, it has been shown that\na vortex driven by an oscillating magnetic \feld will ro-\ntate around its equilibrium position.16{20This gives rise\nto a voltage di\u000berence between the disk edge and cen-\nter as was recently discussed by Ohe et al. [21]. Here,\nwe extend this study by including both the reactive and\nthe dissipative contributions to the voltage, that turn out\nto have a relative phase di\u000berence. This gives rise to a\nphase di\u000berence between the drive \feld and voltage that\nis determined by the so-called \f-parameter.\nII. MODEL\nThe spin-motive force \feld F(~ x) induced by a time-\ndependent magnetization texture that is characterized\nat position ~ xby a unit-vector magnetization direction\nm(~ x;t) is given by13,14\nFi=~\n2[m\u0001(@tm\u0002rim) +\f(@tm\u0001rim)]:(1)arXiv:1103.5858v3  [cond-mat.other]  8 Apr 20112\nThis force \feld acts in this form on the majority spins,\nand with opposite sign on minority spins. In this ex-\npression, the \frst term is the well-known reactive term.1\nThe second term describes dissipative e\u000bects due to spin\nrelaxation13,14and is proportional to the phenomeno-\nlogical\f-parameter, which plays an important role in\ncurrent-induced domain-wall motion.3{8The spin accu-\nmulation\u0016sin the system follows from14\n1\n\u00152\nsd\u0016s\u0000r2\u0016s=\u0000r\u0001F; (2)\nwhere\u0015sd=p\n\u001cDis the spin-di\u000busion length, with \u001c\na characteristic spin-\rip time and Dthe e\u000bective spin-\ndi\u000busion constant. Here, we assume that the spin-\nrelaxation time is much smaller than the timescale for\nmagnetization dynamics. The total electrochemical po-\ntential\u0016that is generated by the spin accumulation due\nto a non-zero current polarization in the system is com-\nputed from14\n\u0000r2\u0016=P(r2\u0016s\u0000r\u0001F); (3)\nwhere the current polarization is given by P= (\u001b\"\u0000\n\u001b#)=(\u001b\"+\u001b#). Note that there is no charge accumulation\nfor\u001b\"=\u001b#, with\u001b\"(\u001b#) the conductivity of the majority\n(minority) spin electrons.\nThe magnetization dynamics is found from the\nLandau-Lifschitz-Gilbert (LLG) equation given by\n@m\n@t=m\u0002\u0012\n\u0000@Emm[m]\n~@m\u0013\n\u0000\u000bm\u0002@m\n@t: (4)\nHere,Emm[m] is the micromagnetic energy functional\nthat includes exchange interaction, anisotropy, and ex-\nternal \feld, and \u000bis the Gilbert damping constant.\nIII. ONE-DIMENSIONAL DOMAIN WALL\nFor one-dimensional problems the voltage di\u000berence\ncan be easily found. For example, an analytic expression\nfor the electric current (which is the open-circuit equiva-\nlent of the chemical-potential di\u000berence) was obtained by\none of us for an analytical model for a one-dimensional\ndriven domain wall.13In this section, we solve the po-\ntential problem for a one-dimensional domain wall and\nobtain the explicit position dependence of the spin accu-\nmulation and the chemical potential.\nA one-dimensional domain wall ( @ym=@zm= 0) is\ndescribed by23\n\u0012(x;t) = 2arctann\neQ[x\u0000X(t)]=\u0015o\n; \u001e (x) = 0;(5)\nwithm= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). Here,Q=\u00061 is\ncalled the topological charge of the domain wall since it\nindicates the way in which an external \feld a\u000bects the\ndomain-wall motion, i.e., a \feld in the direction +^ zwillmove a domain wall in the direction Q^x. Here, we choose\nQ= 1. The domain wall width is indicated by \u0015.\nTo study the time-evolution of a domain-wall, we let\n\u001e(x)!\u001e0(t) so that the wall is described by time-\ndependent collective coordinates fX(t);\u001e0(t)g, called\nposition and chirality, respectively. For external \felds\nsmaller than the Walker-breakdown \feld, there is no\ndomain-wall precession ( i.e., the chirality is constant),\nand the domain wall velocity vis constant so that\n@tm=\u0000v@xm. Since@ym=@zm= 0, we immediately\nsee that the \frst term on the right-hand side of Eq. (1)\nvanishes, and that the force is pointing along the x-axis.\nWe then \fnd that Fx= (\fv~=2)=(\u00152cosh[x=\u0015]2). Due to\nsymmetry we have that @y\u0016=@z\u0016=@y\u0016s=@z\u0016s= 0.\nIn Figs. 1 and 2 we plot the spin accumulation and the\nelectrochemical potential as a function of x.\n-10 -5 5 10x/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΛ\n-1-0.50.512Μs Λ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExtΒv/HBar\nΛsd/Slash1Λ/Equal100Λsd/Slash1Λ/Equal10Λsd/Slash1Λ/Equal12 Λ3\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExtΒvs /HBar/Gradient.F\nFIG. 1: (color online) Spin accumulation as a function of\nposition for several values of the spin-di\u000busion length. The\nblack dotted line gives the value of the source term. The spin\naccumulation tends to zero for x!\u00061 .\n-10 -5 5 10xΛsd/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΛ20.511.522ΜΛ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΒv/HBarP\nΛsd/Slash1Λ/Equal100Λsd/Slash1Λ/Equal10Λsd/Slash1Λ/Equal1\nFIG. 2: (color online) Electrochemical potential as a function\nof position. Note that the potential is proportional to the\npolarization and that on the horizontal axis the position xis\nmultiplied by the spin-di\u000busion length.\nFrom Fig. 2 we see that the total potential di\u000ber-\nence \u0001\u0016=\u0016(x!1 )\u0000\u0016(x!\u00001 ) is independent of3\nthe spin-di\u000busion length and linear in the parameter \f:\n\u0001\u0016=~P\fv=\u0015 . Note that this result is only valid below\nWalker Breakdown.\nTo \fnd the voltage for all \felds Bwe generalize the\nresults for the voltage di\u000berence in Ref. [13] for general\ndomain-wall charge Q. A general expression for the volt-\nage in one dimension is given by13\n\u0001\u0016=\u0000~P\n2jejZ\ndx[m\u0001(@tm\u0002@xm) +\f@tm\u0001@xm]:\n(6)\nWe insert the ansatz [Eq. (5) with \u001e(x) =\u001e0(t)] into\nEq. (6) and \fnd\n\u0001\u0016=\u0000~P\n2jej\"\nQ_\u001e0+\f_X\n\u0015#\n: (7)\nTo \fnd a time-averaged value for the voltage we consider\nthe equations of motion for a domain wall that is driven\nby a transverse magnetic \feld B22{24, contributing to the\nenergy\u0000gB\u0001m, withg >0. The equations of motion\nforX(t) and\u001e0(t) are ultimately derived from the LLG\nequation in Eq. (4), and given by\n(1 +\u000b2)_\u001e0=\u0000gB\n~\u0000\u000bK?\n2~sin(2\u001e0);\n(1 +\u000b2)_X\n\u0015=\u000bQgB\n~\u0000QK?\n2~sin(2\u001e0): (8)\nHere,K?is the out-of-plane anisotropy constant. These\nequations are solved by\nh_\u001e0i=\u0000Sign(B)\n1 +\u000b2Re2\n4s\u0012gB\n~\u00132\n\u0000\u0012\u000bK?\n2~\u001323\n5;\nh_Xi=\u0015Q\n1 +\u000b2 \ngB\n\u000b~+h_\u001e0i\n\u000b!\n(9)\nwhereh::idenotes a time average. It follows that the\nvoltage di\u000berence for general topological charge is\n\u0001\u0016=\u0000Sign(B)Q\n1 +\u000b2~P\n2jej(\n\f\n\u000bgjBj\n~\n\u0000\u0012\n1 +\f\n\u000b\u0013\nRe2\n4s\u0012gB\n~\u00132\n\u0000\u0012\u000bK?\n2~\u001323\n5)\n:(10)\nNote that the overall prefactor Sign( B)Qmakes sense:\ninversion of the magnetic \feld should have the same re-\nsult as inversion of the topological charge.\nIn the above, we used a domain-wall ansatz with mag-\nnetization perpendicular to the wire direction. Using\nthe topological argument by Yang et al.12one can show\nthat the result is more general and holds also for head-\nto-head and tail-to-tail domain walls. Therefore, for a\none-dimensional domain wall, the reactive and dissipa-\ntive contributions, i.e.,the contributions with and with-\nout\fin the above expression, to the voltage always have\nopposite sign.IV. VORTEX DOMAIN WALL\nFor more complicated two-dimensional structures the\nspin-motive force \feld can have rotation and the simpli-\n\fed expression in Eq. (6) is no longer valid so that we\nneed to treat the full potential problem in Eqs. (1-3).\nMotivated by recent experimental results11we consider\nin this section the voltage induced by a moving vortex\ndomain wall.\nWe study the magnetization dynamics using a micro-\nmagnetic simulator25from which we obtain the magneti-\nzation m(~ x;t). This simulator solves the LLG equation in\nEq. (4). For comparison with the experiment by Yang et\nal.,11we simulate a permalloy sample that has the same\ndimensions as this experiment, i.e. 20nm \u0002500nm\u0002\n32\u0016m, which is divided in 1 \u0002128\u00028192 lattice points.\nOn this sample, we drive a head-to-head vortex domain\nwall by means of a magnetic \feld that is pointing from\nright to left, such that the vortex moves from right to left.\nFor several \feld strengths, we obtain the magnetization\nm, and its time-derivative which allows us to compute\nthe force \feld Fat each lattice point. Next, we solve\nthe matrix problem that is the discrete equivalent to the\npotential problem in Eqs. (2) and (3). For details on this\ncalculation, see App. B.\nWe \frst investigate the velocity of the vortex domain\nwall as a function of the applied \feld. We use the value\n\u000b= 0:02 for the Gilbert-damping parameter to obtain\nthe curve in Fig. 3.\n0 0.4 0.8 1.2 1.6\nField/LParen1mT/RParen1050100150200250300Velocity/LParen1m/Slash1s/RParen1\nFIG. 3: Velocity of the vortex domain wall as a function of the\nmagnetic \feld strength for \u000b= 0:02. Above Walker break-\ndown, the velocity is time-averaged. The line is a guide to the\neye.\nThe decrease in velocity for B= 1:5 mT signals\nWalker breakdown. Indeed, up to \felds B= 1:4 mT, the\nvortex moves parallel to the long direction of the sample.\nForB= 1:5 mT, the vortex domain wall motion is more\ncomplicate and has a perpendicular component.26,27We\ntherefore expect that below Walker Breakdown, just like\nfor the one-dimensional domain wall, the vortex domain\nwall only has a dissipative contribution to the voltage.4\nComparison with the experimental results of Ref. [11]\nshows that our velocity is roughly a factor 2 higher.\nThis might be partly caused by a di\u000berence in damping\nand partly by the presence of defects in the experiment\nwhich causes pinning and therefore a decrease of veloc-\nity. The exact value of the Walker breakdown \feld is\nhard to compare, since this depends also on the exact\nvalue of the anisotropy. Nonetheless our value for the\nWalker breakdown \feld is of the same order as Ref. [11].\nMoreover, what is more important is the dependence of\nwall velocity and wall-induced voltage on the magnetic\n\feld normalized to the Walker-breakdown \feld, as these\nresults depend less on system details.\n500\n1000\n1500\n2000length50100\nwidth-505\nΜ/LParen1ΜV/RParen1\n500\n1000\n1500 length\nFIG. 4: Electrochemical potential as a function of position\nfor a moving vortex domain wall on the sample. The num-\nbers on the horizontal axes correspond to lattice points with\nseparation a= 3:9nm. This speci\fc \fgure is for \u000b= 0:02,\nH= 0:8mT (i.e. below Walker breakdown), P= 1 and\n\u0015sd=a. Note that the peak signals the position of the vortex\ncore.\nAn example of a speci\fc form of the electrochemical\npotential on the sample due to a \feld-driven vortex do-\nmain wall is depicted in Fig. 4. We see that there is\na clear voltage drop along the sample, like in the one-\ndimensional model. Additionally, the potential shows\nlarge gradients around the vortex core and varies along\nthe transverse direction of the sample. For each \feld\nstrength, we compute the voltage di\u000berence as a func-\ntion of time. For \feld strengths below Walker Breakdown\nwe \fnd that, as expected, only the dissipative term con-\ntributes and the voltage di\u000berence rapidly approaches a\nconstant value in time. This is understood from the fact\nthat in this regime, the wall velocity is constant after a\nshort time. The dissipative contribution to the voltage\nis closely related to the velocity along the sample, as can\nbe seen in Fig. 5.\nAbove Walker breakdown the reactive term con-\ntributes. We \fnd that for \f=\u000b, the oscillations in\nthe reactive component compensate for the oscillations\n20406080100120140\nTime-2000200400/CapDeltaΜ/Slash1P/LParen1nV/RParen1\nvelocity/CapDeltaΜreac/CapDeltaΜdissFIG. 5: (color online) reactive (blue squares) and dissipative\n(red triangles) contributions to the voltage as a function of\ntime. The numbers on the horizontal axis correspond to time\nsteps of 0.565 ns. The green line gives the velocity along the\nsample, it is scaled to show the correlation with the voltage.\nThese curves are taken for \u000b= 0:02,\f=\u000band \feld strength\nB= 1:6mT.\nin the dissipative component. If we look closely to Fig. 5\nwe see that the length of the periods is not exactly equal.\nThe periods correspond to a vortex moving to the upper\nedge of the sample, or to the lower edge. The di\u000berence\nis due to the initial conditions of our simulation. We\naverage the voltage di\u000berence over time to arrive at the\nresult in Fig. 6. We see that the dissipative contribu-\n0 0.4 0.8 1.2 1.6\nField/LParen1mT/RParen10100200300400/CapDeltaΜ/Slash1P/LParen1nV/RParen1/CapDeltaΜdiss/Plus/CapDeltaΜreac/CapDeltaΜreac/CapDeltaΜdiss\nFIG. 6: (color online) Voltage drop along the sample for \u000b=\n0:02 and\f=\u000b.\ntion becomes smaller for \felds larger than the Walker\nbreakdown \feld, whereas the reactive contribution has\nthe same sign and increases. In fact, for \f=\u000b, the re-\nduction of the dissipative contribution is exactly compen-\nsated by the reactive contribution. The \fdependence is\nillustrated in Fig. 7. The behavior is fundamentally dif-\nferent from the one-dimensional domain-wall situation:\nfor the vortex domain wall, the dissipative contribution\nhas the same sign as the reactive contribution.\nIn order to understand the relative sign, we now dis-\ncuss general vortex domain walls. A single vortex (i.e.5\n0 0.4 0.8 1.2 1.6\nField/LParen1mT/RParen10100200300400/CapDeltaΜ/Slash1P/LParen1nV/RParen1Β/Equal3Α/Slash12Β/EqualΑΒ/EqualΑ/Slash12Β/Equal0\nFIG. 7: (color online) Voltage drop along the sample for \u000b=\n0:02 and several values for \f.\nwith vorticity q= +1) is described by two parameters:\nthe chargep=\u00061 indicates wether the central magnetic\nmoment points in the positive or negative zdirection\nand the chirality Cindicates wether the magnetic mo-\nments align in a clockwise ( C=\u00001) or anti-clockwise\n(C= +1) fashion. We have a vortex that is oriented\nclockwiseC=\u00001. The relative sign is explained from\na naive computation of the voltage above Walker break-\ndown that does not take into account the rotation of the\nspin-motive force \feld\n\u0001\u0016/\u0000Z\ndx[\f@tm\u0001@xm+m\u0001(@tm\u0002@xm)]\n=\fvxZ\ndx(@xm)2+vyZ\ndxm\u0001(@ym\u0002@xm)\n/(\f\u000e+ 1)vx: (11)\nwhere\u000eis a positive number and we used that above\nWalker breakdown m'm(x\u0000vxt;y\u0000vyt) withvx6= 0\nandvy6= 0. Note that if vy= 0 (below Walker\nbreakdown) the reactive term, i.e., the second term in\nthe above expression, indeed vanishes. We used in the\nlast line that above Walker breakdown vx/\u0000pvyandR\ndxm\u0001(@ym\u0002@xm)/ \u0000p. The former equality is\nunderstood from a geometric consideration: consider a\nsample with a vortex characterized by C= 1,p= 1 and\nvxvy<0. By symmetry, this is equivalent to C=\u00001,\np=\u00001 andvxvy>0. It is therefore clear that the sign\nofvxvydepends on either the polarization, or the hand-\nedness of the vortex. Since we know from the vortex\ndomain wall dynamics that reversal of the polarization\nreverses the perpendicular velocity,11we conclude that\nvxvydoes not depend on the handedness of the vortex.\nThe latter equality is understood from a similar argu-\nment:RR\ndxdym\u0001(@ym\u0002@xm) changes sign under the\ntransformation m!\u0000m. During this transformation,\nbothp!\u0000pandC!\u0000C, and therefore their prod-\nuct cannot account for the total sign reversal. Therefore,\nthe integral depends on the polarization12but not on\nthe handedness of the vortex. The positive number \u000eis\nobtained from our numerical simulation, which suggeststhat the magnetic-\feld dependence of the voltage is\n\u0001\u0016=\fB\u0002constant + (1\u0000\f=\u000b)j\u0001\u0016reactivej:(12)\nNote that the sign of the relative contributions can also\nbe obtained using the topological argument by Yang et\nal.12, which gives the same result.\nWe compare our results in Fig. 7 with the experiment\nby Yang et al.11. If we assume that the voltage below\nWalker breakdown lies roughly on the same line as the\nvoltages above Walker breakdown, their results suggest\na slope of 10nV/Oe. For P\u0018 0:8, our results suggest a\nslope of (\f=\u000b)14nV/Oe. Taking into account our higher\nvelocity, we \fnd that \fin the experiment is somewhat\nlarger than \u000b. The decrease in slope of the voltage in\nRef. [11] as Walker breakdown is approached from above\nalso suggests \f >\u000b .\nIn conclusion, the behavior of the voltage around\nWalker breakdown allows us to determine the ratio \f=\u000b.\nIn experiment, the potential di\u000berence as a function of\nthe applied magnetic \feld would show an upturn or\ndownturn around Walker breakdown as in Fig. 7, which\ncorresponds to \f <\u000b and\f >\u000b , respectively.\nV. MAGNETIC VORTEX ON A DISK\nOn small disks (of size \u0016m and smaller) of ferromag-\nnetic material the lowest energy con\fguration is a vor-\ntex. It has been shown that one can let the vortex rotate\naround its equilibrium position by applying an AC mag-\nnetic \feld16{20. This motion gives rise via Eq. (1) to a\nspin motive force on the spins, which induces a voltage on\nthe edge of the disk relative to a \fxed reference voltage,\ne.g. the disk center. Ohe et al.21have shown that the\nreactive contribution to the spin motive force \feld can be\nseen as a dipole that is pointing in the radial direction,\ni.e., the divergence of the force \feld consists of a posi-\ntive and a negative peak along the radial direction (note\nthat the divergence of the force \feld can be seen as an\ne\u000bective charge). Rotation of this dipole gives rise to an\noscillating voltage on the edge of the sample. Here, we\nconsider also the dissipative contribution to the voltage.\nWe consider a vortex on a disk with radius Rthat\nmoves around its equilibrium position ( i.e., the center\nof the disk) at a distance r0from the center of the disk\nwith frequency !. We use as a boundary condition that\nthe magnetization on the edge of the disk is pointing\nperpendicular to the radial direction. In equilibrium, the\nmicro-magnetic energy density of the form \u0000Jm\u0001r2m\u0000\nK?m2\nzis minimized by\nmx(x;y) =\u0000yp\nx2+y2cosh\n2 arctan\u0010\ne\u0000Cp\nx2+y2=\u0014\u0011i\nmy(x;y) =xp\nx2+y2cosh\n2 arctan\u0010\ne\u0000Cp\nx2+y2=\u0014\u0011i\nmz(x;y) =psinh\n2 arctan\u0010\nep\nx2+y2=\u0014\u0011i\n; (13)6\nwhere the center of the vortex is chosen at x=y= 0.\nHere\u0014=p\nK?=Jis the typical width of the vortex core.\nFor permalloy this length scale is of the order \u001810nm.\nThe parameters pandCare de\fned as before, for de\f-\nniteness we choose p= 1,C=\u00001. To describe clockwise\ncircular motion of the vortex around its equilibrium po-\nsition at \fxed radius r0we substitute x!x\u0000r0sin(!t)\nandy!y\u0000r0cos(!t). Note that we assume that the\nform of the vortex is not changed by the motion, which\nis a good approximation for r0\u001cR.\nFrom the magnetization in Eq. (13), we compute the\nforce \feld using Eq. (1). The reactive and dissipative\ncontributions to the divergence of the force \feld are\nshown in Fig. 8. The direction of the dipoles follows\nΠ 0Π 0Π\nAngle00.050.1r/Slash1R\nFIG. 8: The reactive (left) and dissipative (right) contribu-\ntions to the divergence of the force \feld. White means posi-\ntive values, black is negative values. The reactive contribution\ncan be seen as a dipole in the radial direction. The dissipative\ncontribution is a dipole perpendicular to the radial direction.\ndirectly from Eq. (1) if we realize that for our system\n\u0000@tm\u0001~rm=~ v(@^vm)2is always pointing in the direc-\ntion of the velocity which shows that the dissipative con-\ntribution points along the velocity. Likewise the reactive\ncontribution is always pointing perpendicular to the ve-\nlocity.\nFrom the relative orientations of the e\u000bective dipoles,\nwe expect that the peaks in the reactive and dissipative\ncontributions to the voltage on the edge will di\u000ber by a\nphase of approximately \u0019=2 (forr0=R!0 this is exact).\nWe divide our sample in 1000 rings and 100 angles and\nuse the general method in App. B to \fnd the voltage on\nthe edge shown in Fig. 9. To compare with Ref. [21],\nwe take a frequency !=(2\u0019) = 300 MHz and P= 0:8,\nwhich yields amplitudes for the reactive contribution of\n\u0018\u0016V on the edge. However, Ohe et al. suggest that\nvoltage probes that are being placed closer to the vortex\ncore measure a higher voltage. This indeed increases the\nvoltage up to order \u001810\u0016V atr= 2r0. Placing the\nleads much closer to the vortex core does not seem to\nbe realistic because of the size of the vortex. Since the\nvoltage scales with velocity, it can also be increased by a\nlarger radius of rotation, i.e. by applying larger magnetic\n\felds. However, for disks larger than 1 \u0016m, the vortex\n0 Π 2Π 3Π 4Π\nΩt-15-10-50510152 Π /CapDeltaΜ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΩP/LParen1ΜV/Slash1GHz/RParen1\n/CapDeltaΜtotal/CapDeltaΜdiss/Slash1Β/CapDeltaΜreacFIG. 9: (color online) The reactive (red dashed curve) and\ndissipative (blue dashed curve) contributions to the voltage\ndi\u000berence between opposite points on the edge of the disk.\nThe green line gives the total voltage di\u000berence \u0001 \u0016total =\n\u0001\u0016reac+ \u0001\u0016diss, in this example for \f= 0:4. We used r0=\n10\u0015sd,R= 100\u0015sdand\u0014=\u0015sd. For realistic spin-di\u000busion\nlength\u0015sd'5 nm, these parameter values agree with the\nsystem of Ohe et al.21.\nstructure is lost.\nThe dissipative contribution becomes important for\nlarge values of \f. In principle, it is possible to determine\n\fby looking at the shift of the peak in the total volt-\nage with respect to the peak in the reactive contribution,\nwhich is in turn determined by the phase of the applied\nmagnetic \feld. The phase di\u000berence between applied\n\feld and measured voltege then behaves as tan(\u0001 \u001e)/\f.\nVI. DISCUSSION AND CONCLUSION\nWe have investigated the voltage that is induced by a\n\feld-driven vortex domain wall in detail. In contrast to\na one-dimensional model of a domain wall, the reactive\nand dissipative contribution to the voltage have the same\nsign. The qualitative di\u000berences for di\u000berent values of \f\nprovide a way to determine the ratio \f=\u000bexperimentally\nby measuring the wall-induced voltage as a function of\nmagnetic \feld. To this end the experimental results in\nRef. [11] are in the near future hopefully extended to\n\felds below Walker breakdown, which is challenging as\nthe voltages become smaller with smaller \feld.\nWe also studied a magnetic vortex on a disk. When\nthe vortex undergoes a circular motion, a voltage is in-\nduced in the sample. Earlier work computed the reactive\nvoltage on the edge of the disk,21here we include also the\ndissipative contribution to the voltage. We \fnd that the\nphase di\u000berence between voltage and AC driving \feld is\ndetermined by the \f-parameter.7\nAcknowledgments\nThis work was supported by the Netherlands Organiza-\ntion for Scienti\fc Research (NWO) and by the European\nResearch Council (ERC) under the Seventh Framework\nProgram (FP7).\nAppendix A: Boundary conditions\nAs a boundary condition for the potential problems,\nwe demand that the total spin current and charge current\nperpendicular to the upper and lower boundaries is zero:\nj?\ns=j?\n\"\u0000j?\n#= 0 andj?\ns=j?\n\"+j?\n#= 0. Therefore,\nthe majority-and minority spin currents are necessarily\nzero. They are given by j\"=\u001b\"(F\u0000r\u0016\") andj#=\n\u001b#(\u0000F\u0000r\u0016#). From this, the boundary conditions on\nthe derivatives of the potentials follow as @?\u0016s=@?(\u0016\"\u0000\n\u0016#)=2 =Fand@?\u0016=@?(\u0016\"+\u0016#)=2 = 0. We consider\na two-dimensional sample that is in\fnitely long in the\nx-direction, and of \fnite size 2\u0003 in the y-direction the\nboundary conditions are\n@y\u0016s(x;y=\u0006\u0003) =Fy(x;y=\u0006\u0003): (A1)\nTo measure the induced voltage, we also put the deriva-\ntives of the potential at in\fnity to zero so that the bound-\nary conditions for the electrochemical potential are\n@y\u0016(x;y=\u0006\u0003) = 0;\n@x\u0016(x!\u00061;y) = 0: (A2)\nAppendix B: Potential problem on a Lattice\nWe consider a two-dimensional lattice, where we have\nspin accumulation \u0016i;j\nsand an electrochemical potential\n\u0016i;jat sitei;j. Between sites ( i;j) and (i;j+ 1), there\ncan be a particle current density of majority spins\nji;j+1=2\n\";^ \u0011=\u001b\" \nFi;j+1=2\n^ \u0011+\u0016i;j\n\"\u0000\u0016i;j+1\n\"\nai;j+1=2\n^ \u0011!\n=\u001b\"\u0010\nFi;j+1=2\n^ \u0011\u0000\u000e^ \u0011\u0016i;j+1=2\n\"\u0011\n; (B1)\nwithai;j+1=2\n^ \u0011the lattice spacing in the ^ \u0011 direction between\nsites (i;j) and (i;j+ 1), and a particle current density of\nminority spins\nji;j+1=2\n#;^ \u0011=\u001b# \n\u0000Fi;j+1=2\n^ \u0011+\u0016i;j\n#\u0000\u0016i;j+1\n#\nai;j+1=2\n^ \u0011!\n=\u001b#\u0010\n\u0000Fi;j+1=2\n^ \u0011\u0000\u000e^ \u0011\u0016i;j+1=2\n\"\u0011\n; (B2)\nand equivalently for currents in the ^ \u0010 direction. The\nderivative \u000e^ \u0011is de\fned as \u000e^ \u0011Oi;j= (Oi;j+1=2\u0000\nOi;j\u00001=2)=ai;j\n^ \u0011, and likewise for \u000e^ \u0010. Note that upper in-\ndices (i;j) denote a position on the lattices and lower in-\ndices ^ \u0010 or ^ \u0011 denote a direction. We can write \u0016\"=\u0016+\u0016sand\u0016#=\u0016\u0000\u0016s. The continuity-like equations for the\ndensity of majority- and minority spins are (note that\nspins move in the direction of the current)\nAi;jni;j\n\"#\n\u001c=\u0000\u0001(`i;jji;j\n\"#)\njej; (B3)\nwith characteristic spin-\rip time \u001cand with the dimen-\nsionless operator \u0001 given by\n\u0001Oi;j=Oi+1=2;j\n^ \u0010\u0000Oi\u00001=2;j\n^ \u0010+Oi;j+1=2\n^ \u0011\u0000Oi;j\u00001=2\n^ \u0011:\n(B4)\nThese de\fnitions allows for non-square lattices with sides\nat position ( i\u00061=2;j) or (i;j\u00061=2) that have length\n`i\u00061=2;j\n^ \u0010or`i;j\u00061=2\n^ \u0011(lower index denotes the normal di-\nrection), respectively, and the area of the site itself given\nbyAi;j.\nThe equation for the electrochemical potential is ob-\ntained from the continuity equation\n0 =\u0000jejAi;jni;j\n\"+ni;j\n#\n\u001c= \u0001[`i;j(ji;j\n\"+ji;j\n#)] =\n\u001b\"\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\n\")] +\u001b#\u0001[`i;j(\u0000Fi;j\u0000\u000e\u0016i;j\n#)] =\n(\u001b\"+\u001b#)\u0001f`i;j[\u0000\u000e\u0016i;j+P(Fi;j\u0000\u000e\u0016i;j\ns)]g:\n!\u0001(`i;j\u000e\u0016i;j) =P\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\ns)]; (B5)\nwhere the current polarization is given by P= (\u001b\"\u0000\n\u001b#)=(\u001b\"+\u001b#). This result was already obtained for a\ncontinuous system in Ref. [14]. To \fnd an equation for\nthe spin accumulation, we write\n\u0000jejAi;jni;j\n\"\u0000ni;j\n#\n\u001c= \u0001[`i;j(ji;j\n\"\u0000ji;j\n#)] =\n\u001b\"\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\n\")]\u0000\u001b#\u0001[`i;j(\u0000Fi;j\u0000\u000e\u0016i;j\n#)] =\n(\u001b\"+\u001b#)\u0001f`i;j[Fi;j\u0000\u000e\u0016i;j\ns\u0000P\u000e\u0016i;j]g=\n(\u001b\"+\u001b#)(1\u0000P2)\u0001[`i;j(Fi;j\u0000\u000e\u0016i;j\ns)]: (B6)\nIf we compare this in the case of a square lattice to the\nexpression in Ref.14\n1\n\u00152\nsd\u0016s\u0000r2\u0016s=\u0000r\u0001F; (B7)\nwe \fnd that the density of spins that pile up can be\nexpressed in terms of the spin accumulation as ( ni;j\n\"\u0000\nni;j\n#)=\u001c= (\u001b\"+\u001b#)(1\u0000P2)\u0016i;j\ns=(jej\u00152\nsd). 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B 78, 134412 (2008)."    },    {        "title": "1506.01303v3.Antidamping_spin_orbit_torque_driven_by_spin_flip_reflection_mechanism_on_the_surface_of_a_topological_insulator__A_time_dependent_nonequilibrium_Green_function_approach.pdf",        "content": "Antidamping spin-orbit torque driven by spin-\rip re\rection mechanism on the surface\nof a topological insulator: A time-dependent nonequilibrium Green function approach\nFarzad Mahfouzi,1,\u0003Branislav K. Nikoli\u0013 c,2and Nicholas Kioussis1\n1Department of Physics, California State University, Northridge, CA 91330-8268, USA\n2Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2570, USA\nMotivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization\n~ mof a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI),\nwe investigate the origin of the SOT and the magnetization dynamics in such systems. We predict\nthat lateral F/TI bilayers of \fnite length, sandwiched between two normal metal leads, will generate\na large antidamping-like SOT per very low charge current injected parallel to the interface. The\nlarge values of antidamping-like SOT are spatially localized around the transverse edges of the F\noverlayer. Our analysis is based on adiabatic expansion (to \frst order in @~ m=@t ) of time-dependent\nnonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the\napplied bias voltage and by the slow variation of a classical degree of freedom [such as ~ m(t)]. From it\nwe extract formulas for spin torque and charge pumping, which show that they are reciprocal e\u000bects\nto each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula\nfor SOT naturally splits into four components, determined by their behavior (even or odd) under the\ntime and bias voltage reversal. Their complex angular dependence is delineated and employed within\nLandau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability\nof the predicted SOT to e\u000eciently switch ~ mof a perpendicularly magnetized F overlayer.\nPACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nThe spin-orbit torque (SOT) is a recently discovered\nphenomenon1{4in ferromagnet/heavy-metal (F/HM)\nlateral heterostructures involves unpolarized charge cur-\nrent injected parallel to the F/HM interface induces\nswitching or steady-state precession5of magnetization\nin the F overlayer. Unlike conventional spin-transfer\ntorque (STT) in spin valves and magnetic tunnel junc-\ntion (MTJs),6{8where one F layer acts as spin-polarizer\nof electrons that transfer torque to the second F layer\nwhen its free magnetization is noncollinear to the direc-\ntion of incoming spins, heterostructures exhibiting SOT\nuse a single F layer. Thus, in F/HM bilayers, spin-orbit\ncoupling (SOC) at the interface or in the bulk of the\nHM layer is crucial to spin-polarized injected current\nvia the Edelstein e\u000bect (EE)9,10or the spin Hall e\u000bect\n(SHE),11,12respectively.\nThe SOT o\u000bers potentially more e\u000ecient magnetiza-\ntion switching than achieved by using MTJs underlying\npresent STT-magnetic random access memories (STT-\nMRAM).13Thus, substantial experimental and theo-\nretical e\u000borts have been focused on identifying physi-\ncal mechanisms behind SOT whose understanding would\npave the way to maximize its value by using optimal\nmaterials combinations. For example, very recent ex-\nperiments14{16have replaced HM with three-dimensional\ntopological insulators (3D TIs).17The TIs enhance18{20\n(by a factor ~vF=\u000bR, wherevFis the Fermi velocity on\nthe surface of TI and \u000bRis the Rashba SOC strength22?\nat the F/HM interface) the transverse nonequilibrium\nspin density driven by the longitudinal charge current,\nwhich is responsible for the large \feld-like SOT compo-\nnent20,23observed experimentally.14{16\nFIG. 1. (Color online) Schematic view of F/TI lateral bilayer\noperated by SOT. The F overlayer has \fnite length LF\nxand~ m\nis the unit vector along its free magnetization. The TI layer\nis attached to two N leads which are semi-in\fnite in the x-\ndirection and terminate into macroscopic reservoirs. We also\nassume that F and TI layers, as well as N leads, are in\fnite in\nthey-direction. The unpolarized charge current is injected by\nthe electrochemical potential di\u000berence between the left and\nthe right macroscopic reservoirs which sets the bias voltage,\n\u0016L\u0000\u0016R=eVb. We mention that the results do not change if\nthe TI surface is covered by the F overlayer partially or fully.\nFurthermore, recent experiments have also observed\nantidamping-like SOT in F/TI heterostructures with\nsurprisingly large \fgure of merit (i.e., antidamping\ntorque per unit applied charge current density) that sur-\npasses14{16those measured in a variety of F/HM het-\nerostructures. This component competes against the\nGilbert damping which tries to restore magnetization\nto equilibrium, and its large \fgure of merit is, there-\nfore, of particular importance for increasing e\u000eciency of\nmagnetization switching. Theoretical understanding ofarXiv:1506.01303v3  [cond-mat.mes-hall]  24 Jan 20162\nthe physical origin of antidamping-like SOT is crucial to\nresolve the key challenge for anticipated applications of\nSOT generated by TIs|demonstration of magnetization\nswitching of the F overlayer at room temperature (thus\nfar, magnetization switching has been demonstrated only\nat cryogenic temperature15).\nHowever, the microscopic mechanism behind its large\nmagnitude14{16and ability to e\u000eciently (i.e., using as lit-\ntle dc current density as possible) switch magnetization15\nremains under scrutiny. For example, TI samples used\nin these experiments are often unintentionally doped, so\nthat bulk charge carriers can generate antidamping-like\nSOT via rather large24SHE (but not su\u000ecient to explain\nall reported values14,15). The simplistic picture,14in\nwhich electrons spin-polarized by the EE di\u000buse into the\nF overlayer14to deposit spin angular momentum within\nit, cannot operate in technologically relevant F overlayers\nof'1 nm thickness16or explain complex angular depen-\ndence2,15,25typically observed for SOT. The Berry cur-\nvature mechanism25,26for antidamping-like SOT applied\nto lateral F/TI heterostructures predicts its peculiar de-\npendence on the magnetization orientation,27vanishing\nwhen magnetization ~ mis parallel to the F/TI interface.\nThis feature has thus far not been observed experimen-\ntally,15and, furthermore, it makes such antidamping-like\nSOT less e\u000ecient27(by requiring larger injected currents\nto initiate magnetization switching) than standard SHE-\ndriven3,4antidamping-like SOT.\nWe note that the recent experimental14{16and theo-\nretical14,27studies of SOT in lateral F/TI bilayer have\nfocused on the geometry where an in\fnite F overlayer\ncovers an in\fnite TI layer. Moreover, they assume14,27?\npurely two-dimensional transport where only the top sur-\nface of the TI layer is explicitly taken into account by the\nlow-energy e\u000bective (Dirac) Hamiltonian supplemented\nby the Zeeman term due to the magnetic proximity ef-\nfect. On the other hand, transport in realistic TI-based\nheterostructures is always three-dimensional, with unpo-\nlarized electrons being injected from normal metal con-\ntacts, re\rected from the F/TI edge to \row along the sur-\nface of the TI in the yz-plane and then along the bottom\nTI surface in Fig. 1. In fact, electrons also \row within a\nthin layer (of thickness .2 nm in Bi 2Se3as the prototyp-\nical TI material) underneath the top and bottom surfaces\ndue to top and bottom metallic surfaces of the TI doping\nthe bulk via evanescent wave functions.18Therefore, in\nthis study we consider more realistic and experimentally\nrelevant28F/TI bilayer geometries, illustrated in Fig. 1,\nwhere the TI layer of \fnite length LTI\nxand \fnite thick-\nnessLTI\nzis (partially or fully) covered by the F overlayer\nof lengthLF\nx. The two semi-in\fnite ideal N leads are di-\nrectly attached to the TI layer. we should mention that\nthe result does not depend on the length of TI layer that\nis covered by the FM.\nOur principal results are twofold and are summarized\nas follows:\n(i)Theoretical prediction for SOT: We predict that the\ngeometry in Fig. 1 will generate large antidamping-likeSOT per low injected charge current. By studying spatial\ndependence of the SOT (see Fig. 4), we show that in a\nclean FM/TI interface the electrons exert anti-damping\ntorque on the FM as they enter into the interface and un-\nless interfacial roughness or impurities are included the\ntorque remains mainly concentrated around the edge of\nthe interface. Although the exact results show strong\nnonperturbative features, based on second order pertur-\nbation we present two di\u000berent interpretations showing\nthat the origin of the antidamping SOT relies on the\nspin-\rip re\rection of the chiral electrons injected into\nthe FM/TI interface. Its strong angular dependence (see\nFig. 2), i.e., dependence on the magnetization direction\n~ m, o\u000bers a unique signature that can be used to distin-\nguish it from other possible physical mechanisms. By\nnumerically solving the Landau-Lifshitz-Gilbert (LLG)\nequation in the macrospin approximation, we demon-\nstrate (see Figs. 5 and 6) that the obtained SOT is ca-\npable of switching of a single domain magnetization of a\nperpendicularly magnetized F overlayer with bias voltage\nin the oder of the Magneto-Crystaline Anisotropy (MCA)\nenergy.\n(ii)Theoretical formalism for SOT: The widely used\nquantum (such as the Kubo formula25{27,30) and semi-\nclassical (such as the Boltzmann equation31) transport\napproaches to SOT are tailored for geometries where an\nin\fnite F layer covers an in\fnite TI or HM layer. Due to\ntranslational invariance, the nonequilibrium spin density\n~Sinduced by the EE on the surface of TI or HM layer\nhas uniform orientation ~S= (0;Sy;0) [in the coordinate\nsystem in Fig. 1], which then provides reference direc-\ntion for de\fning \feld-like, \u001cf~ m\u0002^y, and antidamping-\nlike,\u001cad~ m\u0002(~ m\u0002^y), components of SOT. In order to\nanalyze spatial dependence of SOT in the device geome-\ntry of Fig. 1, while not assuming anything a priori about\nthe orientation of \feld-like and antidamping-like compo-\nnents of SOT, we employ adiabatic expansion32of time-\ndependent nonequilibrium Green functions (NEGFs)33,34\nto derive formulas for torque, charge pumping35,36and\nGilbert damping37in the presence of SOC. The NEGF-\nbased formula for SOT naturally splits into four compo-\nnents, determined by their behavior (even or odd) under\nthe time and bias voltage reversal. This gives us a general\nframework in quantum mechanics to analyze the dissi-\npative (antidamping-like) and nondissipative (\feld-like)\nforce (torque) vector \felds for a set of canonical variables\n(magnetization directions). Their angular (see Fig. 2)\nand spatial (see Fig. 4) dependence shows that although\n\feld-like and antidamping-like SOTs are predominantly\nalong the~ m\u0002^yand~ m\u0002(~ m\u0002^y) directions, respectively,\nthey are not uniform and can exhibit signi\fcant devia-\ntion from the trivial angular dependence de\fned by these\ncross products [see Fig. 2(h)].\nThe paper is organized as follows. In Sec. II, we present\nthe adiabatic expansion of time-dependent NEGFs, in a\nrepresentation that is alternative to Wigner representa-\ntion34(usually employed for this type of derivation32),\nand derive expressions for torque, charge pumping and3\nGilbert damping. In Sec. III, we decompose the NEGF-\nbased expression for SOT into four components, deter-\nmined by their behavior (even or odd) under the time and\nbias voltage reversal, and investigate their angular de-\npendence. Section IV discusses the angular dependence\nof the zero-bias transmission function which identi\fes the\nmagnetization directions at which substantial re\rection\noccurs. In Sec. V, we study spatial dependence of SOT\ncomponents and discuss their physical origin. Section VI\npresents LLG simulations of magnetization dynamics in\nthe presence of predicted SOT, as well as a switching\nphase diagram of the magnetization state as a function\nof the in-plane external magnetic \feld and SOT. We con-\nclude in Sec. VII.\nII. THEORETICAL FORMALISM\nWe \frst describe the time-dependent Hamiltonian\nmodel, H(t) =H0+U(t), of the lateral F/TI heterostruc-\nture in Fig. 1. Here H0is the minimal tight-binding\nmodel for 3D TIs like Bi 2Se3on a cubic lattice of spacing\nawith four orbitals per site.38The thickness, LTI\nz= 8a\nof the TI layer is su\u000ecient to prevent hybridization be-\ntween its top and bottom metallic surface states.18The\ntime-dependent potential\nU(t) =\u0000\u0001surf1m~ m(t)\u0001~\u001b=2; (1)\ndepends on time through the magnetization of the F over-\nlayer which acts as the slowly varying classical degree of\nfreedom. Here ~ m(t) is the unit vector along the direc-\ntion of magnetization, \u0001 surf= 0:28 eV is the proximity\ninduced exchange-\feld term and 1mis a diagonal matrix\nwith elements equal to unity for sites within the F/TI\ncontact region in Fig. 1 and zero elsewhere. The semi-\nin\fnite ideal N leads in Fig. 1 are taken into account\nthrough the self-energies33,34\u0006L;Rcomputed for a tight-\nbinding model with one spin-degenerate orbital per site.\nThe details of how to properly couple \u0006L;RtoH0, while\ntaking into account that the spin operators for electrons\non the Bi and Se sublattices of the TI are inequivalent,39\ncan be found in Ref. 40.\nWithin the NEGF formalism33,34the advanced\nand lesser GFs matrix elements of the tight-binding\nHamiltonian, H0, are de\fned by Gii0;oo0;ss0(t;t0) =\n\u0000i\u0002(t\u0000t0)hf^cios(t);^cy\ni0o0s0(t0)gi, andG<\nii0;oo0;ss0(t;t0) =\nih^cy\ni0o0s0(t0)^cios(t)i, respectively. Here, ^ cy\nios(^cios) is the\ncreation (annihilation) operator for an electron on site, i,\nwith orbital, o, and spins, respectively,h:::idenotes the\nnonequilibrium statistical average, and ~= 1 to simplify\nthe notation. These GFs are the matrix elements of the\ncorresponding matrices GandG<used throughout the\ntext.\nUnder stationary conditions, the two GFs depend on\nthe di\u000berence of the time arguments, t\u0000t0, and can\nbe Fourier transformed to energy. In the strictly adi-\nabatic limit one can employ41the same retarded GF,Gt(E) = [E\u0000H(t)\u0000\u0006L\u0000\u0006R]\u00001, as under stationary\nconditions, but where the GF depends parametrically on\ntime (denoted by the subscript t) and is computed for the\nfrozen-in-time con\fguration of U(t). However, even for\nslow evolution of ~ m(t) corrections32to the adiabatic GF\nare needed to describe dissipation e\u000bects such as Gilbert\ndamping or the charge current which can be pumped35\nby the dynamics of ~ m(t).\nThe so-called adiabatic expansion, which yields correc-\ntions beyond the strictly adiabatic limit, is traditionally\nperformed using the Wigner representation34in which\nthe fast and slow time scales are easily identi\fable.32The\nslow motion implies that the NEGFs vary slowly with the\ncentral time tc= (t+t0)=2 while they change fast with\nthe relative time tr=t\u0000t0. By expanding the Wigner\ntransformation of NEGFs\nG(<)\nW(E;tc) =Z1\n\u00001dtreiEtrG(<)\u0012\ntc+tr\n2;tc\u0000tr\n2\u0013\n;\n(2)\nin the central time tcwhile keeping only terms contain-\ning \frst-order derivatives @=@tc(due to the slow varia-\ntion withtc) gives the \frst-order correction beyond the\nstrictly adiabatic limit.32This route requires to han-\ndle complicated expressions resulting from the Wigner\ntransform applied to convolutions of the type C(t1;t2) =R\ndt3C1(t1;t3)C2(t3;t2).\nHere we provide an alternative derivation of the \frst-\norder nonadiabatic correction. Namely, we consider t\n(observation time) and t\u0000t0(relative time) as the nat-\nural variables to describe the time evolution of NEGFs\nand then perform the following Fourier transform42\nG(t;t0) =Z1\n\u00001dE\n2\u0019eiE(t\u0000t0)G(E;t): (3)\nThe standard equations of motion for G(t;t0) and\nG<(t;t0) are cumbersome to manipulate42,43or solve\nnumerically,44so they are usually transformed to some\nother representation.35Here we replace G(t;t0) in the\nstandard equations of motion with the rhs of Eq. (3) to\narrive at:\n\u0014\u0012\nE\u0000i@\n@t\u0013\n1\u0000H0\u0000U(t)\u0000\u0006\u0012\nE\u0000i@\n@t\u0013\u0015\nG(E;t) =1;\n(4)\nand\nG<(t;t0) =ZdE\n2\u0019G(E;t)\u0006<(E)Gy(E;t0)eiE(t\u0000t0):(5)\nFor the two-terminal heterostructure in Fig. 1 in the\nelastic transport regime,33,34\u0006(E) =\u0006L(E) +\u0006R(E)\nand\u0006<(E) =ifL(E)\u0000L(E) +ifR(E)\u0000R(E), where\n\u0000L;R=i(\u0006L;R\u0000\u0006y\nL;R). The Fermi-Dirac distribution\nfunctions of electrons in the macroscopic reservoirs into\nwhich the left and right N leads terminate are fL;R(E) =\nf(E\u0000\u0016L;R), where the di\u000berence between the electro-\nchemical potentials, \u0016L;R=EF+eVL;R, de\fnes the bias\nvoltageeVb=\u0016L\u0000\u0016R.4\nUsing the following identity\nX\n\u000b=L;RiG\u0000\u000bGy= (G\u0000Gy) +i@\n@E\u0012\nG@U\n@tGy\u0013\n+O\u0012@2U\n@t2\u0013\n; (6)\nthe lesser GF to \frst order in small @U(t)=@tand in the low bias Vbregime (i.e., the linear-response transport regime)\ncan be expressed as,\nG<(t;t)'ZdE\n2\u00190\n@[G(E;t)\u0000Gy(E;t)]f(E) +X\n\u000b=L;Rf0eV\u000bGt\u0000\u000bGy\nt+if0Gt@U(t)\n@tGy\nt1\nA; (7)\nwhere, the \frst term corresponds to the density matrix of the equilibrium electrons occupying the time dependent\nsingle particle states, while the second and third terms describe the density matrix of the excited (nonequilibrium)\nelectrons occupying the states close to the Fermi energy due to the bias voltage and time dependent term in the\nHamiltonian, respectively. The retarded GF in the \frst term in Eq. (7) expanded to \frst order in @U(t)=@tis of the\nform\nG(E;t)'Gt+i@Gt\n@E@U(t)\n@tGt: (8)\nThe lesser GF determines the time-dependent nonequilibrium density matrix\n\u001a(t) =1\niG<(t;t); (9)\nfrom which we determine the time-dependent expectation values of physical observables, A(t) = Tr [\u001a(t)^A]. In\nparticular, the relevant quantities for the heterostructure in Fig. 1 are the charge current\nI\u000b(t) =eZdE\n2\u0019iTr\b\n\u0000\u000b(E)G<(t;t) +f\u000b(E)\u0000\u000b(E)[G(E;t)\u0000Gy(E;t)]\t\n=e2\n2\u0019Z\ndEf0(E)8\n<\n:X\n\f(V\f\u0000V\u000b)T\u000b\f(E)\u0000\u0001surf\n2eX\ni@mi\n@tT\u000bi(E)9\n=\n;; (10)\nand the spin density\nsi(t) =ZdE\n2\u0019iTr\u0002\n\u001bi1mG<\u0003\n=ZdE\n2\u00198\n<\n:X\n\u000bf\u000b(E)Ti\u000b(E)\u0000\u0001surf\n2X\njf0(E)@mj\n@tTij(E)9\n=\n;; (11)\nwhere\u000b;\f2fL;Rgandi;j2fx;y;zg.\nThe \\trace-formulas\" in Eqs. (10) and (11)\nT\u000b\f(E) = Trh\n\u0000\u000bGt\u0000\fGy\nti\n; (12a)\nT\u000bi(E) = Trh\n1m\u001biGy\nt\u0000\u000bGti\n; (12b)\nTi\u000b(E) = Trh\n1m\u001biGt\u0000\u000bGy\nti\n; (12c)\nTij(E) = Trh\n1m\u001bi(Gy\nt\u0000Gt)1m\u001bj(Gt\u0000Gy\nt)i\n;(12d)\ndetermine charge current45due toVb, charge current\npumped35by the dynamics of ~ m(t) in the presence of\nSOC, the spin torque, and Gilbert damping tensor, re-\nspectively. In the expression for the spin density we ig-\nnore the antisymmetric part of Tijwhich corresponds to\nthe renormalization of the precession frequency of mag-netization dynamics.32Note thatT\u000biin Eq. (12)(b) and\nits time-reversal Ti\u000bin Eq. (12)(c) reveal a reciprocal46\nrelation between charge pumping by magnetization dy-\nnamics in the presence of SOC and current-driven SOT\nateach instant of time t.\nThe spin density in Eq. (11) enters into the LL equa-\ntion for the magnetization dynamics\n@~ m\n@t=\u0000\u0001surf\n2~ m\u0002~ s(t): (13)\nThe \frst term in Eq. (11) generates the spin torque in\nEq. (13) which has three contributions: ( i) an equilib-\nrium component responsible for the interlayer exchange\ninteraction in the presence of a second F layer and/or\nmagneto-crystalline anisotropy (MCA) in the presence of\nSOC; ( ii) a bias-induced \feld-like torque modifying the5\nequilibrium interlayer exchange and MCA \felds; and ( iii)\na damping (antidamping)-like torque describing angular\nmomentum loss (gain) due to the \rux of Fermi surface\nelectrons.\nIII. ANGULAR DEPENDENCE OF SOT\nCOMPONENTS\nIn order to understand the di\u000berent contributions to\nTi\u000b(E), we decompose it into even (e) or odd (o) terms\nunder time-reversal Gt7!Gy\nt:\nTi\u000b\ne(E) = [Ti\u000b(E) +T\u000bi(E)]=2; (14)\nTi\u000b\no(E) = [Ti\u000b(E)\u0000T\u000bi(E)]=2: (15)\nSinceP\n\u000bTi\u000b\no(E) = 0, the contribution of the odd com-\nponent to the equilibrium spin density in Eq. (11) van-\nishes identically, while its nonzero values appear only for\nEwithin the bias window around EF. This motivates\nfurther splitting of Ti\u000b(E) into four components for the\ncase of two-terminal devices\nTi\ne;\u0017(E) =TiL\n\u0017(E) +TiR\n\u0017(E)\n2; (16a)\nTi\no;\u0017(E) =TiL\n\u0017(E)\u0000TiR\n\u0017(E)\n2; (16b)\nwhere the \frst and second subscripts denote their be-\nhavior (even or odd) under bias reversal Vb7!\u0000Vband\ntime reversal, respectively. The corresponding four com-\nponents of torque are determined by\n~Te;\u0017=ZdE\n2\u0019[fL(E) +fR(E)]~ \u001ce;\u0017(E); (16c)\n~To;\u0017=ZdE\n2\u0019[fL(E)\u0000fR(E)]~ \u001co;\u0017(E);(16d)\nwhere the energy-resolved torque is given by\n~ \u001c\u0016;\u0017(E) =\u0000\u0001surf\n2~ m\u0002~T\u0016;\u0017(E); (17)\nand\u0016;\u00172fe;og. The terms ~To;oand~To;eare non-zero\nonly in nonequilibrium driven by Vb6= 0, and depend on\nelectronic states in the bias voltage window around the\nFermi energy (or on the Fermi surface states in the linear-\nresponse regime where integrals are avoided by multiply-\ning integrand by eVb). The term ~Te;o\u00110 is zero, while\n~Te;eis nonzero also in equilibrium and, therefore, depends\non all occupied electronic states.\nFigures 2(a-c) show the netvector \feld (summed over\nall sites of the F overlayer) of ~ \u001c\u0016;\u0017(EF) at zero temper-\nature for di\u000berent directions of ~ mon the unit sphere.\nTheir angular behavior reveals that: ( i)~ \u001co;oshown in\nFig. 4(a) is the \feld-like SOT generated by the EE, with\npredominant orientation along the ~ m\u0002^y-direction; ( ii)\n~ \u001co;ein Fig. 4(b) is the antidamping-like SOT with pre-\ndominant orientation along the ~ m\u0002(~ m\u0002^y)-direction;\nand ( iii)~ \u001ce;ein Fig. 4(c) along the ~ m\u0002^zdirection isthe \feld-like component whose angular dependence be-\nhaves approximately as 2( ~ m\u0001^z)j~ m\u0002^zj\u0011sin(2\u0012) typical\nfor torque components generated by the MCA \feld.2\nThe corresponding angular dependence of the net\n\u001ci\no;o(i2fx;y;zg),j~ \u001co;ej, and\u001ci\ne;ealong the solid trajecto-\nries shown in Figs. 2(a-c) are plotted in Figs. 2(d-f),\nrespectively. We \fnd that the magnitude of the max-\nimum antidamping-like SOT is about a factor of four\nlarger than that of the \feld-like SOT. Additionally, the\n\feld-like SOT peaks when the magnetization is in-plane.\nIn contrast, the antidamping-like SOT peaks when the\nmagnetization is out of plane, which can be attributed to\nthe gap opening of the Dirac cone which in turn enhances\nthe re\rection (see Sec. IV) at the lateral boundaries of\nthe F overlayer.\nNote that the magnitude of the netSOT components\nshown in Figs. 2(g-i) exhibits strong angular dependence\nbecause of the large SOC on the TI surface similar to that\nfound in F/HM heterostructures when the Rashba SOC\nat the interface is su\u000eciently strong.25,26In particular,\nthe signi\fcant deviation of the angular dependence of the\nantidamping-like SOT from the trivial j~ m\u0002(~ m\u0002^y)jbe-\nhavior in the limit of ~ m!^y, shown quantitatively in\nFig. 2(h), indicates its nonperturbative variation with\nrespect to the magnetization direction. A similar nonper-\nturbative angular behavior (i.e., strong deviation from\nthe standard/sin2\u0012dependence on the precession cone\nangle\u0012) has been found for adiabatic charge pumping\nfrom a precessing F overlayer attached to the edge of 2D\nTIs47,48or to the surface of 3D TIs.36\nIV. ANGULAR DEPENDENCE OF\nTRANSMISSION FUNCTION\nFigure 3 shows the transmission function TRL(EF) in\nEq. (12a) for the heterostructure in Fig. 1 versus the ori-\nentation of ~ mon the unit sphere. We \fnd that the charge\ncurrent determined by TRL(E) is smallest49when~ mk^z\nor~ mk^x. This is due to the re\rection of Dirac elec-\ntrons on the TI surface from the lateral boundaries of\nthe F overlayer. Underneath the F overlayer, the ex-\nchange \feld,\u0000\u0001surf~ m\u0001~\u001b=2, induced by the magnetic\nproximity e\u000bect is superimposed on the Dirac cone sur-\nface dispersion. This opens an energy gap \u0001 surfwhen\n~ mk^z(or smaller gap \u0001 surfcos\u0012formz6= 0) at the DP\nof the TI region underneath the F overlayer, which in\nturn gives rise to strong electronic re\rection when ~ mk^z.\nFor~ mk^x, there is no energy gap at the DP and the\nDirac cone e\u000bectively shifts away from the center of the\nBrillouin zone due to proximity exchange \feld. Neverthe-\nless electrons polarized by the EE along the y-axis re\rect\nfrom the magnetization pointing along the x-axis.6\n(a)                                         (b)                                         (c)\n(d)                                         (e)                                         (f)\n(g)                 (h)                                         ( i)\n-0.20.00.2i\ne,e(EF)(1/a)  x\n y\n z\n0 45 90 135 1800.00.10.20.3|e,e(EF)|/sin  2 (1/a)\nAngle  (deg)\n-5-4-3-2-1010-2i\no,o(EF)(1/a) x\n y\n z\n0 45 90 135 180024681010-2|o,o(EF)|/sin  (1/a)\nAngle  (deg)\n0.00.10.2|o,e(EF)|(1/a) =0o\n =45o\n =90o\n0 45 90 135 1800.00.10.20.3|o,e(EF)|/sin   (1/a)\nAngle  (deg)=90o=45o=0o\nFIG. 2. (Color online) (a){(c) The vector \feld of SOT components ~ \u001c\u0016;\u0017(EF), de\fned by Eq. (16), for di\u000berent directions of\n~ mon the unit sphere. The angular behavior in (a) and (c) shows that ~ \u001co;oand~ \u001ce;ebehave as \feld-like torques, while that in\n(b) shows that ~ \u001co;ebehaves as antidamping-like torque. Cartesian components [(d)-(f)] and magnitude of ~ \u001c\u0016;\u0017[(g){(i)] along\nthe trajectories denoted by solid lines in the corresponding panels (a)-(c). The magnitude of ~ \u001c\u0016;\u0017is divided by:j~ m\u0002^yjin (g);\nj~ m\u0002(~ m\u0002^y)jin (h) and 2( ~ m\u0001^z)j~ m\u0002^zjin (i).\nV. SPATIAL DEPENDENCE OF SOT\nCOMPONENTS AND PHYSICAL ORIGIN OF\nANTIDAMPING-LIKE SOT\nFigure 4(a) demonstrates that large values of\nantidamping-like SOT from Fig. 2 are spatially localized\naround the transverse edges of the F overlayer, for Fermi\nenergy inside and outside of the surface state gap induced\nby out-of-plane magnetic exchange coupling. While con-\nductance in this system is close to zero at the Dirac point,\nwe observe that the anti-damping torque does not de-\npend strongly on the Fermi energy. This suggests a high\ne\u000eciency of SOT per injected current for the Fermi en-\nergy close to the Dirac point. For the \feld-like SOT\nin Fig. 4(b) we see the torque independent of the co-\nordinate in the entire F/TI contact region, as expected\nfrom the phenomenology of the EE. In Fig. 4(c) we plot\nthe contribution of the Fermi energy electrons to the\nFM/TI interface induced MCA \feld. Even though to-\ntal\u001ci\ne;o(EF)\u00110, its spatially-resolved value plotted in\nFig. 4(d) is nonzero which can be removed by perform-\ning a proper gauge transformation.\nTo understand the origin of the anti-damping SOT let\nus \fnd an expression for the average of SOT around a\fxed axes~ m=~ m0+(~ m?cos(\u001e)+~ m0\u0002~ m?sin(\u001e))\u000e\u0012with\nsmall cone angle \u000e\u0012. By applying a rotation operator we\ncan align the \fxed axes along z-axis such that ~ m0= ^ez\nand~ m?= ^ex. In this case we have\nhTzi\u001e=\u0001surf\u000e\u0012\n2=Zd\u001e\n2\u0019ZdE\n2\u0019X\n\u000bf\u000bei\u001eT\u0000;\u000b;(18)\nwhere,T\u0000;\u000b=Tx\u000b\u0000iTy\u000band=refers to the imaginary\npart. Expanding the GFs in Eq. 12(b) to the lowest order\nwith respect to \u000e\u0012e\u0000i\u001e, we obtain,\nT\u0000;\u000b=\u0001surf\n4e\u0000i\u001e\u000e\u0012Trh\n1m\u001b\u0000Gt1m\u001b+Gt\u0000\u000bGy\nt(19)\n+1m\u001b\u0000Gt\u0000\u000bGy\nt1m\u001b+Gy\nti\n:\nPlugging this expression into Eq.(18), and using the iden-\ntity,Gt\u0000Gy\nt=iP\n\u000bGt\u0000\u000bGy\nt, in linear bias voltage\nregime we obtain,\nhTzi\u001e=Vb\u00012\nsurf\u000e\u00122\n16Tr[\u001a\"\"\nL1m\u001a##\nR1m\u0000\u001a\"\"\nR1m\u001a##\nL1m];(20)\nwhere,\u001a\u000b=\u0000iG<\n\u000b=Gt\u0000\u000bGy\nt, corresponds to the\ndensity matrix inside the F overlayer for the electrons7\n(a)\nTRL(EF)×10−2\n4.5\n4\n3.5\n3\n𝑥𝑦𝑧\nFIG. 3. (Color online) The transmission function TRL(EF)\nin Eq. (12a) for two-terminal heterostructure shown in Fig. 1\nat di\u000berent directions of magnetization ~ mon the unit sphere.\nThe Fermi energy is set at EF= 3:1 eV (which is 0 :1 eV above\nthe DP).\n02 04 0-1010\n2 04 0-505-10-\n20τ\nx,ze\n,e(EF)=0×10-4×10-2τye\n,e(EF) (1/a)L\nongitudinal Coordinate xF/TI (a)τxe\n,o(EF) (1/a)m=(0,0,1)τ\ny,ze\n,o(EF)=0τy,zo\n,o(EF)=0τ\nx,zo\n,e(EF)=0 EF=3.3eV \nEF=3.1eV(\nd)( c)(b)( a)×\n10-2τyo\n,e(EF) (1/a)×10-2τxo\n,o(EF) (1/a)\nFIG. 4. (Color online) (a){(d) Spatial dependence of SOT\ncomponents, \u001ci\n\u0016;\u0017(EF) (i2fx;y;zgand\u0016;\u00172fe;og), per\nunit length, for Fermi energy inside the magnetization in-\nduced gap around Dirac point ( EF= 3:1eV) and outside\nthe gap (EF= 3:3eV), for~ mk^zin Fig. 1. Their physical\nmeaning is explained in Fig. 2. The range of x-coordinate\ncorresponds to the length LF\nx= 40aof the F overlayer, while\nthe results are independent of the length of the TI layer un-\nderneath,LTI\nx.\n(holes) at the Fermi surface being injected from the lead\n\u000b(\f6=\u000b). The electron-hole analogy can be understood\nby de\fning the hole density matrix, iG>\n\u000b, from the iden-\ntity\u0000i(G<\n\u000b\u0000G>\n\u000b) = 2=(G) =\u001a\u000b+P\n\f6=\u000b\u001a\f. By con-\nsidering left-lead induced holes instead of right-lead in-\nduced electrons, we can interpret Eq.(20) as spin-resolved\nelectron-hole recombination rate, where opposite spins\nhave opposite contributions to the antidamping-like\nSOT. This picture focuses on the energy anti-dissipative\naspect of the phenomena and, since \u001a\u001b\u001b\nL(\u001a\u001b\u001b\nR) cor-\nFIG. 5. (Color online) SOT-induced magnetization trajecto-\nries~ m(t) under di\u000berent Vband~Bext= 0. Higher color inten-\nsity denotes denser bundle of trajectories which start from all\npossible initial conditions ~ m(t= 0) on the unit sphere. Solid\ncurves show examples of magnetization trajectories, while the\nwhite circles denote attractors of trajectories.\nresponds to the spin- \u001bright (left) moving electrons,\nEq. (20) suggests that spin-momentum locking natu-\nrally has a signi\fcant e\u000bect on the enhancement of\nthe antidamping-like SOT magnitude. In particular,\nin the case of F/TI interface, the enhancement of the\nantidamping-like SOT occurs when the spin-up/down is\nalong they-axis (~ m0k^y) which is the spin-polarization\ndirection of electrons passing through the surface of the\nTI induced by the EE. Additionally, in this case the\nantidamping-like SOT gets smaller away from the F/TI\ntransverse edge because the contribution of both of the\nleads to the spin density become identical. Therefore the\nanti-damping torque in this case is more localized around\nthe edge. This e\u000bect is more signi\fcant when the magne-\ntization is out of the plane and the Fermi energy is inside\nthe \u0001 surfcos\u0012gap on the TI surface.\nA alternative interpretation of the results can be\nachieved by considering Gt\u0000Gy\nt=iP\n\u000bGy\nt\u0000\u000bGt. In\nthis case, the average of the antidamping-like SOT is ex-\npressed by\nhTzi\u001e=Vb\n4Tr[T\"#\nLR\u0000T\"#\nRL]; (21)\nwhere the F overlayer induced spin-\rip transmission ma-\ntrix is de\fned as\nT\"#\n\u000b\f= (t\"#\n\u000b\f)yt\"#\n\u000b\f; (22)\nand\nt\"#\n\u000b\f=\u0001surf\u000e\u0012\n2p\n\u0000\u000bGt\u001b+Gtp\n\u0000\f: (23)\nAlthough Eq. (21) is obtained from perturbative con-\nsiderations, it looks identical to the Eq. (8) of Ref. 48\nwhere a spin-\rip re\rection mechanism at the edge of the\nF/2D-TI interface was recognized to be responsible for\nthe giant charge pumping (i.e., anti-damping torque) ob-\nserved in the numerical simulation.48Eq. (23) describes a\ntransmission event in which electrons injected from lead\n\u000b, get spin-\ripped (from up to down) by the FM and\nthen transmit to the lead \f. The path of the electrons\ndescribing this process is shown in Fig. 1. From the k-\nresolved results of the anti-damping torque (not shown8\nhere) we observe that while for the in-plane magnetiza-\ntion electrons moving in the same transverse direction\n(same sign for ky) on both left and right edges of the\nFM/TI interface contribute to the torque, in the case of\nout-of-plane magnetization for the left (right) edge of the\ninterface the local anti-damping torque is induced mostly\nby the electrons with ky>0 (ky<0).\nIt is worth mentioning that due to nonperturbative na-\nture of the SOT induced by the chiral electrons, the ap-\nproximation presented in this section which can as well\nbe obtained from the self energy corresponding to the\nvacuum polarization Feynman diagrams of the electron-\nmagnon coupled system59, does not capture the phenom-\nena accurately. This is evident in the angular dependence\nof the anti-damping torque which in the current section\nis considered up to second order e\u000bect ( \u000e\u00122), while the\ndivergence-like behavior in Figs.2(h) suggest a linear de-\npendence when the magnetization direction is close to the\ny-axis. This signi\fes the importance of the higher order\nterms with respect to \u000e\u0012that can not be ignored. The ap-\nproximation presented in this section also suggests that\nblocking the lower surface leads to the reduction of the\nanti-damping torque. However, in this case an electron\nexperiences multiple spin-\rip re\rections before transmit-\nting to the next lead and in fact it turns out that the ex-\nact results stay intact even if the lower surface is blocked.\nThis is similar to the conclusion made in Ref. 48 which\nshows the redundancy of blocking the lower edge of the\n2D-TI to obtain a nonzero pumped charge current from\nprecessing FM as proposed in Ref. 47.\nAlthough spin-momentum locking of the surface state\nof the TI resembles the 2D Rashba plane, in the case of\nTI surface state the cones with opposite spin-momentum\nlocking reside on opposite surface sides of the TI slab\nwhile in the case of a Rashba plane they are only sepa-\nrated by the SOC energy. This means one can expect a\nsmaller SOT for a FM on top of a 2D Rashba plane due\nto cancellation of the e\u000bects of the two circles with op-\nposite spin-momentum locking, where the nonzero anti-\ndamping torque originates from the electron-hole asym-\nmetry.\nVI. LLG SIMULATIONS OF MAGNETIZATION\nDYNAMICS IN THE PRESENCE OF SOT\nIn order to investigate ability of predicted\nantidamping-like SOT to switch the magnetization\ndirection of a perpendicularly magnetized F overlayer in\nthe geometry of Fig. 1, we study magnetization dynamics\nin the macrospin approximation by numerically solving\nLLG equation at zero temperature supplemented by\nSOT components analyzed in Sec. III\n@~ m\n@t=1\n2\u0019[~ \u001co;e(~ m;EF) +~ \u001co;o(~ m;EF)]eVb+\r~Bext\u0002~ m\n+~ m\u0002\u0014\n\u000b(~ m)\u0001@~ m\n@t\u0015\n+ (~ m\u0001^z)(~ m\u0002^z)\u0001MCA:(24)\nFIG. 6. (Color online) Phase diagram of the magnetization\nstate in lateral F/TI heterostructure from Fig. 1 as a function\nof an in-plane external magnetic Bextk^xandVb(i.e., SOT\n/Vb). Thick arrows on each of the panels (a){(d) show the\ndirection of sweeping of Bext\nxorVbparameter. The small-\nness of central hysteretic region along the Vb-axis, enclosed\nby white dashed line in panel (b) and (d), shows that low\ncurrents are required to switch magnetization from mz>0\ntomz<0 stable states.\nHere\ris the gyromagnetic ratio, \u000b(~ m)ij =\n\u00012\nsurfTij(~ m;EF)=8\u0019is the dimensionless Gilbert damp-\ning tensor, and \u0001 MCA = \u00010\nMCA +j~Te;ej=j(~ m\u0001^z)(~ m\u0002^z)j,\nwhere \u00010\nMCA represents the intrinsic MCA energy of the\nFM. We solve Eq. (24) by assuming that the Gilbert\ndamping is a constant (its dependence on ~ mis relegated\nto future studies) and ignore the dependence of \u0001 MCA\non~ mandVbwhile retaining its out-of-plane direction.\nFigure 5 shows the magnetization trajectories for all\npossible initial conditions ~ m(t= 0) on the unit sphere\nunder di\u000berent Vb. AtVb= 0, the two attractors are\nlocated as the north and south poles of the sphere. At\n\fniteVb, the attractors shift away from the poles along\nthez-axis within the xz-plane, while additional attractor\nappears on the positive (negative) y-axis under negative\n(positive)Vb. Note that the applied bias voltage Vbdrives\ndc current and SOT proportional to it in the assumed\nlinear-response transport regime.\nFigure 6 shows the commonly constructed3,4,15,27\nphase diagram of the magnetization state in the pres-\nence of an external in-plane magnetic \feld Bextk^xand\nthe applied bias voltage Vb(i.e., SOT/Vb). The thick\narrows in each panel of Fig. 6 denote the direction of the\nsweeping variable|in Fig. 6(a) [6(b)] we increase [de-\ncrease]Vbslowly in time, and similarly in Fig. 6(c) [6(d)]\nwe increase [decrease] the external magnetic \feld gradu-\nally. The size of hysteretic region in the center of these\ndiagrams, enclosed by white dashed line in Figs. 6(b) and\n6(d), measures the e\u000eciency of switching.3,4,15,27Since\nthis region, where both magnetization states mz>0 and\nmz<0 are allowed, is relatively small in Figs. 6(a) and9\n6(b), magnetization can be switched by low Bext\nxand\nsmallVb(or, equivalently, small injected dc current), akin\nto the phase diagrams observed in recent experiments.15\nAlthough we considered the FM a single domain, the\nfact that the anti-damping component of the SOT is\nmainly peaked around the edge of the FM/TI interface\nsuggests that it is be more feasible in realistic cases to\nhave the local magnetic moments at the edge of the FM\nswitch \frst and then the total magnetization switches\nby the propagation of the domain walls formed at the\nedge throughout the FM60,61. Therefore, a micromag-\nnetic simulation of the system is required to investigate\nswitching phenomena in large size systems which we rel-\negate to future works.\nVII. CONCLUSIONS\nIn conclusion, by performing adiabatic expansion of\ntime-dependent NEGFs,33,34we have developed a frame-\nwork which yields formulas for spin torque and charge\npumping as reciprocal e\u000bects to each other connected by\ntime-reversal, as well as Gilbert damping due to SOC. It\nalso introduces a novel way to separate the SOT com-\nponents, based on their behavior (even or odd) under\ntime and bias voltage reversal, and can be applied to\narbitrary systems dealing with classical degrees of free-\ndom coupled to electrons out of equilibrium. For the\ngeometry28proposed in Fig. 1, where the F overlayercovers (either partially or fully) the top surface of the TI\nlayer, we predict that low charge current \rowing solely\non the surface of TI will induce antidamping-like SOT on\nthe F overlayer via a physical mechanism that requires\nspin-momentum locking on the surface of TIs|spin-\rip\nre\rection at the lateral edges of a ferromagnetic island\nintroduced by magnetic proximity e\u000bect onto the TI sur-\nface. This mechanism has been overlooked in e\u000borts to\nunderstand why SO-coupled interface alone (i.e., in the\nabsence of SHE current from the bulk of SO-coupled non-\nferromagnetic materials) can generate antidamping-like\nSOT, where other explored mechanisms have included\nspin-dependent impurity scattering at the interface,55\nBerry curvature mechanism,25,26as well as their com-\nbination.56\nThe key feature for connecting experimentally ob-\nserved SOT and other related phenomena in F/TI het-\nerostructures (such as spin-to-charge conversion28,36,58)\nto theoretical predictions is their dependence2,24on the\nmagnetization direction. The antidamping-like SOT pre-\ndicted in our study exhibits complex angular dependence,\nexhibiting \\nonperturbative\" change with the magnetiza-\ntion direction in Fig. 2(h), which should make it possible\nto easily di\u000berentiate it from other competing physical\nmechanisms.\nACKNOWLEDGMENTS\nF. M. and N. 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Svedlindh1 \n1Department of Engineering Sciences , Uppsala University, Box 534, SE -751 21 Uppsala, Sweden  \n2Seagate Technology, BT48  0BF, Londonderry, United Kingdom  \n3Department of Physics and Astronomy, Uppsala University, Box 516, SE -751 20 Uppsala,  \nSweden  \n \nThe effect s of rhenium doping  in the range 0 – 10 at% on the static and dynamic magnetic \nproperties of Fe65Co35 thin films have been studied experimentally as well as with first principles \nelectronic structure calculations  focussing on the change of the saturation magnetization (𝑀𝑠) and \nthe Gilbert damping parameter ( 𝛼) Both experiment al and theoretical results show that 𝑀𝑠 \ndecreases with increasing Re doping level, while at the same time 𝛼 increases. The experimental \nlow temperature saturation magnetic induction exhibits a 2 9% decrease,  from 2.3 1T to 1. 64T, in \nthe investigated doping concentration range , which is more than predicted by the theoretical \ncalculations. The room temperature value of the damping parameter obtained from ferromagnetic \nresonance measurements , correcting for extrinsic contributions  to the damping,  is for the undoped \nsample 2.7×10−3, which is close to the theoretically calculated Gilbert damping parameter . With \n10 at% Re doping , the damping parameter increases to 9.0×10−3, which is in good agreement \nwith the theoretical value of 7.3×10−3. The increase in damping parameter with Re doping is \nexplained by the increase in density of states at Fermi level, mostly contributed by the s pin-up \nchannel of Re. Moreover, both experimental and theoretical values  for the da mping parameter are \nobserved to be weakly decreas ing with decreasing temperature .  \n  1. INTRODUCTION  \nDuring the last decades , thin films of soft magnetic alloys such as NiFe and FeCo  have been in \nfocus  due to possible use in applications such as spin valves ,1,2 magnetic tunneling junctions ,3,4,5 \nspin injectors ,6 magnetic storage technologies  and in particular in magnetic recording write heads .7 \nBeside s spintronic and magnetic memory devices , such materials are useful for shielding \napplications that are necessary in order to reduce the effect of electromagnetic fields created  by \nelectronic devices. The magnetic damping parameter  of the material play s a critical role for the \nperformance of such spintronic and memory devices as well as for shielding applications.  On the \none hand, a low damping parameter is desired in order to get low critical switching current in \nspintronic devices .8,9,10 On the other hand , a high damping parameter is necessary in order to \nreduce the magetization switching time in magnetic memory devices and to be able to operate \ndevices at high speeds .11 FeCo alloys are promising materials for  high frequency spintronic \napplications and magnetic recording devices due to their high saturation magnetization  (𝑀𝑠), high \npermeability, thermal stability and comparably high resistivity .12,13,14 One possible drawback is \nthat FeCo alloy s exhibit  high coercivity  (𝐻𝑐), which is not favorable for such applications , however \nthis problem can be solved by thin film growth  on suitable buffer layer s.15,16,12 Except coercivity \nproblems, the damping parameter of these materials should be increased to make them com patible \nfor high speed devices . \nDynamic properties of  magnetic materials are highly dependent on the damping parameter. This \nparameter is composed of both intrinsic and extrinsic contributions. The intrinsic contribution is \ncalled  the Gilbert damping and  depends  primarily  on the spin-orbit coupling .17 Intrinsic damping \nis explained as scattering of electrons by phonons and magnons .18,19 Beside s electron scattering , \ndue to the close relation between magnetocrystalline  anisotropy and spin-orbit coupling , it can be \nassumed that the intrinsic damping is also related to the magnetocrystalline anisotropy constant .20 \nRegarding extrinsic damping , there can be a number of different contributions. The most common \ncontribution originates  from two magnon scattering  (TMS) .21 However , this contribution vanishes \nwhen ferromagnetic resonance  (FMR) measurements are performed by applying the static \nmagnetic field  along  the film normal  in inplane anisotropic thin films .22 Beside s TMS , there are \nsome other extrinsic contributions to the damping that are not possible to get rid of  by changing \nthe measurement configuration . One of these contributions  is radiative damping , which arises from \ninductive coupling between  the precessing magnetization and the waveguide  used for FMR \nmeasurem ents.23 Another contribution  for metallic ferromagnetic films is the eddy current \ndamping related to microwave magnetic field induced eddy currents in the thin film s during  \nmeasurement s.23,24 \nIn order to make a soft magnetic thin film  suitible for  a specific  applica tion, taking into account  \nrequirements set by the device application , its damping paramete r should be tailored. As mentioned \nabove , an increased damping parameter is necesssary for devices requiring high switching speed . \nSeveral efforts have been made  for enhanching the damping parameter of soft magnetic materials. \nNiFe alloys constitu te one of the most studied systems in this respect . The most common way to enhance  the intrinsic damping of an all oy is to dope it with differ ent elements . Rare earth elements  \nwith large spin-orbit coupling  have revealed  promising results  as dopant s in terms of increas ed \ndampin g parameter .25,26,27 3d, 4d and 5d transition metals  dopants have also been studied \nexperimentally , revealing an increase of the damping parameter .28,29 Beside s experimental results , \ntheoretical calculations support the idea that transition metals and especially 5d elements can \nenhance the damping parameter of NiF e alloys due to   scattering in presence of chemical disorde r \n, as well as due to the effect of spin -orbit coupling .30 \nAlthough NiFe alloys have been the focus in several extensive studies, FeCo alloys have so far not \nbeen studied to the same extent . Attempts have been made  to  dope  FeCo with Yb,20 Dy,31 Gd,32 \nand Si ,33 where in all cases an increase of the  damping parameter  was observed . Apart from  doping \nof alloys , the addition of  adjacent layers to NiFe and CoFe has also been studied . In particular , \nadding layers consisting of rare earth elem ents with large orbital moment s  gave positive results in \nterms of increased damping parameter .34 \nFe65Co35 alloy s are attractive material s because of high 𝑀𝑠 and reduced 𝐻𝑐 values. However , not \nmuch is known about the magnetic damping mechanism s for this composition . Since it is of \ninterest for high data rate magnetic memory devices, the damping parameter should be increased \nin order to make the magnetic switching faster. To the best of our knowledge , systematic doping \nof Fe 65Co35 with 5d elements has not been studied so far  experimentally . Some of us have found \nfrom ab initio  calculations  that 5d transition metal dopants can increase the damping parameter \nand Re is one of the potential candidates.35 Re is particularly interesting as it has a nice compromise \nof having not so much reduced saturation magnetization and a quite enh anced damping parameter. \nIn this work, we have perfomed a systematic ab initio study of Fe65Co35 doped with increasing Re \nconcentration to find an increasing  damping parameter . The theoretical prediction s are confirmed  \nby results obtained from temperature  dependent FMR measurements performed on Re doped  \nFe65Co35 films.  \n \n2. EXPERIMENTAL  AND THEORETIC AL METHOD S \nRhenium  doped Fe 65Co35 samples were prepared by varying the Re concentration from 0 to 10.23  \nat%. All samples were deposited using DC magnetron sputtering on Si/SiO 2 substrate s. First a 3 \nnm thick  Ru seed layer was deposited on the Si/SiO 2 substrate followed by room temperature \ndeposition of  20 nm  and 40 nm  thick Re -doped  Fe65Co35 films by co -sputtering between Fe 65Co35 \nand  Re target s. Finally, a 3 nm thick Ru layer was deposited as a capping layer over  the Re -doped  \nFe65Co35 film. The nominal Re concentration was derived from the calibrated deposition rate used \nin the deposition system. The nominal Re doping concentration s of the Fe65Co35 samples are as \nfollows ; 0, 2.62, 5.45 and  10.23  at%.  \nThe crystalline structure  of the fims were  investigated by utilizing grazing incident X -Ray \ndiffraction  (GIXRD). The i ncidence angle was fixed at  1o during  GIXRD measurements  and a CuKα source was used.  Accurate values for film thickness and interface roughness were  \ndetermined by X -ray reflectivity  (XRR) measurements.   \nBeside XRD , composition and areal density of the films were deduced by Rutherford \nbackscattering spectrometry36 (RBS)  with ion beams of 2 MeV 4He+ and 10 MeV 12C+. The beams \nwere provided by a 5 MV 15SDH -2 tandem accelerator at the Tandem Laboratory at Uppsala \nUniversity. The experiments were performed with the incident beam at 5° with respect to the \nsurface normal and scattering angles of 170° and 120° . The experimental data was evaluated with \nthe SIMNRA program .37 \nIn-plane magnetic hysteresis measurments were performed using a Magnetic Property \nMeasurement System (MPMS, Quantum Design) .  \nFerromagnetic  resonance  measurements were  performed using  two different techniques. First in-\nplane X -band (9.8 GHz) cavity FMR measurements were performed . The setup  is equipped with a \ngoniometer making it possible to rotate the sample with respect  to the applied magnetic field; in \nthis way the in -plane anisotropy fields of the different samples have been determine d. Beside s \ncavity FMR  studies , a setup for broadband out-of-plane FMR measurements have been utilized . \nFor out -of-plane measurements a vector network analyzer (VNA) was used. Two ports of the VNA \nwere connected to a coplanar waveguide (CPW) mounted on a Ph ysical Property Measurement \nSystem (PPMS, Quantum Design) multi -function probe . The PPMS is equipped with a 9 T \nsuperconducting magnet, which is needed to saturate Fe65Co35 films out -plane and to detect the \nFMR signal. The broadband FMR  measurements were carried out a t a fixed microwave frequency \nusing the field -swept mode, repeating the measurement for different f requencies in the range 15 – \n30GHz.  \nThe theoretical calculations are based on spin -polarized relativistic m ultiple scattering theory using \nthe Korringa -Kohn -Rostoker (KKR) formalism implemented in the  spin polarized relativistic \nKKR code  (SPR-KKR) . The Perdew -Burke -Ernzerhof (PBE) exchange -correlation functional \nwithin generalized gradient approximation was used. The equilibrium lattice parameter s were \nobtained by energy minimization for each composition. Substitutional disorder was treated within \nthe Coherent Potential Approximation (CPA). The damping parameters were calcu lated by the \nmethod proposed by Mankovsky et al.,38 based on the ab initio Green's function technique and  \nlinear res ponse formalism where one takes into consi deration scattering processes as well as spin -\norbit coupling built in Dirac's relativistic formulation. The calculations of Gilbert damping \nparameters at finite temperatures were  done using an alloy -analo gy model of atomic displacements \ncorresponding to the thermal average of the root mean square displacement at a given temperature.  \n3. RESULTS AND  DISCUSSION  \nRe concentrations and layer thickness (areal densities) of the 20 nm  doped films were obtained by \nRBS experiments. RBS employing a beam of 2 MeV He primary ions was used to deduce the areal \nconcentration of each layer. Additional measurements with 10 MeV C probing particles permit to resolve the atomic fractions of Fe,  Co and Re. The spectra for the  samples with different Re \nconcentration are shown in Fig. A1 . The measured Re concentrations are 3.0±0.1  at%, 6.6±0.3  at% \nand 12.6±0.5  at%. Moreover, the results for Fe and Co atomic fractions show that there is no \npreferential replacement by Re , implying that the two elements are replaced according to their \nrespective concentration . \nFigure 1 (a) shows GIXRD spectra in the 2𝜃-range from 20o to 120o for the Fe65Co35 films with \ndifferent Re concentration. Diffraction peaks corresponding to the body centered cubic Fe 65Co35 \nstructure have been indexed in the figure; no other diffraction peaks appear in the different spectra.  \nDepending on the Re -dopant concentration shi fts in the peak positions are observed, the diffraction \npeaks are suppressed to lower 2𝜃-values with increasing dopant concentration . The shift for the \n(110) peak for the different dopant concentrations is given as an inset in Fig. 1 (a). Similar shifts \nare observed for the other diffraction peaks. This trend in peak shift is an experimental evidence \nof an increasing amount of Re dopant within the deposited thin films. Since the peaks are shifted \ntowards lower 2𝜃-values  with increasing amount of Re dopant , the lattice parameter increases with \nincreasing Re concentration.39 Figure 1 (b) shows the experimental as well as theoretically \ncalculated lattice parameter versus Re concentration.  The qualitative agreement between theory \nand experiment is obtained. However, t he rate of lattice parameter increase with increasing Re \nconcentration is larger for the theoretically calculated lattice parameter. This is not unexpected as \nthe generalized gradie nt approximation for the exchange -correlation potential has a tendency to \noverestimate the lattice parameter. Another possible explanation for the difference in lattice \nparameter is that the increase of the lattice parameter for the Re -doped Fe 65Co35 films  is held back \nby the compressive strain due to lattice mismatch with Si/SiO 2/Ru. XRR measurements revealed \nthat the surface roughness of the Fe 65Co35 films is less than 1 nm , which cannot affect static and \nmagnetic properties drastically. Results from XRR measurements are given in table 1.  \nRoom temperature normalized magnetization curves for the Re-doped Fe 65Co35 films  are shown  \nin Fig. 2 (a) . The coercivity for  all films is in the range of 2 mT and all films, except for the 1 2.6 \nat% Re doped film that show a slightly rounded hysteresis loop, exhibit rectangular hysteresis \nloops. The low value for the coercivity is expected for seed layer grown films .15 The \nexperimentally determined  low temperature saturation magnetization  together with the \ntheoretically calculated magnetization versus Re concentra tion are shown in Fig. 2 (b). As \nexpected, both experimental and theoretical r esults show that the saturation magnetization \ndecreases with increasing Re concentration . A linear decrease in magnetization is observed in the \ntheoretical calculations whereas a non -linear behavior is seen in the experimental data.  \nAngle resolved cavity FMR measurements were used to study the in -plane magnetic anisotropy . \nThe angular -dependent resonant field ( 𝐻𝑟) data was analyzed using the following equation ,40 \n 𝑓=µ0𝛾\n2𝜋[{𝐻𝑟cos(𝜙𝐻−𝜙𝑀)+𝐻𝑐\n2cos4(𝜙𝑀−𝜙𝐶)+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}{𝐻𝑟cos(𝜙𝐻−\n𝜙𝑀)+𝑀𝑒𝑓𝑓+𝐻𝑐\n8(3+cos4(𝜙𝑀−𝜙𝐶))+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}]12⁄\n,  (1) where  𝑓 is the cavity resonance frequency  and 𝛾 is the gyromagnetic ratio . 𝜙𝐻, 𝜙𝑀, 𝜙𝑢 and 𝜙𝐶 \nare the in -plane directions for the magnetic field,  magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively, with respect to  the [100 ] direction of the Si substrate.   𝐻𝑢=2𝐾𝑢\nµ0𝑀𝑠 and \n𝐻𝑐=4𝐾𝑐\nµ0𝑀𝑠  are the uniaxial and cubic anisotropy fields, where 𝐾𝑢 and 𝐾𝑐 are the uniaxial and cubic \nmagnetic anisotropy constants , and  𝑀𝑒𝑓𝑓 is the effective magnetization.  Fitting parameters were \nlimited to 𝑀𝑒𝑓𝑓,  𝛾 and 𝐻𝑢, since the Hr versus ϕH curves did not give any indication of a cubic \nanisotropy.  \nFigure 3 shows 𝐻𝑟 versus 𝜙𝐻 extracted from the angular -dependent FMR measurements together \nwith fits according to Eq. (1), clearly revealing dominant twofold uniaxial in -plane magnetic \nanisotropy. Extracted anisotropy field and effective magnetization values are given in  Table 2 . The \nresults show that 𝐻𝑢 is within the accuracy of the experiment independent of the Re concentration . \nTemperature dependent o ut-of-plane FMR measurements  were  performed in the temperature range \n50 K to 300 K  recording the complex transmission coefficient 𝑆21. Typical field -swept results for \nthe r eal and imaginary components of  𝑆21 for the undoped and 1 2.6 at% Re-doped samples are \nshown  in Fig. 4. The field -dependent 𝑆21 data was fitted to the following set of equations,41  \n𝑆21(𝐻,𝑡)=𝑆210+𝐷𝑡+𝜒(𝐻)\n𝜒̃0    \n𝜒(𝐻)=𝑀𝑒𝑓𝑓(𝐻−𝑀𝑒𝑓𝑓)\n(𝐻−𝑀𝑒𝑓𝑓)2−𝐻𝑒𝑓𝑓2−𝑖𝛥𝐻 (𝐻−𝑀𝑒𝑓𝑓) .  (2) \nIn these equations 𝑆210 corresponds to  the non-magnetic contribution to the complex transmission \nsignal , 𝜒̃0 is an imaginary function of the microwave frequency and film thickness  and 𝜒(𝐻) is the \ncomplex susceptibility  of the magnetic film. The term 𝐷𝑡 accounts for a linear drift of the recorded \n𝑆21 signal.  𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠, where 𝐻𝑘⫠ is the perpendicular anisotropy field  and 𝐻𝑒𝑓𝑓=2𝑓\n𝛾µ0. The \n𝑆21 spectra were  fitted to Eq. (2 ) in order to extract the linewidth 𝛥𝐻 and 𝐻𝑟 values. Fits t o Eq. (2) \nare shown as solid lines in Fig. 4.  \nThe experimentally measured total d amping parameter ( 𝛼𝑡𝑜𝑡𝑎𝑙 ), including both the intrinsic \ncontribution  (Gilbert damping) and extrinsic contributions , was extracted by fitting 𝛥𝐻 versus \nfrequency to the following expression, 41 \nµ0𝛥𝐻=4𝛼𝑡𝑜𝑡𝑎𝑙 𝑓\n𝛾+µ0𝛥𝐻0 ,   (3) \nwhere 𝛥𝐻0 is the frequency independent linewidth broadening  due to sample inhomogeneity . \nBeside s 𝛼𝑡𝑜𝑡𝑎𝑙 , 𝑀𝑒𝑓𝑓 can also be  extracted by fitting the 𝐻𝑟 versus frequency results to the \nexpression  µ0𝐻𝑟=2𝜋𝑓\n𝛾+µ0𝑀𝑒𝑓𝑓 .   (4) \nTypical temperature dependent results for 𝑓  versus 𝐻𝑟  and 𝛥𝐻 versus 𝑓  are shown in Fig. 5 for \nthe 1 2.6 at% Re -doped Fe65Co35 film. Extracted values of 𝑀𝑒𝑓𝑓  at different temperatures are given \nin Table 3 for all samples . As expected, the results show that 𝑀𝑒𝑓𝑓 decreas es with increasing \ndopant concentration.  Since 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠  and the film thickness is large enough to make a \npossible contribution from out -of-plane anisotropy negligible one can make the justified \nassumption that 𝑀𝑒𝑓𝑓≈𝑀𝑠. The analysis using Eqs. (2) – (4) also give values for the Land é 𝑔-\nfactor ( 𝛾=𝑔µ𝐵\nħ), yielding 2.064 and 2.075 for the undoped and 12.6 at% doped samples, \nrespectively (similar values are obtained at all temperatures).  \nAs indicated above, the d amping parameters extracte d from FMR measurements ( 𝛼𝑡𝑜𝑡𝑎𝑙) include  \nboth intrinsic and extrinsic contributions. One of the most common extrinsic contribution s is TMS , \nwhich is avoided in this study by measuring FMR with the magnetic field applied out of the film \nplane. Except TMS , extrinsic contributions  such as eddy curr ent damping  and radiative damping \nare expected to contribute the measured damping . In a metallic  ferromagnet, which is placed on \ntop of a CPW , precession  of spin waves induces AC currents in the ferromagnet ic film, thereby \ndissipating energy . The radiative  damping has similar origin as the eddy current damping, but here \nthe precession of the magnetization induces microwave -frequency currents in the CPW where \nenergy is dissipated. Thus,  there are two extrinsic  contributions to the measured damping ; one that \nis caused by eddy currents in the ferromagnet ic film (𝛼𝑒𝑑𝑑𝑦) and  another one caused by  eddy \ncurrents in the CPW ( 𝛼𝑟𝑎𝑑).23 In order to  obtain the reduced  damping of the films  (𝛼𝑟𝑒𝑑), which \nwe expect to be close to the intrinsic damping of the films, the extrinsic contributions should be \nsubtracted from  𝛼𝑡𝑜𝑡. We have neglected any contribution to the measured damping originating \nfrom spin -pumping into seed and capping layers. However, since spin -pumping in low spin -orbit \ncoupling materials like Ru with thickness quite less than the spin -diffusion length is quit e small, \nthe assumption  of negligible contribution from spin -pumping  is justified.  The t otal damping can \nthus be given as  𝛼𝑡𝑜𝑡=𝛼𝑟𝑒𝑑+𝛼𝑟𝑎𝑑+𝛼𝑒𝑑𝑑𝑦 .  \nWhen the precession of the magnetization is assumed to be uniform in the sample , the expression \nfor  radiative damping can be given as23 \n𝛼𝑟𝑎𝑑=𝜂𝛾µ02𝑀𝑠𝛿𝑙\n2𝑍0𝑤 ,  (5) \nwhere 𝑍0 =50 Ω is the waveguide impedance, 𝑤=240 µm is the width  of the CPW center \nconductor , 𝜂 is a dimensionless parameter that accounts for FMR mode profile, δ is the thickness \nand 𝑙 is the length of the sample. The l ength of all samples were  4mm and the thickness  20nm for  \nthe undoped and 1 2.6 at% Re-doped films and 40nm for  the 3.0 at% and 6.6 at% Re-doped films. \nTemperature dependent radiative damping contributions for all Fe 65Co35 films are given  in Table \n4. Beside s 𝛼𝑟𝑎𝑑, the 𝛼𝑒𝑑𝑑𝑦 contribution should also be calculated and extracted from 𝛼𝑡𝑜𝑡𝑎𝑙 to extract \nthe reduced  damping parameter. 𝛼𝑒𝑑𝑑𝑦 can be estimated by the expression23 \n𝛼𝑒𝑑𝑑𝑦 =𝐶𝛾µ02𝑀𝑠𝛿2\n16𝜌 ,  (6) \nwhere  𝐶 is a parameter describing the  distribution of eddy current s within the films  and its value \nis 0.5 in our studied samples and 𝜌 is the resistivity of the films. Resistivity is measured for all \nfilms with different dopant concentrations at different temperatures. It is in the range of 8.2×10-8 \nto 5.6 ×10-8 𝛺𝑚 for undoped, 5.7 ×10-7 to 5.3 ×10-7 𝛺𝑚 for 3.0  at% doped , 6.9 ×10-7 to 6.1 ×10-\n7 𝛺𝑚 for 6.6  at% doped and 3.9×10-7 to 3.6 ×10-7 𝛺𝑚 for 12.6 at% doped films. Temperature \ndependent eddy current damping contributions , which are negligible,  for all Fe 65Co35 films are \ngiven in Table  5. \n𝛼𝑡𝑜𝑡 (filled symbols) and 𝛼𝑟𝑒𝑑  (open symbols) versus temperature for the differently Re -doped \nFe65Co35 films are shown in Fig. 6 .  Both  damping parameter s slowly decrease  with decreasing \ntempera ture. Moreover, the damping parameter increases with increasing Re concentration; the \ndamping parameter is 4 times as large for  the 12.6 at% Re -doped sample compared to the undoped \nsample . Since the damping parameter depends both on disorder induced scattering and spin-orbit \ncoupling, the observed enhanc ement of the damping parameter can emerge from the electronic \nstructure of the  alloy and  large spin -orbit coupling of Re.  \nA c omparison between temperature dependent experimental 𝛼𝑡𝑜𝑡 and 𝛼𝑟𝑒𝑑 values and \ntheoretically calculated intrinsic damping parameters is shown in Fig. 7 for the undoped and 12.6 \nat% Re -doped Fe 65Co35 films.  In agreement with the experimental results, the theoretically \ncalculated damping parameters decrease in magnitude with decreasing temperature . It has been \nargued  by Schoen et al., 42 that the contribution to the intrinsic Gilbert damping parameter comes \nprimarily from the strong electron -phonon coupling at high temperatures  due to interband \ntransition whereas at a low temperature, density of states at Fermi level (𝑛(𝐸𝐹)) and spin -orbit \ncoupling  give rise to intraband transition.  In Fig. 8, we show the correspondence between the \ncalculated damping parameter at 10 K with the density of states (spin up +spin down) at Fermi \nlevel for varying Re concentration. The increasing trend in both  properties is obviously seen. The \nincrease in DOS mainly comes from increasing DOS at Re sites in the spin -up channel. In the \ninset, the calculated spin -polarization as a function of Re concentration is shown. Spin polarization \nis defined as   𝜁=𝑛(𝐸𝐹)↑−𝑛(𝐸𝐹)↓\n𝑛(𝐸𝐹)↑+𝑛(𝐸𝐹)↓ where the contribution from both spin channels are seen. It is \nclearly observed that Re doping decreases the spin polarization.  \n \nOne should note that a quantitative comparison between theory and experiment requires more \nrigoro us theoretical considerations. The difference  between experimental and theoretical results \nfor the damping parameter may be  explained by the incompleteness of the model used to calculate \nthe Gilbert damping parameter  by neglecting several complex scatterin g processes. Firstly, the effect of spin fluctuations was neglected, which in principle could be considered in the present \nmethodology if the temperature dependent magnetization and hence information about the \nfluctuations of atomic moments were available from Monte -Carlo simulations. Other effects such \nas non-local damping and more sophisticated treatment of atomic displac ements in terms of \nphonon self -energies40  that may contribute to the relaxation of the magnetization in magnetic thin \nfilm materials  have been neglected . Nevertheless, a qualitative agreement has been achieved where  \nboth experimental  and theoretical results show that there is a significant increase of the damping \nparameter with increasing concentration of Re.  \n \n4. CONCLUSION  \nStatic and dynamic magnetic properties of rhenium doped Fe 65Co35 thin films have been \ninvestigated and clarified in a combined experimental and theoretical study. Results from first \nprinciples theoretical calculations show that the saturation magnetization gradually decreases with \nincreasing Re concentration, from 2.3T  for the undoped sample to 1.95T for the 10% Re -doped \nsample. The experimental results for the dependence of the saturation magnetization on the Re -\ndoping are in agreement with the theoretical results, although indicating a more pronounced \ndecrease of the saturation magnetization for the largest doping concentrations. The theoretical \ncalculations show that the intrinsic Gilbert damping increases with increasing Re concentration; at \nroom temperature the damping parameter  is 2.8×10−3, which increases to 7.3×10−3 for the 10  \nat% Re -doped sample. Moreover, temperature dependent calculations of the Gilbert damping \nparameter reveal a weak decrease of the value with decreasing temperature . At a low temperature, \nour theoretical analysis showed the prominence of intra band contribution arising from an increase \nin the density of states at Fermi level. The experimental results for the damping parameter were \ncorrected for radiative and eddy current contributions to the measured damping parameter and \nreveal similar trends as observed in the theoretical results; the damping parameter increases with \nincreasing Re concentration and the damping parameter value decreases with decreasing \ntemperature. The room temperature value for the reduced damping paramet er was  2.7×10−3 for \nthe undoped sample, which increased to 9.0×10−3 for the 1 2.6 at% Re -doped film. The \npossibility to e nhanc e the damping parameter for Fe65Co35 thin films is a  promising result since \nthese materials are used in magnetic memory applications and higher data rates are achievable if \nthe damping parameter of the material is increased.   \n \nACKNOWLEDGEMENT  \nThis work is supported by the Knut and Alice Wallenberg (KAW) Fou ndation, Grant No. KAW \n2012.0031 and by the Marie Curie Action “Industry -Academia Partnership and Pathways” (ref. \n612170, FP7 -PEOPLE -2013 -IAPP). The authors acknowledge financial support from Swedish \nResearch Council (grant no. 2016 -05366) and Carl Tryggers Stiftelse (grant no. CTS 12:419 and \n13:413). The simulations were performed on resources provided by the Swedish National Infrastructure (SNIC) at National Supercomputer Centre at Link öping University (NSC). M. 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Nembach, O. Eriksson, O. \nKaris, and J. M. Shaw, Nature Physics, 12, 839 –842 (2016).  \n \n     \nFigure 1 (a) GIXRD plot for Fe 65Co35 films with dif ferent Re concentrations. S hift of (110) peak \ndiffraction peak with Re concentration is given as insert . (b) Lattice parameter versus Re \nconcentration.  Circles are lattice parameters extracted from XRD measurements and squares are \ncalculated th eoretical values. Line s are guide to the eye. \n  \nFigure 2 (a) Normalized room temperature magnetization versus magnetic field for Fe 65Co35 \nfilms with different Re concentration . (b) Low temperature saturation magnetization versus Re \nconcentration. Circles  are experimental data and squares  corresponding calculated results.  \nExperimental 𝝁𝟎𝑴𝒔 values were extracted from temperature dependent FMR results. Lines are \nguide s to the eye .  \n  \n \n \nFigure 3 𝝁𝟎𝑯𝒓 versus in -plane angle of magnetic field 𝝓𝑯 for different dopant concentrations  of \nRe. Black squares are experimental  data and red line s are fits to Eq. (1).  \n  \n \n \nFigure 4 Room temperature  real (a and c) and imaginary  (b and d)  𝑺𝟐𝟏 components versus out -\nof-plane magnetic field  for Fe65Co35 thin  films with 0% and 12.6 at% Re recorded at 20GHz . \nBlack squares are data points and red lines are fit s to Eq. (2). \n \n  \n \n \nFigure 5 (a) Frequency versus  𝝁𝟎𝑯𝒓 values at different temperatures for  the Fe65Co35 thin film \nwith 12.6 at% Re. Coloured lines correspond to fits to Eq. ( 4). (b) Linewidth 𝝁𝟎∆𝑯 versus \nfrequency at different temperatures for the same Re doping concentration. Coloured  lines \ncorrespond to  fits to Eq. ( 3). Symbols represent  experimental  data. \n  \n \n \nFigur e 6  𝜶𝒕𝒐𝒕 versus  temperature for Fe 65Co35 thin films with different concentration of Re. \nBesides showing  𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of radiative \ndamping and eddy current damping contributions from  𝜶𝒕𝒐𝒕. Error bars are  given for measured \n𝜶𝒕𝒐𝒕 (same size as symbol size ). \n  \n \nFigur e 7 𝜶𝒕𝒐𝒕 versus  temperature for Fe 65Co35 thin films with 0  at% and 1 2.6 at% concentration \nof Re. Beside s showing  𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of \nradiative damping and eddy current damping contribution s from  𝜶𝒕𝒐𝒕. In addition to \nexperimental results theoretically calculated intrinsic damping parameters  are given for the \nsimilar  concentrations  of Re . Error bars are given for  measured 𝜶𝒕𝒐𝒕 (same size as symbol size) . \n  \n \n \nFigure 8  Calculated density of states at Fermi level (left axis) and damping parameter (right \naxis) are shown as a function of Re concentration. In the inset, spin -polarization is shown  as a \nfunction of Re concentration.  \n  \n0 0.03 0.06 0.09 0.12\nRe concentration0.90.951DOS at EF (States/eV)\n0 0.03 0.06 0.09 0.12\nRe concentration0.350.40.450.50.55Spin polarization\n0123456\nDamping parameter (x 10-3)Re \n(at%)  𝑡𝑅𝑢,𝑐𝑎𝑝 \n(nm)  𝜎  \n(nm)  𝑡𝐹𝑒𝐶𝑜 \n(nm)  𝜎  \n(nm)  𝑡𝑅𝑢,𝑠𝑒𝑒𝑑 \n(nm)  \n  \n(nm)  \n0 2.46 1.89 39.71  0.67 2.74 0.66 \n3.0 2.47 1.80 37.47 0.59 2.45 1.03 \n6.6 1.85 0.50 37.47 0.51 2.13 0.90 \n12.6 2.15 1.49 37.38 0.64 1.89 1.03 \nTable 1 Thickness and roughness (𝝈) values for different layers in films  extracted  from XRR \ndata. Error margin is 0.02nm for all thickness and roughness values.   \n \nRe (at%) 𝜇0𝐻𝑢 (mT) 𝜇0𝑀𝑒𝑓𝑓 (T) \n0 2.20 2.31 \n3.0 2.10 2.12 \n6.6 2.30 1.95 \n12.6 2.20 1.64 \nTable 2 Room temperature  𝝁𝟎𝑴𝒆𝒇𝒇 and  𝝁𝟎𝑯𝒖 values  for Fe 65Co35 films with different \nconcentration  of Re extracted by fitting the angle dependent cavity FMR data to Eq. (1). \n \n \nTemperature  (K) 0% Re  3.0 at% Re 6.6 at% Re 12.6 at% Re \n𝜇0𝑀𝑒𝑓𝑓  (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) \n300 2.29 2.16 1.99 1.61 \n200 2.31 2.16 2.04 1.67 \n150 2.33 2.24 2.06 1.70 \n100 2.36 2.25 2.07 1.72 \n50 2.36 2.27 2.08 1.74 \nTable 3 Temperature dependent  𝝁𝟎𝑴𝒆𝒇𝒇 values for Fe65Co35 films with  different concentrati on \nof Re extracted by fitting broadband out -of-plane FMR data to Eq. (4). Error margin is about  10 \nmT. \n \n \nTemperature(K)  𝛼𝑟𝑎𝑑 (×10-3) \n0% Re  3.0 at% Re 6.6 at% Re 12.6 at% Re \n300 0.218  0.482  0.438  0.154  \n200 0.222  0.494  0.450  0.160  \n150 0.216  0.499  0.454  0.162  \n100 0.225  0.502  0.456  0.219  \n50 0.221  0.505  0.458  0.166  \nTable 4 Temperature dependent r adiative damping contribution to total damping parameter for \nFe65Co35 films with  different concentration of Re  calculated  using Eq.  (5). \n  \nTemperature(K)  𝛼𝑒𝑑𝑑𝑦  (×10-3) \n0% Re  3.3 at% Re 6.6 at% Re 12.6 at% Re \n300 0.038  0.077  0.064  0.006  \n200 0.047  0.081  0.067  0.006  \n150 0.050  0.084  0.070  0.006  \n100 0.055  0.084  0.073  0.007  \n50 0.058  0.086  0.075  0.007  \nTable 5 Temperature dependent eddy current damping contribution to total damping parameter \nfor Fe 65Co35 films with  different concentration of Re calculated using Eq. ( 6).     \n \nFigure A1  RBS spectra for the Re -doped Fe 65Co35 films.  \n"    },    {        "title": "2111.00586v1.Thermally_induced_all_optical_ferromagnetic_resonance_in_thin_YIG_films.pdf",        "content": "1 \n Thermally induced all-optical ferromagnetic resonance in thin YIG films  \nE. Schmoranzerová1*, J. Kimák1, R. Schlitz3, S.T. B. Goennenwein3,6, D. Kriegner2,3, H. Reichlová2,3, Z. Šobáň2, \nG. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1, T. Ostatnický1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany \n6 Department of Physics, University of Konstanz, 78457 Konstanz, Germany \n \nLaser-induced magnetization dynamics is one of the key methods of modern opto-spintronics which aims \nat increasing the spintronic device speed1,2. Various mechanisms of interaction of ultrashort laser pulses \nwith magnetization have been studied, including ultrafast spin-transfer3, ultrafast demagnetization4, \noptical spin transfer and spin orbit torques 5,6,7 , or laser-induced phase transitions8,9. All these effects can \nset the magnetic system out of equilibrium, which can result in precession of magnetization. Laser-induced \nmagnetization precession is an important research field of its own as it enables investigating various \nexcitation mechanisms and their ultimate timescales2. Importantly, it also represents an all-optical analogy \nof a ferromagnetic resonance (FMR) experiment, providing valuable information about the fundamental \nparameters of magnetic materials such as their spin stiffness, magnetic anisotropy or Gilbert damping10. \nThe “all-optical FMR” (AO-FMR) is a local and non-invasive method, with spatial resolution given by the \nlaser spot size, which can be focused to the size of few micrometers.  This makes it particularly favourable \nfor investigating model spintronic devices.  \nMagnetization precession has been induced in various classes of materials including ferromagnetic \nmetals11, semiconductors10, 12, or even in materials with a more complex spin structure, such as non-\ncollinear antiferromagnets13. Ferrimagnetic insulators, with Yttrium Iron Garnet (YIG, Y 3Fe5O12) as the \nprime representative14, are of particular importance for spintronic applications owing to their high spin \npumping efficiency15 and the lowest known Gilbert damping16. However, inducing magnetization dynamics \nin ferrimagnetic garnets using optical methods is quite challenging, as it requires large photon energies 2 \n (bandgap of YIG is Eg ≈ 2.8 eV)17. This spectral region is rather difficult to access with most common ultrafast \nlaser systems, which are usually suited for near-infrared wavelengths. Therefore, methods based mostly \non non-thermal effects, such as inverse Faraday18,19 and Cotton-Mouton effect20 or photoinduced magnetic \nanisotropy21, 22   have been used to trigger the magnetization precession in YIG so far. For these phenomena \nto occur, large laser fluences of tens of mJ/cm2 are required23. In contrast, laser fluences for a thermal \nexcitation of magnetization precession usually do not exceed tens of J/cm2 (Refs. 12, 21, 13). Using the \nlow fluence excitation regime allows for the determination of quasi-equilibrium material parameters, not \ninfluenced by strong laser pulses.  In magnetic garnets, an artificial engineering of the magnetic anisotropy \nvia the inclusion of bismuth was necessary to achieve thermally-induced magnetization precession21.  \nIn this paper, we show that magnetization precession can be induced thermally by femtosecond laser \npulses in a thin film of pure YIG only by adding a metallic capping layer. The laser pulses locally heat the \nsystem, which sets the magnetization out of equilibrium due to the temperature dependence of its \nmagnetocrystalline anisotropy. This way we generate a Kittel (n = 0, homogeneous precession) FMR mode, \nwith a precession frequency corresponding to the quasi-equilibrium magnetic anisotropy of the thin YIG \nfilm10.  We thus prove that the AO-FMR method is applicable for determining micromagnetic parameters \nof thin YIG films. Using the AO-FMR technique we revealed that at low temperature the Kittel mode \ndamping is significantly faster than at room-temperature, in accord with previous FMR experiments24,25.  \nOur experiments were performed on a 50 nm thick layer of pure YIG grown by pulsed-laser deposition on \na gadolinium-gallium-garnet (GGG) (111)-oriented substrate. One part of the film was covered by 8 nm of \nAu capping layer, the other part by Pt capping, both being prepared by ion-beam sputtering. Part of the \nsample was left uncapped as a reference.  X-ray diffraction confirmed the excellent crystal quality of the \nYIG film with a very low level of growth-induced strain, as described in detail in Ref. 26. The magnetic \nproperties were further characterized using SQUID magnetometry and ferromagnetic resonance \nexperiments, showing the in-plane orientation of magnetization (see Supplementary Material, Part 1 and \nFigs. S1 and S2). The deduced low-temperature (20 K) saturation magnetization µ0Ms   180 mT is in \nagreement with results published on qualitatively similar samples27 again confirming a good quality of the \nstudied YIG film. Magnetic anisotropy of the system at 20 K was established from an independent \nmagneto-optical experiment (Ref. 28), the corresponding anisotropy constants for cubic anisotropy of the \nfirst and second order are Kc1 = 4680 J/m3  and Kc2 = 223 J/m3, while the overall uniaxial out-of-plane \nanisotropy is vanishingly small.  3 \n Laser-induced dynamics was studied in a time-resolved magneto-optical experiment in transmission \ngeometry, as schematically shown in Fig. 1(a). An output of a Ti:Sapphire oscillator generating 200 fs laser \npulses was divided into a strong pump beam, with fluences tuned between 70 and 280 µJ/cm2, and a 20-\ntimes weaker probe beam. The beams were focused on a 30 m spot on the sample, which was placed in \na cryostat and kept at cryogenic temperatures (typically 20 K). An external magnetic field (up to 550 mT) \ngenerated by an electromagnet was applied in y direction (see Fig. 1). The wavelength of pump pulses (800 \nnm) was set well below the absorption edge of the YIG layer, as indicated in the transmission spectrum of \nthe sample in Fig. 1(b). The wavelength of probe pulses (400 nm) was tuned to match the maximum of the \nmagneto-optical response of bulk YIG [see inset in Fig. 1(b) and Ref. 29].  \nThe detected time-resolved magnetooptical (TRMO) signal corresponding to the rotation of polarization \nplane of the probe beam Δβ, was measured as a function of the time delay Δt between pump and probe \npulses. In Fig. 1(c), we show an example of TRMO signals observed in uncapped YIG and two YIG/metal \nheterostructures. Clearly, in the presence of the metallic capping layer an oscillatory TRMO signal is \nobserved, whose amplitude depends on the capping metal used. Frequency and damping of the \noscillations, on the other hand, remain virtually unaffected by the type of the capping layer, while no \noscillations are observed in the uncapped YIG sample. \nThe TRMO signals can be phenomenologically described by a damped harmonic function after removing a \nslowly varying background (see Supplementary Material, Part 2 and Fig. S3),12 \n∆𝛽(Δ𝑡)=𝐴cos(2𝜋𝑓𝛥𝑡+𝜑)exp(−𝛥𝑡 𝜏⁄ ),         (1) \nwhere A is the amplitude of precession, f its frequency, φ the phase and τ the damping time. The fits are \nshown in Fig. 1(c) as solid lines.  \nIn order to demonstrate that the TRMO signals result from (laser-induced) magnetization dynamics, we \nvaried the external magnetic field Hext and extracted the particular precession parameters by fitting the \ndetected signals by Eq. (1). As depicted in Fig. 2(a), the experimentally observed dependence of the \nprecession frequency on the applied field is in excellent agreement with the solution of Landau-Lifshitz-\nGilbert (LLG) equation, using the free energy of a [111] oriented cubic crystal [see Supplementary, section \n5, Eq. (S5) and Ref. 28]. This correspondence with the LLG model proves that our oscillatory signals reflect \nindeed the precession of magnetization in uniform (Kittel) mode in YIG. We stress that the precession \nfrequency is inherent to the YIG layer and does not depend on the type of the capping layer. 4 \n The detection of the uniform Kittel mode can be further confirmed by comparing the frequency of the \noscillatory TRMO signal with the frequency of resonance modes observed in a conventional, microwave-\ndriven ferromagnetic resonance (MW-FMR) experiment. The MW-FMR experiment was performed in the \nin-plane ( H = 0°) and out-of-plane ( H = 90°) geometry of the external field. We measured the TRMO signals \nin YIG/Au sample in a range of magnetic field angles H and modelled the angular dependency of f by LLG \nequation with the same parameters that were used in Fig. 2(a). The output of the model is presented in \nFig. 2(b), together with precession frequencies obtained from TRMO and FMR experiments. The MW-FMR \ndata fit well to the overall trend, confirming the presence of uniform magnetization precession [Ref. 30] \nTo find the exact physical mechanism that triggers laser-induced magnetization precession in our \nYIG/metal bilayers, we measured the TRMO signals at different sample temperatures T. For comparison \nwe calculated also the dependence of f on the first order cubic anisotropy constant Kc1   from the LLG \nequation, which is shown in the inset of Fig. 2(c). This graph reveals that f should be directly proportional \nto Kc1 in the studied range of temperatures . In Fig 2(c) we plot f as a function of T, together with the \ntemperature dependence of Kc1 (T) obtained from Ref. 28 and Ref. 32.  Clearly, both Kc1 and f show a similar \ntrend in temperature.  Considering also the temperature dependence of the precession amplitude [see \nFig. S5 (a) and Section 4 of Supplementary Material], we identify the pump pulse-induced heating and \nconsequent modification of the magnetocrystalline anisotropy constant Kc1 as the dominant mechanism \ndriving laser-induced magnetization precession. \nIn order to estimate the pump-induced increase in quasi-equilibrium temperature of the sample, we first \nfit the temperature dependence of the parameter Kc1 reported in literature by a second order polynomial \n[Fig. 2(c)]. Owing to the linear relation between f and Kc1 and the known temperature dependence of f, \nthe measured dependence of f on pump fluence I can be converted to the intensity dependence of the \ntemperature increase T(I), which is shown Fig. 2(d). As expected, higher fluence leads to more \npronounced heating, which results in a decrease of the precession frequency. Note that for the highest \nintensity of 300 J/cm2, the sample temperature can increase by almost 80 K. \nNature of the observed laser-induced magnetization precession was further investigated by comparing \nsamples with different capping layers. In Fig. 3(a) we show the amplitude A of the oscillatory signal in the \nYIG/Pt and YIG/Au layers as a function of I. The difference between the samples is apparent both in the \nabsolute amplitude of the precession and in its increase with I, the YIG/Pt showing stronger precession. \nFurthermore, precession damping is stronger in YIG/Au than in YIG/Pt, as apparent from Fig 3 (b) where \neffective Gilbert damping parameter eff is presented as a function of Hext. These values of eff were 5 \n obtained by fitting the TRMO data by the LLG equation, as described in the Supplementary Material \n(Section 5). Despite the relatively large fitting error, we can still see that YIG/Pt shows slightly lower    \n0.020, while the YIG/Au has   0.025. To understand these differences, we modeled the propagation of \nlaser-induced heat in GGG/YIG/Pt and GGG/YIG/Au multilayers by using the heat equation (see \nSupplementary Material, Section 7). In Fig. 3(c),  T is presented as a function of time delay t after pump \nexcitation for selected depths from the sample surface. In Fig. 3(d), the same calculations are presented \nfor variable depths and fixed t. The model clearly demonstrates that a significantly higher T can be \nexpected in the Pt-capped layer simply due to its smaller reflection coefficient as compared to Au-capping \n(see Supplementary Material, Section 7). This in turn leads to a higher amplitude of the laser-induced \nmagnetization precession in YIG/Pt compared to the YIG/Au, as apparent in Fig. 3(a).  \nAccording to our model, an extreme increase in temperature is induced in the first few picoseconds after \nexcitation, which acts as a trigger of magnetization precession. After approximately 10 ps, precession takes \nplace in quasi-equilibrium conditions. The system returns to equilibrium on a timescale of nanoseconds, \nwhich shows also in the TRMO signals as the slowly varying background (Fig. S3). The precession frequency \nwe detect reflects the quasi-equilibrium state of the system. Therefore, the temperature increase T \ndeduced from the TRMO signal can be compared with our model for large time delays after the excitation \n(t  10 ps). In YIG/Au sample, the experimental values of T = (25  10) K for excitation intensity of 150 \nJ/cm2 [see Fig. 2(d)], while the model gives us  T  5K [Fig. 3 (c)]. Clearly, the values match in the order \nof magnitude but there is a factor of  5 difference. This difference results from the boundary conditions \nof the model that assumes ideal heat transfer between the sample and the holder, which is experimentally \nrealized using a silver glue with less than perfect performance at cryogenic conditions. \nFrom Fig. 3(d) it also follows that large thermal gradients are generated across the 50 nm layer. This could \nlead to significant inhomogeneity in magnetic properties of the layer, that would increase the damping \nparameter  by an extrinsic term. In our TRMO measurements,  is indeed very large for a typical YIG \nsample (TRMO  2-2.5 x10-2) and exceeds the value obtained from room-temperature MW-FMR by almost \nan order of magnitude ( FMR  1x10-3, see Supplementary Material, Section 1b). As the modeled thermal \ngradient alone cannot account for such a large change in Gilbert damping (see Supplementary Material, \nSection 6), we attribute this increase in Gilbert damping to the difference in the ambient temperatures. \nLarge change of Gilbert damping (by a factor of 30) between room and cryogenic (20 K) temperature has \nrecently been reported on a seemingly high quality YIG thin film24. It was explained in terms of the presence \nof rare earth or Fe2+ impurities that are activated at cryogenic temperatures.  It is likely that the same 6 \n process occurs in our sample.  Even though other mechanisms related to the optical excitation can also \ncontribute to the increase in TRMO (see Supplementary Material, Section 6), the all-optical and standard \nFMR generated Kittel modes correspond very well [see Fig. 2(b)]. Furthermore, also the observed sample-\ndependent Gilbert damping is consistent with this explanation.  The YIG/Pt sample is heated to higher \ntemperature by the pump laser pulse [Fig. 3(c), (d)] than the YIG/Au sample, which according to Ref. 24 \ncorresponds to a lower Gilbert damping. It is worth noting that damping parameter can be increased also \nby spin-pumping from YIG to the metallic layer. However, this effect is expected to be significantly higher \nwhen Pt is used as a capping, which does not agree with our observations.  \nIn conclusion, we demonstrated the feasibility of the all-optical ferromagnetic resonance method in 50-\nnm thin films of plain YIG. Magnetization precession can be triggered by laser-induced heating of a metallic \ncapping layer deposited on top of the YIG film. The consequent change of sample temperature modifies \nits magnetocrystalline anisotropy, which sets the system out of equilibrium and initiates the magnetization \nprecession. Based on the field dependence of precession frequency, we identify the induced magnetization \ndynamics as the fundamental (Kittel) FMR mode, which is virtually independent of the type of capping and \nreflects the quasi-equilibrium magnetic anisotropy. The Gilbert damping parameter is influenced by line-\nbroadening mechanism due to low-temperature activation of impurities, which is an important aspect to \nbe taken into account for low-temperature spintronic device applications.  \nRegarding the efficiency of the optical magnetization precession trigger, it was found that the type of \ncapping layer strongly influences the precession amplitude. The precession in YIG/Pt attained almost twice \nthe amplitude of that in YIG/Au under the same conditions. This indicates that a suitable choice of capping \nlayer should be considered in an optimization of this local non-invasive magnetometric method. \n \nAcknowledgments: \nThis work was supported in part by the INTER-COST grant no. LTC20026 and by the EU FET Open RIA \ngrant no. 766566. We also acknowledge CzechNanoLab project LM2018110 funded by MEYS CR for the \nfinancial support of the measurements at LNSM Research Infrastructure and the German Research \nFoundation (DFG SFB TRR173 Spin+X projects A01 and B02 #268565370). \n \n 7 \n LITERATURE \n[1]  A. Hirohata et al., JMMM 509, 16671 (2020)  \n[2]  A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) \n[3] F. Siegries et al., Nature 571, 240–244 (2019) \n[4] E. Beaurepaire  et al., Phys. Rev. Lett. 76, 4250 (1996). \n[5] P. Nemec et al., Nature Physics 8, 411-415 (2012)  \n[6] G.M. Choi et al., Nat. 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B 97, 014422 (2018) \n[22]  A. Stupakiewicz et al., Nature 542, 71 (2017) \n[23] F. Atoneche et al., Phys. Rev. B 81, 214440 (2010) \n[24] C. L. Jermain et al., PRB 95, 174411 (2017) 8 \n [25] H. Maier-Flaig et al.,   Phys. Rev. B 95, 214423 (2017)  \n[26] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) \n[27] J. Mendil et al.: Phys. Rev. Mat. 3, 034403 (2019) \n[28] E. Schmoranzerova et al., ArXiv XXX (2021) \n[29] E. Lišková Jakubisova et al., Appl. Phys. Lett. 108, 082403 (2016) \n[30] We note that the FMR data were obtained at room temperature while the TRMO experiment was \nperformed at 20K. However, as apparent from Fig. 2(c), the precession frequency  varies by less than \n10% between 20 K and 300 K, which is well below the experimental error of   (H). This justifies \ncomparison of the precession frequencies obtained from the TRMO experiment with the FMR data. \n[31] M. Haider et al., J. Appl. Phys. 117, 17D119 (2015) \n[32] N. Beaulieu et al., IEEE Magnetics Letters 9, 3706005 (2018) \n \n \n \nFIGURES \n \n \n9 \n Fig. 1: (a) Schematic illustration of the pump&probe experimental setup, where Eprobe is the probe beam linear \npolarization orientation which is rotated by an angle  after transmission through the sample with respect to the \norientation E’probe. An external magnetic field H ext is applied at an angle H. (b) Absorption spectrum of the studied \nYIG sample, where OD stands for the optical density defined as minus the decadic logarithm of sample \ntransmittance. The red arrow indicates the wavelength of the pump beam PUMP = 800 nm. Inset: Spectrum of Kerr \nrotation K of bulk YIG crystal29. The blue arrow shows the wavelength of the probe beam PROBE = 400 nm. (c) \nTypical time-resolved magneto-optical signals of a plain 50 nm YIG film (black dots), YIG /Pt  (green dots) and \nYIG/Au bilayer (blue dots) at 20 K and 0Hext = 100 mT, applied at an angle H = 40°.  Lines indicate fits by Eq. (1). \nThe data were offset for clarity. \n \n \n \nFig. 2: (a) Frequency f of magnetization precession as a function of magnetic field applied at an angle H = 40°, for \nYIG/Pt (blue dots) and YIG/Au (green triangles) at  T = 20 K and I = 150 J/cm2.  The line is calculated from LLG equation \n(Eq. S3) with the free energy given by (Eq.S5) (b) Field-angle dependence of f in YIG/Au sample for 0Hext = 300 mT \n(blue dots), compared to a model by LLG  model (line) and to frequencies measured by MW-FMR (red stars)32. (c) \nTemperature dependence of f in YIG/Au sample (black points), where 0Hext = 300 mT was applied at H = 40°. The \ntemperature dependence of cubic anisotropy constant Kc1 was obtained from Ref. 28 (red dots) and Ref. 32 (red star, \nT = 20 K). The data were fitted by an inverse polynomial dependence 𝐾ଵ(𝑇)= ଵ\n(ା்ା்మ), with parameters: a = 0.18 \nm2/kJ; b= 9 x 10-4 m2/kJ.K; c = 9 x 10-6m2/kJ.K2. Inset: Dependence f(Kc1) obtained from the LLG equation. (d)  f as a \nfunction of pump pulse fluence I, from which the increase of sample temperature T for the used pump fluences was \nevaluated using the f(T) dependence.  \n \n10 \n  \nFig. 3: Comparison of magnetization precession in YIG/Pt and YIG/Au samples. (a) Precession amplitude A as a \nfunction of pump fluence I (dots) with the corresponding linear fits 𝐴 = 𝑠∙𝐼.  The parameter s Pt = (1.05 0.09)x10-2  \nrad.cm2/J in the YIG/Pt, and s Au = (0.50.1)x10-2  rad.cm2/J in YIG/Au. These dependencies were measured for \n0Hext = 300 mT and T 0 = 20 K. In YIG/Pt sample the as-measured data obtained for H = 40° are shown. In the YIG/Au \nsample, the A(I) dependence was originally measured for H = 21° and recalculated to  H = 40° according to the \nmeasured angular dependence, as described in detail in Supplementary Material, Section 3. (b) Gilbert damping eff \nfor Hext applied at an angle H = 40°. The values of eff result from fitting the TRMO signals to LLG equation; I = 140 \nJ/cm2. (c) and (d) Increase in lattice temperature as a function of time delay between pump and probe pulses for \nselected depths from the sample surface (c) and as a function of depth for fixed time delays (d). I = 140 J/cm2, T0 = \n20 K. The heat capacities and conductivities of individual layers are provided in the Supplementary Material, Section \n7.  \n \n \n  \n11 \n Thermally induced all-optical ferromagnetic resonance in thin YIG films: \nSupplementary Material \nE. Schmoranzerová1*, J. Kimák1, R. Schlitz3  , S.T. B. Goennenwein3,   D. Kriegner2,3, H. Reichlová2,3, ,  Z. \nŠobáň2, G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1 , T. Ostatnický1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany \n \n6 Department of Physics, University of Konstanz, 78464 Konstanz, Germany \n \n \n1. Magnetic characterization  \n \nA. SQUID magnetometry \nA superconducting quantum device magnetometer (SQUID) was used to characterize the magnetic \nproperties of the thin YIG film at several sample temperatures. The magnetic hysteresis loops, detected \nwith magnetic field applied in [2-1-1] crystallographic direction of the YIG layer, are shown in Fig. S1. As \nexpected [t26], the saturation magnetization increases at low temperatures, which is accompanied by a \nslight increase in coercive field. At room temperature, the effective saturation magnetization is estimated \nto be Ms = 95 kA/m. This value is in good agreement with the effective magnetization Meff obtained from \nthe ferromagnetic resonance (FMR) measurement (see Section 1b), which indicates only a weak out-of-\nplane magnetic anisotropy [s1]. However, as discussed in detail in Ref. 26, the Ms from our SQUID \nmeasurement is burdened by a relatively large error. Therefore, mere comparison of SQUID and FMR \nexperiment is not sufficient to evaluate the size of the out-of-plane magnetic anisotropy. An additional \nexperiment such as static magneto-optical measurement [28] is needed in order to get more precise \nestimation of the out-of plane magnetic anisotropy. \n \nB.  FMR measurement  \nThe SQUID magnetometry was complemented by so-called broad band ferromagnetic resonance \nmeasurements using a co-planar waveguide to apply electromagnetic radiation of a variable frequency f \n=/2 to the sample. The measurement was performed at room temperature and further details on the \nmethod can be found in Ref. s2. An exemplary set of spectra showing the normalized microwave \ntransmission | S21|norm obtained at different external fields magnitudes applied in the sample plane, is \nshown in Fig. S2(a). The set of Lorentzian-shape resonances can be fitted by the equation: 12 \n |𝑆ଶଵ|୬୭୰୫=ቀഘ\nమቁమ\nቀഘ\nమഏିഘబ\nమഏቁାቀഘ\nమቁమ+𝑦          (S1) \nWhere f0 =0/2  is the FMR resonance frequency, /2 is the half width half maximum line width, B the \namplitude of the FMR line and y0 a frequency independent offset. From an automated fitting of the set of \nlines obtained at different Hext, we extract the magnetic field dependence of the resonance frequency \n0/2 (Hext) [Fig. S2(b)] and linewidth (Hext) [Fig. S2(c)].  Clearly, the resonance frequencies correspond \nto the fundamental (Kittel) mode, and can correspondingly be fitted by the Kittel formula [s3]:  \nఠబ\nଶగ=ఊ\nଶగඥ𝜇𝐻ୣ୶୲(𝜇𝐻ୣ୶୲+𝜇𝑀ୣ)  (S2) \nWhere Meff is the effective saturation magnetization that includes the out-of-plane anisotropy term, and  \nis gyromagnetic ratio. From this fit, it is possible to evaluate Meff , Kittel  = 94.9 kA/m \nFrom the linewidth dependence (Hext)=2 + 0 we can extract both the inhomogeneous line \nbroadening and the Gilbert damping parameter, as shown in Fig. S2(c) [s2]. In our experiment, the \ninhomogeneous linewidth broadening is 0 = 55.8 MHz, and the Gilbert damping parameter  = 0.001. \nBoth values are on a higher side compared e.g. with YIG prepared by liquid phase epitaxy [s8] but in good \nagreement with typical YIG thin films similar to our layers, which were prepared by pulsed laser deposition \n[27]. This again confirms the good quality of the studied thin YIG films. \n \n2. Processing of time-resolved magneto-optical data \nIn order to extract the parameters describing the precession of magnetization correctly from the time-\nresolved magneto-optical (TRMO) signals, it is first necessary to remove the slowly varying background on \nwhich the oscillatory signals are superimposed. For this purpose, we fitted the measured data by the \nsecond-order polynomial. The fitted curve was then subtracted from the measured signals, as \ndemonstrated in Fig. (S3).  \nFrom the physical point of view, the background can be attributed to a slow return of magnetization to its \nequilibrium state after the pump beam induced heating, which can take place on the timescale of tens of \nnanoseconds [10]. Since both saturation magnetization Ms and magnetocrystalline anisotropy Kc are \ntemperature-dependent, their temporal variation can in principle contribute to the background signal. \nHowever, as explained later in Section 4, the variation of Ms is very weak at cryogenic temperatures. The \nheat-induced modification of Kc, and the resulting change of the magnetization quasi-equilibrium \norientation, is, therefore, more probable origin of the slowly varying background, which is detected in the \nMO experiment by the Cotton-Mouton effect [28]. \n3. Angular dependence of precession amplitude  \nIn order to mutually compare the values of precession amplitudes measured in YIG/Pt and YIG/Au samples \nat different angles of the external magnetic field H, it is necessary to correct their values for the value of \nH. The following procedure was used to correct the data presented in Fig. 3 of the main text . \n 13 \n First, we measured in detail angular dependence of the precession amplitude in the YIG/Au layer, which is \npresented in Fig. S4. Amplitude of the oscillatory signal detected in our experiment does not depend solely \non the amplitude of the magnetization precession but also on the size of the magneto-optical (MO) effect. \nIn our experimental setup, the change of H was achieved by tilting the sample relative to the position of \nelectromagnet poles [see Fig. 1(a)]. The MO response, however, varies also with the angle of incidence \nwhich is modified simultaneously with a change of H [see Fig. 1(a)] . Therefore, it is not straightforward to \ndescribe the A(H) analytically. Instead, we fitted the measured dependence A(H) by a rational function in \na form of y = 1/(A+Bx2), which is the lowest order polynomial function that can describe the signal properly. \nFrom the fit we derived a correction factor of 1.7 by which the amplitudes A measured at H =21° has to \nbe multiplied to correspond to that measured at H =40°. This factor was then used to recalculate the \nintensity dependence of the precession amplitude A(I)  in YIG/Au measured at H =21° to the A(I) at H \n=40°, which could be directly compared to the A(I) dependence detected at YIG/Pt for H =40° - see Fig. \nS4(b). \n \n4. Temperature dependence of precession amplitude \n \nIn order to further investigate the origin of the laser-induced magnetization precession, the amplitude of \nthe oscillatory MO signal was measured as a function of the sample temperature in YIG/Pt sample, see Fig. \nS5(a). In Fig. S5 (b), we show temperature dependence of saturation magnetization Ms, as obtained from \nRef. 32 \nThe only parameter changed within this experiment was the sample temperature. It is reasonable to \nexpect that the size of the magneto-optical effect is not strongly temperature dependent in the studied \ntemperature range between 20 and 50 K (see Ref. 28) Therefore, the dependence A(T) presented in Fig S5 \ncorresponds directly to the temperature dependence of magnetization precession amplitude. By \ncomparing the Ms(T) and A(T) data, it is immediately apparent that the laser-induced heating would not \nmodify Ms enough to account for the large change of the magnetization precession amplitude with the \nsample temperature. Even assuming the most extreme laser-induced temperature increase T  80 K \nshown in Fig. 3(c), the laser-induced Ms variation would be less than 5%, while the precession amplitude \nchanges by more than 50% between 20 and 50 K. In contrast, the magnetocrystalline anisotropy Kc1 \nchanges drastically even in this relatively narrow temperature range [see Fig. 2(c)]. Consequently, the \nchange of Kc1, which leads to a significant change of the position of quasi-equilibrium magnetization \norientation in the studied sample (see Section 5) provides a more plausible explanation for the origin of \nthe laser-induced magnetization precession in the YIG/metal layer. \n \n5.  LLG equation model  \nThe data were modelled by numerical solution of the Landau-Lifshitz-Gilbert (LLG) equation, as defined in \n[s9]: \nௗ𝑴(௧)\nௗ௧= −𝜇𝛾ൣ𝑴(𝑡)×𝑯𝒆𝒇𝒇(𝑡)൧+ఈ\nெೞቂ𝑴(𝑡)×ௗ𝑴(௧)\nௗ௧ቃ, (S3) 14 \n where  is the gyromagnetic ratio,  is the Gilbert damping constant, and MS is saturated \nmagnetization.The effective magnetic field Heff is given by: \n𝑯𝒆𝒇𝒇(𝑡)=డி\nడ𝑴       (S4) \nwhere F is energy density functional that contains contributions from the external magnetic field Hext, \ndemagnetizing field and the magnetic anisotropy of the sample. We consider the form of F including first- \nand second-order cubic terms as defined in Ref. [t24]. The polar angle  is measured with respect to the \ncrystallographic axis [111] and the azimuthal angle  = 0 corresponds to the direction [21ത1ത], with an \nappropriate index referring to the magnetization position (index M) or the direction of the external \nmagnetic field (index H). The resulting functional takes the form (in the SI units): \n \n𝐹= −𝜇𝐻𝑀[sin𝜃ெsin𝜃ு+cos𝜃ெcos𝜃ுcos(𝜑ு−𝜑ெ)]+ቀଵ\nଶ𝜇𝑀ଶ−𝐾uቁsinଶ𝜃ெ \n  +𝐾c1\n12ൣ7cosସ𝜃ெ−8cosଶ𝜃ெ+4−4√2cosଷ𝜃ெsin𝜃ெcos3𝜑ெ൧ \n  +c2\nଵ଼ൣ−24cos𝜃ெ+45cosସ𝜃ெ−24cosଶ𝜃ெ+4−2√2cosଷ𝜃ெsin𝜃ெ(5cosଶ𝜃ெ−\n2)cos3𝜑ெ+cos𝜃ெcos6𝜑ெ൧ ,                       (S5) \n \nwhere 0 is the vacuum permeability and we consider the following values of constants: magnetization M \n= 174 kA/m, first-order cubic anisotropy constant Kc1=4.68 kJ/m3, second-order cubic anisotropy constant \nKc2 = 222 J/m3 [t24].  \nFor modelling the dependence of precession frequency on the external magnetic field Hext  [Fig. 2 (a)] and \non the angle H [Fig. 2 (a)], we assumed that in a steady state magnetization direction is parallel to Hext, i.e. \nM =  H, and M = H. This is surely fulfilled for large enough magnitude of Hext. Since the coercive field is \nvery small, we can assume the procedure to be correct. Further correspondence to experimental data \nEvaluation of the Gilbert damping factor from the as-measured magneto-optical oscillatory data was done \nby fitting signals by a theoretical curve calculated by solving numerically LLG equation [Eq. (S3)]. We \nconsidered the magnetization free energy density in a form of Eq. (S5) using magnitude and direction of \nthe external magnetic field from the experiment. The electron g-factor was set to 2.0 and then the Gilbert \nfactor and five parameters of the fourth-order polynomial to remove the background MO signal were the \nfitting parameters. The resulting dependence of fitted effective Gilbert factors αeff on external magnetic \nfield is displayed in Fig. 3(b) in the main text, from which the field-independent Gilbert factor α can be \nevaluated. \n \n6. Comparison of Gilbert damping parameter from MW-FMR and TRMO experiments \nThe Gilbert damping from the room-temperature FMR measurement on the YIG film   1∙10-3 and the \nresults from fits of the low-temperature pump&probe data   2∙10-2, differ by an order of magnitude. As \ndetailed in the main text, we attribute this difference to the different sample temperatures in the AO-FMR \nand MW-FMR measurements. However, one might also argue that the increased damping in the optical \nexperiments is caused either by a spatial inhomogeneity of the magnetization oscillations or it is the result \nof the perturbation of the YIG surface. 15 \n In the former case, we expect that the spatial inhomogeneity of the temperature distribution [see Fig. \n3(d)] causes the magnetization to oscillate in a form of a superposition of harmonic waves with well-\ndefined in-plane wavevectors. Considering the dispersion of the allowed oscillatory modes [s4] and \nincluding the relevant value of the exchange stiffness [s5], we revealed that neither the inhomogeneity \ndue to the finite cross section of the excitation laser beam nor the temperature gradient perpendicular to \nthe sample surface can cause such a strong decrease of the Gilbert damping factor that is observed \nexperimentally. Here, we provide an estimate on which time scales the mode dispersion influences the \ndecay of the signal if the exchange stiffness is taken into account. Following [s5], the mode dispersion is \ndescribed by the additive exchange field in the form: \n𝜇𝐻ex=𝐷ቈ𝜋ଶ\n𝑑ଶ𝑛ଶ+𝑘∥ଶ , \nwhere D ≈ 5∙10-17 T.m2 is the exchange stiffness, n is the order of the confined magnon mode, d is the YIG \nlayer thickness and k‖ is the in-plane magnon wave vector. We consider here only the n = 0 case since this \nis the only visible harmonic mode observed in the experimental MO data, as proven by the numerical \nfitting. Note that the frequency shift ∆𝜔/2𝜋=|𝛾|𝜇𝐻ex/2𝜋, where the symbol γ stands for the electron \ngyromagnetic ratio, of the n = 1 mode would be 5.5 GHz, which would be then clearly distinguishable from \nthe basic n = 0 mode in the lowest external magnetic fields. The in-plane wave vector k‖ can be calculated \nfrom the FWHM (full width at half maximum) width of the laser spot on the sample L, which is about 30 \nµm in our case, that leads to the order of magnitude k‖ ≈ (2π/L) ≈ 105 m-1. The frequency increase due to \nthe finite laser spot size can be estimated as ∆𝜔/2𝜋=|𝛾|𝐷𝑘∥ଶ/2𝜋≈14 kHz. Inverse of this value (  0.1 \nms) determines the typical time scale at which the magnon dynamics is influenced by their dispersion due \nto the finite laser spot size, which is clearly out of the range of the experimental time scale. \nThe presence of a metallic layer on the top of the YIG sample surface can result into two significant \ndamping processes. First, the magnetization oscillations (and thus oscillations of the macroscopic magnetic \nfield) are coupled to electromagnetic modes which penetrate the surrounding material and can be \neventually radiative for small magnon wave vectors. Penetration into conductive material in turn causes \nenergy dissipation through finite conductivity of such material. We checked the magnitudes of the \nadditional damping caused by the radiative field and energy dissipation in a thin metallic layer and we \nfound that these processes exist but the additional energy loss cannot explain the observed magnitude of \nthe Gilbert damping parameter. The second possible explanation of the increased precession damping due \nto the presence of the metal/YIG surface may be that there is an additional perturbation to otherwise \nhomogeneous sample due to some inhomogeneity through surface roughness or spatially inhomogeneous \nlocal spin pinning. Since both the surface roughness and spin pinning can depend on the composition of \nthe capping layer, it can also cause a minor difference in the resulting damping factor, as observed in Fig. \n3(b). \nOverall, we attribute the experimentally observed difference in Gilbert damping measured by FMR and \npump&probe techniques to the difference in ambient temperatures that were used in these experiments, \nwhich is in accord with the results of Ref. t22. \n \n7.  Heat propagation in YIG/Pt and YIG/Au 16 \n Heat propagation in our sample structures was modelled in terms of the heat equation: \nడ்\nడ௧=ఒ\n∆𝑇 ,        (S4) \nwhere T is the local temperature, λ is the local thermal conductivity, c is the heat capacity and the symbol \nΔ denotes the Laplace operator. The spatio-temporal temperature distribution in the studied sample has \nbeen calculated by a direct integration of Eq. (S4) in a time domain, assuming excitation of the metallic \nlayer by an ultrashort optical pulse [with a temporal duration of 100 fs (FWHM)]. We have taken the whole \nstructure profile of vacuum/metal/YIG/GGG into consideration, assuming that the GGG substrate had a \nperfect heat contact with the cold finger of the cryostat, which has been held on a constant temperature. \nThe respective heat conductivities ( λ) and heat capacities ( c) were set to the following values. Au: λ = 5 \nW/m K [s6], c = 1.3∙104 J/cm3, Pt: λ = 10 W/m K [s7], c = 1.2∙104 J/cm3, YIG: λ = 60 W/m K, c = 6.7∙103 J/cm3, \nGGG: λ = 300 W/m K, c = 2.1∙104 J/m3. \nTo evaluate the initial heat transfer from the optical pulses to the capping metallic layer, we considered \nthe proper geometry of our experiment, i.e. a 8 nm thick metallic layer deposited on the YIG sample, the \nincidence angle of the laser beam of 45 degrees and its p-polarization. We then used optical constants of \ngold and platinum in order to calculate transmission and reflection coefficients of a nanometer-thick \nmetallic layers by means of the transfer matrix method. From those, we estimated the efficiency of power \nconversion from the optical field to heat to be 3% for gold and 6.5% for platinum. The total amount of heat \ndensity was then calculated by multiplication of the pump pulse energy density and the above-mentioned \nefficiency. \nThe data shown in Fig. 3(c)-(d) were then extracted from the full spatio-temporal temperature distribution. \nClearly, the temperature increase in the YIG/Pt sample is approximately twice larger than that of the \nYIG/Au sample as a consequence of twice larger efficiency of the light-heat energy conversion in favour of \nplatinum. Correspondingly, also the amplitudes of the MO oscillations in Fig. 3(a) reveal the ratio  2:1. \n \n \nLITERATURE \n[s1] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) \n[s2] H. Maier-Flaig et al,.PRB 95, 214423 (2017) \n[s3] Ch. Kittel: “Introduction to solid state physics (8th ed.)”. New Jersey: Wiley. (2013). \n \n[s4] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009). \n[s5] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, \nand A. Conca, J. Phys. D: Appl. Phys. 48, 015001 (2014). \n[s6] G. K. White, Proc. Phys. Soc. A 66, 559 (1953). \n[s7] X. Zhang, H. Xie, M. Fujii, H Ago, K. Takahashi, T. Ikuta, T. Shimizu, Appl. Phys. Lett. 86, 171912 (2005) 17 \n  \n[s8] C. Dubs et al., Phys. Rev. Materials 4, 024416 (2020) \n[s9] J. Miltat, G. Albuquerque, and A. Thiaville, An introduction to micromagnetics in the dynamic regime, \nin Spin dynamics in confined magnetic structures I, edited by B. Hillebrands and K. Ounadjela, Springer, \nBerlin, 2002, vol. 83 of Topics in applied physics.  \n \nFIGURES \n \n \n \nFig. S1: Magnetic hysteresis loops measured by SQUID magnetometry with magnetic field Hext applied in \ndirection [2-1-1] at several sample temperatures. The saturation magnetization obtained from SQUID \nmagnetometry measurement at room temperature is roughly Ms = 95 kA/m , assuming a YIG layer thickness \nof 50 nm. \n \n18 \n  \n \nFig. S2: (a) Ferromagnetic resonance spectra measured at room temperature for several different external \nmagnetic field magnitudes µ0Hext from 0 to 540 mT applied in the sample plane. Resonance peaks were \nfitted by Eq. (S1) and the obtained resonance frequencies and  linewidths are plotted as points  in panels \n(b) and (c), respectively. The lines correspond to fit by Kittel formula [Eq. (S2)], which enables to evaluate \neffective magnetization Meff = 94.9 kA/m and Gilbert damping parameter of   = 0.001. \n \n19 \n Fig. S3: Removal of slowly varying background from time-resolved magneto-optical signals. The red dots \ncorrespond to as-measured signals, line indicates the polynomial background that is subtracted from the \nraw signals. Black dots show the signal after background subtraction, black line representing the fit by Eq. \n(1) of the main text. The data were taken at external field of 0Hext = 300 mT, temperature 20 K and pump \nfluence I = 140 J/cm2.. \n \n \nFig. S4: (a) Dependence of the amplitude A of oscillatory magneto-optical signal on the sample tilt \n(different field angles of magnetic field H ) measured in YIG/Au sample . 0Hext = 300 mT, temperature T \n= 20 K and pump fluence I = 150 J/cm2.. (b) Pump intensity dependence of A measured  for YIG/Au \nsample at H =21° (red points), the same dependence recalculated to correspond to H =40 (blue \npoints) where A(I) was measured for YIG/Pt sample (green points); T = 20 K. \n \n \nFig. S5: (a) Temperature dependence of amplitude of the time-resolved magneto-optical signals measured \nfor external field 0Hext = 300 mT applied at an angle H = 30°. (b) Temperature dependence of saturation \nmagnetization Ms obtained from Ref. 32. \n \n"    },    {        "title": "1903.02812v1.Current_induced_motion_of_twisted_skyrmions.pdf",        "content": "1 \n Current -induced  motion of twisted skyrmions  \nChendong Jin1, Chunlei Zhang1, Chengkun Song1, Jinshuai Wang1, Haiyan Xia1, Yunxu \nMa1, Jianing Wang1, Yurui Wei1, Jianbo Wang1,2 and Q ingfang  Liu1,* \n \n1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, \nPeople’ s Republic of China . \n2Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education , Lanzhou University, \nLanzhou 730000, People ’s Republic of China.  \n \nAbstract  \nTwisted skyrmions, whose helicity angles are different from that of Bloch skyrmion s and Né el skyrmion s, have \nalready been demonstrated  in experiments recently. In this work, we first contrast  the magnetic structure and origin of the \ntwisted skyrmion with other three types of skyrmion  including Bloch skyrmion , Né el skyrmion  and antiskyrmion.  \nFollowing, we investigate the dynamics of twisted skyrmions driven by the spin transfer toque (STT) and the spin Hall \neffect (SHE) by using micromagnetic simulations. It is found that the spin Hall angle of the twisted skyrmion is related to \nthe dissipative force tensor and the Gilbert damping both for the motions induced by the  STT and the SHE, especially for \nthe SHE induced motion, the skyrmion Hall angle depends s ubstantially  on the skyrmion helicity. At last, we \ndemonstrate  that the trajectory  of the twisted skyrmion can be controlled in a t wo dimensional plane with  a Gilbert  \ndamping gradient . Our results provide the understanding of current -induced motion of twisted skyrmions, which may \ncontribute to the applications of skyrmion -based racetrack memories.  \n \nKeywords: Twisted skyrmion, spin transfer torque, spin Hall effect  \n \n \n \n \n \n_____________________________  \n*Corresponding author:  Qingfang Liu , liuqf @lzu.edu.cn  2 \n Introduction  \nIt has been recognized that the spin -polarized current -induced the motion and reversal of magnetic structures arises \nas a result of the spin transfer torque (STT) effect [1-4], which has attracted large interests due to the fundamental physics \nand potential applications in spintronic devices, such as magnetic random access memorie s (MRAM s)[5, 6] , racetrack \nmemori es[7, 8] , nano -oscillator s[9-11] and logic device s[12-14]. Recently, it has been reported that the spin Hall effect \n(SHE )[15, 16] , generated by the pure spin currents flowed from the heavy metal  substrate due to t he strong spin -orbit \ncoupling at the interface of ferromagnet/heavy -metal , is an  alternative  efficient method to manipulate the magnetization \ndynamics in magnetic materials [17-20]. Compared with the STT, the SHE  does not require currents flow through the \nmagnetic layer, and then reducing the Joule h eat and electromigration, i.e., avoiding the restricted effect of large current \ndensity in traditional STT devices [21].  \nMagne tic skyrmions are chiral spin magnetization structures with topological properties  and can be divided into the \nfollowing types according  to different types of Dzyaloshinskii -Moriya interaction (DMI) [22-26]: (i) Bloch skyrmmions  \nare first discovered in bulk non -centrosymmetric B20 -type lattice structures such as MnSi [27], FeCoS i[28-30], and \nFeGe [31, 32]  due to the presence of bulk DMI; (ii) Né el skyrmions are observed in multilayered ultrathin films lacking \ninversion symmetry with str ong spin -orbit coupling like Ir (111)/F e[33], Ta/CoFeB [34]  and Pt/Co [35] due to the presence \nof interfacial DMI ; (iii)  Antiskyrmions are reported in Heu sler compound s such as MnPtSn [36] due to the presence of \nanisotropic  DMI [37, 38] . Recently, at the interface of chiral bulk  Cu2OSeO 3 below a certain  thickness, the so -called \ntwisted skyrmions are demonstrated  directly  by the circularly polarized resonant e lastic x -ray scattering, due to the \nbreaking of translational symmetry at the surface of bulk ferromagnet [39, 40] . Up to now, the dynamics of twisted \nskyrmions driven by current have not been reported. Therefore, in this paper,  on the basis of comparing the magnetic \nstructure , origin and topological properties of the above four types of skyrmion,  we focus on the dynamics of twisted \nskyrmions driven by the STT and the SHE and also analysis the simulation results by using Thiele ’s equations [41].   3 \n Micromagnetic simulation details  \nOur magnetic simulation results are performed by using t he Object Oriented MicroMagnetic Framework (OOMMF) \npublic code [42], which includ es the additional modules for bulk DMI, interfacial DMI, anisotropic  DMI  and twisted DMI. \nThe magnetization  dynamics is described  by numerical ly solving the Landau -Lifshitz -Gilbert (LLG) equation  containing \nterms  of the STT and the SHE [17, 20] , as follow : \neff STT SHEd+ + ,m dmm H mdt dt      \n                                                    (1) \nwhere \nm  is the unit vector of the local magnet ization, is the gyromagnetic ratio,   is the Gilbert damping,  \neffH\n  is \nthe effective field including the exchange field, anisotropy field, demagnetization field and DMI  effective field . The STT \nterm is expressed  as \nSTT s s ( ) ( ),mmv m m v mxx    \n                                                        (2) \nwhere is the non -adiabatic factor , and vs is the velocity of the conduction electrons with the form \ns\n0s2PvJeM\n\n , \nwhere J is the current density, e is the electron char ge, P is the spin polarizatio n, \n is the reduced Planck c onstant , 0 is \nthe permeability of  free space , and Ms is the saturation magnetization . The electrons flowing toward + x direction when  vs > \n0. The SHE term is given by  \nSHE SH HM\n0s( ),2m m z jeM L   \n                                                     (3) \nwhere L is the thickness of the magnetic  layer with the value of 1 nm,  SH is the  spin-Hall angle  of Pt substrate with the \nvalue of 0.07 , \nz\n is the unit vectors o f the surface normal direction,  and \nHMj\n  is the current  density  injected into the \nheavy metal . \nIn order to eliminate the influence of the  boundary  effect on the size and dynamics of skyrmions, t he 2D plane  is \nassumed to 500 × 500 ×  1 nm3 (length × width × thickness) with the mesh size  of 1 ×  1 ×  1 nm3, and the initial position \nof the skyrmion is set in the center of the 2D plane. The material parameters  are chosen  similar to  Ref. [8]: saturation \nmagnetization Ms = 580  × 103 A/m, exchange constant A = 1.5  × 10-11 J/m, perpendicular magnetic anisotropy constant 4 \n Ku = 8 × 105 J/m3, and DMI strength DDMI = 2.5 ~  3.5 × 10-3 J/m2.  \nFour types of skyrmions  \nAccording  to the different helicity of skyrmions, there are four types of skyrmions: Bloch skyrmion, Né el skyrmion , \nantiskyrmion and twisted skyrmion as shown in Figs. 1(a)–(d), respectively. Figure s 1(e)–(h) display the corresponding \nspatial profiles of the  local magnetization  across the skyrmions. It can  be seen that the mz of the four types of skyrmions \nare consistent, while the mx and my of the four types of skyrmions are different, which again proves the different \ndistribution of the in-plane  magnetic moments of the four types of skyrmions. We emphasize that the distribut ion of the \nin-plane magnetic moments in the skyrmion structure is determined by the direction of the DMI vector , that is to say, the \nexistence  of the twisted skyrmion in this work is achieved by changing the DMI vector, which is much different as the \nreaso n that observed in the experiments.  Figure s 1(i)–(l) show the four types of DMIs: bulk DMI, interfacial DMI, \nanisotropic  DMI and twisted DMI that promise the existence of the Bloch skyrmion, Né el skyrmion , antiskyrmion and \ntwisted skyrmion, respectively. The four types of DMI considered in C4 symmetry can be written  as: \nˆ ˆ ˆ ˆ BulkDMI\nˆ ˆ ˆ ˆ InterDMI\nˆ ˆ ˆ ˆ AnisoDMI\nTwisDMIˆ ˆ ˆ ˆ ( ),2\nˆ ˆ ˆ ˆ ( ),2\nˆ ˆ ˆ ˆ ( ),2\n(2i i x i x i y i y\ni\ni i x i x i y i y\ni\ni i x i x i y i y\ni\niiDE S S x S x S y S y\nDE S S y S y S x S x\nDE S S y S y S x S x\nDE S S   \n   \n   \n        \n         \n        \n\n\n\nˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ ( ) ( ) ( ) ( ),x i x i y i y\nix y S x y S x y S x y              \n              (4) \nwhere D is the DMI constant representing the DMI strength , \niS\n is the  atomic moment unit vector, \nˆx  and \nˆy  are \nthe unit vectors in the model.  \nTopological properties  of four types of skyrmions  \nA. Helicity, winding number and topological number  \nIn order to better understand the helicity  and winding number of skyrmions , we use the  two-dimensional polar \ncoordinates to describe a general magnetic skyrmion structure, as shown in Fig . 2 which display s a Bloch skyrmion, as 5 \n example, in the polar coordinates with azimuthal angle ( ) and radial coordinate ( ). Therefore , the unit vector of  the \nlocal magnetization  mx, my and mz in the C artesian coordinates  can be written as [26, 43, 44] : \nsin ( )cos ( ),\nsin ( )sin ( ),\ncos ( ),x\ny\nzm\nm\nm   \n   \n\n\n\n                                                                     (5) \nwhere () is the radial profile of the perpendicular component of the magnetization, and fro m the center  to the boundary, \nits value chan ges from 0.5 to 0.5; () is the angle between the magnetic moment and the radial coordinate . The \nvorticity of skyrmions is obtained by calculat ing the full turns of the transverse magnetic moments on the perimeter and \nis defined by the winding num ber[45] \n2\n01d ( )2W\n . Therefore, the winding number W = 1 for twisted \nskyrmion , Bloch skyrmion  and Né el skyrmion  as shown in  Fig. 3(a), and W = 1 for antiskyrmion  as shown in  Fig. 3(b). \nThe helicity of a skyrmion is given by \n( ) ( 0) W           with the value ranging  form  to, that is, for \nthe Bloch skyrmion, 0.5; for Né el skyrmion , 0 or ; for twisted skyrmion , 0.5, 0 and , and the helicity \n of the twi sted skyrm ion shown in  Fig. 1 (d) equals to 0.25; for antiskyrmion  as shown in Fig.  1(c), . The \ntopological  number Q relates to the winding number and counts how many times the unit vector  along the magnetic \nmoment  wraps the unit sphere with the form [26] \n1,,4mmQ qdxdy q mxy     \n                                                      (6) \nwhere q is the topological density. Figure s 3(c) and (d) show the topological densities corresponding to the magnet ic \nskyrmions shown in  Figs.  3(a) and (b), respectivel y. It can be seen that Q = 1 in Fig.  3(c) and Q = 1 in Fig.  3(d), i.e., Q \n= W when the spins point down in the central region and  point up in the boundary region.  \nB. Skyrmion size  and d issipative force tensor  \nThe diameter of twisted skyrmion size ( d) is usually defined as the distance from in -plane to in -plane magnetization, \ni.e., the distance between the region mz = 0, as shown in the inset of Fig.  4. The dissipative force tensor  D is used to \ndescribe the effect of the dissipative forces on the moving skyrmion [46-48]. For a single twisted skyrmion, D is given by  6 \n \n0 14 , ,0 4mmdxdyxx    \nDDDD                                                    (7) \nwher e D is the diagonal element of the dissipative tensor  and also called dissipative  parameter. The dissipative  parameter \nD is determined by t he diameter  and domain wall width of the twisted skyrmion. Therefore, both d and D are affected by  \nDMI strength as shown in Fig.  4. With  the increase in DDMI from 2.5  to 3.5 mJ/m2, d increases from 7.9 to 34.8 nm and D \nincreases from 1.0577 to 1.961, respectively, for the twisted skyrmion.  \nDynamics of twisted skyrmion driven by the STT  \nTo understand the STT-induced motion of the twisted skyrmion s, we first use the Thiele equation [41] to describe the \ndynamics  of the four kinds of skyrmions  mentioned above  by casting  the L LG Eqs. (1) and (2)  to the following \nequation [46, 47] : \ns d s d( - ) ( ) 0,v v v v      D G\n                                                             (8) \nwhere G is the gyrovector with the form G = (0 0 G) = (0 0 4Q), and vd is the drift velocity of the skyrmion. When the \nvelocity of the conduction electrons vs applied along the x direction, vd = (vx, vy) is derived from Eq.  (8) as  \n2\nxs 2 2 2\nys 2 2 2()+,()\n().QvvQ\nv Q vQ  \n  \n\n     D\nD\nD\n                                                              (9) \nIt can be seen that the direction of the skyrmion deviates from the direction of the conduction elect rons when, and \nthis phenomenon  is called the skyrmion Hall effect and can be further defined  by the skyrmion Hall angle  \n \nx\nSky y22\nxy= sign( ) arccos( ),vv\nvv \n                                                          (10) \nwhich defines the angle in the range from 180o to 180o. For the situation of STT -induced skyrmion motion, the sign of \nthe vx is always  the same with vs, i.e., the skyrmion Hall angle is in the range of (-90o, 90o), and therefore the Eq. (10) can \nbe reduced to \nSky 22()= arctan( )Q\nQ\nD\nD .  \nThe trajectories  of the f our types of skyrmion driven by the in -plane STT with  vs = 100 m/s,  = 0.4,  = 0.2 and 7 \n DDMI = 3 mJ/m2 is shown in Fig.  5. The positions of the skyrmions are obtain ed by solving  the guiding center ( Rx, Ry) \nwith the form [49, 50]  \nxy ,xqdxdy yqdxdy\nR = , R =\nqdxdy qdxdy \n \n                                                         (11) \nwhere q is the topological density. One can see that the antiskyrmion deflects to the y direction, while for Bloc h \nskyrmion, Né el skyrmion  and twisted skyrmion  deflect to the y direction, i.e., θSky of the skyrmions with Q =1 \n(antiskyrmion) and Q =1 (Bloch,  Né el and twisted skyrmion ) equal to 12.89o and 12.89o, respectively. Following we \nfocus  on the STT -induced motion of twisted skyrmion with differen t conditions, as shown in Fig.  6. Figure s 6 (a) and (b) \nshow the vx and vy as a function of vs for different  with  = 0.2 and DDMI = 3 mJ/m2, respectively.  It can be seen that vx \nand vy both increase linearly with the increase in  vs for different α, it should be also note that vy is a negative value for  < \n, a positive value  for  > , and zero for  = . Then we chose the situation of vs = 100 m/s to investigate the skyrmion \nHall angle of the twisted skyrmion as a function of vs, as shown in Fig.  6(c), the skyrmion Hall angel Sky remains almost \nunchanged with the increase in vs. Figure 6(d) shows the simulation and calculation of Sky as a function of  with  = 0.2 , \nthe skyrmion Hall angle θSky decreases from 13.7 o to 12.89 o with the  increasing from 0. 01 to 0.4. According to Eq s. \n(9) and (10), both the velocity and the skyrmion Hall angle Sky are affected by the dissipative  parameter D, and the \ndissipative parameter D is determined by the DMI strength DDMI. Therefore, it is necessary to investigate the dynamics of \nthe twisted skyrmion under different DDMI, as shown in Figs.  6 (e) and (f) wit h vs = 100 m/s,  = 0.4 and  = 0.2 . vx \nincreases at first and then decreases w ith DDMI increasing from 2.5 to 3.5 mJ/m2, while vy keeps decreasing (the a bsolute \nvalue  of vy is continuously increasing ), and both simulation and calculation results support that the corresponding \nskyrmion Hall angle Sky decreases from  11.4 o to 16.8 o ( the absolute value of Sky is proportional to the DDMI). \nWe have known that the STT -induced twisted skyrmion motion is affected by the damping in the previous paragraph . \nFollowing, we investigate the dynamics of twisted skyrmion induced by the STT under a damp ing gradient, as shown in \nFig. 7. Figure 7(a) shows t he position along the y axis of the twisted skyrmion as a functio n of distance along  the x axis 8 \n with vs = 100 m/s,  = 0.4 and DDMI = 3 mJ/m2. The damping  decreases from 0.5 to 0.25 linearly from 0 to 50 nm along \nthe x axis, as indicated by the color code.  Figure 7 (b) sho ws the skyrmion Hall angle Sky of the twisted skyrmion as a \nfunction of its position along the x axis. In the region  > , the twisted skyrmion moves along the x axis direction  from 0 \nnm and deflects in the –y direction  until moving to the x axis of  20 nm , where  =  = 0.4 ; from the  region of 20 to 50 \nnm along x axis, the twisted skyrmion begins to deflect in the + y direction because of  < . Therefore , the trajectory of \ntwisted skyrmion induced by the ST T can be controlled  under a damping gradient.  \nDynamics of twisted skyrmion driven by the SHE  \nSHE -induced motion of antiskyrmion has already been studied  in Ref. [38], which demonstrates that the \nantiskyrmion Hall angle depends  on the direction of the current strongly. In thi s section, we focus on  the SHE -induced \nmotion s of the skyrmio ns whose winding number W = 1(Bloch, Né el and twisted skyrmion ). The LLG Eqs. (1) and (3)  \ncan be cast into the following form : \nd d HM 4 ( ) 0 v v B J      D GR\n                                                         (12) \nwhere G = (0 0 4) due to Q = 1, B is linked to  the SHE , and the sign of B is determined by th e SHE angle;  R(χ) is the \nin-plane rotation matrix with the form \ncos sin()sin cosR [49, 51] . When the current  JHM injected into the \nheavy metal  along the x direction, vd = (vx, vy) is derived from Eq.  (12) as  \nx HM 22\ny HM 22cos sin,1\nsin cos.1v B J\nv B J  \n\n  \n    D\nD\nD\nD\n                                                            (13) \nThe skyrmion Hall angle Sky can be obtained by the Eq.  (10), which is in the range of  180o to 180o.  \n The Eq. (13) suggests that the direction of motion of the skyrmions depends on their helicities. Therefore, we first \ninvestigate the trajectories of skyrmions driven by the SHE with  JHM = 10 ×  1010 A/m2,  = 0.2 and DDMI = 3 mJ/m2 for \ndifferent  helicities of skyrmions, as shown in Fig.  8. These  skyrmions with different helicities are achieved by changing \nthe direction of DMI vector. The simulation results in Fig.  8(a) show that the skyrmion Hall angles Sky are 150.4o, 9 \n 165.4o, 121.4o, 75.6o, 29.6o, 14.6o, 58.6o and 104.4o for the helicities χ = 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75 \nand , respective ly. Figure 8(b) shows the skyrmion Hall angle as a function of the helicity both supported by simulations \nand calculations . Following we take the case of χ = 5 (the twisted skyrmion  shown in Fig. 1(d) ) and investigate the \nmotion induced by the SHE , as shown in Fig.  9. Figure 9(a) shows the simulation results of vx and vy of the twisted \nskyrmion as a function of JHM with  = 0.2 and DDMI = 3 mJ/m2. It can be seen that vx and vy both increase linearly with \nthe increase in  JHM, and the corresponding skyrmion Hall angle Sky is shown in Fig.  9(b). The skyrmion Hall angle  Sky \nalmost remains at 29.6o when JHM is no more than 200 ×  1010 A/m2, while for the case JHM =500 ×  1010 A/m2, the \nskyrmion Hall angle  Sky decreases to 28.9o. This is because  the size of the twisted skyrmion, i.e., the dissipative  \nparameter D, increases slightl y with JHM increasing to 500 ×  1010 A/m2, the skyrmion Hall angle  Eq. (10) can be reduced \nto  \nSky1= arctan( ).1+D\nD\n                                                                     (14) \nFor χ = 5, which indicates that the skyrmion Hall angle  Sky decreases with the increase in D. Figure 9(c) shows the \nsimulation results of vx and vy of the twisted skyrmion as a function of  with JHM =100 ×  1010 A/m2 and DDMI = 3 mJ/m2, \nvx first increases and then decreases w ith  increasing from 0.01 to 1, while vy keeps decreasing (the a bsolute value  of vy \ndecreases at first and then increases), and therefore the corresponding  skyrmion Hall angle  Sky decreases from  44.3o to \n8.2o (the trend of Sky is consistent with  vy), which also supported by  calculation, as shown in Fig.  9(d). Figure 9(e) \nshows that vx and vy both increases with DDMI increasing  from 2.5 to 3.5 mJ/m2 when JHM = 100 ×  1010 A/m2 and  = 0.2.  \nFigure 9(f) shows that the corresponding skyrmion Hall angle Sky decreases with the increase in DDMI, which is similar \nto the res ults by calculating the Eq.  (14) with the increase in D.  \nIn contrast to  the STT -induced twisted skyrmion motion under a damping gradient , we investigate th e dynamics of \ntwisted skyrmion driven  by the S HE under a damping gradient , as shown in Fig.  10. Figure 10 (a) shows the trajectory of \nthe twisted skyrmion as a functio n of its position  along  the x axis with JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2. The \ndamping  increases from 0.2 to 1.2 linearly from 0 to 20 0 nm along the x axis, as indicated by the color code. Figure 10 (b) 10 \n shows the corresponding skyrmion Hall angle Sky as a function of its position along the x axis. The Eq. (14)  implies that:  \nin the region 1D > 0, the twisted skyrmion moves along the x axis direction  from 0 nm and deflects in the y direction  \nuntil moving to the x axis of  114 nm where 1D = 0; in the region 1D < 0, i.e., from 114 to 200 nm along x axis, the \ntwisted skyrmion deflects in the y direction . Therefore , the trajectory of the SHE -induced motion of twisted skyrmion \ncan also be controlled  by a damping gradient.  \nConclusions  \nIn summary,  we first introduce the magnetic structure and the corresponding DMI of the twisted skyrmion in contr ast \nto that of Bloch skyrmion , Né el skyrmion  and antiskyrmion. Furthermore, we discuss and calculate the helicity, winding \nnumber, topological number, size and dissipative force tensor  of the twisted skyrmion, which pave the way for the \nfollowing study of the dynamics  of twisted skyrmion driven by the STT and the SHE . For the STT -induced motion of \ntwisted skyrmion , it is found that the skyrmion Hall angle is determined by the topological number, the dissipative force \ntensor  and the difference  between  the Gilbert damping  and the non -adiabatic factor . 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We only intercept the central region of the 2D plane with the size of 50 nm × 50 nm. The red, white \nand blue represent where the z comp onent of the magnetization is positive, zero and negative, respectively. The black arrows  denote  \nthe distribution of the in -plane magnetization. (e)–(h) are the spatial profiles of the  local magnetization  corresponding to the yellow \ndotted line which marked in the Fig.  1(a). (i)–(l) are the configuration s of bulk DMI, interfacial DMI ，anisotropic DMI and twisted \nDMI , respectively. The orange arrows denote  the directions of the DMI vector.   \n \n \n14 \n  \nFIG. 2. Schematic of a general  skyrmion in two-dimensional polar co ordinates ., ,  and  () indica te the radial coordinate, \nazimuthal angle, skyrmion helicity and the angle between the magnetic moment and the radial coordinate, respectively.  \n \n \n \n \n \n \n \n \n15 \n  \nFIG. 3. (a) display s the magnetization distribution s of twisted skyrmion , Bloch skyrmion and Né el skyrmion  with W = 1. (b)  display s \nthe magnetization distribution  of anti skyrmion  with W = 1. (c) and (d) show the distribution s of topological density  corresponding to \nthe magnetizations shown in (a) and (b) with Q = 1 and Q = 1, respectively.  \n \n \n \n \n \n \n \n \n \n16 \n  \nFIG. 4. Skyrmion  diameter  (d) and the diagonal element of the dissipative tensor  (D) as a function of DMI stre ngth. The inset is the \nspatial profile of mz across the twisted skyrmion. It should be note that the twisted skyrmion exist s stably  in region of 250 nm × 250 \nnm, the diagram only show the central part of 50 nm × 50 nm . \n \n \n \n \n \n \n \n \n \n17 \n  \nFIG. 5. The trajectories of four types of skyrmion driven by the STT. The initial  position of the skyrmions is at  the center  of the 2D \nmagnetic film, the size of 2D plane is 250 nm × 250 nm, vs = 100 m/s  in x direction ,  = 0.4,  = 0.2 and DDMI = 3 mJ/m2. The big \nyellow  solid arrow and white dotted arrow s represent the direction of conduction electrons and the trajectories of skyrmions, \nrespectively. It should be note that the four types of skyrmions are enlarged with the purpose to see their helicities clearly . The actual \nsizes of the four skyrmions are almost the same as the skyrmion at the center position.  \n \n \n \n \n \n \n \n \n \n18 \n  \nFIG. 6. The STT-induced motion of the twisted skyrm ion (0.25). (a) and (b) display  the vx and vy as a function of vs for  = 0.01, \n0.1, 0.2, 0.3 and 0.4 with  = 0.2 and DDMI = 3 mJ/m2, respectively. (c) T he skyrmion Hall an gle Sky as a function of vs corresponding \nto the situation of  = 0.4 shown in Figs.  (a) and (b). (d) The skyrmion Hall a ngle Sky as a function of α corresponding to the situation \nof vs = 100 m/s shown in Figs.  6 (a) and (b). (e) and (f) display  the skyrmon velocity and the skyrmion Hall angle as a function of DDMI \nwith vs = 100 m/s,  = 0.4 and  = 0.2 , respectively.  \n \n \n \n19 \n  \nFIG. 7. The STT -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance  (y axis) of \nthe skyrmion  and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively.  The initial  position \nof the skyrmions is  defined as 0 nm both in x and y axis, vs = 100 m/s,  = 0.4  and DDMI = 3 mJ/m2. The c olor code  represents that the \ndamping  decreases from 0.5 to 0.25 linearly in the region from 0 to 50 nm along the x direction. The red dotted line represents the \nposition w here  = .  \n20 \n  \nFIG. 8. The SHE -induced motion of skyrm ions with different  (a) The trajectories of eight types of  skyrmions with χ = 0.75, 0.5, \n0.25, 0, 0.25, 0.5, 0.75 and  driven by the SHE. The initial  position of the  eight  skyrmions is in the center of the 2D magnetic \nfilm whose size is 250 nm × 250 nm,  = 0.2 and DDMI = 3 mJ/m2. The big yellow  solid arrow denote s the direction of current JHM = 10 \n×  1010 A/m2. The wh ite dotted arrow s represent the trajectories of skyrmions. It also should be note here that the eight  types of \nskyrmi ons are enlarged to see their helicities clearly . The actual sizes of the eight skyrmions are almost the same as them at the center \nposition. (b) The skyrmion Hall angle Sky as a function of  the helicitiy . The black solid squares c orrespond to the e ight types of \nskyrmion in Fig.  8(a), and the black  hollow squares are calculated by the equation.  \n \n21 \n  \nFIG. 9. The SHE -induced motion of the twisted skyrm ion (0.25). (a) and (b) display  the skyrmion velocity and skyrmion Hall \nangle Sky as a function of JHM with  = 0.2 and DDMI = 3 mJ/m2, respectively. (c) and (d) denote the skyrmion velocity and skyrmion \nHall angle Sky as a function of  with JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2, respectively. (e ) and (f) represent the skyrmion \nvelocity and skyrmion Hall angle Sky as a function of DDMI with JHM = 100 ×  1010 A/m2 and  = 0.2, respectively.  \n \n \n \n \n22 \n  \nFIG. 10. The SHE -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance  (y axis) \nof the skyrmion  and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively.  The initial  \nposition of the skyrmions is  defined as 0 nm both in x and y axis, JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2. The c olor code  \nrepresents that the damping  increases from 0.2 to 1.2 linearly in the region from 0 to 200 nm along the x direction. The red dotted \nline represents the position w here 1D = 0, i.e., the skyrmion Hall angle Sky = 0o. \n"    },    {        "title": "1406.6225v2.Interface_enhancement_of_Gilbert_damping_from_first_principles.pdf",        "content": "arXiv:1406.6225v2  [cond-mat.mtrl-sci]  16 Nov 2014Interface enhancement of Gilbert damping from first-princi ples\nYi Liu,1,∗Zhe Yuan,1,2,†R. J. H. Wesselink,1Anton A. Starikov,1and Paul J. Kelly1\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n2Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Ma inz, Staudingerweg 7, 55128 Mainz, Germany\n(Dated: June 6, 2018)\nThe enhancement of Gilbert damping observed for Ni 80Fe20(Py) films in contact with the non-\nmagnetic metals Cu, Pd, Ta and Pt, is quantitatively reprodu ced using first-principles scattering\ncalculations. The “spin-pumping” theory that qualitative ly explains its dependence on the Py thick-\nness is generalized to include a number of extra factors know n to be important for spin transport\nthrough interfaces. Determining the parameters in this the ory from first-principles shows that inter-\nface spin-flipping makes an essential contribution to the da mping enhancement. Without it, a much\nshorter spin-flip diffusion length for Pt would be needed than the value we calculate independently.\nPACS numbers: 85.75.-d, 72.25.Mk, 76.50.+g, 75.70.Tj\nIntroduction. —Magnetizationdissipation, expressedin\ntermsofthe Gilbert dampingparameter α, is akeyfactor\ndetermining the performance of magnetic materials in a\nhost of applications. Of particular interest for magnetic\nmemorydevicesbasedupon ultrathin magneticlayers[ 1–\n3] is the enhancement of the damping of ferromagnetic\n(FM) materials in contact with non-magnetic (NM) met-\nals [4] that can pave the way to tailoring αfor particu-\nlar materials and applications. A “spin pumping” theory\nhas been developed that describes this interface enhance-\nment in terms of a transverse spin current generated by\nthe magnetization dynamics that is pumped into and ab-\nsorbed by the adjacent NM metal [ 5,6]. Spin pumping\nsubsequently evolved into a technique to generate pure\nspin currents that is extensively applied in spintronics\nexperiments [ 7–9].\nA fundamental limitation of the spin-pumping the-\nory is that it assumes spin conservation at interfaces.\nThis limitation does not apply to a scattering theoret-\nical formulation of the Gilbert damping that is based\nupon energy conservation, equating the energy lost by\nthe spin system through damping to that parametrically\npumped out of the scattering region by the precessing\nspins [10]. In this Letter, we apply a fully relativistic\ndensity functional theory implementation [ 11–13] of this\nscattering formalism to the Gilbert damping enhance-\nment in those NM |Py|NM structures studied experimen-\ntally in Ref. 4. Our calculated values of αas a function\nof the Py thickness dare compared to the experimental\nresults in Fig. 1. Without introducing any adjustable pa-\nrameters, we quantitatively reproduce the characteristic\n1/ddependence aswellasthe dependenceofthe damping\non the NM metal.\nTo interpret the numerical results, we generalize the\nspin pumping theory to allow: (i) for interface [ 14–16]\nas well as bulk spin-flip scattering; (ii) the interface mix-\ning conductance to be modified by spin-orbit coupling;\n(iii) the interface resistance to be spin-dependent. An\nimportant consequence of our analysis is that withoutinterface spin-flip scattering, the value of the spin-flip\ndiffusion length lsfin Pt required to fit the numerical\nresults is much shorter than a value we independently\ncalculate for bulk Pt. A similar conclusion has recently\nbeen drawn for Co |Pt interfaces from a combination of\nferromagnetic resonance, spin pumping and inverse spin\nHall effect measurements [ 17].\nGilbert damping in NM |Py|NM.—We focus on the\nNM|Py|NM sandwiches with NM = Cu, Pd, Ta and Pt\nthat were measured in Ref. 4. The samples were grown\non insulating glass substrates, the NM layer thickness\nwas fixed at l=5 nm, and the Py thickness dwas vari-\nable. To model these experiments, the conventional NM-\nlead|Py|NM-lead two-terminal scattering geometry with\nsemi-infinite ballistic leads [ 10–13] has to be modified\nbecause: (i) the experiments were carried out at room\n0 2 4 6 8 10 \nd (nm) 00.02 0.04 0.06 0.08 0.10 _Pt |Py|Pt \nPd|Py|Pd \nTa|Py|Ta \nCu|Py|Cu Calc. Expt. NM \r\n(l)NM \r\n(l)\nLead \n Lead Py \r\n(d)\nFIG. 1. (color online). Calculated (solid lines) Gilbert da mp-\ning of NM |Py|NM (NM = Cu, Pd, Ta and Pt) compared to\nexperimental measurements (dotted lines) [ 4] as a function of\nthe Py thickness d. Inset: sketch of the structure used in the\ncalculations. The dashed frame denotes one structural unit\nconsisting of a Py film between two NM films.2\ntemperature so the 5 nm thick NM metals used in the\nsamples were diffusive; (ii) the substrate |NM and NM |air\ninterfaces cannot transmit charge or spin and behave ef-\nfectively as “mirrors”, whereas in the conventional scat-\ntering theory the NM leads are connected to charge and\nspin reservoirs.\nWe start with the structural NM( l)|Py(d)|NM(l) unit\nindicated by the dashed line in the inset to Fig. 1that\nconsists of a Py film, whose thickness dis variable, sand-\nwichedbetween l=5nm-thick diffusiveNM films. Several\nNM|Py|NM units are connected in series between semi-\ninfinite leads to calculate the total magnetization dissi-\npation of the system [ 10–13] thereby explicitly assuming\na “mirror” boundary condition. By varying the number\nof these units, the Gilbert damping for a single unit can\nbe extracted [ 18], that corresponds to the damping mea-\nsured for the experimental NM( l)|Py(d)|NM(l) system.\nAs shown in Fig. 1, the results are in remarkably\ngood overall agreement with experiment. For Pt and\nPd, where a strong damping enhancement is observed for\nthin Py layers, the values that we calculate are slightly\nlower than the measured ones. For Ta and Cu where\nthe enhancement is weaker, the agreement is better. In\nthe case of Cu, neither the experimental nor the calcu-\nlated data shows any dependence on dindicating a van-\nishinglysmalldampingenhancement. Theoffsetbetween\nthe two horizontal lines results from a difference between\nthe measured and calculated values of the bulk damping\nin Py. Acareful analysisshowsthat the calculated values\nofαare inversely proportional to the Py thickness dand\napproach the calculated bulk damping of Py α0=0.0046\n[11] in the limit of large dfor all NM metals. However,\nextrapolation of the experimental data yields values of\nα0ranging from 0.004 to 0.007 [ 19]; the spread can be\npartly attributed to the calibration of the Py thickness,\nespecially when it is very thin.\nGeneralized spin-pumping theory. —In spite of the very\ngood agreement with experiment, our calculated re-\nsults cannot be interpreted satisfactorily using the spin-\npumping theory [ 5] that describes the damping enhance-\nment in terms of a spin current pumped through the\ninterface by the precessing magnetization giving rise to\nan accumulation of spins in the diffusive NM metal,\nand a back-flowing spin current driven by the ensuing\nspin-accumulation. The pumped spin current, Ipump\ns=\n(/planckover2pi12A/2e2)Gmixm×˙m, is described using a “mixing con-\nductance” Gmix[20] that is a property of the NM |FM\ninterface [ 21,22]. Here, mis a unit vector in the di-\nrection of the magnetization and Ais the cross-sectional\narea. The theory only takes spin-orbit coupling (SOC)\ninto account implicitly via the spin-flip diffusion length\nlsfof the NM metal and the pumped spin current is con-\ntinuous across the FM |NM interface [ 5].\nWith SOC included, this boundary condition does not\nhold. Spin-flip scattering at an interface is described by\nthe “spin memory loss” parameter δdefined so that thespin-flip probability of a conduction electron crossing the\ninterface is 1 −e−δ[14,15]. It alters the spin accumula-\ntion in the NM metal and, in turn, the backflow into the\nFM material. To take δand the spin-dependence of the\ninterface resistance into account, the FM |NM interface\nis represented by a fictitious homogeneous ferromagnetic\nlayerwithafinitethickness[ 15,16]. Thespincurrentand\nspin-resolved chemical potentials (as well as their differ-\nenceµs, the spin accumulation) are continuous at the\nboundaries of the effective “interface” layer. We impose\nthe boundary condition that the spin current vanishes at\nNM|air or NM |substrate interfaces. Then the spin accu-\nmulation in the NM metal can be expressed as a function\nof the net spin-current Isflowing out of Py [ 23], which\nis the difference between the pumped spin current Ipump\ns\nand the backflow Iback\ns. The latter is determined by the\nspin accumulation in the NM metal close to the inter-\nface,Iback\ns[µs(Is)]. Following the original treatment by\nTserkovnyak et al. [ 5],Isis determined by solving the\nequation Is=Ipump\ns−Iback\ns[µs(Is)] self-consistently. Fi-\nnally, the total damping of NM( l)|Py(d)|NM(l) can be\ndescribed as\nα(l,d) =α0+gµB/planckover2pi1\ne2MsdGmix\neff=α0+gµB/planckover2pi1\ne2Msd\n×/bracketleftbigg1\nGmix+2ρlsfR∗\nρlsfδsinhδ+R∗coshδtanh(l/lsf)/bracketrightbigg−1\n.(1)\nHere,R∗=R/(1−γ2\nR) is an effective interface spe-\ncific resistance with Rthe total interface specific resis-\ntance between Py and NM and its spin polarization,\nγR= (R↓−R↑)/(R↓+R↑) is determined by the con-\ntributions R↑andR↓from the two spin channels [ 16].ρ\nis the resistivity of the NM metal. All the quantities in\nEq. (1) can be experimentally measured [ 16] and calcu-\nlated from first-principles [ 24]. If spin-flip scattering at\nthe interface is neglected, i.e., δ= 0, Eq. ( 1) reduces to\nthe original spin pumping formalism [ 5]. Eq. (1) is de-\nrived using the Valet-Fert diffusion equation [ 25] that is\nstill applicable when the mean free path is comparable\nto the spin-flip diffusion length [ 26].\nMixing conductance. —Assuming that SOC can be\nneglected and that the interface scattering is spin-\nconserving, the mixing conductance is defined as\nGmix=e2\nhA/summationdisplay\nm,n/parenleftbig\nδmn−r↑\nmnr↓∗\nmn/parenrightbig\n, (2)\nin terms of rσ\nmn, the probability amplitude for reflection\nofaNMmetalstate nwithspin σintoaNM state mwith\nthesamespin. UsingEq.( 2), wecalculate GmixforPy|Pt\nand Py|Cu interfaces without SOC and indicate the cor-\nresponding damping enhancement gµB/planckover2pi1Gmix/(e2MsA)\non the vertical axis in Fig. 2with asterisks.\nWhen SOC is included, Eq. ( 2) is no longer applicable.\nWecanneverthelessidentify aspin-pumpinginterfaceen-\nhancement Gmixas follows. We artificially turn off the3\n0 2 4 6 8 10\nd (nm)00.050.100.15αd (nm)Pt\nCuWithout backflow\nWith backflow\nFIG. 2. (color online). Total damping calculated for Pt |Py|Pt\nand Cu|Py|Cu as a function of the Py thickness. The open\nsymbols correspond to the case without backflow while the\nfull symbols are the results shown in Fig. 1where backflow\nwas included. The lines are linear fits to the symbols. The as-\nterisks on the yaxis are the values of Gmixcalculated without\nSOC using Eq. ( 2).\nbackflow by connecting the FM metal to ballistic NM\nleads so that any spin current pumped through the in-\nterface propagatesawayimmediately and there is no spin\naccumulation in the NM metal. The Gilbert damping αd\ncalculated without backflow (dashed lines) is linear in\nthe Py thickness d; the intercept Γ at d= 0 represents\nan interface contribution. As seen in Fig. 2for Cu, Γ\ncoincides with the orange asterisk meaning that the in-\nterface damping enhancement for a Py |Cu interface is,\nwithin the accuracy of the calculation, unchanged by in-\ncluding SOC because this is so small for Cu, Ni and Fe.\nBy contrast, Γ and thus Gmix=e2MsAΓ/(gµB/planckover2pi1) for the\nPy|Pt interface is 25% larger with SOC included, con-\nfirming the breakdown of Eq. ( 2) for interfaces involving\nheavy elements.\nThe data in Fig. 1for NM=Pt and Cu are replotted\nas solid lines in Fig. 2for comparison. Their linearity\nmeans that we can extract an effective mixing conduc-\nTABLE I. Different mixing conductances calculated for\nPy|NM interfaces. Gmixis calculated using Eq. ( 2) without\nSOC.Gmixis obtainedfrom theinterceptofthetotal damping\nαdcalculated as a function of the Py thickness dwith SOCfor\nballistic NM leads. The effective mixing conductance Gmix\neffis\nextracted from the effective αin Fig.1in the presence of 5 nm\nNM on either side of Py. Sharvin conductances are listed for\ncomparison. All values are given in units of 1015Ω−1m−2.\nNM GSh GmixGmixGmix\neff\nCu 0.55 0.49 0.48 0.01\nPd 1.21 0.89 0.98 0.57\nTa 0.74 0.44 0.48 0.34\nPt 1.00 0.86 1.07 0.95tanceGmix\neffwith backflow in the presence of 5 nm dif-\nfusive NM metal attached to Py. For Py |Pt,Gmix\neffis\nonly reduced slightly compared to Gmixbecause there is\nvery little backflow. For Py |Cu, the spin current pumped\ninto Cu is only about half that for Py |Pt. However, the\nspin-flipping in Cu is so weak that spin accumulation in\nCu leads to a backflow that almost exactly cancels the\npumped spin current and Gmix\neffis vanishingly small for\nthe Py|Cu system with thin, diffusive Cu.\nThe values of Gmix,GmixandGmix\neffcalculated for all\nfour NM metals are listed in Table I. Because Gmix(Pd)\nandGmix(Pt) are comparable, Py pumps a similar spin\ncurrent into each of these NM metals. The weaker spin-\nflipping and larger spin accumulation in Pd leads to a\nlarger backflow and smaller damping enhancement. The\nrelatively low damping enhancement in Ta |Py|Ta results\nfrom a small mixing conductance for the Ta |Py interface\nrather than from a large backflow. In fact, Ta behaves\nas a good spin sink due to its large SOC and the damp-\ning enhancement in Ta |Py|Ta can not be significantly\nincreased by suppressing the backflow.\nThickness dependence of NM. —In the following we fo-\ncus on the Pt |Py|Pt system and examine the effect of\nchanging the NM thickness lon the damping enhance-\nment, a procedure frequently used to experimentally de-\ntermine the NM spin-flip diffusion length [ 27–31].\nThe total damping calculated for Pt |Py|Pt is plotted\nin Fig.3as a function of the Pt thickness lfor two thick-\nnessesdof Py. For both d= 1 nm and d= 2 nm,\nαsaturates at l=1–2 nm in agreement with experiment\n0 10 20 30 40 50 l (nm)0.00.51.0\n0 1 2 3 4 5\nl (nm)0.000.050.100.15 αd=1 nm\nd=2 nmPt(l)|Py(d)|Pt( l)G↑↑/G↑\nG↑↓/G↑Pt@RT\nl↑=7.8±0.3 nm\nFIG. 3. αas a function of the Pt thickness lcalculated for\nPt(l)|Py(d)|Pt(l). The dashed and solid lines are the curves\nobtained by fitting without and with interface spin memory\nloss, respectively. Inset: fractional spin conductances G↑↑/G↑\nandG↑↓/G↑when a fully polarized up-spin current is injected\ninto bulk Pt at room temperature. Gσσ′is (e2/htimes) the\ntransmission probability of a spin σfrom the left hand lead\ninto a spin σ′in the right hand lead; G↑=G↑↑+G↑↓. The\nvalue of the spin-flip diffusion length for a single spin chann el\nobtained by fitting is lσ= 7.8±0.3 nm.4\n[17,28–31]. AfitofthecalculateddatausingEq.( 1)with\nδ≡0 requires just three parameters, Gmix,ρandlsf. A\nseparate calculation gives ρ= 10.4µΩcm at T=300 K in\nvery good agreement with the experimental bulk value of\n10.8µΩcm [32]. Using the calculated Gmixfrom Table I\nleaves just one parameter free; from fitting, we obtain\na valuelsf=0.8 nm for Pt (dashed lines) that is consis-\ntent with values between0.5and 1.4nm determined from\nspin-pumping experiments [ 28–31]. However, the dashed\nlines clearly do not reproduce the calculated data very\nwell and the fit value of lsfis much shorter than that\nextracted from scattering calculations [ 11]. By injecting\na fully spin-polarized current into diffusive Pt, we find\nl↑=l↓= 7.8±0.3nm, asshownin theinsettoFig. 3, and\nfrom [25,33],lsf=/bracketleftbig\n(l↑)−2+(l↓)−2/bracketrightbig−1/2= 5.5±0.2 nm.\nThis value is confirmed by examining how the current\npolarization in Pt is distributed locally [ 34].\nIf we allow for a finite value of δand use the in-\ndependently determined Gmix,ρandlsf, the data in\nFig.3(solid lines) can be fit with δ= 3.7±0.2 and\nR∗/δ= 9.2±1.7 fΩm2. The solid lines reproduce the\ncalculateddatamuch better than when δ= 0 underlining\nthe importance of including interface spin-flip scattering\n[17,35]. The large value of δwe find is consistent with a\nlow spin accumulation in Pt and the corresponding very\nweak backflow at the Py |Pt interface seen in Fig. 2.\nConductivity dependence. —Many experiments deter-\nmining the spin-flip diffusion length of Pt have reported\nPt resistivities that range from 4.2–12 µΩcm at low tem-\nperature [ 35–38] and 15–73 µΩcm at room temperature\n[17,39–41]. The large spread in resistivity can be at-\ntributed to different amounts of structural disorder aris-\ning during fabrication, the finite thickness of thin film\nsamples etc. We can determine lsfandρ≡1/σfrom\nfirst principles scattering theory [ 11,12] by varying the\ntemperature in the thermal distribution of Pt displace-\nments in the range 100–500 K. The results are plot-\nted (black solid circles) in Fig. 4(a).lsfshows a lin-\near dependence on the conductivity suggesting that the\nElliott-Yafet mechanism [ 42,43] dominates the conduc-\ntion electron spin relaxation. A linear least squares fit\nyieldsρlsf= 0.61±0.02fΩm2that agrees very well with\nbulk data extracted from experiment that are either not\nsensitive to interface spin-flipping [ 37] or take it into ac-\ncount [17,35,38]. For comparison, we plot values of lsf\nextracted from the interface-enhanced damping calcula-\ntions assuming δ= 0 (empty orange circles). The result-\ning values of lsfare very small, between 0.5 and 2 nm, to\ncompensate for the neglect of δ.\nHaving determined lsf(σ), we can calculate the\ninterface-enhanced damping for Pt |Py|Pt for different\nvalues of σPtand repeat the fitting of Fig. 3using Eq. ( 1)\n[44]. The parameters R∗/δandδare plotted as a func-\ntion of the Pt conductivity in Fig. 4(b). The spin mem-\nory lossδdoes not show any significant variation about0102030R*/δ (fΩ m2)\n0 0.1 0.2 0.3\nσ (108Ω-1m-1)024\nδ1020lsf (nm)Rojas-Sánchez\nNiimi\nNguyen\nKurt50 20 10 7 54ρ (µΩ cm)\nδ=0(a)\n(b)\nFIG. 4. (a) lsffor diffusive Pt as a function of its conductivity\nσ(solid black circles) calculated by injecting a fully polar ized\ncurrent into Pt. The solid black line illustrates the linear\ndependence. Bulk values extracted from experiment that are\neithernotsensitivetointerface spin-flipping[ 37]ortakeitinto\naccount [ 17,35,38] are plotted (squares) for comparison. The\nempty circles are values of lsfdetermined from the interface-\nenhanced damping using Eq. ( 1) withδ= 0. (b) Fit values of\nR∗/δandδas a function of the conductivity of Pt obtained\nusing Eq. ( 1). The solid red line is the average value (for\ndifferent values of σ) ofδ=3.7.\n3.7, i.e., it does not appear to depend on temperature-\ninduced disorder in Pt indicating that it results mainly\nfrom scattering of the conduction electrons at the abrupt\npotential change of the interface. Unlike δ, the effective\ninterfaceresistance R∗decreaseswithdecreasingdisorder\nin Pt and tends to saturate for sufficiently ordered Pt. It\nsuggests that although lattice disorder at the interface\ndoes not dissipate spin angular momentum, it still con-\ntributestotherelaxationofthemomentumofconduction\nelectrons at the interface.\nConclusions. —We have calculated the Gilbert damp-\ning for Py |NM-metal interfaces from first-principles and\nreproduced quantitatively the experimentally observed\ndamping enhancement. To interpret the numerical re-\nsults, we generalized the spin-pumping expression for\nthe damping to allow for interface spin-flipping, a mix-\ning conductance modified by SOC, and spin dependent\ninterface resistances. The resulting Eq. ( 1) allows the\ntwo main factors contributing to the interface-enhanced\ndamping to be separated: the mixing conductance that\ndeterminesthespincurrentpumpedbyaprecessingmag-\nnetization and the spin accumulation in the NM metal\nthat induces a backflow of spin current into Py and low-\ners the efficiency of the spin pumping. In particular, the\nlatter is responsible for the low damping enhancement\nfor Py|Cu while the weak enhancement for Py |Ta arises\nfrom the small mixing conductance.\nWe calculate how the spin-flip diffusion length, spin5\nmemory loss and interface resistance depend on the con-\nductivity of Pt. It is shown to be essential to take ac-\ncount of spin memory loss to extract reasonable spin-\nflip diffusion lengths from interface damping. This has\nimportant consequences for using spin-pumping-related\nexperiments to determine the Spin Hall angles that char-\nacterize the Spin Hall Effect [ 17].\nAcknowledgments. —We are grateful to G.E.W. Bauer\nfor a critical reading of the manuscript. Our work was\nfinancially supported by the “Nederlandse Organisatie\nvoor Wetenschappelijk Onderzoek” (NWO) through the\nresearch programme of “Stichting voor Fundamenteel\nOnderzoek der Materie” (FOM) and the supercomputer\nfacilities of NWO “Exacte Wetenschappen (Physical Sci-\nences)”. It was also partly supported by the Royal\nNetherlands Academy of Arts and Sciences (KNAW). Z.\nYuan acknowledges the financial support of the Alexan-\nder von Humboldt foundation.\n∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany\n†zyuan@uni-mainz.de\n[1] See the collection of articles in Handbook of Spin Trans-\nport and Magnetism , edited by E. Y. Tsymbal and\nI.ˇZuti´ c (Chapman and Hall/CRC Press, Boca Raton,\n2011).\n[2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees,\nNature Materials 11, 391 (2012) .\n[3] A. Brataas, A. D. Kent, and H. Ohno,\nNature Materials 11, 372 (2012) .\n[4] S. Mizukami, Y. Ando, and T. Miyazaki,\nJpn. J. Appl. Phys. 40, 580 (2001) ;\nJ. Magn. & Magn. Mater. 226–230 , 1640 (2001) .\n[5] Y. Tserkovnyak, A. Brataas, and G. E. W.\nBauer, Phys. Rev. Lett. 88, 117601 (2002) ;\nPhys. Rev. B 66, 224403 (2002) .\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. 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Turnbull (Academic, New York, 1963) pp. 1–98.\n[44] Experiment [ 8] and theory [ 45] indicate at most a weak\ntemperature dependence of the spin mixing conductance.\n[45] K.Nakata, J. Phys.: Condens. Matter 25, 116005 (2013) ."    },    {        "title": "2308.08331v1.Discovery_and_regulation_of_chiral_magnetic_solitons__Exact_solution_from_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Discovery and regulation of chiral magnetic solitons: Exact solution from Landau-Lifshitz-Gilbert\nequation\nXin-Wei Jin,1, 2Zhan-Ying Yang,1, 2,∗Zhimin Liao,3Guangyin Jing,1, †and Wen-Li Yang2, 4\n1School of Physics, Northwest University, Xi’an 710127, China\n2Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n3School of Physics, Peking University, Beijing, 100871,China\n4Insititute of Physics, Northwest University, Xi’an 710127, China\n(Dated: August 17, 2023)\nThe Landau-Lifshitz-Gilbert (LLG) equation has emerged as a fundamental and indispensable framework\nwithin the realm of magnetism. However, solving the LLG equation, encompassing full nonlinearity amidst\nintricate complexities, presents formidable challenges. Here, we develop a precise mapping through geometric\nrepresentation, establishing a direct linkage between the LLG equation and an integrable generalized nonlinear\nSchr ¨odinger equation. This novel mapping provides accessibility towards acquiring a great number of exact\nspatiotemporal solutions. Notably, exact chiral magnetic solitons, critical for stability and controllability in\npropagation with and without damping effects are discovered. Our formulation provides exact solutions for the\nlong-standing fully nonlinear problem, facilitating practical control through spin current injection in magnetic\nmemory applications.\nIntroduction.— The seminal 1935 work by Landau and Lif-\nshitz, which laid down the foundational dynamical equation\ngoverning magnetization based on phenomenological insights\n[1–3], and the subsequent introduction of a damping term by\nGilbert [4], the amalgamation of these concepts has given\nrise to the renowned Landau-Lifshitz-Gilbert (LLG) equation.\nOver the years, this equation has emerged as a fundamental\nand indispensable framework within magnetism field. Its con-\ntemporary significance has been amplified through remarkable\nadvancements, most notably the incorporation of an additional\nterm that facilitates the explication of spin torque phenomena\nin spintronics [5–10], spin waves [11–18], magnetic solitons\n[19–28], spatio-temporal patterns [29, 30], and even chaotic\nbehavior [31]. Further advancements have paved the way\nfor applications in next-generation magnetic storage [32–34],\nneural networks [35–37], and logic gates [38–42].\nDespite its deceptively simple appearance, solving the LLG\nequation poses an exceptional challenge [29, 30], rendering\nit a persistently unresolved problem for nearly nine decades.\nThis complexity emanates from its intricate nature as a vector-\nbased highly nonlinear partial differential equation. In real-\nworld scenarios, the LLG equation encompasses a myriad of\ncomplex interactions among the components of the magneti-\nzation vector [4]. Consequently, solutions often necessitate\nrecourse to linearization, approximations, and asymptotic\ntechniques such as the Holstein-Primakoff (HP) transforma-\ntion [43, 44], reductive perturbation scheme [45, 46], and long\nwavelength approximation. Nonetheless, these techniques\nprove utterly ineffectual in regions of large amplitudes or\nstrong nonlinearity. Therefore, exact solution of the LLG\nequation emerges as a potent bridge, overcoming these gaps\nand revealing profound revelations regarding magnetization\ndynamics, thereby furnishing insightful understandings for\nsimulating and comprehending intricate magnetic systems.\nIn this Letter, through a geometric representation [47],\nwe establish an exact mapping of the LLG equation onto\nan integrable generalized nonlinear Schr ¨odinger equation,\nfree of any approximation. This novel mapping provides\naccessibility towards acquiring a great number of exact\nspatiotemporal solutions of the original equation. Notably, we\nunveil an analytical formulation for chiral magnetic solitons,encompassing a spectrum ranging from left-handed, neutral to\nright-handed configurations, determined by a defined chirality\nfactor. The derived exact solution indicates the potential for\narbitrary manipulation of magnetic soliton motion through the\ninjection of spin current —a discovery that aligns seamlessly\nwith our numerical findings. To encapsulate the realism of\ndissipative devices, we incorporate Gilbert damping into the\ndynamics of these chiral magnetic solitons, thereby estimating\ntheir dynamic propagation.\nModeling.— We consider an isotropic ferromagnetic\nnanowire with spin-polarized current flowing along the axis\nof nanowire as depicted in Fig. 1. A “nanowire” as defined\nhere is a planar ferromagnetic stripe of length Lx, width\nLy, and thickness Lzalong ˆx,ˆy, and ˆz, respectively, with\nLx≫Ly>Lz.\nFigure 1: Schematic diagram of 1D ferromagnetic structure.\nMagnetic soliton excitation driven by spin-polarized currents.\nHere∆represents the width of magnetic solitons.\nThe magnetization dynamics is described by the famous\nLLG equation\n∂m\n∂t=−γm×Heff+α/parenleftbigg\nm×∂m\n∂t/parenrightbigg\n+τb, (1)\nwhere m=M/Ms= (mx,my,mz)is the unit magnetization\nvector with Msbeing the saturated magnetization. The first\nterm on the right-hand side represents the torque contributed\nby the effective field Heff(including applied, demagnetizing,\nanisotropy, and exchange fields), γis the gyromagnetic\nconstant. The second term describes the Gilbert damping\ntorque, parameterized by a dimensionless damping factor α.arXiv:2308.08331v1  [nlin.PS]  16 Aug 20232\nFigure 2: Spatial structure and classification of chiral magnetic solitons. (a) Vertical views of the left-handed and right-handed\nmagnetic solitons. (b) Schematic plot of the chirality defined by the azimuth angle change. The pair of red arrows delineate the\nazimuthal directional changes of the left and right chiral magnetic solitons across the distribution axis. Their discrepancies in\nazimuthal variation are denoted by ∆ϕ′\nLHand∆ϕ′\nRH. (c)-(e) Spatial spin structures of left-handed magnetic soliton, neutral\nmagnetic soliton, and right-handed magnetic soliton. (f)-(h) illustrate the azimuthal, polar angle, and phase gradient flow of the\nthree kinds of chiral solitons.\nThe last term τbrepresents the spin-transfer torque (STT),\nwhich comprises dual components that can be written as τb=\n−bJ(ˆJ·∇)M+βbJM×(ˆJ·∇)M. Here ˆJis the unit vector in\nthe direction of the current. These two components are most\ncommonly termed adiabatic and non-adiabatic spin torques,\nrespectively, with bJ=P jeµB/(eMs)andβdefined as the non-\nadiabatic torque coefficient. Wherein, Prepresents the spin\npolarization of current, jeis the electric current density, µBis\nthe Bohr magneton, and eis the magnitude of electron charge.\nIn what follows, we take only adiabatic STT into consideration\nfor two reasons: one is that the most widely agreed upon\ninteraction between a spin-polarized current and a magnetic\nsoliton is adiabatic STT; and the other is that the magnitude\nof the nonadiabatic spin torque is about 2 orders of magnitude\nsmaller than adiabatic torque (β≈10−2). Let us begin by\nexamining the most elementary effective field, encompassing\nsolely exchange fields, i.e. Heff= (2A/Ms)∇2m, where Ais\nthe exchange stiffness constant.\nThe spatiotemporal transformation τ=γµ0Mst/(1+α2)\nandζ=λex·xare introduced to recast the LLG equation\ninto the dimensionless Landau-Lifshitz form (Note that λex=/radicalbig\n2A/(µ0M2s)is the exchange length):\nmτ=−m×mζζ−αm×/parenleftbig\nm×mζζ/parenrightbig\n+Qmζ, (2)\nwhereQ=bJ(1+αβ)//radicalbig\n2Aγ2µ0, a dimensionless number\nmeasuring the ratio of external spin current over exchange\ninteraction strength. This dimensionless STT-LLG model\n(2) effectively describes the dynamics of nonlinear excita-\ntions, such as magnetic solitons, occurring in ferromagnetic\nnanowires upon spin injection. Moreover, it exhibits qualita-tive reproduction much of the behavior seen experimentally.\nFor a permalloy nanowire, the standard material parameters\nare:γ=1.76×1011rad/s·T,Ms=8×105A/m,A=1.3×\n10−11J/m,P=0.5. As a result, the units in time and space\nafter rescaling are 1 τ≈5.70 ps ,1ζ≈5.68 nm.\nChiral magnetic soliton.— The high nonlinearity of STT-\nLLG model (2) presents a great challenge for comprehensive\nanalytical research and restricts the exploration of novel\nspin textures to the realm of micromagnetic simulation or\nweak nonlinearity. In this context, to obtain an analytical\ndepiction of large-amplitude magnetic textures, we exactly\nmap the STT-LLG equation (2) into a generalized nonlinear\nSchr ¨odinger (GNLS) equation (See Supplementary Material\nfor further details on the spatial curve mapping procedure),\ndevoid of any approximations. For no damping, the GNLS\nequation reads\niΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (3)\nExact cycloidal chiral magnetic soliton solutions can be\nconstructed by applying the Darboux transformation (DT) [48,\n49]. Indeed, using the mapping relationship between equation\nsolutions, the specific expression of three components of\nmagnetization are obtained (See Supplementary Material for3\ndetailed calculations):\nmx=2a\na2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ),\nmy=2a\na2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ),\nmz=1−2a2\na2+b2sech2(Ξ),\n(4)\nwith\nΞ=2a/bracketleftbigg\nx+/integraldisplay\n(Q+4b)dt/bracketrightbigg\n,\nΓ=2b/bracketleftbigg\nx+/integraldisplay\n(Q−2(a2−b2)/b)dt/bracketrightbigg\n,\nwhere aandbdescribe the wave number and the velocity of\nthe magnetic soliton.\nFig. 2(a) depicts the spin textures of the obtained\nmagnetic solitons. Evidently, these solitons showcase a mirror\nsymmetry relative to the wave vector axis, indicating their\ninherent chirality. The chirality can be characterized by\nvariations of the azimuth angle. To clarify, we denote the polar\nand azimuthal angles of mbyθandϕ, respectively (as shown\nin Fig. 1), such that M+=mx+imy=M0sin(θ)exp(iϕ),mz=\nM0cos(θ). Note that the azimuthal angle exhibits a periodic\nbackground, which arises from the variation of magnetization\nin space, as revealed by the solution (4). The oscillation\nstructures present in the azimuthal angle profiles are related\nto the small oscillation of mν(ν=x,y), even though they\nmay not be readily visible in Fig. 2(a). By eliminating the\nmeaningless periodic background phase, the real phase jumps\nof the chiral magnetic solitons are obtained by calculating the\nintrinsic argument ϕ′(x) =argM′\n+, where M′\n+=M+exp(iΓ).\nAs a result, the azimuthal angles of both chiral solitons\nare demonstrated in Fig. 2(b). Two red arrows span\nbetween the blue dashed line representing negative infinity and\nthe corresponding positive infinity, delineating the azimuthal\nevolution of magnetic solitons across the distribution axis. The\ndistinction in azimuthal variation for chiral magnetic solitons\nare denoted as ∆ϕ′\nLHand∆ϕ′\nRH. Notably, the two classes\nof chiral magnetic solitons exhibit opposite phase jumps,\ncorresponding to two distinct chiralities.\nThe total phase change is defined as ∆ϕ′=ϕ′(x→+∞)−\nϕ′(x→ −∞). In general, the phase change of arbitrary\nmagnetic solitons can be determined by integrating the phase\ngradient flow. Insight can be gained from combining both\nargument and the phase gradient flow ∇ϕ′(x). Starting from\nM′\n+that constructed from exact solutions, we obtain\n∇ϕ′(x) =2b[sech(2Ξ)+1]/parenleftig\na2−b2\na2+b2/parenrightig\nsech(2Ξ)−1. (5)\nOne can observe that the denominator of the aforementioned\nexpression is consistently a non-positive value, which indi-\ncates that “ +” and “ −” families of phase gradient flow are\ncharacterized by the opposite signs of b. Here, we define\na chirality factor C=sgn(b) =±1, which determines the\nchirality of magnetic solitons. It is straightforward to verify\nthat the nonzero phase variation is characterized by a simple\nFigure 3: Coupling between chiral magnetic solitons and spin\ncurrent injection. (a) Velocities of three distinct classes of\nchiral magnetic solitons plotted against spin-polarized\ncurrents ( j). (b)-(d) Controlled manipulation of right-handed\nmagnetic solitons under varying current strengths, enabling\nforward, backward, and arrested motion.\nexpression: ∆ϕ′=2Carctan (|a/b|).Thus, the chirality of\nthe chiral magnetic soliton is entirely determined by this\nchirality factor C. When b=0, a special case naturally\noccurs, where the chirality factor cannot be defined, and\nchirality disappears, corresponding to the neutral cycloidal\nmagnetic solitons. Finally, we can now classify the exact\nsolution (4) into three categories based on the chirality factor,\ncorresponding to neutral, left-handed, and right-handed chiral\nmagnetic solitons. Figs. 2(c)–2(h) depict the typical spin\ntextures, azimuthal angles, polar angles, and phase gradient\nflow at t=0 when no spin current is applied.\nSpin-current coupling and damping effect.— We now move\nto study the coupling between chiral magnetic soliton and the\ninjection of spin current. It has been demonstrated that the\nspin transfer torque is capable of driving the domain wall\nor skyrmion [7, 26], eliciting their prompt movement at a\nconsiderable velocity upon the application of spin current.\nHere we report a comparable phenomenon on the chiral\nmagnetic soliton from both theoretical and simulation results.\nThe numerical simulation results depicting the relationship\nbetween the velocities of three categories of chiral magnetic\nsolitons and the injected spin currents are illustrated in Fig.\n3(a), and are in direct agreement with those obtained from\nthe analytical solutions (4). These linear correlations can be\nrealized from the equivalent GNLS equation (3), wherein the\nspin current term can be normalized to resemble the “driving\nvelocity”, as supported by the dimensional analysis of Q.\nFigs. 3(b)–3(d) exemplify the current manipulation for right-\nhanded magnetic soliton that comprise a series of transient\nsnapshots captured during the magnetization evolution process\n(the model parameters are shown in the caption). The above4\nFigure 4: Transmission of magnetic solitons in damped ferromagnetic nanowires and the anti-damping effect of non-adiabatic\nSTT. (a) Schematic diagram of magnetization dissipation under damping. (b) Propagation of right-handed magnetic solitons in\nferromagnetic nanowires with Gilbert-damping constant α=0.05. (c) Temporal evolution of the magnetization component mz\nin the absence of non-adiabatic STT, where damping constant α=0.01. (d) Temporal evolution of the magnetization component\nmzwith spin-polarized current j=3.7×107A·cm−2. (e) Gilbert-damping dependence of lifetime and moving distance.\nresults highlight two notable aspects. Firstly, the chiral\nmagnetic soliton possesses an inherent velocity linked to its\ninitial magnetization state. Secondly, the solitons’ motion can\nbe stimulated by a spin-polarized current, while preserving\ntheir chirality. The external injection of spin current offers a\nmeans to manipulate chiral magnetic solitons, granting control\nover their forward, backward, and frozen motion.\nUntil now, our analysis is based on a perfect ferromagnetic\nwire in the absence of damping. Strictly speaking, in\nrealistic nanowires, magnetic solitons cannot move over a\nlarge distance due to Gilbert damping. The existence of\ndamping introduces a small torque field, which dissipates the\nenergy of the system during magnetization dynamics, and\nleads to a helical precession of the magnetization towards\nthe direction of the effective field, i.e. the minimum energy\nstate (See Fig. 4(a)). To understand this damping effect in\ngreater detail we performed numerical simulations of single-\nsoliton dynamics. Fig. 4(b) shows the evolution of a right-\nhanded magnetic soliton in a nanowire with Gilbert-damping\nconstant α=0.05. It can be seen that the chiral magnetic\nsoliton degenerates to a homogeneous magnetized state after\npropagating for about 223.8 ps. During the whole process, the\nmagnetic soliton undergoes continuous deformation. This has\ntwo consequences: magnetic soliton spreading and slowing of\ninternal oscillations. In order to characterize the presence of a\nchiral magnetic soliton, we define the soliton polarization Ps=\n[1−min(mz)]\n2. The soliton is deemed to have dissipated when its\npolarization is lower than 5% in comparison to the maximum\nmagnetization. The movement of magnetic solitons within\nferromagnetic nanowires, subject to varying damping, resultsin distinct lifetimes. Fig. 4(e) depicts line graphs illustrating\nthe relationship between the damping coefficient, lifetime,\nand moving distance. The dissipation of solitons due to\ndamping is a challenge to circumvent, and one approach is to\nseek ferromagnetic materials with low damping coefficients.\nHere, we explored the potential anti-damping effect of non-\nadiabatic STT, as depicted in Figs. 4(c) and 4(d). In the\nabsence of external spin current, the mzcomponent of the\nmagnetic soliton diminishes during transmission. However,\nupon injecting an appropriate spin current, the incorporation\nof non-adiabatic STT enables the chiral magnetic soliton to\npropagate uniformly in its original velocity, resulting in a\nsignificantly extended lifetime.\nConclusions.— In this Letter, we have shown that the\ndimensionless LLG equation containing STT is entirely\nequivalent to the generalized nonlinear Schr ¨odinger equation\nwithout any approximation. This remarkable integrable\nsystem enables us to predict novel exact spatiotemporal\nmagnetic solitons. By applying the Darboux transformation,\nwe obtain exact solution of chiral magnetic solitons, emerging\nwithin an isotropic ferromagnetic nanowire. Our analytical\nformulation establishes a distinct correlation between chiral\nmagnetic solitons and the infusion of spin currents, corrobo-\nrating our numerical findings. This interrelation underscores\nthe potential for arbitrary manipulation of magnetic soliton\nmotion through spin current injection. The inherent chirality\nof the micromagnetic structure plays a pivotal role in soliton\nmotion: a reversal in chirality leads to a shift in motion\ndirection. To encapsulate the realism of dissipative devices,\nwe investigate the influence of Gilbert damping on the motion5\nof chiral magnetic solitons. The results reveal that in\nthe presence of damping, chiral magnetic solitons gradually\nevolve toward a uniformly magnetized state. This implies\na topological equivalence between the two magnetization\nstates. We propose selecting an appropriate spin current\nintensity to introduce an anti-damping effect, thereby ensuring\nthe long-distance transmission of the chiral magnetic soliton.\nThese results present new possibilities for developing chiral\nmagnetic soliton-based racetrack memory.\nThe authors thank Prof. H. M. Yu, Prof. L. C. Zhao, Prof. J.\nLiu and Prof. C. P. Liu for their helpful discussions. 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Guo, Communications in Nonlinear\nScience and Numerical Simulation 32, 285 (2016).Supplementary Materials for “Discovery and regulation of chiral magnetic solitons: Exact solution\nfrom Landau-Lifshitz-Gilbert equation”\nXin-Wei Jin and Zhan-Ying Yang∗\nSchool of Physics, Northwest University, Xi’an 710127, China and\nPeng Huanwu Center for Fundamental Theory, Xi’an 710127, China\nZhimin Liao\nSchool of Physics, Peking University, Beijing, 100871,China\nGuangyin Jing†\nSchool of Physics, Northwest University, Xi’an 710127, China\nWen-Li Yang\nPeng Huanwu Center for Fundamental Theory, Xi’an 710127, China and\nInsititute of Physics, Northwest University, Xi’an 710127, ChinaarXiv:2308.08331v1  [nlin.PS]  16 Aug 20232\nIn this supplementary material, we will show more details on the exact geometric mapping between Landau-Lifshitz-Gilbert\nand generalized nonlinear Schr ¨odinger equation, and calculation details of solving the magnetic solitons.\nA. Exact mapping between LLG and GNLS equation: Geometric Representation\nWe identify the magnetization state of ferromagnetic nanowire at any instant of time with a moving space curve in Euclidean\nthree-dimensional space E3. This is achieved by mapping the unit magnetization vector m(x,t)on the unit tangent vector e1\nassociated with the curve. Thus the dimensionless STT-LLG equation (in the absence of damping) becomes\ne1t=e1×e1xx+Qe1x. (A.1)\nIn the usual way, the normal and binormal vectors of the moving space curve are constructed by taking e2in the direction of e′1\nande3=e1×e2. The spatial variations of these orthogonal unit vectors is determined by the Serret-Frenet equations\n\ne1\ne2\ne3\n\nx=\n0κ0\n−κ0τ\n0−τ0\n\ne1\ne2\ne3\n, (A.2)\nwhere κ(x,t)andτ(x,t)are the curvature and torsion of the space curve. In view of (A.1) and (A.2) alongside the orthogonality\nof the three unit vectors, it is easy to obtain\n\ne1\ne2\ne3\n\nt=\n0 −κτ+Qκ κ x\nκτ−Qκ 0 −τ2+Qτ+κ−1κxx\n−κxτ2−Qτ−κ−1κxx 0\n\ne1\ne2\ne3\n. (A.3)\nThe compatibility conditions∂\n∂t/parenleftig\n∂ei\n∂x/parenrightig\n=∂\n∂x/parenleftig\n∂ei\n∂t/parenrightig\n,i=1,2,3, between Eqs. (A.2) and (A.3) lead to the following evolution\nequations for κandτ\nκt=−(κτx+2κxτ)−Qκx, (A.4a)\nτt=/parenleftbig\nκ−1κxx−τ2/parenrightbig\nx+κκx−Qτx. (A.4b)\nOn making the complex transformation Ψ=1\n2κexp/parenleftbig\ni/integraltext\nτdx/parenrightbig\n, we finally arrive at a generalized nonlinear Schr ¨odinger (GNLS)\nequation (it is easy to verify that the real and imaginary parts of (A.5) is equivalent to (A.4a) and (A.4b), respectively)\niΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (A.5)\nThus we have proved that the STT-LLG equation can be exactly mapped into the integrable GNLS equation.\nB. Lax Representation and Darboux Transformation\nWe now turn to establish the connection between the solutions of the LLG equation and the GNLS equation. Using the Pauli\nmatrices ( σ1,σ2,σ3), the LLG equation can be rewriten into the matrix form\n/hatwidemt=1\n2i[/hatwidem,/hatwidemxx]+Q/hatwidemx, (B.1)\nwhere/hatwidem=mxσ1+myσ2+mzσ3and[·,·]denotes the Lie bracket of the matrices. For this equation, the boundary condition is\ngiven by lim\nx→±∞/hatwidem=σ3, i.e., lim\nx→±∞m= (0,0,1). Considering the Lax representation of the GNLS (A.5)\n∂Φ\n∂x=UΦ,∂Φ\n∂t=VΦ, (B.2)\nwhere U=U0+λU1,V=V0+λV1+λ2V2andλis the spectral parameter,\nU0=/parenleftigg\n0Ψ\n−Ψ∗0/parenrightigg\n,U1=−σ3,V0=/parenleftigg\ni|Ψ|2iΨx+QΨ\niΨ∗\nx−QΨ∗−i|Ψ|2/parenrightigg\n,V1=−2iU0−Qσ3,V2=2iσ3. (B.3)3\nSuppose Φ1(x,t,λ)andΦ2(x,t,λ)are two linear independence eigenvectors of Lax pair (B.2), then Ω= (Φ1,Φ2)also satisfies\nEq. (B.2). Let g(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle\nλ=0, we have gx=U0g,gt=V0g. From transformation /hatwideΦ=g−1Φ, we obtain\n∂/hatwideΦ\n∂x=/hatwideU/hatwideΦ,∂/hatwideΦ\n∂t=/hatwideV/hatwideΦ, (B.4)\nwhere\n/hatwideU=g−1Ug−g−1gx=g−1(U−U0)g=λg−1U1g,\n/hatwideV=g−1Vg−g−1gt=g−1(V−V0)g=λg−1V1g+λ2g−1V2g,(B.5)\nLet/hatwidem=−g−1σ3gbe a solution of Eq. (B.1), then\n/hatwidem/hatwidemx=−g−1σ3U0σ3g+g−1σ2\n3U0g=2g−1U0g, (B.6)\nSubstitute (B.3) into (B.5), together with the definition of /hatwidemand (B.6), matrices (B.5) of new Lax pair (B.2) can be rewritten as\n/hatwideU=λ/hatwidem,/hatwideV=λQ/hatwidem−iλ/hatwidem/hatwidemx−2iλ2/hatwidem, (B.7)\nUsing the factor /hatwidem2=I, the compatibility condition /hatwideUt−/hatwideVx+ [/hatwideU,/hatwideV] =0 exactly yields the matrix form LLG equation (B.1).\nThus we proved the Lax gauge equivalence of the GNLS equation (A.5) and the dimensionless STT-LLG equation. Through the\nestablished gauge equivalence detailed above, it becomes evident that given a non-zero solution Φof the GNLS equation (A.5),\nthe corresponding eigenfunctions can be derived via Lax pair (B.2). This process thereby elucidates the determination of the\ninvertible matrices Ω(x,t,λ)andg(x,t). Further through the transformation /hatwideΦ=g−1Φand/hatwidem=−g−1σ3g, we are enabled to\nacquire the solution /hatwidemfor (B.1). Finally, the three components of magnetization, namely mx,my, and mz, can be obtained from\nthe definition of /hatwidem, constituting the non-trivial solution to the original STT-LLG equation.\nTo obtain the dynamical magnetic soliton in the ferromagnetic nanowire, we are going to construct the Darboux transformation\nof (A.5). Let Φ[0]\n1(x,t,λ)andΦ[0]\n2(x,t,λ)be the eigenfunction of the Lax pair (B.2) corresponding to the zero solution of\nthe GNLS equation (A.5). Demonstrating the reciprocity of the Lax pair solution with respect to spectral parameters, it is\nstraightforward to establish that if/parenleftbig\nΦ[0]\n1(x,t,λ1),Φ[0]\n2(x,t,λ1)/parenrightbigTrepresents the solution for Lax pair (B.2) corresponding to the\nspectral parameter λ1, then/parenleftbig\nΦ[0]∗\n2(x,t,λ∗\n1),−Φ[0]∗\n1(x,t,λ∗\n1)/parenrightbigTconstitutes the solution for the corresponding spectral parameter\nλ∗\n1. Denote\nH1=/parenleftigg\nΦ[0]\n1Φ[0]∗\n2\nΦ[0]\n2−Φ[0]∗\n1/parenrightigg\n,Λ1=/parenleftigg\nλ10\n0λ∗\n1/parenrightigg\n, (B.8)\nwhere/parenleftbig\nΦ[0]\n1,Φ[0]\n2/parenrightbigT=/parenleftbig\nexp[−λ1x+2iλ2\n1t−λ1/integraltext\nQdt],exp[λ1x−2iλ2\n1t+λ1/integraltext\nQdt]/parenrightbigT. The Darboux matrix is acquired through\nthe standard procedure\nT[1]=λI−H1Λ1H−1\n1, (B.9)\nleading to the solution Φ[1]=T[1]Φof new spectral problem. Therefore the Darboux transformation is written as\nΨ[1](x,t) =Ψ[0](x,t)−2(λ1+λ∗\n1)Φ[0]\n1Φ[0]∗\n2\n|Φ[0]\n1|2+|Φ[0]\n2|2. (B.10)\nTaking λ1=a+ibwe get the soliton solution of GNLS equation\nΨ[1](x,t) =−2asech[2a(x+4bt+Qt)]exp/bracketleftbig\n2i/bracketleftbig\n(2a2−2b2−Q)t−bx/bracketrightbig/bracketrightbig\n, (B.11)\nand the corresponding eigenfunction Φ[1](x,t,λ) = (Φ[1]\n1,Φ[1]\n2)T. Substitute the above results into Ω(x,t,λ) =/parenleftigg\nΦ[1]\n1Φ[1]∗\n2\nΦ[1]\n2−Φ[1]∗\n1/parenrightigg\n,\ng(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle\nλ=0,/hatwidem=−g−1σ3gin sequence, we finally obtain exact cycloidal chiral magnetic soliton solution in\nferromagnetic nanowires under the influence of spin current injection4\nmx=/hatwidem12+/hatwidem21\n2=2a\na2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ),\nmy=/hatwidem21−/hatwidem12\n2i=2a\na2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ),\nmz=/hatwidem11=1−2a2\na2+b2sech2(Ξ),(B.12)\nwith\nΞ=2a/bracketleftbigg\nx+/integraldisplay\n(Q+4b)dt/bracketrightbigg\n,Γ=2b/bracketleftbigg\nx+/integraldisplay\n(Q−2(a2−b2)/b)dt/bracketrightbigg\n,\nwhere aandbdescribe the wave number and the velocity of the chiral soliton.\nTo gain deeper insight into the interaction dynamics between two chiral magnetic solitons, we continue to utilize gauge\ntransformation (2.2.67) to construct two-soliton solutions based on the above single soliton solutions. The second-order Darboux\nmatrix is expressed as follows\nH2=/parenleftigg\nΦ[1]\n1Φ[1]∗\n2\nΦ[1]\n2−Φ[1]∗\n1/parenrightigg\n,Λ2=/parenleftigg\nλ20\n0λ∗\n2/parenrightigg\n,T[2]=λI−H2Λ2H−1\n2, (B.13)\nand the specific Darboux transformation form of the two-soliton solution is subsequently obtained\nΨ[2](x,t) =Ψ[1](x,t)−2(λ2+λ∗\n2)Φ[1]\n1Φ[1]∗\n2\n|Φ[1]\n1|2+|Φ[1]\n2|2. (B.14)\nTaking λ1=a1+ib1,λ2=a2+ib2, after tedious simplification, we get two-soliton solution of GNLS equation\nΨ[2](x,t) =4η1eiβ2cosh(α2)+η2eiβ1cosh(α1)+iη3/parenleftbig\neiβ1sinh(α1)−eiβ2sinh(α2)/parenrightbig\nη4cosh(α1+α2)+η5cosh(α1−α2)+η6cos(β1−β2), (B.15)\nwhere\nα1=2a1(x+4b1t+/integraldisplay\nQdt),β1=4(a2−b2)t−2b2(x+/integraldisplay\nQdt),\nα2=2a2(x+4b2t+/integraldisplay\nQdt),β2=4(a2−b2)t−2b2(x+/integraldisplay\nQdt),\nη1= [(a2\n2−a2\n1)−(b2−b1)2]a1,η4=−(a2−a1)2−(b2−b1)2,η3=2a1a2(b2−b1),\nη2= [(a2\n1−a2\n2)−(b1−b2)2]a2,η5=−(a2+a1)2−(b2−b1)2,η6=4a1a2.\nContinuing with the same approach in the previous text, we are able to provide a precise expression for the three-component of\nthe magnetization mfor the dynamic chiral magnetic two-solitons. Owing to the complexity of its explicit expression, we opt\nto omit it and solely showcase the corresponding figure. Two typical solutions for the interaction between two chiral magnetic\nsolitons are shown in Fig.(I).\nC. Chirality of cycloidal chiral magnetic soliton in Bloch sphere\nAs shown in the main text, we denote the polar and azimuthal angles of the vector masθandϕ, respectively. This notation\nallows us to express M+=mx+imy=M0sin(θ)exp(iϕ)andmz=M0cos(θ). By employing the three-component analytical\nformulation, we can infer the inverse solution for θandϕ, which in turn can be mapped onto the Bloch unit sphere. This\napproach yields a trajectory map delineating the movement of chiral magnetic solitons across the unit sphere. Consequently,\nwithin the magnetization unit sphere, a chiral magnetic soliton traces a closed curve encompassing a single pole. The trajectories\nof motion for the two distinct types of chiral magnetic soliton solutions on the Bloch spheres can indirectly manifest their\nchirality. Commencing from negative infinity, which corresponds to the pole of ground state, the left and right-handed chiral5\n(a)\n (b)\nFigure I: The interaction between two chiral magnetic solitons. (a) Interaction between left-handed and right-handed magnetic\nsoliton. (b) Bound states formed by two right-handed magnetic Solitons.\nFigure II: Trajectories of chiral magnetic solitons on the Bloch sphere at time t=0. (a) Left-handed chiral magnetic soliton\na=1,b=1, (b) Right-handed chiral magnetic soliton a=1,b=−1.\nmagnetic solitons will give rise to enclosed paths, one proceeding in a clockwise direction and the other counterclockwise. This\nmotion pattern eventually in mirror-symmetrical trajectories.\n∗zyyang@nwu.edu.cn\n†jing@nwu.edu.cn"    },    {        "title": "1301.2114v1.First_principles_calculation_of_the_Gilbert_damping_parameter_via_the_linear_response_formalism_with_application_to_magnetic_transition_metals_and_alloys.pdf",        "content": "arXiv:1301.2114v1  [cond-mat.other]  10 Jan 2013APS/123-QED\nFirst-principles calculation of the Gilbert damping param eter via the linear response\nformalism with application to magnetic transition-metals and alloys\nS. Mankovsky1, D. K¨ odderitzsch1, G. Woltersdorf2, and H. Ebert1\n1University of Munich, Department of Chemistry,\nButenandtstrasse 5-13, D-81377 Munich, Germany and\n2Department of Physics, Universit¨ at Regensburg, 93040 Reg ensburg, Germany\n(ΩDated: September 9, 2018)\nA method for the calculations of the Gilbert damping paramet erαis presented, which based on\nthe linear response formalism, has been implemented within the fully relativistic Korringa-Kohn-\nRostoker band structure method in combination with the cohe rent potential approximation alloy\ntheory. To account for thermal displacements of atoms as a sc attering mechanism, an alloy-analogy\nmodel is introduced. This allows the determination of αfor various types of materials, such as\nelemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for\ndisordered magnetic alloys at T= 0 K and above. The effects of spin-orbit coupling, chemical a nd\ntemperature induced structural disorder are analyzed. Cal culations have been performed for the\n3dtransition-metals bcc Fe, hcp Co, and fcc Ni, their binary al loys bcc Fe 1−xCox, fcc Ni 1−xFex,\nfcc Ni 1−xCoxand bcc Fe 1−xVx, and for 5 dimpurities in transition-metal alloys. All results are in\nsatisfying agreement with experiment.\nPACS numbers: 72.25.Rb 71.20.Be 71.70.Ej 75.78.-n\nI. INTRODUCTION\nDuringthe lastdecadesdynamicalmagneticproperties\nhave attracted a lot of interest due to their importance in\nthe development of new devices for spintronics, in par-\nticular, concerning their miniaturization and fast time\nscale applications. A distinctive property of such devices\nis the magnetization relaxation rate characterizing the\ntime scale on which a system being deviated from the\nequilibrium returns to it, or how fast the device can be\nswitched from one state to another. In the case of dy-\nnamics of a uniform magnetization /vecMthis property\nis associated with the Gilbert damping parameter ˜G(M)\nused first in the phenomenologicalLandau-Lifshitz (LL)1\nand Landau-Lifshitz-Gilbert (LLG) theory2describing\nthe magnetization dynamics processes by means of the\nequation:\n1\nγdM\ndτ=−M×Heff+M×/bracketleftBigg˜G(M)\nγ2M2sdM\ndτ/bracketrightBigg\n,(1)\nwhereMsis the saturation magnetization, γthe gy-\nromagnetic ratio and Heff=−∂MF[M(r)] being the\neffective magnetic field. Sometimes it is more conve-\nnient to use a dimensionless Gilbert damping parame-\nterαgiven byα=˜G/(γMs) (see, e.g.3–5). Safonov\nhas generalized the Landau-Lifshitz equation by intro-\nducing a tensorial form for the Gilbert damping parame-\nter with the diagonal terms characterising magnetization\ndissipation6. Beingintroducedasaphenomenologicalpa-\nrameter, the Gilbert damping is normally deduced from\nexperiment. In particular, it can be evaluated from the\nresonant line width in ferromagnetic-resonance (FMR)\nexperiments. The difficulty of these measurements con-\nsists in the problem that there exist several different\nsources for the broadening of the line width, which havebeen discussed extensively in the literature7–13. The line\nwidth that is observed in ferromagnetic resonance spec-\ntra is usually caused by intrinsic and extrinsic relax-\nation effects. The extrinsic contributions are a conse-\nquence of spatially fluctuating magnetic properties due\nto sample imperfections. Short range fluctuations lead\nto two magnon scattering while long range fluctuations\nlead to an inhomogeneous line broadening due a super-\nposition of local resonances14. In order to separate the\nintrinsic Gilbert damping from the extrinsic effects it is\nnecessary to measure the frequency and angular depen-\ndence of the ferromagnetic resonance line width, e. g.\ntwo magnon scattering can be avoided when the mag-\nnetization is aligned along the film normal11(perpen-\ndicular configuration). Usually one finds a linear fre-\nquency dependence with a zero frequency offset and one\ncan write ∆ H(ω) =αω\nγ+∆H(0). When such measure-\nments are performed over a wide frequency range the\nslope of ∆ Has a function of frequency can be used\nto extract the intrinsic Gilbert damping constant. In\nmetallic ferromagnets Gilbert damping is mostly caused\nby electron magnon scattering. In addition Gilbert-like\ndamping can be caused by eddy currents. The magni-\ntude of the eddy current damping is proportional to d2,\nwheredis the sample thickness10. In sufficiently thin\nmagnetic films ( d≤10 nm) the eddy current damping\ncan be neglected10. However, for very thin films relax-\nation mechanisms that occur at the interfaces can also\nincrease and even dominate the damping. Such effects\nare spin pumping15,16and the modified electronic struc-\nture at the interfaces. In the present work spin pumping\nand the modified interface electronic structure are not\nconsidered and we assume that bulk-like Gilbert damp-\ning dominates.\nMuch understanding of dynamical magnetic properties\ncould in principle be obtained from the simulation of2\nthese processes utilizing time-dependent first-principles\nelectronic structure calculations, that in turn would pave\nthe way to developing and optimizing new materials for\nspintronic devices. In spite of the progress in the de-\nvelopment of time-dependent density functional theory\n(TD-DFT) during the last decades17that allows to study\nvariousdynamicalprocessesin atomsand molecules from\nfirstprinciples, applicationstosolidsarerare. Thisisdue\nto a lack of universally applicable approximations to the\nexchange-correlation kernel of TD-DFT for solids. Thus,\natthemoment, atractableapproachconsistsintheuseof\nthe classical LLG equations, and employing parameters\ncalculated within a microscopic approach. Note however\nthat this approach can fail dealing with ultrafast mag-\nnetization dynamics, which is discussed, for instance, in\nRefs. [18 and 19], but is not considered in the present\nwork.\nMost of the investigations on the magnetization dissi-\npation have been carried out within model studies. Here\none has to mention, in particular, the so-called s-dor\np-dexchange model20–23based on a separate considera-\ntion of the localized ’magnetic’ d-electrons and delocal-\nizeds- andp-electrons mediating the exchange interac-\ntions between localized magnetic moments and responsi-\nble for the magnetization dissipation in the system. As\nwas pointed out by Skadsem et al.24, the dissipation pro-\ncess in this case can be treated as an energy pumping\nout of thed-electron subsystem into the s-electron bath\nfollowed by its dissipation via spin-flip scattering pro-\ncesses. This model gave a rather transparent qualita-\ntive picture for the magnetization relaxation in diluted\nalloys, e.g. magnetic semiconductors such as GaMnAs.\nHowever, it fails to give quantitative agreement with ex-\nperiment in the case of itinerant metallic systems (e.g.\n3d-metalalloys),wherethe d-statesareratherdelocalised\nand strongly hybridized with the sp-electrons. As a con-\nsequence the treatment of allvalence electrons on the\nsame footing is needed, which leads to the requirement\nof first-principles calculations of the Gilbert damping go-\ning beyond a model-based evaluation.\nVarious such calculations of the Gilbert damping pa-\nrameter are already present in the literature. They usu-\nally assume a certain dissipation mechanism, like Kam-\nbersky’sbreathingFermisurface(BFS)25,26, ormoregen-\neral torque-correlation models (TCM)3,27. These models\ninclude explicitly the spin-orbit coupling (SOC), high-\nlighting its key role in the magnetization dissipation pro-\ncesses. However, the latter methods used for electronic\nstructure calculations cannot take explicitly into account\ndisorder in the system that in turn is responsible for the\naforementioned spin-flip scattering process. Therefore,\nthis has to be simulated by using external parameters\ncharacterizing the finite lifetime of the electronic states.\nThisweakpointwasrecentlyaddressedbyBrataas et al.4\nwho described the Gilbert damping by means of scatter-\ning theory. This development supplied the formal basis\nfor the first parameter-free investigations on disordered\nalloys for which the dominant scattering mechanism ispotential scattering caused by chemical disorder5.\nTheoretical investigations of the magnetization dissi-\npation by means of first-principles calculations of the\nGilbert damping parameter already brought much un-\nderstanding of the physical mechanisms responsible for\nthis effect. First of all, key roles are played by two ef-\nfects: the SOC of the atomic species contained in the\nsystem and scattering on various imperfections, either\nimpurities or structural defects, phonons, etc. Account-\ning for the crucial role of scattering processes respon-\nsible for the energy dissipation, different types of scat-\ntering phenomena have to be considered. One can dis-\ntinguish between the ordered-compound or pure-element\nsystems for which electron-phonon scattering is a very\nimportant mechanism for relaxation, and disordered al-\nloys with dominating scattering processes resulting from\nrandomly distributed atoms of different types. In the\nfirst case, the Gilbert damping behavior is rather differ-\nent at low and high temperatures. At high temperature\natomic displacements create random potentials leading\ntoSOC-inducedspin-flipscattering. At lowtemperature,\nwhere the magnetization dissipation is well described via\nthe BFS (Breathing Fermi-surface) mechanism25,26, the\nspin-conserving electron-phonon scattering is required to\nbring the electronic subsystem to the equilibrium at ev-\nery step of the magnetization rotation, i.e. to reoccupy\nthe modified electronic states.\nIn this contribution we describe a formalism for the\ncalculation of the Gilbert damping equivalent to that\nof Brataas et al.4, however, based on the linear re-\nsponce theory28as implemented in fully relativistic mul-\ntiple scattering based Korringa-Kohn-Rostoker (KKR)\nformalism. It will be demonstrated that this allows to\ntreat elegantly and efficiently the temperature depen-\ndence ofαin pure crystals as well as disordered alloys.\nII. THEORETICAL APPROACH\nTo have direct access to real materials and to obtain\na deeper understanding of the origin of the properties\nobserved experimentally, the phenomenological Gilbert\ndamping parameter has to be treated on a microscopic\nlevel. This implies to deal with the electrons responsible\nforthe energydissipation in the magnetic dynamicalpro-\ncesses. Thus, one equates the corresponding expressions\nfor the dissipation rate obtained in the phenomenologi-\ncal and microscopic approaches ˙Emag=˙Edis. Although\na temporal variation of the magnetization is a required\ncondition for the energy dissipation to occur, the Gilbert\ndamping parameter is defined in the limit ω→0 (see\ne.g., Ref. [24]) and therefore can be calculated within the\nadiabatic approximation.\nIn the phenomenological LLG theory the time depen-\ndent magnetization M(t) is described by Eq. (1). Ac-\ncordingly, the time derivative of the magnetic energy is3\ngiven by:\n˙Emag=Heff·dM\ndτ=1\nγ2(˙ˆm)T[˜G(M)˙ˆm](2)\nwhereˆm=M/Msdenotes the normalized magnetiza-\ntion. To represent the Gilbert damping parameter in\nterms of a microscopic theory, following Brataas et al.4,\nthe energy dissipation is associated with the electronic\nsubsystem. The dissipation rate upon the motion of the\nmagnetization ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\n, is determined by the under-\nlying Hamiltonian ˆH(τ). Assuming a small deviation of\nthe magnetic moment from the equilibrium the normal-\nized magnetization ˆm(τ) can be expanded around the\nequilibrium magnetization ˆm0\nˆm(τ) =ˆm0+u(τ), (3)\nresulting in the expression for the linearized time depen-\ndent Hamiltonian for the system brought out of equilib-\nrium:\nˆH=ˆH0(ˆm0)+/summationdisplay\nµuµ∂\n∂uµˆH(ˆm0).(4)\nDue tothe smalldeviation from the equilibrium, ˙Ediscan\nbe obtained within the linearresponseformalism, leading\nto the expression4:\n˙Edis=−π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν/angbracketleftBigg\nψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ej),(5)\nwhereEFis the Fermi energy and the sums run over\nall eigenstates of the system. As Eq. (5) characterizes\nthe rate of the energy dissipation upon transition of the\nsystem from the tilted state to the equilibrium, it can\nbe identified with the corresponding phenomenological\nquantity in Eq. (2), ˙Emag=˙Edis. This leads to an ex-\nplicit expression for the Gilbert damping tensor ˜Gor\nequivalently for the damping parameter α=˜G/(γMs)\n(Ref. [4]):\nαµν=−/planckover2pi1γ\nπMs/summationdisplay\nij/summationdisplay\nµν/angbracketleftBigg\nψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ej),(6)\nwhere the summation is running over all states at the\nFermi surface EF.\nIn full analogy to the problem of electric\nconductivity29, the sum over eigenstates |ψi/angbracketrightmay be\nexpressed in terms of the retarded single-particle Green’s\nfunction Im G+(EF) =−π/summationtext\ni|ψi/angbracketright/angbracketleftψi|δ(EF−Ei). This\nleads for the parameter αto a Kubo-Greenwood-like\nequation:\nαµν=−/planckover2pi1γ\nπMsTrace\n/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc(7)with/angbracketleft.../angbracketrightcindicating a configurational average in case of\na disordered system.\nThe most reliable way to account for spin-orbit cou-\npling as the source of Gilbert damping is to evaluate\nEq. (7) using a fully relativistic Hamiltonian within the\nframework of local spin density formalism (LSDA)30:\nˆH=cα·p+βmc2+V(r)+βσ·ˆmB(r).(8)\nHereαiandβare the standard Dirac matrices, σde-\nnotes the vector of relativistic Pauli matrices, and pis\nthe relativistic momentum operator31. The functions\nV(r) andB=σ·ˆmB(r) are the spin-averaged and\nspin-dependent parts, respectively, of the LSDA poten-\ntial. The spin density ms(r) as well as the effective ex-\nchange field B(r) are assummed to be collinear within\nthe unit cell and aligned along the z-direction in the\nequilibrium (i. e. ms,0(r) =ms(r)ˆm0=ms(r)ezand\nB0(r) =B(r)ˆm0=B(r)ez). Tilting of the magnetiza-\ntion direction by the angle θaccording to Eq. (3), i.e.\nms(r) =ms(r)ˆm=ms(r)(sinθcosφ,sinθsinφ,cosθ)\nandB(r) =B(r)ˆmleads to a perturbation term in the\nHamiltonian\n∆V(r) =βσ·(ˆm−ˆm0)B(r) =βσ·uB(r),(9)\nwith (see Eq. (4))\n∂\n∂uµˆH(ˆm0) =βσµB(r). (10)\nThe Green’s function G+in Eq. (7) can be obtained in\na very efficient way by using the spin-polarized relativis-\ntic version of multiple scattering theory30that allows us\nto treat magnetic solids:\nG+(r,r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(r,E)τnm\nΛΛ′(E)Zm×\nΛ′(r′,E)\n−δnm/summationdisplay\nΛ/bracketleftbig\nZn\nΛ(r,E)Jn×\nΛ′(r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(r,E)Zn×\nΛ′(r′,E)Θ(rn−r′\nn)/bracketrightbig\n.(11)\nHerer,r′refer to site nandm, respectively, where\nZn\nΛ(r,E) =ZΛ(rn,E) =ZΛ(r−Rn,E) is a function\ncentered at site Rn. The four-component wave functions\nZn\nΛ(r,E) (Jn\nΛ(r,E)) are regular (irregular) solutions to\nthe single-site Dirac equation labeled by the combined\nquantum numbers Λ (Λ = ( κ,µ)), withκandµbeing\nthe spin-orbit and magnetic quantum numbers31. The\nsuperscript ×indicates the left hand side solution of the\nDirac equation. τnm\nΛΛ′(E) is the so-called scattering path\noperator that transfers an electronic wave coming in at\nsiteminto a wave going out from site nwith all possible\nintermediate scattering events accounted for.\nUsing matrix notation with respect to Λ, this leads to\nthe following expression for the damping parameter:\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(12)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the4\ntotal magnetic moment µtot=µspin+µorb, ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and with the energy argument EFomit-\nted. The matrix elements in Eq. (12) are identical to\nthose occurring in the context of exchange coupling32:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(r)/bracketleftbigg∂\n∂uµˆH(ˆm0)/bracketrightbigg\nZn\nΛ(r)\n=/integraldisplay\nd3rZn×\nΛ′(r) [βσµBxc(r)]Zn\nΛ(r).(13)\nThe expression in Eq. (12) for the Gilbert-damping\nparameterαis essentially equivalent to the one obtained\nwithin the torque correlation method (see e.g. Refs. [33–\n35]). However, in contrast to the conventional TCM the\nelectronicstructureishererepresentedusingtheretarded\nelectronic Green function giving the present approach\nmuch more flexibility. In particular, it does not rely on\na phenomenological relaxation time parameter.\nThe expression Eq. (12) can be applied straightfor-\nwardly to disordered alloys. This can be done by de-\nscribing in a first step the underlying electronic struc-\nture (forT= 0 K) on the basis of the coherent po-\ntential approximation (CPA) alloy theory. In the next\nstep the configurational average in Eq. (12) is taken fol-\nlowing the scheme worked out by Butler29when dealing\nwith the electrical conductivity at T= 0 K or residual\nresistivity, respectively, of disordered alloys. This im-\nplies in particular that so-called vertex corrections of the\ntype/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat\naccount for scattering-in processes in the language of the\nBoltzmann transport formalism are properly accounted\nfor.\nOne has to note that the factorg\nµtotin Eq. (12) is sep-\narated from the configurational average /angbracketleft.../angbracketrightc, although\nboth values, gandµtot, have to represent the averageper\nunit cell doing the calculations for compounds and al-\nloys. This approximation is rather reasonable in the case\nof compounds or alloys where the properties of the ele-\nments of the system are similar (e.g. 3 d-element alloys),\nbut can be questionable in the case of systems containing\nelements exhibiting significant differences (3 d-5d-, 3d-4f-\ncompounds, etc), or in the case of non-uniform systems\nas discussed by Nibarger et al36.\nThermal vibrations as a source of electron scattering\ncan in principle be accounted for by a generalization of\nEqs. (7) – (13) to finite temperatures and by including\nthe electron-phonon self-energy Σ el−phwhen calculating\nthe Green’s function G+. Here we restrict our considera-\ntion to elastic scattering processes by using a quasi-static\nrepresentation of the thermal displacements of the atoms\nfrom their equilibrium positions. The atom displaced\nfrom the equilibrium position in the lattice results in a\ncorresponding variation ∆ tn=tn−tn\n0of the single-site\nscattering matrix in the global frame of reference37,38. A\nsingle-site scattering matrix tn(the underline denotes a\nmatrix in an angular momentum representation Λ) for\nthe atomndisplaced by the value sn\nνfrom the equilib-\nrium position in the lattice can be obtained using thetransformation matrices37,39\nUn\nLL′(sν,E) = 4π/summationdisplay\nL′′il′′+l−l′\n×CLL′L′′jl′′(sn\nν√\nE)YL′′(ˆsn\nν).(14)\nHeremeis the electron mass, jla spherical Bessel func-\ntion,CLL′L′′stands for the Gaunt coefficients, and a\nnon-relativistic angular momentum representation with\nL= (l,ml) has been used. Performing a Clebsch-Gordon\ntransformation for the transformation matrix Un\nLL′to\nthe relativistic Λ representation, the tmatrixtnfor the\nshifted atom can be obtained from the non-shifted one\ntn\n0from the expression\ntn\nν= (Un\nν)−1tn\n0Un\nν. (15)\nTreating for a discrete set of displacements sn\nνeach\ndisplacement as an alloy component, we introduce an\nalloy-analogy model to average over the set sn\nνthat\nis chosen to reproduce the thermal root mean square\naverage displacement/radicalbig\n/angbracketleftu2/angbracketrightTfor a given temperature\nT. This in turn may be set according to /angbracketleftu2/angbracketrightT=\n1\n43h2\nπ2mkΘD[Φ(ΘD/T)\nΘD/T+1\n4] with Φ(Θ D/T) the Debye func-\ntion,hthe Planck constant, kthe Boltzmann constant\nandΘ DtheDebyetemperature40. Ignoringthezerotem-\nperature term 1 /4 and assuming a frozen potential for\nthe atoms, the situation can be dealt with in full analogy\nto the treatment of disordered alloys on the basis of the\nCPA (see above).\nFor small displacements the transformation Eq. (14)\ncan be expanded with respect to sn\nν(see Ref. [39]) re-\nsulting in a linear dependence on sn\nνfor non-vanishing\ncontributions with ∆ l=|l−l′|=±1. This leads, in\nparticular, in the presence of atomic displacements for\ntransition-metals (TM), for which an angular momen-\ntum cut-off of lmax= 2 in the KKR multiple scattering\nexpansion is in general sufficient for an undistorted lat-\ntice, to an angular momentum expansion up to at least\nlmax= 3. However, this is correct only under the assump-\ntion of very small displacements allowing linearisation of\nthe transformation Uwith respect to the displacement\namplitudes. Thus, since the temperature increase leads\nto a monotonous increaseof s, the cut-off lmaxshould also\nbe increased.\nIII. MODEL CALCULATIONS\nIn the following we present results of calculations for\nwhich single parameters have artificially been manipu-\nlated in the first-principles calculations in order to sys-\ntematically reveal their role for the Gilbert-damping.\nThis approach is used to disentangle competing influ-\nences on the Gilbert-damping parameter.5\nA. Vertex corrections\nThe impact of vertex corrections is shown in Fig. 1\nfor two different cases: Fig. 1(a) represents the Gilbert\ndamping parameter for an Fe 1−xVxdisordered alloy as a\nfunction of concentration, while Fig. 1(b) gives the cor-\nresponding value for pure Fe in the presence of temper-\nature induced disorder and plotted as a function of tem-\nperature. Both figures show results calculated with and\nwithout vertexcorrectionsallowingforcomparison. First\nof all, a significant effect of the vertex corrections is no-\nticeable in both cases, although the variation depends on\nincreasingconcentrationofVinthebinaryFe 1−xVxalloy\nand the temperature in the case of pure Fe, respectively.\nSome differences in their behavior can be explained by\nthe differences of the systems under consideration. Deal-\ning with temperature effects via the alloy analogy model,\nthe system is considered as an effective alloy consisting\nof a fixed number of components characterizing different\ntypes of displacements. Thus, in this case the tempera-\nture effect is associated with the increase of disorder in\nthe system caused only by the increase of the displace-\nment amplitude, or, in other words – with the strength\nof scattering potential experienced by the electrons rep-\nresented by tn(T)−tn\n0. In the case of a random alloy\ntheA1−xBxvariation of the scattering potential, as well\nas the difference tn\nA−tn\nB, upon changing the concentra-\ntions is less pronounced for small amounts of impurities\nBandtheconcentrationdependenceisdeterminedbythe\namount of scatterers of different types. However, when\nthe concentration of impurities increases, the potentials\nof the components are also modified (this is reflected,\ne.g., in the shift of electronic states with respect to the\nFermi level, that will be discussed below) and this can\nlead to a change of the concentration dependence of the\nvertex corrections. An important issue which one has to\nstress that neglect of the vertex corrections may lead to\nthe unphysical result, α <0, as is shown in Fig. 1(a).\nIn terms of the Boltzmann transport formalism, this is\nbecause of the neglect of the scattering-in term41lead-\ning obviously to an incomplete description of the energy\ntransfer processes.\nB. Influence of spin-orbit coupling\nAswasalreadydiscussedabove,thespin-orbitcoupling\nfor the electrons of the atoms composing the system is\nthe main driving force for the magnetization relaxation,\nresultingintheenergytransferfromthemagneticsubsys-\ntem to the crystal lattice. Thus, the Gilbert damping pa-\nrameter should approach zero upon decreasing the SOC\nin the system. Fig. 2 shows the results for Py+15%Os,\nwhere√αis plotted as a function of the scaling param-\neter of the spin-orbit coupling42applied to all atoms in\nthe alloy. As one can see,√αhas a nearly linear depen-\ndence on SOC implying that αvaries in second order in\nthe strength of the spin-orbit coupling43.00.10.2 0.3 0.4 0.5\nconcentration xV02040α × 103without vertex corrections\nwith vertex correctionsFe1-xVx\n(a)\n0100200 300 400 500\ntemperature (K)051015202530α × 103without vertex corrections\nwith vertex correctionsbcc Fe\n(b)\nFIG. 1. The Gilbert damping parameter for (a) bcc Fe 1−xVx\n(T= 0 K) as a function of V concentration and (b) for bcc-Fe\nas a function of temperature. Full (open) symbols give resul ts\nwith (without) the vertex corrections.\n0 0.5 1 1.5 2\nSOC scaling parameter00.10.20.30.4  α1/2 Py+15%Os\nFIG. 2. The Gilbert damping parameter for Py+15%Os as\na function of the scaling parameter of spin-orbit coupling a p-\nplied to all atoms contained in the alloy. Red dashed line in\nplot – linear fit. The values 0 and 1 for the SOC scaling pa-\nrameter correspond to the scalar-relativistic Schr¨ oding er-like\nand fully relativistic Dirac equations, respectively.6\nIV. RESULTS AND DISCUSSIONS\nA. 3dtransition-metals\nWe have mentioned above the crucial role of scatter-\ning processes for the energy dissipation in magnetiza-\ntion dynamic processes. In pure metals, in the absence\nof any impurity, the electron-phonon scattering mecha-\nnism is of great importance, although it plays a different\nrole in the low- and high-temperature regimes. This was\ndemonstrated by Ebert et al.28using the alloy analogy\napproach,aswellasbyLiu44et al.usingthe ’frozenther-\nmal lattice disorder’ approach. In fact both approaches\nare based on the quasi-static treatment of thermal dis-\nplacements. However, while the average is taken by the\nCPA within the alloy analogy model the latter requires\na sequence of super-cell calculations for this purpose.\nAs a first example bcc Fe is considered here. The cal-\nculations have been performed accounting for the tem-\nperature induced atomic displacements from their equi-\nlibrium positions, according to the alloy analogy scheme\ndescribed in section II. This leads, even for pure systems,\nto a scattering process and in this way to a finite value\nforα(see Fig. 3(a)). One can see that the experimen-\ntal results available in the literature are rather different,\ndepending on the conditions of the experiment. In par-\nticular, the experimental results Expt. 2 (Ref. [45]) and\nExpt. 3 (Ref. [46]) correspondto bulk while the measure-\nments Expt. 1 (Ref. [47]) have been done for an ultrathin\nfilm with 2 .3 nm thickness. The Gilbert damping con-\nstant obtained within the present calculations for bcc Fe\n(circles,a= 5.44 a.u.) is compared in Fig. 3(a) with\nthe experiment exhibiting rather good agreement at the\ntemperature above 100 K despite a certain underestima-\ntion. One can also see a rather pronounced increase of\nthe Gilbert damping observed in the experiment above\n400 K (Fig. 3(a), Expt. 2 and Expt. 3), while the theo-\nretical value shows only little temperature dependent be-\nhavior. Nevertheless, the increase of the Gilbert damp-\ning with temperature becomes more pronounced when\nthe temperature induced lattice expansion is taken into\naccount, that can be seen from the results obtained for\na= 5.45 a.u. (squares). Thick lines are used to stress\nthe temperature regions for which corresponding lattice\nparameters are more appropriate. At low temperatures,\nbelow 100 K, the calculated Gilbert damping parameter\ngoes up when the temperature decreases, that was ob-\nserved only in the recent experiment47. This behavior is\ncommonly denoted as a transition from low-temperature\nconductivity-like to high-temperature resistivity-like be-\nhavior reflecting the dominance of intra- and inter-band\ntransitions, respectively3. The latter are related to the\nincrease of the smearing of electron energy bands caused\nby the increase of scattering events with temperature.\nNote that even a small amount of impurities reduces\nstrongly the conductivity-like behavior28,45, leading to\nthe more pronounced effect of impurity-scattering pro-\ncesses due to the increase of scattering events caused bychemical disorder. Large discrepancies between the lat-\nter experimental data47and theoretical results of the α\ncalculationsforbcc Fe arerelatedto the verysmall thick-\nness of the film investigated experimentally, that leads to\nan increase of spin-transfer channels for magnetization\ndissipation as was discussed above.\nResults for the temperature dependent Gilbert-\ndamping parameter αfor hcp Co are presented in Fig.\n3(b) which shows, despite certain underestimation, a rea-\nsonable agreement with the experimental results45. The\ngeneral trends at low and high temperatures are similar\nto those seen in Fe.\nThe results for pure Ni are given in Fig. 3(c) that show\nin full accordance with experiment a rapid decrease of α\nwith increasing temperature until a regime with a weak\nvariation of αwithTis reached.\nNote that in the discussions above we have treated α\nas a scalar instead of a tensorial quantity ignoring a pos-\nsible anisotropy of the damping processes. This approx-\nimation is reasonable for the systems considered above\nwith the magnetization directions along a three- or four-\nfold symmetry axis (see, e.g., the discussions in Ref. [48\nand 49]). For a more detailed discussion of this issue\nthe anisotropy of the Gilbert damping tensor α(M) has\nbeen investigated for bcc Fe. To demonstrate the depen-\ndence ofαon the magnetization direction M, the cal-\nculations have been performed for M= ˆz|M|with the\nˆzaxis taken along the /angbracketleft001/angbracketright,/angbracketleft111/angbracketrightand/angbracketleft011/angbracketrightcrystallo-\ngraphic directions. Fig. 4 presents the temperature de-\npendence of the diagonal elements αxxandαyy. As to be\nexpected for symmetry reasons, αxxdiffers from αyyonly\nin the case of ˆ z/bardbl/angbracketleft011/angbracketright. One can see that the anisotropic\nbehavior of the Gilbert damping is pronounced at low\ntemperatures. With an increase of the temperature the\nanisotropy nearly disappears, because of the smearing of\nthe energy bands caused by thermal vibrations49. A sim-\nilar behavior is caused by impurities with a random dis-\ntribution, aswasobservedforexamplefortheFe 0.95Si0.05\nalloy system. The calculations of the diagonal elements\nαxxandαyyfor two different magnetization directions\nalong/angbracketleft001/angbracketrightand/angbracketleft011/angbracketrightaxes giveαxx=αyy= 0.00123 in\nthe first case and αxx= 0.00123 and αyy= 0.00127 in\nthe second, i.e. the damping is nearly isotropic.\nThe damping parameter αincreases very rapidly with\ndecreasing temperature in the low temperature regime\n(T≤100 K) for all pure ferromagnetic 3 dmetals, Fe,\nCo, and Ni (see Fig. 3), leading to a significant discrep-\nancy between theoretical and experimental results in this\nregime. The observed discrepancy between theory and\nexperiment can be related to the exact limit ω= 0 taken\nintheexpressionfortheGilbertdampingparameter. Ko-\nrenmann and Prange13have analyzedthe magnon damp-\ning in the limit of small wave vector of magnons q→0,\nassuming indirect transitions in the electron subsystem\nand taking into account the finite lifetime τof the Bloch\nstates due to electron-phonon scattering. They discuss\nthe limiting cases of low and high temperatures showing\nthe analogy of the present problem with the problem of7\n0 200 400 600\nTemperature (K)0246810α × 103Theory: a = 5.42 a.u.\nTheory: Fe+0.1% Vac.\nTheory: a = 5.45 a.u.\nExpt. 1\nExpt. 2\nExpt. 3bcc Fe\n(a)\n0100200 300 400 500600\nTemperature (K)05101520α × 103Expt\nTheory\nTheory: Co + 0.03% Vac\nTheory: Co + 0.1% Vachcp Co\n(b)\n0100200 300 400 500\nTemperature (K)00.050.10.150.20.25 αExpt\nTheory\nTheory: Ni + 0.1%Vacfcc Ni\n(c)\nFIG. 3. Temperature variation of the Gilbert damping pa-\nrameter of pure systems. Comparison of theoretical results\nwith experiment: (a) bcc-Fe: circles and squares show the re -\nsults for ideal bcc Fe for two lattice parameters, a= 5.42 a.u.\nanda= 5.45 a.u.; stars show theoretical results for bcc Fe\n(a= 5.42 a.u.) with 0.1% of vacancies (Expt. 1 - Ref. [47],\nExpt. 2 - Ref. [45], Expt. 3 - Ref. [46]); (b) hcp-Co: circles\nshow theoretical results for ideal hcp Co, stars - for Co with\n0.03% of vacancies, and ’pluses’ - for Co with 0.1% of vacan-\ncies (Expt. Ref. [45]); and (c) fcc-Ni (Expt. Ref. [45]).extreme cases for the conductivity leading to the normal\nand anomalous skin effect. On the basis of their result,\nthe authors point out that the expression for the Gilbert\ndamping obtained by Kambersky25, withα∼τis cor-\nrect in the limit of small lifetime (i.e. qvFτ≪1, in their\nmodel consideration, where qis a magnon wave vector\nandvFis a Fermi velocity of the electron). In the low-\ntemperature limit the lifetime τincreases with decreas-\ningTand one has to use the expression corresponding\nto the ’anomalous’ skin effect for the conductivity, i.e.\nα∼tan−1(qvFτ)/qvF, leading to a saturation of αupon\nthe increase of τ.\nAnother possible reason for the low-temperature be-\nhavior of the Gilbert damping observed experimentally\ncan be structural defects present in the material. To\nsimulate this effect, calculations have been performed for\nfcc Ni and bcc Fe with 0.1% of vacancies and for hcp\nCo with 0.1% and 0.03% of vacancies. Fig. 3(a)-(c)\nshows the corresponding temperature dependence of the\nGilbert dampingparameterapproachingafinite value for\nT→0. The remaining difference in the T-dependent be-\nhavior can be attributed to the non-linear dependence of\nthe scattering cross section at low temperatures as is dis-\ncussed in the literature for transport properties of metals\nand is not accounted for within the present approxima-\ntion.\nB. 3dTransition-metal alloys\nAs is mentioned above, the use of the linear response\nformalism within multiple scattering theory for the elec-\ntronicstructurecalculationsallowsustoperformthenec-\nessary configurational averaging in a very efficient way\navoiding supercell calculations and to study with mod-\nerate effort the influence of varying alloy composition\nonα. The corresponding approach has been applied to\n0100200 300 400 500\nTemperature (K)051015202530α × 103Theory: z||<001>: αxx=αyy\nTheory: z||<011>: αxx\nTheory: z||<011>: αyy\nTheory: z||<111>: αxx=αyybcc Fe\nFIG. 4. Temperature variation of the αxxandαyycom-\nponents of the Gilbert damping tensor of bcc Fe with the\nmagnetization direction taken along different crystallogr aphic\ndirections: M= ˆz|M|/bardbl/angbracketleft001/angbracketright(circles), M/bardbl/angbracketleft011/angbracketright(squares),\nM/bardbl/angbracketleft111/angbracketright(diamonds).8\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\nconcentration xCo0123456 α × 103bcc, CPA\nCsCl: CPA (partial ord)\nbcc, NL CPA (ord)\nbcc, NL CPA (disord)\nn(EF)Fe1-xCox\nn(EF)\nn(EF) (sts./Ry)\n102030405060\n0\n(a)\n-8-6-4 -2 0 2\nenergy (eV)00.511.52n↑tot(E) (sts./eV)00.511.52n↓tot(E) (sts./eV)x = 0.01\nx = 0.5EF\n(b)\nFIG. 5. (a) Theoretical results for the Gilbert damping pa-\nrameter of bcc Fe 1−xCoxas a function of Co concentration:\nCPA results for the bcc structure (full circles) describing\nthe random alloy system, results for the partially ordered\nsystem (opened square) for x= 0.5 (i.e. for Fe 1−xCox\nalloy with CsCl structure and alloy components randomly\ndistributed in two sublattices in the following proportion s:\n(Fe0.9Co0.1)(Fe0.1Co0.9), the NL-CPA results for random al-\nloy with bcc structure (opened circles) and the NL-CPA re-\nsults for the the system with short-range order within the\nfirst-neighbor shell (opened diamonds). The dashed line rep -\nresents the DOS at the Fermi energy, EF, as a function of Co\nconcentration. (b) spin resolved DOS for bcc Fe 1−xCoxfor\nx= 0.01 (dashed line) and x= 0.5 (solid line).\nthe ferromagnetic 3 d-transition-metal alloy systems bcc\nFe1−xCox, fcc Ni 1−xFex, fcc Ni 1−xCoxand bcc Fe 1−xVx.\nFig. 5(a) shows as an example results for the Gilbert\ndamping parameter α(x) calculated for bcc Fe 1−xCox\nforT= 0 K at different conditions. Full circles rep-\nresent the results of the single-cite CPA calculations\ncharacterizing the random Fe-Co alloy. These results\nare compared to those obtained employing the non-local\nCPA52,53(NL-CPA) assuming no short-range order in\nthe system (opened circles). Dealing in both cases (CPA\nand NL-CPA), with completely disordered system, the\nNL-CPA maps the alloy problem on that of an impurity\ncluster embedded in a translational invariant effective\nmedium determined selfconsistently, thereby accounting0 0.2 0.4 0.6\nconcentration xCo0246 α × 103Exp.\nTheoryFe1-xCox\n(a)\n0 0.2 0.4 0.6\nconcentration xCo010203040 α × 103Exp.\nTheory (Starikov et al.)\nTheoryNi1-xCox\n(b)\n0 0.2 0.4 0.6\nconcentration xFe01020 α × 103Theory (Starikov et al.)\nTheory\nExpt 1\nExpt 2Ni1-xFex\n(c)\n0 0.2 0.4 0.6\nconcentration xV012345 α × 103Expt.\nTheory, T = 0 K\nTheory, T = 300 KFe1-xVx\n(d)\nFIG. 6. The Gilbert damping parameter for Fe 1−xCox(a)\nNi1−xCox(b) and Ni 1−xFex(c) as a function of Co and Fe\nconcentration, respectively: present results within CPA ( full\ncircles), experimental data by Oogane50(full diamonds). (d)\nResults for bcc Fe 1−xVxas a function of V concentration:\nT= 0 K (full circles) and T= 300 K (open circles). Squares:\nexperimental data51. Open circles: theoretical results by\nStarikov et al.5.9\nfor nonlocal correlations up to the range of the cluster\nsize. The present calculations have been performed for\nthe smallestNL-CPAclusterscontainingtwositesforbcc\nbased system, accountingfor the short-rangeorder in the\nfirst-neighborshell. As onecan see, this results in a small\ndecrease of the αvalue in the region of concentrations\naroundx= 0.5 (opened diamonds), that is in agreement\nwith the results obtained for partially ordered system\n(opened square) for x= 0.5. The latter have been calcu-\nlated for the Fe 1−xCoxalloy having CsCl structure and\nalloycomponentsrandomlydistributed intwosublattices\nin the following proportions: (Fe 0.9Co0.1)(Fe0.1Co0.9).\nBecause the moments and spin-orbit coupling strength\ndo not differ very much for Fe and Co, the variation of\nα(x) should be determined in the concentrated regime\nessentially by the electronic structure at the Fermi en-\nergyEF. As Fig. 5(a) shows, there is indeed a close\ncorrelation with the density of states n(EF) that may be\nseen as a measure for the number of available relaxation\nchannels. The change of α(x) due to the increase of the\nCo concentration is primarily determined by an appar-\nent shift of the Fermi energy also varying with concen-\ntration (Fig. 5(b)). The alloy systems considered have\nthe common feature that the concentration dependence\nofαis governed by the concentration dependent density\nof statesn(EF).\nA comparison of theoretical αvalues with the experi-\nmentforbccFe 1−xCoxisshowninFig. 6(a),demonstrat-\ning satisfying agreement. In the case of Ni 1−xFexand\nNi1−xCoxalloysshown in Fig. 6, (b) and (c), the Gilbert\ndampingdecreasesmonotonouslywith the increaseofthe\nFe and Co concentration, in line with experimental data.\nAt all concentrations the experimental results are under-\nestimated by theory approximately by a factor of 2. The\ncalculated dampingparameter α(x) is found in verygood\nagreementwith theresultsbasedonthe scatteringtheory\napproach5demonstrating numerically the equivalence of\nthe two approaches. An indispensable requirement to\nachieve this agreement is to include the vertex correc-\ntions mentioned above. As suggested by Eq. (12) the\nvariation of α(x) with concentration xmay also reflect\nto some extent the variation of the averagemagnetic mo-\nment of the alloy, µtot.\nThepeculiarityoftheFe 1−xVxalloywhencomparedto\nthose discussed above is that V is a non-magnetic metal\nand has only an induced spin magnetic moment. De-\nspite that, the concentration dependence of the Gilbert\ndamping parameter at T= 0 K for small amounts of\nV (see Fig. 6(d)) displays the same trend as the pre-\nviously discussed alloys shown in Fig. 6(a)-(c). Taking\ninto account a finite temperature of T= 300 K changes\nαvalue significantly at small V concentrations leading\nto an improved agreement with experiment for pure Fe,\nwhile it still compares poorly with the experimental data\natxV= 0.27. One should stress once more that the con-\ncentration dependent behavior of the Gilbert damping\nparameter of the alloys discussed above is different for\nan increased amount of impurities (more than 10%), as aresultofadifferentvariationoftheDOS n(EF)causedby\na concentration dependent modification of the electronic\nstates and shift of the Fermi level.\nC. 5dimpurities in 3 dtransition metals\nAs discussed in our recent work28investigating the\ntemperature dependent Gilbert damping parameter for\npure Ni and for Ni with Cu impurities, αis primarily\ndetermined by the thermal displacement in the regime of\nsmall impurity concentrations. This behavior can also be\nseen in Fig. 7, where the results forFe with 5 d-impurities\nare shown. Solid lines represent results for T= 0 K for\nan impurity concentration of 1% (full squares) and 5%\n(full circles). As one can see, at smaller concentrations\nthe maximum of the Gilbert damping parameter occurs\nfor Pt. With increasing impurity content the αparame-\nter decreases in such a way that at the concentration of\n5% a maximum is observed for Os.\nThe reason for this behavior lies in the rather weak\nscattering efficiency of Pt atoms due to a small DOS\nn(EF) of the Pt states when compared for example for\nOsimpurities (see Fig. 9). This results in a slowdecrease\nofαat small Pt concentration when the BFS mechanism\nis mostly responsible for the energy dissipation. A con-\nsequence of this feature can be seen in the temperature\ndependence of α(T= 300 K, opened squares): a most\npronounced temperature induced decrease of the αvalue\nis observed for Pt and Au. When the concentration of\n5d-impurities is increased up to 5%, the maximum in α\noccurs for the element with the most efficient scatter-\ning potential resulting in spin-flip scattering processes\nresponsible for dissipation. The temperature effect at\nthis concentration is very small.\nConsidering in more detail the temperature dependent\nbehavior of the Gilbert damping parameter for Fe with\nOs and Pt impurities, shown in Fig. 8, one can also ob-\nserve the consequence of the features mentioned above.\nAt 1% of Pt impurities αdecreases much steeper upon\nincreasing the temperature, as compared to the case of\nOs impurities. Therefore, in the first case the role of the\nscattering processes due to atomic displacements is much\nmore pronounced than in the second case with rather\nstrong scattering on the Os impurities. When the con-\ncentration increases to 5% the dependence of αon the\ntemperature in both cases becomes less pronounced.\nThe previousresults can be comparedto the results for\nthe 5d-impurities in the permalloy Fe 80Ni20(Py), which\nhas been investigated also experimentally54. This system\nshows some difference in the concentration dependence\nwhen compared with pure Fe, because Py is a disordered\nalloy with a finite value of the αparameter. Therefore,\na substitution of 5 dimpurities leads to a nearly linear\nincrease of the Gilbert damping with impurity content,\njust as seen in experiment54.\nThe total damping for 10% of 5 d-impurities shown in\nFig.10(a)variesroughlyparabolicallyoverthe 5 dTMse-10\nTa W Re Os Ir Pt Au0246810 α × 103x = 0.05; T = 0 K\nx = 0.05; T = 300 K \nx = 0.01; T = 0 K\nx = 0.01; T = 300 K\nFIG. 7. Gilbert damping parameter for bcc Fe with 1%\n(squares) and 5% (circles) of 5 dimpurities calculated for\nT= 0K (full symbols) and for T= 300K (opened sysmbols).\nries. This variation of αwith the type of impurity corre-\nlateswellwith the densityofstates n5d(EF) (Fig. 10(b)).\nAgain the trend of the experimental data is well repro-\nduced by the calculated values that are however some-\nwhat too low.\nV. SUMMARY\nIn summary, aformulationforthe Gilbert dampingpa-\nrameterαin terms of linear response theory was derived\nthatledtoaKubo-Greenwood-likeequation. Thescheme\nwas implemented using the fully relativistic KKR band\nstructuremethod incombinationwiththe CPAalloythe-\nory. This allows to account for various types of scat-\ntering mechanisms in a parameter-free way, that might\nbe either due to chemical disorder or to temperature-\ninduced structural disorder (i.e. electron-phonon scat-\ntering effect). The latter has been described by using\nthe so-called alloy-analogy model with the thermal dis-\nplacement of atoms dealt with in a quasi-static manner.\nCorresponding applications to pure metals (Fe, Co, Ni)\naswellastodisorderedtransition-metalalloysledto very\ngood agreement with results based on the scattering the-\nory approach of Brataas et al.4and well reproduces the\nexperimental results. The crucial role of vertex correc-\ntions for the Gilbert damping is demonstrated both in\nthe case of chemical as well as structural disorder and\nthe accuracy of finite-temperature results is analyzed via\ntest calculations.\nFurthermore, the flexibility and numerical efficiency\nof the present scheme was demonstrated by a study\non metallic systems on a series of binary 3 d-alloys\n(Fe1−xCox, Ni1−xFex, Ni1−xCoxand Fe 1−xVx), 3d−5d\nTM systems, the permalloy-5 dTM systems. The agree-\nment between the present theoretical and experimental\nresults is quite satisfying, although one has to stress\na systematic underestimation of the Gilbert damping\nby the numerical results. This disagreement could be\ncaused either by the idealized system considered theoret-\nically (e.g., the boundary effects are not accounted for0100200 300 400 500\ntemperature (K)12345α × 103Fe0.99Me0.01Pt\nOs\n(a)\n0100200 300 400 500\ntemperature (K)22.533.54α × 103Fe0.95Me0.05\nPtOs\n(b)\nFIG. 8. Gilbert damping parameter for bcc Fe 1−xMxwith\nM= Pt (circles) and M= Os (squares) impurities as a func-\ntion of temperature for 1% (a) and 5% (b) of the impurities.\n-8-6-4 -2 0 2\nenergy (eV)00.81.62.43.2n↑(E) (sts./eV)00.81.62.43.2n↓(E) (sts./eV)Pt\nOsEF\nEF\nFIG. 9. DOS for Pt in Fe 1−xPtx(full line) and Os in\nFe1−xOsx(dashed line) for x= 0.01.11\nTa W Re OsIr Pt Au02468α × 102Expt.\nTheory\n(a)\nTa W Re OsIr Pt Au051015n5d(EF) (Sts/Ry)\n(b)\nFIG. 10. 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Physics 101,\n033911 (2007)"    },    {        "title": "1002.4958v1.Correlation_Effects_in_the_Stochastic_Landau_Lifshitz_Gilbert_Equation.pdf",        "content": "arXiv:1002.4958v1  [cond-mat.mes-hall]  26 Feb 2010Correlation Effects in Stochastic Ferromagnetic Systems\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗\n(Dated: June 16, 2018)\nAbstract\nWe analyze the Landau-Lifshitz-Gilbert equation when the p recession motion of the magnetic\nmoments is additionally subjected to an uniaxial anisotrop y and is driven by a multiplicative cou-\npled stochastic field with a finite correlation time τ. The mean value for the spin wave components\noffers that the spin-wave dispersion relation and its damping is strongly influenced by the deter-\nministic Gilbert damping parameter α, the strength of the stochastic forces Dand its temporal\nrangeτ. The spin-spin-correlation function can be calculated in t he low correlation time limit by\nderiving an evolution equation for the joint probability fu nction. The stability analysis enables us\nto find the phase diagram within the α−Dplane for different values of τwhere damped spin wave\nsolutions are stable. Even for zero deterministic Gilbert d amping the magnons offer a finite life-\ntime. We detect a parameter range wherethe deterministic an d the stochastic damping mechanism\nare able to compensate each other leading to undamped spin-w aves. The onset is characterized by\na critical value of the correlation time. An enhancement of τleads to an increase of the oscillations\nof the correlation function.\nPACS numbers: 75.10.Hk, 05.40.-a, 75.30.Ds,72.70.+m,76.60.Es\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nMagnetism can be generally characterized and analyzed on different length and time scales.\nThe description of fluctuations of the magnetization, the occurre nce of damped spin waves\nand the influence of additional stochastic forces are successfully performed on a mesoscopic\nscale where the spin variables are represented by a continuous spa tio-temporal variable [1].\nIn this case a well established approach isbased uponthe Landau-L ifshitz equation [2] which\ndescribes the precession motion of the magnetization in an effective magnetic field. This\nfield consists of a superposition of an external field and internal fie lds, produced by the in-\nteracting magnetic moments. The latter one is strongly influenced b y the isotropic exchange\ninteraction and the magnetocrystalline anisotropy, for a recent r eview see [3]. The studies\nusing this frame are concentrated on different dynamical aspects as the switching behav-\nior of magnetic nanoparticles which can be controlled by external tim e-dependent magnetic\nfields [4] and spin-polarized electric currents [5, 6]. Such a current- induced spin transfer\nallows the manipulation of magnetic nanodevices. Recently, it has bee n demonstrated that\nan electric current, flowing through a magnetic bilayer, can induce a coupling between the\nlayers [7]. Likewise, such a current can also cause the motion of magn etic domain walls in\na nanowire [8]. Another aspect is the dynamical response of ferrom agnetic nanoparticles\nas probed by ferromagnetic resonance, studied in [9]. In describing all this more complex\nbehavior of magnetic systems, the Landau-Lifshitz equation has t o be extended by the in-\nclusion of dissipative processes. A damping term is introduced pheno menologically in such a\nmanner, that the magnitude of the magnetization /vectorSis preserved at any time. Furthermore,\nthe magnetization should align with the effective field in the long time limit. A realization\nis given by [2]\n∂S\n∂t=−γ[S×Beff]−ε[S×(S×Beff)]. (1)\nThe quantities γandεare the gyromagnetic ratio and the damping parameter, respectiv ely.\nAn alternative equation for the magnetization dynamics had been pr oposed by Gilbert [10].\nThe Gilbert equation yields an implicit form of the evolution of the magne tization. A com-\nbination of both equations, called Landau-Lifshitz-Gilbert equation (LLG) will be used as\nthe basic relation for our studies, see Eq. (2). The origin of the dam ping term as a non-\nrelativistic expansion of the Dirac equation has been discussed in [11 ] and a generalization\nof the LLG for conducting ferromagnetics is offered in [12]. The form of the damping seems\n2to be quite general as it has been demonstrated in [13] using symmet ry arguments for fer-\nroelectric systems.\nAs a new aspect let us focus on the influence of stochastic fields. Th e interplay between\ncurrent and magnetic fluctuations and dissipation has been studied recently in [14]. Via the\nspin-transfer torque, spin-current noise causes a significant en hancement of the magnetiza-\ntion fluctuations. Such a spin polarized current may transfer mome ntum to a magnet which\nleads to a spin-torque phenomenon. The shot noise associated with the current gives rise to\na stochastic force [15]. In our paper we discuss the interplay betwe en different dissipation\nmechanism, namely the inherent deterministic damping in Eq. (1) and t he stochastic mag-\nnetic field originated for instance by defect configurations giving ris e to a different coupling\nstrength between the magnetic moments. Assuming further, tha t the stochastic magnetic\nfield is characterized by a finite correlation time, the system offers m emory effects which\nmight lead to a decoherent spin precession. To that aim we analyze a f erromagnet in the\nclassical limit, i.e., the magnetic order is referred to single magnetic at oms which occupy\nequivalent crystal positions, and the mean values of their spins exh ibit a parallel orientation.\nThe last one is caused by the isotropic exchange interaction which will be here supplemented\nby a magneto-crystalline anisotropy that defines the direction of t he preferred orientation.\nEspecially, we discuss the influence of an uniaxial anisotropy. The co upling between differ-\nent dissipation mechanisms, mentioned above, leads to pronounced correlations, which are\ndiscussed below. Due to the multiplicative coupling of the stochastic fi eld and the finite\ncorrelation time the calculation of the spin-spin correlation function is more complicated.\nTo that aim we have to derive an equivalent evolution equation for the joint probability\ndistribution function. Within the small correlation time limit this approa ch can be fulfilled\nin an analytical manner. Our analysis is related to a recent paper [16] in which likewise the\nstochastic dynamics of the magnetization in ferromagnetic nanopa rticles has been studied.\nFurther, we refer also to a recent paper [17] where the mean first passage time and the\nrelaxation of magnetic moments has been analyzed. Different to tho se papers our approach\nis concentrated on the correlation effects in stochastic system wit h colored noise.\nOur paper is organized as follows: In Sec.II we discuss the LLG and ch aracterize the ad-\nditional stochastic field. The equations for the single and the two pa rticle joint probability\ndistribution are derived in Sec.III. Using these functions we obtain t he mean value of the\nspin wave variable and the spin-spin correlation function. The phase diagram, based on the\n3stability analysis, is presented in Sec.IV. In Sec.V we finish with some co nclusions.\nII. MODEL\nIn order to develop a stochastic model for the spin dynamics in ferr omagnetic systems let\nus first consider the deterministic part of the equation of motion. W e focus on a description\nbased upon the level of Landau-Lifshitz phenomenology [2], for a r ecent review see [3]. To\nfollow this line we consider a high spin systems in a ferromagnet sufficien tly below the\nCurie temperature. In that regime the dynamics of the magnet are dominated by transverse\nfluctuationsofthespatio-temporalvaryinglocalmagnetization. Theweakexcitations, called\nspin waves or magnons, are determined by a dispersion relation, the wavelength of which\nshould be large compared to the lattice constant a, i.e., the relation q·a≪1 is presumed to\nbe satisfied, where qis the wavenumber. In this limit the direction of the spin varies slowly\nwhile its magnitude |S|=msremains constant in time. A proper description for such a\nsituation is achieved by applying the Landau-Lifshitz-Gilbert equatio n (LLG) [4, 10, 18].\nThe spin variable is represented by S=msˆ n, whereˆ n(r,t) is a continuous variable which\ncharacterizes the local orientation of the magnetic moment. The e volution equation for that\nlocal orientation reads\n∂ˆ n\n∂t=−γ\n1+α2ˆ n×[Beff+α[ˆ n×Beff]]. (2)\nThe quantities γandαare the gyromagnetic ratio and the dimensionless Gilbert damping\nparameter, respectively, where αis related to εintroduced in Eq. (1). Beffis the effective\nmagnetic field that drives the motion of the spin density. Generally, it consists of an internal\npart originated by the interaction of the spins and an external field . This effective field is\nrelated to the Hamiltonian of the system by functional variation with respect to ˆ n\nBeff=−m−1\nsδH\nδˆ n. (3)\nIn absence of an external field the Hamiltonian can be expressed as [19, 20]\nH=/integraldisplay\nd3r{wex+wan},with\nwex=1\n2msκ(∇ˆ n)2andwan=1\n2msΓ sin2θ.(4)\nThereby, the constants κand Γ denote the exchange energy density and the magneto-\ncrystalline anisotropy energy density. To be more precise, κ∝Ja2,Jbeing the coupling\n4strength that measures theinteraction between nearest neighb ors inthe isotropic Heisenberg\nmodel [21]. Once again ais thelattice constant. Notice that the formof theexchange ener gy\nin the Hamiltonian (4) arises from the Heisenberg model in the classica l limit. The quantity\nθrepresents the angle between ˆ nand the anisotropy axis ˆν= (0,0,1), where ˆνpoints\nin the direction of the easy axis in the ground state in the case of zer o applied external\nfield. Thus, the constant Γ >0 characterizes anisotropy as a consequence of relativistic\ninteractions (spin-orbital and dipole-dipole ones [20]). In deriving Eq . (4) we have used\nˆ n2= 1. Although it is more conventional to introduce the angular coord inates (θ,Φ) [2, 4],\nwe find it more appropriate to use Cartesian coordinates. To proce ed, we divide the vector\nˆ ninto a static and a dynamic part designated by µandϕ, respectively. In the linearized\nspin wave approach let us make the ansatz\nˆ n(r,t) =µ(r)+ϕ(r,t) =µˆν+ϕ, µ= const., (5)\nwhereˆ n2= 1 is still valid. The effective field can now be obtained from Eqs. (3) an d (4).\nThis yields\nBeff=κ∇2ϕ−Γϕ′;ϕ′= (ϕ1,ϕ2,0). (6)\nEq. (2) together with Eqs. (3) and (4) represent the determinist ic model for a classical\nferromagnet. In order to extent the model let us supplement the effective magnetic field in\nEq. (6) by a stochastic component yielding an effective random field Beff=Beff+η(t). The\nstochastic process η(t) is assumed to be Gaussian distributed with zero mean and obeying\na colored correlation function\n˜χij(t,t′) =∝an}b∇acketle{tηi(t)ηj(t′)∝an}b∇acket∇i}ht=˜Dij\n˜τijexp/bracketleftbigg\n−|t−t′|\n˜τij/bracketrightbigg\n. (7)\nHere,˜Dijand ˜τijare the noise strength and the finite correlation time of the noise η.\nDue to the coupling of the effective field to the spin orientation ˆ nthe stochastic process\nis a multiplicative one. Microscopically, such a random process might be originated by\na fluctuating coupling strength for instance. The situation associa ted with our model is\nillustrated in Fig. 1 and can be understood as follows: The stochastic vector fieldη(t) is able\nto change the orientation of the localized moment at different times. Therefore, fixed phase\nrelations between adjacent spins might be destroyed. Moreover, theη(tk) are interrelated\ndue to the finite correlation time τ. The anisotropy axis defines the preferred orientation of\nthe mean value of magnetization. Due to the inclusion of η(t) the deterministic Eq. (2) is\n5xyz\nanisotropy axis ˆν\nexchange ∝Jaη(t1)η(t2)η(t3)random field at\ndifferent times ti\nFIG. 1. Part of a ferromagnetic domain influenced by stochast ic forces for the example of cubic\nsymmetry with lattice constant a. The black spin in the center only interacts with its nearest\nneighbors (green), where Jis a measure for the exchange integral.\ntransformed into the stochastic LLG. Using Eq. (5) it follows\n∂ϕ\n∂t=−γ\n1+α2(µ+ϕ)×[Beff+α[(µ+ϕ)×Beff]]. (8)\nThe random magnetic field is defined by\nBeff=κ∇2ϕ−Γϕ′+η(t), (9)\nwhereϕ′is given in Eq. (6). With regard to the following procedure we suppose the random\nfield to be solely generated dynamically, i.e., ˆ n×η(t) =ϕ×η(t). So far, the dynamics\nof our model (Eqs. (8) and (9)) are reflected by a nonlinear, stoc hastic partial differential\nequation (PDE). Using Fourier transformation, i.e., ψ(q,t) =F{ϕ(r,t)}and introducing\nthe following dimensionless quantities\nβ= (l0q)2+1, l2\n0=κ\nΓ, ω=γΓ,¯t=ωt ,λ(t) =η(t)\nΓ,(10)\nthe components ψi(q,t) fulfill the equation\nd\ndtψi(q,t) = Ωi(ψ(q,t))+Λ ij(ψ(q,t))λj(t). (11)\n6The quantity l0is the characteristic magnetic length [22]. The vector Ωand the matrix Λ\nare given by\nΩ=ξµβ\n−(αµψ1+ψ2)\nψ1−αµψ2\n0\n, ξ=1\n1+α2, (12)\nand\nΛ =ξ\nαµψ3ψ3−(ψ2+αµψ1)\n−ψ3αµψ3ψ1−αµψ2\nψ2−ψ1 0\n. (13)\nFor convenience we have substituted ¯t→tagain. The statistical properties of λ(t) are\nexpressed as ∝an}b∇acketle{tλ(t)∝an}b∇acket∇i}ht= 0 and\nχkl(t,t′) =∝an}b∇acketle{tλk(t)λl(t′)∝an}b∇acket∇i}ht=Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\nτkl→0− −− →2Dklδklδ(t−t′).(14)\nIncidentally, in the limit τ→0 the usual white noise properties are recovered. We empha-\nsize that although we regard the long-wavelength limit ( a·q≪1), wave vectors for which\nl0·q≫1 (in Eq. (10)) can also occur [22]. But this case is not discussed in the present\npaper and will be the content of future work. Whereas, in what follo ws we restrict our\nconsiderations to the case q→0 so that, actually, l0·q≪1 is fulfilled. Hence, we can set\nβ= 1 approximately in Eq. (10). Due to the anisotropy the spin wave dis persion relation\noffers a gap at q= 0. Owing to this fact ψis studied at zero wave vector. For this situation\nthe assumption of a space-independent stochastic force ηi(t), compare Eq. (7), is reasonable.\nFor non-zero wave vector the noise field should be a spatiotempora l fieldηi((r,t). Because\nour model is based on a short range interaction we expect that the corresponding noise\ncorrelation function is δ-correlated, i.e. instead of (14) we have\nχkl(r,t;r′,t′) =Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\n2Mδ(r−r′),\nwhereMis the strength of the spatial correlation. Using this relation we are able to study\nalso the case of small qwhich satisfies l0·q≪1. In the present paper we concentrate on\nthe case of zero wave vector q= 0.\n7III. CORRELATION FUNCTIONS\nIn the present section let us discuss the statistical behavior of th e basic Eqs. (11)-(14). They\ndescribe a non-stationary, non-Markovian process attributed t o the finite correlation time.\nDue to their common origin both characteristics can not be analyzed separately. In the\nlimitτ→0, Eq. (11) defines a Markovian process which provides also station arity by an\nappropriate choice of initial conditions [23]. However, the present s tudy is focused on the\neffectofnonzerocorrelationtimes. Tothatpurposeweneedapro perprobabilitydistribution\nfunction which reflects the stochastic process defined by Eqs. (1 1)-(14). In deriving the\nrelevant joint probability distribution function we follow the line given in [24], where the\ndetailedcalculationshadbeencarriedout, seealsothereferences citedtherein. Inparticular,\nit has been underlined in those papers that in order to calculate corr elation functions of type\n∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hta single probability distribution function P(ψ,t) is not sufficient. Instead of\nthat one needs a joint probability distribution of the form P(ψ,t;ψ′,t′). Before proceeding\nlet us shortly summarize the main steps to get the joint probability dis tribution function.\nTo simplify the calculation we assume τkl=τδklandDkl=Dδkl. Notice that our system\nhas no ergodic properties what would directly allow us to relate the st ochastic interferences\nwith temperature fluctuations by means of a fluctuation-dissipatio n theorem. Based on\nEq. (11) the appropriate joint probability distribution is defined by [2 4, 25], for a more\ngeneral discussion compare also [26]:\nP(ψ,t;ψ′,t′) =∝an}b∇acketle{tδ(ψ(t)−ψ)δ(ψ(t′)−ψ′)∝an}b∇acket∇i}ht. (15)\nHere the average is performed over all realizations of the stochas tic process. In defining the\njoint probability distribution function we follow the convention to indic ate the stochastic\nprocess by the function ψ(t) whereas the quantity without arguments ψstands for the\nspecial values of the stochastic variable. These values are even re lalized with the probaility\nP(ψ,t;ψ′,t′). The equation of motion for this probability distribution reads acco rding to\n8[24]\n∂\n∂tP(ψ,t;ψ′,t′)\n=−∂\n∂ψit/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t)\nδλk(t1)/bracketrightbigg\nψ(t)=ψ·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1\n−∂\n∂ψ′it′/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t′)\nδλk(t1)/bracketrightbigg\nψ(t′)=ψ′·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1,(16)\nwhere Novikov’s theorem [27] has been applied. Expressions for the response functions\nδψi(t)/δλk(t1) andδψi(t′)/δλk(t1) can be found by formal integration of Eq. (11) and\niterating the formal solution. After a tedious but straightforwar d calculation including the\ncomputation of the response functions to lowest order in ( t−t1) and (t′−t1) and the\nevaluation of several correlation integrals referring to χklfrom Eq. (14), Eq. (16) can be\nrewritten in the limit of small correlation time τas\n∂\n∂tPs(ψ,t;ψ′,t′) =/braceleftbig\nL0(ψ,τ)\n+exp[−(t−t′)/τ]D∂\n∂ψiΛik(ψ)∂\n∂ψ′\nnΛnk(ψ′)/bracerightbigg\nPs(ψ,t;ψ′,t′).(17)\nThereby, transient terms and terms of the form ∝τexp[−(t−t′)/τ] (these terms would lead\nto terms of order τ2in Eq. (22)) have been neglected. The result is valid in the stationary\ncase characterized by t→ ∞andt′→ ∞but finites=t−t′. In Eq. (17) L0is the operator\nappearing in the equation for the single probability density. Following [2 4, 28] the operator\nreads\nL0(ψ,τ) =−∂\n∂ψiΩi(ψ)+∂\n∂ψiΛik(ψ)∂\n∂ψn/braceleftigg\nD/bracketleftbig\nΛnk(ψ)−τMnk(ψ)/bracketrightbig\n+D2τ/bracketleftbigg\nKnkm(ψ)∂\n∂ψlΛlm(ψ)+1\n2Λnm(ψ)∂\n∂ψlKlkm(ψ)/bracketrightbigg/bracerightigg\n,(18)\nwith\nMnk= Ωr∂Λnk\n∂ψr−Λrk∂Ωn\n∂ψr\nKnlk= Λrk∂Λnl\n∂ψr−∂Λnk\n∂ψrΛrl.(19)\nThe equation of motion for the expectation value ∝an}b∇acketle{tψi∝an}b∇acket∇i}htscan be evaluated from the single\nprobability distribution in the stationary state\n∂\n∂tPs(ψ,t) =L0Ps(ψ,t). (20)\n9One finds\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi∝an}b∇acket∇i}hts+D/angbracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/angbracketrightbigg\ns−D2τ/braceleftigg/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/angbracketrightbigg\ns/bracerightigg\n.(21)\nThe knowledge of the evolution equation of the joint probability distr ibutionP(ψ,t;ψ′,t′)\ndue to Eqs. (17) and (18) allows us to get the corresponding equat ion for the correlation\nfunctions. Following again [24], it results\nd\ndt∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi(ψ(t))ψj(t′)∝an}b∇acket∇i}hts+D/angbracketleftbigg/bracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n−D2τ/braceleftigg/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns/bracerightigg\n+Dexp/bracketleftbigg\n−t−t′\nτ/bracketrightbigg\n∝an}b∇acketle{tΛik(ψ(t))Λjk(ψ(t′))∝an}b∇acket∇i}hts,(22)\nwhere the symbol [ ...]tdenotes the quantity [ ...] at timet. As mentioned above the result\nis valid for t, t′→ ∞whiles=t−t′>0 remains finite. The quantities MnkandKklmare\ndefined in Eq. (19). The components Ω iand Λ ijare given in Eqs. (12) and (13). Performing\nthe summation over double-indices according to Eqs. (21) and (22) we obtain the evolution\nequations for the mean value and the correlation function\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t)∝an}b∇acket∇i}hts, (23)\nand\nd\ndsCij(s) =d\nds∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t′+s)ψj(t′)∝an}b∇acket∇i}hts\n+Dexp/bracketleftig\n−s\nτ/bracketrightig\n∝an}b∇acketle{tΛik(ψ(t′+s))Λjk(ψ(t′))∝an}b∇acket∇i}hts.(24)\nNotice, that in the steady state one gets Cij(t,t′) =Cij(s) withs=t−t′. The matrix\ncomponents of Gikare given by\nGik=\n−A1A20\n−A2−A10\n0 0 −A3\n, (25)\n10where\nA1=−D2τ(6µ2α2−1)ξ4+2µ2αDτξ3−D(µ2α2−2)ξ2+µ2αξ\nA2=1\n2µαD2τ/parenleftbig\n11−3µ2α2/parenrightbig\nξ4+µDτ/parenleftbig\nµ2α2−1/parenrightbig\nξ3+3µDαξ2−µξ\nA3= +D2τ/parenleftbig\n3µ2α2+1/parenrightbig\nξ4−4µ2αDτξ3+2Dξ2,(26)\nandξis defined in Eq. (12). At this point let us stress that in the case t′= 0 the term\n∝exp[−(t−t′)/τ] on the rhs. in Eqs. (22) and (24), respectively, would vanish in the steady\nstate, i.e.\n∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts∝ne}ationslash=∝an}b∇acketle{tψi(s)ψj(0)∝an}b∇acket∇i}hts.\nTheoccurrenceofsuchatermisastrongindicationforthenon-st ationarityofourmodel. An\nexplicit calculation shows, that in general this inequality holds for non -stationary processes\n[23].\nIV. RESULTS\nThe solution of Eq. (23) can be found by standard Greens function methods and Laplace\ntransformation. As the result we find\n∝an}b∇acketle{tψ(t)∝an}b∇acket∇i}hts=\ne−A1tcos(A2t)e−A1tsin(A2t) 0\n−e−A1tsin(A2t)e−A1tcos(A2t) 0\n0 0 e−A3t\n·∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts, (27)\nwhere∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts=∝an}b∇acketle{tψ(t= 0)∝an}b∇acket∇i}htsare the initial conditions. The parameters A1,A3andA2defined\nin Eqs. (26) play the roles of the magnon lifetime and the frequency o f the spin wave at\nzero wave vector, respectively. As can be seen in Eq. (26) all of th ese three parameters are\naffected by the correlation time τand the strength Dof the random force. Moreover, the\nGilbert damping parameter αinfluences the system as well. The solution of Eq. (24) for\nthe correlation function in case of t′= 0 is formal identical to that of Eq. (27). The more\ngeneral situation t′∝ne}ationslash= 0 allows no simple analytic solution and hence the behavior of the\ncorrelation function C(s) is studied numerically. In order to analyze the mean values and\nthe correlation function let us first examine the parameter range w here physical accessible\nsolutions exist. In the following we assume ∝an}b∇acketle{tψ1(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ2(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ0∝an}b∇acket∇i}htand∝an}b∇acketle{tψ3(0)∝an}b∇acket∇i}ht= 0, since\nthe solutions for ψ1(t) andψ2(t) on the one hand and ψ3(t) on the other hand are decoupled\n11in Eq. (27). Therefore, spin wave solutions only exists for non-zer o averages ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htand\n∝an}b∇acketle{tψ2(t)∝an}b∇acket∇i}ht. The existence of such non-trivial solutions are determined in depe ndence on the\nnoise parameters Dandτand the deterministic damping parameter α. Notice, that the\ndimensionless quantity D=˜D/Γ, i.e.,Dis the ratio between the strength of the correlation\nfunction (Eq. (7)) and the anisotropy field in the original units. The stability of spin wave\nsolutions is guaranteed for positive parameters A1andA3. According to Eqs. (26) the\nphase diagrams are depicted in Fig. 2 within the α−Dplane for different values of the\ncorrelation time τ. The separatrix between stable and unstable regions is determined by\nthe condition A1= 0. The second condition A3= 0 is irrelevant due to the imposed initial\nconditions. As the result of the stability analysis the phase space dia gram is subdivided into\nfour regions where region IV does not exist in case of τ= 0, see Fig. 2(a). For generality,\nwe take into account both positive and negative values of Dindicating correlations and\nanti-correlations of the stochastic field. Damped spin waves are ob served in the areas I and\nIV, whereas the sectors II and III reveal non-accessible solutio ns. In those regions the spin\nwave amplitude, proportional to exp[ −A1t], tends to infinity which should not be realized,\ncompare Figs. 2(b)-2(d). Actually, a reasonable behavior is obser ved in regions I and IV. As\nvisible from Fig. 2 damped spin waves will always emerge for D>0 even in the limit of zero\ndamping parameter αand vanishing correlation time τ. This behavior is shown in Fig. 3,\nwhere theevolution of ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htis depicted for different valuesof α. As canbeseen in Fig.2(a)\nthesolutionfor D<0isunlimitedandconsequently, itshouldbeexcludedfurther. Contr ary\nto this situation, additional solutions will be developed in region IV in ca se ofτ >0 and\nsimultaneously α= 0, see Figs. 2(b)-2(d). Thereby the size of area IV grows with inc reasing\nτ. Likewise, the extent of region I decreases for an enhanced τ. However, in the limit of\nD= 0 and consequently for τ= 0, too, only damped spin waves are observed. Immediately\non the separations line undamped periodic solutions will evolve, compa re the sub-figures in\nFig. 2. This remarkable effect can be traced back to the interplay be tween the deterministic\ndamping and the stochastic forces. Both damping mechanism are co mpensated mutually\nwhich reminds of a kind of resonance phenomenon. The difference to conventional resonance\nbehavior consists of the compensation of the inherent determinist ic Gilbert damping and the\nstochastic one originated from the random field. This statement is e mphasized by the fact\nthat undamped periodic solutions do not develop in the absence of st ochastic interferences,\ni.e.,D= 0. The situation might be interpreted physically as follows: the requ ired energy\n12(a)τ= 0 (b)τ= 0.1\n(c)τ= 1 (d)τ= 10\nFIG. 2.α−Dplane for fixed magnetization µ= 0.9 and different values of τ.\nthat enables the system to sustain the deterministic damping mecha nisms is delivered by\nthe stochastic influences due to the interaction with the environme nt. To be more precise, in\ngeneral, the Gilbert damping enforces the coherent alignment of th e spin density along the\nprecession axis. Contrary, the random field supports the dephas ing of the orientation of the\nclassical spins. Surprisingly, the model predicts the existence of a critical value τ=τc≥0\n13FIG. 3. Evolution of the mean value ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}ht, withµ= 0.9,D= 0.1 andτ= 0.αvaries from 0\n(dash-dotted line), 0 .05 (solid line), 0 .5 (dotted line) and 1 (dashed line).\ndepending on αandDwhich determines the onset of undamped periodic solutions. Notice,\nthat negative values of τcare excluded. The critical value is\nτc=−[µ2(α3−Dα2+α)+2D](1+α2)2\n2Dµ2(α3−3Dα2+α)+D2. (28)\nHence, this result could imply the possibility of the cancellation of both damping processes.\nExamples according to the damped and the periodic case are displaye d in Fig. 4. An increas-\ningτfavors the damping process as it is visible in Fig. 4(a). Based on estima tions obtained\nfor ferromagnetic materials [29] and references therein, the Gilbe rt damping parameter can\nrange between 0 .04<α<0.22 in thin magnetic films, whereas the bulk value for Co takes\nαb≈0.005. The phase space diagram in Fig. 2 offers periodic solutions only fo r values of\nαlarger than those known from experiments. Therefore such perio dic solutions seem to be\nhard to see experimentally. We proceed further by analyzing the be havior of the correlation\nfunction by numerical computation of the solution of Eq. (24) with E qs. (25) and (26). As\ninitial values we choose Cik(t=t′,t′) =Cik(s= 0) =C0for every combination i,k={1,2,3}.\nThe results are depicted in Figs. 5 and 6. Inspecting Figs. 5(a)-5(c ) one recognizes that an\nenhancement of the correlation time τleads to an increase of the oscillations within the\ncorrelation functions C1k,k={1,2,3}. Moreover, Fig. 5(d) reveals that the oscillatory\n14(a) (b)\nFIG. 4. Evolution of the mean values ∝an}b∇acketle{tψ1,2(t)∝an}b∇acket∇i}ht, withµ= 0.9. (a):D= 0.1,α= 0.005 andτvaries\nfrom 10 (solid line), 1 (dotted line) and 0 (dash-dotted line ). (b):D= 2,α= 1 andτ=τc≈1.79\n(Eq. (28)). The solid line represents ∝an}b∇acketle{tψ1∝an}b∇acket∇i}htand the dash-dotted line is ∝an}b∇acketle{tψ2∝an}b∇acket∇i}ht.\nbehavior of C31seems to be suppressed. Obviously, the decay of the correlation f unction is\nenhanced if τgrowths up. The pure periodic case for τ=τc, corresponding to Fig. 4(b),\nis depicted in Fig. 6. Exemplary, C12andC31are illustrated. The behavior of the latter is\nsimilar to the damped case, displayed in Fig. 5(d), unless slight oscillatio ns occur. However,\nif one compares the form of C12in Fig. 5(b) and Fig. 6 the differences are obvious. The am-\nplitude of the correlation function for the undamped case grows to the fourfold magnitude\nin comparison with C0, whereas the damped correlation function approaches zero. Fur ther,\na periodic behavior is shown in Fig. 6, and therefore the correlation w ill oscillate about zero\nbut never vanish for all s=t−t′>0.\nV. CONCLUSIONS\nIn this paper we have analyzed the dynamics of a classical spin model with uniaxial\nanisotropy. Aside from the deterministic damping due to the Landau -Lifshitz-Gilbert\nequation the system is subjected to an additional dissipation proce ss by the inclusion of a\nstochastic field with colored noise. Both dissipation processes are a ble to compete leading to\n15(a) (b)\n(c) (d)\nFIG. 5. Correlation functions Cik(s) forµ= 0.9,D= 0.1 andα= 0.005.τtakes 0 (dotted line),\n1 (solid line) and 10 (dash-dotted line).\na more complex behavior. To study this one we derive an equation for the joint probability\ndistribution which allows us to find the corresponding spin-spin-corr elation function. This\nprogram can be fulfilled analytically and numerically in the spin wave appr oach and the\nsmall correlation time limit. Based on the mean value for the spin wave c omponent and\n16FIG. 6. Correlation functions Cik(s) forτ=τc≈1.79 (Eq. (28)), µ= 0.9,D= 2 andα= 1. The\ndotted line represents C12and the solid line is C31.\nthe correlation function we discuss the stability of the system in ter ms of the stochastic\nparameters, namely the strength of the correlated noise Dand the finite correlation time\nτ, as well as the deterministic Gilbert damping parameter α. The phase diagram in the\nα−Dplane offers that the system develops stable and unstable spin wave solutions due to\nthe interplay between the stochastic and the deterministic damping mechanism. So stable\nsolutions evolve for arbitrary positive Dand moderate values of the Gilbert damping α.\nFurther, we find that also the finite correlation time of the stochas tic field influences the\nevolution of the spin waves. In particular, the model reveals for fix edDandαa critical\nvalueτcwhich characterizes the occurrence of undamped spin waves. The different situa-\ntions are depicted in Fig. 2. Moreover, the correlation time τaffects the damped spin wave\nwhich can be observed in regions I and IV in the phase diagram. If the parameters Dand\nαchanges within these regions, an increasing τleads to an enhancement of the spin wave\ndamping, cf. Fig. 4(a). The influence of τon the correlation functions is similar as shown\nin Figs. 5(a)-5(c). The study could be extended by the inclusion of fi nite wave vectors and\nusing an approach beyond the spin wave approximation.\n17ACKNOWLEDGMENTS\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which is\nsupported by the Saxony-Anhalt State, Germany.\n18[1] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of continuous media (Pergamon\nPress, Oxford, 1989).\n[2] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halpe rin, Rev. Mod. Phys. 77, 1375\n(2005).\n[4] A. Sukhov and J. Berakdar, J. Phys. - Cond. 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Daniel and M. Lakshmanan, Physica A 120, 125 (1983).\n[19] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984).\n[20] V. G. Bar’Yakhtar, M. V. Chetkin, B. A. Ivanov, andS. N. G adetskii, Dynamics of Topological\nMagnetic Solitons: Experiment and Theory (Springer Tracts i n Modern Physics) (Springer,\n1994).\n[21] M. Lakshmanan, T. W. Ruijgrok, and C. J. Thompson, Physi ca A84, 577 (1976).\n[22] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep .194, 117 (1990).\n[23] A. Hernandez-Machado and M. San Miguel, J. Math. Phys. 25, 1066 (1984).\n[24] A. Hernandez-Machado, J. M. Sancho, M. San Miguel, and L . Pesquera, Zeitschr. f. Phys. B\n52, 335 (1983).\n19[25] N. G. van Kampen, Braz. J. Phys. 28, 90 (1998).\n[26] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster-\ndam, 1981).\n[27] E. A. Novikov, Sov. Phys. JETP 20, 1290 (1965).\n[28] H. Dekker, Phys. Lett. A 90, 26 (1982).\n[29] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev . Lett.88, 117601 (2002).\n20"    },    {        "title": "1405.6354v2.Spin_Hall_phenomenology_of_magnetic_dynamics.pdf",        "content": "Spin Hall phenomenology of magnetic dynamics\nYaroslav Tserkovnyak and Scott A. Bender\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n(Dated: July 2, 2021)\nWe study the role of spin-orbit interactions in the coupled magnetoelectric dynamics of a fer-\nromagnetic \flm coated with an electrical conductor. While the main thrust of this work is phe-\nnomenological, several popular simple models are considered microscopically in some detail, includ-\ning Rashba and Dirac two-dimensional electron gases coupled to a magnetic insulator, as well as a\ndi\u000busive spin Hall system. We focus on the long-wavelength magnetic dynamics that experiences\ncurrent-induced torques and produces \fctitious electromotive forces. Our phenomenology provides\na suitable framework for analyzing experiments on current-induced magnetic dynamics and recip-\nrocal charge pumping, including the e\u000bects of magnetoresistance and Gilbert-damping anisotropies,\nwithout a need to resort to any microscopic considerations or modeling. Finally, some remarks are\nmade regarding the interplay of spin-orbit interactions and magnetic textures.\nPACS numbers: 85.75.-d\nI. INTRODUCTION\nSeveral new directions of spintronic research have\nopened and progressed rapidly in recent years. Much\nenthusiasm is bolstered by the opportunities to initiate\nand detect spin-transfer torques in magnetic metals1and\ninsulators,2which could be accomplished by variants of\nthe spin Hall e\u000bect,3along with the reciprocal electromo-\ntive forces induced by magnetic dynamics. The spin Hall\ne\u000bect stands for a spin current generated by a transverse\napplied charge current, in the presence of spin-orbit in-\nteraction. From the perspective of angular momentum\nconservation, the spin Hall e\u000bect allows angular momen-\ntum to be leveraged from the stationary crystal lattice\nto the magnetic dynamics. A range of nonmagnetic ma-\nterials from metals to topological insulators have been\ndemonstrated to exhibit strong spin-orbit coupling, thus\nallowing for e\u000ecient current-induced torques.\nFocusing on quasi-two-dimensional (2D) geometries,\nwe can generally think of the underlying spin Hall phe-\nnomena as an out-of-equilibrium magnetoelectric e\u000bect\nthat couples planar charge currents with collective mag-\nnetization dynamics. In typical practical cases, the rel-\nevant system is a bilayer heterostructure, which consists\nof a conducting layer with strong spin-orbit coupling and\nferromagnetic layer with well-formed magnetic order. In\nthis case, the current-induced spin torque re\rects a spin\nangular momentum \rux normal to the plane, which ex-\nplains the spin Hall terminology.\nThe microscopic interplay of spin-orbit interaction and\nmagnetism at the interface translates into a macroscopic\ncoupling between charge currents and magnetic dynam-\nics. A general phenomenology applicable to a variety of\ndisparate heterostructures can be inferred by considering\na course-grained 2D system, which both conducts and\nhas magnetic order as well as lacks inversion symmetry\n(or else the pseudovectorial magnetization would not cou-\nple linearly to the vectorial current density). In a bilayer\nheterostructure, the latter is naturally provided by the\nbroken re\rection symmetry with respect to its plane.II. GENERAL PHENOMENOLOGY\nLet us speci\fcally consider a bilayer heterostructure\nwith one layer magnetic and one conducting, as sketched\nin Fig. 1. The nonmagnetic layer can be tailored to\nenhance spin-orbit coupling e\u000bects in and out of equi-\nlibrium. Phenomenologically, we have a quasi-2D sys-\ntem along the xyplane, which will for simplicity be\ntaken to be isotropic and mirror-symmetric in plane while\nbreaking re\rection symmetry along the zaxis. In other\nwords, the structural symmetry is assumed to be that\nof a Rashba 2D electron gas (although microscopic de-\ntails could be more complex), subject to a spontaneous\ntime-reversal symmetry breaking due to the magnetic\norder. Common examples of such heterostructures in-\nclude a thin transition-metal1or magnetic-insulator2\flm\ncapped by a heavy metal, or a layer of 3D topological in-\ny\nz\nj˙njsHaNaFxFN\nFIG. 1. Heterostructure consisting of a magnetic top layer\nand conducting underlayer. The charge current jinduces a\ntorque \u001cacting on the magnetic dynamics, which quanti\fes\nthe spin angular-momentum transfer in the zdirection. This\ncan be thought of as a spin current jsentering the ferro-\nmagnet at the interface. Reciprocally, magnetic dynamics _n\ninduces a motive force \u000facting on the itinerant electrons in\nthe conductor.arXiv:1405.6354v2  [cond-mat.mes-hall]  25 Jul 20142\nsulator doped on one side with magnetic impurities.4\nThe course-grained hydrodynamic variables used to de-\nscribe our system are the three-component collective spin\ndensity (per unit area) s(r;t) =sn(r;t)\u0011(snx;sny;snz)\nand the two-component 2D current density (per unit\nlength) j(r;t)\u0011(jx;jy) in thexyplane. Considering\nfully saturated magnetic state well below the Curie tem-\nperature, we treat the spin density as a directional vari-\nable, such that its magnitude sis constant and orienta-\ntional unit vector nparametrizes a smooth and slowly-\nvarying magnetic texture. We will be interested in slow\nand long-wavelength agitations of the ferromagnet cou-\npled to the electron liquid along with reciprocal motive\nforces. Perturbed out of equilibrium, the temporal evolu-\ntion of the heterostructure is governed by the forces that\ncouple to the charge \row and magnetic dynamics: the\n(planar) electric \feld and magnetic \feld, respectively.\nA. Decoupled dynamics\nA uniform electric-current carrying state in the isolated\nconducting \flm, subject to a constant external vector\npotential A, has the free-energy density\nF(p;A) =F0(p)\u0000p\u0001A\nc+O(A2); (1)\nwhereF0=Lp2=2 is the free-energy density in terms of\ntheparamagnetic current p(i.e., the current de\fned in\nthe absence of the vector potential A), andLis the local\nself-inductance of the \flm (including inertial and elec-\ntromagnetic contributions). According to time-reversal\nsymmetry, in equilibrium p= 0 when A= 0. The gauge\ninvariance (which requires that the minimum of F, as a\nfunction of p, is independent of A), furthermore, dictates\nthe following form of the free energy:\nF=L\n2\u0012\np\u0000A\ncL\u00132\n: (2)\nTherefore, the phenomenological expression for the full\ncurrent density is\nj\u0011\u0000c\u000eAF=p\u0000A\ncL; (3)\nwith\u000estanding for the 2D functional derivative of the\ntotal electronic free energy F[p] =R\nd2rF(p). We con-\nclude, based on Eqs. (2) and (3), that j=L\u00001\u000epF, which\nis thus the force thermodynamically conjugate to Lp.\nGeneral quasistatic equilibration5of a perturbed electron\nsystem can now be written as\nL_p=\u0000^%j; (4)\nor, in terms of the physical current:\nL_j+ ^%j=E; (5)where E\u0011\u0000@tA=cis the electric \feld, and ^ %is identi\fed\nas the resistivity tensor. This is the familiar Ohm's law,\nwhich, in steady state, reduces to\nj= ^gE; (6)\nin terms of the conductivity tensor ^ g\u0011^%\u00001. Based on\nthe axial symmetry around z, we can generally write ^ g=\ng+gHz\u0002, wheregis the longitudinal (i.e., dissipative)\nandgHHall conductivities.\nThe isolated magnetic-\flm dynamics, on the other\nhand, are described by the Landau-Lifshitz-Gilbert\nequation:6\ns(1 +\u000bn\u0002)_n=H\u0003\u0002n; (7)\nwhere H\u0003\u0011\u000enF[n] is the e\u000bective magnetic \feld gov-\nerned by the magnetic free-energy functional F[n] =R\nd2rF(n). The (dimensionless) Gilbert damping \u000bcap-\ntures the (time-reversal breaking) dissipative processes in\nthe spin sector.\nThe total dissipation power in our combined, but still\ndecoupled, system is given by\n\u0000_F=\u0000Z\nd2r(L_p\u0001j+_n\u0001H\u0003) =Z\nd2r\u0000\n%j2+\u000bs_n2\u0001\n;\n(8)\nwhere%=g=(g2+g2\nH) is the longitudinal resistiv-\nity. According to the \ructuation-dissipation theorem,5\n\fnite-temperature \ructuations are thus determined by\nhji(r;t)ji0(r0;t0)i= 2gkBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0) and\nhhi(r;t)hi0(r0;t0)i= 2\u000bskBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0). Hav-\ning mentioned this for completeness, we will not pursue\nthermal properties any further.\nB. Coupled dynamics\nHaving recognized ( Lp;j) and ( n;H\u0003) as two pairs\nof thermodynamically conjugate variables, their coupled\ndynamics must obey Onsager reciprocity.5Charge cur-\nrent \rowing through our heterostructure in general in-\nduces a torque \u001con the magnetic moment and, vice\nversa, magnetic dynamics produce a motive force \u000facting\non the current, de\fned as follows:\ns(_n+n\u0002^\u000b_n) =H\u0003\u0002n+\u001c; (9)\nL_j+ ^%j=E+\u000f; (10)\nwhereL_j=L_p+E, according to Eq. (3). In gen-\neral, due to the spin-orbit interaction at the interface,\nGilbert damping7^\u000band resistivity tensor8^%can acquire\nanisotropic n-dependent contributions. Let us start by\nexpanding the motive force, according to the assumed\nstructural symmetries, in the Cartesian components of\nn:\n\u000f= [(\u0011+#n\u0002)_n]\u0002z; (11)3\nwhere\u0011is the reactive and#thedissipative coe\u000ecients\ncharacterizing spin-orbit interactions in our coupled sys-\ntem. While \u0011and#can generally depend on n2\nz, we will\nfor simplicity be focusing our attention on the limit when\nthey are mere constants. The dimensionless parameter\n\f\u0011#=\u0011describes their relative strengths. The Onsager\nreciprocity then immediately dictates the following form\nof the torque:\n\u001c= (\u0011+#n\u0002)(z\u0002j)\u0002n: (12)\nIn line with the existing nomenclature,1,2we can write\nthe dissipative coe\u000ecient as\n#\u0011~\n2eaNtan\u0012; (13)\nin terms of a length scale aN, which we take to corre-\nspond to the normal-metal thickness,9and dimensionless\nparameter\u0012identi\fed as the e\u000bective spin Hall angle at\nthe interface. The coe\u000ecient \u0011in Eq. (12) parametrizes\nthe so-called \feld-like torque, which could arise, for ex-\nample, as a manifestation of the interfacial Edelstein ef-\nfect.10\nAnother important e\u000bect of the nonmagnetic layer on\nthe ferromagnet is the enhanced damping of the magne-\ntization dynamics by spin pumping,11such that\n\u000b=\u000b0+a\"#\naF: (14)\n\u000b0is the bulk damping, which is thickness aFindepen-\ndent, anda\"#parametrizes the strength of angular mo-\nmentum [as well as energy, according to Eq. (8)] loss at\nthe interface. Spin pumping into a perfect spin reser-\nvoir corresponds to11a\"#=~g\"#\nr=4\u0019S, whereg\"#\nris the\n(real part of the dimensionless) interfacial spin-mixing\nconductance per unit area and S\u0011s=aFis the 3D spin\ndensity in the ferromagnet. In reality, a\"#depends on\nthe spin-relaxation e\u000eciency in the normal metal as well\nas the spin-orbit interaction at the interface, and may\ndepend on aNin a nontrivial manner (see Ref. 12 for\na di\u000busive model), so long as aN.\u0015N, where\u0015Nis\nthe spin-relaxation length in the normal metal.13With\nthese conventions in mind and focusing on the limit of\naN\u001d\u0015Nand, in the case of a metallic ferromagnet,\naF\u001d\u0015F, we will suppose that the coe\u000ecients \u0012,\f, and\na\"#de\fned above are thickness independent.14\nUnless otherwise stated, we will disregard anisotropies\nin\u000b, which may in general depend on the directions of n\nand_n, subject to the reduced crystalline symmetries and\nthe lack of re\rection asymmetry at the interface.15In the\nsame spirit, with the exception of Sec. III C, we will not\nconcern ourselves much with the n-dependent interfacial\nmagnetoresistance/proximity e\u000bects,8which would enter\nthrough the resistivity tensor ^ %(n) in Eq. (10).\nWe remark that while we considered a nonequilibrium\nmagnetoelectric coupling in terms of torque \u001cand force\n\u000fin Eqs. (9) and (10), we had retained the decoupled\nform of the free-energy density, F(p)+F(n). We excludethe possibility of a linear coupling of pto the magnetic\norder, since it would suggest a nonzero electric current in\nequilibrium.\nC. Current-induced instability\nEquations (9) and (10) encapsulate rich nonlinear dy-\nnamics. Of particular interest are the current-induced\nmagnetic instabilities and switching. For a \fxed current\nbiasj, it is convenient to multiply Eq. (9) by (1 \u0000\u000bn\u0002)\non the left to obtain\ns(1 +\u000b2)_n=h\u0002n\u0000\u000bn\u0002h0\u0002n: (15)\nHere,\nh\u0011H\u0003+ (\u0011+#\u000b)z\u0002j;h0\u0011H\u0003+ (\u0011\u0000#=\u000b)z\u0002j(16)\nare the e\u000bective Larmor and damping \felds, respectively.\nA magnetic instability (bifurcation) at an equilibrium\n\fxed point may occur, for example, when either the ef-\nfective \feld or e\u000bective damping change sign.\nTo illustrate this, consider a simple case, where a con-\nstant current is applied in the xdirection: j=jx, while\nan external magnetic \feld parametrized by His applied\nalong theyaxis: H\u0003=Hy+Knzz, where we also in-\nclude an easy-plane magnetic anisotropy K. Equations\n(16) then become\nh= [H+ (\u0011+#\u000b)j]y+Knzz; (17)\nh0= [H+ (\u0011\u0000#=\u000b)j]y+Knzz: (18)\nIn equilibrium, when j= 0:n=\u0000y. Whenjis ramped\nup, however, this \fxed point may become unstable. Let\nus consider two extreme limits: First, suppose the mag-\nnetoelectric coupling (12) is purely reactive, i.e., #= 0.\nThe e\u000bect of the torque can thus be fully captured by\na rede\fnition of the applied \feld as H!H+\u0011j. We\nthus see that when \u0000jexceedsH=\u0011, the e\u000bective \feld\nswitches sign, and the stable magnetic orientation \rips\nfrom\u0000ytoy.\nIf, on the other hand, the magnetoelectric coupling\n(12) is purely dissipative, i.e., \u0011= 0, thenH!H+\n#\u000bj according to Eq. (17), whereas H!H\u0000(#=\u000b)j\naccording to Eq. (18). Supposing, furthermore, that \u000b\u001c\n1, as is nearly always the case, the e\u000bect of #onhis\nnegligible in comparison to its e\u000bect on h0. We thus\nrewrite Eqs. (17) and (18) as\nh\u0019Hy+Knzz;h0= [H\u0000(#=\u000b)j]y+Knzz:(19)\nA simple stability analysis gives for the critical current\nat which n=\u0000ybecomes unstable:\njc=\u000b\n#\u0012\nH+K\n2\u0013\n: (20)\nIn the presence of comparable reactive and dissipative\ntorques, i.e., \f\u00181 so that\u0011\u0018#, while still \u000b\u001c1,4\nhremains essentially una\u000bected by currents of order jc\n(unlessK&H=\u000b\u001dH), so that the above dissipative\nmagnetic instability at jcis maintained. We could thus\nexpect Eq. (20) to rather generally describe the leading\nspin-torque instability threshold16for the monodomain\ndynamics.\nIt is instructive to obtain from Eq. (20) the intrinsic in-\nstability threshold for thin magnetic \flms, aF\u001ca\"#=\u000b0,\nfor which the bulk contribution, \u000b0, to the damping (14)\ncan be neglected:\nj(0)\nc=2e\n~a\"#\ntan\u0012aN\naF\u0012\nH+K\n2\u0013\n: (21)\nWriting, furthermore, j(0)\nc=J(0)\ncaN, in terms of the 3D\ncurrent density J(0)\nc;a\"#=~g\"#\nr=4\u0019S, in terms of the\ne\u000bective spin-mixing conductance g\"#\nr(including the ef-\nfects of spin back\row from the normal layer,12in case\nof an imperfect spin sink); and converting e\u000bective \feld\nto physical units: H=!BaFSandK=!KaFS, where\n!B=\rBin terms of the gyromagnetic ratio \rand ap-\nplied \feldB,!K= 4\u0019\rMswithMs=\rS, in case of\nonly the shape anisotropy, we obtain\nJ(0)\nc=e\n2\u0019g\"#\nr\ntan\u0012\u0010\n!B+!K\n2\u0011\n: (22)\nWe recall that the Kittel formula for the ferromagnetic-\nresonance frequency is !=p\n!B(!B+!K). Using\nquantities characteristic of the Pt jYIG compound:1,2\u0012\u0018\n0:1,g\"#\nr\u00185 nm\u00002,!K\u00184\u00021010s\u00001, we would get for\nthe intrinsic instability threshold (in the absence of an\napplied \feld B):J(0)\nc\u00183\u00021010A\u0001m\u00002. (Threshold\ncurrents at this order were also evaluated in Ref. 17.) In\nthe opposite limit of thick magnetic \flms, aF\u001da\"#=\u000b0\n(\u00181=2\u0016m for YIG, using \u000b0\u001810\u00004), the bulk Gilbert\ndamping dominates magnetic dissipation, and\nJc\u0019\u000b0aF\na\"#J(0)\nc=2e\n~\u000b0aFS\ntan\u0012\u0010\n!B+!K\n2\u0011\n(23)\nincreases linearly with aFbeyond the intrinsic threshold.\nIII. SIMPLE MODELS\nEquations (9)-(14) provide a general phenomenological\nframework for exploring the coupled magnetoelectric dy-\nnamics in thin-\flm magnetic heterostructures, which we\nverify by considering several simple microscopic models\nin the following.\nA. Rashba Hamiltonian\nOne of the simplest models engendering the phe-\nnomenology of interest is based on a 2D electron gas ata re\rection-asymmetric interface, which, at low energies,\nis described by the (single-particle) Rashba Hamiltonian:\n^HR=p2\n2m+vp\u0001z\u0002^\u001b: (24)\nVelocityvhere parametrizes the spin-orbit interaction\nstrength due to structural asymmetry; ^\u001bis a vector of\nPauli matrices. When the \frst (nonrelativistic) term in\nHamiltonian (24) dominates over the second (relativistic)\nterm (i.e.,v\u001cvF, the Fermi velocity), we can treat v\nperturbatively.\nTo zeroth order in v, the velocity operator is @p^HR=\np=m, such that the current density is j=\u0000enhpi=m,\nin terms of the particle-number density n=k2\nF=2\u0019=\nm2v2\nF=2\u0019~2and the positron charge e>0. On the other\nhand, to \frst order in v, Eq. (24) results in the steady-\nstate spin density\n\u001a=mv\n2\u0019~z\u0002hpi=\u0000m2v\n2\u0019~enz\u0002j; (25)\nrecalling that the 2D density of states (which de\fnes the\nspin susceptibility) is given by m=2\u0019~2. Equation (25)\nre\rects the Edelstein e\u000bect.10\nExchange coupling this Rashba 2DEG to an adjacent\nferromagnet according to the local Hamiltonian\nH0=\u0000Z\nd2r[J(nx\u001ax+ny\u001ay) +J?nz\u001az]; (26)\nwhereJandJ?are respectively the in-plane and out-of-\nplane exchange constants, we get for the torque:\n\u001c=\u000enH0\u0002n=\u0000[J(\u001axx+\u001ayy) +J?\u001azz]\u0002n:(27)\nEvaluating this torque to leading (i.e., \frst) order in the\nexchange, we need to \fnd \u001ato zeroth order, which is\ngiven by Eq. (25). We thus have:\n\u001c=\u0011(z\u0002j)\u0002n; (28)\nwhere\n\u0011=m2vJ\n2\u0019~en=~\nevJ\nv2\nF: (29)\nThe dissipative (i.e., spin Hall) coe\u000ecient #vanishes in\nthis model at this level of approximation. We should,\nhowever, expect #to arise at quadratic order in J\n[whereas at \frst order in J, it must vanish for arbitrarily\nlargev, since, in the absence of magnetism, Eq. (25) here\ndescribes the general form of spin response to dc current].\nB. Dirac Hamiltonian\nIn the opposite extreme, the spin-orbit interaction in\nEq. (24) dominates over the nonrelativistic piece, which\nformally corresponds to sending m! 1 . The corre-\nsponding 2D Dirac Hamiltonian\n^HD=vp\u0001z\u0002^\u001b (30)5\narises physically on the surfaces of strong 3D topological\ninsulators.18\nExchange coupling electrons to a magnetic order n, ac-\ncording to Eq. (26), gives the single-particle Hamiltonian\n^H0=\u0000~\n2[J(nx^\u001bx+ny^\u001by) +J?nz^\u001bz]; (31)\nwhich can be combined with Eq. (30) as follows:\n^HD+^H0=v(p\u0000A\u0003)\u0001z\u0002^\u001b\u0000m\u0003^\u001bz: (32)\nHere,\nA\u0003\u0011~J\n2vz\u0002nandm\u0003\u0011~J?\n2nz (33)\nare \fctitious vector potential and mass. The correspond-\ning electromotive force (recalling that the electron charge\nis\u0000e) is\n\u000f=@tA\u0003\ne=\u0000~J\n2ev_n\u0002z; (34)\nsuch that, according to Eq. (11),\n\u0011=\u0000~J\n2ev; (35)\nwhich is of opposite sign to Eq. (29). Note that unlike\nthe latter result, Eq. (35) is derived nonperturbatively.\nThe reciprocal torque (12) with this \u0011gives:\n\u001c=\u0011(z\u0002j)\u0002n: (36)\nUsing the helical identity between the current and spin\ndensities,\nj=\u00002ev\n~z\u0002\u001a; (37)\naccording to the velocity operator @p^HD=vz\u0002^\u001b, we\nrecognize in Eq. (36) the torque (27) due to the planar\nexchangeJ. The above relations mimic the structure of\nthe preceding Rashba model. For a vanishing chemical\npotential, the mass term opens a gap, in which case the\nlong-wavelength conductivity tensor is given by the half-\nquantized Hall response:19^g=\u0000sgn(m\u0003)(e2=4\u0019~)z\u0002. In\naddition to the in-plane spin density z\u0002\u001a\u0002zentering\nEq. (36), the out-of-plane component \u001azshould also exert\na torque/J?, according to the exchange coupling (27).\nAt the leading order, the latter contributes to the out-of-\nplane magnetic anisotropy K, which is absorbed by the\nmagnetic free-energy density F(n).20At a \fnite doping,\ntheJ?interaction could in general be also expected to\ngive rise to a dissipative coupling #.\nC. Di\u000busive spin Hall system\nThe previous two models naturally produced the reac-\ntive coupling \u0011between planar charge current and mag-\nnetic dynamics. Here, we recap a di\u000busive spin Hallmodel8,21that results in both \u0011and#, which is based\non a \flm of a featureless isotropic normal-metal conduc-\ntor in contact with ferromagnetic insulator. If electrons\ndi\u000buse through the conductor with weak spin relaxation,\nwe can develop a hydrodynamic description based on con-\ntinuity relations both for spin and charge densities. We\n\frst construct bulk di\u000busion equations and then impose\nspin-charge boundary conditions, which allows us to solve\nfor spin-charge \ruxes in the normal metal and torque on\nthe ferromagnetic insulator.\nThe relevant hydrodynamic quantities in the normal-\nmetal bulk are 3D charge and spin densities, \u001a(r;t) and\n\u001a(r;t), respectively. The associated thermodynamic con-\njugates are the electrochemical potential, \u0016\u0011\u0000e\u000e\u001aF,\nand spin accumulation, \u0016\u0011~\u000e\u001aF, whereF[\u001a;\u001a] is the\nfree-energy functional of the normal metal. Supposing\nonly a weak violation of spin conservation (due to mag-\nnetic or spin-orbit impurities), we phenomenologically\nwrite spin-charge continuity relations as\n@t\u001a=\u0000@{J{; @t\u001a|=\u0000@{J{|\u0000\u0000\u0016|; (38)\nwhere{and|label Cartesian components of real and\nspin spaces, respectively, and the summation over the\nrepeated index {is implied. \u0000 = ~N=2\u001cs, in terms of\nthe (per spin) Fermi-level density of states Nand spin-\nrelaxation time \u001cs.J{are the components of the 3D\nvectorial charge-current density and J{|of the tensorial\nspin-current density, which can be expanded in terms of\nthe thermodynamic forces governed by \u0016and\u0016:\nJ{=\u001b\ne@{\u0016\u0000\u001b0\n2e\u000f{|k@|\u0016k; (39)\n2e\n~J{|=\u0000\u001b+\n2e@{\u0016|\u0000\u001b\u0000\n2e@|\u0016{\u0000\u001b0\ne\u000f{|k@k\u0016; (40)\nwhere\u001bis the (isotropic) electrical conductivity and \u001b0\nthe spin Hall conductivity of the normal-metal bulk. The\nlast terms of Eqs. (39) and (40) are governed by the same\ncoe\u000ecient\u001b0due to the Onsager reciprocity. The bulk\nspin Hall angle \u00120is conventionally de\fned by\ntan\u00120\u0011\u001b0\n\u001b: (41)\nBulk di\u000busion equations (39), (40) are complemented\nby the boundary conditions\nJz= 0 atz=\u0000aN;0 (42)\nfor the charge current, where z=\u0000aNcorresponds to\nthe normal-metal interface with vacuum and z= 0 to\nthe interface with the ferromagnet, and11\nJz=1\n4\u0019(0 at z=\u0000aN \u0010\ng\"#\ni+g\"#\nrn\u0002\u0011\n~\u0016\u0002natz= 0;(43)\nfor the spin current, with Jzstanding for Jz|. Here,\n~\u0016\u0011\u0016\u0000~n\u0002_ncaptures contributions from the spin-\ntransfer torque and spin pumping, respectively.6\nHaving established the general structure of the coupled\nspin and charge di\u000busion, let us calculate the steady-\nstate charge-current density jdriven by a simultaneous\napplication of a uniform electric \feld in the xyplane,\nr\u0016!eE, and magnetic dynamics, _n:\nJ=\u001bE\u0000\u001b0\n2er\u0002\u0016: (44)\nThe spin accumulation \u0016is found by solving\n\u0010\u001b+\n\u001b+\u001b\u0000\n\u001b\u000ezj\u0011\n@2\nz\u0016j=\u0016j\nl2s; (45)\nwherels\u0011p\n~\u001b=4e2\u0000 is the spin-di\u000busion length. Using\nDrude formula for the conductivity \u001b, we get the famil-\niarls=l=p\n3\u000f, wherelis the scattering mean free path\nand\u000f\u0011\u001c=\u001cs\u001c1 is the spin-\rip probability per scatter-\ning (\u001cis the transport mean free time). The boundary\nconditions are\n\u001b0z\u0002E\u0000\u001b+\n2e@z\u0016\u0000\u001b\u0000\n2er\u0016z\n=e\nh(0 at z=\u0000aN \u0010\ng\"#\ni+g\"#\nrn\u0002\u0011\n~\u0016\u0002natz= 0;(46)\nwhereh= 2\u0019~is the Planck's constant.\nIn the limit of vanishing spin-orbit coupling, \u001b+!\u001b,\n\u001b\u0000!0, and\u00120!0. For small but \fnite spin-orbit\ninteraction, we may expect ( \u001b+\u0000\u001b)\u0018\u001b\u0000\u0018O(\u001202). In\nthe following, we will neglect these quadratic terms and\napproximate tan \u00120\u0019\u00120\u001c1, in the spirit of the present\nconstruction.\nIn the limit of ls\u001caN, the spin accumulation decays\nexponentially away from the interface as \u0016(z) =\u00160ez=ls,\nwhere\n\u00160= (\u0018i+\u0018n\u0002) [~_n\u00002els\u00120(z\u0002E)\u0002n] + 2els\u00120z\u0002E:\n(47)\nHere,\u0018\u0011\u001f(1 +\u0010+\u00102\ni) and\u0018i\u0011\u001f\u0010\u0010i, in terms of\n\u0010\u0011\u001b=gQg\"#\nrls,\u0010i\u0011g\"#\ni=g\"#\nr,\u001f\u00001\u0011(1 +\u0010)2+\u00102\ni, and\nthe quantum of conductance gQ\u00112e2=h. The spin accu-\nmulation \u00160consists of the decoupled spin-pumping and\nspin Hall contributions. Integrating the resultant charge-\ncurrent density (44) over the normal-layer thickness aN,\nwe \fnally get for the 2D current density in the \flm:\nj=\u001b\u0012\naNE\u0000\u00120\n2ez\u0002\u00160\u0013\n= ^gfE+ [(\u0011+#n\u0002)_n]\u0002zg;\n(48)\nwhere\n^g\n\u001b= ~aN+ls\u001202\b\n\u0018inz(z\u0002)\u0000\u0018[n2\nz+ (z\u0002n\u0002z)n\u0001]\t\n(49)\nis the anisotropic 2D conductivity tensor (~ aN\u0011aN+\nls\u001202\u0019aN), which is referred in the literature to as the\nspin Hall magnetoconductance,8and\n\u0011\u0019~\n2eaN\u00120\u0018i; #\u0019~\n2eaN\u00120\u0018; (50)neglecting corrections that are cubic in \u00120. If\u0010i\u001c1,\nwhich is typically the case,22we have#\u001d\u0011. It could be\nnoted that restoring \u001b\u0000\u0018O(\u001202) in Eqs. (45) and (46)\nwould a\u000bect ^ gonly at orderO(\u001203).\nThe above spin accumulation can also be used to calcu-\nlate the spin-current density injected into the ferromag-\nnet atz= 0:\nJz=~\u001b\n2e\u0012\n\u00120z\u0002E\u0000\u00160\n2els\u0013\n\u0019\u0000sn\u0002^\u000b_n+ (\u0011+#n\u0002)(z\u0002j)\u0002n; (51)\nwhere\n^\u000b=~2\u001b\n4e2lss(\u0018\u0000\u0018in\u0002); (52)\nand we dropped terms that are cubic in \u00120, as be-\nfore. The corresponding magnetic equation of motion\ns_n=H\u0003\u0002n+Jzreproduces Eq. (10), with the current-\ndriven torque of the form (12) that is Onsager recipro-\ncal to the motive force in Eq. (48). Writing the Gilbert\ndamping/\u0018in Eq. (52) as a\"#=aFidenti\fes the inter-\nfacial damping enhancement in Eq. (14). In the formal\nlimit\u001b!1 (while keeping all other parameters, includ-\ningls, \fxed), which reproduces the perfect spin sink,\nthis givesa\"#=~g\"#\nr=4\u0019S. In the general case, \u0018also\ncaptures the spin back\row from the normal layer.12An\nanisotropic contribution to the Gilbert damping would\nbe produced at the cubic order in \u00120, had we not made\nany approximations in Eq. (51).\nIV. MAGNETIC TEXTURES\nFor completeness, we also provide some rudimentary\nremarks regarding the e\u000bect of directional magnetic inho-\nmogeneities, such as those associated with, for example,\nmagnetic domain walls.23Expanding the 2D magnetic\nfree-energy density to second order in spatial derivatives,\nwe have for a \flm with broken re\rection symmetry in\nthexyplane (see Sec. II for a detailed description of the\nstructure shown in Fig. 1):24\nF(n) =n\u0001H+K\n2n2\nz+ \u0000 (nz@ini\u0000ni@inz) +A\n2(@in)2;\n(53)\nwhere summation over Cartesian coordinates i=x;yis\nimplied and the dot products are in the 3D spin space.\n\u0000 here parametrizes the strength of the Dzyaloshinski-\nMoriya (DM) interaction and Ais the magnetic exchange\nsti\u000bness. A nonzero \u0000 requires macroscopic breaking of\nthe re\rection symmetry as well as a microscopic spin-\norbit interaction that breaks the spin-space isotropicity.\nEquation (53) can be rewritten in a more compact form\nas@xn(y\u0002n)\u0000@yn(x\u0002n) =\u0000nx@xnz+nz@xnx\u0000\nny@ynz+nz@yny\nF(n) =n\u0001H+~K\n2n2\nz+A\n2(Din)2; (54)7\nwhere\nDi\u0011@i+Q(z\u0002ei)\u0002 (55)\nis the so-called chiral derivative,25Q\u0011\u0000=A, and ~K\u0011\nK\u0000\u00002=A.Qis the wave number of the magnetic spiral\nthat minimizes the texture-dependent part of the free\nenergy.\nThe DM interaction of the form (53) arises natu-\nrally from the Rashba Hamiltonian (24). In a min-\nimal model,25where electrons with the single-particle\nHamiltonian (24) magnetically order due to their spin-\nindependent (e.g., Coulombic) interaction, the spin-orbit\nterm/vcan be gauged out at the \frst order in vby a\nposition-dependent rotation in spin space. To see this,\nwe \frst rewrite Eq. (24) as\n^HR=p2\n2m+vp\u0001z\u0002^\u001b=(p+mvz\u0002^\u001b)2\n2m\u0000mv2:(56)\nIt then immediately follows that\n^Uy^HR^U=p2\n2m+O(v2);where ^U=e\u0000iQRr\u0001z\u0002^\u001b=2;\n(57)\nde\fning\nQR\u00112mv\n~: (58)\n^Uis the operator of spin rotation around axis r\u0002z\nby anglerQR(recalling that r2xyplane), such that\nthe electron spin precesses by angle 2 \u0019over distance\nlso\u00112\u0019=QR=h=2mv(the spin-precession length).\nSince the transformed Hamiltonian (57) would describe\nmagnetic order that is spin isotropic, the corresponding\nfree energy is given simply by ( A=2)(@in)2(neglecting ex-\nternal and dipolar \felds). In the original frame of refer-\nence with Rashba Hamiltonian (56), the free-energy den-\nsity is then given by F(n) = (A=2)(@i~n)2, where n=^R~n\nand ^R(r) is the natural SO(3) representation of ^U(r).\nDi\u000berentiating @i~n=^RT(@i+^R@i^RT)n, we \fnally ob-\ntainF(n) = (A=2)(Din)2, where\nDi=@i+^R@i^RT=@i+QR(z\u0002ei)\u0002 (59)\nindeed reproduces Eq. (55) with Q!QR. In Ref. 20, the\nfree-energy density (53) was also obtained for the Dirac\nmodel of Sec. (III B), with the result:\n\u0000D\u0018\u0000~\n8\u0019vJJ?: (60)\nAs was pointed out in Ref. 25, the chiral derivative (55)\nis also expected to govern the nonequilibrium magnetic-\ntexture properties such as the current-driven torque \u001c\nand the spin-motive force \u000f. This can either be derived\nmicroscopically or understood on purely phenomenolog-\nical symmetry-based grounds. For example, the hydro-\ndynamic (advective) spin-transfer torque (along with its\nOnsager-reciprocal motive force)26\n\u001c/(j\u0001r)n; (61)which arises due to spin-current continuity in a model\nwithout any spin-orbit interactions and frozen magnetic\nimpurities, would be modi\fed by replacing r!Din\nthe perturbative treatment of the above Rashba model.\nHowever, while this simpli\fes a phenomenological con-\nstruction of various terms, in general, there is no funda-\nmental reason why the same Qshould de\fne the chiral\nderivatives entering in di\u000berent physical properties (such\nas free energy and spin torque).\nV. CONCLUSIONS\nIn summary, we have developed a phenomenology for\nslow long-wavelength dynamics of conducting quasi-2D\nmagnetic \flms and heterostructures, subject to struc-\ntural symmetries and Onsager reciprocity. The formal-\nism could address both small- and large-amplitude mag-\nnetic precession (assuming it is slow on the characteristic\nelectronic time scales), including, for example, magnetic\nswitching and domain-wall or skyrmion motion. Owing\nto the versatility of available heterostructures, including\nthose based on magnetic and topological insulators, we\nhave focused our discussion on the case of a ferromag-\nnetic/nonmagnetic bilayer, which serves two purposes:\nIt naturally has a broken inversion symmetry, and the\nspin-orbit and magnetic properties could be separately\noptimized and tuned in one of the two layers.\nIn the case when the spin-relaxation length in the nor-\nmal layer is short compared to its thickness, we can asso-\nciate the interplay between spin-orbit and exchange inter-\nactions to a narrow region in the vicinity of the interface,\nfor which we de\fne the kinetic coe\u000ecients such as the\ninterfacially enhanced Gilbert damping parametrized by\na\"#and the spin Hall angle parametrized by #. Such (sep-\narately measurable) phenomenological coe\u000ecients, which\nenter in our theory, must thus be viewed as joint proper-\nties of both of the bilayer materials as well as structure\nand quality of the interface.\nWe demonstrate the emergence of our phenomenology\nout of three microscopic models, based on Rashba, Dirac,\nand di\u000busive normal-metal \flms, all in contact with a\nmagnetic insulator. In addition to Onsager-reciprocal\nspin-transfer torques and electromotive forces, our\nphenomenology also accommodates arbitrary Gilbert-\ndamping and (magneto)resistance anisotropies, which are\ndictated by the same structural symmetries and may mi-\ncroscopically depend on the same exchange and spin-\norbit ingredients as the reciprocal magnetoelectric cou-\npling e\u000bects.\nACKNOWLEDGMENTS\nWe acknowledge stimulating discussions with G. E. W.\nBauer, S. T. B. Goennenwein, and D. C. Ralph. This\nwork was supported in part by FAME (an SRC STAR-\nnet center sponsored by MARCO and DARPA), the NSF8\nunder Grant No. DMR-0840965, and by the Kavli Insti- tute for Theoretical Physics through Grant No. NSF\nPHY11-25915.\n1K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008); I. M. Miron, G. Gaudin, S. Au\u000bret, B. Rod-\nmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella,\nNature Mater. 9, 230 (2010); I. M. Miron, K. Garello,\nG. Gaudin, P.-J. Zermatten, M. V. Costache, S. Au\u000bret,\nS. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella,\nNature 476, 189 (2011); L. Liu, T. Moriyama, D. C.\nRalph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601\n(2011); L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph,\nand R. A. Buhrman, Science 336, 555 (2012).\n2Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. 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Magn.\n40, 3443 (2004).\n7More precisely, it is only the symmetric part of ^ \u000bthat\nshould be identi\fed with a generalized Gilbert damping.\nIndeed, Onsager reciprocity requires ^ \u000bT(n) = ^\u000b(\u0000n),\nwhile the dissipative (i.e., time-reversal symmetry break-\ning) character dictates ^ \u000b(\u0000n) = ^\u000b(n), which together lead\nto ^\u000bT(n) = ^\u000b(n). The antisymmetric component of ^ \u000b, on\nthe other hand, contributes to the e\u000bective, matrix-valued\ngyromagnetic ratio.\n8H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel,\nS. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennen-\nwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013);Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n9When the ferromagnet is insulating, tan \u0012so de\fned de-\nscribes the conversion between 3D current density in the\nnormal metal and the spin-current density absorbed by the\nferromagnetic insulator. In a simple limit of weak spin-\norbit interaction at the interface, such tan \u0012may corre-\nspond to the bulk spin Hall angle of the normal metal.\nWhen the thickness aNis larger than the spin-relaxation\nlength\u0015Nin the normal metal, it is natural to expect \u0012\nde\fned by Eq. (13), as well as \f\u0011#=\u0011, to be essentially\nthicknessaNindependent.\n10V. M. Edelstein, J. Phys.: Condens. Matter 7, 1 (1995).\n11Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002); Y. Tserkovnyak, A. Brataas,\nG. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77,\n1375 (2005).\n12Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. 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Martinez, and\nG. S. D. Beach, Nature Mater. 12, 611 (2013); K.-S. Ryu,9\nL. Thomas, S.-H. Yang, and S. Parkin, Nature Nanotech.\n8, 527 (2013).\n24A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138,\n255 (1994).25K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys.\nRev. Lett. 111, 216601 (2013).\n26Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater. 320, 1282 (2008)."    },    {        "title": "2002.06858v2.Self_similar_shrinkers_of_the_one_dimensional_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Self-similar shrinkers of the one-dimensional\nLandau–Lifshitz–Gilbert equation\nSusana Gutiérrez1and André de Laire2\nAbstract\nThe main purpose of this paper is the analytical study of self-shrinker solutions of the\none-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics\nfor the spin in ferromagnetic materials. We show that there is a unique smooth family of\nbackward self-similar solutions to the LLG equation, up to symmetries, and we establish\ntheir asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of\nthe self-similar profiles converge to great circles on the sphere S2, at an exponential rate.\nIn particular, the results presented in this paper provide examples of blow-up in finite\ntime, where the singularity develops due to rapid oscillations forming limit circles.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, self-similar expanders, backward\nself-similar solutions, blow up, asymptotics, ferromagnetic spin chain, heat flow for harmonic\nmaps, quasi-harmonic sphere.\n2010Mathematics Subject Classification: 82D40; 35C06; 35B44; 35C20; 53C44; 35Q55;\n58E20; 35K55.\n82D40;35C06;35B44; 35C20;53C44;35Q55;58E20;35K55\n1 Introduction\n1.1 The Landau–Lifshitz–Gilbert equation: self-similar solutions\nIn this paper we continue the investigation started in [32, 33] concerning the existence and prop-\nerties of self-similar solutions for the Landau–Lifshitz–Gilbert equation (LLG). This equation\ndescribes the dynamics for the magnetization or spin in ferromagnetic materials [43, 27] and is\ngiven by the system of nonlinear equations\n∂tm=βm×∆m−αm×(m×∆m), (LLG)\nwherem= (m 1,m2,m3) :RN×I−→S2is the spin vector, I⊂R,β≥0,α≥0,×denotes\nthe usual cross-product in R3, and S2is the unit sphere in R3. This model for ferromagnetic\nmaterials constitutes a fundamental equation in the magnetic recording industry [53]. The\nparameters β≥0andα≥0are, respectively, the so-called exchange constant and Gilbert\ndamping, and take into account the exchange of energy in the system and the effect of damping\non the spin chain. By considering a time-scaling, one can assume without loss of generality that\nthe parameters αandβsatisfy\nα∈[0,1]andβ=/radicalbig\n1−α2.\n1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.\nE-mail: s.gutierrez@bham.ac.uk\n2Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille, France.\nE-mail: andre.de-laire@univ-lille.fr\n1arXiv:2002.06858v2  [math.AP]  21 May 2020From a purely mathematical point of view, the LLG equation is extremely interesting since\nit interpolates between two fundamental geometric evolution equations, the Schrödinger map\nequation and the heat flow for harmonic maps, via specific choices of the parameters involved.\nPrecisely, we recall that in the limit case α= 1(and, consequently, β= 0), (LLG) reduces to\nthe heat flow for harmonic maps onto S2,\n∂tm−∆m=|∇m|2m, (HFHM)\nand, ifα= 0(no damping), it reduces to the Schrödinger map equation\n∂tm=m×∆m. (SM)\nWhen 0<α< 1, (LLG) is of parabolic type. We refer the reader to [40, 29, 32, 33, 12, 14, 15, 13]\nand the references therein for more details and surveys on these equations.\nA natural question, that has proved relevant to the understanding of the global behavior\nof solutions and formation of singularities, is whether or not there exist solutions which are\ninvariant under scalings of the equation. In the case of the LLG equation it is straightforward\nto see that the equation is invariant under the following scaling: If mis a solution of (LLG),\nthenmλ(t,x) =m(λx,λ2t), for any positive number λ, is also a solution. Associated with this\ninvariance, a solution mof (LLG) defined on I=R+orI=R−is called self-similar if it is\ninvariant under rescaling, that is\nm(x,t) =m(λx,λ2t),∀λ>0,∀x∈RN,∀t∈I.\nFixingT∈Rand performing a translation in time, this definition leads to two types of self-\nsimilar solutions: A forward self-similar solution or expander is a solution of the form\nm(x,t) =f/parenleftbiggx√\nt−T/parenrightbigg\n,for (x,t)∈RN×(T,∞), (1.1)\nand a backward self-similar solution or shrinker is a solution of the form\nm(x,t) =f/parenleftbiggx√\nT−t/parenrightbigg\n,for (x,t)∈RN×(−∞,T), (1.2)\nfor some profile f:RN−→S2. In this manner, expanders evolve from a singular value at time\nT, while shrinkers evolve towards a singular value at time T.\nSelf-similar solutions have received a lot of attention in the study of nonlinear PDEs because\nthey can provide important information about the dynamics of the equations. While expanders\nare related to non-uniqueness phenomena, resolution of singularities and long time description\nof solutions, shrinkers are often related to phenomena of singularity formation (see e.g. [26, 18]).\nOn the other hand, the construction and understanding of the dynamics and properties of self-\nsimilar solutions also provide an idea of which are the natural spaces to develop a well-posedness\ntheory that captures these often very physically relevant structures. Examples of equations for\nwhichself-similarsolutionshavebeenstudiedinclude,amongothers,theNavier–Stokesequation,\nsemilinear parabolic equations, and geometric flows such as Yang–Mills, mean curvature flow\nand harmonic map flow. We refer to [37, 48, 36, 51, 5] and the references therein for more\ndetails.\nAlthough the results that will be presented in this paper relate to self-similar shrinkers of\nthe one-dimensional LLG equation (that is, to solutions m:R×I−→S2of LLG), for the sake\nof context we describe some of the most relevant results concerning maps from RN×IintoSd,\nwithN≥2andd≥2. In this setting one should point out that the majority of the works in the\n2literature concerning the study of self-similar solutions of the LLG equation are confined to the\nheat flow for harmonic maps equation, i.e. α= 1. In the case when α= 1, the main works on the\nsubject restrict the analysis to corotational maps taking values in Sd, which reduces the analysis\nof (HFHM) to the study of a second order real-valued ODE. Then tools such as the maximum\nprinciple or the shooting method can be used to show the existence of solutions. We refer to\n[19, 21, 23, 7, 8, 6, 22] and the references therein for more details on such results for maps taking\nvalues in Sd, withd≥3.Recently, Deruelle and Lamm [17] have studied the Cauchy problem\nfor the harmonic map heat flow with initial data m0:RN→Sd, withN≥3andd≥2, where\nm0is a Lipschitz 0-homogeneous function, homotopic to a constant, which implies the existence\nof expanders coming out of m0.\nWhen 0<α≤1, the existence of self-similar expanders for the LLG equation was recently\nestablished by the authors in [33]. This result is a consequence of a well-posedness theorem for\nthe LLG equation considering an initial data m0:RN→S2in the space BMO of functions of\nbounded mean oscillation. Notice that this result includes in particular the case of the harmonic\nmap heat flow.\nAs mentioned before, in the absence of damping ( α= 0), (LLG) reduces to the Schrödinger\nmap equation (SM), which is reversible in time, so that the notions of expanders and shrinkers\ncoincide. For this equation, Germain, Shatah and Zeng [24] established the existence of ( k-\nequivariant) self-similar profiles f:R2→S2.\n1.2 Goals and statements of main results\nThe results of this paper aim to advance our understanding of self-similar solutions of the one-\ndimensional LLG equation. In order to contextualize and motivate our results, we continue to\nprovide further details of what is known about self-similar solutions in this context.\nIn the 1d-case, when α= 0, (SM) is closely related to the Localized Induction Approximation\n(LIA), and self-similar profiles f:R→S2were obtained and analyzed in [34, 35, 41, 10]. In the\ncontext of LIA, self-similar solutions constitute a uniparametric family of smooth solutions that\ndevelop a singularity in the shape of a corner in finite time. For further work related to these\nsolutions, including the study of their continuation after the blow-up time and their stability,\nwe refer to the reader to [4, 3]. At the level of the Schrödinger map equation, these self-similar\nsolutions provide examples of smooth solutions that develop a jump singularity in finite time.\nIn the general case α∈[0,1], the analytical study of self-similar expanders of the one-\ndimensional (LLG) was carried out in [32]. Here, it was shown that these solutions are given by\na family of smooth profiles {fc,α}c,α, and that the corresponding expanders are associated with\na discontinuous (jump) singular initial data. We refer to [32, 33] for the precise statement of this\nresult, and the stability of these solutions, as well as the qualitative and quantitative analysis\nof their dynamics with respect to the parameters candα.\nIt is important to notice that in the presence of damping ( α > 0), since the LLG equation\nis not time-reversible, the notion of expander is different from that of shrinker. It is therefore\nnatural to ask the following question: What can be said about shrinker solutions for the one-\ndimensional LLG equation?\nAnswering this question constitutes the main purpose of this paper. Precisely, our main goals\nare to establish the classification of self-similar shrinkers of the one-dimensional LLG equation\nof the form (1.2) for some profile f:R→S2, and the analytical study of their properties. In\nparticular, we will be especially interested in studying the dynamics of these solutions as ttends\nto the time of singularity T, and understanding how the dynamical behavior of these solutions\nis affected by the presence of damping. Since, as it has been already mentioned, the case α= 0\n3has been previously considered in the literature (see [4, 31]), in what follows we will assume that\nα∈(0,1].\nIn order to state our first result, we observe that if mis a solution to (LLG) of the form (1.2)\nfor some smooth profile f, thenfsolves the following system of ODEs\nxf/prime\n2=βf×f/prime/prime−αf×(f×f/prime/prime),onR, (1.3)\nwhich recasts as\nαf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime\n2= 0,onR, (1.4)\ndue to the fact that ftakes values in S2.\nIn the case α∈(0,1), it seems unlikely to be able to find explicit solutions to (1.4), and even\ntheir existence is not clear (see also equation (1.16)). Nevertheless, surprisingly we can establish\nthe following rigidity result concerning the possible weak solutions to (1.4) (see Section 2 for the\ndefinition of weak solution).\nTheorem 1.1. Letα∈(0,1]. Assume that fis a weak solution to (1.4). Thenfbelongs to\nC∞(R;S2)and there exists c≥0such that|f/prime(x)|=ceαx2/4, for allx∈R.\nTheorem 1.1 provides a necessary condition on the possible (weak) solutions of (1.4): namely\nthe modulus of the gradient of any solution mustbeceαx2/4, for somec≥0. We proceed now to\nestablish the existence of solutions satisfying this condition for any c>0(notice that the case\nwhenc= 0is trivial).\nTo this end, we will follow a geometric approach that was proven to be very fruitful in similar\ncontexts (see e.g. [41, 46, 42, 34, 16]), including the work of the authors in the study of expanders\n[32]. As explained in Subsection 3.1, this approach relies on identifying fas the unit tangent\nvectorm:=fof a curveXminR3parametrized by arclength. Thus, assuming that fis\na solution to (1.4) and using the Serret–Frenet system associated with the curve Xm, we can\ndeduce that the curvature and the torsion are explicitly given by\nk(x) =ceαx2/4,andτ(x) =−βx\n2, (1.5)\nrespectively, for some c≥0(see Subsection 3.1 for further details). In particular, we have\n|m/prime(x)|=k(x) =ceαx2/4, in agreement with Theorem 1.1. Conversely, given c≥0and denoting\nmc,αthe solution of the Serret–Frenet system\n\n\nm/prime(x) =k(x)n(x),\nn/prime(x) =−k(x)m(x) +τ(x)b(x),\nb/prime(x) =−τ(x)n(x),(1.6)\nwith curvature and torsion as in (1.5), and initial conditions (w.l.o.g.)\nm(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1), (1.7)\nwe obtain a solution to (1.4). Moreover, we can show that the solutions constructed in this\nmanner provide, up to symmetries, all the solutions to (1.3). The precise statement is the\nfollowing.\nProposition 1.2. The set of nonconstant solutions to (1.3)is{Rmc,α:c >0,R∈SO(3)},\nwhereSO(3)is the group of rotations about the origin preserving orientation.\n4The above proposition reduces the study of self-similar shrinkers to the understanding of the\nfamilyofself-similarshrinkersassociatedwiththeprofiles {mc,α}c,α. Thenextresultsummarizes\nthe properties of these solutions.\nTheorem 1.3. Letα∈(0,1],c > 0,T∈Randmc,αbe the solution of the Serret–Frenet\nsystem(1.6)with initial conditions (1.7),\nk(x) =ceαx2/4andτ(x) =−/radicalbig\n1−α2x\n2.\nDefine\nmc,α(x,t) =mc,α/parenleftbiggx√\nT−t/parenrightbigg\n, t<T. (1.8)\nThen we have the following statements.\n(i) The function mc,αbelongs toC∞(R×(−∞,T);S2), solves(LLG)fort∈(−∞,T), and\n|∂xmc,α(x,t)|=c√\nT−teαx2\n4(T−t),\nfor all (x,t)∈R×(−∞,T).\n(ii) The components of the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)satisfy that m1,c,αis even,\nwhilem2,c,αandm3,c,αare odd.\n(iii) There exist constants ρj,c,α∈[0,1], Bj,c,α∈[−1,1],andφj,c,α∈[0,2π),forj∈{1,2,3},\nsuch that we have the following asymptotics for the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)\nand its derivative:\nmj,c,α(x) =ρj,c,αcos(cΦα(x)−φj,c,α)−βBj,c,α\n2cxe−αx2/4\n+β2ρj,c,α\n8csin(cΦα(x)−φj,c,α)/integraldisplay∞\nxs2e−αs2/4ds+β\nα5c2O(x2e−αx2/2),(1.9)\nand\nm/prime\nj,c,α(x) =−cρj,c,αsin(cΦα(x)−φj,c,α)eαx2/4\n+β2ρj,c,α\n8cos(cΦα(x)−φj,c,α)eαx2/4/integraldisplay∞\nxs2e−αs2/4ds+β\nα5cO(x2e−αx2/4),(1.10)\nfor allx≥1, where\nΦα(x) =/integraldisplayx\n0eαs2\n4ds.\nMoreover, the constants ρj,c,α,Bj,c,α, andφj,c,αsatisfy the following identities\nρ2\n1,c,α+ρ2\n2,c,α+ρ2\n3,c,α= 2, B2\n1,c,α+B2\n2,c,α+B2\n3,c,α= 1andρ2\nj,c,α+B2\nj,c,α= 1, j∈{1,2,3}.\n(iv) The solution mc,α= (m 1,c,α,m2,c,α,m3,c,α)satisfies the following pointwise convergences\nlim\nt→T−(mj,c,α(x,t)−ρj,c,αcos/parenleftbigcΦα/parenleftbigx√\nT−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx>0,\nlim\nt→T−(mj,c,α(x,t)−ρ−\nj,c,αcos/parenleftbigcΦα/parenleftbig−x√\nT−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx<0,(1.11)\nforj∈{1,2,3}, whereρ−\n1,c,α=ρ1,c,α,ρ−\n2,c,α=−ρ2,c,αandρ−\n3,c,α=−ρ3,c,α.\n5(v) For anyϕ∈W1,∞(R;R3), we have\nlim\nt→T−/integraldisplay\nRmc,α(x,t)·ϕ(x)dx= 0.\nIn particular, mc,α(·,t)→0ast→T−, as a tempered distribution.\nIt is important to remark that Theorem 1.3 provides examples of (smooth) solutions to the 1d-\nLLG equation that blow up in finite time. In order to see this, let us first recall that the existence\nof smooth solutions to (LLG) on short times can be established as in the case of the heat flow\nfor harmonic maps [45], using that (LLG) is a strongly parabolic system [30, 2]. In particular, in\nthe one-dimensional case, for any initial condition m0∈C∞(R,S2), there exists a maximal time\n0< T max≤∞such that (LLG) admits a unique, smooth solution m∈C∞(R×[0,Tmax);S2).\nMoreover, if Tmax<∞, then\nlim\nt→T−\nmax/bardbl∂xm(·,t)/bardblL∞(R)=∞.\nNext, observe that for any c>0andT∈R, the solution of the initial value problem associated\nwith (LLG) and with initial condition mc,α(·)at timeT−1is given by mc,αin Theorem 1.3,\nfort∈[T−1,T), and blows up at time T. Indeed, from (i)in Theorem 1.3 , we have that\nlim\nt→T−|∂xmc,α(x,t)|= lim\nt→T−c√\nT−teαx2\n4(T−t)=∞,\nforc>0and for allx∈R.\nNotice also that from the asymptotics in part (iii)and the symmetries of the profile estab-\nlished in part (ii), we obtain a precise description of the fast oscillating nature of the blow up of\nthe solution (1.8) given in Theorem 1.3. In this setting, we observe that part (iii)of the above\ntheorem provides the asymptotics of the profile mc,αat infinity, in terms of a fast oscillating\nprincipal part, plus some exponentially decaying terms. Notice that for the integral term in\n(1.9), we have (see e.g. [1])\n/integraldisplay∞\nxs2e−αs2/4ds=2xe−αx2/4\nα/parenleftBig\n1 +2\nαx2−4\nα2x4+···/parenrightBig\n,asx→∞,\nand that using the asymptotics for the Dawson’s integral [1], we also get\nΦα(x) =2eαx2/4\nαx/parenleftBig\n1 +2\nαx2+12\nα2x4+···/parenrightBig\n,asx→∞.\nIt is also important to mention that the big- Oin the asymptotics (1.9) does not depend on the\nparameters, i.e. there exists a universal constant C > 0, such that the big- Oin (1.9) satisfies\n|O(x2e−αx2/2)|≤Cx2e−αx2/2,for allx≥1.\nIn this manner, the constants multiplying the big- Oare meaningful and in particular, big- O\nvanishes when β= 0(i.e.α= 1).\nIn Figure 1 we have depicted the profile mc,αforα= 0.5andc= 0.5, where we can see their\noscillating behavior. Moreover, the plots in Figure 1 suggest that the limit sets of the trajectories\nare great circles on the sphere S2whenx→±∞. This is indeed the case. In our last result we\nestablish analytically that mc,αoscillates in a plane passing through the origin whose normal\nvector is given by B+\nc,α= (B1,c,α,B2,c,α,B3,c,α), andB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α)asx→+∞\nandx→−∞, respectively.\n6m1m2m3\nB+\nc,α\nm1m2m3\n-1.0-0.50.00.51.0-1.0-0.50.00.51.0\nm1m2\nFigure 1: Profile mc,αforc= 0.5andα= 0.5. The figure on the left depicts profile for x∈R+\nand the normal vector B+\nc,α≈(−0.72,−0.3,0.63). The figure on the center shows the profile for\nx∈R; the angle between the circles C±\nc,αisϑc,α≈1.5951. The figure on the right represents the\nprojection of limit cycles C±\nc,αon the plane.\nTheorem 1.4. Using the constants given in Theorem 1.3, let P±\nc,αbe the planes passing through\nthe origin with normal vectors B+\nc,αandB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α), respectively. Let C±\nc,αbe\nthe circles in R3given by the intersection of these planes with the sphere, i.e. C±\nc,α=P±\nc,α∩S2.\nThen the following statements hold.\n(i) For all|x|≥1, we have\ndist(mc,α(x),C±\nc,α)≤30√\n2β\ncα2|x|e−αx2/4. (1.12)\nIn particular\nlim\nt→T−dist(mc,α(x,t),C+\nc,α) = 0,ifx>0,\nlim\nt→T−dist(mc,α(x,t),C−\nc,α) = 0,ifx<0.(1.13)\n(ii) Letϑc,α= arccos(1−2B2\n1,c,α)be the angle between the circles C±\nc,α. Forc≥β√π/√α, we\nhave\nϑc,α≥arccos/parenleftBigg\n−1 +2πβ2\nc2α/parenrightBigg\n. (1.14)\nIn particular\nlimc→∞ϑc,α=π,for allα∈(0,1], and lim\nα→1ϑc,α=π,for allc>0.(1.15)\nThe above theorem above establishes the convergence of the limit sets of the trajectories\nof the profile mc,αto the great circles C±\nc,αas shown in Figure 1. Moreover, (1.12) gives us\nan exponential rate for this convergence. In terms of the solution mc,αto the LLG equation,\nTheorem 1.4 provides a more precise geometric information about the way that the solution\nblows up at time T, as seen in (1.13). The existence of limit circles for related ferromagnetic\nmodels have been investigated for instance in [52, 9] but to the best of our knowledge, this is the\nfirst time that this type of phenomenon has been observed for the LLG equation. In Figure 1\ncan see that ϑc,α≈1.5951forα= 0.5andc= 0.5, where we have chosen the value of csuch\nthat the angle is close to π/2.\nFinally, (1.14) and (1.15) in Theorem 1.4 provide some geometric information about behavior\nof the limit circles with respect to the parameters candα. In particular, formulae (1.15) states\n7that the angle between the limiting circles C+\nc,αandC−\nc,αisπasc→∞, for fixedα∈(0,1], and\nthe same happens as α→1, for fixedc>0. In other words, in these two cases the circles C±\nc,α\nare the same (but differently oriented).\n1.3 Comparison with the limit cases α= 0andα= 1\nIt is well known that the Serret–Fenet system can be written as a second-order differential\nequation. Forinstance, if (m,n,b) = (mj,nj,bj)3\nj=1isasolutionof (1.5)–(1.6), usingLemma3.1\nin [32], we have that new variable\ngj(s) =e1\n2/integraltexts\n0k(σ)ηj(σ)dσ,withηj(x) =nj(x) +ibj(x)\n1 +mj(x),\nsatisfies the equation, for j∈{1,2,3},\ng/prime/prime\nj(x)−x\n2(α+iβ)g/prime\nj(x) +c2\n4eαx2/2gj(x) = 0. (1.16)\nThen, in the case α= 1, it easy to check (see also Remark 3.3) that the profile is explicitly given\nby the plane curve\nmc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0). (1.17)\nIn particular, we see that the asymptotics in Theorem 1.3 are satisfied with\nρ1,c,1= 1, ρ 2,c,1= 1, ρ 3,c,1= 0, φ 1,c,1= 0, φ 2,c,1= 3π/2, φ 3,c,1∈[0,2π).\nThe caseα= 0is more involved, but using (1.16), the solution {mc,0,nc,0,bc,0}of the system\n(1.6)canstillbeexplicitlydeterminedintermsofconfluenthypergeometricfunctions. Thisleads\nto the asymptotics [34, 32, 20]\nmc,0(x) =Ac−2c\nxBccos/parenleftBigg\nx2\n4+c2ln(x) +π\n2/parenrightBigg\n+O/parenleftbigg1\nx2/parenrightbigg\n, (1.18)\nasx→∞, for some vectors Ac∈S2andBc∈R3. In particular, we see that mc,0(x)converges\nto some vector Ac, asx→∞. Hence, there is a drastic change in the behavior of the profile\nin the cases α= 0andα > 0: In the first case mc,0converges to a point at infinity, while in\nthe second case (1.12) tells us that mc,αconverges to a great circle. In this sense, there is a\ndiscontinuity in the behavior of mc,αatα= 0.\nAlso, from equation (1.16), we can formally deduce that the difference between the expanders\nand shrinkers corresponds to flipping the sign in the parameters α→−αandβ→−β. Notice\nthat the exponential coefficient in (1.16) is proportional to the square of the curvature, given by\nce−αx2/4for the skrinkers, and ceαx2/4for the expanders. We used equation (1.16) (with flipped\nsigns) to obtain the asymptotics of the expanders in [32], relying on the fact the exponential\nterm in equation vanishes as x→∞. However, the exponential grow in the case of skrinkers in\n(1.16) changes the behavior of the solution and we cannot use the methods introduced in [32].\nGoing back to Theorem 1.3, it is seems very difficult to get asymptotics for the constants\nin (1.9). Our strategy for the constants appearing in the asymptotics for the expanders in [32]\nrelied on obtaining uniform estimates and using continuity arguments. In particular, using the\nfact that the constants in (1.18) are explicit, we were able to get a good information about the\nconstants in the asymptotics when αwas close to 0. Due to the above mentioned discontinuity\n8ofmc,αatα= 0, it seems unlikely that the use of continuity arguments will provide information\nfor the constants in the asymptotics for the shrinkers.\nFinally, let us also remark that we cannot use continuation arguments to find the behavior\nof the circles for csmall. This is expected since m0,α(x) = (1,0,0)for allx∈R, whenc= 0\n(see (4.6)). In Section 4 we give some numerical simulations for csmall.\nStructure of the paper. The outline of this paper is the following. In Section 2, we study\n(1.4) as an elliptic quasilinear system and prove the rigidity result Theorem 1.1. By using the\nSerret–Frenet system, we prove there existence and uniqueness of solution, up to a rotation, in\nSection 3. We also use this system to obtain the asymptotics of the self-similar profiles. Finally,\nSection 4 is devoted to the proof of Theorem 1.4.\n2 Rigidity result. Theorem 1.1\nThe purpose of this section is to prove the rigidity result stated in Theorem 1.1 concerning\n(weak) solutions of the system\nxf/prime\n2=βf×f/prime/prime−αf×(f×f/prime/prime),onR. (2.1)\nWe start by introducing the notion of weak solution of the above system. To this end, we first\nobserve that the system (2.1) recasts as\nαf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime\n2= 0, (2.2)\nusing the following vector identities for a (smooth) function fwith|f|= 1:\nf×f/prime/prime= (f×f/prime)/prime,\n−f×(f×f/prime/prime) =f/prime/prime+|f/prime|2f.(2.3)\nWe prefer to use the formulation (2.2) since it is simpler to handle in weak sense. Indeed, we\nsay thatf= (f1,f2,f3)∈H1\nloc(R,S2)is aweak solution to the system (2.2) if\n/integraldisplay\nR/parenleftBig\n−αf/prime·ϕ/prime+α|f/prime|2f·ϕ−β(f×f/prime)·ϕ/prime−x\n2f/primeϕ/parenrightBig\ndx= 0, (2.4)\nfor allϕ= (ϕ1,ϕ3,ϕ3)∈C∞\n0(R).\nUsing (2.3), we can recast (2.2) as,\nαf/prime/prime\n1+α|f/prime|2f1+β(f2f/prime/prime\n3−f3f/prime/prime\n2)−x\n2f/prime\n1= 0, (2.5a)\nαf/prime/prime\n2+α|f/prime|2f2+β(f3f/prime/prime\n1−f1f/prime/prime\n3)−x\n2f/prime\n2= 0, (2.5b)\nαf/prime/prime\n3+α|f/prime|2f3+β(f1f/prime/prime\n2−f2f/prime/prime\n1)−x\n2f/prime\n3= 0. (2.5c)\nThus we see that the weak formulation (2.4) can be written as\n/integraldisplay\nRA(f(x))f/prime(x)·ϕ/prime(x) =/integraldisplay\nRG(x,f,f/prime)ϕ(x),for allϕ∈C∞\n0(R), (2.6)\nwith\nA(u) =\nα−βu3βu2\nβu3α−βu1\n−βu2βu1α,\nandG(x,u,p) =\nαu1|p|2−xp1\n2\nαu2|p|2−xp2\n2\nαu3|p|2−xp3\n2\n,\n9whereu= (u1,u2,u3)andp= (p1,p2,p3). We want now to invoke the regularity theory for\nquasilinear elliptic system (see [39, 25]). To verify that the system is indeed uniformly elliptic,\nwe can easily check that\nA(u)ξ·ξ=α|ξ|2,for allξ,u∈R3.\nIn addition, Ghas quadratic growth on bounded domains, i.e.\n|G(x,u,p)|≤√\n3(M|p|2+R|p|),\nfor all|u|≤Mand|x|≤R. Since a weak solution fto (2.6) belongs by definition to H1\nloc(R;S2),\nwe have by the Sobolev embedding theorem that fis Hölder continuous with |f(x)|= 1.\nTherefore we can apply the results in Theorem 1.2 in [25] (see also Lemma 8.6.3 in [38] or\nTheorem 2.4.3 in [49] for detailed proofs), to conclude that f∈H2\nloc(R)∩W1,4\nloc(R), and so\nthatf∈C1,γ\nloc(R), for someγ∈(0,1). We get that G(x,f(x),f/prime(x))belongs toC0,γ\nloc(R), which\nallows us to invoke the Schauder regularity theory (see e.g. Theorem A.2.3 in [38]) to infer that\nf∈C2,γ\nloc(R). This implies that G(x,f(x),f/prime(x))belongs toC1,γ\nloc(R), as well as the coefficients of\nA(u), so the Schauder estimates yield that f∈C3,γ\nloc(R). By induction, we this argument shows\nthatf∈C∞(R).\nWe are now in position to complete the proof of Theorem 1.1. Indeed, let first remark that\ndifferentiating the relation |f|2= 1, we have the identities\nf·f/prime= 0, (2.7)\nf·f/prime/prime=−|f/prime|2. (2.8)\nBy taking the cross product of fand (2.2), and using (2.3), we have\nβf/prime/prime+β|f/prime|2f−α(f×f/prime)/prime+x\n2f×f/prime= 0. (2.9)\nThus, by multiplying (2.2) by α, (2.9) byβ, and recalling that α2+β2= 1, we get\nf/prime/prime+|f/prime|2f−x\n2(αf/prime−βf×f/prime) = 0.\nTaking the scalar product of this equation and f/prime, the identity (2.7) allow us to conclude that\n1\n2(|f/prime|2)/prime−αx\n2|f/prime|2= 0. (2.10)\nIntegrating, we deduce that there is a constant C≥0such that|f/prime|2=Ceαx2/2. This completes\nthe proof of Theorem 1.1.\nWe conclude this section with some remarks.\nRemark 2.1. A similar result to the one stated in Theorem 1.1 also holds for the expanders\nsolutions. Precisely, any weak solution to (2.1), withxf/prime/2replaced by−xf/prime/2in the l.h.s., is\nsmooth and there exists c≥0such that|f/prime(x)|=ce−αx2/4, for allx∈R.\nRemark 2.2. Let us mention that in the case α= 1, a nonconstant solution u:RN→Sdto\nequation\n∆u+|∇u|2u−x·∇u\n2= 0,onRN, (2.11)\n10is usually called quasi-harmonic sphere , since it corresponds to the Euler–Lagrange equations of\na critical point of the (so-called) quasi-energy [44]\nEquasi(u) =/integraldisplay\nRN|∇u(y)|2e−|y2|/4dy.\nIt has been proved in [19] the existence of a (real-valued) function hsuch that\nu(x) =/parenleftBigx\n|x|sin(h(|x|)),cos(h(|x|)/parenrightBig\nis a solution to (2.11)with finite quasi-energy for 3≤N=d≤6. In addition, there is no\nsolution of this form if d≥7[8]. Both results are based on the analysis of the second-order\nODE associated with h. We refer also to [21] for a generalization of the existence result for\nN≥3of other equivariant solutions to (2.11). In the case N= 1andd= 2, the solution to\n(2.11)is explicitly given by (1.17), and its associated quasi-energy is infinity, as remarked in\n[54].\n3 Existence, uniqueness and properties\n3.1 Existence and uniqueness of the self-similar profile. Proposition 1.2\nIn the previous section we have shown that any solution to the profile equation\nαm/prime/prime+α|m/prime|2m+β(m×m/prime)/prime−xm/prime\n2= 0, (3.1)\nis smooth and that there is c≥0such that\n|m/prime(x)|=ceαx2/4,for allx∈R. (3.2)\nWe want to give now the details about how to construct such a solution by using the Serret–\nFrenet frame, which will correspond to the profile mc,αin Theorem 1.3. The idea is to identify\nmas the tangent vector to a curve in R3, so we first recall some facts about curves in the space.\nGivenm:R→S2a smooth function, we can define the curve\nXm(x) =/integraldisplayx\n0m(s)ds, (3.3)\nso thatXmis smooth, parametrized by arclenght, and its tangent vector is m. In addition,\nif|m/prime|does not vanish on R, we can define the normal vector n(x) =m/prime(x)/|m/prime(x)|and the\nbinormal vector b(x) =m(x)×n(x). Moreover, we can define the curvature and torsion of Xm\nask(x) =|m/prime(x)|andτ(x) =−b/prime(x)·n(x). Since|m(x)|2= 1,for allx∈R, we have that\nm(x)·n(x) = 0, for allx∈R, that the vectors {m,n,b}are orthonormal and it is standard to\ncheck that they satisfy the Serret–Frenet system\n\n\nm/prime=kn,\nn/prime=−km+τb,\nb/prime=−τn.(3.4)\nLet us apply this construction to find a solution to (3.1). We define curve Xmas in (3.3), and\nremark that equation (3.1) rewrites in terms of {m,n,b}as\nx\n2kn=β(k/primeb−τkn)−α(−k/primen−kτb).\n11Therefore, from the orthogonality of the vectors nandb, we conclude that the curvature and\ntorsion ofXmare solutions of the equations\nx\n2k=αk/prime−βτkandβk/prime+αkτ= 0,\nthat is\nk(x) =ceαx2\n4andτ(x) =−βx\n2, (3.5)\nfor somec≥0. Of course, the fact that k(x) =ceαx2/4is in agreement with the fact that we\nmust have|m/prime(x)|=ceαx2/4.\nNow, given α∈[0,1]andc>0, consider the Serret–Frenet system (3.4) with curvature and\ntorsion function given by (3.5) and initial conditions\nm(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1). (3.6)\nThen, by standard ODE theory, there exists a unique global solution {mc,α,nc,α,bc,α}in\n(C∞(R;S2))3, and these vectors are orthonormal. Also, it is straightforward to verify that\nmc,αis a solution to (3.1) satisfying (3.2).\nThe above argument provides the existence of solutions in the statement of Proposition 1.2.\nWe will now complete the proof of Proposition 1.2 showing the uniqueness of such solutions, up\nto rotations.\nTo this end, assume that ˜mis a weak nontrivial solution to (3.1). By Theorem 1.1, ˜mis\ninC∞(R,S2)and there exists c >0such that|˜m/prime(x)|=ceαx2/4, for allx∈R. Following the\nabove argument, the curve X˜m(defined in (3.3)), has curvature ceαx2/4and torsion−βx/2.\nSince the curve Xmc,αassociated with mc,α, andX˜mhave the same curvature and torsion,\nusing fundamental theorem of the local theory of space curves (see e.g. Theorem 1.3.5 in [47]),\nwe conclude that both curves are equal up to direct rigid motion, i.e. there exist p∈R3and\nR∈SO(3)such thatX˜m(x) =R(Xmc,α(x))+p, for allx∈R3. By differentiating this identity,\nwe finally get that ˜m=Rmc,α, which proves the uniqueness of solution, up to a rotation, as\nstated in Proposition 1.2.\n3.2 Asymptotics of the self-similar profile\nThe rest of this section is devoted to establish properties of the family of solutions {mc,α}c,α,\nfor fixedα∈(0,1]andc >0. Due to the self-similar nature of these solutions, this analysis\nreduces to study the properties of the associated profile mc,α, or equivalently, of the solution\n{mc,α,nc,α,bc,α}of the Serret–Frenet system (3.4) with curvature and torsion given in (3.5),\nand initial conditions (3.6).\nIt is important to mention that the recovery of the properties of the trihedron {m,n,b},\nand in particular of the profile m, from the knowledge of its curvature and torsion is a difficult\nquestion. This can be seen from the equivalent formulations of the Serret–Frenet equation in\nterms of a second-order complex-valued highly non-linear EDO, or in terms of a complex-valued\nRiccati equation (see e.g. [11, 50, 42, 32]). For this reason, the integration of the trihedron can\noften only be done numerically, rather than analytically.\nSince the Serret–Frenet equations are decoupled, we start by analyzing the system for the\n12scalarfunctionsmc,α,nc,αandbc,α\n\n\nm/prime\nc,α(x) =ceαx2\n4nc,α(x),\nn/prime\nc,α(x) =−ceαx2\n4mc,α(x)−βx\n2bc,α(x),\nb/prime\nc,α(x) =βx\n2nc,α(x),(3.7)\nwithinitialconditions (mc,α,bc,α,nc,α)(0),thatwesupposeindependentof candα,andsatisfying\nmc,α(0)2+bc,α(0)2+nc,α(0)2= 1.\nThen by ODE theory, the solution is smooth, global and satisfies\nmc,α(x)2+bc,α(x)2+nc,α(x)2= 1,for allx∈R. (3.8)\nMoreover, the solution depends continuously on the parameters c>0andα∈(0,1].\nTo study the behavior of the solution of the system (3.7), we need some elementary bounds\nfor the non-normalized complementary error function.\nLemma 3.1. Letγ∈(0,1]. The following upper bounds hold for x>0\n/integraldisplay∞\nxe−γs2ds≤1\n2γxe−γx2and/integraldisplay∞\nxse−γs2ds=1\n2γe−γx2. (3.9)\nAlso, forγ∈(0,1]andx≥1,\n/integraldisplay∞\nxs2e−γs2ds≤x\nγ2e−γx2,and/integraldisplay∞\nxs3e−γs2ds≤x2\nγ2e−γx2. (3.10)\nProof.We start recalling some standard bounds the complementary error function (see e.g. [1,\n28])\nxe−x2\n2x2+ 1≤/integraldisplay∞\nxe−s2ds≤e−x2\n2x,forx>0. (3.11)\nThe first formula in (3.9) follows by scaling this inequality. The second formula in (3.9) follows\nby integration by parts.\nTo prove the first estimate in (3.10), we use integration by parts and (3.9) to show that\n/integraldisplay∞\nxs2e−γs2ds=xe−γx2\n2γ+1\n2γ/integraldisplay∞\nxe−γs2ds≤e−γx2/parenleftbiggx\n2γ+1\n4γ2x/parenrightbigg\n≤xe−γx2/parenleftbigg1\n2γ+1\n4γ2/parenrightbigg\n,∀x≥1.\nSinceγ∈(0,1], we haveγ2≤γand thus we conclude the estimate for the desired integral. The\nsecond inequality in (3.10) easily follows from the identity\n/integraldisplay∞\nxs3e−γs2ds=1 +γx2\n2γ2e−γx2,∀x∈R,\nnoticing that 1 +γx2≤x2(1 +γ)≤2x2, sincex≥1andγ∈(0,1].\nNow we can state a first result on the behavior of {mc,α,nc,α,bc,α}.\n13Proposition 3.2. Letα∈(0,1]andc>0, and define\nΦα(x) =/integraldisplayx\n0eαs2\n4ds.\nThen the following statements hold.\ni) For allx∈R,\nbc,α(x) =Bc,α+βx\n2ce−αx2/4mc,α(x) +β\n2c/integraldisplay∞\nx/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4mc,α(s)ds,(3.12)\nwhere\nBc,α=bc,α(0)−β\n2c/integraldisplay∞\n0/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4mc,α(s)ds. (3.13)\nIn particular, for all x≥1\n|bc,α(x)−Bc,α|≤6β\ncαxe−αx2/4. (3.14)\nii) Setting wc,α=mc,α+inc,α, for allx∈R, we have\nwc,α(x) =e−icΦα(x)/parenleftBig\nWc,α−βx\n2ceicΦα(x)−αx2/4bc,α(x)\n−β\n2c/integraldisplay∞\nxeicΦα(s)−αs2/4/parenleftbigβs2\n2nc,α(s) +/parenleftbig1−αs2\n2/parenrightbigbc,α(s)/parenrightbigds/parenrightBig\n,(3.15)\nwhere\nWc,α=wc,α(0) +β\n2c/integraldisplay∞\n0eicΦα(s)−αs2/4/parenleftbigβs2\n2nc,α(s) +/parenleftbig1−αs2\n2/parenrightbigbc,α(s)/parenrightbigds.(3.16)\nIn particular, for all x≥1,\n|wc,α(x)−e−icΦα(x)Wc,α|≤10β\ncα2xe−αx2/4. (3.17)\nFurthermore, the limiting values Bc,αandWc,αare separately continuous functions of (c,α)for\n(c,α)∈(0,∞)×(0,1].\nProof.For simplicity, we will drop the subscripts candαif there is no possible confusion. From\n(3.7), we get\nb(x)−b(0) =/integraldisplayx\n0b/prime(s)ds=β\n2c/integraldisplayx\n0se−αs2\n4m/prime(s)ds\n=β\n2c/parenleftBig\nxe−αx2\n4m(x)−/integraldisplayx\n0/parenleftbig1−αs2\n2/parenrightbige−αs2\n4m(s)ds/parenrightBig\n,(3.18)\nwhere we have used integration by parts. Notice that/integraltext∞\n0(1−αs2/2)e−αs2/4m(s)dsis well-\ndefined, since α∈(0,1]andmis bounded. Therefore, the existence of B:= limx→∞b(x)follows\nfrom (3.18). Moreover,\nB:=b(0)−β\n2c/integraldisplay∞\n0/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4m(s)ds.\nFormula (3.12) easily follows from integrating b/primefromx∈Rto∞and arguing as above.\n14To prove (3.14), it is enough to observe that by Lemma 3.1, for x≥1and0<α≤1,\n/integraldisplay∞\nxe−αs2/4ds≤2\nαxe−αx2\n4≤2\nαxe−αx2\n4,and/integraldisplay∞\nxs2e−αs2/4ds≤16\nα2xe−αx2\n4.(3.19)\nSettingw=m+inand using (3.7), we obtain that wsatisfies the ODE\nw/prime+iceαx2/4w=−iβx\n2b(x), (3.20)\nor, equivalently,/parenleftBig\neicΦα(x)w/parenrightBig/prime\n=−iβx\n2b(x)eicΦα(x). (3.21)\nIntegrating (3.21) from 0tox>0, and writing\neicΦα(x)=−i\nc/parenleftBig\neicΦα(x)/parenrightBig/prime\ne−αx2/4,\nintegrating by parts, and using once again (3.7), we get\neicΦα(x)w(x) =w(0)−β\n2cxb(x)eicΦα(x)−αx2/4\n+β\n2c/integraldisplayx\n0eicΦα(s)−αs2/4/parenleftBigβ\n2s2n(s) + (1−αs2\n2)b(s)/parenrightBig\nds.\nSinceα∈(0,1], from the above identity it follows the existence of\nW:= limx→∞eicΦα(x)w(x),\nand formula (3.16) for W.\nFormula (3.15) now follows from integrating (3.21) from x >0to∞and arguing as in the\nprevious lines. The estimate in (3.17) can be deduced as before, since the bounds in (3.19) imply\nthat\n|wc,α(x)−e−icΦα(x)Wc,α|≤β\n2cxe−αx2/4/parenleftbigg\n1 +16(α+β)\n2α2+2\nα/parenrightbigg\n≤10β\ncα2xe−αx2/4,\nwhere we used that α+β≤2andα≤1.\nTo see that the limiting values Bc,αandWc,αgiven by (3.13) and (3.16) are continuous\nfunctions of (c,α), for (c,α)∈(0,∞)×(0,1], we recall that by standard ODE theory, the func-\ntionsmc,α(x),nc,α(x)andbc,α(x)are continuous functions of x,candα. Then, the dominated\nconvergence theorem applied to the formulae (3.13) and (3.16) yield the desired continuity.\nRemark 3.3. As mentioned before, the shrinkers of the 1d-harmonic heat flow can be computed\nexplicitly, because if α= 1, the system (1.5)-(1.6)-(1.7)can be solved easily. Indeed, in this case\nβ= 0, so that we obtain\nmc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0),\nnc,1(x) = (−sin(cΦ1(x)),cos(cΦ1(x)),0),\nbc,1(x) = (0,0,1),\nfor allx∈R.\nIn order to obtain a better understanding of the asymptotic behavior of {mc,α,nc,α,bc,α},\nwe need to exploit the oscillatory character of the function eicΦα(s)in the integrals (3.12) and\n(3.15). In our arguments we will use the following two lemmas.\n15Lemma 3.4. Let0<α≤1. Forσ∈R\\{0}andx∈R, the limit\n/integraldisplay∞\nxseiσΦα(s)ds:= limy→∞/integraldisplayy\nxseiσΦα(s)ds\nexists. Moreover, for all x≥1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nxseiσΦα(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤11x\n|σ|αe−αx2/4, (3.22)\nand/integraldisplay∞\nxseiσΦα(s)ds=ix\nσeiσΦα(x)−αx2/4+O/parenleftBigg\nx2\nσ2e−αx2/2/parenrightBigg\n. (3.23)\nProof.Letx∈Rand takey≥x. Then, integrating by parts,\n/integraldisplayy\nxseiσΦα(s)ds=1\niσ/integraldisplayy\nxs(eiσΦα(s))/primee−αs2/4ds\n=s\niσeiσΦα(s)−αs2/4/vextendsingle/vextendsingle/vextendsingle/vextendsingley\nx−1\niσ/integraldisplayy\nxeiσΦα(s)−αs2/4/parenleftbig1−αs2\n2/parenrightbigds. (3.24)\nThe existence of the improper integral/integraltext∞\nxseiσΦα(s)dsfollows taking the limit as ygoes to∞\nin the above formula, and bearing in mind that α>0. The estimate (3.22) follows from (3.19)\nand the fact that x≥1and0<α≤1. Finally, integrating by parts once more, we have\niσ/integraldisplay∞\nxeiσΦα(s)−αs2/4/parenleftbig1−αs2\n2/parenrightbigds=−eiσΦα(x)−αx2/2/parenleftbig1−αx2\n2/parenrightbig−/integraldisplay∞\nxeiσΦα(s)−αs2/2/parenleftbigα2s3\n2−2αs/parenrightbigds.\nHence, using Lemma 3.1 and (3.24), we obtain (3.23).\nLemma 3.5. Letσ∈R\\{0},γ∈R,α>0and set ˜γ=γ+α/4. If0<˜γ≤1, then forx≥1,\n/integraldisplay∞\nxeiσΦα(s)−γs2ds=O/parenleftBigg\ne−˜γx2\n|σ|/parenrightBigg\n,/integraldisplay∞\nxseiσΦα(s)−γs2ds=O/parenleftBigg\nxe−˜γx2\n|σ|˜γ)/parenrightBigg\n,\n/integraldisplay∞\nxs2eiσΦα(s)−γs2ds=O/parenleftBigg\nx2e−˜γx2\n|σ|˜γ/parenrightBigg\n.(3.25)\nProof.Forn∈{0,1,2}, we set\nIn=/integraldisplay∞\nxsneiσΦα(s)−γs2ds.\nIn=1\niσ/parenleftbigg\n−xneiσΦα(x)−˜γx2−/integraldisplay∞\nxeiσΦα(s)−˜γs2/parenleftBig\nnsn−1−2˜γsn+1/parenrightBig\nds/parenrightbigg\n.\nThen the desired asymptotics follow from Lemma 3.1.\nUsing previous lemmas, we can now improve the asymptotics in Proposition 3.2 and obtain\nexplicitly the term decaying as e−αx2/4(multiplied by a polynomial).\nCorollary 3.6. With the same notation as in Proposition 3.2, the following asymptotics hold\nforx≥1\nbc,α(x) =Bc,α+βx\n2ce−αx2/4Re(e−icΦα(x)Wc,α) +β\nc2α3O(x2e−αx2/2), (3.26)\nwc,α(x) =e−icΦα(x)/parenleftBig\nWc,α−βBc,α\n2cxeicΦα(x)−αx2/4+iβ2Wc,α\n8c/integraldisplay∞\nxs2e−αs2/4ds/parenrightBig\n(3.27)\n+β\nc2α5O(x2e−αx2/2).\n16Proof.As usual, we drop the subscripts candαin the rest of the proof. Recalling that w=\nm+in, we have from (3.17),\nm= Re(e−icΦα(x)W) +β\ncα2O(xe−αx2/4).\nThus, replacing in (3.12),\nb(x) =B+βx\n2ce−αx2/4Re(e−icΦα(x)W) +β2\nc2α2O(x2e−αx2/2) +Rb(x),(3.28)\nwith\nRb(x) =β\n2cRe/parenleftBigg\nW/integraldisplay∞\nx/parenleftbig1−αs2\n2/parenrightbige−icΦα(s)−αs2/4ds+/integraldisplay∞\nx/parenleftbig1−αs2\n2/parenrightbigO/parenleftbigse−αs2/2\ncα2/parenrightbigds/parenrightBigg\n.\nByusingLemmas 3.1and3.5toestimatethefirstandsecondintegrals, respectively, weconclude\nthat\nRb(x) =β\nc2α3O/parenleftbigx2e−αx2/2/parenrightbig. (3.29)\nBy putting together (3.28) and (3.29), we obtain (3.26). To establish (3.27) we integrate (3.21)\nfromx≥1and∞, and use (3.26) and Lemma 3.1 to get\neicΦw(x)−W=I1(x) +I2(x) +I3(x) +β2\nc2α5O(xe−αx2/2), (3.30)\nwith\nI1(x) =iβB\n2/integraldisplay∞\nxseicΦα(s)ds, I 2(x) =iβ2W\n8c/integraldisplay∞\nxs2e−αs2/4ds,and\nI3(x) =iβ2¯W\n8c/integraldisplay∞\nxs2e2icΦα(s)−αs2/4ds,\nwhere we have used that Re(z) = (z+ ¯z)/2. The conclusion follows invoking again Lemmas 3.1,\n3.4 and 3.5.\nIn Figure 2 we depict the first components of the trihedron {mc,α,nc,α,bc,α}forc= 0.5and\nα= 0.5, andx>0. As described in Corollary 3.6 (recall that wc,α=mc,α+inc,α), in the plots\nin Figure 2 one can observe that, while both m1,c,αandb1,c,αoscillate highly for large values of\nx>0, the component b1,c,αconverges to a limit B1,c,α≈−0.72asx→+∞.\n246810\n-1.0-0.50.51.0\n(i)m1,c,α\n (ii)n1,c,α\n246810\n-1.0-0.8-0.6-0.4-0.2 (iii)b1,c,α\nFigure 2: Functions m1,c,α,n1,c,αandb1,c,αforc= 0.5andα= 0.5onR+. The limit at infinity\nin (iii) isB1,c,α≈−0.72.\n173.3 Proof of Theorem 1.3\nFor simplicity, we will drop the subscripts candαin the proof of Theorem 1.3.\nProof of Theorem 1.3. Let{m,n,b}be the solution of the Serret–Frenet system (3.4)–(3.5)\nwith initial condition (3.6). By ODE theory, we have that the solution {m,n,b}is smooth, is\nglobal, and satisfies\n|m(x)|=|n(x)|=|b(x)|= 1,for allx∈R, (3.31)\nand the orthogonality relations\nm(x)·n(x) =m(x)·b(x) =n(x)·b(x) = 0,for allx∈R. (3.32)\nDefine\nm(x,t) =m/parenleftbiggx√\nT−t/parenrightbigg\n, t<T.\nThen, it is straightforward to check that mis a smooth solution of the profile equation (3.1),\nand consequently msolves (LLG) for t∈(−∞,T). Moreover, using the Serret–Frenet system\n(3.4)–(3.5), we have\n|∂xm(x,t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√\nT−tm/prime/parenleftbiggx√\nT−t/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=c√\nT−teαx2\n4(T−t).\nThis shows part (i).\nNotice also that, since {mj,nj,bj}forj∈{1,2,3}solves (3.7) with initial condition (1,0,0),\n(0,1,0)and(0,0,1)respectively, the relation (3.8) is satisfied, i.e.\nm2\nj(x) +n2\nj(x) +b2\nj(x) = 1,for allx∈R, j = 1,2,3. (3.33)\nPart(ii)follows from the uniqueness given by the Cauchy–Lipschitz theorem for the solution\nof (3.1)–(3.5) with initial condition (3.6), and the invariance of (3.1)–(3.5)–(3.6) under the\ntransformations\nm(x) = (m1(x),m2(x),m3(x))→(m1(−x),−m2(−x),−m3(−x)),\nn(x) = (n1(x),n2(x),n3(x))→(−n1(−x),n2(−x),n3(−x)),\nb(x) = (b1(x),b2(x),b3(x))→(−b1(−x),b2(−x),b3(−x)),\nSettingWj=ρjeiφj, withρj≥0andφj∈[0,2π], the asymptotics for mjand it derivative\nm/prime\nj, forj∈{1,2,3}in part(iii)are a direct consequence of the asymptotic behavior of the\nprofile established in Proposition 3.2 and Corollary 3.6 (recall also that m/prime(x) =ceαx2\n4n(x)and\nw=m+in). In particular the following limits exist:\nWj:= limx→∞eicΦα(x)(mj+inj)(x), Bj:= limx→∞bj(x).\nNotice also that the relations in (3.33) implies that ρ2\nj+B2\nj= 1, and the fact that bis unitary\nimplies that B2\n1+B2\n2+B2\n3= 1, and thatρ2\n1+ρ2\n2+ρ2\n3= 2.\nThe convergence in part (iv)is an immediate consequence of the definition of min terms of\nthe profilem, and the asymptotics established in part (iii). The relations between ρ−\njandρj,\nforj∈{1,2,3}follow from the parity relations for the profile mestablished in part (ii).\nIt remains to prove part (v). Forϕ∈W1,∞(R), bearing in mind (3.15) and (3.17), it suffices\nto show that\nlim\nt→0+/integraldisplay\nRe−icΦα(x/√\nt)ϕ(x)dx= 0,and lim\nt→0+/integraldisplay\nR/parenleftbig1 +|x|√\nt/parenrightbige−αx2\n4t|ϕ(x)|dx= 0.(3.34)\n18The second limit is a direct consequence of following the explicit computations:\n/integraldisplay\nRe−αx2\n4tdx=/radicalbiggπt\nα,and/integraldisplay\nR|x|e−αx2\n4tdx=4t\nα. (3.35)\nFor the first limit in (3.34), we integrate by parts to obtain\nic/integraldisplay\nRe−icΦα(x/√\nt)ϕ(x)dx=−√\nt/integraldisplay\nR/parenleftbige−icΦα(x/√\nt))/primee−αx2\n4tϕ(x)dx\n=/integraldisplay\nRe−icΦα(x/√\nt)−αx2\n4t/parenleftbigg√\ntϕ/prime(x)−αx\n2√\ntϕ(x)/parenrightbigg\ndx.\nSince|ϕ|and|ϕ/prime|are bounded on R, the conclusion follows again by using (3.35).\n4 Limiting behaviour of the trajectories of the profiles\nIn this section we prove Theorem 1.4. In what follows we denote C±\nc,αthe great circleCc,α=\nP±\nc,α∩S2, withP±\nc,αbeing the planes passing through the origin with (unitary) normal vectors\nB+\nc,α= (B1,c,α,B2,c,α,B3,c,α)andB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α),\ngiven by Theorem 1.3.\nWe will show that the trajectories of the profiles mc,αconverge to the great circles C±\nc,αas\nx→±∞, and study the behavior of the limit circles C±\nc,αwith respect to the parameters cand\nα, analyzing the angle ϑc,α∈[0,π]between their normal vectors B+\nc,αandB−\nc,α.\nWe start this section stating a corollary of Theorem 1.3 that will be used in what follows.\nPrecisely, recalling that w=m+in, from (3.17) and using the constants defined in Theorem 1.3,\nwe have the following\nCorollary 4.1. Forx≥1andj∈{1,2,3}, we have\nmj,c,α(x) =ρj,c,αcos(cΦα(x)−φj) +Rj(x), nj,c,α(x) =−ρj,c,αsin(cΦα(x)−φj) +˜Rj(x),\nfor some functions Rjand ˜Rjsatisfying the bounds |Rj(x)|,|˜Rj(x)|≤10β/(cα2)xe−αx2/4.\nWe will also use the two lemmas below in the proof of Theorem 1.4. The first one establishes\nsome relations between the constants appearing in the asymptotics of the profile mc,αthat can\nbe deduced by using some geometric properties of the Serret–Frenet system.\nLemma 4.2. The constants given by Theorem 1.3 satisfy the following identities\nB1,c,α=ρ2,c,αρ3,c,αsin(φ3,c,α−φ2,c,α), B2,c,α=ρ1,c,αρ3,c,αsin(φ1,c,α−φ3,c,α),\nB3,c,α=ρ1,c,αρ2,c,αsin(φ2,c,α−φ1,c,α),(4.1)\nand\nB1,c,αρ1,c,αeiφ1,c,α+B2,c,αρ2,c,αeiφ2,c,α+B3,c,∞ρ3,c,αeiφ3,c,α= 0, (4.2)\nρ2\n1,c,αe2iφ1,c,α+ρ2\n2,c,αe2iφ2,c,α+ρ2\n3,c,αe2iφ3,c,α= 0. (4.3)\n19Proof.Dropping the subscripts candα, using the relation b=m×nand the asymptotics in\nCorollary 4.1, we get for x≥1,\nb1(x) =−ρ2ρ3/parenleftbigcos(cΦα(x)−φ2) sin(cΦα(x)−φ3)−cos(cΦα(x)−φ3) sin(cΦα(x)−φ2)/parenrightbig+o(1),\nb2(x) =−ρ3ρ1/parenleftbigcos(cΦα(x)−φ3) sin(cΦα(x)−φ1)−cos(cΦα(x)−φ1) sin(cΦα(x)−φ3)/parenrightbig+o(1),\nb3(x) =−ρ1ρ2/parenleftbigcos(cΦα(x)−φ1) sin(cΦα(x)−φ2)−cos(cΦα(x)−φ2) sin(cΦα(x)−φ1),/parenrightbig+o(1),\nwhereo(1)is a function of x, which depends on the parameters αandc, that converges to 0as\nx→∞. Noticing that, for k,j∈{1,2,3},\n−cos(cΦα(x)−φj) sin(cΦα(x)−φk) + cos(cΦα(x)−φk) sin(cΦα(x)−φj) = sin(φk−φj),\nwe obtain\nb1(x) =ρ2ρ3sin(φ3−φ2) +o(1),\nb2(x) =ρ1ρ3sin(φ1−φ3) +o(1),\nb3(x) =ρ1ρ2sin(φ2−φ1) +o(1).\nLettingx→∞, in the above identities we obtain (4.1).\nTo establish the other identities, we first recall that the vectors m,nandbsatisfy the\nfollowing relations\nm·n=n·b=b·m= 0and|m|=|n|=|b|= 1.\nNow, from the asymptotics in Corollary 4.1 and the identity\n(m+in)·b= 0,\nwe have\nb1(x)ρ1e−i(cΦα(x)−φ1)+b2(x)ρ2e−i(cΦα(x)−φ2)+b3(x)ρ3e−i(cΦα(x)−φ3)=o(1),\nfori∈{1,2,3}. Dividing by e−icΦα(x)and letting x→∞, we obtain formula (4.2). Finally,\nformula (4.3) follows easily using a similar argument, bearing in mind the orthogonality relation\nm·n= 0,and that 2 cos(y) sin(y) = Im(e2iy), fory∈R.\nRemark 4.3. Although we do not use (4.3)in this work, this relation could be helpful in estab-\nlishing further properties of the solutions.\nNext we study the angle between the normal vectors to the great circles C±\nc,α, given by\nϑc,α= arccos(2B2\n1,c,α−1),withϑc,α∈[0,π]. We have the following.\nLemma 4.4. Forc≥β√π/√α, we have\nϑc,α≥arccos/parenleftBigg\n−1 +2πβ2\nc2α/parenrightBigg\n. (4.4)\nProof.Using the formula (3.12) in Proposition 3.2 for b1,c,αwithx= 0, we get\n|b1,c,α(0)−B1,c,α|≤β\n2c/integraldisplay∞\n0/parenleftBigg\n1 +αs2\n2/parenrightBigg\ne−αs2/4.\n20Noticing that b1,c,α(0) = 0, and that\n/integraldisplay∞\n0/parenleftBigg\n1 +αs2\n2/parenrightBigg\ne−αs2/4=2√π√α,\nwe conclude that\n2B2\n1,c,α≤2πβ2\nc2α.\nSincec≥β√π/√α, we have−1 + 2πβ2/(c2α)∈[−1,1], so we can use that the function arccos\nis decreasing to obtain (4.4).\nWe continue to prove Theorem 1.4.\nProof of Theorem 1.4. As usual, we omit the subscripts candαwhen there is no confusion.\nIn view of the symmetries established in Theorem 1.3, it is enough to prove the theorem for\nx→∞. We start noticing that |Bc,α|= 1, so that the distance between mandP+is given by\ndist(m(x),P+) =|m1(x)B1+m2(x)B2+m3(x)B3|.\nTo compute the leading term, we notice that using (4.2), we have\n3/summationdisplay\nj=1ρjBjcos(cΦα(x)−φj) = Re/parenleftBigg\neicΦα(x)/parenleftBigg3/summationdisplay\nj=1ρjBje−iφj/parenrightBigg/parenrightBigg\n= 0.\nThus the estimates in Corollary 4.1 give us\ndist(m(x),P+))≤30β\ncα2xe−αx2/4,forx≥1, (4.5)\nLet us fix x≥1and letQ(x)be the orthogonal projection of m(x)on the planeP+, so\nthat dist(m(x),P+) = dist(m(x),Q(x)). We also set C(x)∈C+such that dist(m(x),C+) =\ndist(m(x),C(x)). By the Pythagorean theorem,\n|Q(x)|2= 1−dist(m(x),Q(x))2\nand\ndist(m(x),C(x))2= dist(m(x),Q(x))2+ (1−|Q(x)|)2.\nTherefore\ndist(m(x),C(x))2= 2−2(1−dist(m(x),Q(x))2)1/2,\nand using the elementary inequality√1−y≥1−y, fory∈[0,1], we get\ndist(m(x),C(x))≤√\n2 dist(m(x),Q(x)).\nCombing this estimate with (4.5), we conclude that\ndist(m(x),C+)) = dist(m(x),C(x)))≤√\n2 dist(m(x),P+))≤30√\n2β\ncαxe−αx2/4.\nThe limits in (1.13) follow at once using the definition of mc,αin (1.8) (recall that ϑc,α∈[0,π]).\nThe statement in (ii)is an immediate consequence of Lemma 4.4. This finishes the proof of\nTheorem 1.4.\n21The limit limc→∞ϑc,α=πin part (ii) of Theorem 1.4 helps us to understand the angle\nbetween the great circles as c→∞. However, the behavior of the limit circles C±\nc,αforcsmall\nis much more involved. We conclude this section with some reflections on the behavior of the\nlimit circlesC±\nc,αwhencis small.\nLet us remark that when c= 0, the explicit solution to (1.6)–(1.7) is given by\nm0,α(x) = (1,0,0),\nn0,α(x) = (0,cos(βx2/4),−sin(βx2/4)),\nb0,α(x) = (0,sin(βx2/4),cos(βx2/4)).(4.6)\nThus we see that in this limit case, there is a change in the behavior of the solution: There is\nno limit circle and the vector b0,αdoes not have a limit at infinity. On the other hand, we know\nthat (mc,α(x),nc,α(x),bc,α(x))are continuous with respect to the to c,αandx. Therefore,\nlim\nc→0b1,c,α(x) =b1,0,α(x) = 0,for allx∈R. (4.7)\nOf course, we cannot conclude from (4.7) estimates for B1,c,α, ascgoes to 0. For this reason,\nwe performed some numerical simulations for different values of csmall. In Figures 3 and 4, we\nshow one of these simulations, in the case α= 0.5andc= 0.01, where we see that m1,c,α≈1\nandb1,c,α≈0on[0,2]in agreement with (4.6) and the continuous dependence on c. On the\nother hand, for x≥8.5, we haveb1,c,α(x)≈−1.\nHowever, it seems difficult to infer from our simulations the behavior of Bc,αascgoes to zero.\nFor instance, we have computed numerically Bc,α, and it is not clear that this quantity converges\nforcsmall. Forinstance, wehaveobtained B1,c,α=−0.99215, forc= 10−12,B1,c,α=−0.992045,\nforc= 10−14, andB1,c,α=−0.991965, forc= 10−16.\n2468100.20.40.60.81.0\n(i)m1,c,α\n246810\n-1.0-0.8-0.6-0.4-0.2 (ii)b1,c,α\nFigure 3: Functions m1,c,αandb1,c,αforc= 0.01andα= 0.5. The limit at infinity in (ii) is\nB1,c,α≈−0.996417.\nAcknowledgements. S. Gutiérrez was partially supported by ERCEA Advanced Grant 2014\n669689 - HADE. The Université de Lille also supported S. Gutiérrez’s research visit during July\n2018 through their Invited Research Speaker Scheme. 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Phys. 13, 448 (2017)] on magnetic-\feld-driven annihilation of two magnetic\ndomain walls (DWs) but, furthermore, we predict that this setup additionally generates highly un-\nusual pumping of electronic spin currents in the absence of any bias voltage. Prior to the instant of\nannihilation, their power spectrum is ultrabroadband , so they can be converted into rapidly changing\nin time charge currents, via the inverse spin Hall e\u000bect, as a source of THz radiation of bandwidth\n'27 THz where the lowest frequency is controlled by the applied magnetic \feld. The spin pump-\ning stems from time-dependent \felds introduced into the quantum Hamiltonian of electrons by the\nclassical dynamics of localized magnetic moments (LMMs) comprising the domains. The pumped\ncurrents carry spin-polarized electrons which, in turn, exert backaction on LMMs in the form of\nnonlocal damping which is more than twice as large as conventional local Gilbert damping. The\nnonlocal damping can substantially modify the spectrum of emitted SWs when compared to widely-\nused micromagnetic simulations where conduction electrons are completely absent . Since we use fully\nmicroscopic (i.e., Hamiltonian-based) framework, self-consistently combining time-dependent elec-\ntronic nonequilibrium Green functions with the Landau-Lifshitz-Gilbert equation, we also demon-\nstrate that previously derived phenomenological formulas miss ultrabroadband spin pumping while\nunderestimating the magnitude of nonlocal damping due to nonequilibrium electrons .\nIntroduction .|The control of the domain wall (DW)\nmotion1{3within magnetic nanowires by magnetic \feld\nor current pulses is both a fundamental problem for\nnonequilibrium quantum many-body physics and a build-\ning block of envisaged applications in digital memories.4\nlogic5and arti\fcial neural networks.6Since DWs will be\nclosely packed in such devices, understanding interaction\nbetween them is a problem of great interest.7For ex-\nample, head-to-head or tail-to-tail DWs|illustrated as\nthe left (L) or right (R) noncollinear texture of local-\nized magnetic moments (LMMs), respectively, in Fig. 1|\nbehave as free magnetic monopoles carrying topological\ncharge.8The topological charge (or the winding number)\nQ\u0011\u00001\n\u0019R\ndx@x\u001e, associated with winding of LMMs as\nthey interpolate between two uniform degenerate ground\nstates with \u001e= 0 or\u001e=\u0019, is opposite for adjacent\nDWs, such as QL=\u00001 andQR= +1 for DWs in Fig. 1.\nThus, long-range attractive interaction between DWs\ncan lead to their annihilation , resulting in the ground\nstate without any DWs.9{12This is possible because to-\ntal topological charge remains conserved, QL+QR= 0.\nThe nonequilibrium dynamics13triggered by annihilation\nof topological solitons is also of great interest in many\nother \felds of physics, such as cosmology,14gravitational\nwaves,15quantum13and string \feld16theories, liquid\ncrystals17and Bose-Einstein condensates.18,19\nThe recent experiment20has monitored annihilation\nof two DWs within a metallic ferromagnetic nanowire by\nobserving intense burst of spin waves (SWs) at the mo-\nment of annihilation. Thus generated large-amplitude\nSWs are dominated by exchange, rather than dipolar,\ninteraction between LMMs and are, therefore, of short\nwavelength. The SWs of \u001810 nm wavelength are cru-cial for scalability of magnonics-based technologies,21,22\nlike signal transmission or memory-in-logic and logic-\nin-memory low-power digital computing architectures.\nHowever, they are di\u000ecult to excite by other methods\ndue to the requirement for high magnetic \felds.23,24\nThe computational simulations of DW annihila-\ntion,9,10,20together with theoretical analysis of generic\nfeatures of such a phenomenon,11have been based exclu-\nsively on classical micromagnetics where one solves cou-\npled Landau-Lifshitz-Gilbert (LLG) equations25for the\ndynamics of LMMs viewed as rotating classical vectors of\n\fxed length. On the other hand, the dynamics of LMMs\ncomprising two DWs also generates time-dependent \felds\nwhich will push the surrounding conduction electrons out\nof equilibrium. The nonequilibrium electrons comprise\npumped spin current26{28(as well as charge currents if\nthe left-right symmetry of the device is broken28,29) in\nthe absence of any externally applied bias voltage. The\npumped spin currents \row out of the DW region into\nthe external circuit, and since they carry away excess an-\ngular momentum of precessing LMMs, the backaction of\nnonequilibrium electrons on LMMs emerges26as an ad-\nditional damping-like (DL) spin-transfer torque (STT).\nThe STT, as a phenomenon in which spin angu-\nlar momentum of conduction electrons is transferred\nto LMMs when they are not aligned with electronic\nspin-polarization, is usually discussed for externally in-\njected spin current.30But here it is the result of compli-\ncated many-body nonequilibrium state in which LMMs\ndrive electrons out of equilibrium which, in turn, ex-\nertbackaction in the form of STT onto LMMs to\nmodify their dynamics in a self-consistent fashion.27,31\nSuch e\u000bects are absent from classical micromagneticsarXiv:1908.03194v5  [cond-mat.mes-hall]  24 Jun 20212\nFIG. 1. Schematic view of a ferromagnetic nanowire modeled\nas a 1D tight-binding chain whose sites host classical LMMs\n(red arrows) interacting with spins (blue arrow) of conduc-\ntion electrons. The nanowire is attached to two NM leads\nterminating into the macroscopic reservoirs kept at the same\nchemical potential. The two DWs within the nanowire carry\nopposite topological charge,8QL=\u00001 for the L one and\nQR= +1 for the R one. They collide with the opposite ve-\nlocities VL\nDWandVR\nDWand annihilate, upon application of an\nexternal magnetic \feld Bextparallel to the nanowire, thereby\nmimicking the setup of the experiment in Ref. 20.\nor atomistic spin dynamics25because they do not in-\nclude conduction electrons. This has prompted deriva-\ntion of a multitude of phenomenological expressions32{39\nfor the so-called nonlocal (i.e., magnetization-texture-\ndependent) and spatially nonuniform (i.e., position-\ndependent) Gilbert damping that could be added into\nthe LLG equation and micromagnetics codes40{42to cap-\nture the backaction of nonequilibrium electrons while not\nsimulating them explicitly. Such expressions do not re-\nquire spin-orbit (SO) or magnetic disorder scattering,\nwhich are necessary for conventional local Gilbert damp-\ning,43{45but they were estimated33,36to be usually a\nsmall e\u000bect unless additional conditions (such as narrow\nDWs or intrinsic SO coupling splitting the band struc-\nture33) are present. On the other hand, a surprising\nresult40of Gilbert damping extracted from experiments\non magnetic-\feld-driven DW being several times larger\nthan the value obtained from standard ferromagnetic res-\nonance measurements can only be accounted by including\nadditional nonlocal damping.\nIn this Letter, we unravel complicated many-body\nnonequilibrium state of LMMs and conduction elec-\ntrons created by DW annihilation using recently de-\nveloped27,46{49quantum-classical formalism which com-\nbines time-dependent nonequilibrium Green function\n(TDNEGF)50,51description of quantum dynamics of con-\nduction electrons with the LLG equation description of\nclassical dynamics of LMMs on each atom.25Such TD-\nNEGF+LLG formalism is fully microscopic, since it re-\nquires only the quantum Hamiltonian of electrons and the\nclassical Hamiltonian of LMMs as input, and numerically\nexact . We apply it to a setup depicted in Fig. 1 where\ntwo DWs reside at time t= 0 within a one-dimensional\n(1D) magnetic nanowire attached to two normal metal\n(NM) leads, terminating into the macroscopic reservoirs\nwithout any bias voltage.\nOur principal results are: ( i) annihilation of two DWs\n[Fig. 2] pumps highly unusual electronic spin currents\nwhose power spectrum is ultrabroadband prior to the in-\nFIG. 2. (a) Sequence of snapshots of two DWs, in the course\nof their collision and annihilation in the setup of Fig. 1; and\n(b) the corresponding time-dependence of the z-component\nof LMMs where blue and orange line mark t= 6:9 ps (when\ntwo DWs start vanishing) and t= 7:2 ps (when all LMMs\nbecome nearly parallel to the x-axis) from panel (a). A movie\nanimating panels (a) and (b) is provided in the SM.58Spatio-\ntemporal pro\fle of: (c) angle \u000eeq\niand (d) \\nonadiabaticity\"\nangle\u000eneq\ni\u0000\u000eeq\ni, with the meaning of \u000eneq\niand\u000eeq\niillustrated in\nthe inset above panel (c); (e) DL STT [Eq. (3)] as electronic\nbackaction on LMMs; (f) ratio of DL STT to conventional\nlocal Gilbert damping [Eq. (2)]; and (g) ratio of the sum of\nDL STT to the sum of conventional local Gilbert damping\nover all LMMs.\nstant of annihilation [Fig. 3(d)], unlike the narrow peak\naround a single frequency for standard spin pumping;26\n(ii) because pumped spin currents carry away excess\nangular momentum of precessing LMMs, this acts as\nDL STT on LMMs which is spatially [Figs. 2(e) and\n4(b)] and time [Fig. 2(g)] dependent, as well as '2:4\ntimes larger [Fig. 2(f)] than conventional local Gilbert\ndamping [Eq. (2)]. This turns out to be remarkably\nsimilar to'2:3 ratio of nonlocal and local Gilbert\ndamping measured experimentally in permalloy,40but\nit is severely underestimated by phenomenological the-\nories32,33[Fig. 4(a),(b)].\nModels and methods .|The classical Hamiltonian for3\n≃ 27 THz\nFIG. 3. Time dependence of: (a){(c) electronic spin currents pumped into the right NM lead during DW collision and annihila-\ntion; (e){(g) SW-generated contribution to spin currents in panels (a){(c), respectively, after spin current carried by SW from\nFig. 2(b) is stopped at the magnetic-nanowire/nonmagnetic-NM-lead interface and converted (as observed experimentally20,61)\ninto electronic spin current in the right NM lead. Vertical dashed lines mark times t= 6:9 ps andt= 7:2 ps whose snapshots\nof LMMs are shown in Fig. 2(a). For easy comparison, gray curves in panels (f) and (g) are the same as the signal in panels (b)\nand (c), respectively, for post-annihilation times t\u00157:2 ps. Panels (d) and (h) plot FFT power spectrum of signals in panels\n(c) and (g), respectively, before (red curve) and after (brown curves) completed annihilation at t= 7:2 ps.\nLMMs, described by unit vectors Mi(t) at each site iof\n1D lattice, is chosen as\nH=\u0000JX\nhijiMi\u0001Mj\u0000KX\ni(Mx\ni)2\n+DX\ni(My\ni)2\u0000\u0016BX\niMi\u0001Bext; (1)\nwhereJ= 0:1 eV is the Heisenberg exchange coupling\nbetween the nearest-neighbor LMMs; K= 0:05 eV is the\nmagnetic anisotropy along the x-axis; andD= 0:007 eV\nis the demagnetizing \feld along the y-axis. The last term\nin Eq. (1) is the Zeeman energy ( \u0016Bis the Bohr magne-\nton) describing the interaction of LMMs with an external\nmagnetic \feld Bextparallel to the nanowire in Fig. 1 driv-\ning the DW dynamics, as employed in the experiment.20\nThe classical dynamics of LMMs is described by a system\nof coupled LLG equations25(using notation @t\u0011@=@t)\n@tMi=\u0000gMi\u0002Be\u000b\ni+\u0015Mi\u0002@tMi\n+g\n\u0016M\u0010\nTih\nIS\u000b\nexti\n+Ti[Mi(t)]\u0011\n: (2)\nwhere Be\u000b\ni=\u00001\n\u0016M@H=@Miis the e\u000bective magnetic\n\feld (\u0016Mis the magnitude of LMMs); gis the gy-\nromagnetic ratio; and the magnitude of conventional\nlocal Gilbert damping is speci\fed by spatially- and\ntime-independent \u0015, set as\u0015= 0:01 as the typi-\ncal value measured40in metallic ferromagnets. The\nspatial pro\fle of a single DW in equilibrium, i.e.,\nat timet= 0 as the initial condition, is given by\nMi(Q;X DW) =\u0000\ncos\u001ei(Q;X DW);0;sin\u001ei(Q;X DW)\u0001\n,\nwhere\u001ei(Q;X DW) =Qarccos [tanh ( xi\u0000XDW)];Q\nis the topological charge; and XDW is the positionof the DW. The initial con\fguration of two DWs,\nMi(t= 0) = Mi(QL;XL) +Mi(QR;XR), positioned at\nsitesXL= 15 andXR= 30 harbors opposite topological\nchargesQR=\u0000QL= 1 around these sites.\nIn general, two additional terms32,33,52in Eq. (2) ex-\ntend the original LLG equation|STT due to externally\ninjected electronic spin current,30which is actually ab-\nsentTih\nIS\u000b\nexti\n\u00110 in the setup of Fig. 1; and STT due to\nbackaction of electrons\nTi[Mi(t)] =Jsd(h^siineq(t)\u0000h^siieq\nt)\u0002Mi(t); (3)\ndriven out of equilibrium by Mi(t). HereJsd= 0:1 eV\nis chosen as the s-dexchange coupling (as mea-\nsured in permalloy53) between LMMs and electron\nspin. We obtain \\adiabatic\"54,55electronic spin density,\nh^siieq\nt= Tr [ \u001aeq\ntjiihij\n\u001b], from grand canonical equilib-\nrium density matrix (DM) for instantaneous con\fgura-\ntion of Mi(t) at timet[see Eq. (5)]. Here \u001b= (^\u001bx;^\u001by;^\u001bz)\nis the vector of the Pauli matrices. The nonequilibrium\nelectronic spin density, h^siineq(t) = Tr [ \u001aneq(t)jiihij\n\u001b],\nrequires to compute time-dependent nonequilibrium DM,\n\u001aneq(t) =~G<(t;t)=i, which we construct using TD-\nNEGF algorithms explained in Refs. 56 and 57 and com-\nbine27with the classical LLG equations [Eq. (2)] using\ntime step\u000et= 0:1 fs. The TDNEGF calculations require\nas an input a quantum Hamiltonian for electrons, which\nis chosen as the tight-binding one\n^H(t) =\u0000\rX\nhiji^cy\ni^ci\u0000JsdX\ni^cy\ni\u001b\u0001Mi(t)^ci: (4)\nHere ^cy\ni= (^cy\ni\";^cy\ni#) is a row vector containing operators\n^cy\ni\u001bwhich create an electron of spin \u001b=\";#at the sitei,4\nand ^ciis a column vector that contains the correspond-\ning annihilation operators; and \r= 1 eV is the nearest-\nneighbor hopping. The magnetic nanowire in the setup\nin Fig. 1 consists of 45 sites and it is attached to semi-\nin\fnite NM leads modeled by the \frst term in ^H. The\nFermi energy of the reservoirs is set at EF= 0 eV. Due\nto the computational complexity of TDNEGF calcula-\ntions,51we use magnetic \feld jBextj= 100 T to complete\nDW annihilation on \u0018ps time scale (in the experiment20\nthis happens within \u00182 ns).\nResults .|Figure 2(a) demonstrates that\nTDNEGF+LLG-computed snapshots of Mi(t)fully\nreproduce annihilation in the experiment,20including \f-\nnalewhen SW burst is emitted at t'7:2 ps in Fig. 2(b).\nThe corresponding complete spatio-temporal pro\fles\nare animated as a movie provided in the Supplemental\nMaterial (SM).58However, in contrast to micromagnetic\nsimulations of Ref. 20 where electrons are absent,\nFig. 2(d) shows that they generate spin density h^siineq(t)\nwhich is noncollinear with either Mi(t) orh^siieq\nt. This\nleads to \\nonadiabaticity\" angle ( \u000eneq\ni\u0000\u000eeq\ni)6= 0 in\nFig. 2(d) and nonzero STT [Eq. (3) and Fig. 2(e)] as\nself-consistent backaction of conduction electrons onto\nLMMs driven out of equilibrium by the dynamics of\nLMMs themselves. The STT vector, Ti=TFL\ni+TDL\ni,\ncan be decomposed [see inset above Fig. 2(e)] into: ( i)\neven under time-reversal or \feld-like (FL) torque, which\na\u000bects precession of LMM around Be\u000b\ni; and ( ii) odd\nunder time-reversal or DL torque, which either enhances\nGilbert term [Eq. (2)] by pushing LMM toward Be\u000b\nior\ncompetes with it as antidamping. Figure 2(f) shows that\nTDL\ni[Mi(t)] acts like an additional nonlocal damping\nwhile being'2:4 times larger than conventional local\nGilbert damping \u0015Mi\u0002@tMi[Eq. (2)].\nThe quantum transport signature of DW vanishing\nwithin the time interval t= 6:9{7:2 ps in Fig. 2(a) is the\nreduction in the magnitude of pumped electronic spin\ncurrents [Fig. 3(a){(c)]. In fact, ISx\nR(t)!0 becomes\nzero [Fig. 3(a)] at t= 7:2 ps at which LMMs in Fig. 2(a)\nturn nearly parallel to the x-axis while precessing around\nit. The frequency power spectrum [red curve in Fig. 3(d)]\nobtained from fast Fourier transform (FFT) of ISz\nR(t), for\ntimes prior to completed annihilation and SW burst at\nt= 7:2 ps, reveal highly unusual spin pumping over an\nultrabroadband frequency range. This can be contrasted\nwith the usual spin pumping26whose power spectrum\nis just a peak around a single frequency,59as also ob-\ntained [brown curve in Fig. 3(d)] by FFT of ISz\nR(t) at\npost-annihilation times t>7:2 ps.\nThe spin current in Fig. 3(a){(c) has contributions\nfrom both electrons moved by time-dependent Mi(t) and\nSW hitting the magnetic-nanowire/NM-lead interface.\nAt this interface, SW spin current is stopped and trans-\nmuted47,48,60into an electronic spin current \rowing into\nthe NM lead. The transmutation is often employed ex-\nperimentally for direct electrical detection of SWs, where\nan electronic spin current on the NM side is converted\ninto a voltage signal via the inverse spin Hall e\u000bect.20,61\n~10-7FIG. 4. Spatial pro\fle at t= 6:9 ps of: (a) locally pumped\nspin current ISx\ni!j47between sites iandj; and nonlocal damp-\ning due to backaction of nonequilibrium electrons . Solid lines\nin (a) and (b) are obtained from TDNEGF+LLG calcula-\ntions, and dashed lines are obtained from SMF theory phe-\nnomenological formulas.32,33,69(c){(e) FFT power spectra22\nofMz\ni(t) where (c) and (d) are TDNEGF+LLG-computed\nwith\u0015= 0:01 and\u0015= 0, respectively, while (e) is LLG-\ncomputed with backaction of nonequilibrium electrons re-\nmoved, Ti[Mi(t)]\u00110, in Eq. (2). The dashed horizontal\nlines in panels (c){(e) mark frequencies of peaks in Fig. 3(d).\nWithin the TDNEGF+LLG picture, SW reaching the\nlast LMM of the magnetic nanowire, at the sites i= 1\nori= 45 in our setup, initiates their dynamics whose\ncoupling to conduction electrons in the neighboring left\nand right NM leads, respectively, leads to pumping47of\nthe electronic spin current into the NM leads. The prop-\nerly isolated electronic spin current due to transmutation\nof SW burst, which we denote by IS\u000b;SW\np , is either zero\nor very small until the burst is generated in Fig. 3(e){\n(g), as expected. We note that detected spin current in\nthe NM leads was attributed in the experiment20solely\nto SWs, which corresponds in our picture to considering\nonlyIS\u000b;SW\np while disregarding IS\u000bp\u0000IS\u000b;SW\np .\nDiscussion .|A computationally simpler alternative to\nour numerical self-consistent TDNEGF+LLG is to \\in-\ntegrate out electrons\"31,62{65and derive e\u000bective expres-\nsions solely in terms of Mi(t), which can then be added\ninto the LLG Eq. (2) and micromagnetics codes.40{42\nFor example, spin motive force (SMF) theory69gives\nISx\nSMF(x) =g\u0016B~G0\n4e2[@M(x;t)=@t\u0002@M(x;t)=@x]xfor the\nspin current pumped by dynamical magnetic texture, so\nthatM\u0002D\u0001@tMis the corresponding nonlocal Gilbert\ndamping.32,33Here M(x;t) is local magnetization (as-\nsuming our 1D system); D\u000b\f=\u0011P\n\u0017(M\u0002@\u0017M)\u000b(M\u0002\n@\u0017M)\f(using notation \u000b;\f;\u00172 fx;y;zg) is 3\u00023\nspatially-dependent damping tensor; and \u0011=g\u0016B~G0\n4e2\nwithG0=G\"+G#being the total conductivity. We\ncompare in Fig. 4: ( i) spatial pro\fle of ISx\nSMF(x) to locally\npumped spin current ISx\ni!j47from TDNEGF+LLG calcu-5\nlations [Fig. 4(a)] to \fnd that the former predicts negli-\ngible spin current \rowing into the leads, thereby missing\nultrabroadband spin pumping predicted in Fig. 3(d); ( ii)\nspatial pro\fle of M\u0002D\u0001@tMto DL STT TDL\nifrom TD-\nNEGF+LLG calculations, to \fnd that the former has\ncomparable magnitude only within the DW region but\nwith substantially di\u000bering pro\fles. Note also that47\n[P\niTi(t)]\u000b=~\n2eh\nIS\u000b\nL(t) +IS\u000b\nR(t)i\n+P\ni~\n2@h^ s\u000b\niineq\n@t, which\nmakes the sum of DL STT plotted in Fig. 2(g) time-\ndependent during collision, in contrast to the sum of lo-\ncal Gilbert damping shown in Fig. 2(g). The backaction\nof nonequilibrium electrons viaTi[Mi(t)] can strongly\na\u000bect the dynamics of LMMs, especially for the case of\nshort wavelength SWs and narrow DWs,32,33,41,42as con-\n\frmed by comparing FFT power spectra of Mz\ni(t) com-\nputed by TDNEGF+LLG [Fig. 4(c),(d)] with those from\nLLG calculations [Fig. 4(e)] without any backaction .\nWe note that SMF theory69is derived in the \\adi-\nabatic\" limit,2,54which assumes that electron spin re-\nmains in the the lowest energy state at each time. \\Adi-\nabaticity\" is used in two di\u000berent contexts in spintron-\nics with noncollinear magnetic textures|temporal and\nspatial.2In the former case, such as when electrons in-\nteract with classical macrospin due to collinear LMMs,\none assumes that classical spins are slow and h^siineq(t)\ncan \\perfectly lock\"2to the direction Mi(t) of LMMs.\nIn the latter case, such as for electrons traversing thick\nDW, one assumes that electron spin keeps the lowest en-\nergy state by rotating according to the orientation of\nMi(t) at each spatial point, thereby evading re\rection\nfrom the texture.2The concept of \\adiabatic\" limit is\nmade a bit more quantitative by considering2ratio of\nrelevant energy scales, Jsd=~!\u001d1 orJsd=\u0016BjBextj\u001d1,\nin the former case; or combination of energy and spa-\ntial scales,JsddDW=~vF=JsddDW=\ra\u001d1, in the latter\ncase (where vFis the Fermi velocity, ais the lattice spac-\ning anddDWis the DW thickness). In our simulations,\nJsd=\u0016BjBextj\u001910 andJsddDW=\ra\u00191 fordDW\u001910a\nin Fig. 2(a). Thus, it seems that fair comparison of our\nresults to SMF theory requires to substantially increase\nJsd. However, Jsd= 0:1 eV (i.e.,\r=Jsd=\u001810, for typical\n\r\u00181 eV which controls how fast is quantum dynam-\nics of electrons) in our simulations is \fxed by measured\nproperties of permalloy.53\nLet us recall that rigorous de\fnition of \\adiabaticity\"\nassumes that conduction electrons within a closed quan-\ntum system54at timetare in the ground state j\t0i\nfor the given con\fguration of LMMs Mi(t),j\t(t)i=\nj\t0[Mi(t)]i; or in open system55their quantum state is\nspeci\fed by grand canonical DM\n\u001aeq\nt=\u00001\n\u0019Z\ndEImGr\ntf(E): (5)\nwhere the retarded GF, Gr\nt=\u0002\nE\u0000H[Mi(t)]\u0000\u0006L\u0000\u0006R\u0003\u00001, and \u001aeq\ntdepend parametrically66{68(or implic-\nitly, so we put tin the subscript) on time via instanta-\nneous con\fguration of Mi(t), thereby e\u000bectively assum-\ning@tMi(t) = 0. Here Im Gr\nt= (Gr\nt\u0000[Gr\nt]y)=2i;\u0006L;R\nare self-energies due to the leads; and f(E) is the Fermi\nfunction. For example, the analysis of Ref. 69 assumes\nh^siineq(t)kh^siieq\ntto reveal the origin of spin and charge\npumping in SMF theory|nonzero angle \u000eeq\nibetween\nh^siieq\ntandMi(t) with the transverse component scaling\njh^siieq\nt\u0002Mi(t)j=\u0000\nh^siieq\nt\u0001Mi(t)\u0001\n/1=Jsdas the signature\nof \\adiabatic\" limit. Note that our \u000eeq\ni.4\u000e[Fig. 2(c)]\nin the region of two DWs (and \u000eeq\ni!0 elsewhere). Ad-\nditional Figs. S1{S3 in the SM,58where we isolate two\nneighboring LMMs from the right DW in Fig. 1 and\nput them in steady precession with frequency !for sim-\nplicity of analysis, demonstrate that entering such \\adi-\nabatic\" limit requires unrealistically large Jsd&2 eV.\nAlso, our exact55result [Figs. S1(b), S2(b) and S3(b) in\nthe SM58] showsjh^siieq\nt\u0002Mi(t)j=\u0000\nh^siieq\nt\u0001Mi(t)\u0001\n/1=J2\nsd\n(instead of/1=Jsdof Ref. 69). Changing ~!|which,\naccording to Fig. 3(c), is e\u000bectively increased by the\ndynamics of annihilation from ~!'0:01 eV, set ini-\ntially by Bext, toward ~!'0:1 eV|only modi\fes scal-\ning of the transverse component of h^siineq(t) withJsd\n[Figs. S1(a), S2(a), S3(a), S4(b) and S4(d) in the SM58].\nThe nonadiabatic corrections55,66{68take into account\n@tMi(t)6= 0. We note that only in the limit Jsd!1 ,\u0000\nh^siineq(t)\u0000h^siieq\nt\u0001\n!0. Nevertheless, multiplication\nof these two limits within Eq. (3) yields nonzero geo-\nmetric STT,54,55as signi\fed by Jsd-independent STT\n[Figs. S1(c), S2(c) and S3(c) in the SM58]. Otherwise,\n\\nonadiabaticity\" angle is always present ( \u000eneq\ni\u0000\u000eeq\ni)6= 0\n[Fig. 2(d)], even in the absence of spin relaxation due to\nmagnetic impurities or SO coupling,70and it can be di-\nrectly related to additional spin and charge pumping48,70\n[see also Figs. S1(f), S2(f) and S3(f) in the SM58].\nConclusions and outlook .|The pumped spin current\nover ultrabroadband frequency range [Fig. 3(d)], as our\ncentral prediction, can be converted into rapidly chang-\ning transient charge current via the inverse spin Hall ef-\nfect.71{73Such charge current will, in turn, emit electro-\nmagnetic radiation covering \u00180:03{27 THz range (for\njBextj\u00181 T) or\u00180:3{27:3 THz range (for jBextj\u001810\nT), which is highly sought range of frequencies for variety\nof applications.72,73\nACKNOWLEDGMENTS\nM. D. P., U. B., and B. K. N. was supported by the\nUS National Science Foundation (NSF) Grant No. ECCS\n1922689. P. P. was supported by the US Army Research\nO\u000ece (ARO) MURI Award No. W911NF-14-0247.6\n\u0003bnikolic@udel.edu\n1G. Tatara, H. Kohno, and J. Shibata, Microscopic ap-\nproach to current-driven domain wall dynamics, Phys.\nRep.468, 213 (2008).\n2G. 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Dang2, Enzo Rongione2, James Boust2, Sophie Collin2, Jean-Marie\nGeorge2, Henri-Jean Drouhin3, Yannis Laplace3, Romain Grasset3, Jingwei Dong3, Juliette\nMangeney1, Jerome Tignon1, Henri Ja\u000br\u0012 es2, Luca Perfetti3\u0003and Sukhdeep Dhillon1\n1Laboratoire de Physique de l'Ecole normale sup\u0013 rieure, ENS, Universit\u0013 e PSL,\nCNRS, Sorbonne Universit\u0013 e, Universit\u0013 e de Paris, F-75005 Paris, France\n2Unit\u0013 e Mixte de Physique, CNRS, Thales, Universit\u0013 e Paris-Sud,\nUniversit\u0013 e Paris-Saclay, F-91767 Palaiseau, France and\n3Laboratoire des Solides Irradi\u0013 es, CEA/DRF/lRAMIS, Ecole Polytechnique,\nCNRS, Institut Polytechnique de Paris, F-91128 Palaiseau, France\nTerahertz (THz) spin-to-charge conversion has become an increasingly important process for THz\npulse generation and as a tool to probe ultrafast spin interactions at magnetic interfaces. However,\nits relation to traditional, steady state, ferromagnetic resonance techniques is poorly understood.\nHere we investigate nanometric trilayers of Co/X/Pt (X=Ti, Au or Au0:85W0:15) as a function of\nthe 'X' layer thickness, where THz emission generated by the inverse spin Hall e\u000bect is compared\nto the Gilbert damping of the ferromagnetic resonance. Through the insertion of the 'X' layer\nwe show that the ultrafast spin current injected in the non-magnetic layer de\fnes a direct spin\nconductance, whereas the Gilbert damping leads to an e\u000bective spin mixing-conductance of the\ntrilayer. Importantly, we show that these two parameters are connected to each other and that spin-\nmemory losses can be modeled via an e\u000bective Hamiltonian with Rashba \felds. This work highlights\nthat magneto-circuits concepts can be successfully extended to ultrafast spintronic devices, as well\nas enhancing the understanding of spin-to-charge conversion processes through the complementarity\nbetween ultrafast THz spectroscopy and steady state techniques.\nPACS numbers:\nI. INTRODUCTION\nWhen a pure spin current pass through materials with\nlarge spin-orbit coupling, it can generate a transverse\ncharge current1,2by means of the Inverse Spin-Hall-E\u000bect\n(ISHE). A \rurry of activity on this topic has been mo-\ntivated by the intimate relation between ISHE and the\ndirect SHE3. The latter can be very e\u000eciently employed\nto generate a spin transfer torque capable of switching\nthe magnetization of ferromagnetic thin \flms4,5. Most\nexperiments in this \feld have been performed by spin-\npumping viaferromagnetic resonance while some works\nhave investigated the ultrafast regime6,7. More recently,\nother authors have proven that ISHE can be employed to\ngenerate an intense THz radiation. This breakthrough\nhighlighted that interfaces leading to large spin trans-\nfer torque are also excellent emitters of electromagnetic\nwaves8{12. Theoretical simulations based on superdi\u000bu-\nsive transport equations have successfully reproduced the\nobserved emission13,14. However, these frameworks do\nnot cover the impact of the electronic transmission at in-\nterfaces, neither the discussion of the particular role of\nthe interfacial spin-orbit \felds originating from charge\ntransfer and symmetry breaking15. To this end, a tighter\nconnection with steady state spintronics is highly desir-\nable. For example, the magnetocircuits analogies are\n\u0003The authors to whom the correspondence show be addressed are\nluca.perfetti@polytechnique.edu, henri.ja\u000bres@cnrs-thales.fr and\nsukhdeep.dhillon@phys.ens.frwidely employed to de\fne the e\u000eciency of the spin-to-\ncharge conversion16,17. An extension of such formalism\nto impulsive excitations has been discussed in the case\nof ultrafast spin-Seebeck e\u000bect18but not yet for the spin\ncurrent injected from ferromagnetic transition metals.\nIn the \feld of spintronics, the optimal e\u000eciency of spin\norbit torque (SOT) requires the engineering of metallic\ninterfaces favoring a higher spin-current generation. Re-\ncently, many authors have tackled this issue by inserting\ntransion metals19,20or noble metals21between cobalt (or\nCoFe, CoFeB) and platinum. Experiments with di\u000ber-\nent interlayers have shown clear correlations between the\nspin-transfer-torque, magnetoresistance19, perpendicular\nanisotropy15,19and spin memory loss22. Here, we in-\nvestigate this topic by comparing the THz emission ef-\n\fciency detected by Time Domain Spectroscopy (TDS)\nwith Ferro-Magnetic Resonance (FMR) spectroscopy of\ntrilayers Co/X/Pt. The insertion of an atomically thick\ninterlayer of X=Ti, Au or Au 0:85W0:15modi\fes the ca-\npability of the interface to generate spin currents. This\nproperty a\u000bects, on the same footing, the emission of\nelectromagnetic radiation as well as the Gilbert damping\nof the multilayer. Our result show that the THz-TDS\nemission spectroscopy is contactless and non-destructive\nmethod that can give an accurate and reliable esti-\nmate of the spin-injection e\u000eciency at spintronic in-\nterfaces. We discuss the data in the magnetocircuit\nformalism16{18, by assuming that spin-currents follow the\nevolution of magnetic \ructuations. The average spin-\nconductance ( g\"+g#)=2 characterizes the ultrafast cur-\nrents in platinum18whereas an e\u000bective spin conductancearXiv:2103.09557v1  [cond-mat.mes-hall]  17 Mar 20212\ng\"#\ne\u000bdescribes the damping torque in the ferromagnetic\nmaterial ( in our case is cobalt )16. These two quantities\nwould be proportional to each other if the spin \row was\nconserved at the interface17. In reality, strong spin-orbit\nassisted scattering processes generate a sink of angular\nmomentum and limit the spin \row that can propagate\nin platinum23{26. The comparison between ( g\"+g#)=2\nandg\"#\ne\u000bshows that the fraction of spin current lost at the\ninterface25is proportional to the spin conductance and\nmay exceed 40% in the Co/Pt bilayer. Our measure-\nments highlight that passivation of the interface by dif-\nferent compounds follows a common trend and suggests\nthe existence of a general relation between spin memory\nloss and spin-conductance. We prove this claim by choos-\ning inter-layer materials with very di\u000berent properties:\nTi is more chemically reactive and has small spin-orbit\ncoupling whereas Au and Au 0:85W0:15are less chemically\nreactive and hosts a larger spin-orbit interaction (espe-\ncially the Au:W alloy).\nFIG. 1: A) Detection of spin-to-charge conversion in a spin-\ntronic emitter. The cobalt layer has magnetization ~Mpar-\nallel to the external magnetic \feld ~Hand is in contact with\nthe platinum layer. An ultrashort laser pulse photoexcites\nthe sample generates, in the Pt side of the interface, a spin\ncurrent~Jsthat is proportional to the direct spin conductance\n(g\"+g#)=2. The inverse spin Hall e\u000bect of Pt leads to a trans-\nverse charge current ~Jc. Being shorter than one picosecond,\nthe~Jcpulse emits radiation in the THz spectral range. B)\nE\u000bect of the platinum layer on the ferromagnetic resonance of\nthe underlying cobalt. The magnetization precession driven\nby a radiofrequency \feld ~hrfinduces a spin current. The in-\ncrease of Gilbert damping due to the ~Jsinjection in the Pt\nlayer is proportional to the e\u000bective spin-mixing conductance.\nII. GENERAL FRAMEWORK OF SPINTRONIC\nTHZ EMISSION\nA framework building on few hypothesis connects\nthe spin conductance to the emitted THz radiation.In the thin \flm limit, the THz electric \feld of a\nplane wave at the surface of the sample is given\nby~ET(!) =eZR~Jc(!;z)dz. This expression links\n~ET(!) to the charge current density ~Jcvia an e\u000bective\nimpedance8,27,28:\nZ=Z0\n1 +n+Z0R\n\u001b(z)dz; (1)\nwherenis the refractive index of the substrate, zis the\ncoordinate perpendicular to the interface, Z0= 377\n is\nthe vacuum impedance andR\n\u001b(z)dzis the local conduc-\ntivity integrated over the total thickness of the multi-\nlayer. The charge current ~Jc(z) arises in platinum be-\ncause of the inverse-spin-Hall-e\u000bect acting on the spin\ncurrent \row ~Js(z) along the normal direction to the \flm\nplane. The latter decreases exponentially over a distance\nequal to the spin di\u000busion length. It follows that:\nZdPt\n0+~Jc(z)dz=Js(0+)(~ en\u0002~ es)\u0015Pt\ns\u0002stanhdPt\n2\u0015Pts;(2)\nwhere~ enis a unitary vector normal to the interface, ~ es\nis the polarization direction of the spin current, dPtis\nthe thickness of platinum layer, \u0015Pt\nsis the spin di\u000busion\nlength in platinum, \u0002 sis the spin-Hall-angle of platinum\nandJs(0+) is the magnitude of spin current density gen-\nerated in the ferromagnet, penetrating into the heavy\nmetal, and thus responsible for the charge current os-\ncillations at the platinum side of the interface. At this\nstage, it is important to recall that the magnitude of\ntheJspropagating in platinum can be smaller than the\none generated in the ferromagnet. The discontinuity of\nspin current between the two sides of the active interface\nis generally ascribed to the spin-decoherence induced by\nlocal spin-orbit \felds (also known as spin-memory-loss)23\nand has been recently proved viare\fned spin-orbit torque\nexperiments15.\nEmission over an ultrabroad spectral range8,18and\ntheoretical modeling13,14,18show thatJsevolves on a\ntimescale comparable to the energy and momentum re-\nlaxation of hot electrons. We make use of the magneto-\ncircuit formalism to write to the spin current in terms of\nspin conductance parameters17. The ultrafast generation\nof a spin accumulation on the ferromagnetic side leads to\na longitudinal component whereas the spin accumulation\non the Pt side induces a transverse component18. The\nresulting expression reads:\n~Js=~\n4\u0019\u0012g\"+g#\n2h@tM\nM^mi+g\"#h^m\u0002@t^mi\u0013\n;(3)\nwhereMis the magnetization magnitude in the very\nproximity of the interface, ^mis the local magnetization\ndirection,g\"(g#) is the spin conductance parallel (an-\ntiparallel) to the magnetization and g\"#is the spin mix-\ning conductance. The longitudinal component is propor-\ntional to (g\"+g#)=2 and to the relative demagnetiza-\ntion@tM=M . This term is driven by the quasi-ballistic3\nFIG. 2: A) THz waveforms emitted by a set of di\u000berent tri-\nlayers Co/Au 0:85W0:15(d)/Pt. B) Spin current generated in\nCo/Ti(d)/Pt, Co/Au( d)/Pt and Co/Au 0:85W0:15(d)/Pt tri-\nlayers of di\u000berent thickness. The Co and Pt layer have \fxed\nthickness of 2 nm and 5 nm, respectively. The thickness dof\nthe X=Ti,Au,Au 0:85W0:15layer is instead varied between 0\nnm and 2 nm. The parameter \u0011(d) has been extracted from\nthe THz signal via Eqn. [5] and can be considered as a nor-\nmalized spin current density in the platinum layer.\ntransport of highly excited electrons from the Cobalt to\nPlatinum and it represents the dominant contribution in\nthe case of the spin current that are generated by ultra-\nfast laser pulse14. Accordingly, the THz emission from\nCo/Pt is many orders of magnitude more intense8than\nthe one observed from an interface where the longitudinal\ncomponent is inactive18.\nOwing to the quasi-ballistic nature of the injection, the\nspin current arises the spin accumulation taking place\non a length scale 1 :4 nm18,27,28. As a consequence, the\nstrength of the emitted THz radiation scales as the en-\nergy density injected by the pump pulse8,27,28\nj@tM\nMj/ABFI\nd+dPt+dCo; (4)\nwhereABis the absorbed fraction of pump pulse in the\nmultilayer, FIis the incident \ruence of the pump pulse,\ndPt= 5 nm is the thickness of platinum layer, dCo= 2\nnm is the thickness of cobalt layer, dis the thickness\nof the X = Ti, Au or Au 0:85W0:15layer. To investigate\nthis, we have prepared Co/X( d)/Pt trilayers on glass and\nhighly resistive Si(111) substrates by sputtering deposi-\ntion at room temperature with standard experimental\nconditions. The Au 0:85W0:15material has been obtained\nvia the evaporation of a rod containing 85% of gold and\n15% of tungsten. The thickness dof the interlayer is typ-\nically varied between 0 and 2 nm. Within this range of\nd, theABcoe\u000ecient can be considered constant8,27,28.\nMorever the incident laser \ruence FIhas been kept \fxed\nand stable.The THz TDS system is placed in a re\rection geom-\netry where the generated THz pulses are collected from\nthe same surface of the spin-emitter as the excitation\n(i.e. no beam passes through the substrate). The emit-\nters are mounted with small magnetic \feld parallel ( \u0018=10\nmT) to the spin interface. We veri\fed that a switch-\ning of the ~Morientation reverses the direction of the\nemitted THz \feld, thereby con\frming that charges cur-\nrents arise from the ISHE. Fig. 2A) displays a set of THz\ntraces emitted from Co/Au 0:85W0:15(d)/Pt multilayers\nwith di\u000berent values of the Au 0:85W0:15thicknessd. The\nTHz traces recorded for di\u000berent values of dhold nearly\nidentical waveforms (see also supplementary information\n\fle29). SinceET(t;d)\u0018=ET(d)f(t) (and equivalently\nET(!;d)\u0018=ET(d)f(!)), we assume that spin \ructua-\ntions, spin mixing conductance and spin-Hall-angle have\nnegligible frequency dependence within the bandwidth of\nthe detected THz. As observed experimentally, the drop\nof THz signal as a function of dis mainly due to a de-\ncreasing spin-conductance. The latter is related to the\ndetection of the THz \feld via Eqs. [1-4]. By solving for\nthe spin conductance we obtain:\n\u0011(d) =g\"(d) +g#(d)\ng\"(0) +g#(0)=ET(d)\nET(0)Z(0)\nZ(d)d+dPt+dCo\ndPt+dCo;(5)\nwhere the impedance Z(d) has been calculated by as-\nsuming the THz conductivity in thin \flms27,28,30\u001bCo=\n3\u0002106S/m,\u001bPt= 4\u0002106S/m,\u001bAu= 4\u0002106S/m,\n\u001bAu:W= 1:2\u0002106S/m and\u001bTi= 0:5\u0002106S/m. Di\u000ber-\nences of these conductivities with respect to bulk values\nare due to strong charge scattering at the landscape of\nthe interface and to the formation of small grains30. As\na matter of facts, the factor Z(0)=Z(d) remains close to\nunity, owing to the fact that metallic interlayers with\nnanometric thickness have small parallel conductivity.\nIII. DATA ANALYSIS AND DISCUSSION\nThe parameter \u0011(d) of Eqn. [5] re\rects the relative\nreduction of the spin-injection e\u000eciency in Pt if an in-\nterlayer of thickness dis grown between Co and Pt. As\nshown by Fig. 2B), \u0011(d) follows nearly an exponential de-\ncay exp(\u0000d=lX), with characteristic length lAu= 4 nm\nfor X=Au or X=Au 0:85W0:15andlTi= 1:5 nm for X=Ti.\nAs can be observed, Ti a\u000bects the spin mixing conduc-\ntance much more e\u000bectively than Au or Au 0:85W0:15do.\nRecent experiments have shown that a submonolayer\nof Ti can indeed substantially modify the spin-transfer\ntorque of the CoFeB/Pt20and Co/Pt31interfaces. The\ninsertion of the chemically reactive Ti alters the spin\ndependent transmission/re\rection probabilities that fa-\nvor the transport of one spin \ravor with respect to the\nother. Furthermore, the surface passivation by Ti atoms\nmay modify the spin-\rip scattering potential at the in-\nterface. Although the microscopic mechanisms leading to\nthe large reduction of spin conductance is still debated,4\nFIG. 3: A) Derivative of the spin susceptibility vs inten-\nsity of the static magnetic \feld Hin the reference bi-\nlayer Co/Pt. The di\u000berent curves correspond to hrffre-\nquencies of 4-18 GHz, with step of 2 GHz. B) Varia-\ntion of resonance frequency as a function the static mag-\nnetic \feldHin the Co/Au 0:85W0:15(d)/Pt trilayers. C) Full\nwidth at half maximum of the ferromagnetic resonance in\nCo/Au 0:85W0:15(d)/Pt trilayers. The Co layer has thick-\nness of 15 nm, the Pt layer has thickness of 5 nm and the\nAu0:85W0:15layer has thickness dvarying between 0 nm and\n1.5 nm.\na systematic investigation of spin orbit torque with dif-\nferent transition metals concluded that the d-orbital \fll-\ning has a stronger in\ruence on charge-to-spin conversion\nthan the atomic number19. Our measurements corrob-\norate this \fnding: the passivation of Co/Pt interface is\nmore e\u000bective in the case of a transition metal with in-\ncomplete 3d-shell like titanium than in the case of an\nalloy with larger atomic number but closed 5 dshell like\nAu. Moreover, the larger spin-orbit interaction of W in\nthe Au 0:85W0:15does not seem to make any appreciable\ndi\u000berence with respect to pure gold. Our model in the\nlast section of this article will further clarify this, some-\nhow surprising, result.\nNext, we discuss the e\u000bective spin mixing conduc-\ntance that is measured by means of FerroMagnetic Res-\nonance (FMR)12. Samples made with 5 nm of Pt and\nthicker Co \flms (15nm) were deposited on highly resis-\ntive Si/SiO2(111) substrates before lithography pattern-\ning. The thicker ferromagnetic layer provides a clearer\nFIG. 4: A) Gilbert damping and spin conductance in\nCo/Ti(d)/Pt, Co/Au( d)/Pt and Co/Au 0:85W0:15(d)/Pt tri-\nlayers as a function of thickness d. B) Gilbert damping and\nspin conductance of the two trilayer set plot against the \u0011pa-\nrameter extracted from the emitted THz. The green dot cor-\nresponding to vanishing THz emission is the intrinsic Gilbert\ndamping measured on cobalt capped by 2 nm of alumina.\nA model that includes the spin memory loss is calculated via\nEq. [9] and superimposed (solid line) to the experimental data.\nresonance spectrum compared to a 2 nm layer. Fig. 3A)\ndisplays the di\u000berential susceptibility of the Co/Pt bi-\nlayer as a function of the external magnetic \feld H.\nCurves of di\u000berent colors stand for increasing frequency\nof radiofrequency \feld hrf. We show in Fig. 3B) the reso-\nnance frequency !ras a function of Hfor the multilayers\nCo/Au 0:85W0:15(d)/Pt. The FMR theory predicts:\n!r=\r\u00160p\nH(H+M); (6)\nwhere\ris the gyromagnetic ratio and \u00160vacuum perme-\nability and Mis the saturation magnetization. By \ftting\nthe data with Eq. [6], it is possible to extract the satura-\ntion magnetization M= 1500\u000650 emu/cm3. The damp-\ning term can be quanti\fed by measuring the half width\nat half maximum \u0001 Hof FMR linewidth. As shown by\nFig. 3C) the linear regression\n\u0001H= \u0001H0+!r\u000b\n\r\u00160; (7)5\nprovides the Gilbert damping \u000b(d) for the\nCo/Au 0:85W0:15(d)/Pt series. Likewise, this proce-\ndure is applied to extract the Gilbert damping of\nCo/Ti(d)/Pt trilayers. Moroever, the larger thickness of\ncobalt layer ( dCo= 15 nm in FMR experiments instead\nofdCo= 2 nm chosen for the THz emission experiment)\nminimize the extra contribution of two-magnons scatter-\ning to the\u000bvalue. Since two-magnon scattering scales\nas 1=d2\nCo, the associated damping term26should not\nexceed 8\u000210\u00004and it has been neglected. Therefore,\n\u000bdi\u000bers from the intrinsic \u000b0only by a term arising\nfrom the injected spin current. The e\u000bective spin mixing\nconductance g\"#\ne\u000bis obtained via23,25:\n\u0001\u000b=\u000b\u0000\u000b0=g\u0016B\n4\u0019Md Cog\"#\ne\u000b; (8)\nwheregstands for Land\u0013 e factor of the electron and \u0016B\nis the Bohr magnetron. The value \u000b0= 5\u000210\u00003is\nobtained by measuring the Gilbert damping of a 15 nm\ncobalt capped by 2 nm of alumina.\nFig. 4A) shows \u000bandg\"#\ne\u000bfor the two trilayer series as a\nfunction of interlayer thickness d. Similarly to THz mea-\nsurements, the drop of spin mixing conductance is faster\nin Co/Ti(d)/Pt than in Co/Au 0:85W0:15(d)/Pt samples.\nThis \fnding highlights the \frst important outcome of\nthis work: an intimate connection between ( g\"+g#)=2\nobtained by ultrafast currents in the THz spectral range,\nwithg\"#\ne\u000bextracted from the FMR damping linewidth.\nWe \fnd phenomenologically the universal relation:\ng\"#\ne\u000b/\u0011(d)\n1\u0000\u0018(d): (9)\nThe solid line of Fig. 4B is calculated from Eq. [9] with\nparameters g\"#\ne\u000b(0) = 75nm\u00002and\u0018(d) = 0:4\u0011(d). From\ntheir dependence on the transmission coe\u000ecient at the\ninterface17, we evince that g\",g#and (g\"+g#)=2<\ng\"#\u0018=g\"should scale as \u0011(d) upon the insertion of\nthe interlayer. Namely, we assume that g\"(d)=g\"(0) =\ng#(d)=g#(0) =g\"#(d)=g\"#(0) =\u0011(d). Moreover, we set\ng\"#= (1\u0000\u0018)g\"#\ne\u000b, where the parameter \u0018 <1 arises from\nthe spin-memory-loss22,23,25. Due the spin scattering at\nthe interface, the spin-current leading to THz emission\nin platinum is 1\u0000\u0018times smaller than the spin current\na\u000becting the ~Mprecession. The second important result\nof our work is that \u0018is proportional to the spin conduc-\ntance at the interface. The more e\u000ecient the generation\nof spin current, the higher the spin memory loss. When\nexpressed in terms of relative variation of spin conduc-\ntance, the spin memory loss \u0018appears to be insensitive to\nthe compound and thickness that has been employed to\nperform the passivation of the interface. We now turn on\nto the modeling of the spin memory loss through Rashba\n\felds at the interface.IV. MODELING OF SPIN MEMORY LOSS\nTHROUGH RASHBA SPIN-ORBIT\nINTERACTION AT THE INTERFACE.\nA. Electronic quantum transmission with\nspin-orbit interaction\nThe insertion of an interlayer X at the Co/Pt interface\nhas two mains e\u000bects: i)the formation of a thin potential\nbarrier is accompanied by smaller the spin-transmission\nvsCo/Pt. Indeed Co/Pt is known to build an excellent\nmatching for the majority spin channel near the Fermi\nlevel whereas a larger chemical mismatch may take place\nin the case of Co/X/Pt with X=Ti, Au or Au 0:85W0:15\nand; ii)since the Ti or pure Au lack the open 5 dshell of\nPt, the presence of an interlayer has to reduce spin orbit\ninteraction (SOI) at the interface32.\nIn the following, we consider a simpli\fed SOI assisted\nquantum transmission model that has been recently im-\nplemented with success for the description of SOT33{37.\nThis model will \frst highlight the role of i)andii)in the\ndescription of our data. The interface is treated as an\nideal trilayer structure Co/X/Pt with a spin current Js\npropagating along the ~ endirection, normal to the layers\n(CPP geometry). Jsis computed from the propagation\nof selected plane waves with in-plane conserved wavevec-\ntorkk, and normal wavector kzalong~ en. The quantum\ntransmission is summed hereafter over the Fermi surface,\nas it is required within an extended Landauer treatment.\nWe obtain theJs(z) pro\fle across the interface viaa\nre\fned model involving a Rashba-like term33{36. We re-\nstrict the electronic states to two electron bands with spin\npolarized states. The partitioned Hamiltonian in Co and\nPt reads:\n^H=^p2\n2m\u0003+ \u0001 e^m\u0001^\u001b+^V (10)\nwhere ^p=\u0000i~rzis the impulsion operator, m\u0003is the\ne\u000bective mass, ^mis the magnetization direction, \u0001 e'\n2 eV is the exchange coupling for Co, and ^V=^VCo= 0\nrepresents the energy position of the bottom of the spin-\naveraged 3 dCo bands. Along the same lines, we set for\nPt an exchange coupling \u0001 e= 0 and ^V=^VPt'\u00001 eV.\nThe potential di\u000berence ^VCo\u0000^VPtis representative of\nthe workfunction o\u000bset between the two metals.\nThe addition of an interlayer is simulated by an inter-\nfacial potential ^VSthat is expressed by34,36,37:\ntI^VS\u000e(z) =tIh\nVX+\u000bR\n~\u0010\n^~ p\u0002~ en\u0011\n\u0001^\u001bi\n\u000e(z); (11)\nwherezis the coordinate along the direction ~ en, the func-\ntion\u000e(z) is Dirac delta function and tIis the e\u000bective\ninterface thickness. The operator ^VSis de\fned via:VX\nis the average interface of an unpolarized potential bar-\nrier and\u000bRis the strength of Rashba interaction. We\nintroduce the two parameters having the dimension of\ninverse length. The quantity kX=VXtIm\u0003=~2tunes the\ntransmission trough the barrier and kso=\u000bRkFtIm\u0003=~26\nFIG. 5: A) Pro\fle of spin current Jsin the Co/X/Pt surface\nat the vicinity of the Co/Pt interface for 3 di\u000berent cases:\nno scattering potential (black curve), Rashba scattering only\nwithkso= 2\u0017A\u00001(red curve) and Rashba scattering plus a\npotential barrier ( kX= 3\u0017A\u00001). The spin memory loss \u0018=\n(Js(0\u0000)\u0000Js(0+))=Js(0\u0000) is the relative discontinuity of Js\nat the interface. B) Spin memory loss \u0018as a function of\npotential barrier kXfor three di\u000berent strengths of the Rashba\nscattering. C) Spin memory loss \u0018as a function of e\u000bective\nspin-conductance ge\u000bobtained by varying kXand with spin\norbit parameter equal to kso= 2\u0017A\u00001(blue circles). As a term\nof comparison we also show the relation extracted from the\nexperimental data (green dashed line).\nrule the strength of the spin-orbit scattering (see also\nsupplementary information \fle29).\nB. Results of the model\nOur model provides the pro\fle of a normalized spin-\npolarized current originating from Co (where it is normal-\nized to unity) and propagating through a Co/X/Pt tri-\nlayer. Figure 5A) depicts three speci\fc cases, correspond-\ning to: no interfacial potentials ( kX= 0 andkso= 0),\na pure Rashba interaction ( kX= 0 andkso= 2\u0017A\u00001)\nand, both a potential barrier and a Rashba interaction\n(kX= 3\u0017A\u00001andkso= 2\u0017A\u00001). The spin current is always\nmaximal in the bulk of Co, while it goes towards zero\nwhen penetrating in the non-magnetic Pt layer and mov-ing away from the interface. In the absence of the scat-\ntering potential ^VS(black curve in Fig. 5A)) the Js(0)\nvalue at the Co/Pt interface results from an equilibrium\ncondition between bulk spin-\rip rates in the two regions.\nThe spin-current is continuous everywhere (no spin-orbit\nscattering) and its value Js(0)\u00190:6 coincides with the\nprediction of a pure di\u000busive spin-model. This agreement\ncorroborates the validity of our quantum transmission\nmodel in the absence of any ^VSscattering.\nAdding a Rashba interaction kso= 2\u0017A\u00001(red curve\nin Fig. 5A)) leads to the spin-memory loss. Indeed\nthe Rashba \felds are not collinear to the incoming spin\nand induce a local spin-precession. Only a fraction of\nspin current coming from the Co reservoir is injected\ninto the Pt layer so that Js(z) displays a sizable dis-\ncontinuity at the interface22. In order to quantify this\ne\u000bect, we introduce the memory loss parameter \u0018=\n(Js(0\u0000)\u0000Js(0+))=Js(0\u0000), where 0\u0000and 0 +are the lim-\niting values reached by approaching the interface from\nthe Co and Pt side, respectively. From the chosen pa-\nrameters we extract \u0018= 0:6, which is only 50% higher\nthan our experimental value and in agreement with pre-\nvious FRM estimates23.\nThe presence of an additional unpolarized scattering\npotential with kX= 3\u0017A\u00001(Blue curve in Fig. 5A) has\ntwo main e\u000bects. On one hand, the larger back\row of Js\nin the Co layer leads to a smaller ejection of spin-current\nfrom the ferromagnet. On the other hand, an unchanged\nstrength of the Rashba \feld results in a smaller jump of\ntheJscurrent at the interface. As shown in Fig. 5B), the\nmonotonic reduction of spin memory loss as a function\nofkXtakes place for two representatives values of the\ninverse spin length kso.\nWe extract the e\u000bective spin conductance from the\nrescaled ratio between the spin current Js(0\u0000) obtained\nin the presence of an interlayer (i.e. for kX>0) and\ntheJs(0\u0000) obtained for the bare Co/Pt interface (i.e.\nforkX= 0). Figure 5B) shows the calculated \u0018vs.ge\u000b\nwhen the potential barrier kXis increased linearly to\n5\u0017A\u00001while the value ksois kept \fxed to 2 \u0017A\u00001. Note\nthat the spin memory loss display the same trend of the\ncurve that is extracted by combining FMR-spin-pumping\nand THz methods (green dashed line). This shows that\nan interposition of Ti, Au or Au 0:85W0:15introduces a\nchemical barrier at the interface. The enhanced back-\nward di\u000busion of electrons has the e\u000bect of decreasing\nboth the spin mixing conductance and the spin memory\nloss. This e\u000bect takes place even if the spin dependent\nscatteringksoremains equal to the pristine value.\nV. CONCLUSIONS AND\nACKNOWLEDGMENTS.\nIn conclusion, we report that the spin-conductance can\nbe extracted from broadband THz spectroscopy. The in-\nvestigation of Co/X( d)/Pt trilayers with X=Ti, Au and\nAu0:85W0:15show that in all cases, an interlayer reduces7\nthe spin-to-charge conversion. THz experiments have\nbeen bench-marked with the e\u000bective spin-mixing con-\nductance extracted by FerroMagnetic Resonance mea-\nsurements. A model including spin memory loss show\nthat the relative drop of spin current at the interface is\nproportional to the spin conductance and attains \u0018= 0:4\nat the Co/Pt interface. The simulations indicate that\nmodi\fed spin transmission probabilities at the interface\ncan explain this correlation. Our \fndings are very gen-\neral and show that a combination of THz emission with\nFMR spectroscopy can bring accurate characterizations\nand provide new insights into spintronic multilayers.\nWe acknowledge E. Jacquet for his contribution in\nthe thin \flm growth and M. Cosset-Cheneau for hishelp in the FMR experiments. We are very thank-\nful to Tobias Kampfrath and Marco Battiato for the\nenlightening discussions on the interpretation of THz\nemission mechanism. Synchrotron Soleil hosts a THz\nsetup where some transmission measurements have been\ndone. Financial support has been provided by the DGA\nproject ITEHR (No. 2018600074) as well as ANR Project\nTOPRISE No. ANR-16-CE24-0017. We acknowledge\nthe Horizon2020 Framework Programme of the European\nCommission under FET-Proactive Grant agreement No.\n824123 (SKYTOP). This project has received funding\nfrom the H2020 research and innovation programme s-\nNebula under grant agreement No.0863155.\n1E. Saitoh, M. Ueda, H. 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Stiles, \\Interfa-\ncial spin-orbit torques,\" Journal of Applied Physics 128,\n151101 (2020)."    },    {        "title": "1701.02475v1.Magnetic_properties_in_ultra_thin_3d_transition_metal_alloys_II__Experimental_verification_of_quantitative_theories_of_damping_and_spin_pumping.pdf",        "content": "1  Magnetic properties in ultra -thin 3d transition metal alloys \nII: Experimental verification of quantitative theories of \ndamping and spin -pumping    \n \nMartin A. W. Schoen,1,2* Juriaan Lucassen,3 Hans T. Nembach,1 Bert Koopmans,3 T. J. Silva,1 Christian \nH. Bac k,2 and Justin M. Shaw1 \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO \n80305 , USA  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, \nGermany  \n3Department of Applied P hysics, Eindhoven University of Technology, 5600 MB Eindhoven, The \nNetherlands  \n \n \nDated: 01/05/2017  \n \n *Corresponding author: martin1.schoen@physik.uni -regensburg.de  \n \n \n \n \nAbstract  \nA systematic experimental study of Gilbert damping is performed via ferromagnet ic \nresonance for the disordered  crystalline  binary 3 d transition metal alloys Ni -Co, Ni -Fe and \nCo-Fe over the full range of alloy compositions. After accounting for inhomogeneous \nlinewidth  broadening , the damping shows clear evidence of both  interfacial da mping \nenhancement (by spin pumping) and radiative damping. We quantify these two extrinsic  \ncontributions  and thereby determine the intrinsic damping. The comparison of the intrinsic \ndamping to multiple theoretical calculation s yields good qualitative  and q uantitative  \nagreement  in most cases . Furthermore, the values of the damping obtained in this study are \nin good agreement with  a wide range of published experimental and theoretical values . \nAdditionally, we find a compositional dependence of the spin mixing  conductance.  \n \n \n \n \n 2  \n \n \n \n1  Introduction  \nThe magnetization dynamics in f erromagnetic films are phen omenologically well described \nby the Landau -Lifshitz -Gilbert formalism (LLG)  where  the damping  is described by a \nphenomenological damping parameter α.4,5 Over the past four decades, there have been \nconsiderable efforts to derive the phenomenological d amping parameter from first principles \ncalculations and to do so in a quantitative manner. One of the early promising  theories was that of \nKamberský, who introduced the so -called breathing Fermi surface model6–8. The name “breathing \nFermi  surface ” stems from the picture that the precessing magnetization, due to spin -orbit coupling, \ndistorts the Fermi surface. Re -populating the Fermi surface is delayed by the scattering time, \nresulting in a phase lag between the precession  and the Fermi sur face distortion. This  lag leads to  a \ndamping that is proportional to the scattering time. Although this approach describes the so-called \nconductivity -like behavior of the damping at low temperatures, it fails to describe  the high \ntemperature behavior of so me materials . The high temperature or resistivity -like behavior is \ndescribed by the so-called  “bubbling Fermi surface ” model. In the case of energetically shifted  \nbands , thermal  broaden ing can lead to a significant  overlap  of the spin-split bands  in 3d \nferromagnets.  A precessing magnetization can induce elec tronic transitions between such \noverlapping bands , leading to spin-flip process es. This process scales with the amount of band \noverlap. Since s uch overlap is further increased with the band broadening th at results from the finite \ntemperature of the sample, this contribution  is expected to increase as the temperature is increased. \nThis model for interband transition mediated damping describes the resistivity -like behavior of the \ndamping at higher temperatu res (shorter scattering times).  These two damping processes are  \ncombined in a torque correlation model by Gilmore , et al.9, as well as Thonig , et al.10, that  describes \nboth the low -temperature  (intra band transitio ns) and high -temperature  (inter band transitions) \nbehavior  of the damping . Another app roach via  scattering theory was successfully implemented by \nBrataas , et al.11 to describe damping in transition metals. Starikov , et al. ,2 applied the scattering \nmatrix approach to calculate the damping  of NixFe1-x alloys and  Liu, et al. ,12 expanded the formalism \nto include the influence of electron -phonon interaction s.  \nA numerical realization of the torque correlation model was performed  by Mankovsky , et \nal., for NixCo1-x, Ni xFe1-x, CoxFe1-x, and FexV1-x1. More recently , Turek , et al. ,3 calculated the \ndamping for NixFe1-x and CoxFe1-x alloys with the torque -correlation model, u tilizing non -local \ntorque correlators. It is important to stress that a ll of these  approaches  consider  only the intrinsic \ndamping. This complicates the quantitative comparison of calculated values for t he damping to \nexperimental data  since there are many extrinsic contributions to the damping that result from \nsample  structure , measurement geometr y, and/or sample properties . While some extrinsic \ncontributions to the damping  and linewidth  were discovered in the 1960 ’s and 1970 ’s, and are well \ndescribed  by the ory, e.g. eddy -current damping13,14, two -magnon scattering15–17, the slow rel axer \nmechanism18,19, or radiative damping20,21, interest in these mechanisms has been re -ignited \nrecently22,23. Further contributions, such as  spin-pumping, both extrinsic24,25 and intrinsic24,26, have 3 been discovered more recently and a re subject to extensive research27–31 for spintronics application.  \nTherefore , in order to allow a quantitative comparison to theoretical calculations  for intrins ic \ndamping, both the measurement and sample geometry must  be designed  to allow both the \ndetermination and possibl e minimization  of all additional contributions to the measured damping.  \nIn this study , we demonstrate methods to determine significant extrins ic contributions to the \ndamping , which  includ es a measurement of the effective spin mixing conductance for both the pure \nelements and select alloys. By precisely accounting for all of these extrinsic contributions, we \ndetermine the intrinsic damping  parame ters of the binary alloys Ni xCo1-x, Ni xFe1-x and Co xFe1-x and \ncompare them  to the calculation s by Mankovsky , et al. ,1, Turek , et al. , and Starikov , et al.2. \nFurthermore, we present the concentration -dependence of the  inhomogeneous linewidth \nbroadening, which for most alloys shows exceptionally small values, indicative  of the high \nhomogeneity of our samples.  \n2 Samples and method  \nWe deposited  NixCo1-x, Ni xFe1-x and Co xFe1-x alloys of varying composition (all compositio ns \ngiven in atomic percent)  with a thickness of 10 nm  on an oxidized (001)  Si substrate with  a Ta(3 \nnm)/Cu(3 nm) seed layer and  a Cu(3 nm) /Ta(3 nm) cap layer. In order to investigate interface \neffects , we also deposited multiple thickness series at 10 nm, 7 nm, 4 nm, 3 nm , and 2 nm of  both \nthe pure elements and select alloy s.  Structural characterization  was performed using X-ray \ndiffraction (XRD).  Field swept vector -network -analyzer  ferromagnetic resonance spectroscopy \n(VNA -FMR) was used in the out -of-plane geometry  to determine the total damping parameter αtot. \nFurther d etails of the deposition conditions, XRD, FMR measurement and fitting of the complex \nsusceptibility to the measured S21 parameter are reported in  Ref [66]. \nAn example of susceptibility fits t o the complex S21 data is shown in Fig.  1 (a) and (b). All \nfits were constrained  to a 3×  linewidth ΔH field window around the resonance field in order to \nminimize the influence of measurement drifts on the error in the susceptibility fits. The total \ndampin g parameter αtot and the inhomogeneous linewidth broadening Δ H0 are then determined  from \na fit to  the linewidth Δ H vs. frequency f plot22, as shown in Fig.  1 (c). \n ∆𝐻= 4𝜋𝛼𝑡𝑜𝑡𝑓\n𝛾𝜇0+ ∆𝐻0, (1) \nwhere γ=gμB/ħ is the gyro -magnetic ratio, μ0 is the vacuum permeability, μB is the Bohr -magneton , \nħ is the reduced Planck constant,  and g is the spectroscopic g-factor reported in Ref [ 66].  \n  4  \nFigure 1: (a) and (b) show respectively the real and imaginary part of the S21 transmission \nparameter (black squares)  measured at 20 GHz  with the complex susceptibility fit (red lines)  for \nthe Ni 90Fe10 sample . (c) The linewidths  from the suscept ibility fits  (symbols) and linear fits (solid \nlines) are plotted against frequency for different Ni -Fe compositions. Concentrations are denoted \non the right -hand  axis. The damping α and the inhomogeneous linewidth broadening Δ H0 for each \nalloy can be extracted from the fits via Eq.  (1). \n3 Results  \nThe first contribution t o the linewidth we discuss  is the  inhomogeneous linewidth broadening \nΔH0, which  is presumably  indicative of  sample inhomogeneity32,33. We plot Δ H0 for all the alloy \nsystem s against the respective concentrations in Fig.  2. For all alloys , ΔH0 is in the range of a few \nmT to 10  mT . There are only a limited  number of reports  for ΔH0 in the literature with which to \ncompare . For Permalloy (Ni 80Fe20) we measure Δ H0 = 0.35 mT, which is close to other reported \nvalues .34 For the  other  NixFe1-x alloys , ΔH0 exhibits a significant peak  near the fcc-to-bcc (face -\ncentered -cubic to body -centered -cubic) phase transition  at 30  % Ni , (see Fig.  2 (b)) which is easily \nseen in the raw data  in Fig. 1 (c). We speculate  that this  increase of  inhomogeneous broadening  in \nthe NixFe1-x is caused by the coexistence of the bcc and fcc  phases at the phase transition.  However,  \nthe CoxFe1-x alloys do not exhibit an increase in ΔH0 at the equivalent phase transition  at 70  % Co . \nThis suggests that the bcc and fcc phases of NixFe1-x tend to segregate near the phase transition, \nwhereas the same phases for CoxFe1-x remain intermixed throughout the transition.  \n5 One possible explanation for inhomogeneous broadening is magnetic anisotropy , as originally \nproposed  in Ref. [35]. However, this explanation does not account for our measured dependence of \nΔH0 on alloy concentration, since the perpendicular magnetic anisotropy , described in Ref [ 66] \neffectively  exhibits opposite behavior with alloy concentration.  For our alloys Δ H0 seems to roughly \ncorrelate to the inverse exchange constant36,37, which co uld be a starting point for future \ninvestigation of a quantitative theory of inhomogeneous broadening.  \n \n \n \n  \nFigure 2: The inhomogeneous linewidth -broadening Δ H0 is plotted vs. alloy composition for (a) \nNi-Co, (b) Ni -Fe and (c) Co -Fe. The alloy phases are denoted by color code described in Ref [ 66] \n \nWe plot the total measured damping αtot vs. composition for NixCo1-x, NixFe1-x and CoxFe1-x in \nFig. 3 (red crosses). The total damping of the NixCo1-x system increases monotonically  with \nincreased Ni content . Such smooth  behavior in the damping is  not surprising  owing to the absence \nof a phase transition for this alloy . In the NixFe1-x system , αtot changes very little from pure Fe to \napproximately 25 % Ni  where the bcc to fcc phase transition occurs . At the phase  transition,  αtot \nexhibits a step, increasing  sharply by approximately 30  %. For higher Ni concentrations , αtot \nincreases monotonically  with increasing Ni concentration . On the other hand, t he CoxFe1-x system \nshows a different behavior in the damping and d isplays a sharp minimum of (2.3 ± 0.1)×10-3 at 25 \n6 % Co as previously reported38. As the system changes to an fcc phase ( ≈ 70 % Co), αtot become \nalmost  constant.  \nWe compare our data to previously published values in Table I. However, direct c omparison \nof our data to previous report s is not trivial , owing to the variation in measurement conditions  and \nsample characteristic s for all the reported measurements . For example , the damping  can depend  on \nthe temperature .9,39 In addition, multiple intrinsic and intrinsic contributions to the total damping  \nare not always accounted for in the literature . This can be seen in the fact that the reported  damping \nin Ni80Fe20 (Permalloy) varies  from α=0.0 055 to α=0.04  at room tem perature  among studies . The \nlarge variation  for these reported data  is possibly the result of  different uncontrolled contributions  \nto the extrinsic damping  that add  to the total damping in the different experiments , e.g. spin-\npumping40–42, or roughness41. Therefore , the value for the intrinsic damping of Ni 20Fe80 is expected \nto be at the low end of this scatter . Our measured value of α=0.007 2 lies within the range of reported \nvalues. Similarly , many of our measured damping values for different alloy compositions lie within \nthe range  of reported values22,43 –48. Our measured  damping of the pure elements and the Ni 80Fe20 \nand Co 90Fe10 alloys is compared to room temperature values found in literature in Table 1, Col umns  \n2 and 3 . Column 5 contains theoretically calculated values .   \n \nTable 1: The total measured damping α tot (Col. 2)  and the intrinsic damping (C ol. 4) f or Ni 80Fe20, \nCo90Fe10, and the pure elements are compared to both experimental (Col. 3) and theoretical (Col. \n5) values from the literature . All values of the damping  are at room temperature if not noted \notherwise . \nMaterial  αtot (this study)  \n Liter ature values  αint (this study)  Calculated literature \nvalues  \nNi 0.029 (fcc)  0.06444 \n0.04549 0.024 (fcc)  0.0179 (fcc) at 0K  \n0.02212 (fcc) at 0K  \n0.0131 (fcc)  \nFe 0.0036 (bcc)  0.001944 \n0.002746 0.0025 (bcc)  0.00139 (bcc) at 0K  \n0.001012 (bcc)  at 0K  \n0.00121 (bcc) at 0K  \nCo 0.0047 (fcc)  0.01144  \n 0.0029 (fcc)  0.00119 (hcp)  at 0K  \n0.0007312 (hcp)  at 0K  \n0.0011 (hcp)  \nNi80Fe20 0.0073 (fcc)  0.00844 \n0.008 -0.0450 \n0.007848 \n0.00751 \n0.00652 \n0.00647 \n0.005553 0.0050 (fcc)  0.00462,54 (fcc) at 0K  \n0.0039 -0.00493 (fcc)  at 0K  \nCo90Fe10 0.0048 (fcc)  0.004344 \n0.004855 0.0030 (fcc)   7  \n \n \n \n  \nFigure 3: (color online) The measured damping αtot of all the alloys is plotted against the alloy \ncompositi on (red crosses) for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (the data in (c) are taken from \nRef.[38]). The black squares  are the intrinsic damping  αint after correction for spin pumping and \nradiative contributions to the measured damping. The blue line is the intr insic damping calculated \nfrom the Ebert -Mankovsky theory ,1 where the blue circles are the values for the pure elements  at \n300K . The green  line is the calculated damping for the Ni -Fe alloys  by Starikov , et al.2 The inset \nin (b) depicts the damping in a smaller concentration window in order to better depict the small \nfeatures in the damping around the ph ase transition. The damping  for the Co -Fe alloys, calculated \nby Turek et al.3 is plotted as the orange  line. For the Ni -Co alloys the damp ing calculated by th e \nspin density of the respective alloy weighted bulk damping55 (purple dashed line).  \n8  \n \nThis scatter in the experimental data reported in the literature and its divergence from calculated \nvalues of the damping shows th e necessity to determine  the intrinsic damping  αint by quantification \nof all extrinsic contributions to the measured total damping α tot.  \nThe first extrinsic  contribution to the damping  that we consider  is the radiative damping α rad, \nwhich is caused by ind uctive coupling between sample and  waveguide , which results in energy \nflow from the sample back into the microwave circuit.23 αrad depends directly  on the measurement \nmethod and geometry. The effect is easily understood , since the strength of the inductive coupling \ndepends on the inductance of the FMR mode itself , which is in turn determined by the saturation \nmagnetization, sampl e thickness, sample  length, and waveguide width.  Assuming a homogeneous \nexcitation field , a uniform magnetization profile throughout the sample , and negligible spacing \nbetween the waveguide and sample , αrad is well approximated by23  \n 𝛼rad=𝛾𝑀𝑠𝜇02𝛿𝑙\n16 𝑍0𝑤𝑐𝑐, (2) \nwhere l (= 10 mm  in our case) is the sample length on the waveguide, wcc (= 100µm) is the width \nof the co -planar wave guide center conductor and Z0 (= 50 Ω), the impedance of the waveguide. \nThough inh erently small for most thin films , αrad can become  significant  for alloys with \nexceptionally small intrinsic damping and /or high saturation magnetization.  For example, it plays \na significant role (values of αrad ≈ 5x10-4) for the whole composition range of  the Co -Fe alloy system \nand the Fe -rich side of the Ni -Fe system. On the other hand, for pure Ni and Permalloy (Ni 80Fe20) \nαrad comprises only 3 % and 5  % of αtot, respectively.  \nThe second non -negligible contribution to the damping  that we consider  is the interfacial \ncontribution to the measured  damping , such as  spin-pumping into the adjacent Ta/Cu bilayers . Spin \npumping  is proportional to the reciprocal sample thickness as described in24 \n 𝛼sp= 2𝑔eff↑↓𝜇𝐵𝑔\n4𝜋𝑀s𝑡. (3) \nThe spectroscopic  g-factor and the saturation magnetization Ms of the alloys were reported  in \nRef [66] and the factor of 2 accounts for the presence of two nominally identical interfaces of the \nalloys in the cap and seed layers.  In Fig.  4 (a)-(c) we plot the damping dependence on reciprocal \nthickness  1/t for select alloy concentrations, which allows us to determine  the effective spin mixing \nconductance 𝑔eff↑↓  through fits to Eq. (3) . The effective spin m ixing conductance contains details of \nthe spin transport in the adjacent non -magnetic layers, such as the interfacial spin mixing \nconductance, both the conductivity and spin diffusion for all the non -magnetic layers with a non -\nnegligible spin accumulation,  as well as the details of the spatial profile for the net spin \naccumulation .56,57 The values of 𝑔eff↑↓, are plotted versus the alloy concentration in Fig. 4 (d), and are \nin the range of previously reported values  for samples prepare d under similar growth conditions55–\n59. Intermediate values of 𝑔eff↑↓ are determined by a guide to the eye interpolation [ grey lines, Fig.  4 \n(d)] and αsp is calculated for all alloy concentrations utilizing  those interpolated values.  \nThe data for 𝑔eff↑↓  in the NixFe1-x alloys shows approximately a factor two  increase of 𝑔eff↑↓ between \nNi concentrations of 30  % Ni and 50  % Ni, which we speculate to occur at the fcc to bcc phase \ntransition around 30  % Ni. According to this line of speculation , the previously mentioned step  \nincrease in the measured total damping at the NixFe1-x phase transition can be fully attributed to the \nincrease in spin pumping at the phase transition.  In CoxFe1-x, the presence of a step in  𝑔eff↑↓   at the \nphase transition is not confirmed, given the measurement precision, although  we do observe an \nincrease in the effective spin mixing conductance when transitioning from the bcc to fcc phase.  The 9 concentration dependence of 𝑔eff↑↓  requires further thorough investigation and we therefore restrict \nourselves to reporting the expe rimental findings.  \n \n \n           \n \nFigure 4: The damping for the thickness series at select alloy compositions vs. 1/ t for (a) Ni -Co, \n(b) Ni -Fe and (c) Co -Fe (data points, concentrations denoted in the plots), with linear fits to Eq. \n(3) (solid lines). (d) The extracted effective spin mixing conductance 𝑔eff↑↓  for the measured alloy \nsystems, where the gray lines show the  linear  interpolations for intermediate alloy concentrations. \nThe data for the Co -Fe system are taken from Ref.[38].  \n \n \n Eddy -current damping13,14 is estimated  by use of  the equations given in Ref.  [23]  for films  \n10 nm thick or less . Eddy currents are neglected because they are found to be less than 5  % of the \ntotal damping. Two -magnon scattering  is disregarded  because the mechanism is largely e xcluded \nin the out -of-plane measurement geometry15–17. The total measured damping  is therefore well \napproximated as the sum   \n 𝛼tot≅𝛼int+𝛼rad+𝛼sp, (4) \nWe determine  the intrinsic damping of the material by subtracting α sp and α rad from the measured \ntotal damping , as shown in Fig. 3 .  \n10 The intrinsic damping increases monotonically with Ni concentration  for the NixCo1-x alloys . \nIndicative of the importance of extrinsic sources of damping, αint is approximately 40 % smaller \nthan αtot for the Fe -rich alloy, though the difference decreases to only 15 % for pure Ni. This \nbehavior is expected, given that both αrad and αsp are proportional to  Ms. A comparison of αint to the \ncalculations by Mankovsky , et al. ,1 shows excellent quantitative agreement  to within 30 %.  \nFurthermore, w e compare αint of the NixCo1-x alloys to the spin density weighted average of the \nintrinsic damping of Ni and C o [purple dashed line in Fig. 3 (a)] , which gives good agreement with \nour data, as previously reported .55  \nαint for NixFe1-x (Fig. 3 (b)) also increases with Ni concentration after a small initial decrease \nfrom pure Fe to the first NixFe1-x alloys. The step increase found in αtot at the bcc to fcc phase \ntransition is fully attributed to αsp, as detailed in the previous section, and therefore does not occur \nin αint. Similar to the NixCo1-x system αint is significantly lower than αtot for Fe -rich alloys. With in \nerror bars, a  comparison to the calculations by Mankovsky , et al.1 (blue line) and Starikov , et al.2 \n(green line) exhibit excellent agreement in the fcc phase, with marginally larger deviations in the \nNi rich regime. Starikov , et al.2 calculated the damping over the ful l range of compositions, under \nthe assumption of continuous  fcc phase. This calcu lation deviates further from  our measured αint in \nthe bcc phase exhibiting qualitatively different behavior.  \nAs previously reported, t he dependence of αint on alloy compositio n in the CoxFe1-x alloys \nexhibits strongly non -monotonic behavior, differing from the two previously discussed alloys.38  \nαint displays a minimum at 25 % Co concentration with a, for conducting ferromagnets \nunprecedented, low value of int (5±1.8)  × 10-4. With increasing Co concentration , αint grows up \nto the phase transition, at which point  it increases by 10  % to 20 % unt il it reaches  the value for \npure Co.  It was shown  that αint scales with the density of states  (DOS)  at the Fermi energy n( EF) in \nthe bcc phase38, and the DOS also exhibits a sharp  minimum for Co 25Fe75. This scaling is \nexpected60,61 if the damping is dominated by the breathing Fermi surface process. With the \nbreathing surface model,  the intraband scattering  that leads to damping  directly scales with n( EF). \nThis scaling is particularly pronounc ed in the Co -Fe alloy system due to the small concentration  \ndependence of the spin -orbit coupling on alloy composition. The special properties of the CoxFe1-x \nalloy system are discussed  in greater detail in Ref.[38]. \nComparing αint to the calculations by Mankovsky  et al.1, we find good quantitative \nagreement with the value of the minimum. However, t he concentration of the minimum is \ncalculated to occur  at approximately 10 % to  20% Co, a slightly lower value than 25 % Co t hat we \nfind in this study. Furthermore , the strong concentration dependence around the minimum is not \nreflected in the calculations. More recent calculations by Turek et al.3, for the bcc CoxFe1-x alloys \n[orange line in Fig. 3 (c)] find the a minimum of the damping of 4x10-4 at 25 % Co concentration \nin good agreement with our experiment, but there is some deviation in concentration dependenc e \nof the damping around the minimum. Turek et al.3 also reported on the damping in the NixFe1-x \nalloy system, with similar qualitative and  quantitative results as the other two presented quantitative \ntheories1,2 and the results are therefore not plotted  in Fig.  3 (b) for the sake of comprehensibility of \nthe figure.  For both NixFe1-x and the CoxFe1-x alloys , the calculated spin density weighte d intrinsic \ndamping of the pure elements (not plotted) deviates significantly from the determined intrinsic \ndamping of the alloys, in contrary to the  good agreement  archived  for the  CoxNi1-x alloys. We \nspeculate that this difference between the alloy syste ms is caused by the non -monotonous \ndependence of the density of states at the Fermi Energy in the CoxFe1-x and NixFe1-x systems.  \nOther  calculated damping values for the pure elements and the Ni80Fe20 and Co 90Fe10 alloys \nare compared to the determined intr insic damping in Table 1. Generally , the calculations \nunderestimate the damping significantly, but our data are in good agreement with more recent \ncalculations for Permalloy ( Ni80Fe20).   11 It is important to point out that n one of the theories  considered he re include thermal \nfluctuations . Regardless, we find exceptional agreement with the calculations to αint at intermediate \nalloy concentrations . We speculate that the modeling of atomic disorder in the alloys in the \ncalculations, by the coherent potential approximation (CPA)  could be responsible for this \nexceptional agreement. The effect of disorder on the  electronic band structure possibly dominates \nany effect s due to nonzero temperature. Indeed, both effects cause a broadening of the bands due \nto enhanced  momentum scattering rates. This directly correlates to a change of the damping \nparameter  according to  the theory of Gilmore and Stiles9. Therefore , the inclusion of the inherent  \ndisorder  of solid -solution alloys  in the calculations  by Mankovsky et al1 mimic s the effects of \ntemperature on damping  to some extent . This argument is corroborated by the fact that the \ncalculations  by Mankovsky et al1 diverge for diluted alloys and pure elements  (as shown in Fig. 2 \n(c) for pure Fe) , where no  or to little disorder is introduced to account for temperature effects. \nMankovsky et al.1 performed temperature dependent calculations of the damping for pure bcc Fe, \nfcc Ni and hcp Co  and the values for 300 K are shown in Table 1 and Fig. 3. These calculations  for \nαint at a temperature of 300 K  are approximately a factor of two  less than our measured values , but \nthe agreement is  significantly improved relative to those  obtained by calculations that neglect \nthermal fluctuations . \n \n \n Figure 5: The intrinsic damping α int is plotted against ( g-2)2 for \nall alloys. We do not observe a proportionality between α int and \n(g-2)2. \n12 Finally, i t has been  reported45,64 that there is a  general proportionality between αint and (g-\n2)2 , as contained in the original microscopic BFS model proposed by  Kambersky .62 To examine \nthis relationship, w e plot αint versus (g-2)2 (determined in Ref [66]) for all samples measured here  \nin Figure 5 . While some samples with large values for ( g-2)2 also exhib it large αint, this is not a \ngeneral trend for all the measured samples . Given  that the damping is not purely a function of the \nspin-orbit strength, but also depends on the details of the band structure , the result  in Fig.  5 is \nexpected .  For example , the amount of  band overlap  will determine the amount of interband \ntransition leading to that damping channel.  Furthermore, the density of states at the Fermi energy \nwill affect the intraband contribution to the damping9,10.   Finally , the ratio of inter - to intra -band \nscattering that mediate s damping contributions at a fixed temperature (RT for our measurements) \nchanges  for different elements9,10 and therefore with alloy concentration. None of these f actors are \nnecessarily proportional to the spin -orbit coupling .  Therefore , we conclude that this simple \nrelation, which originally traces to an order of magnitude estimate  for the case of spin relaxation \nin semiconductors65, does not hold for  all magnetic systems  in general.   \n \n4  Summary  \nWe determined  the damping for the full compositi on range of the binary  3d transition  metal all oys \nNi-Co, Ni -Fe, and Co -Fe and showed that the measured damping can be explained by three \ncontributions to the damping: Intrinsic damping, radiative damping and damping due to spin \npumping. By quantifying all  extrinsic contributions to the measured damping, we determine  the \nintrinsic damping over the whole range of alloy compositions .  These values are compared  to \nmultiple theoretical calculations and yield excellent qualitative and good quantitative agreement for \nintermediate alloy concentrations. 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Schoen, Martin A. W. , Lucassen, Juriaan , Nembach, Hans T. , Silva, T. J. , Koopmans, Bert , Back, Christian H. , \nShaw, Justin M.  Magnetic properties of ultra -thin 3d transition -metal binary alloys I: spin and orbital mo ments, \nanisotropy, and confirmation of Slater -Pauling behavior . arXiv:1701.02177  (2017 ).  \n \n \n "    },    {        "title": "1706.04670v2.Temperature_dependent_Gilbert_damping_of_Co2FeAl_thin_films_with_different_degree_of_atomic_order.pdf",        "content": "1 \n Temperature -dependent Gilbert damping of Co 2FeAl thin films with different degree of \natomic order  \nAnkit Kumar1*, Fan  Pan2,3, Sajid  Husain4, Serkan  Akansel1, Rimantas  Brucas1, Lars \nBergqvist2,3, Sujeet Chaudhary4, and Peter  Svedlindh1# \n \n1Department of Engineering Sciences, Uppsala University, Box 534, SE -751 21 Uppsala, \nSweden \n2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of \nTechnology, Electrum 229, SE -16440 Kista, Sweden \n3Swedish e -Science Research Center, KTH Roy al Institute of Technology, SE -10044 \nStockholm, Sweden \n4Department of Physics, Indian Institute of Technology Delhi, New Delhi -110016, India  \n \nABSTRACT  \nHalf-metallicity and low magnetic damping are perpetually sought for in spintronics materials \nand full He usler alloys in this respect provide outstanding properties . However, it is \nchallenging to obtain the well -ordered half-metallic phase in as -deposited full Heusler alloys \nthin films and theory has struggled to establish  a fundamentals understanding of the \ntemperature dependent Gilbert damping in these systems. Here we present a study of the \ntemperature dependent Gilbert damping of differently ordered as -deposited Co 2FeAl full \nHeusler alloy thin films. The sum of inter - and intraband electron scattering in conjunction \nwith the finite electron lifetime in Bloch states govern the Gilbert damping for the well -\nordered phase in contrast to the damping of partially -ordered and disordered phases which is \ngoverned by interband electronic scat tering alone. These results, especially the ultralow room \ntemperature intrinsic damping observed for the well -ordered phase provide new fundamental \ninsight s to the physical origin of the Gilbert damping in full Heusler alloy thin films.   \n 2 \n INTRODUCTION \nThe Co-based full Heusler alloys have gained massive attention over the last decade due to \ntheir high Curie temperature and half-metallicity; 100% spin polarization of the density of \nstates at the Fermi level [1 -2]. The room temperature half- metallicity and lo w Gilbert \ndamping make them ideal candidates for magnetoresistive and thermoelectric spintronic \ndevices [3]. Co2FeAl (CFA), which is one of the most studied Co-based Heusler alloys , \nbelongs to the 𝐹𝐹𝐹𝐹 3𝐹𝐹 space group, exhibits half-metallicity  and a high C urie temperature \n(1000 K) [2, 4]. In CFA, half-metallicity is the result of hybridization between the d orbitals \nof Co and Fe. The d orbitals of Co hybridize resulting in bonding (2e g and 3t 2g) and non-\nbonding hybrids (2e u and 3t 1u). The bonding hybrids of Co further hybridise with the d \norbitals of Fe yielding bonding and anti -bonding hybrids. However, the non-bonding hybrids \nof Co cannot hybridise with the d orbitals of Fe. The half-metallic gap arises from the \nseparation of non-bonding states, i.e. the conduction band of e u hybrids and the valence band \nof t 1u hybrids [5, 6]. However, chemical or atomic disorder modifies the band hybridization \nand results in a reduc ed half-metallicity in CFA. The ordered phase of CFA is the L2 1 phase, \nwhich is half -metallic [7]. The partially ordered B2 phase forms when the Fe and Al atoms \nrandomly share their sites, while the disordered phase forms when Co, Fe, and Al atoms \nrandomly share all the sites [5-8]. These chemical disorders strongly influence the physical \nproperties and result in additional states at the Fermi level therefore reducing the half-\nmetallicity or spin polarization [7, 8]. It is challenging to obtain the ordered L2 1 phase of \nHeusler alloys in as-deposited films, which is expecte d to possess the lowest Gilbert damping \nas compared to the other phases [4, 9-11]. Therefore, in the last decade several attempts have \nbeen made to grow the ordered phase of CFA thin films employing different methods [4, 9-\n13]. The most successful attempts  used post -deposition annealing to reduce the anti -site \ndisorder by a thermal activation process [4]. The observed value of the Gilbert damping for \nordered thin films was found to lie in the range of 0.001-0.004 [7-13]. However, the \nrequirement of post -deposition annealing might not be compatible with the process constraints \nof spintronics and CMOS devices. The annealing treatment requirement for the formation of \nthe ordered phase can be circumvented by employing energy enhanced growth mechanisms \nsuch as io n beam sputtering where the sputtered species carry substantially larger  energy, ~20 \neV , compared to other deposition techniques [14, 15]. This higher energy of the sputtered \nspecies enhances the ad -atom mobility during coalescence of nuclei in the initial  stage of the \nthin film growth, therefore enabling the formation of the ordered phase. Recently we have 3 \n reported growth of the ordered CFA phase on potentially advantageous Si substrate  using ion \nbeam sputtering. The samples deposited in the range of 300°C to 500°C substrate temperature \nexhibited nearly equivalent I(002)/I(004) Bragg diffraction intensity peak ratio, which \nconfirms at least B2 order ed phase as it is difficult to identify  the formation of the L2 1 phase \nonly by X -ray diffraction analysis [16] .  \nDifferent theoretical approaches have been employed to calculate the Gilbert damping in Co -\nbased full Heusler alloys, including first principle calculations on the ba sis of (i) the torque \ncorrelation  model [17], (ii) the fully relativistic Korringa -Kohn-Rostoker model in \nconjunction with the coherent potential approximation and the linear response formalism [8] , \nand (iii) an approach considering different exchange correlation effects using both the local \nspin density approximation including the Hubbard  U and the local spin density approximation \nplus the dynamical mean field theory approximation [7]. However, very little is known about \nthe temperature dependence of the Gilbert damping in differently ordered Co-based Heusler \nalloys and a unifying conse nsus between theoretical and experimental results is still lacking. \nIn this study we report the growth of differently ordered  phases, varying from disordered to \nwell-ordered phases, of as -deposited CFA thin films grown on Si employing ion beam \nsputtering a nd subsequently the detailed temperature dependent measurements of the Gilbert \ndamping. The observed increase in intrinsic Gilbert damping with decreasing temperature in \nthe well -ordered sample is in contrast to the continuous decrease in intrinsic Gilbert  damping \nwith decreasing temperature observed for partially ordered and disordered phases. These \nresults are satisfactorily explained by employing spin polarized relativistic Korringa -Kohn-\nRostoker band structure calculations in combination with the local spin density \napproximation.  \nSAMPLES & METHODS  \nThin films of CFA were deposited on Si substrates at various growth temperatures using ion \nbeam sputtering system operating at 75W RF ion-source power ( 𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖). Details of the \ndeposition process as well as st ructural and magnetic properties of the films have been \nreported elsewhere [16]. In the present work to study the temperature dependent Gilbert \ndamping of differently ordered phases (L2 1 and B2) we have chosen CFA thin films deposited \nat 573K, 673K and 773K substrate temperature ( 𝑇𝑇𝑆𝑆) and the corresponding samples are named \nas LP573K, LP673K and LP773K, respectively. The sample thickness was kept constant at 50 \nnm and the samples were capped with a 4 nm thick Al layer. The capping layer protects the \nfilms by forming a 1.5 nm thin protective layer of Al 2O3. To obtain the A2 disordered CFA 4 \n phase, the thin film was deposited at 300K on Si employing 100W ion-source power, this \nsample is referred to  as HP300K. Structural and magnetic properties of this film are presented \nin Ref. [18]. The absence of the (200) diffraction peak in the HP300K sample [18] reveals that \nthis sample exhibits the A2 disordered structure. The appearance of the (200) pea k in the LP \nseries samples clearly indicates at least formation of B2 order [16]. Employing the Webster model along with the analysis approach developed by Takakura et al. [19] we have calculated \nthe degree of B2 ordering in the samples, S\nB2= �I200 I220⁄\nI200full orderI220full order�� , where \nI200 I220⁄  is the experimentally obtained intensity ratio of the (200) and (220) diffractions and \nI200full orderI220full order⁄  is the theoretically calculated intensity ratio for fully ordered B2 structure \nin polycrystalline films  [20]. The estimated values of SB2 for the LP573, LP673, and LP773 \nsamples are found to be ∼  90 %, 90% and 100%, respectively , as presented in Ref.  [20]. The \nI200 I400⁄  ratio of the (200) and (400) diffraction peaks for all LP series samples is ∼  30 %, \nwhich compares well with  the theoretical value for perfect B2 order [21, 22]. Here it is \nimportant to note that the L21 ordering parameter, SL21, will take different values depending \non the degree of B2 ordering. S L21 can be calculated from the I111 I220⁄  peak ratio in \nconjunction with the SB2 ordering parameter [19]. However, in the recorded grazing incident \nXRD spectra on the polycrystalline LP samples (see Fig. 1 of Ref. [16]) we did not observe \nthe (111) peak. This could be attributed to the fact t hat theoretical intensity of this peak is only \naround two percent of the (220) principal peak. The appearance of this peak is typically \nobserved in textured/columnar thicker films [19, 23 ]. Therefore, here using the experimental \nresults of the Gilbert damping, Curie temperature and saturation magnetization, in particular employing the temperature dependence of the Gilbert damping that is very sensitive to the \namount of site disorder in CFA films, and comparing with corresponding results obtained \nfrom first principle calculations, we provide a novel method for determining the type of \ncrystallographic ordering in full Heusler alloy thin films.  \nThe observed values of the saturation magnetization ( µ0MS) and coercivity ( µ0Hci), taken \nfrom Refs. [16, 18] are presented in Table I. The temperature dependence of the magnetization \nwas recorded in the high temperature region (300–1000K) using a vibrating sample \nmagnetometer i n an external magnetic field of 𝜇𝜇 0𝐻𝐻=20 mT. An ELEXSYS EPR \nspectrometer from Bruker equipped with an X -band resonant cavity was used for angle \ndependent in-plane ferromagnetic resonance (FMR) measurements . For studying the 5 \n temperature dependent spin dynamics in the magnetic thin films, an in-house built out -of-\nplane FMR setup was used. The set up, using a Quantum Design Physical Properties \nMeasurement System covers the temperature range 4 – 350 K and the magnetic field range \n±9T. The system employs an Agilent N5227A PNA network analyser covering the frequency \nrange 1 – 67 GHz and an in-house made coplanar waveguide. The layout of the system is \nshown in Fig. 1. The complex transmission coefficient ( 𝑆𝑆21) was recorded as a function of \nmagnetic field for different frequencies in the range 9-20 GHz and different temperatures in \nthe range 50-300 K.  All FMR measurements were  recorded keeping constant 5 dB power.  \nTo calculate the Gilbert damping, we have the used the torque –torque correlation model [7, \n24], which includes both intra - and interband transitions. The electronic structure was \nobtained from the spin polarized relativistic Korringa -Kohn-Rostoker (SPR- KKR) band \nstructure method [24, 25] and the local spin density approximation (LSDA) [26] was used for \nthe exchange correlation potential. Relativistic effects were taken into account by solving the Dirac equation for the electronic states, and the atomic sphere approximation (ASA) was employed for the shape of potentials. The experimental bulk value of the lattice constant [27] \nwas used. The angular momentum cut -off of 𝑙𝑙\n𝑚𝑚𝑚𝑚𝑚𝑚 =4 was used in the mu ltiple -scattering \nexpansion.  A k-point grid consisting of ~1600 points in the irreducible Brillouin zone was \nemploye d in the self -consistent calculation while a substantially more dense grid of ~60000  \npoints was employe d for the Gilbert damping calculation.  The exchange parameters 𝐽𝐽 𝑖𝑖𝑖𝑖 \nbetween the atomic magnetic moments were calculated using the magnetic force theorem \nimplemented in the Liechtenstein -Katsnelson -Antropov-Gubanov (LKAG) formalism [28, 29] \nin order to construct a parametrized mod el Hamiltonian. For the B2 and L2 1 structures, the \ndominating exchange interactions were found to be between the Co and Fe atoms, while in A2 the Co-Fe and Fe -Fe interactions are of similar size. Finite temperature properties such as the \ntemperature dependent magnetization was obtained by performing Metropolis Monte Carlo \n(MC) simulations [30] as implemented in the UppASD software [31,  32] using the \nparametrized Hamiltoni an. The coherent potential approximation (CPA) [33, 34] was ap plied \nnot only for the treatment of the chemical disorder of the system, but also used to include the \neffects of quasi -static lattice displacement and spin fluctuations in the calculation of the \ntemperature dependent Gilbert damping [35–37] on the basis of linear response theory [38].  \nRESULTS & DISCUSSION  \nA. Magnetization vs. temperature measurements  6 \n Magnetization measurements were performed with the ambition to extract values for the \nCurie temperature ( 𝑇𝑇𝐶𝐶) of CFA films with different degree of atomic order; the results a re \nshown in Fig. 2(a). Defining 𝑇𝑇𝐶𝐶 as the inflection point in the magnetization vs. temperature \ncurve, the observed values are found to be 810  K, 890 K and 900 K for the LP573K, LP773K \nand LP673K samples, respectively. The 𝑇𝑇𝐶𝐶 value for the HP300K sample is similar to the \nvalue obtained for LP573. Using the theoretically calculated exchange interactions, 𝑇𝑇𝐶𝐶 for \ndifferent degree of atomic order in CFA varying from B2 to L2 1 can be calculated using MC \nsimul ations. The volume was kept fixed as the degree of order varied between B2 and L2 1 \nand the data presented here represent the effects of differently ordered CFA phases. To obtain \n𝑇𝑇𝐶𝐶 for the different phases, the occupancy of Fe atoms on the Heusler alloy 4a sites was varied \nfrom 50% to 100%, corresponding to changing the structure from B2 to L2 1. The estimated \n𝑇𝑇𝐶𝐶 values , cf. Fig. 2 (b), monotonously increases from 𝑇𝑇 𝐶𝐶=810 K (B2) to 𝑇𝑇𝐶𝐶=950 K \n(L2 1). A direct comparison between experimental and calculated  𝑇𝑇𝐶𝐶 values is hampered by the \nhigh temperature (beyond 800K) induced structural transition from well -ordered to partially -\nordered CFA phase which interferes with the magnetic transition [39, 40]. The irreversible \nnature of the recorded magnetization vs . temperature curve  indicates a distortion of structure \nfor the ordered phase during measurement , even though interface alloying at elevated \ntemperature cannot be ruled out . The experimentally observe d 𝑇𝑇𝐶𝐶 values are presented in \nTable I.  \nB. In-plane angle dependent FMR measurements  \nIn-plane angle dependent FMR measurements were performed at 9.8 GHz frequency for all \nsamples; the resonance field 𝐻𝐻𝑟𝑟 vs. in -plane angle  𝜙𝜙𝐻𝐻 of the applied magnetic field is plotted \nin Fig. 3. The  experimental results have been fitted using the expression [41],  \n𝑓𝑓=\n𝑔𝑔∥𝜇𝜇𝐵𝐵𝜇𝜇0\nℎ��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−𝜙𝜙𝑀𝑀)+2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−\n𝜙𝜙𝑀𝑀)+ 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒+𝐾𝐾𝑐𝑐\n2𝜇𝜇0𝑀𝑀𝑠𝑠(3+cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2 (𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��12�\n, (1) \nwhere 𝑓𝑓 is resonance frequency , 𝜇𝜇𝐵𝐵 is the Bohr magneton  and ℎ is Planck constant . 𝜙𝜙𝑀𝑀, 𝜙𝜙𝑢𝑢 \nand 𝜙𝜙𝑐𝑐 are the in -plane directions of the magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively , with respect to the [100] direction of the Si substrate . 𝐻𝐻𝑢𝑢=2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠 and \n𝐻𝐻𝑐𝑐=2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠 are the in-plane uniaxial and cubic anisotropy fields , respectively, and 𝐾𝐾𝑢𝑢 and 𝐾𝐾𝑐𝑐 7 \n are the uniaxial and cubic magnetic anisotrop y constant s, respectively, 𝑀𝑀𝑠𝑠 is the saturation \nmagnetization and 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 is the effective magnetization . By considering 𝜙𝜙 𝐻𝐻 ∼ 𝜙𝜙𝑀𝑀, 𝐻𝐻𝑢𝑢 and 𝐻𝐻𝑐𝑐 \n<<𝐻𝐻𝑟𝑟<< 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, equation (1) can be simplified as:  \n𝐻𝐻𝑟𝑟=�ℎ𝑒𝑒\n𝜇𝜇0𝑔𝑔∥𝜇𝜇𝐵𝐵�21\n𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒−2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝐻𝐻−𝜙𝜙𝑐𝑐)−2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝐻𝐻−𝜙𝜙𝑢𝑢) . (2) \nThe extracted cubic anisotropy fields µ 0Hc ≤ 0.22mT are negligible for all the samples. The \nextracted in -plane Landé splitting factors g∥ and the uniaxial anisotropy fields  µ0Hu are \npresented in T able I. The purpose of the angle dependent FMR measurements was only to \ninvestigate the symmetry of the in -plane magnetic anisotropy. Therefore, care was not taken \nto have the same in -plane orientation of the samples during angle dependent FMR \nmeasurements, which explains why the maxima appear at diffe rent angles for the different \nsamples.  \nC. Out-of-plane FMR measurements  \nField -sweep out -of-plane FMR measurements were performed at different constant \ntemperatures in the range 50K – 300K and at different constant frequencies in the range of 9-\n20 GHz. Figure 1(b) shows the amplitude  of the complex transmission coefficient 𝑆𝑆21(10 \nGHz) vs. field measured for the LP673K thin film at different temperatures. The recorded \nFMR spectra were fitted using the equation [42],  \n𝑆𝑆21=𝑆𝑆�∆𝐻𝐻2��2\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐴𝐴�∆𝐻𝐻2��(𝐻𝐻−𝐻𝐻𝑟𝑟)\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐷𝐷∙𝑡𝑡,  (3) \nwhere 𝑆𝑆 represents the coefficient describing the transmitted microwave power, 𝐴𝐴 is used to \ndescribe a waveguide induced phase shift  contribution  which is, however, minute , 𝐻𝐻 is \napplied magnetic field, ∆𝐻𝐻 is the full-width of half maxim um, and 𝐷𝐷∙𝑡𝑡 describes the linear \ndrift in time  (𝑡𝑡) of the recorded signal. The extracted ∆ 𝐻𝐻 vs. frequency at different constant  \ntemperatures  are shown in Fig. 4 for all the samples. For brevity only data at a few \ntemperatures are plotted. The Gilbert damping was estimated using the equation [42 ],  \n∆𝐻𝐻=∆𝐻𝐻0+2ℎ𝛼𝛼𝑒𝑒\n𝑔𝑔⊥𝜇𝜇𝐵𝐵𝜇𝜇0   (4) \nwhere ∆𝐻𝐻0 is the inhomogeneous line -width broadening, 𝛼𝛼 is the experimental Gilbert \ndamping constant , and 𝑔𝑔⊥ is the Landé  splitting factor measured employing out -of-plane \nFMR. The insets in the figures show the temperature dependence of  𝛼𝛼. The effective 8 \n magnetization ( 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) was estimated  from the 𝑓𝑓 vs. 𝐻𝐻𝑟𝑟 curves using out -of-plane Kittel’s \nequation [43],  \n𝑓𝑓=𝑔𝑔⊥𝜇𝜇0𝜇𝜇𝐵𝐵\nℎ�𝐻𝐻𝑟𝑟−𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒�,     (5) \nas shown in Fig. 5 . The temperature dependence of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 and 𝜇𝜇0∆𝐻𝐻0 are shown as  insets  in \neach figure . The observed room temperature values of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 are closely equal  to the  𝜇𝜇0𝑀𝑀𝑠𝑠 \nvalues obtained  from static magnetizat ion measurements, presented in T able I. The extracted \nvalues of g⊥ at different temperatures are within error limits  constant for all samples. \nHowever , the difference between estimated values of g ∥ and g⊥ is ≤ 3%. This difference c ould \nstem from the limited frequency range  used since  these values are quite sensitive to the value \nof 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, and even a minute uncertainty in this quantity can result in the observed small \ndifference between  the g∥ and g⊥ values.  \nTo obtain  the intrinsic Gilbert damping (𝛼𝛼 𝑖𝑖𝑖𝑖𝑖𝑖) all extrinsic contributions to the experimental 𝛼𝛼 \nvalue  need to be  subtracted. In metallic ferromagnets , the intrinsic Gilbert damping is mostly \ncaused by electron magnon scattering, but  several other extrinsic co ntributions can also \ncontribute to the experimental value of the damping constant. One  contribution is two -\nmagnon scattering which is however minimized for the  perpendicular geometry used in this \nstudy and therefore this contribution is disregarded [44].  Another contribution is spin-\npumping into the  capping layer as the LP573K, LP673K and LP773K samples are capped \nwith 4 nm of Al that naturally forms a thin top layer consisting of  Al2O3. Since spin pumping \nin low spin-orbit coupling materials with  thickness less than the spin-diffusion length is quite \nsmall this contribution is also disregarded in all samples. However, the HP300K sample is \ncapped with Ta and therefore a spin-pumping contribution have been subtracted from the \nexperimental 𝛼𝛼 value ; 𝛼𝛼𝑠𝑠𝑠𝑠= 𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(with  Ta capping )−𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(without capping )≈\n1×10−3. The third contribution arises  from the inductive coupling between  the precessi ng \nmagnetization and the CPW , a reciprocal phenomenon of FMR, known as radiative damping \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 [45]. This damping is directly proportional to the magnetization and thickness of the thin \nfilms samples and therefore usually dominates in thicker and/or high magnetization samples. \nThe l ast contribution is eddy current damping ( αeddy) caused  by eddy current s in metallic \nferromagnetic  thin films [ 45, 46]. As per Faraday’s law the time varying magnetic flux density  \ngenerate s an AC voltage in the metallic ferromagnetic  layer and therefore result s in the eddy 9 \n current damping . Thi s damping is directly proportional to the square of the film thickness  and \ninversely proportional to the resistivity of the sample [ 45].  \nIn contrast to eddy -current damping, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 is independent of the conductivity of the \nferromagnetic layer, hence this damping mechanism is also operati ve in ferromagnetic \ninsulators. Assuming a uniform magnetization of the sample the radiative damping can be \nexpressed as [45],  \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 𝜂𝜂𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿𝛿𝛿\n2 𝑍𝑍0𝑤𝑤 ,   (6) \nwhere 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵ℏ� is the gyromagnetic ratio, 𝑍𝑍0 = 50 Ω is the waveguide impedance, 𝑤𝑤 = 240 \nµm is the width of the waveguide, 𝜂𝜂 is a dimensionless parameter which accounts for the \nFMR mode profile and depends on boundary conditions, and 𝛿𝛿 and 𝑙𝑙 are the thickness and \nlength of the sample on the waveguide, respectively. The strength of this inductive coupling \ndepends on the inductance of the FMR mode which is determined by the waveguide width, \nsample length over waveguide, sample saturation magnetization and sample thickness.  The \ndimensions of the LP573K, LP673K  and LP773K samples were 6.3×6.3 mm2, while the \ndimensions of the HP300K sample were  4×4 mm2. Th e 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 damping was estimated  \nexperimentally as explained by Schoen et al.  [45] by  placing a 200 µm thick glass spacer \nbetween  the waveguide and the sample , which decreases  the radiative damping by more than \none order magnitude as shown in Fig. 6(a). The measured radiative damping by placing the \nspacer  between the waveguide and the LP773 sample, \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟=𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ𝑖𝑖𝑢𝑢𝑖𝑖  𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟  −𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟≈ (2.36 ±0.10×10−3) − (1.57 ±0.20×10−3)=\n0.79±0.22×10−3. The estimated value matches  well with the calculated value using Eq. \n(6); 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 0.78 ×10−3. Our results are also analogous to previously reported results  on \nradiative damping [45].  The estimated temperature dependent radiative damping values for all \nsamples are shown in Fig. 6(b).   \nSpin wave precession in ferromagnetic layers induces an AC current in the conducting \nferromagnetic layer which results in  eddy current damping. It can be  expressed as [45, 46], \n𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 = 𝐶𝐶𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿2\n16 𝜌𝜌 ,   (7) \nwhere 𝜌𝜌 is the resistivity of the sample and 𝐶𝐶 accounts for the eddy current distribution in the \nsample ; the smaller the value of 𝐶𝐶 the larger is the localization of eddy currents in the sample. \nThe measured  resistivity values between 300  K to 50 K temperature range fall in the ranges \n1.175 – 1.145  µΩ-m, 1 .055 – 1.034 µΩ -m, 1 .035 – 1.00 µΩ -m, and 1. 45 – 1.41 µΩ -m for the \nLP573K, LP673 , LP773 and HP300K samples, respect ively. The parameter 𝐶𝐶 was obtained 10 \n from thickness dependent experimental Gilbert damping constants measured for B2 ordered  \nfilms,  by line ar fitting of 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs. 𝛿𝛿2 keeping other parameters  constant  (cf. Fig. \n6(c)). The fit to the data yield ed 𝐶𝐶 ≈ 0.5±0.1. These results are concurrent to those  \nobtained for permalloy  thin films [45]. Since the variations of the resistivity and \nmagnetization for the samples  are small , we have used the same 𝐶𝐶 value  for the estimation of \nthe eddy current damping in all the samples. The estimated temperature dependent values of \nthe eddy current damping are presented in Fig. 6(d).  \nAll these contributions have  been subtracted from the experimentally observed values of 𝛼𝛼. \nThe estimated intrinsic Gilbert damping 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values so obtained are plotted in Fig. 7(a) for all \nsamples.   \nD. Theoretical results: first principle calculations  \nThe calculated temperature dependent intrinsic Gilbert damping for Co 2FeAl phases with \ndifferent degree of atomic order are shown in Fig. 7(b). The temperature dependent Gilbert \ndamping indicates that the lattice displacements and spin fluctuations contribute differently in \nthe A2, B2 and L2 1 phases. The torque correlation model [47, 48] describes qualitatively two \ncontributions to the Gilbert damping. The first one is the intraband scattering where the band \nindex is always conserved. Since it has a linear dependence on the electron lifetime, in the \nlow temperature regime this term increases rapid ly, it is also known as the conductivity like \nscattering. The second mechanism is due to interband transitions where the scattering occurs \nbetween bands with different indices. Opposite to the intraband scattering, the resistivity like \ninterband scattering with an inverse depe ndence on the electron lifetime  increases with \nincreas ing temperature. The sum of the intra - and interband electron scattering contributions \ngives rise to a non-monotonic dependence of the Gilbert damping on temperature for the L2 1 \nstructure. In contrast to the case for L2 1, only interband scattering is present in the A2 and B2 \nphases, which results in a monotonic increase of the intrinsic Gi lbert damping with increas ing \ntemperature. This fact is also supported by a previous study [37 ] which showed that even a \nminute chemical disorder can inhibit the intraband scattering of the system. Our theoretical results manifest that the L2\n1 phase has the lowest Gilbert damping around 4.6 × 10−4 at 300 \nK, and that the value for the B2 phase is only slightly larger at room temperature. According \nto the torque correlation model, the two main contributions to damping are the spin orbit \ncoupling and the density of states (DOS) at the Fermi level [47 , 48]. Since the spin orbit \nstrength is the same for the different phases it is enough to focus the discussion on the DOS 11 \n that provide s a qualitative explanation  why damping is found lower in B2 and L2 1 structure s \ncompare d to A2 structure. The DOS at the Fermi level of the B2 phase (24.1 states/Ry/f.u;  f.u \n= formula unit ) is only slightly  larger to that of the L2 1 phase (20.2 states/Ry/f.u.) , but both \nare significantly smaller than for the A2 phase (59.6 states/Ry/f.u.) as shown in Fig. 8. The \ngap in the mi nority spin channel of the DOS for the B2 and L2 1 phases indicate half-\nmeta llicity, while the A2 phase is metallic. The atomically resolved spin polarized DOS \nindica tes that the Fermi -level states mostly have  contr ibutions from Co and Fe atoms. For \ntransition elements such as Fe and Ni, it has been reported that  the intrinsic Gilbert damping \nincreases significantly below 100K with decreas ing temperature [37]. The present electronic \nstructure calculations were performed using Green’s functions, which do rely on a \nphenomenological relaxation time parameter,  on the expense that the different contributions to \ndamping cannot  be separated eas ily. The reported results in Ref. [37] are by some means \nsimilar to our findings of the temperature dependent Gilbert damping in full Heusler alloy films with different degr ee of atomic order. The intermediate states of B2 and L2\n1 are more \nclose to the trend of B2 than L2 1, which indicates that even a tiny atomic orde r induced by the \nFe and Al site disorder will inhibit the conductivity -like channel in  the low temperature \nregion. The theoretically calculated Gilbert damping constants are matching qualitatively with \nthe experimentally observed  𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values as shown in Fig. 7. However, the theoretically \ncalculated 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 for the L2 1 phase increases rapidly below 100K, in co ntrast to the \nexperimental results for the well -ordered CFA thin film (LP673K ) indicating that  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 saturates at low temperature. This discrepancy between the theoretical and experimental \nresults can be understood taking into account the low temperature behaviour of the life time τ \nof Bloch states. The present theoretical model assum ed that the Gilbert damping has a linear \ndependence on the electron lifetime in intraband transitions which is however correct only in \nthe limit of small lifetime, i.e., 𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏≪1, where q is the magnon wave vector and 𝑣𝑣𝐹𝐹 is the \nelectron Fermi velocity. However, in the low temperature limit the lifetime  𝜏𝜏 increases and as \na result of the anomalous skin effect the intrinsic Gilbert damping saturates  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖∝ tan−1𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏𝑞𝑞𝑣𝑣𝐹𝐹�  at low temperature [37], which is evident from our experimental \nresults.  \nRemaining discrepancies between theoretical and experimental values of the intrinsic Gilbert \ndamping might stem from the fact that the samples used in the present study are 12 \n polycrystalline and because of sample imperfections these fil ms exhibit significant \ninhomogenous line -width broadening due to superposition of local resonance fields.  \nCONCLUSION  \nIn summary , we report temperature dependent FMR measurements on as -deposited Co 2FeAl \nthin films with different degree of atomic order. The degree of atomic ordering is established \nby comparing experimental and theoretical results for the temperature dependent intrinsic \nGilbert damping constant. It is evidenced that the experimentally observed intrinsic Gilbert \ndamping in samples with atomic  disorder (A2 and B2 phase samples) decreases with \ndecreasing temperature. In contrast, the atomically well -ordered sample, which we identify at \nleast partial L21 phase, exhibits an intrinsic Gilbert damping constant that increases with \ndecreasing temperat ure. These temperature dependent results are explained employing the \ntorque correction model including interband transitions and both interband as well as \nintraband transitions for samples with atomic disorder and atomically ordered phases, \nrespectively.  \nACKNOWLEDGEMENT \nThis work is supported by the Knut and Alice Wallenberg (KAW) Foundation,  Grant No.  \nKAW 2012.0031  and from Göran Gustafssons Foundation (GGS), Grant No.  GGS1403A. The \ncomputations were performed on resources provided by SNIC (Swedish National \nInfrastructure for Computing) at NSC (National Supercomputer Centre) in Linköping, \nSweden.  S. 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Lett. 99 , 027204 (2007) .  \n  15 \n Table I P arameters describing magnetic properties of the different CFA samples.  \nSample  𝜇𝜇0𝑀𝑀𝑆𝑆(𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) \n(T) 𝜇𝜇0𝐻𝐻𝑐𝑐𝑖𝑖 \n(mT)  𝑔𝑔∥(𝑔𝑔⊥) \n 𝜇𝜇0𝐻𝐻𝑢𝑢 \n(mT)  𝑇𝑇𝐶𝐶 \n(K) 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 \n(× 10-3) \nLP573K  1.2±0.1 (1.09 1±0.003)  0.75 2.06 (2.0)  1.56 810 2.56 \nLP673K  1.2±0.1 (1.11 0±0.002)  0.57 2.05 (2.0)  1.97 >900  0.76 \nLP773K  1.2±0.1 (1.081 ±0.002)  0.46 2.05 (2.0)  1.78 890 1.46 \nHP300K  0.9±0.1 (1.066 ±0.002)  1.32 2.01 (2.0)  3.12 -- 3.22  \n 16 \n Figure 1  \nFig. 1.  (a) Layout of the in -house made VNA-based out -of-plane ferromagnetic resonance  \nsetup . (b) Out -plane ferromagnetic resonance spectra recorded for the well -ordered LP673K \nsample at different temperatures  𝑓𝑓=10 GHz . \n  \n \n17 \n Figure 2  \nFig. 2. (a) Magnetization vs. temperature plots measured on the CFA films with different \ndegree of atomic order. (b) Theoretically calculated magnetization vs. temperature curves for \nCFA phases with different degree of atomic order, where 50 % (100 %) Fe atoms on Heusler \nalloy 4a sites indicate B2 (L2 1) ordered phase, and the rest are intermediate B2 & L2 1 mixed \nordered  phases.  \n  \n \n \n18 \n Figure 3  \nFig. 3.  Resonance field vs. in -plane orientation of the applied magnetic field of (a) 𝑇𝑇𝑆𝑆=\n300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇 𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited films. Red lines \ncorrespond to fits to the data using Eq. (1).  \n    \n \n \n19 \n  \nFigure 4  \nFig. 4.   Line-width vs. frequency of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to fits to the data to extract the \nexperimental Gilbert damping constant and inhomogeneous line -width. Respective insets \nshow the experimentally determined temperature dependent Gilbert damping constants.  \n  \n  \n \n \n20 \n Figure 5  \nFig. 5.  Frequency vs. applied field of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=\n400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to Kittel’s fits to the data. Respective \ninsets show the temperature dependent effective magnetization a nd inhomogeneous line -width \nbroadening values.  \n  \n  \n \n21 \n Figure 6  \nFig. 6.  (a) Linewidth vs. frequency with and without a glass spacer between the waveguide \nand the sample.  Red lines correspond to fits using Eq. (4).  (b) Temperature dependent values \nof the radiative damping using Eq.  (6). The lines are guide to the eye. (c) 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 \nvs 𝛿𝛿2. The red line corresponds to a fit  using Eq. (7) to extract the value of the correction \nfactor  𝐶𝐶. (d) Temperature dependent values of eddy current dampi ng using Eq. (7). The lines \nare guide to the eye.  \n  \n  \n \n \n22 \n Figure 7  \nFig. 7.  Experimental (a) and theoretical (b) results for the temperature dependent intrinsic \nGilbert damping constant for CFA samples with different degree of atomic order . The B2 & \nL21 mixed phase corresponds to the 75 % occupancy of Fe atoms on the Heusler alloy 4a \nsites. The lines are guide to the eye.  \n  \n \n \n23 \n Figure 8  \nFig. 8.  Total and atom -resolved spin polarized density of states plots for various \ncompositional CFA phases; (a) A2, (b) B2 and (c) L2 1. \n  \n \n \n"    },    {        "title": "1908.11084v1.Enhancement_of_ultrafast_demagnetization_rate_and_Gilbert_damping_driven_by_femtosecond_laser_induced_spin_currents_in_Fe81Ga19_Ir20Mn80_bilayers.pdf",        "content": "1 \n Enhancement of u ltrafast demagnetization  rate and Gilbert damping driven by \nfemtosecond  laser -induced spin currents in Fe81Ga 19/Ir20Mn 80 bilayers  \nWei Zhang1,2, Qian Li u3, Zhe Yuan3, Ke Xia3, Wei He1, Qing -feng Zhan4, Xiang -qun \nZhang1, and Zhao -hua Cheng1,2,5* \n1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina  \n2School o f Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China  \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China  \n4State Key Laboratory of Precision Spectroscopy, School of Phy sics and Materials \nScience, East China Normal University, Shanghai 200241, China  \n5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China  \n \n \n \n \n 2 \n Abstract  \nIn spintronics application s, ultrafast spin dynamics have  to be controlled at  \nfemtosecond (fs) timescale s via f s-laser radiation. At such ultrafast timescale s, the \neffect of the Gilbert damping factor α on ultrafast demagnetization time\nM should  be \nconsidered.  In previous explorations for the relationship between these two parameters, \nit was found that the theoretical calculations based on the local spin -flip scattering \nmodel do not agree  with the experimental results.  Here,  we find that in \nFe81Ga19(FeGa) /Ir20Mn 80(IrMn) bilayers, the unconventional IrMn  thickness \ndependence of α results  from  the competition between spin currents pumped from the \nferromagnetic (FM) FeGa layer to the antiferromagnetic (AFM) IrMn layer and those \npumped from the AFM layer to the FM layer. More importantly , we establish a \nproportional relationship between the change of the ultrafast demagnetization rate and \nthe enhancement of Gilbert damping induced by the spin currents via interfacial spin \nchemical potential\ns  . Our work build s a bridge to connect the ul trafast \ndemagnetization time and Gilbert damping in ultrafast photo -induced spin currents  \ndominated systems , which  not only explains the disagreement  between experimental \nand theoretical results in the relation  of 𝜏𝑀 with α, but pr ovides  further insight into \nultrafast spin dynamics  as well .    \nPACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p  \n*To whom all correspondence should be addressed.  zhcheng@iphy.ac.cn  \n 3 \n I. INTRODUCTION  \nThe understanding of spin dynamics from nanosecond  (ns) down to femtosecond  \n(fs) timescales is an essential task toward s the realization of ultrafast spintronic devices \nin the frequency range  from GHz to THz  [1,2] . The study of ultrafast demagnetization \ntime,\nM, is one of the most challenging problem s in laser -induced ultrafast spin \ndynamics . The Gilbert damping factor , α , is of the utmost importance for high \nfrequency switching of spintronic devices.  Since both \nM  and α require a transfer of \nangular momentum from the electronic system to the lattice, the unification of these \ntwo seemingly unrelated parameters can facilitate the exploration of the microscopic \nmechanism of laser -induced ultrafast spin dynamics. An inverse ly proportional \nrelation ship between  \nM and α was predicted  by theoretical calculations  based on the \nlocal phonon -mediated Elliott -Yafet scattering mechanism  [3-5] as well as the \nstochastic Landau -Lifshitz -Bloch  (LLB) model  [6]. However,  the relationship between \nM\nand α has been debated for over one decade  [7]. Until now, all experimental results  \nhave show n that \nM increases with α [8-12].  \nApart from the local spin -flip scattering mechanism  [13], we proposed that the \nnon-local spin current s should be taken into account to coordinate the contradiction in \nthe relationship between 𝜏𝑀 and α. Previous work suggest ed that the superdiffusive \nspin current contribut ed to ultrafast demagnetization  [14], whilst the Gilbert damping \ncould  also be enhanced via non -local spin currents in ferromagnetic  (FM)/nonmagnetic  \n(NM) [15 ] and FM/antiferromagnetic  (AFM) heterostructures  [16]. Femtosecond laser 4 \n irradiation of  ferromagnetic thin films is a  fascinating novel approach to create large \nspin currents  [17,18 ]. Figure 1(a) shows that i n the case of time -resolved magneto -\noptical Kerr effect  (TRMOKE) experiments, hot electrons excited by fs -laser pulses \ncan travel at high velocities and over tens of  nanometers through the films. The \ndifference of mean free path between spin majority and spin minority hot electrons in \nferromagnetic thin film s generates superdiffusive spin currents on fs timescales. Such \nspin current s dissipated at the interface of the heterostructure result in the out -of-\nequilibrium spin accumulation represented by spin chemical potential  𝜇𝑠. Moreover,  \nfigure 1(b) shows the damped magnetization precession around the effective field  could \nbe influenced via spin current.  Tveten et al.  [19] predicted that the ultrafast \ndemagnetization time \nM could be described in the language of spin current -induced \ndamping  𝛼𝑠𝑝 in magnetic heterostructures based on the  electron -magnon scattering \ntheory. However, the experimental evidence on the connection of ultrafast \ndemagnetization  time with damping driven by f s laser-induced spin currents  is not yet \nunderstood.   \n  II. RESULTS  \nA. Sample properties   \n    Ir20Mn 80 (tIrMn)/Fe 81Ga19 (10 nm) bilayers [20 ] were deposited on optically \ntransparent single -crystalline MgO (001) substrates in a magnetron sputtering system \nwith a base pressure below 3×10−7 Torr. The substrates were annealed at 700 °C for 1 h \nin a vacuum chamber and then held  at 250 °C during deposition. FeGa layers were 5 \n obliquely deposited at an incidence angle of 45°. The IrMn layers were deposited while \ncontinuously rotating the substrates. In order to induce an exchange bias (EB) along the \nFeGa [010] direction, a magnetic field of 500 Oe provided by a permanent magnet was \napplied along the MgO [110] axis during growth. After deposition, a 3 nm protective \nTa layer was deposited on the samples to avoid oxidation. The static longitudinal Kerr \nloops  of Fe 81Ga19 (10 nm)/Ir 20Mn 80 (𝑡𝐼𝑟𝑀𝑛) along FeGa  [010] direction with various \nAFM IrMn thicknesses  (𝑡𝐼𝑟𝑀𝑛) at room temperature were acquired using a laser diode \nwith a wavelength of 650 nm.  \n     Figure 2(a) shows the longitudinal Kerr loops  of Fe 81Ga19 (10 nm)/Ir 20Mn 80 (\nIrMnt\nnm) along FeGa  [010] direction with various AFM IrMn thicknesses  (\nIrMnt ) at room \ntemperature, whereas the thickness of FM FeGa layer was fixed at 10 nm. For\nnm2IrMnt\n, the width of the hysteresis loops is enlarged with no obvious shift along  \nthe x-axis, implying that the thickness of IrMn layer is too thin to form an \nantiferromagnetic order for pinning the magnetization reversal of FeGa  [21] (Insert in \nFig. 2(b) (left)). For\nnm2IrMnt , the antiferromagnetic orders are well established, and \nconsequently the antiferromagnetic moments pin FM ones reversal to induce a \nunidirectio nal anisotropy (Insert in Fig. 2 (b) (Right)). The loops therefore exhibit \nevidently exchange bias behavior. The exchange bias field achieves a value of about 60 \nOe when \nnm2IrMnt , whilst the largest value of coercivity (~72 Oe) occurs at \nIrMnt\n2 nm.  \nB. TRMOKE measurements for ultrafast demagnetization and Gilbert damping   6 \n We performed the polar TRMOKE experiment to measure ultrafast \ndemagnetization time under a saturated  applied field of 20 kOe in the normal direction \nof the samples  [22]. The details of the TRMOKE experiment are described in \nAPPENDIX A.  Figure 3(a) shows th e demagnetization curves  for various IrMn \nthicknesses with a maximum magnetization quenching  of ~10%  [23,24 ]. The temporal \nchanges of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just \nbefore the pump laser excitation. The time evolution of magnetization on sub -\npicosecond timescales can be fitted according to Eq. (1) in terms of the three -\ntemperature model (3TM)  [17]. \n−∆𝑀(𝑡)\n𝑀={{[𝐴1\n(𝑡𝜏0+1⁄)0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝑀−𝜏𝐸(𝐴1−𝐴2)\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺)}∗𝐺(𝑡,𝜏𝐺)        (1)                                                                        \nwhere \n),(*GtG represents the convolution product with the Gaussian laser pulse \nprofile, \nG  is the full width at half maximum (FWHM) of the laser pulses , \n)(t  is a \nstep function , \n)(t  is the Dirac delta function . 𝐴1 represents the value of ∆𝑀(𝑡)\n𝑀 after \nequilibrium between electrons, spins , and lattices . 𝐴2 is proportional to the initial \nelectrons temperature rise. Here, we used the 780 nm laser as the pump pulse to excite \nthe magnetic system out of equilibrium, while the 390 nm laser pulse was used as a \nprobe beam. Therefore, i n Eq. (1), t he state filling e ffects during pump probe \nexperiment are neglected due to the different wavelength of pump and probe beams \nused in this study. The cooling time by heat diffusion is described by  𝜏0, which should \nbe about one order of magnitude larger than  𝜏𝐸 representing the timescale of electron -\nphonon interactions. The best -fitted value of 𝜏𝐸=500 𝑓𝑠 for all samples is in good 7 \n agreement with that of previous reports  [18]. The fitting parameters in Eq. (1) are shown \nin Table I, from which one notes the pulse  width is 350 fs for all the samples. In our \nexperimental setup, the time -resolution is about 80 fs. In order to obtain a high time \nresolution, we measured the ultrafast demagnetization with very fine step of time delay \n(15 fs). The values of ultrafast dem agnetization time (120 -220 fs) obtained from Eq. (1) \nare defined as the time needed for the magnetization to reach a level of 𝑒−1 of its \nmaximum demagnetization. The time needed for magnetization to reach its maximum \ndemagnetization (>500fs) should be longer than the time extracted from Eq. (1). A \nsimilar result was reported by B. V odungbo et al .[25]. The very large temporal \nstretchi ng of the laser pulse up to 430 fs was attributed to the conversion of the incident \nlaser pulse into a cascade of hot electrons. This could be one of the possible reasons \nresulting in the spread of laser pulse up on the samples in this study. Via changing the \nsingle parameter ， , we can accurately reproduce the experimental results for various \nsamples.  The ultrafast demagnetization time\nM was observed to decrease from  220±\n10 fs for \nIrMnt   = 0 nm to 120±10 fs for \nIrMnt   = 2 nm, then increase with  further \nincreasing 𝑡𝐼𝑟𝑀𝑛  [Fig. 3(b)].  \nThe precessional frequency and damping factor can be derived by means of the \nTRMOKE signal s as well [26, 27]. Figure 4(a) shows the typical time evolution of the \npolar component of magnetization after pump laser excitation at different fields applied \nalong  with the [110] direction of F eGa for 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. It is observed clearly that \nthe spin precession process can be influenced obviously by applied fields. The exact \nM8 \n values for 𝑓 with various applied fields can be obtained using the damped harmonic \nfunction added to an ex ponential -decaying background : \n𝛥𝑀(𝑡)=𝐴+𝐵𝑒𝑥𝑝(−𝑣𝑡)+𝐶𝑒𝑥𝑝(−𝑡\n𝜏)𝑠𝑖𝑛 (2𝜋𝑓𝑡+𝜑)                     (2)                   \nwhere 𝐴 and 𝐵 are the background magnitudes, and 𝑣 is the background recovery \nrate. 𝐶,𝜏,𝑓 and 𝜑are the magnetization precession amplitude, relaxation time, \nfrequency and phase, respectively. The field dependence of frequency 𝑓 extracted \nfrom the fitting procedure is shown in Fig.  4(b). We note that the experimental  f-H \nrelation can be reproduced very well by Kittel equation ( 3) [27 ]. \n             .                                                 (3) \nwith                                                          and             \n                                            . \nAnd γ=𝛾𝑒𝑔2⁄ is the gyromagnetic ratio. 𝜑𝑀 and 𝜑𝐻 are the angles of in -plane \nequilibrium M and H respect to the FeGa [010] easy axis. 𝐾1,𝐾𝑢,𝐾𝑒𝑏 and 𝐾𝑂𝑢𝑡 are \nthe in -plane magneto crystalline, uniaxial , unidirectional and out -of-plane magnetic \nanisotrop y constants of FeGa films,  respectively . The value of magnetocrystalline \nanisotropy  constant  is 𝐾1=4.5×105 𝑒𝑟𝑔/𝑐𝑚3 for the samples with various AFM \nlayer thickness during the fitting procedure and the uniaxial  magnetic anisotropy \nconstant𝐾𝑢=(1.5±0.3)×105𝑒𝑟𝑔/𝑐𝑚3. For 𝑡𝐼𝑟𝑀𝑛=3 𝑛𝑚 and 5 nm, the \nunidirectional magnetic anisotropy constant of  𝐾𝑒𝑏=3×104𝑒𝑟𝑔/𝑐𝑚3 has to be \nincluded for more accurate fitting, although it is one order magnitude smaller than those \n2 1 22 1)2( HHMf\ns\nM eb H M s M M u s O K HM KK K M K H      cos ) cos( 2sin 2 cos2 4 2-2\n1 12 2\nut 1    \nM eb H M s M u M K HM K K H     cos ) cos( 2cos2 4cos21 2    9 \n of magneto crystalline and uniaxial anisotropy .   \nThe effective G ilbert damping factor  𝛼𝑒𝑓𝑓 shown in Fig. 4 (c) is determined \nfrom the relaxation time 𝜏 by Eq. ( 4) [28]: \n \n) (/22 1H Heff                                              (4) \nSince the overall effective damping factor 𝛼𝑒𝑓𝑓 consists of intrinsic damping and \nextrinsic damping whereby the second one arises from  both the two-magnon -scattering \nand the dephasing effect  in the sample s, the overall effective  Gilbert damping factor \ndecreases monotonously to a constant  value  with increasing the applied field (Fig . 4(c)). \nAs one of the main ly extrinsic contributions , the two -magnon -scattering induced \ndamping has been extensively studied in exchange biased heterostructures  [29-34]. The \nmature theory was developed to explain  the two-magnon scattering process due to \nspatial fluctuations of anisotropy and exchange bias field [30,35 ]. The two -magnon \nscattering process comes from  the scatterings of the uniform ( 𝑘=0) precession mode \ninto nonuniform modes ( k≠0 magnons) that  are degenerate in frequency. This \nprocess is described by the Hamiltonian, in which the spatial fluctuation in the exchange \ncoupling caused by interface roughness determines the scattering strength. The \nroughness gives rise to a large fluctuating field because the FM magnetization interacts \nalternatively with one or the other AF sublattice via the atomic exchange coupling. It is \na well -known relaxation mechanism effective in exchange biased heterostructur es due \nto the interface roughness occurring on the short length scales. When a low external \nfield comparable with the exchange bias field was applied, the two-magnon scattering 10 \n effect will result in the increase of Gilbert damping with the exchange bias fi eld \naccording to previous reports  [33, 34 ]. However, as shown in Ref. 36, a strong enough \napplied field can be used to exclude the contributions from the two-magnon -scattering, \nwhere the value of Gilbert dam ping factor keeps as a constant with various two -\nmagnon -scattering strength. Based on this  result , a similar method using strong enough \nexternal fields was applied in this study to exclude the two -magnon -scattering effect. \nMoreover, previous works show that the two -magnon -scattering induced damping \nincreases with precession frequency because of the increased degeneracy of spin waves  \n[37, 38]. Our work demonstrated that the damping factor keeps almost a constant value \nat high enough applied fields, i ndicating the minor contributions from the two-magnon -\nscattering to Gilbert damping. Besides, it has been demonstrated previously that the \ntwo-magnon -scattering contri butions decrease monotonously with increasing the film \nthickness  [33, 34 ]. This again disagrees with the tendency of thickness dependence of \ndamping at high applied field shown in Fig. 5(c). Therefore, in this study, the two -\nmagnon -scattering strength was suppressed effectively by applying a high enough \nexternal field. On the other hand, inhomogeneities in FeGa thin film may cause \nvariations in the local magnetic anisotropy field. It leads to the variations of spin \norientations when the external field is not large enough, and gives rise to t he enhanced \ndamping arising from spin dephasing e ffect [28]. However, an applied field (~ kOe) \nmuch la rger than the anisotropy field makes the spin orientation uniform, as a result, \nthe dephasing effect was suppressed largely.  Based on the above analysis, t he intrinsic \npart of damping is independent of t he external field or precession frequency, while the 11 \n extrinsic part  including both the depha sing effect and the two-magnon -scattering effect  \nare field-dependent.  In order to avoid the effect of the extrinsic damping factor , the \nintrinsic damping factors were obtained by fitting the overall damping factor as the \nfunction of applied fields with the Eq.  (5) [39, 40 ] shown as the red line in Fig. 4(c): \n0/\n1HH\neff e\n                                                   (5) \nwhere α and 𝛼1𝑒−𝐻𝐻0⁄ are the intrinsic and extrinsic parts of the damping factor, \nrespectively.   \nFor the derivation of spin precessional frequency as well as the Gilbert damping, \nthe similar producers as shown above were adapted to various samples. Fig ure 5(a) \nshows the precessional frequency  from oscillation curves with various IrMn thicknesses. \nSince the exchange bias field and coercivity are much weaker tha n applied fields, the  f-\nH curves  of FeGa films are therefore slightly different  with various AFM layer \nthicknesses , which is in contra st to the observation that the enhanced uniaxial \nanisotropy of Fe/CoO bilayers  [28] increases the precessional frequency largely.  More \nimportantly, w e find th e effective damping factor \neff decreases with applied fields \n[Fig. 5(b)]. The solid lines represent the fittin g expression shown as the Eq. (5 ). \nInterestingly, the effective Gilbert damping factors drop to a nearly constant value as \nthe intrinsic damping factor when the applied fields increase strong enough to suppress \nthe extrinsic contributions  as stated above .  \nThe value s of the intrinsic damping factor as a function of the thickness of the \nIrMn layer  are illustrated in Fig. 5(c ). It increases firstly  and reaches the maximum 12 \n value when the thickness of the IrMn layer at \nIrMnt = 2 nm, and finally  decreases with \nfurther increasing the thickness of the IrMn AFM layer. A drastic change of 2.5 times \nfor damping occurs at \nIrMnt = 2 nm . Similarly, S. Azzawi et al. showed around 2 times \nenhancement of damping in NiFe/Pt bilayers when a continuous Pt capping layer is just \nforming at 0.6 nm  by TRMOKE measurements  [41]. Moreover, once a continuous IrMn \nlayer is forming at 2 nm, the accompanied strong intrinsic anisotropy of AFM would \ncontribute partly t o the damping enhancement superimposed to  spin pumping effect. \nThis has been demonstrated previously by W. Zhang et al where the damping of \nPy/IrMn bilayers is 3 or 4 times larger than that in the Py/Cu/IrMn samples  [42]. Based \non the discussions  in Fig. 4 , we can exclude the  extrinsic mechanisms such as the two-\nmagnon -scattering and the dephasing effect as the dominant contributions to the \ndamping process when the external fields are high enough  [43]. Besides,  FeGa alloys \nare particularly interesting because of their magneto -elastic properties  [44]. The \nacoust ic waves are possible to be tri ggered by ultrashort laser and as a result, spin \nprecession would be excited  non-thermally via a magnetoelastic effect  [45]. However, \nthis effect  can be  excluded based on the following reasons: firstly, the external field has \nto be applied along with the hard axis of FeGa, otherwise , the magnetization precession \ncannot be induced. It agrees with the fact that  the canted  magnetization from the easy \naxis is necessary when the spin precession arising  from instantaneous anisotropy \nchange  accompanied by ultrafast demagnetization occurs  [26]. In contrast,  the \noccurrence of spin precession from the magnetoelastic effect is independent of  initial \nmagnetization orientation. Secondly,  in order to check the contribution of resonance 13 \n mode from the magnetoelastic effect, we per formed a fast Fourier transform  in \nAPPENDIX B . Only the uniform  field-dependent  precession mode  was excited at present \nstudy.  This is not the expected behavior for the acoustically induced modulation of the \nmagneto -optical effects.  Therefore, the magnetoelastic effect of FeGa was suppressed \nlargely in this study. It  probably because the laser fluence of around 1 mJ/cm2 is not \nhigh enough to induce a large amplitude of strain pulse. According to Ref. 45 , the \noscillations amplitude of acoustic mode  increases linearly with the laser energy density \nwithin the probed range . Moreover, the FeGa material with a thickness as thic k as 60 \nnm is preferred to induce an obvious magnetoelastic behavior  [46], while 10 nm  at the \npresent experiment is  probably too thin.  As a result, t he intrinsic damping can be \ninfluenced by the following  paramenters : (1) magnetocrystalline anisotropy  of FM [47]; \n(2) exchange  bias field  [30, 31, 36 ], and (3) spin pumping effect  at the interface between \nFM and  the AFM [ 15, 16, 42, 48 ]. In the case of FeGa/IrMn bilayers , the \nmagnetocrystalline anisotropy constant of FeGa\n1K  =\n3 5/ 105.4 cm erg  , which is \nobtained fro m Fig. 4 and Fig. 5 , is invariant  with AFM layer thickness. Moreover , \nreferring to Fig. 2 (b), it seems that there is no direct relationship between  the intrinsic \ndamping factor and the exchange  bias field Heb. When the applied field is far higher \nthan the exchange bias field, both the precessional frequency and the damping factor \nshow independence of excha nge bias field  [36]. Therefore, the IrMn  thickness \ndependence of the intrinsic damping is not attributed to the magnetocrystalline \nanisotropy and the exchange bias field . Due to the strong spin-orbit coupling of the \nheavy metal  (HM) Ir in the IrMn alloy, the contribution of spin pumping to the damping 14 \n factor must be taken into account. It is noteworthy that the IrMn thickness dependence \nof damping in FeGa/IrMn is different from  that in other normal FM/HM bilayers, where \nthe damping factor increases monotonically with the thickness  of HM layer and \napproa ches a saturation valu e [49]. However,  the damping of FeGa ferromagnetic layer \ndecreases again after reaching a peak value at 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. The change of the \ndamping  factor  is always accompanied by the spin currents transfer between FM and \nAFM layers.  More spin currents absorbed by the neighbor ing layer result in larger \ndamping in the FM layer. A n unconventional  decrease of the damping factor implies \nthat not only the effect of heavy metal Ir in IrMn alloy has to be taken into account, but \nalso the ant iferromagnetic magnetization. The heavy metal Ir serves as a perfect spin \nsink to absorb the spin currents , and consequently increases the damping in FeGa, while \nthe antiferromagnetic magnetization in IrMn serves as a new source to compensate the \ndissipati on of magnetization precession  and decrease the damping of FeGa.  \nC. First -principle calculations for IrMn layer thickness dependence of \nGilbert damping  \nTo understand the behavior of the IrMn thickness -dependent damping factor, we \ncalculated the damping factor  using the scattering theory of magnetization dissipation \ncombined with the first -principles electronic structure  [50]. The calculated FM/AFM \nbilayer structure  shown in Fig. 6(a) are the same as that in  the experiment.  Here, the \nmagnetic moments of AFM sublattices serve as not only a spin sink to absorb the spin \ncurrent pumped from the adjacent FM layer, but also a spin current emitter to partly 15 \n cance l the spin pumping effect of the FM. The interfacial exchange coupling force s the \nmagnetic moments of  the IrMn sublattices in a few layers near the interface to preces s \nfollowing the adjacent FM , generat ing spin current s back into the FM layer  [Fig. 6 (b)]. \nBased on this model, t he enhancement of damping due to the spin current  𝛼𝑠𝑝=∆𝛼=\n𝛼𝑡𝐼𝑟𝑀𝑛−𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚 as a function of IrMn thickness was calculated  and shown as the \nsolid circle in Fig. 6(c). It increases firstly to a peak value at \nIrMnt = 2 nm, and then \ndrops with further increasing  the IrMn layer thickness . When \nIrMnt  2 nm , the \nthickness of the IrMn  layer is too thin to establish the antiferromagnetic order, which \ncan be supported by the negligible exchang e bias as shown in Fig. 2 (b). In this case, the \npumped spin current from the AFM back into the FM to partially cancel the spin \npumping effect by  the FM is largely reduced because of the disorder of the \nantiferromagnetic moments as illust rated on  the left side in Fig. 6 (b). In this re gion, \ntherefore, the magnetic moments in the AFM serve as a perfect spin sink to absorb the \nspin current pumped from the adjacent FM  resulting in a significant enhancement in \nthe damping  factor . For the samples with the thickness of IrMn  𝑡𝐼𝑟𝑀𝑛>2𝑛𝑚, however , \nthe antiferromag netic order is well established and the accompanied exchange bias is \nremarkably large ( See Fig . 2(b) and its insert ). Because of the exchange coupling \nbetween FM and AFM at the interface, the magneti c moments of the AFM sublattices \nin a few layers near the interface is forced to precess following the magnetic moment \nof the FM, while those far away from the interface would keep static. Such an exchange \nspring effect at the interface caused spin precession in the AFM layer,  and consequently , \nspin currents would be transfe rred from AFM to the FM layer. Moreover, these  spin 16 \n current s from the AFM would be enhanced due to the coherent precession of \nmagnetization in different sublattices as illustr ated in the right side of Fig. 6 (b). The \nexchange  spring effect induced  precession of the AFM has two effects: (1) the AFM \nhas intrinsic damping that increases the overall dam ping of the FM/AFM bilayer. (2) \nthe precessional motion of magnetic moments in AFM sublattices pumps spin current s \ninto the FM, which cancels partly the spin pumping by the FM. As a result, the overall \ndamping of the bilayers is reduced. From  the solid circles in Fig. 6(c), one can find that \nthe damping decreases with increasing\nIrMnt  when \nIrMnt 2 nm, indicating that the \nlatter effect of the pumped spin currents is dominant over the intrinsic damping.  Besides, \nby comparing the calculated and experimental values  [Fig. 6(c) and (d)], one can find \nthat the calculated Gilbert damping  is larger than the experiment al one for \nIrMnt 1 nm . \nThe reason for t he deviation is the assumption of a  perfect ly flat  FeGa/ IrMn interface \nin the  calculation, which leads to a larger spin current pumped from the FM . \nUnfortunately, it is almost impossible to fabricate the perfect ly flat film when the \nthickness is less than 1  nm.  \nIn order to separate the contribution of the precession of the magnetic moment of \nthe AFM sublattice to damping, we also calculated the damping by assuming perfectly \nstatic AFM ordered IrMn without precession (solid diamonds  in Fig. 6 (c)) and a \nparamagnetic IrMn layer with vanishing Néel order  (solid triangles  in Fig. 6(c)). The \ncalculated results demonstrate that if the magnetic moments of the AFM sublattice \neither do not precess or align randomly, the IrMn layers serve only as a perfect spin 17 \n sink to absorb the spin current s pumped from the adjacent FM  resulting in a significant \nenhancement of damping . The damping increases monotonically to a saturation value \nwith IrMn thickness, which is similar to that of heavy metals  [49].  \nD. Relationship between ultrafast demagnetization rate and Gilbert \ndamping induced by non -local spin currents  \nThe central strategy of our study is to establish a direct correlation between \nM\nand α. According to Fig. 3(b ) and Fig. 5(c), we find that the femtosecond laser -induced \nultrafast demagnetization time \nM and the Gilbert damping  α show an opposite IrMn \nthickness dependence  in FeGa/IrMn bilayers.  By plotting \nM versus α as shown in \nFig. 7 (a), one can clearly observe that the value of \nM decreases with  α, suggesting \nthat spin transport plays an additional dissipation channel for accelerating the ultrafast \ndemagnetization and enhanc ing the d amping.  The damping  factor  𝛼𝑡𝐼𝑟𝑀𝑛 for \nIrMnt 0 \nnm is ascribed to the spin pumping effect induced by various AFM thicknesses 𝛼𝑠𝑝 \nand the contribution from the FM itself 𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚.To give further insight into th e \nrelationship, we replo tted Fig.  7(a) by using t he change of the ultrafast demagnetization \nrate ∆1\n𝜏𝑀=1\n𝜏𝑀|𝑡𝐼𝑟𝑀𝑛−1\n𝜏𝑀|𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚  versus the enhancement of Gilbert damping  \n𝛼𝑠𝑝=∆𝛼=𝛼𝑡𝐼𝑟𝑀𝑛−𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚  induced by the spin current.  An approximately \nlinear relationship is  confirmed and shown in Fig. 7 (b), which can be fitted using Eq. \n(6): (For the de rivation of Eq. (6 ), please see APPENDIX D  for details)  \n∆1\n𝜏𝑀=𝜇𝑠\nℏ∆𝛼,                                                      (6) 18 \n Where ∆1\n𝜏𝑀 , ∆α  represents the enhancement of ultrafast demagnetization rate and \nGilbert damping induced by the  spin current, respectively, 𝜇𝑠 is the spin chemical \npotential,  and ℏ is the Planck constant. A reasonable value of  𝜇𝑠≈1 𝑒𝑉 which is \nsimilar to that of spin splitting in 3d transition metals was obtained  by the linear fitting \nusing Eq. (6).  \n    The spin chemical potential 𝜇𝑠  is proportional to spin accumulation s at the \ninterface between different layers . It contributes largely to ultrafast demagnetization \naccording to the model of laser -induced ultrafast superdiffusive spin transpor t in \nlayered heterostructures  [14, 5 1]. There is a large difference in velocities or lifetimes \nfor spin -dependent hot electrons  [52]. As a result,  the transport properties of ho t \nelectrons are spin -dependent. For instance , the minority -spin electrons excited by \nultrashort laser survive  for a quite short time an d they decay to non -mobile bands \napproximately at the position  they were excited. Instead, majority -spin electrons have \nlonger lifetimes and higher velocities. So they leave  fast from the excitation reg ion after \nbeing created, resulting in  part of the demagnetization process. Because the directions \nof motion for all the electrons are random , they can obtain a velocity directed back \ntowards the ferromagnetic film . A second part of the demagnetization is ascribed to  the \nbackflow of spin -minority elec trons from the substrate or the neighbor layer. Spin -\nmajority electrons entering the ferromagnetic  layer will find good transport properties \nand continu e diffusing without severely decay ing. However , spin-minority electrons \nexperience a considerable worsen ing of the transport properties as soon as they enter 19 \n the ferromagnetic layer. The consequence is that they are trapped at the entrance of the \nferromagnetic layer , giving rise to  the spin accumulations  at the interface . Nevertheless, \nthe quantitative description for spin accumulations during ultrashort laser -induced \ndemagnetization in heterostructures  is still lacking. This work aims at filling this gap \nby relating ultrafast demagnetization time and Gilbert damping. A de tailed calculation \nfor the value of 1 eV for spin chemical potential obtained in this experiment is highly \ndesirable.    \nThe non -local spin current s dissipated at the interface of FeGa/IrMn open an \nadditional channel to accelerate the ultrafast demagnetization and enhance  the Gilbert  \ndamping. However , in the case of the sample with \nIrMnt = 0 nm without the assistant \nAFM layer , both the local spin -flip and non -local spin transport mechanism s probably \ncontribute to the ultrafast demagnetization in the ferromagnetic layer. For instance, \nbased on the breathing Fermi -surface model of the Gilbert damping and the Elliott -\nYafet relation for the spin -relaxation time, a relation shown as Eq. (7 ) is established \nbetween the conductivity -like Gilbert damping \n and ultrafast demagnetization time  \n𝜏𝑀 [10]. \n                                                (7) \nTaking the values of \nnm tMIrMn 0 and \nnm tIrMn 0 are 220  fs and 0.004, respectively , a \nvalue of 𝛼𝜏𝑀⁄=1.8×1010𝑠−1 is derived.  This value is reasonable and agrees well \nwith that  of 3d transition metal  Ni calculated by  the breathing Fermi -surface model [ 53], \n2pbFM\nelM20 \n indicating that  the ultrafast demagnetization of ferromagnetic FeGa film  itself  is mainly \ngoverned by the local spin -flip scattering events . Nonetheless, we have to address that, \nultrafast demagnetization in the ferromagnetic layer was accelerated and the Gilbert \ndamping was enhanced via the interfacial spin  accumulations o nce the IrMn layer was \nattached .  \nIII. CONCLUSIONS  \n The unconventional IrMn thickness dependence of α  is attributed to the \ncancellation of the spin currents pumped from the AFM IrMn layer to the FM FeGa \nlayer.  We establish a  proportional  relationship between the change of ultrafast \ndemagnetization rate and the enhancement of Gilbert damping induced by the spin \ncurrents via interfacial spin chemical potential . This result  can facili tate the utilization \nof ultrafast spintronic devices in the THz region.  \n \n \n \n \n \n \n 21 \n Acknowledgments  \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701,  and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212) and \nthe Key Research Program of Fro ntier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Normal University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. \n61774017,  61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03).  The work at East China Normal University  is partly supported by the \nNational Natural Sciences F oundation of China  (Grant No. 11874150).   \n \n \n \n \n \n \n \n 22 \n APPENDIX A: TRMOKE MEASUREMENTS  \nIn this study, the dynamical process of fast and ultrafast spin dynamics was \nmeasured by TRMOKE. The experiments were carried out using an all -optical pump -\nprobe technique. A  train of optical pulses with a wavelength of 780 nm, 55 fs duration , \nand 100 nJ/pulse is generated at 5.2 MHz repetition rate by a Ti: sapphire oscillator \n(FEMTOLASER, XL -100). A 200  μm thickness BBO crystal was used to double the \nfrequency of femtosecond laser. The laser beam from the source is split into both 780 \nnm and 390 nm beams. We use the 780 nm laser as the pump pulse to excite the \nmagnetic system out of equilibrium, while the 3 90 nm laser pulse was used as a probe \nbeam to measure the subsequent magnetization dynamics with the timescale from sub -\npicosecond to nanosecond. The pump laser beam is much stronger than the probe with \nan intensity ratio of about 100 for all the measureme nts. Both the pump and probe \nbeam s are incident along the normal axis (z -axis) of the sample s. The detection \ngeometry is only sensitive to the out -of-plane component of the magnetization Mz. For \nfast spin dynamics, we applied various external fields along the Fe 81Ga19 [110] direction \nto trigger the spin precession, while a large enough field about 20 kOe  was applied \nalong the Fe 81Ga19 [001] direction to obtain the ultrafast demagnetization curves. W e \nadjusted the pump laser fluence from 1 mJ/cm2 to 1.25 mJ/cm2 to obtain the same \nmaximum quenching for various samples. The pump and probe beams are focused onto \nthe sample s with spot diameters of ~10 μm  and ~5μm  via an objective lens, \nrespectively. For the spin precession measurements, t he scheme  of the TRMOKE 23 \n experiment is illustrated in Fi g. 8. The signal s are sensitive with the polar component \nof magnetization after pump laser excitation at different fields applied along the [110] \ndirection of FeGa .   \n \nAPPENDIX B: FAST FOURIER TRANSFORM ANALYSIS  \n    The ferromagnetic FeGa is a famous material for its magneto -elastic properties. \nAfter femtosecond laser irradiation, an external field -independent resonance mode \nwould be triggered due to the excitation of coherent acoustic phonons. However, only \none f ield-dependent resonance mode was excited in this study according to fast Fourier \ntransform analysis  in Fig.  9.   \n \nAPPENDIX C: FIRST -PRINCIPLE CALCULATIONS  \nThe electronic structure of FeGa /IrMn bilayer is calculated self -consistently using \nthe local density approximation of the density functional theory. The spin -dependent \npotentials, charge and spin densities are obtained with the minimal basis of tight -\nbinding linear muffin -tin orbitals. In the calculation of the total damping, the scattering \nregion  consisting of the repeated FeGa /IrMn bilayers are connected to two semi -infinite \nCu leads. We have introduced the thermal lattice disorder into a 4x4 supercell a nd \ndisplaced the atoms in the scattering region randomly away from their equilibrium \npositions  with a Gaussian distribution. The root -mean -square atomic displacements of \nthe Gaussian distribution are determined using a simple Debye model with the Debye 24 \n temperature of 470 K. The two -dimensional Brillouin zone of the supercell is sampled \nby a 24x24 k -mesh corresponding to the 96x96 mesh for the Brillouin zone for the 1x1 \nunit cell. The effect of magnons in the FM FeGa is neglected in our calculation. This is \nbecause the magnetic damping is dominated by electrons at the Fermi level in metals, \nwhich can  efficiently transfer spin angular momentum into the orbital motion via spin -\norbit interaction. In metals and alloys, the influence of magnon -phonon coupling is \nnegligible except for near the Curie temperature  [54]. \nIf magnetization precession occurs only in the FM FeGa layer, the calculated \ndamping enhancement does not sensitively depend on the specific order of the AFM \nIrMn. Here we take two limits: the perfectly antiferromagnetic ordered IrMn and the \nparamagnetic IrMn (the magnetic moments of Mn are rand omly distributed such that \nboth the Néel order and total magnetization vanish). The damping enhancements \ncalculated for the two cases are nearly identical, where the damping factor is enhanced \nand saturates at the thickness of 2 nm. It indicates that the p umped spin current by the \nprecessional FeGa is immediately absorbed by the IrMn layer. The large moment on the \nMn atom can absorb the pumped transverse spin current efficiently. On the other hand, \nthe AFM IrMn is forced to precess due to the interfac ial exchange coupling,  however,  \nthe efficient of the spin current generation by AFM depends on its specific order . It is \nsuppressed largely in the case of paramagnetic  IrMn because of the cancellation via \nmagnetic moments with various orientations shown on the  left side of Fig. 6(b) in the \nmain text . In contrast, the efficient of the spin current generation by the AFM is \nenhanced remarkably by the coherent precession of the ordered magnetic moments 25 \n shown in the right side of Fig. 6(b) in the main text . And the cone angle of precessional \nIrMn is modeled to exponentially decay from the interface with a typical decay length \nof 2 nm. The precessional AFM has mainly two contributions to the damping \nenhancement of the bilayer. First, the AFM has intrinsic damping that increases the total \nenergy loss during the magnetization dynamics. The second effect is that the \nprecessional AFM pumps spin current into the FM that cancels partly the spin pumping \nby the FM and decreases the damping enhancement.  \n \nAPPENDIX D: DERIV ATION OF EQ. (6 ) IN THE MAIN TEXT  \nIt is well known that the magnetic moment \nsM  is proportional to the spin angular \nmomentum \nS via gyromagnetic ratio \nBg ： \nS Ms\n                                                          (8) \nwhere\ng is Lande factor, \nB is Bohr magneton. Normally, we take \nsMVm as the \ntotal magnetic moments , where \nV is the volume of the atom.  \n    \nM is the ultrafast demagnetization time. Therefore, t he value of \nM1 is taken as \nthe demagnetization rate. The demagnetization is alw ays accompanied by the \ndissipation of spin angular momentum, and hence the rate of spin angular momentum \ndissipation is :  \nMm\n1\n.                                                           (9) \nOn the other hand, the spin current 𝑗𝑠⃗⃗⃗  of per unit area generated by spin pumping effect 26 \n reads:  \nj𝑠⃗⃗ =1\n4𝜋𝑔𝑒𝑓𝑓𝜇𝑠⃗⃗⃗ ,                                                     (10)  \nwhere 𝑔𝑒𝑓𝑓is the effective interfacial spin -mixing conductance including the influence \nof the backflow spin current from the AFM IrMn to FeGa , \ns is the spin accumulation -\ndriven chemical potential. The pumped spin current across the interface is  𝐼𝑠⃗⃗⃗ =𝑗𝑠⃗⃗⃗ 𝐴, \nwhere\nA is the area of the interface.    \n𝑔𝑒𝑓𝑓=4𝜋𝑀𝑠𝑑∆𝛼\n𝑔𝜇𝐵,                                                    (11) \nwhere 𝑑is the thickness of the ferromagnetic layer, \nnm t tIrMn IrMn 0  is the \nenhancement of Gilbert damping induced by the absorption and generation of spin \ncurrent via various IrMn thicknesses.  \nTherefore, if we correlate the spin angular momentum dissipated by the ultrafast \ndemagnetization and that induced by spin pumping, the relationship reads:  \n \ns\nMIm1                                                        (12) \nAnd then we take Eq. (10) into Eq. 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Rev. B. 86, 024404 (2012).  \n[52] V . P. Zhukov, E. V . Chulkov, and P. M. Echenique, Phys. Rev. B . 73, 125105 (2006).  32 \n [53] M. Fähnle, J. Seib, and C. Illg, Phys. Rev. B. 82, 144405 (2010).  \n[54] Y . W. Zhao, Y . Liu, H. M. Ta ng, H. H. Jiang, Z. Yuan, and K.  Xia, Phys. Rev. B, \n98, 174412 (2018).   \n \nFigure Captions  \nTable. I . Values of the main fit parameters of ultrafast demagnetizations curves for \nvarious thicknesses of the samples.  \nFIG.1. (color online ) Basic concept of both ultrafast demagnetization and spin \nprecession induced  by spin currents . (a) The excitation of fs laser pulse transforms slow \nmajority -spin d electrons (red) into fast sp electrons, thereby launching a sp in current \ntoward s the AFM layer. The spin current crossing the interface results in the spin \naccumulation at the interface represented by spin chemical potential  𝜇𝑠. (b) The typical \ntime evolu tion of magnetization after femtosecond laser irradiation measured by \nTRMOKE experiment.  \nFIG. 2. ( Color online) Static magnetic properties of of MgO/Fe 81Ga19 (10 nm)/Ir 20Mn 80 \n(t nm) bilayers. (a) Longitudinal -MOKE loops with various thicknesses of IrMn layer  \nIrMnt\n. (b) Coercivity Hc and exchange bias field \nebH as a function of IrMn layer \nthickness \nIrMnt . \nFIG.3. (Color online ) Ultrafast demagnetization. (a) Ultrafast demagnetization curves \nwith various IrMn layer thickness es. The solid lines represent  the fitting results by Eq.  \n(1) in the text.  The insert  shows the configuration of the measurement for ultrafast 33 \n demagnetization. (b)  Ultrafast demagnetization time  as a function of IrMn layer \nthickness.  \nFIG.4. (Color Online)  Spin precession.  (a) TRMOKE signals of FeGa/IrMn bilayers \nwith \nMnIrt = 2 nm in various applied fields.  (b) Precessional frequency as a function of \napplied fields . (c) Effective Gilbert damping constant as a function of applied fields.  \nFIG. 5 . (Color Online)  Frequency and damping of spin precession. (a) Fr equency of \nspin precession as a function of applied fields with various IrMn thickness. The solid \nlines represent the fitting results by Kittle equations.  (b) Effective Gilbert damping \nconstants as a function  of applied fields with various IrMn thicknesses.  (c) Intrinsic \nGilbert damping as a function of IrMn thickness.  \nFIG.6. (Color online) Results of First -principle  calculation s. (a) Illustration of the \nferromagnet (FM)/antiferromagnet  (AFM) structure employe d to investigate the spin \ntransport. (b) The configuration of the IrMn magnetic moments located at the first layer \nnear the interface. (c)  The calculated damping enhancement as a function of the \nthickness of the antiferromagnetic IrMn. The solid circles sh ow the calculated damping \nenhancement with the precession of AFM magnetic moments. The solid diamonds \nshow the calculated damping enhancement with perfectly static AFM ordered IrMn \nwithout precession, while the  solid  triangles correspond to the calculated values using \na static paramagnetic IrMn layer with vanishing Néel order.  (d) The experimental \ndamping enhancement as a function of the thickness of antiferromagnetic IrMn.  34 \n FIG.7. (Color online ) (a) Ultrafast demagnetization time as a function of Gilbert  \ndamping . (b) The variation of ultrafast demagnetization rate as a function of Gilbert \ndamping  enhancement . The red line indicates the fitting via Eq. (6 ) in the text.  \nFIG. 8. (Color online ) Scheme of TRMOKE experiment for spin precession dynamics.  \nFIG. 9. (Color online ) Fourier transform spectra measured between 0.85 kOe and 3.0 \nkOe for 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. \n  35 \n  \n \nTable I  \n \n \n \n \n \n. \n \n \n \n tIrMn (nm)                      A1       A2           \n0  220±10 500    5  350  0.8       2      \n1  160±10 500    6  350  0.8       2      \n2  120±10 500    7  350        0.8       2      \n3  145±10 500    4  350  0.8       2       \n5  200±10 500    5  350  0.8       2       \n)(fsM\n)fsE（\n)(0ps\n)(fsG36 \n  \n  \n \n \n \n \n \n \n \n \n \nFIG.1.  \n \n \n \n  \n37 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 2.  \n \n \n  \n38 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG.3. \n \n \n  \n39 \n  \n \n \n \n \n \n \n \n \n \n \nFIG.4. \n \n \n \n \n  \n40 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 5 . \n \n \n \n  \n41 \n  \n \n \n \n \n \n \n \n \nFIG.6. \n \n \n \n \n \n \n  \n42 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG.7. \n \n \n  \n43 \n  \n \n \n  \n \n \n \n \nFIG. 8.  \n \n \n \n \n \n \n \n  \n44 \n  \n \n \n \n \n \n \n \n \n \n \n \nFIG.9.  \n \n"    }]